pricing options and equity-indexed annuities in...

Title Pricing options and equity-indexed annuities in regime-switching models by trinomial tree method

Author(s) Yuen, Fei-lung; 袁飛龍

Citation

Issue Date 2011

URL http://hdl.handle.net/10722/133208

Rights unrestricted

PRICING OPTIONS AND

EQUITY-INDEXED ANNUITIES IN

REGIME-SWITCHING MODELS BY

TRINOMIAL TREE METHOD

by

YUEN FEI LUNG

A thesis submitted in partial fulfillment of the requirements for

the Degree of Doctor of Philosophy

at The University of Hong Kong.

December 2010

Abstract of thesis entitled

PRICING OPTIONS AND

EQUITY-INDEXED ANNUITIES IN

REGIME-SWITCHING MODELS BY

TRINOMIAL TREE METHOD

Submitted by

YUEN FEI LUNG

for the degree of Doctor of Philosophy

at The University of Hong Kong

in December 2010

Starting from the well-known paper in pricing vanilla European call and put

options by Black and Scholes in 1973, there are many different research papers

on option valuation. The basic assumption of the Black-Scholes model is that the

price of the underlying asset, usually the stock, is a geometric Brownian motion.

The Markov regime-switching model (MRSM) introduced by Hamilton (1989)

improves the adaptability of the Black-Scholes model by allowing the parameters

of the stock price process change according to the financial situation. However,

due to the additional uncertainty brought by the Markov process, option pricing

in the MRSM is complicated and is usually done by simulation, or by solving a

system of partial differential equations.

Lattice model (or tree model) provides a simple way to price derivatives. In the

existing literature, additional branches are required in the lattice for derivative

pricing under the MRSM. In this case, the lattice is not recombining and its

efficiency is reduced significantly. Based on the trinomial tree model introduced

by Boyle (1988), a multi-state trinomial tree is introduced to price various options

in the MRSM. The key idea is to use the same lattice to accommodate all different

regimes by adjusting the probability measure. The method is simple and efficient.

The numerical results of the option prices obtained by this method are analyzed.

There are different MRSMs. The stock price in the MRSM of Elliott, Chan

and Siu (2005) is a continuous process while the price of derivatives jumps when

the regime switches. In order to study the jump risk of assets in details, the jump

diffusion model of Naik (1993) is studied and extended. The multi-state trinomial

tree is used to price options in this jump diffusion model and the nature of jump

risk is discussed.

Asian option is a strong path-dependent option. The payoff of Asian option

depends on the whole path of the asset price process and pricing Asian options

is not an easy task in the MRSM. Equity-indexed annuities (EIAs) are derivative

products linked to the performance of an equity index and is now a popular

product in the market. The payoff structure of EIAs can take different forms

according to the needs of the investors. The multi-state trinomial tree model is

modified using the idea of Hull and White (1993) and applied to Asian options.

The problem of quadratic approximation suggested by Hull and White (1993) is

identified and solved. Using the result of Asian option, the price of Asian-option-

related EIAs is obtained by an iterative equation.

Declaration

I declare that this thesis represents my own work, except where due acknowl-

edgements are made, and that it has not been previously included in a thesis,

dissertation or report submitted to this University or to any other institution for

a degree, diploma or other qualification.

Signed

YUEN FEI LUNG

i

Acknowledgements

I would like to express my most sincere gratitude to my supervisor, Prof.

Yang, Department of Statistics and Actuarial Science, the University of Hong

Kong, for his guidance. He always spends a large part of his valuable time to

teach us and monitor our progress. I would also like to express my thanks to

all of the department members for the help and support in these four years of

study.

ii

Table of Contents

Declaration i

Acknowledgements ii

Table of Contents iii

List of Tables v

1 Introduction 1

2 Multi-state Trinomial Tree 7

2.1 Introduction 7

2.2 Multi-state Trinomial Lattice 8

2.3 Numerical Results and Analysis 15

2.4 Alternative Models 27

2.5 Hedging Risk of Regime Switching 36

2.6 Conclusions 42

3 Pricing Regime-switching Risk 44

3.1 Introduction 44

3.2 Jump Diffusion Model 45

3.3 Arrival Rates of Jumps under Risk Neutral Measure 54

3.4 Trinomial Tree Pricing under Jump Diffusion Model 59


3.5.1 Jump Risks are Not Priced 63

3.5.2 Jump Risks are Priced 73

3.6 Conclusions 77

iii

4 Pricing Asian Option and Related EIAs 78

4.1 Introduction 78

4.2 A Modified Trinomial Lattice 79

4.3 Pricing Equity-Indexed Annuities 84


4.5 Conclusions 102

5 Concluding Remarks 107

References 110

iv

List of Tables

2.1 Comparison of different methods in pricing Euroean call option in

MRSM 16

2.2 Pricing European call option with trinomial tree 18

2.3 Pricing European put option with trinomial tree 18

2.4 Pricing American call option with trinomial tree 20

2.5 Pricing American put option with trinomial tree 21

2.6 Pricing down-and-out barrier call option with trinomial tree 22

2.7 Pricing double barrier call option with trinomial tree 23

2.8 Price of double barrier call options with different barrier levels 24

2.9 Pricing European call option with trinomial tree: great derivation

in volatilities 25

2.10 Pricing European call option under model with three regimes 27

2.11 Pricing European call option under model with three regimes using

trinomial tree 34

2.12 Pricing European call option under model with three regimes using

combined trinomial tree 35

3.1 Pricing European call option when jump risk is not priced 65

3.2 Pricing European put option when jump risk is not priced 65

3.3 Comparison of European call option prices in jump and non jump

models 67

3.4 Comparison of European put option prices in jump and non jump

models 67

3.5 Comparison of American call option prices in jump and non jump

models 70

v

3.6 Comparison of American put option prices in jump and non jump

models 71

3.7 Comparison of down-and-out barrier call option prices in jump and

non jump models 72

3.8 Comparison of European call option prices with priced and non-

priced jump risk 74

3.9 Comparison of European put option prices with priced and non-

priced jump risk 75

3.10 Comparison of American call option prices with priced and non-

priced jump risk 75

3.11 Comparison of American put option prices with priced and non-

priced jump risk 76

3.12 Comparison of down-and-out call option prices with priced and

non-priced jump risk 76

4.1 Comparison of the prices of (Eurpean type) average price call op-

tions in simple BS model (linear approximation of representative

value) 93

4.2 Comparison of the prices of (American type) average price call

options in simple BS model (linear approximation of representative

value) 94

4.3 Comparison of the prices of (Eurpean type) average price call op-

tions in simple BS model (quadratic and modified quadratic ap-

proximation, simple average asset price) 96

4.4 Comparison of the prices of (American Type) average price call

options in simple BS model (quadratic and modified quadratic ap-

proximation, simple average asset price) 97

vi

4.5 Comparison of the price of average price call options in MRSM

found by different methods I 100

4.6 Comparison of the price of average price call options in MRSM

obtained by different methods II 101

4.7 Price of average price call options with early exercise option in

MRSM 104

4.8 Price of one-year EIA in MRSM 105

4.9 Price of unit annual reset EIA in MRSM 106

vii

Chapter 1

Introduction

In the past decades, there are a lot of researches on option pricing and many

models have been proposed for the underlying asset. Markov regime-switching

model (MRSM), which allows the parameters of the market model controlled by

a Markov process, have become one of the popular models recently. It is found

to be consistent with the market data and has gained its popularity because it

can reflect the information of the market environment which cannot be modelled

by linear Gaussian process solely. Markov process can ensure the parameters

changing according to the market environment and preserve the simplicity of the

model. It is also consistent with the efficient market hypothesis, all the effects of

the information about the stock price are reflected on the current stock price.

Merton (1969) uses stochastic differential equation (SDE) to study the continuous-

time portfolio theory; since then geometric Brownian motion (GBM) becomes one

of the most commonly used stochastic process in financial mathematics because of

its highly random nature and simplicity. Black and Scholes (1973), based on the

work of Merton (1969), give the no-arbitrage price of an European option when

the price of the underlying assets is a GBM. The development of the formulae of

various derivatives is easy based on this model. However, when the parameters

1

of the SDE are not constant but controlled by a Markov process, the price of the

options cannot be found easily.

There are many papers about option pricing in multi-state models such as

MRSM. Some of them use lattice model. Boyle (1988) uses a pentanomial tree

lattice to find the price of derivatives with two states. Kamrad and Ritchken

(1991) suggest a (2k + 1)-branch model for k sources of uncertainty. Bollen

(1998) constructs a pentanomial tree which is excellent in finding a fair price of

European option and American option in two-regime situation. Aingworth, Das

and Motwani (2006) use a lattice with 2k branches to study the k-state model.

The increasing number of branches reduces the efficiency of the tree models and

so some other methods are used by different researchers to price derivatives.

Buffington and Elliott (2002) find the price of European option and American

option using partial differential equations (PDE). Mamon and Rodrigo (2005)

find an explicit solution to European options in regime-switching economy by

considering the solution of the associated PDE. Elliott, Chan and Siu (2005) use

Esscher transform to find the explicit price formula for European option. Boyle

and Draviam (2007) find the price of exotic options under regime switching using

PDE. PDE has become the focus of most researchers for option valuation in

MRSM as it is flexible.

Since the introduction of binomial tree model by Cox, Ross and Rubinstein

2

(1979), lattice model is one of the most popular methods to calculate the price of

simple options like European option and American option. Various lattice models

are suggested after that, see, for example, Jarrow and Rudd (1983) and Boyle

(1986). Trinomial tree model of Boyle (1986) is highly flexible. The extra middle

branch of it gives one degree of freedom to the lattice and that makes the lattice

very useful in regime-switching model. Boyle and Tian (1998) use this property

of the trinomial tree to price double barrier option and Bollen (1998) uses the

similar idea to construct an efficiently recombining tree. In Chapter 2, a new

method is introduced so that a trinomial tree can be used to find a fair price

of options under MRSM efficiently. The MRSM used by Buffington and Elliott

(2002) is a good and popular choice, and it is used to illustrate the idea of the

multi-state trinomial tree model.

Regime-switching market is not complete. There are many different ways

to find a fair price of the options. Miyahara (2001) uses the minimal entropy

martingale measure to find the price which maximizes the exponential utility.

Elliott, Chan and Siu (2005) use Esscher transform to obtain a fair price. Guo

(2001) introduces change-of-state (COS) contracts to complete the market. Naik

(1993) shows that the price of options can also be found by the given market

price of risks. In the MRSM of Buffington and Elliott (2002), stock price is a

continuous process and jump risk seems not a systematic risk in a certain sense.

3

In Chapter 3, the two-regime model of Naik (1993) is extended to k states and the

option prices under this market model are tested using the multi-state trinomial

tree.

Various modern insurance products are introduced into the market in these

years, including equity-indexed annuities (EIAs) and variable annuities (VAs).

Their payoffs are linked to the preformance of some assets and indices. Different

special features such as minimum yearly return, ceiling rate and participation

rate make the products more flexible and more complicated. Tiong (2000) gives a

details study on EIAs and prices the products using Esscher transform. Lee (2002,

2003) proposes several new designs of EIAs and finds their explicit formulae under

the Black-Scholes framework. Lin and Tan (2003) consider an Asian-option-

related EIA and price it under a stochastic interest rate model. Asian option is

a strong path-dependent option of which the value of payoff depends on the path

of the asset price process. Its valuation is complex under MRSM. In Chapter

4, the multi-state trinomial tree model is used to price Asian options in MRSM

using the idea of Hull and White (1993). The problem of quadratic approximation

suggested by Hull and White (1993) is identified and solved. Using the Markovian

property of the regime-switching process, the price of Asian-option-related EIA

can be obtained by an iterative equation with the price of Asian option.

The theoretical MRSM of Buffington and Elliott (2002) is presented here for

4

completeness.

We consider the real world probability space (Ω,F , P ). We let T be the

time interval [0, T ] that is being considered. W (t)t∈T is a standard Brownian

motion on (Ω,F , P ). X(t)t∈T is a continuous-time Markov process with finite

state space X := x1, x2, . . . , xk, which represents the economic condition and is

observable. A set of unit vector e1, e2, . . . , ek where xi = ei = (0, . . . , 1, . . . , 0) ∈

Rk is used to denote the current state of the Markov process. For simplicity, the

state ei is called the state i. We denote the set of states to be K := 1, 2, . . . , k.

Let A(t) = [aij(t)]i,j=1,...,k be the generator of the Markov process. By the

semi-martingale representation theorem,

X(t) = X(0) +

∫ t

0

X(s)A(s)ds+M(t), (1.1)

where M(t)t∈T is a Rk-valued martingale with respect to the P -augmentation

of the natural filtration generated by X(t)t∈T .

There are two basic investment tools in the model, one is bond and the other

is stock. The market interest rate is denoted by r(t,X(t))t∈T which depends

on the current state of economy only,

r(t) := r(t,X(t)) = 〈r,X(t)〉, (1.2)

where r := (r1, r2, . . . , rk); ri > 0 for all i ∈ K and 〈·, ·〉 denotes the inner product

in Rk.

5

The bond price process B(t)t∈T satisfies the equation

dB(t) = r(t)B(t)dt, B(0) = 1. (1.3)

The rate of return and the volatility of the stock price process are denoted

by µ(t,X(t))t∈T and σ(t,X(t))t∈T , respectively. Similar to the interest rate

process, they are only affected by the state of economy,

µ(t) := µ(t,X(t)) = 〈µ,X(t)〉, σ(t) := σ(t,X(t)) = 〈σ,X(t)〉, (1.4)

where µ := (µ1, µ2, . . . , µk) and σ := (σ1, σ2, . . . , σk) with σi > 0 for all i ∈ K.

The stock price process S(t)t∈T is a Markov-modulated geometric Brownian

motion. Z(t) is the cumulative rate of return of the stock over time interval [0, t],

that is, Z(t) = ln(S(t)/S(0)). Then, we have

S(t) = S(u) exp(Z(t)− Z(u)), (1.5)

dS(t) = µ(t)S(t)dt+ σ(t)S(t)dW (t), (1.6)

Z(t) =

∫ t

0

(µ(s)− 1

2σ2(s)

)ds+

∫ t

0

σ(s)dW (s). (1.7)

6

Chapter 2

Multi-state Trinomial Tree

2.1 Introduction

Since the binomial tree model was introduced by Cox, Ross and Rubinstein

(1979), the lattice model has become a popular way to calculate the price of

simple options like the European option and the American option. It is mainly

because the lattice method is simple and easy to implement. Various lattice mod-

els have been suggested after that, see, for example, Jarrow and Rudd (1983) and

Boyle (1986). The trinomial lattice of Boyle (1986) is highly flexible, and has

some important properties that the binomial lattice lacks. The extra branch of

the trinomial tree gives one degree of freedom to the lattice and makes it very

useful in the regime-switching model. Boyle and Tian (1998) use this property

of the trinomial tree to price double barrier options, and propose an interesting

method to eliminate the error in pricing barrier options. Bollen (1998) uses a

similar idea to construct an efficiently recombining tree. There are many other

researches using tree methods for derivative pricing in multi-state model. Boyle

(1988) uses a pentanomial tree model to calculate the price of derivatives with

two states. Kamrad and Ritchken (1991) suggest a 2k + 1-branch model for k

7

sources of uncertainty. Aingworth, Das and Motwani (2006) uses a lattice with

2k branches to study the k-state regime-switching model. However, when the

number of states is large, the tree models mentioned above are not efficient. We

propose a multi-state trinomial tree to price the options in a regime-switching

model. The trinomial tree that we propose is recombining. Instead of increasing

the number of branches in the tree for different regimes, we use different sets

of risk neutral probabilities for different regimes. Since it is a recombining tree,

option valuation is fast, simple and efficient using this method.

2.2 Multi-state Trinomial Lattice

In CRR binomial tree, when σ is the asset’s volatility and ∆ is the size of time

step, the ratios of change are given by eσ√

∆ and e−σ√

∆, the risk neutral proba-

bilities of getting up and down are specified so that the expected rate of return

of the stock matches the risk-free interest rate. In the trinomial model, with

constant risk-free interest rate and volatility, the stock price is allowed to remain

unchanged, go up or go down by a ratio. The upward ratio must be greater than

eσ√

∆ to ensure that a risk neutral probability measure exists. If πu, πm, πd are

the risk neutral probabilities of the stock price increases, remains unchanged and

decreases in the tree, r is the risk-free interest rate, then, for a constant λ, we

8

have

πueλσ√

∆ + πm + πde−λσ√

∆ = er∆, (2.1)

(πu + πd)λ2σ2∆ = σ2∆ (2.2)

λ should be greater than 1 so that the risk neutral probability measure exists.

In the literature, λ is usually taken to be√

3 (Figlewski and Gao (1999), Baule

and Wilkens (2004)) or√

1.5 (Boyle (1988), Kamrad and Ritchken (1991)). After

fixing the value of λ, the risk neutral probabilities can be found and the whole

lattice can be constructed.

However, in the Markov regime-switching model (MRSM), the risk-free inter-

est rate and the volatility are not constant. They change according to the Markov

process. More branches can be introduced into the lattice so that extra regimes

and information can be incorporated in the tree, for example, Boyle and Tian

(1988), Kamrad and Ritchken (1991) and Bollen (1998). The increasing number

of branches makes the lattice model more complex. Bollen (1998) suggests an

excellent recombining tree to solve the option prices in two-regime case, but the

multi-regime problem still cannot be solved effectively.

Here, we propose a different way to construct the tree. Instead of increasing

the number of branches in the tree, we change the risk neutral probability measure

under different regimes so that a recombining tree allows more regimes. The

9

method relies greatly on the flexibility of the trinomial tree model and the core

idea of the multi-state trinomial tree model here is to change the probability

measure to accommodate different regimes in the same recombining lattice.

Assuming that there are k regimes in the MRSM, the corresponding risk-free

interest rate and volatility of price of the underlying asset under these regimes

be r1, r2, . . . , rk and σ1, σ2, . . . , σk, respectively. The up-jump ratio of the lattice

is taken to be eσ√

∆. For a lattice which can be used by all regimes,

σ > max1≤i≤k

σi. (2.3)

For the regime i, let πiu, πim, π

id are the risk neutral probabilities of the stock

price increases, remains unchanged and decreases in the branch of the tree. Then,

similar to the simple trinomial tree model, the following set of equations can be

obtained, for all i ∈ K,

πiueσ√

∆ + πim + πide−σ√

∆ = eri∆, (2.4)

(πiu + πid)σ2∆ = σ2

i ∆. (2.5)

If λi is defined to be σ/σi for each i, then, λi > 1 and the values of πiu, πim, π

id

10

can be found, in terms of λi,

πim = 1− σ2i

σ2= 1− 1

λ2i

, (2.6)

πiu =eri∆ − e−σ

√∆ − (1− 1/λ2

i )(1− e−σ√

∆)

eσ√

∆ − e−σ√

∆, (2.7)

πid =eσ√

∆ − eri∆ − (1− 1/λ2i )(e

σ√

∆ − 1)

eσ√

∆ − e−σ√

∆. (2.8)

Therefore, the set of risk neutral probabilities depends on the value of σ. In

order to ensure that σ is greater than all σi, we might take

σ = max1≤i≤k

σi + (√

1.5− 1)σ, (2.9)

where σ is the arithmetic mean of σi. Root mean square is another suitable choice

of σ. The most efficient choice of σ is unknown. In this section, σi are assumed to

be not greatly different from each other; and the selection of σ is not important

as long as it is comparable with the volatilities of different regimes.

After the whole lattice is constructed, the main idea of the pricing method is

presented here. We let T be the expiration time of the option, N be the number

of time steps, then ∆ = T/N . At time step t, there are 2t+1 nodes in the lattice,

the node is counted from the lowest stock price level, and St,n denotes the stock

price of the nth node (it starts from the 0th node, for convenience) at time step

t. As all the regimes are sharing the same lattice and the regime state cannot be

reflected by the position of the nodes, each of the nodes has k possible derivative

11

prices corresponding to the regime state. Let Vt,n,j be the value of the derivative

at the nth node at time step t under the jth regime state.

The transition probability of the Markov process can be found by the generator

matrix. The generator matrix is assumed to be constant and taken to be A. We

define pij(∆) as the transition probability from regime state i to regime state

j for the time interval with length ∆; and for simplicity, it is denoted by pij.

The transition probability matrix, denoted by P , can be found by the following

equation,

P (∆) =

p11 · · · p1k

.... . .

...

pk1 · · · pkk

= eA∆ = I +∞∑l=1

(∆)lAl/l!. (2.10)

With the transition probability matrix, the price of a derivative at each node

can be found by iteration. We start from the expiration time, for example, for

an European call option with strike price K,

VN,n,i = (SN,n −K)+ for all states i, (2.11)

where SN,n = S0 exp[(n−N)σ√

∆].

We assume that the Markov process is independent of the Brownian motion

under the real market measure and the transition probabilities are not affected by

the use of risk neutral measure. With the derivative payoff at expiration, using

12

the following equation recursively,

Vt,n,i = e−ri∆

[k∑j=1

pij(πiuVt+1,n+2,j + πimVt+1,n+1,j + πidVt+1,n,j)

], (2.12)

the price of the option under all regimes can be obtained.

Regime switching is another source of risk because we do not know the time

of regime switching before it takes place. Moreover, due to regime switching,

the market is incomplete and the derivatives do not have a unique no-arbitrage

price. There are many ways to treat the additional risk from regime switching,

for example, not pricing the regime-switching risk (Bollen (1998)), or introducing

change-of-state (COS) contracts into the model (Guo (2001)). The first way is

used in the previous calculation. Some derivatives benefit while some are suffered

by the regime switching which depends on the initial regime, the transition prob-

abilities and the structure of the derivatives. It is also hard to make a compromise

in choosing appropriate transition probabilities if the market is not complete. It

is a reasonable choice of not pricing the regime-switching risk as long as there is

no arbitrage opportunity in the market and the Markov process is independent of

the Brownian motion. New securities COS can be introduced into the model to

complete the market. However, the regime is referring to the macroeconomic con-

dition, hedging or pooling regime-switching risk is complicated in the incomplete

market and insurance companies might not be willing to take the risk. We as-

sume that there are not suitable COS securities in the market. The risk premium

13

comes from the risk of Brownian motion only. We focus on the multi-state trino-

mial tree model in this chapter and the regime-switching risk will be discussed in

details in the other parts.

If we have to price American option, the value of the option at each node

under different regimes can be compared with the payoff of exercising the op-

tion immediately; and the larger value is used as the price for iteration. The

calculation is similar to the valuation of American option in simple lattice model.

For barrier option, the idea of Boyle and Tian (1998) can be applied. The

whole lattice is constructed from the lower barrier. As the initial price of the

underlying asset is not necessarily at the grid, a quadratic approximation is used

to calculate the price of the down-and-out option. The price of a down-and-in

option can be found using the idea that the sum of down-and-out option and

down-and-in option is a vanilla option. For a double barrier option, we have used

the flexibility of trinomial tree lattice but the value of σ is in fact not fixed. We

can set both of the upper and lower barriers on the node level by a fine adjustment

of the lattice parameter σ. The price of curved barrier option and discrete-time

barrier option can also be found, using a similar method suggested by Boyle and

Tian (1998).

The regime is observable, the payoff of the derivatives can depend on the

regime state, because the prices of the derivative under all regimes are found in

14

each node, the model is also applicable to price this kind of derivatives.

2.3 Numerical Results and Analysis

Based on the model introduced in the last section, we calculate the prices of

various options in different regimes. In this section we study the European option,

the American option, the down-and-out barrier option, the double barrier option,

and their prices are calculated by the multi-state trinomial tree. The results give

us some insights into the price of derivatives in the MRSM and the effects of

regime switching. First of all, the model is tested by comparing with the results

given by Boyle and Draviam (2007).

Table 2.1 shows that the option price obtained by using the trinomial lattice

is very close to the value obtained by using the analytical solutions derived in

Naik (1993), and also close to those obtained using partial differential equations in

Boyle and Draviam (2007). This verifies that the trinomial tree method proposed

in this chapter is applicable.

We now study the values of different types of options in the MRSM. The

underlying asset is assumed to be a stock with initial price of 100, following a

geometric Brownian motion of a two-regime model with no dividend. In Regime

1, the risk-free interest rate is 4% and the volatility of stock is 0.25; in Regime

15

Table 2.1: Comparison of different methods in pricing Euroean call option in

MRSM

European Call Option I

Regime 1 Regime 2

S0 Naik B&D Lattice Naik B&D Lattice

94 5.8620 5.8579 5.8615 8.2292 8.2193 8.2297

96 6.9235 6.9178 6.9229 9.3175 9.3056 9.3181

98 8.0844 8.0775 8.0827 10.4775 10.4647 10.4772

100 9.3401 9.3324 9.3369 11.7063 11.6929 11.7049

102 10.6850 10.6769 10.6828 13.0008 12.9870 13.0001

104 12.1127 12.1045 12.1108 14.3575 14.3436 14.3571

106 13.6161 13.6082 13.6143 15.7729 15.7591 15.7725

European Call Option II

Regime 1 Regime 2

S0 Naik B&D Lattice Naik B&D Lattice

94 6.2748 6.2705 6.2760 7.8905 7.8844 7.8943

96 7.3408 7.3352 7.3422 8.9747 8.9680 8.9789

98 8.5001 8.4938 8.5010 10.1335 10.1264 10.1374

100 9.7489 9.7423 9.7489 11.3641 11.3568 11.3673

102 11.0820 11.0755 11.0833 12.6631 12.6659 12.6674

104 12.4937 12.4877 12.4959 14.0267 14.0197 14.0317

106 13.9777 13.9726 13.9805 15.4510 15.4446 15.4565

†S0 is the initial stock price and the strike price is set to be 100. The volatilities of the stock

in Regime 1 and Regime 2 are 0.15 and 0.25 respectively. The option lasts for 1 year and the

lattice is set to have 1000 time steps. The generators of the regime-switching process are −0.5 0.5

0.5 −0.5

and

−1 1

1 −1

for the above two sets of data respectively.

16

2, the risk-free interest rate is 6% and the volatility of stock is 0.35. All options

expire in one year with strike price equal to 100. The generator for the regime-

switching process is taken to be −0.5 0.5

0.5 −0.5

.

The transition probabilities of the branch of state up, middle and down with 20

time steps are 0.177003, 0.641304 and 0.181693 in Regime 1; 0.351844, 0.296956

and 0.351200 in Regime 2, respectively. These values depend on the size of time

step, but the values with other sizes of time step are not much different from

these values because the time step is small in general. The values in 20-step case

can already give the idea of the size of the risk neutral probabilities. We study

the numerical results to see if there are any special characteristics of the prices

of these derivatives and the convergence properties of the model.

Tables 2.2 and 2.3 show that the convergence rate of the European call and

the European put options is fast. We know that the price of European call options

and European put options found by the CRR model converges with order 1, that

is, the error of the price is halved if the number of time steps is doubled (Baule

and Wilkens (2004), Omberg (1987)). We can see from the tables that most of

the ratios shown in the tables are close to 0.5. However, it is not the case for

the European call option when the number of iterations is large for Regime 2.

17

Table 2.2: Pricing European call option with trinomial tree

European Call Option

Regime 1 Regime 2

N Price Diff Ratio Price Diff Ratio

20 12.6282 0.0654 0.4954 15.7560 0.0043 0.5581

40 12.6936 0.0324 0.5000 15.7603 0.0024 0.5417

80 12.7260 0.0162 0.5000 15.7627 0.0013 0.4615

160 12.7422 0.0081 0.4938 15.7640 0.0006 0.6667

320 12.7503 0.0040 0.5000 15.7646 0.0004 0.2500

640 12.7543 0.0020 0.5000 15.7650 0.0001 1.0000

1280 12.7563 0.0010 0.5000 15.7651 0.0001 1.0000

2560 12.7573 0.0005 15.7652 0.0001

5120 12.7578 15.7653

†N is the number of time steps used in calculation. Diff is referring to the difference in price

calculated using various numbers of time steps and ratio is the ratio of the difference.

Table 2.3: Pricing European put option with trinomial tree

European Put Option

Regime 1 Regime 2


20 8.37107 0.05781 0.4959 10.2660 0.0119 0.5210

40 8.42888 0.02867 0.4977 10.2779 0.0062 0.5000

80 8.45755 0.01427 0.4989 10.2841 0.0031 0.5161

160 8.47182 0.00712 0.5000 10.2872 0.0016 0.5000

320 8.47894 0.00356 0.5000 10.2888 0.0008 0.5000

640 8.48250 0.00178 0.5000 10.2896 0.0004 0.5000

1280 8.48428 0.00089 0.4944 10.2900 0.0002 0.5000

2560 8.48517 0.00044 10.2902 0.0001

5120 8.48561 10.2903

18

This is because the approximation errors for the two regimes are different and

the round-off error. Boyle (1988) shows that using the trinomial tree model, the

approximation error is smaller if the three risk neutral probabilities of the tree are

almost equal with same number of time steps. In our case, we can see that the risk

neutral probabilities of Regime 1 are not as close as those of Regime 2. Therefore,

in Regime 2, the change in prices is smaller which implies a smaller approximation

error as shown in the numerical results in the tables. The differences between

the price changes for Regime 2 are less than one-tenth of that for Regime 1 most

of the time. However, the prices of the asset in both regimes affect one another.

The larger pricing error in Regime 1 affects the accuracy of the price in Regime 2.

The result is that the value in Regime 2 converges in a faster, but more unstable

way. On the other hand, the error in Regime 2 is small compared with that in

Regime 1; thus the convergence patterns in Regime 1 are more stable. Moreover,

the change of prices in Regime 2 is small when the number of time steps is large.

The round-off error then becomes significant.

When we apply the put-call parity to each of the regimes, the interest rate

found in the two regimes are 4.37% and 5.63% respectively using the result of

5120 time steps. It is reasonable because both of them are between 4% and 6%,

the interest rate found by Regime 1 data is close to Regime 1 rate while the same

is true for Regime 2. Interestingly, the deviations between the current interest

19

Table 2.4: Pricing American call option with trinomial tree

American Call Option

Regime 1 Regime 2


20 12.6282 0.0654 0.4954 15.7560 0.0043 0.5581

40 12.6936 0.0324 0.5000 15.7603 0.0024 0.5417

80 12.7260 0.0162 0.5000 15.7627 0.0013 0.4615

160 12.7422 0.0081 0.4938 15.7640 0.0006 0.6667

320 12.7503 0.0040 0.5000 15.7646 0.0004 0.2500

640 12.7543 0.0020 0.5000 15.7650 0.0001 1.0000

1280 12.7563 0.0010 0.5000 15.7651 0.0001 1.0000

2560 12.7573 0.0005 15.7652 0.0001

5120 12.7578 15.7653

rate and the interest rate found by put-call parity in both regimes are equal

to 0.37%. It is due to the symmetry of two regimes in terms of the transition

probabilities. The mechanism behind and the meaning of it will be discussed in

the next chapter.

The result of the American option is similar to that of the CRR model. The

prices of the American call option found by the trinomial tree is the same as

the European call option. It is consistent with the fact that it is not optimal to

exercise an American call option before expiration if the underlying asset pays

no dividend. We know that this result is also true for the MRSM. The prices

of the American put option in the table are larger than those of the European

option, meaning that early exercise of the option is preferred and there are some

20

Table 2.5: Pricing American put option with trinomial tree

American Put Option

Regime 1 Regime 2


20 8.80315 0.05236 0.5107 10.8942 0.0007 2.5714

40 8.85551 0.02674 0.4862 10.8949 0.0018 0.1111

80 8.88225 0.01300 0.4869 10.8967 0.0002 0.5000

160 8.89525 0.00633 0.4945 10.8969 0.0001 0.0000

320 8.90158 0.00313 0.4984 10.8970 0.0000 N/A

640 8.90471 0.00156 0.4936 10.8970 0.0000 N/A

1280 8.90627 0.00077 0.4935 10.8970 0.0000 N/A

2560 8.90704 0.00038 10.8970 0.0000

5120 8.90742 10.8970

situations in which we have to exercise the American put option before expiration.

The convergence pattern of the American put option is more complicated than

the European one. The rate of convergence for Regime 2 is very fast, even faster

than that of the European put option. The American put option is optimal

to be exercised somewhere before the maturity, so the approximation error is

smaller than that of the European option. The convergence pattern of Regime

2 is unstable, which is consistent with the results for the European option case;

larger initial pricing error in Regime 1 and round-off error affect the convergence

of the price in Regime 2.

For the down-and-out barrier call option, the prices found in both regimes

are smaller than those of the European call option due to the presence of the

21

Table 2.6: Pricing down-and-out barrier call option with trinomial tree

Down-and-out Barrier Call Option

Regime 1 Regime 2


20 8.97860 -0.01239 -0.6917 9.73967 -0.02790 0.0487

40 8.96621 0.00857 -0.4831 9.71177 -0.00136 4.9779

80 8.97478 -0.00414 0.1304 9.71041 -0.00677 0.3840

160 8.97064 -0.00054 -0.2778 9.70364 -0.00260 0.3269

320 8.97010 0.00015 -0.4667 9.70104 -0.00085 1.2588

640 8.97025 -0.00097 -0.3505 9.70019 -0.00107 0.0748

1280 8.96928 0.00034 -0.2059 9.69912 -0.00008 2.1250

2560 8.96962 -0.00007 9.69904 -0.00017

5120 8.96955 9.69887

The barrier level is set to be 90.

down-and-out barrier. The prices in the two regimes are closer to each other

compared with the case of European option. Although the volatility in Regime 2

is greater and has a higher chance to achieve a higher value at expiration, the high

volatility also increases the chance of hitting the down-and-out barrier and thus

eliminates its advantage. The convergence pattern of barrier option is complex.

It is difficult to get any conclusions from the data. However, we can see that

apart from converging uniformly in one direction, the values of the option found

in Regime 1 oscillate and the differences still have a decreasing trend in absolute

value. It is likely due to the effect of quadratic approximation.

The price of the double barrier option can also be obtained by the trinomial

22

Table 2.7: Pricing double barrier call option with trinomial tree

Double Barrier Call Option

Regime 1 Regime 2


20 6.15869 -0.15826 0.7097 4.54096 -0.13822 0.6130

40 6.00043 -0.11232 0.7314 4.40274 -0.08473 0.5189

80 5.88811 -0.04845 0.4111 4.31801 -0.04397 0.3834

160 5.83966 -0.01992 0.5954 4.27404 -0.01686 0.6109

320 5.81974 -0.01186 0.5320 4.25718 -0.01030 0.5029

640 5.80788 -0.00631 0.6133 4.24688 -0.00518 0.6120

1280 5.80157 -0.00387 0.1731 4.24170 -0.00317 0.2145

2560 5.79770 -0.00067 4.23853 -0.00068

5120 5.79703 4.23785

The barrier level is set to be 70 and 150.

model. The method suggested by Boyle and Tian (1998) is adopted here. The

lattice is built from the lower barrier and touches the upper barrier by controlling

the value of σ used in the lattice. Table 2.7 shows the price of the double barrier

option with different numbers of time steps being used. The lower barrier is 70 and

the upper barrier is 150. The values decrease progressively and converge. Table

2.8 summarizes the values of the double barrier options with different barrier

levels using 1000 time steps. When the difference between the upper and lower

barriers is smaller, the price of the options is reduced as there is a higher chance

of touching the barrier and becoming out of value. The effect of barriers is

more significant for Regime 2 because the stock has a higher volatility in Regime

23

Table 2.8: Price of double barrier call options with different barrier levels

Double Barrier Call Option in Regime 1

90 80 70 60 50

110 0.00063 0.0249 0.0498 0.0544 0.0546

120 0.10229 0.4310 0.5773 0.5952 0.5970

130 0.71002 1.6257 1.9120 1.9422 1.9451

140 1.88418 3.4101 3.8049 3.8446 3.8463

150 3.30481 5.3336 5.8019 5.8474 5.8490

200 7.87455 10.8888 11.4649 11.5163 11.5183

Double Barrier Call Option in Regime 2

90 80 70 60 50

110 0.00004 0.0049 0.0202 0.0285 0.0297

120 0.01567 0.1446 0.2909 0.3385 0.3440

130 0.01933 0.7381 1.1160 1.2117 1.2210

140 0.73257 1.8882 2.5051 2.6410 2.6515

150 1.62095 3.4224 4.2422 4.4065 4.4181

200 6.65432 10.5198 11.7835 11.9909 12.0042

The price of the double barrier options with lower barrier of 90, 80, 70, 60, 50 and upper

barrier of 110, 120, 130, 140, 150, 200 in the two regimes are calculated using 1000 time steps.

24

Table 2.9: Pricing European call option with trinomial tree: great derivation in

volatilities


Regime 1 Regime 2


20 9.07428 0.37247 0.5368 19.9973 -0.0409 0.4572

40 9.44675 0.19995 0.4475 19.9564 -0.0187 0.4706

80 9.64670 0.08948 0.4641 19.9377 -0.0088 0.5000

160 9.73618 0.04153 0.4869 19.9289 -0.0044 0.4773

320 9.77771 0.02022 0.4936 19.9245 -0.0021 0.5238

640 9.79793 0.00998 0.4971 19.9224 -0.0011 0.4545

1280 9.80791 0.00496 0.5000 19.9213 -0.0005 0.6000

2560 9.81287 0.00248 19.9208 -0.0003

5120 9.81535 19.9205

The volatilities of the two regimes are 0.10 and 0.50 respectively.

2, hence it has a greater chance of reaching the barriers. When the difference

between the barriers increases, their effect on the barrier options is reduced and

the options in Regime 2 with a larger volatility have a higher price than that of

the same option in Regime 1. Their prices are lower than those of the vanilla call

option, which has prices of 12.7557 and 15.7651 in the two regimes, respectively,

found by trinomial tree with 1000 time steps.

We now consider a few more examples. We predict that the convergence rate

reduces if the volatilities of different regimes are largely different from each other.

We want to find if the prediction is true. All the other conditions are assumed

to be the same, but the volatilities of the asset under the two regimes become

25

0.10 and 0.50. The prices of the European call option are found. The risk neutral

probabilities of Regime 1 of 20 time steps case in the three branches are 0.0224138,

0.968941, 0.00864505, respectively. Most of the probabilities are distributed on

the middle branch.

The price of European option is positively related to the volatility and so

the value in Regime 1 decreases while the value in Regime 2 increases, when

compared with the results of previous example. The pricing error in Regime 1

is larger compared with the results in the previous example as a large σ is being

used to construct lattice. We can make use of the fact that the price of European

option converges with order 1 so that a better result can be obtained even with

a smaller number of time steps.

Next we consider a three-regime example. This example is used to examine

the efficiency of the trinomial tree under multi-state market. The interest rate and

the volatility in the three regimes are 4%, 5%, 6% and 0.20, 0.30, 0.40, respectively.

The initial price and strike price are both set as 100 and the generator matrix is

taken as −1 0.5 0.5

0.5 −1 0.5

0.5 0.5 −1

. (2.13)

The numerical results are shown in Table 2.10. They show that the conver-

26

Table 2.10: Pricing European call option under model with three regimes


Regime 1 Regime 2 Regime 3

N Price Diff Price Diff Price Diff

20 11.9484 0.1196 14.2232 0.0510 16.6246 -0.0143

40 12.0680 0.0582 14.2742 0.0255 16.6103 -0.0065

80 12.1262 0.0289 14.2997 0.0126 16.6038 -0.0031

160 12.1551 0.0143 14.3123 0.0064 16.6007 -0.0015

320 12.1694 0.0071 14.3187 0.0031 16.5992 -0.0008

640 12.1765 0.0036 14.3218 0.0016 16.5984 -0.0004

1280 12.1801 0.0018 14.3234 0.0008 16.5980 -0.0002

2560 12.1819 14.3242 16.5978


calculated using various numbers of time steps.

gence pattern is similar to that of the two-regime case. That is, the convergence

rate is still order 1 even for the three-regime case. The convergence property is

very useful as it can help us approximate the price of vanilla options even with a

small number of time steps.

2.4 Alternative Models

Several amendments can be made to improve the rate of convergence or adapt-

ability of the model under other situations. In the last section, it is assumed that

the generator of the Markov process is a constant matrix and the volatilities of

different regimes do not greatly deviate from the others. These two constraints

27

can be relaxed in some situations.

The generator process can be a function of time. If it is continuous, an

approximation can be used in the branches of each time point, for example, at the

branches at time t to t+∆, the transition probability matrix can be approximated

by the following equation,

P (t,∆) =

pt,11(∆) · · · pt,1k(∆)

.... . .

...

pt,k1(∆) · · · pt,kk(∆)

≈ eA(t)∆. (2.14)

The value of the options found by the lattice still converges to the value of the

options under a continuous-time model. Apart from using I +∑∞

l=1(∆)lA(t)l/l!

to approximate the value of transition probability matrix, another expression can

also be used,

P (t,∆) ≈ limn→∞

(I +A(t)∆

n)n = lim

n→∞(I +

A(t)∆

2n)2n

. (2.15)

This expression has also a good performance in approximating the value of P (t,∆)

using recursion in computer. It is important because the transition probability

matrix has to be calculated for each time step. A good approximation method

can greatly improve the efficiency of computation.

When the number of regime states is large, the volatilities of the asset in

different regimes might not be close to each other. The lattice in the last section

is constructed by a value, σ, which is larger than the asset’s volatilities in all

28

regimes, so that all regimes can be incorporated into this recombining lattice.

This simplifies calculations. However, when the volatilities in different regimes

largely deviate from one another, volatilities are relatively small in some regimes.

But since the model still has to accommodate the largest σi, the σ used in the

model is large. For those regimes with small volatilities, due to the up and down

ratios used in the tree are large, a high risk neutral probability has to be assigned

to the middle branch. The initial error of these regimes is relatively larger. A

recombining trinomial tree can be used to solve the problem.

When we confront a number of regimes corresponding to quite different volatil-

ities, we can divide the regimes into groups according to their size of volatility.

The regimes with large volatility are grouped together, and so are the regimes

with small volatility. The trinomial model can be applied to each group with

regimes whose volatilities are close to each other. The trinomial lattices are then

combined to form a multi-branch lattice, which is similar to the model suggested

by Kamrad and Ritchken (1991) in the (2k + 1)-branch model. More branches

can be introduced to handle more complex situations in the market. All of them

share the same middle branch. The problem is that the parameters σ in differ-

ent trinomial lattices do not necessarily match. When the lattices are combined,

the branches in each of the lattices need not meet each other, that is, the ratios

used in one lattice are not multiples of the other lattices and the simplicity of

29

the model disappears because the branches do not recombine in the whole lattice

efficiently and the number of nodes in the tree is very large.

In order to preserve the simplicity of the model and improve the rate of con-

vergence for the low-volatility regimes at the same time, a similar idea used in

the lattices by Bollen (1998) can be adopted. All regimes are divided into two

groups. In fact, they can be separated into more than two groups, but for pur-

poses of illustration, we only use two groups here. Again, the σ used in trinomial

lattice by the group with larger volatility is not necessarily a multiple of the σ

used by the other group. The problem can be solved by adjusting the value of

σ in either group or even both of the groups, depending on the situation. The

volatility of the group with large volatility should be at least double that of the

small volatility group; otherwise the multi-state trinomial tree in the previous

section should be good enough for pricing. If the ratio between the two values is

larger than 2, the values of lattice parameters σ in both groups are adjusted so

that their ratio is set to 2. In practice, the ratio should not be very large. This

model should be able to handle real data.

Similar to the model that we propose in the last section, assume that there are

k regimes and they are divided into two groups, k1 of them in the low volatility

group and k2 of them in the high volatility group. The states of economy are

30

arranged in ascending order of volatility and so

σ1 ≤ σ2 ≤ . . . ≤ σk1 ≤ . . . ≤ σk.

We now construct the combined trinomial tree in which the stock can increase

with factors e2σ√

∆ and eσ√

∆, remain unchanged, or decrease with factors e−σ√

∆

and e−2σ√

∆. At time step t, there are 4t + 1 nodes in the lattice, the node is

counted from the lowest stock price level, and St,n denotes the stock price of

the nth node at time step t. Each of the nodes has k possible derivative prices

corresponding to the regime states. Let Vt,n,j be the value of the derivative at the

nth node at time step t in the jth regime state. The regimes of group 1 use the

middle three branches with ratios eσ√

∆, 1, and e−σ√

∆. The regimes of group 2

use the branches with ratios e2σ√

∆, 1, and e−2σ√

∆.

We have to ensure that the combined trinomial tree can accommodate all

regimes so that the risk neutral probabilities of all regimes exist. That is

σ > max1≤i≤k1

σi and 2σ > maxk1+1≤i≤k

σi. (2.16)

For the regime i, πiu, πim, and πid are the risk neutral probabilities for up,

middle and bottom branches of the tree, respectively. Similar to the trinomial

tree model, the following set of equations can be obtained. For 1 ≤ i ≤ k1,

πiueσ√


∆ = eri∆, (2.17)


i ∆; (2.18)

31

for k1 + 1 ≤ i ≤ k,

πiue2σ√

∆ + πim + πide−2σ√

∆ = eri∆, (2.19)

(πiu + πid)(2σ)2∆ = σ2i ∆. (2.20)

Solving the equations above, we have,

for 1 ≤ i ≤ k1,

πim = 1− σ2i

σ2, (2.21)

πiu =eri∆ − e−σ

√∆ − πim(1− e−σ

√∆)

eσ√

∆ − e−σ√

∆, (2.22)

πid =eσ√

∆ − eri∆ − πim(eσ√

∆ − 1)

eσ√

∆ − e−σ√

∆, (2.23)

and for k1 + 1 ≤ i ≤ k,

πim = 1− σ2i

4σ2, (2.24)

πiu =eri∆ − e−2σ

√∆ − πim(1− e−2σ

√∆)

e2σ√

∆ − e−2σ√

∆, (2.25)

πid =e2σ√

∆ − eri∆ − πim(e2σ√

∆ − 1)

e2σ√

∆ − e−2σ√

∆. (2.26)

With the payoff of a derivative in different regimes at expiration, the price of the

derivative at different regimes can be found using the following two equations

recursively.

For 1 ≤ i ≤ k1,

Vt,n,i = e−ri∆

[k∑j=1

pij(πiuVt+1,n+3,j + πimVt+1,n+2,j + πidVt+1,n+1,j)

], (2.27)

32

For k1 + 1 ≤ i ≤ k,

Vt,n,i = e−ri∆

[k∑j=1

pij(πiuVt+1,n+4,j + πimVt+1,n+2,j + πidVt+1,n,j)

]. (2.28)

A simple example is given here to illustrate the idea. We assume that there are

three regimes in the market. The corresponding volatilities and risk-free interest

rates in these regimes are 0.15, 0.40, 0.45 and 4%, 6%, 8%, respectively. The

generator matrix of the regime-switching process is−1 0.5 0.5

0.5 −1 0.5

0.5 0.5 −1

. (2.29)

Under the trinomial model in Section 2, the suggested value of σ is 0.524915 and

the risk neutral probabilities of Regime 1 under the up, middle and down state

with 20 time steps used are 0.0469448, 0.918341, 0.0347143, respectively. The

convergence rate of the price of derivatives in this regime is affected due to the

volatility difference. If the three regimes are divided into two groups, Regime 1

forms the low volatility group and Regimes 2 and 3 form the high volatility group.

By (2.9), the corresponding σ value in each of the trinomial trees is suggested to

be

σ(1) = 0.15 + (√

1.5− 1)0.15 = 0.183712,

σ(2) = 0.45 + (√

1.5− 1)(0.40 + 0.45)/2 = 0.545517.

33

Table 2.11: Pricing European call option under model with three regimes using

trinomial tree




20 11.9872 0.2082 17.5029 0.0167 19.0695 -0.0226

40 12.1954 0.0966 17.5196 0.0088 19.0469 -0.0103

80 12.2920 0.0468 17.5284 0.0045 19.0366 -0.0050

160 12.3388 0.0232 17.5329 0.0022 19.0316 -0.0024

320 12.3620 0.0114 17.5351 0.0012 19.0292 -0.0012

640 12.3734 0.0058 17.5363 0.0006 19.0280 -0.0006

1280 12.3792 0.0028 17.5369 0.0003 19.0274 -0.0003

2560 12.3820 17.5372 19.0271

Note that σ(2) is about three times of σ(1). In order to make it adaptive to the

combined trinomial tree model, we must make adjustments to their values. For

example, we can take σ(1) to be 0.272758, which is half of σ(2). That is, the

value of σ used by group 1 is 0.272758. The risk neutral probabilities with 20

time steps for Regime 1 in the combined tree are 0.163008, 0.697569, 0.139423.

Tables 2.11 and 2.12 present the price of the European call option using the

trinomial tree and the combined trinomial tree. The pricing error in the combined

trinomial tree for Regime 1 in which the stock has a small volatility is smaller than

that in the trinomial tree. For the combined tree, the approximation errors of the

three regimes are closer to each other compared with those of the trinomial tree,

which is consistent with the result of Boyle (1998). However, we note that if N

34

Table 2.12: Pricing European call option under model with three regimes using

combined trinomial tree




20 12.2024 0.0928 17.5325 0.0032 19.0964 -0.0346

40 12.2952 0.0452 17.5357 0.0010 19.0618 -0.0475

80 12.3404 0.0223 17.5367 0.0004 19.0443 -0.0087

160 12.3627 0.0111 17.5371 0.0002 19.0356 -0.0044

320 12.3738 0.0056 17.5373 0.0001 19.0312 -0.0022

640 12.3794 0.0027 17.5374 0.0000 19.0290 -0.0011

1280 12.3821 0.0014 17.5374 0.0000 19.0279 -0.0006

2560 12.3835 17.0574 19.0273

time steps are used, the number of nodes of the combined tree is (2N+1)(N+1);

it is about double of that ((N + 1)2) of a trinomial tree; and the pricing error

of the combined trinomial tree in Regime 3 is greater than that of the trinomial

tree. These suggest that the trinomial tree is more effective than the combined

trinomial tree, especially when the diffusion volatilities of different regimes are

comparable. Therefore, in most of the situations, the simple trinomial tree model

should be good enough and there is no need to use this combined trinomial

tree. These also suggests that trinomial tree is in some sense better than the

pentanomial tree of Bollen (1998) even in the two-regime case.

35

2.5 Hedging Risk of Regime Switching

We assume that there is only one risky underlying asset and one risk-free asset.

The market is not complete and the risk neutral probability is not unique. Our

model is different from the jump-diffusion model in which the price of the un-

derlying asset has jumps. There are some works regarding the pricing of options

in incomplete markets and the choice of risk neutral probability measure. For

example, Follmer and Sondermann (1986), Follmer and Schweizer (1991) and

Schweizer (1996) identify an equivalent martingale measure by minimizing the

variance of the hedging loss under the basic measure. In fact, the quadratic

loss of the hedge position can be related to the concept of a quadratic utility

(Boyle and Wang (2001)). Davis (1997) proposes the use of a traditional eco-

nomic approach of pricing, called the marginal rate of substitution, for pricing

options in incomplete markets. He determines a specific pricing measure, and

hence a fair price, of an option by solving a utility maximization problem. An-

other popular method in the literature is by minimizing entropy. Cherny and

Maslov (2003) justify the use of the Esscher transform for option valuation in

a general discrete-time financial model with multiple underlying risky assets by

maximizing exponential utility under the minimal entropy martingale measure.

They also highlight the duality between the exponential utility maximization and

the minimal entropy martingale measure (Frittelli (2000)). We assume that the

36

Markov process is independent of the Brownian motion under the real measure

and the pricing measure. In our model, when the regime changes, the volatility

of the underlying stock changes (and the risk-free rate also changes), the price of

the stock does not jump as the dynamic of the stock price is a continuous pro-

cess. The change in (expected) volatility changes the option price. For different

corresponding volatilities, the option prices are different. This means that the

option price jumps when the regime state changes. We think that the nature of

regime-switching risk is somehow different from that of the market risk. There-

fore, we can use the real transition probability for pricing. That is, we should not

price the regime-switching risk.

From the very basic concept of valuation, we know that in a complete market,

the risk neutral probability is the probability measure which determines the no-

arbitrage price of all assets in the market by taking discounted expectation using

the risk-free interest rate as the discounting rate. The ultimate tool that helps

us in finding the price of derivative assets is still the assumption of no arbitrage

in the market, which is useful in complete, and incomplete markets. As long

as there is no arbitrage, the price of a derivative asset can be anything. If we

want to price a derivative, it is rational to do it by comparing it with other

related securities in the market. As we know, the price of assets in the market is

determined by people, who have different views on the future and have different

37

risk preferences. Securities are traded in the market according to their investment

characteristics, and an equilibrium price is achieved in the market. In our model,

the real transition probability is assumed to be known, but it needs not be the

transition probability that is used by us in valuation. In practice, if the MRSM

is applied as the dynamic of risky assets in the market, the transition probability

matrix is not known and our estimation of this matrix is important. When a new

derivative is traded in the market with a price that the traders think suitable,

people trade this derivative in the market and an equilibrium price is achieved.

However, we can only observe the price of the derivatives in the current regime. If

we do not have the price information of the assets in all regimes, the no-arbitrage

price of the assets found is not unique.

In finance, when the price of a derivative is considered, the required return

of the derivative should be related to the risk involved. However, the measure of

risk and return, the exact relation between risk and return are still not clear. The

capital asset pricing model (CAPM) suggests that the risk premium of the asset is

proportional to its market risk measure β, which is useful and easy to understand

and therefore widely accepted. The price of the stocks is a continuous process in

the MRSM in this Chapter. Stocks of a company can be viewed as parts of its

business, where the business is something that can earn money by selling things

with a higher price than their costs. They generate values by transforming raw

38

materials into a more useful and valuable form. Derivatives are not present in

the market naturally, but introduced by some financial institutions. They are

just a form of betting; its outcome is related to the price of the underlying assets.

The trading of a derivative is a zero sum game. Therefore, when the regime-

switching risk of derivatives is considered, the issuers should not be rewarded

even it seems to bear the market risk. The regimes refer to the market situation,

but regime switching is a diversifiable risk. The price is unfair if either the issuers

or the buyers are rewarded by taking this jump risk. Hence, we suggest that the

original transition probability can be used in pricing in this model.

Under the continuous-time Markov regime-switching model, due to the regime-

switching risk, the market is incomplete. Guo (2001) uses the change-of-state

contracts to complete the market and price the options. In a model of k regimes,

there are k − 1 possible jumps for derivatives and thus there are k − 1 indepen-

dent derivatives at most in the market. Here, independent refers to the linear

independence of the jump sizes of the derivatives. Therefore, we can add k − 1

derivatives into the market and complete the market. The idea of having a risk

neutral transition probability emerges. There are k2 entries in the transition

probability matrix with k(k − 1) degrees of freedom. If all of the derivatives are

independent in terms of their jump sizes, each of the derivatives has k price infor-

mation for the k regimes, therefore k− 1 independent derivatives are required to

39

complete the market. There is a unique risk neutral transition probability that is

used by all the k−1 derivatives for pricing. In fact, all the other derivatives in the

market should also be priced using this unique risk neutral transition probability

to avoid arbitrage. Theoretically, when the price information of k − 1 additional

independent derivatives in all different regimes at each time point are known, the

market is completed and the unique risk neutral transition probability matrix

exists. The risk neutral transition probability matrix is the only matrix process

which is consistent with the price process of all assets. However, it is not easy

to construct the risk neutral transition probability matrix, especially when the

number of regimes is large.

We know that, although the jumps of derivative price correspond to the change

of regimes which indicates the change of market situation, the regime-switching

risk is different from the market risk in nature. If we really want to price the

jump risks of derivatives due to regime switching, regime-switching risk cannot

be eliminated by diversification in the market and hence it is a fundamental risk.

That is, stock prices should jump when the regime switches. Naik (1993) presents

a good and simple model under this framework. The prices of risk due to the

fluctuation of the Brownian motion and the risk of jump due to regime switching

are defined and used to find the risk neutral transition probability matrix. It will

be studied in details next chapter.

40

We consider a new situation. Apart from the bond, the risky asset and its

derivatives, there are some other assets in the market. If the total market value

has a significant change during regime switching, which implies that there ex-

ist some fundamental assets, rather than derivatives, having jump during regime

switching. Regime-switching risk is a market risk as it cannot be eliminated by

diversification and then it is appropriate to price this risk even if the risky under-

lying asset that we are now considering does not jump during regime switching.

Guo (2001) considers the jump risk premium in a continuous two-state regime-

switching asset price model and identifies a risk neutral measure by changing

the regime transition rates. Siu, Yang and Lau (2008) uses Esscher transform

to obtain the risk neutral measure in which the transition rates change. Their

results are consistent with pricing method used in jump-diffusion model in Naik

(1993) and Yuen and Yang (2009), where changing the transition rates is the

key to obtain the risk neutral measure. Again, our goal is to identify a pricing

transition probability.

Under the current asset price model, we suggest that if the real transition

probability is given, it can be used to price the first and all the other derivatives

of the asset; however, if the prices of the derivatives are already available in

the market, we should try to price the newly developed one using a transition

probability which is consistent with the current prices of all the assets. The real

41

transition probability would no longer be the one used in pricing but the risk

neutral transition probability would take its role and this is parallel to the idea

of risk neutral measure in the Black-Scholes-Merton model.

2.6 Conclusions

MRSM is gaining its popularity in the area of derivative valuation. However, the

difficulties in pricing and hedging derivatives in MRSM limit its development. In

this chapter, we introduced the multi-state trinomial tree lattice. Option pricing

under the MRSM using this lattice is now as simple as that in the Black-Scholes

model using binomial tree. The options which can be priced using the CRR model

under the simple Black-Scholes framework can also be priced using the trinomial

tree under the MRSM of Buffington and Elliott (2002).

The nature of jump risk has been discussed in details. The market is not

complete in the MRSM. As long as there is no arbitrage, jump risk (regime-

switching risk) is suggested not to be priced because the regime-switching process

is independent of the Brownian motion under the real and pricing measures and

jump risk during regime switching is not a kind of market risk in this asset price

model. If the real (market) transition probability is given, it can be used directly

to find an appropriate option price. If prices of derivatives in the market are

available, these prices can be used in order to determine a risk neutral transition

42

probability and the corresponding price of jump risk.

43

Chapter 3

Pricing Regime-switching Risk

3.1 Introduction

The regime-switching market is not complete. Therefore, many researches have

been done in order to find a fair price of the derivatives. Based on the result

of the last chapter, if all the underlying assets in the market have a continuous

price dynamic, regime-switching risk can be diversified and is not a systematic

risk. The real transition probability can be used in pricing derivatives. In general,

underlying assets and the market can have jump in value during regime switching.

A jump diffusion type MRSM is required to illustrate the dynamic of these assets’

price.

Naik (1993) considers this problem in depth and develops a two-state stock

price model. A specific risk neutral transition probability and hence a fair price

of each derivative can be found if the prices of jump risk under different states are

known. In this chapter, this idea is extended to a k-state market. The multi-state

trinomial tree is found to be useful in this jump diffusion model.

44

3.2 Jump Diffusion Model

The model in this section is based on the work of Naik (1993). We make use of

the same notation and similar assumptions as in Chapter 1, but the stock price

process jumps during regime switching.

We consider the real-world probability space (Ω,F , P ). Let T be the time

interval [0, T ] that is being considered. W (t)t∈T is a standard Brownian motion

on (Ω,F , P ). X(t)t∈T is a continuous-time Markov Chain with finite state

space e1, e2, . . . , ek, where ei = (0, . . . , 1, . . . , 0) ∈ Rk, which is a unit vector

representing the economic condition and is independent of the Brownian motion

under P . For simplicity, the state ei is called the state i.

Let A(t) = [aij(t)]i,j=1,...,k be the generator of the Markov chain. By the

semi-martingale representation theorem, we have

X(t) = X(0) +

∫ t

0

X(s)A(s)ds+M(t), (3.1)

where M(t)t∈T is a Rk-valued martingale with respect to the P -augmentation

of the natural filtration generated by X(t)t∈T . Point processes N(t; j)t∈T ,j∈K

count the number of state transitions to state j over time 0 to time t. Therefore,

if we assume that X(t−) = i, and i is not equal to j, the arrival rate of the point

process N(t; j) is just the corresponding entry aij(t) of the generator matrix.

Obviously, when X(t−) = j, the arrival rate of N(t; j) is zero. A new matrix is

45

set up so that the arrival rate of jump can be summarized in the following matrix,

A(t) =

0 a12(t) · · · a1k(t)

a21(t) 0 · · · a2k(t)

......

. . ....

ak1(t) ak2(t) · · · 0

= A(t)− diag(a11(t), a22(t), · · · , akk(t)), (3.2)

where diag(a11(t), a22(t), · · · , akk(t)) represents the diagonal matrix with elements

given by the vector (a11(t), a22(t), · · · , akk(t)). If the generator matrix A(t) is

assumed to be constant, the matrix of arrival rate A(t) would also be constant

and it is denoted by A = [aij]k×k.

We assume that the risk-free interest rate depends on the current state of the

economy only and therefore

r(t) := r(X(t)) = 〈r,X(t)〉 (3.3)

where r := (r1, r2, . . . , rk); ri > 0 for all i = 1, 2, . . . , k and 〈·, ·〉 denotes the inner

product in Rk.

Given the interest rate process, the bond price process B(t)t∈T satisfies the

equations,

dB(t) = r(t)B(t)dt, B(0) = 1. (3.4)

The rate of return and the volatility of the stock price process are denoted

by µ(t,X(t))t∈T and σ(t,X(t))t∈T , respectively. Similar to the interest rate

46

process, they are affected by the state of economy only,

µ(t) := µ(X(t)) = 〈µ,X(t)〉, σ(t) := σ(X(t)) = 〈σ,X(t)〉, (3.5)

where µ := (µ1, µ2, . . . , µk) and σ := (σ1, σ2, . . . , σk) with σi > 0 for all i =

1, 2, . . . , k.

The stock price process is assumed to jump during the transition of states

to ensure that the risk of state transition is a non-diversifiable risk. We assume

that the jump size depends on the state before and after the state change and

the current stock price only. If exp(yij)−1 denotes the ratio of jump of the stock

price during the state transition from i to j to the current stock price, then, the

stock price process S(t) is assumed to satisfy

dS(t)

S(t−)= µ(t−)dt+ σ(t−)dW (t)

+k∑j=1

(exp(yX(t−),j)− 1)(N(dt; j)− aX(t−),jdt). (3.6)

We denote N(t; j)−N(t−; j) by N(dt; j) and we find that yiji,j∈K has an impor-

tant property which makes the trinomial tree method applicable to price deriva-

tives in this model, that is,

yil + ylj = yij for all i, j, l ∈ K. (3.7)

The equation holds because of the Markovian property of X(t). For a Markov

chain, all information about the process in the past are represented by the current

47

information. We consider two situations; one is that the state of economy changes

from state i to state j, and the other is that the economic state changes from

state i to state l and goes to state j immediately. All of the other conditions in

these two situations are assumed to be the same. Due to the Markovian property,

the past information is not useful given the current state information. A person

cannot distinguish the difference in value of these two stock price processes under

these two situations. The prices of stocks in these two cases are the same and

equation (3.7) should hold. If j and l in equation (3.7) are taken to be i,

yii + yii = yii ⇒ yii = 0, (3.8)

for all i ∈ K. We should also have the following condition,

yij + yji = yii = 0 ⇒ yij = −yji, (3.9)

for all i, j ∈ K.

Here, we base on the works of Elliott, Chan and Siu (2005) on the double-

indexed σ-algebra. Let FXt t∈T and FZt t∈T be the the natural filtration of

X(t)t∈T and Z(t)t∈T , respectively. We define Gt to be the σ-algebra FXt ∨FZt

and Gt,s to be the double indexed σ-algebra FXt ∨FZs . A risk neutral probability

measure can be obtained so that the price of the derivatives can be calculated

easily. Let Q denote the risk neutral probability measure. We write ξ(T ) =

dQ/dP , ξ(t) = E(ξ(T )|Gt). For a derivative of the asset S(t) with a final payoff

48

of g(S(T )), its price at time t can be calculated by

V g(t) = EQ

[exp

(−∫ T

t

r(s)ds

)g(S(T ))

∣∣∣Gt] (3.10)

= E

[exp

(−∫ T

t

r(s)ds

)ξ(T )

ξ(t)g(S(T ))

∣∣∣Gt] , (3.11)

where exp(−∫ t

0r(s)ds

)ξ(t) is known as the state-price density. Because ξ(t) =

E(ξ(T )|Gt), ξ(t) is a martingale under the probability measure P . By the

martingale representation theorem, we have

dξ(t)

ξ(t−)= η0(t−)dW (t) +

k∑j=1

η(t−; j)(N(dt; j)− aX(t−),jdt) (3.12)

for some predictable processes η0(t−) and η(t−; j), where η(t−; j) > −1. Ac-

cording to Naik (1993), η0(t−) and η(t−; j) can be interpreted as the price of

continuous risk generated by the Brownian motion and the price of discontinuous

risk of the change in volatility, respectively. We assume that the risk characteris-

tic of the investors depends on the economic state only. Then, η0(t−) depends on

X(t−), that is, η0(t−) = η0(X(t−)); and the value of η(t−; j) depends on X(t−)

and j only, that is, η(t−; j) = η(X(t−), j). We write η(t−; j) as ηX(t−),j. The pro-

cess

exp(−∫ t

0r(s)ds

)ξ(t)S(t)

is a martingale under the probability measure

49

P . By Ito’s formula, we have

d[S(t)ξ(t)] = S(t−)dξ(t) + ξ(t−)dS(t) + 〈dS(t), dξ(t)〉

= S(t−)ξ(t−)

[µ(t−)dt+ σ(t−)dW (t) + η0(t−)dW (t)

+k∑j=1

(exp(yX(t−),j)− 1

)(N(dt; j)− aX(t−),jdt

)

+k∑j=1

η(t−; j)(N(dt; j)− aX(t−),jdt

)+ σ(t−)η0(t−)dt

+k∑j=1


)η(t−; j)N(dt; j)

]

= S(t−)ξ(t−)

[µ(t−)dt+ σ(t−)dW (t) + η0(t−)dW (t)

+k∑j=1



)

+k∑j=1


)+ σ(t−)η0(t−)dt

+k∑j=1


)η(t−; j)

(N(dt; j)− aX(t−),jdt

)

+k∑j=1


)η(t−; j)aX(t−),jdt

]. (3.13)

At the same time, r(t) is bounded and is continuous almost everywhere on T ,

thus we have

∫ t

0

r(s)ds =

∫ t

0

r(s−)ds, (3.14)

and therefore,

exp(−∫ t

0r(s−)ds

)ξ(t)S(t)

is also a martingale. By Ito’s for-

50

mula again,

d

[exp

(−∫ t

0

r(s−)ds)S(t)ξ(t)

]= exp

(−∫ t

0

r(s−)ds)S(t−)ξ(t−)

[µ(t−)dt+ σ(t−)dW (t)

+k∑j=1



)− r(t−)dt

+k∑j=1


)+ η0(t−)dW (t)

+k∑j=1


)η(t−; j)

(N(dt; j)− aX(t−),jdt

)

+k∑j=1


)η(t−; j)aX(t−),jdt+ σ(t−)η0(t−)dt

]. (3.15)

Apart from the time point of jump, we have

µ(t−)− r(t−) + σ(t−)η0(t−) +k∑j=1


)η(t−; j)aX(t−),j = 0. (3.16)

We rearrange the equation above and we have

µ(t−)− r(t−) = −σ(t−)η0(t−)−k∑j=1


)aX(t−),jη(t−; j). (3.17)

The risk premium of the stock has two components: the risk premium from the

Brownian motion and the risk premium from the jump of stock price. From the

capital asset pricing model under a pure diffusion formulation, the risk premium

is proportional to the volatility of the stock price. Here, the risk premium of a

jump due to a transition of the Markov chain is proportional to the size of jump

51

exp(yX(t−),j) − 1 and the arrival rate of jump aX(t−),j. As mentioned in Naik

(1993), −η0(t−) and −η(X(t−), j) represent the price of risk of diffusion volatility

and jump, respectively.

−η0(t−) =Risk Premium from Brownian Motion Risk

σ(t−)(3.18)

−η(X(t−), j) =Risk Premium from State j Jump Risk

(exp(yX(t−),j)− 1)aX(t−),j

. (3.19)

To ensure that the risk premiums of these two sources of risk are non-negative,

we have

η0(t−) ≤ 0 and (3.20)

η(X(t−), j)(exp(yX(t−),j)− 1) ≤ 0 for all X(t−), j ∈ K. (3.21)

Since exp(yX(t−),j)− 1 and yX(t−),j have the same sign, the second condition can

be simplified,

η(X(t−), j)yX(t−),j ≤ 0 for all X(t−), j ∈ K. (3.22)

From (3.6) and (3.12) and Ito’s formula, it is not difficult to see that the

52

expressions of S(T ) and ξ(T ) are given by

S(T ) = S(0)exp

[ ∫ T

0

µ(t−)dt+

∫ T

0

σ(t−)dW (t)− 1

2

∫ T

0

σ2(t−)dt

+k∑j=1

∫ T

0

yX(t−),jN(dt; j)−k∑j=1

∫ T

0


)aX(t−),jdt

],

(3.23)

ξ(T ) = exp

[ ∫ T

0

η0(t−)dW (t)− 1

2

∫ T

0

η20(t−)dt

+k∑j=1

∫ T

0

ln(1 + ηX(t−),j

)N(dt; j)−

k∑j=1

∫ T

0

aX(t−),jηX(t−),jdt

].(3.24)

Because there are jump processes N(t; j) in the expression, we apply Taylor

expansion to equation (3.23) and (3.24). We can verify that

dS(t) = S(t−)

[µ(t−)dt+ σ(t−)dW (t)− 1

2σ2(t−)dt+

k∑j=1

yX(t−),jN(dt; j)

−k∑j=1


)aX(t−),jdt+

1

2σ2(t−)dt

+1

2

k∑j=1

y2X(t−),jN(dt; j) +

∞∑l=3

1

l!

k∑j=1

ylX(t−),jN(dt; j)

]

= S(t−)

[µ(t−)dt+ σ(t−)dW (t)

+k∑j=1



)], (3.25)

53

dξ(t) = ξ(t−)

[η0(t−)dW (t)− 1

2η2

0(t−)dt+k∑j=1

ln(1 + ηX(t−),j

)N(dt; j)

−k∑j=1

aX(t−),jηX(t−),jdt+1

2η2

0(t−)dt

+∞∑l=2

1

l!

k∑j=1

[ln(1 + ηX(t−),j

)]lN(dt; j)

]

= ξ(t−)

[η0(t−)dW (t) +

k∑j=1


)]. (3.26)

Equations (3.23) and (3.24) satisfy the stochastic differential equations of S(t)

and ξ(t) and they are useful for us to understand the dynamic of stock price and

the change in probability measure.

3.3 Arrival Rates of Jumps under Risk Neutral

Measure

In the geometric Brownian motion model, Girsanov theorem is used to change

the real probability measure to the risk neutral probability measure. Under a

risk neutral probability, the expected rate of return of the risky asset becomes

the risk-free rate. The same idea is used in this section. A risk neutral probability

can be obtained such that the underlying security and all of its derivatives have

the risk-free interest rate as the expected return. The volatility of the diffusion

part of the risky asset under the risk neutral measure is the same as that under

54

the real-world measure, but the arrival rates of jumps change when the measure

changes. The arrival rates of jumps, which are the same as the rates of state

transitions, will be discussed in this section.

To understand the probabilistic properties of the standard Brownian motion

W (t)t∈T and the Markov chain X(t)t∈T under the risk neutral measure, the

stochastic differential equations governing the evolution of these two processes

over time can be considered. ξ(T ) is the Radon Nikodym derivative, then,

d(ξ(t)W (t)) = ξ(t−)dW (t) +W (t)dξ(t) + 〈dW (t), dξ(t)〉

= ξ(t−)dW (t) +W (t)dξ(t) + ξ(t−)η0(t−)dt. (3.27)

W (t) and ξ(t) are (G,P)-martingales, and G := Gtt∈T . Using the above equa-

tion, another process can be studied,

d

[ξ(t)(W (t)−

∫ t

0

η0(s−)ds)

]= ξ(t−)

[dW (t)− η0(t−)dt

]+(W (t)−

∫ t

0

η0(s−)ds)dξ(t) +[

dW (t)− η0(t−)dt]dξ(t)

= ξ(t−)dW (t) +(W (t)−

∫ t

0

η0(s−)ds)dξ(t). (3.28)

Therefore, we know thatξ(t)

(W (t)−

∫ t0η0(s−)ds

)is a (G,P)-martingale.

Obviously, the quadratic variation of (W (t)−∫ t

0η0(s−)ds) is t. By Levy’s theorem,

it is a standard Brownian motion under Q.

55

Similarly, the Markov chain can be studied using stochastic differential equa-

tion, but instead of studying the Markov chain directly, the point processes

N(t; j)t∈T ,j∈K derived from the Markov chain are our focus. If the regime

is at state X(t−) just before time t, the point process N(t; j) has arrival rate

equal to aX(t−),j, then

d

(ξ(t)N(t; j)

)= ξ(t−)N(dt; j) +N(t−; j)dξ(t) +N(dt; j)dξ(t)

= ξ(t−)N(dt; j) +N(t−; j)dξ(t) + ξ(t−)η(t−; j)N(dt; j)

= ξ(t−)(1 + η(t−; j))(N(dt; j)− aX(t−),jdt) +N(t−; j)dξ(t)

+ξ(t−)(1 + η(t−; j))aX(t−),jdt. (3.29)

As X(t−) is a left continuous function of t, N(t−; j) represents the number of

state transitions to state j from time 0 to just before time t. We now consider

56

another process,

d

[ξ(t)

(N(t; j)−

∫ t

0

(1 + η(s−; j))aX(s−),jds)]

= ξ(t−)(N(dt; j)− (1 + η(t−; j))aX(t−),jdt)

+(N(t−; j)−

∫ t

0

(1 + η(s−; j))aX(s−),jds)dξ(t)

+dξ(t)(N(dt; j)− (1 + η(t−; j))aX(t−),jdt)

= ξ(t−)(N(dt; j)− aX(t−),jdt)

+(N(t−; j)−

∫ t

0

(1 + η(s−; j))aX(s−),jds)dξ(t)

−ξ(t−)η(t−; j)aX(t−),jdt+ ξ(t−)η(t−; j)N(dt; j)

= ξ(t−)(1 + η(t−; j))(N(dt; j)− aX(t−),jdt)

+(N(t−; j)−

∫ t

0

(1 + η(s−; j))aX(s−),jds)dξ(t). (3.30)

By the definition of the Radon Nikodym derivative ξ(T ), which is always positive,

the risk neutral probability measure Q is equivalent to the real probability mea-

sure P . N(t; j) is still a jump process under Q. Furthermore, if we divide a time

interval [s, t] into m pieces with mesh Π, and ∆N(tl; j) represents the change of

57

N(t; j) in the corresponding lth small time interval, then

EQ[

limΠ→0

m∑l=0

∆N(tl; j)∆N(tl; j)]

= E[ξ(T ) lim

Π→0

m∑l=0

∆N(tl; j)∆N(tl; j)]

= E[ξ(T ) lim

Π→0

m∑l=0

∆N(tl; j)]

= EQ[

limΠ→0

m∑l=0

∆N(tl; j)]. (3.31)

This shows that N(t; j) is a point process under Q. From stochastic differential

equation (3.30), the arrival rate of N(t; j) is (1 + η(t−; j))aX(t−),j under Q. That

is, under the risk neutral probability measure Q, the arrival rate matrix A∗ is

[a∗ij]k×k =

0 (1 + η12)a12 · · · (1 + η1k)a1k

(1 + η21)a21 0 · · · (1 + η2k)a2k

......

. . ....

(1 + ηk1)ak1 (1 + ηk2)ak2 · · · 0

. (3.32)

and the corresponding generator matrix A∗ is

[a∗ij]k×k =

a∗11 (1 + η12)a12 · · · (1 + η1k)a1k

(1 + η21)a21 a∗22 · · · (1 + η2k)a2k

......

. . ....

(1 + ηk1)ak1 (1 + ηk2)ak2 · · · a∗kk

(3.33)

where

a∗ii = −∑j 6=i

(1 + ηij)aij = −∑j 6=i

(1 + ηij)aij. (3.34)

58

3.4 Trinomial Tree Pricing under Jump Diffu-

sion Model

Based on the result of the last section, we are able to find the price of the deriva-

tives using trinomial tree that developed in Chapter 2. The life of the derivative

T can be divided into N time steps with length ∆. Some values of σ that sat-

isfied equation (2.3), for example, the one that suggested by (2.9), can be used.

eσ√

∆ is used as the ratio in the lattice. Each of the nodes accommodates price

information of k states. qij(∆) denotes the probability of state transition from

state i to state j in a time step ∆ under the risk neutral probability Q, which

can be found by the generator matrix A∗ using the matrix exponential suggested

by (2.10). That is,

[qij(∆)]k×k =

q11(∆) · · · q1k(∆)

.... . .

...

qk1(∆) · · · qkk(∆)

= eA∗∆. (3.35)

With no ambiguity, we write qij(∆) as qij.

For the regime i, πiu, πim and πid are the risk neutral probabilities corresponding

to stock price increases, remains the same and decreases, respectively. Due to the

presence of jumps in stock price, the values of these probabilities change. For

59

each i ∈ K, we have,

(πiue

σ√


∆)( k∑

r=1

qijeyij

)= eri∆, (3.36)


i ∆. (3.37)

Equation (3.36) can be rearranged and expressed as

πiueσ√


∆ = exp(ri∆− ln

( k∑r=1

qijeyij

)). (3.38)

If λi is defined as σ/σi for each i, then, λi > 1 and the value of πiu, πim, π

id can

be calculated in terms of λi,

πim = 1− σ2i

σ2= 1− 1

λ2i

, (3.39)

πiu =eri∆−ln(

∑kr=1 qije

yij ) − e−σ√

∆ − (1− 1/λ2i )(1− e−σ

√∆)

eσ√

∆ − e−σ√

∆, (3.40)

πid =eσ√

∆ − eri∆−ln(∑k

r=1 qijeyij ) − (1− 1/λ2

i )(eσ√

∆ − 1)

eσ√

∆ − e−σ√

∆. (3.41)

As all the regimes share the same lattice and the regime state cannot be

reflected by the position of the nodes, each of the nodes has k possible prices

corresponding to the regime state at the node. At time step t, there are 2t + 1

nodes in the lattice, the node is counted from the lowest stock price level, and

St,n,j denotes the stock price of the nth node at time step t under regime j. The

stock prices in different regimes sharing the same node are different to the others

and the difference correspond to the ratio of jump that is given by yij. So, we

60

have

St,n,i = St,n,jeyji for all i, j ∈ K. (3.42)

Given the initial stock price and regime state, the stock prices at each node of the

lattice in different regimes can be found. Let Vt,n,j be the value of the derivative

at the nth node at time step t under the jth regime state. The trinomial model

can price European options, American options and barrier options. If we consider

a European call option at time T , we have

VN,n,i = (SN,n,i −K)+ for all i ∈ K, all n. (3.43)

Now, with the derivative prices in all regimes at expiration, by conditioning

on the economic state and the stock price level after one time step, we can apply

the following equation recursively,

Vt,n,i = e−ri∆

[k∑j=1

qij(πiuVt+1,n+2,j + πimVt+1,n+1,j + πidVt+1,n,j)

], (3.44)

and the price of the option can be obtained.

The price of American option can be calculated using the method suggested

in Chapter 2, by comparing the value of option at different regimes and its value

when it is exercised immediately at each node.

However, there is a problem when we price barrier options. Boyle and Tian

(1998) observes that the price of the barrier option obtained by the lattice model

61

is more accurate when the grid of the lattice touches the barrier level. Under our

jump diffusion model, the prices of stock at different regimes are not necessarily

the same even if they are in the same node. The stock price of all regimes might

not touch the barrier level at grids. We apply the method of Boyle and Tian

(1998) to start the lattice at the barrier level at a particular regime so that the

grid can touch the barrier level at least at one regime. In the two-regime case,

the value of σ can be adjusted so that the grid can touch the barrier level in

the lattice for both regimes. When more regimes are involved, it seems hard to

ensure that the grid touches the barrier level in the lattice at all regimes. It is

even harder for the stock prices at grid to touch both barriers for a double barrier

option in two or more regimes. More time steps in the lattice are required to

reduce the error.

We have to be careful about the difference between the model without jump

and the jump diffusion model under the risk neutral probability. Under the risk

neutral probability measure Q, the stock price process is given by

S(t) = S(0) exp

[ ∫ t

0

r(s−)ds+

∫ t

0

σ(s−)dW (s) +k∑j=1

∫ t

0

yX(s−),jN(ds; j)

−1

2

∫ t

0

σ2(s−)ds−k∑j=1

∫ t

0

(exp(yX(s−),j)− 1

)a∗X(s−),jds

]. (3.45)

The stock price jumps when there is regime switching. Given the same volatility

for the diffusion part, the jump process gives extra variability to the stock price

62

and can result in a higher option price than that in the model with no jump.

The stock prices of different regimes are different even if they share the same

node. When the option prices of different regimes are calculated, apart from

the effects of jumps on volatility, the risk-free interest rate, the intensity and

magnitude of jumps, the price of the underlying assets should also be considered

as they need not be the same at different regimes under this model.


In this section, we consider some examples and use them to study the properties

of our jump diffusion model. We study the convergence property and the effect

of the regime-switching risk. The value of the European options, the American

options and the down-and-out barrier options are considered. We first focus on

the case when the jump risk is not priced, then the option pricing when jump

risk is priced.

3.5.1 Jump Risks are Not Priced

Similar conditions are used so that the data obtained in this section can be

easily compared with those of the last chapter. We consider the market has two

regimes. The underlying asset is assumed to be a stock with initial price of 100

63

in Regime 1, following a geometric Brownian motion of the two-regime model

with no dividend. In Regime 1, the risk-free interest rate is 4% and the diffusion

volatility of stock is 0.25; in Regime 2, the risk-free interest rate is 6% and the

diffusion volatility of stock is 0.35. All options expire in one year with strike price

equal to 100. The generator for the regime-switching process under P is taken as −0.5 0.5

0.5 −0.5

.

The jump parameters yiji,j∈K are summarized in the following matrix,

Y =

0 y12

y21 0

=

0 .1

−.1 0

(3.46)

From this, we can obtain the price of the stock in all regimes at each node in the

lattice using (3.42). For example, initial stock price in Regime 2 is 100exp(0.1).

The price of jump risk is taken to be zero so we have

η =

0 η12

η21 0

=

0 0

0 0

. (3.47)

With the value of η, under the risk neutral probability measure Q, the arrival

rate matrix is

A∗ =

0 (1 + η12)a12

(1 + η21)a21 0

=

0 0.5

0.5 0

. (3.48)

With all the information given above, we can use the trinomial tree to find the

prices of options.

64

Table 3.1: Pricing European call option when jump risk is not priced


Regime 1, 100 Regime 2, 110.5171


20 12.9940 0.0862 0.2575 23.2553 0.0316 -0.6709

40 13.0802 0.0222 0.6532 23.2869 -0.0212 0.1981

80 13.1024 0.0145 0.7586 23.2657 -0.0041 -1.0714

160 13.1169 0.0110 0.2636 23.2615 0.0045 -0.5111

320 13.1279 0.0029 0.9310 23.2660 -0.0023 -0.4348

640 13.1308 0.0027 0.2593 23.2637 0.0010 -0.6000

1280 13.1335 0.0007 0.7143 23.2647 -0.0006 0.0000

2560 13.1342 0.0005 23.2641 0.0000

5120 13.1347 23.2641


calculated using various numbers of time steps and ratio is the ratio of the difference. The

value next to the regime number is the initial stock price at this regime.

Table 3.2: Pricing European put option when jump risk is not priced

European Put Option

Regime 1, 100 Regime 2, 110.5171


20 8.73688 0.07863 0.2336 7.24824 0.03917 -0.4447

40 8.81551 0.01837 0.6892 7.28741 -0.01742 0.1355

80 8.83388 0.01266 0.7915 7.26999 -0.00236 -2.3178

160 8.84654 0.01002 0.2445 7.26763 0.00547 -0.3400

320 8.85656 0.00245 1.0082 7.27310 -0.00186 -0.7043

640 8.85901 0.00247 0.2146 7.27124 0.00131 -0.4122

1280 8.86148 0.00053 0.9623 7.27255 -0.00054 -0.1296

2560 8.86201 0.00051 7.27201 0.00005

5120 8.86252 7.27208

65

In Table 3.1 and Table 3.2, we can see the convergence patterns, they are not

as smooth as those in the non-jump model shown in Table 2.2 and Table 2.3.

The jumped stock price during the state transition makes the situation more

complicated.

In the following, we will investigate the effect of the stock price jump by

comparing the numerical results for the models with and without jumps. In the

non-jump model, we assume that the risk-free rate and the diffusion volatility at

each regime are the same as that in the model with jump, but the dynamic of

the stock price process is a regime-switching geometric Browian motion without

jump. We also assume the same generator for both regime-switching processes.

Since we have different initial prices for the stock at different regimes in the

model with jumps, we use different initial prices at different regimes for the

model without jump to compare the prices of options in these two models. The

results are given in Tables 3.3 and 3.4. For the same initial stock price, diffusion

volatility, risk-free interest rate and keeping all the other assumptions, the jump

model gives a higher price. This is due to the jump. We can observe that the

convergence behavior in the jump diffusion model is more complex that that in

the continuous model. We can also observe that, at both regimes, the differences

between the call option price and put option price in the two models converge

to the same value. This is because of the put-call parity. This is shown in the

66

Table 3.3: Comparison of European call option prices in jump and non jump

models

European Call Options

Regime 1, 100 Regime 2, 110.5171

N Jump Non Jump Diff Jump Non Jump Diff

20 12.9940 12.6282 0.3658 23.2553 23.0144 0.2409

40 13.0802 12.6936 0.3688 23.2869 23.0464 0.2405

80 13.1024 12.7260 0.3764 23.2657 23.0033 0.2624

160 13.1169 12.7422 0.3747 23.2615 22.9917 0.2698

320 13.1279 12.7502 0.3776 23.2660 22.9965 0.2695

640 13.1308 12.7503 0.3765 23.2637 22.9915 0.2722

1280 13.1335 12.7563 0.3772 23.2647 22.9926 0.2721

2560 13.1342 12.7573 0.3769 23.2641 22.9913 0.2728

5120 13.1347 12.7578 0.3769 23.2641 22.9911 0.2730

†Jump is referring to the jump diffusion model and Non Jump is referring to the original

model. Diff is the difference of option prices between these two models.

Table 3.4: Comparison of European put option prices in jump and non jump

models

European Put Options

Regime 1, 100 Regime 2, 110.5171


20 8.73688 8.37107 0.36581 7.24824 7.00727 0.24097

40 8.81551 8.42888 0.38663 7.28741 7.04691 0.24050

80 8.83388 8.45755 0.37633 7.26999 7.00756 0.26243

160 8.84654 8.47182 0.37472 7.26763 6.99784 0.26979

320 8.85656 8.47894 0.37762 7.27310 7.00357 0.26953

640 8.85901 8.48250 0.37651 7.27124 6.99910 0.27214

1280 8.86148 8.48428 0.37720 7.27255 7.00044 0.27211

2560 8.86201 8.48517 0.37684 7.27201 6.99920 0.27281

5120 8.86252 8.48561 0.37691 7.27208 6.99912 0.27296

67

following. The prices of risk from Brownian motion and from Markov chain are

fixed. The risk neutral probability measure and thus the corresponding prices of

the derivatives can be found. We consider the situation that the stock pays no

dividend. Let c and p be the prices of the European call and the put option,

respectively. In the two models, by the put-call parity, for any given regimes, we

have,

c− p = S0 −KEQ[exp(−

∫ T

0

r(s)ds)]. (3.49)

The generator matrices and thus the arrival rate matrices are the same for both

models as the jump risk is not priced. The expected discounting factor for the

exercise price K shown in equation (3.49) is the same for the two models. The

result indicates that when the call option has a higher price due to the extra

variability from the jump of stock, the put option price increases by the same

amount if regime-switching risk is not priced.

If we substitute the values of the European call option and the European put

option of the two models obtained by iterations to equation (3.49), the value of

the expected discounted factor in the equation can also be obtained at the two

regimes. The value of the expected discounted factor can also be obtained by

the generator matrix. The probability of the regime changing from state i to

state j after a period of time t is given by the ij entry of etA∗. The generator

matrix under Q is the same as the one under the probability measure P because

68

the jump risk is not priced. The expected discounted factor can be obtained by

considering the expected amount of time that the Markov chain spends on each

regime. If A∗ was non-singular matrix, we have,∫ T

0

exA∗dx = (A∗)−1

(eTA

∗ − I)

if det A∗ 6= 0. (3.50)

However, by equation (3.34), let 1k and 0k be the column vector of size k with

all entries equal to 1 and 0, respectively. We know that,

A∗(1k) = 0(1k) = 0k. (3.51)

Therefore, 0 is an eigenvalue of A∗ and thus A∗ is a singular matrix. Equation

(3.50) cannot be used to find the expected amount of time that the Markov chain

spends on each regime. Given we are in state i, the expected amount of time that

the Markov chain is in state j over the period of length T is defined as τij. From

the values obtained using 5120 time steps, at each regime, we can calculate the

fixed interest rate that is equivalent to the stochastic interest rate being used.

Each fixed interest rate, in turn, reflects the expected amount of time that the

market spends on the corresponding regime, given the current state.

For state 1 : r1 =−1

1ln

(100− 13.1347 + 8.86252

100

)= 4.37%

⇒ Expected proportion of time in Regime 1 =6− 4.37

6− 4= 81.5%.

For state 2 : r2 =−1

1ln

(100e1.1 − 23.2641 + 7.27208

100

)= 5.63%

⇒ Expected proportion of time in Regime 2 =6− 5.63

6− 4= 18.5%.

69

Table 3.5: Comparison of American call option prices in jump and non jump

models

American Call Options

Regime 1, 100 Regime 2, 110.5171


20 12.9940 12.6282 0.3658 23.2553 23.0144 0.2409

40 13.0802 12.6936 0.3688 23.2869 23.0464 0.2405

80 13.1024 12.7260 0.3764 23.2657 23.0033 0.2624

160 13.1169 12.7422 0.3747 23.2615 22.9917 0.2698

320 13.1279 12.7502 0.3776 23.2660 22.9965 0.2695

640 13.1308 12.7503 0.3765 23.2637 22.9915 0.2722

1280 13.1335 12.7563 0.3772 23.2647 22.9926 0.2721

2560 13.1342 12.7573 0.3769 23.2641 22.9913 0.2728

5120 13.1347 12.7578 0.3769 23.2641 22.9911 0.2730

The expected proportion of time that the Markov chain stays in each state ob-

tained here is very close to the values obtained by numerical estimation, which is

given by

∫ 1

0

exp

−0.5x 0.5x

0.5x −0.5x

dx ≈

0.816060 0.183940

0.183940 0.816060

. (3.52)

The approximation errors of equation (3.52) should be much smaller than that by

using the information of option prices. For the continuous model, similar results

can be found in Elliott, Chan and Siu (2005).

Tables 3.5 and 3.6 show the American call and put results. From the tables, we

can observe that the prices of the American call option found by the lattice model

70

Table 3.6: Comparison of American put option prices in jump and non jump

models

American Put Options

Regime 1, 100 Regime 2, 110.5171


20 9.12138 8.80315 0.31823 7.59267 7.36070 0.23197

40 9.19402 8.85551 0.33851 7.62430 7.39476 0.22954

80 9.21489 8.88225 0.33264 7.60882 7.36040 0.24842

160 9.22818 8.89525 0.33293 7.60721 7.35298 0.25423

320 9.23700 8.90125 0.33542 7.61088 7.35672 0.25416

640 9.23974 8.90471 0.33503 7.60932 7.35306 0.25626

1280 9.24192 8.90627 0.33565 7.61021 7.35395 0.25626

2560 9.24254 8.90704 0.33550 7.60970 7.35286 0.25684

5120 9.24298 8.90742 0.33556 7.60971 7.35275 0.25696

shown in Table 3.5 are exactly the same as the European call option prices. This

is consistent with the theoretical result that the American call option should not

be exercised before expiration. If we use C to denote the price of the American

call, for any given regimes, we would have the following inequality at time t before

expiration,

C ≥ c ≥ St −KEQ[

exp(−∫ T

t

r(s)ds)]≥ St −K. (3.53)

The American call is more valuable than the European call because of its early

exercise option. By equation (3.49), we use the fact that European put option

should have a non-negative value, we have the second inequality. The risk-free

interest rate under all regimes should be positive and thus the expected discounted

71

Table 3.7: Comparison of down-and-out barrier call option prices in jump and

non jump models

Down-and-out Call Options

Regime 1, 100 Regime 2, 110.5171


20 9.19243 8.97860 0.21383 19.8270 19.5065 0.3205

40 9.59712 8.96621 0.63091 20.3482 19.4800 0.8682

80 9.20153 8.97478 0.22675 19.7730 19.4623 0.3107

160 9.11647 8.97064 0.14583 19.6408 19.4521 0.1887

320 9.20268 8.97010 0.23258 19.7587 19.4489 0.3098

640 9.11665 8.97025 0.14640 19.6336 19.4468 0.1868

1280 9.20246 8.96928 0.23318 19.7546 19.4452 0.3094

2560 9.11626 8.96962 0.14664 19.6316 19.4451 0.1862

5120 9.11352 8.96955 0.14397 19.6272 19.4448 0.1824

factor should also be less than 1 unless t = T . Therefore, we obtain the third

inequality. The value St−K is just the value of the American call if it is exercised

immediately with a positive value. The price of the American call can be equal

to its immediately exercise price only if t = T . In all the other situations, the

price of the American call option is greater than its immediately exercise value.

Early exercise is always not optimal for American call option.

The price of the American put option is higher than that of the European

put option and the option price in the jump model is higher than that in the

continuous model. Both observations are consistent with our intuition and the

theoretical results in finance.

72

In Table 3.7, we can find that the convergence speed of the down-and-out

barrier call option is not stable, compared with the data of non-jump model,

even we have chosen an optimal σ value to ensure that the stock prices touch the

barrier level in both regimes, especially when N is small.

3.5.2 Jump Risks are Priced

We consider the option pricing when the jump risk is priced in this subsection.

We assume that all the set-up and conditions are the same as those in the last

subsection, except that the price of the jump risk is not 0 here. The price of the

jump risk is taken as

η =

0 η12

η21 0

=

0 −0.1

0.1 0

. (3.54)

With the value of η, under risk neutral probability measure Q, the arrival rate

matrix A∗ is

[a∗ij] =

0 (1 + η12)a12

(1 + η21)a21 0

=

0 0.45

0.55 0

. (3.55)

We can apply the trinomial tree method and find the price of the options. The

values are compared with the values of which the jump risk is not priced as shown

in the last subsection.

When jump risk is priced, we might expect that the prices of the options is

73

Table 3.8: Comparison of European call option prices with priced and non-priced

jump risk

European Call Options

Regime 1, 100 Regime 2, 110.5171

N Priced Non Priced Diff Priced Non Priced Diff

20 12.9940 12.8789 -0.1151 23.2553 23.1855 -0.0698

40 13.0802 12.9619 -0.1183 23.2869 23.2153 -0.0716

80 13.1024 12.9845 -0.1179 23.2657 23.1951 -0.0706

160 13.1169 12.9990 -0.1179 23.2615 22.1911 -0.0704

320 13.1279 13.0095 -0.1184 23.2660 22.1953 -0.0707

640 13.1308 13.0125 -0.1183 23.2637 22.1931 -0.0706

1280 13.1335 13.0151 -0.1184 23.2647 22.1941 -0.0706

2560 13.1342 13.0158 -0.1184 23.2641 22.1935 -0.0706

5120 13.1347 13.0163 -0.1184 23.2641 22.1935 -0.0706

†Priced is referring to the priced jump risk prices and Non Priced is referring to the prices

that jump risk is not priced. Diff is the option price of priced model minus non priced model.

greater. However, in Table 3.8 to Table 3.12, we calculate the prices of various

options for the two situations, the prices of all types of options are lower if jump

risk is being priced. The reason is that, under the risk neutral probability, the

only difference in pricing between the two models is the arrival rate matrix. When

the jump risk is priced, under the risk neutral probability measure, the market

spends more time on the first regime in which the underlying asset has a smaller

diffusion volatility and a smaller risk-free interest rate. This results in a lower

option price when the jump risk is priced.

74

Table 3.9: Comparison of European put option prices with priced and non-priced

jump risk

European Put Options

Regime 1, 100 Regime 2, 110.5171


20 8.73688 8.65535 -0.08153 7.24824 7.21182 -0.03642

40 8.81551 8.73153 -0.08398 7.28741 7.24995 -0.03746

80 8.83388 8.75078 -0.08310 7.26999 7.23391 -0.03608

160 8.84654 8.76353 -0.08301 7.26763 7.23193 -0.03570

320 8.85656 8.77321 -0.08335 7.27310 7.23725 -0.03585

640 8.85901 8.77575 -0.08326 7.27124 7.23555 -0.03569

1280 8.86148 8.77814 -0.08334 7.27255 7.23683 -0.03572

2560 8.86201 8.77874 -0.08327 7.27201 7.23633 -0.03568

5120 8.86252 8.77920 -0.08332 7.27208 7.23640 -0.03568

Table 3.10: Comparison of American call option prices with priced and non-priced

jump risk

American Call Options

Regime 1, 100 Regime 2, 110.5171


20 12.9940 12.8789 -0.1151 23.2553 23.1855 -0.0698

40 13.0802 12.9619 -0.1183 23.2869 23.2153 -0.0716

80 13.1024 12.9845 -0.1179 23.2657 23.1951 -0.0706

160 13.1169 12.9990 -0.1179 23.2615 22.1911 -0.0704

320 13.1279 13.0095 -0.1184 23.2660 22.1953 -0.0707

640 13.1308 13.0125 -0.1183 23.2637 22.1931 -0.0706

1280 13.1335 13.0151 -0.1184 23.2647 22.1941 -0.0706

2560 13.1342 13.0158 -0.1184 23.2641 22.1935 -0.0706

5120 13.1347 13.0163 -0.1184 23.2641 22.1935 -0.0706

75

Table 3.11: Comparison of American put option prices with priced and non-priced

jump risk

American Put Options

Regime 1, 100 Regime 2, 110.5171


20 9.12138 9.03967 -0.08171 7.59267 7.55410 -0.03857

40 9.19402 9.10973 -0.08429 7.62430 7.58468 -0.03962

80 9.21489 9.13110 -0.08379 7.60882 7.57031 -0.03851

160 9.22818 9.14429 -0.08389 7.60721 7.56895 -0.03826

320 9.23700 9.15286 -0.08414 7.61088 7.57252 -0.03836

640 9.23974 9.15563 -0.08411 7.60932 7.57108 -0.03824

1280 9.24192 9.15774 -0.08418 7.61021 7.57194 -0.03827

2560 9.24254 9.15838 -0.08416 7.60970 7.57147 -0.03823

5120 9.24298 9.15882 -0.08416 7.60971 7.57149 -0.03822

Table 3.12: Comparison of down-and-out call option prices with priced and

non-priced jump risk

Down-and-out Call Options

Regime 1, 100 Regime 2, 110.5171


20 9.19243 9.16036 -0.03207 19.8270 19.8061 -0.0209

40 9.59712 9.57579 -0.02133 20.3482 20.3321 -0.0161

80 9.20153 9.16939 -0.03214 19.7730 19.7497 -0.0233

160 9.11647 9.08205 -0.03441 19.6408 19.6162 -0.0246

320 9.20268 9.17050 -0.03218 19.7587 19.7347 -0.0240

640 9.11665 9.08220 -0.03445 19.6336 19.6086 -0.0250

1280 9.20246 9.17026 -0.03220 19.7546 19.7305 -0.0241

2560 9.11626 9.08178 -0.03448 19.6316 19.6063 -0.0250

5120 9.11352 9.07897 -0.03455 19.6272 19.6021 -0.0251

76

3.6 Conclusions

Naik’s (1993) Markov regime-switching model (MRSM) has been re-examined and

extended to k-state. The trinomial tree method developed in last chapter was

shown to be useful for finding the price of European options, American options

and barrier options by using the Markovian property of the regime-switching

process. Unlike the MRSM used in Elliott, Chan and Siu (2005), Naik’s model

allows the stock to jump when the regime switches. The jump of stock price

in Naik’s model theoretically gives ground to price regime-switching risk. If the

stock price dynamic of all underlying assets does not have a jump term, we think

that the regime-switching risk is not a fundamental risk, and need not be priced.

Therefore, Naik’s model is a good starting point if we want to price the regime-

switching risk.

We first considered the market where the jump risk during regime switching

was not priced. It is found that the possibility of stock price jump could result

in a greater option price as it increases the total volatility of the stock. For

the market with priced jump risk, the risk-free transition probability was found

by modifying the real transition probability and used to calculate the price of

derivatives. Therefore, the effect of priced jump risk can be understood more

easily by considering its effect on the pricing transition probability.

77

Chapter 4

Pricing Asian Option and Related EIAs

4.1 Introduction

In Chapter 2, we introduce the multi-state trinomial tree model to price the

European options, American options and barrier options in the Markov regime-

switching model (MRSM) of Buffington and Elliott (2002). The model is extended

to a MRSM with jump in Chapter 3. For strong path-dependent options, like

Asian option, the payoff of the derivatives depends on the path of the underlying

asset price process. We cannot apply the tree model directly to calculate their

price.

Equity-indexed annuities (EIA) is a popular insurance product in recent years.

The design of EIA is flexible and that makes the valuation of EIA a challenging

task. We use the method in Hull and White (1993) to price the Asian option in

our tree model and use the results to price some Asian-option-related EIAs by

iterative equations. We also identify the problem of quadratic approximation as

suggested by Hull and White (1993) and introduce a simple solution to it.

78

4.2 A Modified Trinomial Lattice

When we want to price Asian options, the average stock price plays a role which

makes the pricing of the Asian option more complex. The average asset price

depends on the path of the stock price process and can take a lot of different values

which cannot be reflected by the node position directly in the lattice model. This

causes the pricing of this strong path-dependent option a difficult problem in the

past. Hull and White (1993) presents an idea of representative sets of values, the

prices of the Asian options with the average stock price equal to representative

sets of values are calculated; when the average stock price level is not in the sets,

linear approximation is used to obtain the option price. The idea is a natural

extension of the lattice model and can be used in this chapter.

We assume T to be the expiration time of the option, N to be the number of

time steps, then ∆ = T/N . At time step t, there are 2t+ 1 nodes in the lattice,

the node is counted from the lowest stock price level, and St,n denotes the stock

price of the nth node at time step t. We let S0 = S0,0 be the initial stock price,

u = eσ√

∆ and d = u−1.

At a particular node, the highest average stock price can be achieved when the

stock price increases continuously and then decreases and reaches this node, and

similarly, the stock price decreases continuously and then increases can result in

79

the lowest average stock price value. Instead of calculating their values directly,

we use recursive method to find the highest average and lowest average of each

node in the lattice. We consider the node (t, n). If n equals to 0 or 2t, in all

time intervals, the stock price only drops or rises, the maximum average and

minimum average stock price level would be the same in these two situations.

We let ASmax(t, n) and ASmin(t, n) be the highest and lowest average stock price

at node (t, n). We have

ASmin(t, 0) = ASmax(t, 0) =S0

t

[1− dt+1

1− d− 1

2(1 + dt)

], (4.1)

ASmin(t, 2t) = ASmax(t, 2t) =S0

t

[ut+1 − 1

u− 1− 1

2(1 + ut)

]. (4.2)

The weight of the first and last node is half of the others and we base on this

method to find the average stock price in this section. We can also use the

approximation that each time step has the same weight. These approximations

are consistent with the discrete lattice model used in Hull and White (1993).

For an arbitrary node in the lattice, say (t, n), the path of stock price results in

lowest average comes from (t−1, n−2) just before reaching (t, n) and the highest

average comes from (t−1, n). This provides us with an easy way to calculate the

lowest and highest averages recursively at each node. However, we note that the

argument has a little problem, for the node (t, 1), there is no (t − 1,−1) for the

minimum path, and similarly, there is no (t − 1, 2t − 1) for the maximum path

of node (t, 2t − 1). Because we are now using a trinomial tree, the stock price

80

is allowed to remain unchange in any time interval. The lowest average for (t, 1)

is achieved when the stock price remains unchanged in the last step, so is the

highest average for (t, 2t− 1). Therefore, we have

ASmin(t, 1) = ASmin(t, 0) +1

2tS0d

t(u− 1), (4.3)

ASmax(t, 1) = ASmin(t, 1) +1

tS0(1− dt−1), (4.4)

ASmax(t, 2t− 1) = ASmax(t, 2t)−1

2tS0u

t(1− d), (4.5)

ASmin(t, 2t− 1) = ASmax(t, 2t− 1)− 1

tS0(ut−1 − 1). (4.6)

For the other nodes where n is not equal to 0, 1, 2t− 1, 2t, we have

ASmin(t, n) =1

t

[(t− 1)ASmin(t− 1, n− 2) +

1

2S0(un−2−t+1 + un−t)

],(4.7)

ASmax(t, n) =1

t

[(t− 1)ASmax(t− 1, n) +

1

2S0(un−t+1 + un−t)

]. (4.8)

We can calculate the number of representative values of each node based on

the idea in Hull and White (1993). The representative levels are taken as the form

S0emh where h is a constant and m is an integer (can be negative or zero) known

as the representative value. We first find the largest and smallest representative

values at each node. Let Mmin(t, n) and Mmax(t, n) represent the minimum and

the maximum value of m in node (t, n), we have

Mmin(t, n) = bln(ASmin(t, n))/hc , (4.9)

Mmax(t, n) = dln(ASmax(t, n))/he , (4.10)

81

where b. . .c and d. . .e represent the floor and the ceiling integral value, respec-

tively. Then, Mmin(t, n) and Mmax(t, n) are the possible range of representative

values of node (t, n) and we can calculate the price of the Asian option at each

node with the average stock price equals to all the representative levels of the

form S0emh. That is, the value m represents the average price level.

As all the regimes share the same lattice and the regime state cannot be re-

flected by the position of the nodes, each node has k possible derivative prices

corresponding to the current regime state at that node. The derivative price also

depends on the path of the stock which is now summarized using the represen-

tative value and each node can take one of the representative values between

Mmin(t, n) and Mmax(t, n) and obviously cannot be shown directly by the node

position. At the nth node of time step t under the jth regime state, let Vt,n,m,j

be the value of the derivative with representative value m, and V a(t, n, s, j) be

the value of the derivative with average stock price s. By definition, we have

Vt,n,m,j = V a(t, n, S0emh, j).

We assume that the Markov chain is independent of the Brownian motion

under the real market measure P , and the transition probabilities are not af-

fected by changing the probability measure from the physical probability to the

risk neutral probability. The transition probability does not change when the

probabilities assigned to the three branches change from the real measure to a

82

risk neutral measure and they can be considered separately during the calcula-

tion of the expectation. The price of the derivative at each node can be found by

iteration. We start from the expiration time, for example, for an average price

call option with strike price K,

VN,n,m,i = (S0emh −K)+ for all states i and all nodes n. (4.11)

Similar to the previous chapters, qij denotes the pricing transition probability

from regime i to regime j, πiu, πim, πid denote the risk neutral probabilities of the

stock price increases, remains unchanged and decreases, respectively, in the tree.

With the derivative payoff at expiration, using the following equation recursively

can help obtain the price of the Asian option,

Vt,n,m,i = V a(t, n, S0emh, i)

= e−ri∆k∑j=1

qij

[πiuV a(t+ 1, n+ 2,

tS0emh + St,n(1 + u)/2

t+ 1, j)

+πimV a(t+ 1, n+ 1,tS0e

mh + St,nt+ 1

, j)

+πidV a(t+ 1, n,tS0e

mh + St,n(1 + d)/2

t+ 1, j)].(4.12)

When the average price level is not at a representative level, we use a linear

approximation. If we have m = bln(AVE)/hc where AVE denotes the average

83

stock price value, then

V a(t, n, AVE, j) =AVE − S0e

mh

S0emh(eh − 1)V a(t, n, S0e

(m+1)h, j)

+S0e

(m+1)h − AVES0emh(eh − 1)

V a(t, n, S0emh, j)

=AVE − S0e

mh

S0emh(eh − 1)Vt,n,m+1,j +

S0e(m+1)h − AVE

S0emh(eh − 1)Vt,n,m,j

(4.13)

and the price of the option in all regimes can be obtained.

For an average strike option, the above method is still valid. Taking an average

strike call option as an example, we have

VN,n,m,i = (SN,n − S0emh)+ for all states i and all nodes n. (4.14)

Using the above recursive method, we can obtain the option price.

4.3 Pricing Equity-Indexed Annuities

Equity-indexed Annuities (EIAs) can protect the investors from a drop in equity

index and at the same time allow them to have a profit when the index appreci-

ates, which is an attractive feature for the investors. For the people who want to

do some investments in order to support their living after retirement, they do not

want to expose to the high risk of stocks but at the same time fear that the high

grade bonds cannot offer them a significant real return. The guarantee return of

84

EIA allow them to prepare for their living after retirement more easily as they

know at least how much they can obtain from EIA, and the EIA has a potential

to gain upside profits when the price of the reference asset appreciates.

For a long term investment, it is obvious that the expected return rate, the

volatility of stock prices and equity indices are not constant and it is more rea-

sonable to apply some stochastic models for the return rate and volatility when

we price an EIA as we know a small derivation in volatility can result in a large

error in EIA pricing.

Now we treat the security process S(t) as the equity index process and A(t)

is the average index level over time 0 to time t, that is

A(t) =1

t

∫ t

0

S(u)du. (4.15)

Then, we consider a general expression of a point-to-point Asian EIA which is

similar to the one used in Lin and Tan (2003) and has a cumulative return equal

to

C(t) = max[min[1 + αRt, (1 + ζ)t], (1 + g)t], (4.16)

where Rt = A(t)/S0 − 1. Rt is the average return of the equity index over the

period from time 0 to t; α is the participation rate that shows the extra return

received by the investors per unit of the average return of the equity index; ζ is

the cap rate which is the maximum annual return that can be enjoyed by the

85

investors and g is the guarantee rate which is the minimum annual return of the

EIA contract. Apart from the guarantee rate, the payment of this EIA contact has

some favorite features for the investors. Rt is the average return over the period

which can provide the investors a more stable return and can better reflecting the

performance of the equity index over the entire period. Various values of α can

be chosen by the investors according to their own risk preference and they can

expose to different levels of equity index risk. Due to these protection features,

the participation rate which is the potential extra return that can be obtained by

the investors might be small, if the investors require a high guarantee return. ζ

is a upper bound or the maximum return rate. By introducing this ceiling rate,

the investors can choose a higher participation rate while keeping the level of

guarantee return and the price of EIA.

The cumulative return of the EIA is very similar to a collar,

max[min[1 + αRt, (1 + ζ)t], (1 + g)t]

= 1 + αRt − [1 + αRt − (1 + ζ)t]+

+[(1 + g)t − (1 + αRt) + [1 + αRt − (1 + ζ)t]+]+

= 1 + αRt − [1 + αRt − (1 + ζ)t]+ + [(1 + g)t − (1 + αRt)]+. (4.17)

The last equality holds because when 1 +αRt > (1 + ζ)t, the value of the second

option is zero when ζ > g. Therefore, we can delete the redundant term [1 +

αRt − (1 + ζ)t]+ in the last expression. Furthermore, we can write the return of

86

EIA in terms of standard Asian options,

[1 + αRt − (1 + ζ)t]+ =α

S0

[S0

α+ A(t)− S0 −

S0

α(1 + ζ)t

]+

=α

S0

[A(t)− S0(1 +

(1 + ζ)t − 1

α)

]+

(4.18)

and

[(1 + g)t − (1 + αRt)]+ =

α

S0

[S0(1 +

(1 + g)t − 1

α)− A(t)

]+

. (4.19)

From the characteristics of EIA, we know that it consists of two options. The

minimum return guarantee results in a long position of a put option and the cap

return results in a short position of a call option. Because we are now considering

the return of EIA and thus a factor S−10 is used to obtain the proportional return

of the options. α is the participation rate. We notice that the participation

rate might not be equal to one, that means the EIA being considered does not

necessarily change by the same amount as what the average equity index does,

even the average return of the equity index is within the guarantee level and

the cap level. The strikes of both options are also adjusted according to the

participation rate.

Investors can demand a guarantee return every year rather than just a guaran-

tee return for the whole contract period so that they can lock the return annually

to avoid the fluctuation of the equity index in the future. In this way, they can

have a better forecast about what they get at the expiration of contract. This

87

kind of EIA is called the annual reset EIA or ratchet EIA and its cumulative

return is given by,

C(t) =t∏

k=1

max[min(1 + αR′k, 1 + ζ), 1 + g], (4.20)

where R′k =∫ kk−1

S(u)du/S(k−1)−1, which is the average index return of the kth

year. If the equity index follows the Black-Scholes framework, the appreciation

rate of the index in a time interval is independent of the return rate in the previous

intervals due to the independent increment property of Brownian motion and

so the expected return of the whole contract period is equal to the product of

expected return in each year. However, in our MRSM, the future return rate

and volatility of the index are affected by the current data due to the presence of

regime switching. For example, if the return at this time point is low, there is a

higher probability that we are now in a regime with low expected return and thus

the expected return in the next time period will be low because it is likely that

we are still in this low return regime state. Fortunately, the regime-switching

process is a Markov process, we are able to determine the expected return in the

year with the regime information at the very beginning of the year. Therefore,

we are able to solve this problem by considering a conditional expectation.

Similar to the geometric Brownian motion model, the risk neutral probability

can be obtained so that the value of the derivatives can be calculated as the risk-

free discounted expectation of the payoff of the derivatives. Let Q denote the risk

88

neutral probability measure. The expected risk neutral discounted value of the

unit ratchet EIA for t years with initial regime i is denoted by Vr(t, i) which is

equal to

EQ

[exp(−

∫ t

0

r(u)du)t∏

k=1

max[min(1 + αR′k, 1 + ζ), 1 + g]

∣∣∣∣∣X(0) = xi

]

= EQ

[exp(−

∫ 1

0

r(u)du) max[min(1 + αR′1, 1 + ζ), 1 + g] EQ[

exp(−∫ t

1

r(u)du)

×t∏

k=2

max[min(1 + αR′k, 1 + ζ), 1 + g]∣∣∣G1, X(0) = xi

]∣∣∣∣∣X(0) = xi

]

= Vr(1, i)k∑

j1=1

pij1(1)Vr(t− 1, j1)

= Vr(1, i)

[k∑

j1=1

pij1(1)Vr(1, j1)

×[ k∑j2=1

· · ·Vr(1, jt−2)[ k∑jt−1=1

pjt−2jt−1(1)Vr(1, jt−1)]· · ·]]. (4.21)

Therefore, with the values of 1-year Asian option of different initial regimes,

together with the 1-year transition probability, we are able to find the value of

the ratchet EIA recursively using the above equation.

We can include the mortality component into the EIA contact. We assume

that the ratchet EIA is payable at the end of the year that the investor dies or

the EIA contract expires. There is no selection effect and the future lifetime

random variable is independent of the Brownian motion and regime-switching

process under the real and pricing measures. Furthermore, we assume that the

89

numbers of participants of this EIA contract in different ages are large and so

the insurance companies can diversify their mortality risk and are risk neutral

towards this risk. We denote the future lifetime, the real probability of death,

survival in one year of a x-year-old person by T (x), qx and px respectively. Let

FMt t∈T be the natural filtration of the future lifetime random variable. We

define Ht to be the σ-algebra FXt ∨ FZt ∨ FMt , using the same notations as in

Chapter 2. The expected risk neutral discounted value of the unit life dependent

ratchet EIA for t years with initial regime i for a x-year-old investor is denoted

90

as Vm,x(t, i) which is equal to

EQ

[exp

[−∫ min(t,dT (x)e)

0

r(u)du]

×min(t,dT (x)e)∏

k=1

max[min(1 + αR′k, 1 + ζ), 1 + g]

∣∣∣∣∣X(0) = xi

]

= EQ

[exp

[−∫ 1

0

r(u)du]

max[min(1 + αR′1, 1 + ζ), 1 + g]

×[I(T (x) ≤ 1) + I(T (x) > 1)EQ

[exp[−

∫ min(t,dT (x+1)+1e)

1

r(u)du]

×min(t,dT (x+1)+1e)∏

k=2

max[min(1 + αR′k, 1 + ζ), 1 + g]∣∣∣H1, X(0) = xi

]]∣∣∣∣∣X(0) = xi

]

= Vm,x(1, i)

[qx + px

k∑j1=1

pij1(1)Vm,x+1(t− 1, j1)

]

= Vm,x(1, i)

[qx + px

k∑j1=1

pij1(1)Vm,x+1(1, j1)

[qx+1 + px+1

k∑j2=1

· · ·

×Vm,x+t−2(1, jt−2)[qx+t−2 + px+t−2

k∑jt−1=1

pjt−2jt−1(1)Vm,x+t−1(1, jt−1)]· · ·]].

(4.22)

If we have the values of ratchet EIAs with different expirations, together with the

mortality information, we can find the life dependent ratchet EIA easily,

Vm,x(1, i) = Vr(1, i) for all i and all x, (4.23)

Vm,x(t, i) =t−1∑n=0

npx qx+n Vr(n+ 1, i) + tpx Vr(t, i), (4.24)

where npx is the probability of a x-year-old person survives for n years. Because

the EIA contracts can be expressed in terms of Asian options, their values can be

91

calculated using the trinomial tree method or with the above iterative equations.


The trinomial tree introduced in this chapter is both efficient and easy to use for

pricing Asian options and the other related derivatives like EIA. In the beginning

of this section, we compare our results with those of Hull and White (1993) to

show the similarity between our trinomial tree and the binomial tree without

regime switching. We study the properties of the option prices found by the tree

model. We also compare our results with those of Boyle and Draviam (2007) in

order to show the validity of this model under regime-switching model. Other

numerical results are presented so that we can have a better understanding of this

pricing method and the characteristics of Asian options and related derivatives.

In Table 4.1 and Table 4.2, we use different methods to approximate the price

of European type and American type average price call option. The adjusted

average takes the weights of the first and the last nodes in the lattice be half

of the others while the simple average takes all nodes with the same weight. It

shows that the option price obtained by using the trinomial lattice is very close to

the value obtained by Hull and White (1993), and that means trinomial tree has

a similar performance in pricing Asian options as binomial tree of Hull and White

(1993). The results of using simple average are closer to those of adjusted average

92

Table 4.1: Comparison of the prices of (Eurpean type) average price call options

in simple BS model (linear approximation of representative value)

Average Price Call Option (Eurpean type)

Number of Time Steps (N)

h Results 20 40 60 80

0.100 Adjusted Average 4.6756 4.6870 4.6904 4.6919

Simple Average 4.6419 4.6705 4.6792 4.6839

Hull and White 4.663 4.679 4.685 4.687


Simple Average 4.5794 4.6011 4.6083 4.6124

Hull and White 4.588 4.605 4.612 4.614


Simple Average 4.5119 4.5274 4.5242 4.5377

Hull and White 4.517 4.530 4.536 4.539


Simple Average 4.5079 4.5194 4.5243 4.5274

Hull and White 4.513 4.522 4.526 4.529


Simple Average 4.5073 4.5177 4.5217 4.5240

Hull and White 4.512 4.520 4.523 4.525

†The initial stock price and strike price are 50, the risk free interest rate is 10% and the

volatility is 0.30.

93

Table 4.2: Comparison of the prices of (American type) average price call options

in simple BS model (linear approximation of representative value)

Average Price Call Option (American Type)


h Results 20 40 60 80


Simple Average 5.0583 5.2179 5.2826 5.3207

Hull and White 5.197 5.311 5.360 5.377


Simple Average 4.9405 5.0524 5.1006 5.1286

Hull and White 4.971 5.080 5.124 5.145


Simple Average 4.8202 4.9033 4.9402 4.9608

Hull and White 4.823 4.906 4.941 4.962


Simple Average 4.8138 4.8904 4.9231 4.9414

Hull and White 4.815 4.892 4.924 4.942


Simple Average 4.8127 4.8875 4.9189 4.9357

Hull and White 4.814 4.890 4.920 4.936


volatility is 0.30.

94

when N is larger, which is obvious as more time steps being used, the effect of

individual time step is smaller. Increasing number of time steps increases the

numbers of nodes, and at the same time the possible range of average asset price

for nodes. Because using the adjusted average to do the calculation can improve

the consistency of approximating, we use this adjusted average approximation.

Since the calculation in Hull and White (1993) is based on simple averages, both

approximations are used to find the option prices so that the comparisons between

different methods shown in the tables are more meaningful.

The option prices are calculated for the set of different representative values

and that imposes further approximation error on option valuation. Hull and

White (1993) suggest that quadratic interpolation can be used to improve the

accuracy of the approximation, but our numerical results are different from what

we predicted. In Table 4.3 and Table 4.4, the prices of the options calculated

by quadratic approximation are significantly lower than the results found by

linear approximation, especially when h and N are large and the options have

early exercise option. Our goal of using quadratic interpolation is to reduce the

approximation error when h is large so that the method can be of high efficiency.

However, the quadratic approximation tends to underestimate the price of the

Asian options rather than provides a more accurate result when h is large.

The payoff of the Asian option is linear to the average asset price to a certain

95

Table 4.3: Comparison of the prices of (Eurpean type) average price call options

in simple BS model (quadratic and modified quadratic approximation, simple

average asset price)

Average Price Call Option (Eurpean type)


h Method 20 40 60 80

0.100 Modified Quad. 4.5949 4.6021 4.6059 4.6083

Quadratic Appro. 4.3660 4.3720 4.3736 4.3754

Hull and White 4.663 4.679 4.685 4.687

0.050 Modified Quad. 4.5353 4.5456 4.5493 4.5512

Quadratic Appro. 4.4779 4.4869 4.4903 4.4920

Hull and White 4.588 4.605 4.612 4.614

0.010 Modified Quad. 4.5075 4.5185 4.5219 4.5237

Quadratic Appro. 4.5058 4.5158 4.5192 4.5209

Hull and White 4.517 4.530 4.536 4.539

0.005 Modified Quad. 4.5068 4.5170 4.5206 4.5224

Quadratic Appro. 4.5066 4.5165 4.5200 4.5217

Hull and White 4.513 4.522 4.526 4.529

0.003 Modified Quad. 4.5068 4.5168 4.5203 4.5221

Quadratic Appro. 4.5067 4.5166 4.5201 4.5219

Hull and White 4.512 4.520 4.523 4.525


volatility is 0.30.

96

Table 4.4: Comparison of the prices of (American Type) average price call options

in simple BS model (quadratic and modified quadratic approximation, simple

average asset price)



h Method 20 40 60 80

0.100 Modified Quad. 4.9529 5.0615 5.1114 5.1352

Quadratic Appro. 4.5826 4.5321 4.5076 4.5053

Hull and White 5.197 5.311 5.360 5.377

0.050 Modified Quad. 4.8644 4.9268 4.9655 4.9733

Quadratic Appro. 4.7590 4.7932 4.8162 4.8238

Hull and White 4.971 5.080 5.124 5.145

0.010 Modified Quad. 4.8128 4.8875 4.9178 4.9336

Quadratic Appro. 4.8112 4.8848 4.9149 4.9307

Hull and White 4.823 4.906 4.941 4.962

0.005 Modified Quad. 4.8119 4.8861 4.9169 4.9331

Quadratic Appro. 4.8117 4.8857 4.9164 4.9325

Hull and White 4.815 4.892 4.924 4.942

0.003 Modified Quad. 4.8120 4.8859 4.9166 4.9329

Quadratic Appro. 4.8120 4.8858 4.9165 4.9327

Hull and White 4.814 4.890 4.920 4.936


volatility is 0.30.

97

extent. When the average price is below the strike, the payoff is zero. Once the

average price is greater than the strike, the payoff increases linearly with respect

to the average price. Therefore, the value of the Asian option at expiration is two

joined linear equations with a kink at strike. Normally quadratic interpolation

is a better approximation than linear approximation because the linear equation

is just a special case of a quadratic equation. However, when the average asset

price is close to the strike, as the payoff is in fact two joined linear equations,

it is obvious that linear interpolation provides an exact solution while quadratic

interpolation which takes three points to do the calculation underestimates the

payoff. Underestimation of the payoff takes effect at various nodes and makes

the final result lower than the true value. We can use linear approximation when

the average asset value is close to the strike and use quadratic approximation

elsewhere. This modification provides a better result. However, we know that

linear approximation tends to overestimate the price while quadratic approxima-

tion tends to underestimate the price, this kind of mixture effects might cause a

chaotic convergent pattern.

We then compare our method with Boyle and Draviam (2007). Assume that

the underlying asset with initial price of S0, and there are two regimes. At Regime

1, the diffusion volatility of the asset price is 15%; and at Regime 2, the diffusion

volatility of the asset price is 25%. The risk-free interest rate at both regimes

98

is 5%. The average price call option lasts for a year and the generator matrix

of the Markov chain regime switching process is

a11 a12

a21 a22

=

−1 1

1 −1

.

We compare the values of the options with different initial asset prices S0 and

different strikes K.

We can see that the results using simple average asset price are very close

to those of Boyle and Draviam (2007) in Table 4.5. Boyle and Draviam (2007)

use partial differential equations to solve the problem in which simple average

and linear approximation are used during discretization. Because the price of the

Asian option is a convex function of the average asset price, linear approximation

overestimates the price of options, and therefore, the modified quadratic approx-

imation tends to be smaller compared to the results of the other two methods.

The same observation is found when we are considering the price of options under

different sets of transition probabilities in Table 4.6.

Comparing with other valuation methods, lattice model has an inborn advan-

tage of obtaining the price of derivative with early exercise option easily. We

consider the same situation as we did in Table 4.5 but for the price of a average

price call option with early exercise option in Table 4.7.

Using equation (4.17), we are able to find the fair value of the EIA with the

Asian option prices obtained from the trinomial tree. In order to consider the

99

Table 4.5: Comparison of the price of average price call options in MRSM found

by different methods I

Average Price Call Option (European Type)

Regime 1 (σ1 = 0.15, r1 = 5%) Regime 2 (σ2 = 0.25, r2 = 5%)

S0 Method K = 90 K = 100 K = 110 K = 90 K = 100 K = 110

90 Simple 4.6104 1.1134 0.1969 5.8747 2.1804 0.6685

B&D 4.6204 1.1172 0.1966 5.8747 2.1808 0.6694

Modified 4.5964 1.0970 0.1899 5.8655 2.1688 0.6600

95 Simple 8.1076 2.6203 0.5802 9.1482 3.9854 1.4278

B&D 8.1132 2.6288 0.5809 9.1475 3.9850 1.4281

Modified 8.0937 2.6014 0.5668 9.1380 3.9731 1.4165

100 Simple 12.3354 5.1227 1.4520 13.0385 6.5274 2.7007

B&D 12.3374 5.1338 1.4574 13.0381 6.5274 2.7010

Modified 12.3253 5.1071 1.4331 13.0294 6.5172 2.6876

105 Simple 16.9513 8.5765 3.0859 17.3581 9.7668 4.6045

B&D 16.9523 8.5831 3.0956 17.3580 9.7659 4.6041

Modified 16.9453 8.5608 3.0651 17.3506 9.7554 4.5911

110 Simple 21.7336 12.7213 5.6350 21.9427 13.5779 7.1801

B&D 21.7353 12.7242 5.5472 21.9435 13.5774 7.1802

Modified 21.7306 12.7091 5.6179 21.9372 13.5675 7.1689

†B&D refers to the results of Boyle and Draviam (2007). For the lattice methods, N and h

are set to be 200 and 0.005 respectively. Simple refers to results using equal weight on each

node and linear approximation. Modified refers to results using adjusted weight on each node

and modified quadratic approximation.

100

Table 4.6: Comparison of the price of average price call options in MRSM ob-

tained by different methods II

Average Price Call Option (European Type)


S0 Method K = 90 K = 100 K = 110 K = 90 K = 100 K = 110

0 Simple 12.1595 4.7007 1.0821 13.2292 6.8674 3.0211

B&D 12.1617 4.7135 1.0864 13.2286 6.8684 3.0228

Modified 12.1515 4.6864 1.0638 13.2187 6.8561 3.0073

0.5 Simple 12.2631 4.9493 1.3001 13.1168 6.6663 2.8315

B&D 12.2651 4.9609 1.3053 13.1165 6.6669 2.8325

Modified 12.2538 4.9340 1.2817 13.1073 6.6559 2.8183

1 Simple 12.3354 5.1227 1.4520 13.0385 6.5274 2.7007

B&D 12.3374 5.1338 1.4574 13.0381 6.5274 2.7010

Modified 12.3253 5.1071 1.4331 13.0294 6.5172 2.6876

1.5 Simple 12.3877 5.2480 1.5609 12.9819 6.4279 2.6072

B&D 12.3899 5.2589 1.5667 12.9812 6.4273 2.6069

Modified 12.3772 5.2325 1.5421 12.9729 6.4177 2.5940

2 Simple 12.4267 5.3414 1.6417 12.9396 6.3542 2.5380

B&D 12.4291 5.3523 1.6480 12.9387 6.3531 2.5372

Modified 12.4160 5.3261 1.6231 12.9306 6.3438 2.5247

†The generator of the Markov chain is taken to be

−λ λ

λ −λ

.

101

effect of the interest rate in regime-switching model, we change the interest rate

in the Regime 2, r2, from 5% to 7% in the following analysis. Table 4.8 shows the

value of Vr(1, i) as defined in equation (4.21) with different guarantee and ceiling

rates; the participation rate α is equal to 1. We find that two approximation

methods are very close to each other. With the value of Vr(1, i), we can obtain

the price of ratchet EIAs easily using recursive equation. We make use of the

previous results and consider the case when ceiling and guarantee rates are equal

to 20% and 3%, respectively. The prices of unit annual reset EIA with different

expirations are shown in Table 4.9.

4.5 Conclusions

In this chapter, we have applied a modified trinomial tree method to obtain the

price of Asian option under the Markov regime-switching model (MRSM). We

have extended the trinomial tree model in Chapter 2 using the idea of represen-

tative values of Hull and White (1993) and obtained the fair value of various

Asian options in the MRSM. From the numerical results we found that quadratic

approximation as suggested by Hull and White (1993) significantly underesti-

mate values of the Asian options when average price level is close to the strike

price. We suggested to use linear approximation when average price level is close

to the strike price and use quadratic approximation elsewhere so that the un-

102

derestimation problem can be solved, and at the same time we can take the

advantage of quadratic approximation. The weights of the asset price at the

start and expiration can be taken as half of the other time points to improve the

approximation of average asset price. However, these improvements are not sig-

nificant when the number of time steps is large and the representative constant is

small. The structure of the lattice model can help find the price of Asian options

with early exercise option. The prices of Asian options can also be used to price

point-to-point EIA and ratchet EIA by using a simple iterative equation with the

Markovian property of the regime-switching process.

103

Table 4.7: Price of average price call options with early exercise option in MRSM



S0 Method K = 90 K = 100 K = 110 K = 90 K = 100 K = 110

90 Simple 5.0528 1.1510 0.1993 6.5088 2.2931 0.6845

Modified 5.0370 1.1333 0.1921 6.5067 2.2815 0.6757

95 Simple 9.2242 2.7757 0.5928 10.4691 4.2914 1.4812

Modified 9.2197 2.7548 0.5787 10.4824 4.2813 1.4694

100 Simple 14.1866 5.6142 1.5066 15.2530 7.2320 2.8513

Modified 14.1980 5.5967 1.4861 15.2867 7.2297 2.8385

105 Simple 19.2650 9.7290 3.2779 20.3650 11.1434 4.9701

Modified 19.2790 9.7214 3.2542 20.4078 11.1561 4.9590

110 Simple 24.3479 14.6381 6.1757 25.5137 15.8374 7.9552

Modified 24.3641 14.6491 6.1563 25.5604 15.8711 7.9526

104

Table 4.8: Price of one-year EIA in MRSM

Price of One-year EIA at Regime 1

Guarantee Ceiling Return

Return Method 5% 10% 15% 20% 25%

0% Simple 0.96881 0.98304 0.99085 0.99476 0.99659

Modified 0.96885 0.98307 0.99085 0.99470 0.99650

1% Simple 0.97292 0.98715 0.99496 0.99887 1.00070

Modified 0.97294 0.98716 0.99494 0.99879 1.00059

2% Simple 0.97741 0.99164 0.99946 1.00336 1.00520

Modified 0.97743 0.99165 0.99943 1.00328 1.00507

3% Simple 0.98229 0.99652 1.00434 1.00824 1.01008

Modified 0.98230 0.99652 1.00430 1.00815 1.00995

Price of One-year EIA at Regime 2

Guarantee Ceiling Return

Return Method 5% 10% 15% 20% 25%

0% Simple 0.96076 0.97743 0.98878 0.99610 1.00062

Modified 0.96079 0.97747 0.98880 0.99611 1.00061

1% Simple 0.96498 0.98166 0.99300 1.00032 1.00484

Modified 0.96500 0.98168 0.99301 1.00032 1.00482

2% Simple 0.96947 0.98615 0.99749 1.00481 1.00934

Modified 0.96949 0.98617 0.99750 1.00481 1.00931

3% Simple 0.97424 0.99092 1.00226 1.00958 1.01411

Modified 0.97425 0.99093 1.00227 1.00957 1.01407

105

Table 4.9: Price of unit annual reset EIA in MRSM

Regime 1

Length of Ratchet EIA (years)

Method 1 2 3 4 5 10 15 20

Simple 1.00824 1.01714 1.02619 1.03534 1.04457 1.09195 1.14149 1.19328

Modified 1.00815 1.01699 1.02599 1.03508 1.04425 1.09135 1.14057 1.19201

Regime 2

Length of Ratchet EIA (years)

Method 1 2 3 4 5 10 15 20

Simple 1.00958 1.01867 1.02776 1.03693 1.04617 1.09363 1.14325 1.19511

Modified 1.00957 1.01861 1.02765 1.03676 1.04595 1.09312 1.14243 1.19395

106

Chapter 5

Concluding Remarks

Option, which is expected to be a tool for hedging, is sometimes also used for

speculative purposes. It is very popular among some investors because of its high

leverage. Selling options can be a profitable business for the investment banks

and other financial institutions. However, option pricing is difficult and requires

profound knowledge in financial economics and long-time experience of trading

in markets. A simple option pricing method not only promotes the efficiency in

trading options but also reduces the cost for those who really wants to use options

to hedge their investment risks as options can be sold at a more reasonable price.

Stock price models become more complicated and various complex derivatives are

introduced into the market. Derivative pricing is challenging and is mainly done

by PDE and simulations. Simple methods like lattice model lose their glory.

Lattice models are simple and easy to understand. It is a good news if lattice

models can be used to find the price of options even in more advanced models.

Multi-state trinomial tree model was introduced to find a fair price of various

derivatives in MRSMs.

In Chapter 2, we focused on Buffington and Elliott’s (2002) model. We intro-

duced the multi-state trinomial tree model and used it to price different options.

107

The nature of regime-switching risk was discussed in details and we suggested

that this risk need not be priced under this model. Although pricing jump risk

need not result in arbitrage, jump risk can be eliminated in the market if the

fundamental asset price processes do not jump during regime switching.

To justify pricing the regime-switching risk, the model of Naik (1993) has

been studied and extended in Chapter 3 in which stock price jumps when regime

switches. By using the Markovian property of the regime-switching process, tri-

nomial tree is still efficient for option pricing under this model. The numerical

results were analyzed and it was found that some properties that are true in one-

state market are also true for the MRSM, for example, early exercise is also not

optimal for American call option in the MRSM if the underlying asset pays no

dividend.

Asian option is strongly path-dependent and we cannot directly use the ap-

proach in Chapter 2 and Chapter 3 to price Asian option. In Chapter 4, we

applied the method of Hull and White (1993) to price Asian option in Buffington

and Elliott’s (2002) MRSM. The problem of quadratic approximation suggested

in Hull and White (1993) was identified and a simple method was introduced to

solve this problem. The fair price of Asian-option-related EIAs can then be found

using iterative equations.

With the method to calculate the price of derivatives in MRSM, a more com-

108

plicated problem emerges, it is how to find an appropriate risk neutral probability

(and transition probability) from the market price of the options. One of the ma-

jor difficulties is that we can only have the price information of the current regime.

Further researches can focus in this area.

109

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