eval prod struct

8/17/2019 Eval Prod Struct

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Valuation of Structured

ProductsPricing of Commodity Linked Notes

Shahid Jamil, Stud nr: SJ80094

Academic Advisor: Jochen Dorn

Department of Business Studies

Aarhus School of Business, University of Aarhus

February 2011


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Abstract

Structured products including commodity linked structured products have complex

composition. These are suitable for those investors who want a complete or partial

protection of their investment. A typical structured product is a combination of a risk

free bond and an option. The bond part guarantees capital protection while the option

part provides the possibility of payoff. The option pricing is the tricky part of these

products and a wide range of theories are available to price them. In this thesis the well

known Black & Scholes option pricing frame work is applied and the theoretical

estimated price of the selected commodity linked structured notes are compared with

their issue price to evaluate if these products are offered to the investors at fair price.


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Table of Contents

1. Introduction .......................................................................................................................... 5

2. Structured Products .................................................................................................................. 8

2.1 Structured products are suitable for investors who .......................................................... 9

2.2 Disadvantage of structured Products ................................................................................ 10

2.3 Difference between a Conventional Bond and a Commodity Linked Bond ...................... 10

2.4 Commodity –Linked Bonds, a brief history ....................................................................... 11

2.5 Classification ...................................................................................................................... 12

2.5.1 Classic Products .......................................................................................................... 12

2.5.2 Corridor Products ....................................................................................................... 13

2.5.3 Guarantee Products ................................................................................................... 13

2.5.4 Turbo Products ........................................................................................................... 13

2.6 Products with exotic option components ......................................................................... 13

2.6.1 Barrier Products ......................................................................................................... 13

2.6.2 Rainbow Products ...................................................................................................... 14

2.7 Structure of structured products ...................................................................................... 14

2.7.1 The Bond Component ................................................................................................ 15

2.7.2 The Option Component .............................................................................................. 17

2.7.3 Swaps.......................................................................................................................... 17

2.7.4 Participation Rate ....................................................................................................... 18

3 Understanding Options ............................................................................................................ 19

3.1 Exotic Options ................................................................................................................... 19

3.2 Path dependent options .................................................................................................... 19

3.2.1 Asian options .............................................................................................................. 19

3.2.2 Lookback options ....................................................................................................... 21

3.2.3 Ladder options............................................................................................................ 21

3.2.4 Barrier options............................................................................................................ 21

3.3 Time dependent options ................................................................................................... 22

3.4 Multifactor options ........................................................................................................... 23

3.5 Payoff modified options .................................................................................................... 23

4 Option Pricing Theory ............................................................................................................... 25

4.1 Assumptions ...................................................................................................................... 254.2 Stochastic Process ............................................................................................................. 26


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4.2.1 Properties of a stochastic process.................................................................................. 26

4.2.2 The Markov Property ................................................................................................. 26

4.2.3 Wiener Process ........................................................................................................... 26

4.2.4 Generalized Wiener Process ...................................................................................... 27

4.2.4 Geometric Brownian motion ...................................................................................... 28

4.2.6 Ito’s Lemma ................................................................................................................ 29

4.2.7 Risk Neutral Valuation ................................................................................................ 30

5 The Black- Scholes Equation (BS) ............................................................................................. 31

5.1 Options on dividend paying stock ..................................................................................... 35

5.2 Commodity Options .......................................................................................................... 35

5.3 Options on many underlying ............................................................................................. 36

5.4 Black- Scholes Pricing Formulas ........................................................................................ 37

5.5 Upper and Lower bounds for the call option .................................................................... 38

5.6 Forward Contract .............................................................................................................. 39

5.7 Futures contracts .............................................................................................................. 40

5.8 Futures Options ................................................................................................................. 42

5.9 Pricing of European futures options ................................................................................. 43

6 An overview of the selected products ..................................................................................... 45

6.1 DB Råvarer 2013 Basel (the “Notes”) ................................................................................ 46

6.1.1 Payoff Structure ......................................................................................................... 46

6.1.2 Risk Factors ................................................................................................................. 47

6.1.3 Issuance costs ............................................................................................................. 47

6.1.4 Embedded option ....................................................................................................... 47

6.1.5 The underlying asset ........................................................................................... 48

6.2 Analysis of Råvarer Basis 2010 .......................................................................................... 48

6.2.1. Payoff Structure ........................................................................................................ 48

6.2.2 Risk factors ................................................................................................................. 49

6.2.3 Issuance costs ............................................................................................................. 50

6.2.4 Embedded option ....................................................................................................... 50

6.2.5 Underlying Asset ........................................................................................................ 50

7 Pricing of the selected products ............................................................................................... 51

7.1 Pricing of DB Råvarer 2013 Basel ...................................................................................... 52

7.1.1 Deriving the Zero Coupons ......................................................................................... 52

7.2 Term Structure for Råvare.r Basis 2010 ............................................................................ 53


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7.3 Option Pricing .................................................................................................................... 55

7.4 Estimating Volatility .......................................................................................................... 55

7.5 Monte Carlo Simulation .................................................................................................... 56

7.6 Variance Reduction Technique ......................................................................................... 57

7.7 Generating Random numbers ........................................................................................... 57

7.8 DB Råvarer Basel 2013 ...................................................................................................... 58

7.9 Pricing of Råvarer Basel 2010 ............................................................................................ 59

8 Evaluation of the Model ........................................................................................................... 61

8.1 Possible extensions to the thesis ...................................................................................... 62

9 Conclusion ................................................................................................................................ 65

References ................................................................................................................................... 67


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1. Introduction

Structured products are drawing more and more investors now a day. Retail and

institutional investors alike are piling into these products. Structured products are

suitable for defensive or conservative investors because investments in structured

products assure a complete or partial protection of their invested capital but at the same

time, they can take advantage of the economic exposure to the growth potential of the

selected underlying. The underlying asset can be a single or a basket of the underlying

assets.

The popular structured products offer exposure to the equities, foreign exchange,

indices, volatility indices, commodities and commodity indexes such as S&P

commodity index, the Dow Jones-AIG commodity index or the Rogers International

commodities index. Commodity indices differ from the equity indices. The underlying

investments are not shares, bonds or the commodities themselves but “futures” contracts

on a single or a basket of commodities (a contract to buy or sell an asset at a given

future date for a set price). Futures contracts generally expire after three months;

therefore the so called rolling principal is applied for futures index where the index

sponsor replaces near to expiry contracts with the longer maturity contracts.

Structured products have become very advanced too in their structure. The complex

mechanisms within their structures are sometimes difficult to understand by the

investors and even sometimes by the financial managers too.

So, the theme of this thesis is to present an in depth analysis of the selected structured

products. The following will describe exactly what this thesis will be answering

•

What are structured products and their composition?

• How the individual components of a structured product are valued. That is, how

are bonds priced (coupon and zero coupon bonds), and how the underlying

embedded options are priced (plain vanilla and exotic options)?

• What is the theoretical fair value of the selected structured products and whether

these products are offered to the investors at fair value, overvalued or

undervalued?


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Structured products can be found in a wide variety of underlying assets. The underlying

assets can be from equities to equity indices, foreign exchange, interest rates,

commodities or commodity indices. This thesis will mainly focus on valuing

commodity linked structured products.

The contents of the thesis are split into two parts. The first section mainly focuses on the

theory behind the structured products. This section consist of chapters 2, 3, 4 and 5.

Section two consists of the valuation of the selected products and comparison of the

theoretical price with the issuing price of the selected products. Chapters 6, 7, 8 and 9

will be part of the second part of this thesis.

Chapter 2 begins with the introduction and definition of structured products and thendefining the commodity linked structured products. It will also describe in general how

structured products are engineered. This chapter will also discuss the advantages and

disadvantages of commodity linked products, their brief history, the difference between

a conventional bond and a structured bond and explanation of different concepts within

structured products.

Structured products normally consist of two components i.e. the bond and an embedded

option. The option component is generally the most tricky and complex part of these

products. Chapter three, four and five consist of the option theory including their

classification and the underlying concepts involved in option pricing in particular

stochastic process, geometric Brownian motion and generalized Wienner process. Black

and Scholes option pricing framework along with underlying assumptions and the

concept of risk neutral world will be discussed in chapter five.

Chapter six and seven deal with the pricing issues of the selected products. Chapter six begins with the analysis of prospectus of the selected products. While chapter seven

starts with pricing of the bond and option part of these structured products. Valuation of

the option components start with an overview of the Monte Carlo simulation. The

qualitative and quantitative comparison of theoretical and issue price of the products is

performed. Chapter eight starts with a critical evaluation of the underlying assumptions

and their effect on valuation model. This thesis will give an understanding of how the

structured products are priced and therefore will trigger readers interest to find

improvements in the pricing model and possibly to use other pricing models too. So, a


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brief description of the alternative option pricing models is also discussed in chapter 8.

Finally, the thesis ends up with a brief conclusion in chapter 9.

Although, structured products are available with a wide variety of underlying assets, but

this thesis will focus on commodity linked products. The products are selected from the

Danish market, therefore, interest rates will also be considered from the Danish market.

The valuation will be performed by applying the well known Black & Scholes option

pricing theory because the main theme of this thesis is the valuation of the selected

products and not to evaluate the performance of different option pricing models.

Therefore, other advanced option pricing models are not considered in this thesis. In

order to price the option component in Black & Scholes frame work, Monte Carlo

simulations are applied which follows the principal of risk neutral random walk. Tax

issues will not be considered in the valuation process. Default risk of the issuing firm

will be also disregarded because the selected products in this thesis are from the issuers

with very good credit ratings.

The data for this thesis has been downloaded directly from the Dow Jones UBS

commodity index web site, while the data for deriving zero coupon term structure was

down loaded from data stream.

Finally, I would like to thank my advisor Jochen Dorn for his use full guidance and

patience during the thesis.


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2. Structured Products

Structured products have emerged as an important instrument in financial markets. A

structured product can be defined as a security that combines the features of a fixed

income security with the characteristics of a derivative transaction. Generally, a

structured product contains two components i.e. a fixed income security (a zero coupon

bond that guarantees full or part of the invested capital) and an option or forward – like

instrument which has a specific class of the asset as an underlying. The underlying

assets can be equity, interest rate, an index, inflation, foreign exchange, commodities or

credit. The underlying can be a single asset or a basket of multiple assets. The additional

payoff of a structured product depends on the performance of the underlying asset.

Structured products are also said to be a ‘marriage of a fixed income security and anoption like instrument1

When the underlying in a structured product is a commodity or a basket of commodities

or a commodity index then they are called commodity- linked structured products. The

underlying commodities can be for example crude oil, gas oil, metal (gold, silver,

copper, and precious metals), energy etc.

’.

Commodity indices are different from the other indices. The underlying investments are

not bonds or shares or the commodities themselves but it is ‘futures’ contracts on the

commodities. A futures contract is defined as the contract which gives its holder the

right but not the obligation to buy or sell an asset (commodities, equity, foreign

exchange etc) at a given future date at an agreed price. Futures have normally three

months expiry date. These expiry dates are normally standardized. The index sponsor is

therefore required to replace the expiring futures contracts with the new ones traded on

the futures market (every three months). This is called the ‘rolling’ principle. When afutures contract expires, the index will treat it as sold and the proceeds are reinvested in

the new futures contract that will again expire after the next three months.

1 BNP Paribas equities & Derivatives handbook


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The index level takes care, the price movements in the underlying commodities and

takes into account the price difference between the old and the new futures contract that

are rolled.2

Commodity linked structured products are available in a wide variety of range. One of

them is Commodity- Linked bonds/ Notes, which is also the topic of this thesis.

Commodity linked notes or bonds are classified into two classes, i.e. The Forward type

and the option type. In a forward type bond, the coupon and or principle payment to the

bearer are linearly related to the price of the stated reference commodity i.e. it allows

the holder to receive either the nominal face value or the designated commodity amount

at maturity . While, an option type bond, the coupon payments are similar to that of a

conventional bond but at maturity, the bearer receives the face value plus an option to

buy or sell a predetermined quantity of the commodity at specified price

3

2.1 Structured products are suitable for investors who

. In literature

both the terms (bonds and note) are used interchangeably.

4

1. Want protection of their invested capital by hedging the risk of existing

investments.

2.

Want to enhance the return from their investment while controlling risks,

whereby the structure is designed to enhance equity return with leverage.

3. Want to diversify with the adjustable risk/ return profiles and market cycle

optimization capabilities of structured products.

4. Want to exploit their market view with more freedom and flexibility.

5. Want a growth by capitalizing on the market upside while protecting the

downside.

6.

Want to benefit from periodic returns with limited risks. This income type ofstructure is built to deliver coupons while protecting capital

2 Barclays Wealth, Light Energy Commodity Plan

2,3 Joseph Atta- Mensah. Commodity- Linked Bonds

4 BNP Paribas equities & Derivatives handbook


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2.2 Disadvantage of structured Products

Despite the fact, that structured products including commodity linked products provide

capital protection and a possible payoff from the option component, the investor still

loses the payoff associated with a traditional risk free bond. In a structured product, the

investor receives only the invested capital if the option expires out of the money, but in

a traditional bond, the investor receives a risk free interest of his invested capital along

with the invested capital. This lost risk free interest or profit is called the Opportunity

cost and can be defined as the “forgone risk-free rate of return that could have been

achieved if the principal would have been invested in the safe fixed- income securities

such as Treasury bills”5

For example, if an investor invests 100 DKK in treasury bonds for one year with a 5%

interest rate. He will receive DKK105 at maturity while if he invests in a structured

product, he will only get DKK 100 at maturity plus a possible payoff from the option

embedded in it, because he will actually invest 100*1,05^-1 = DKK 95 in the risk free

investment and DKK 5 will go to buy a call option plus administration fee . This DKK 5

from investment in risk free bond will be the opportunity cost that he will miss if he

would invest in a structured product. The payoff of a traditional bond will exceed as

long as the option component of a structured product is out of the money, at the money

or if it is in the money but still below DKK105.

.

2.3 Difference between a Conventional Bond and a Commodity Linked

Bond6

Commodity linked bonds are different from conventional bonds in many aspects. Some

of the key differences between are

1. In conventional bonds, the investor receives fixed coupon payments i.e. interest

payments during the life cycle of the bond (annually or semiannually etc) and

the face value at the maturity of the bond. While the holder of a commodity

linked bond receives the physical units of the underlying commodity or

equivalent of its monetary value. Similarly the coupon payments may or not be

5 Lehman Brothers, A guide to Equity_Linked Notes


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in units of the underlying commodity (it depends upon the performance of the

underlying).

2. The nominal return of a conventional bond held to maturity is known while the

real return is not known because of inflation. On the other hand both the real and

the nominal return of a commodity linked bond are unknown.

3. The results of Atta- Mensah study also show that the coupon rate for a

conventional bonds are greater than that of the commodity linked bonds whose

terminal payoff is greater of the face value and monetary value of a pre-specified

unit of a commodity.

4. The coupon rate of a conventional bond is less than that of a commodity linked

bond that pays its holders on maturity the minimum of the face value and the

monetary value of a pre-specified unit of a commodity.

2.4 Commodity –Linked Bonds, a brief history

The concept of structured note is considered to be relatively new in the financial

markets. In reality these products have been in existence for a considerable time. For

example callable notes and equity linked securities i.e. convertibles and debt with equity

warrants are the precursors to the structured note products that are common place today.

Commodity linked bonds were introduced during 1970’s when the oil backed bonds

were issued by the Mexican Government in the financial market. These bonds were

called Petrobonds. Each 1000 Peso bond was linked to 1.95353 barrels of oil with a

coupon payment of 12. 658% annually and had three years to maturity. Later on the

French Government issued gold backed bonds during 1973. They were known as

‘Giscard’ in the financial markets. These bonds have 7% coupon rate and redemption

value was indexed to the one kilogram bar of gold. In 1981 Eco Bay Mines Company ofCanada also issued gold warrants. Commodity linked bond with sliver as an underlying

was issued during 1983 and again in 1985 by the Sunshine Mining Company in USA.

The objective was to hedge against the fluctuations in the price of silver.

Later on, bonds indexed to other commodities like nickel, copper, silver, cobalt and

platinum were issued during 1984 by Inco. (a leading producer of metals). Now a days,

commodity linked bond are issued by many investment banks around the world. These

bonds are linked to the performance of a basket of specific commodities or commodity


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index for example Goldman Sachs Commodity Index (GSCI), London Metal Exchange

(LME), S & P commodity index or Dow Jones UBS commodity index.

2.5 Classification

Structured products including commodity linked structured products are available with

a wide variety of product characteristics and heterogeneous characteristics in the

market. However, Pavel A Stoimenov and Sascha Wilkens, in their article about the

equity linked structured products in the German market, “Are Structured Products

Fairly Priced”? have classified them according to the underlying option components

embedded in the product. As is shown in the figure 1, structured products can be

divided into two categories i.e. Plain vanilla Option- Component and exotic Option-

component. Plain vanilla products are further classified into Classic, Corridor,

Guarantee and Turbo while Exotic components products are further classified into

Barrier and rainbow. Barrier structured products are further divided into Knock- in,

Partial Knock- in and knock – out products. The brief definitions of these products are

discussed below.

Figure 1

2.5.1 Classic Products

A classic structured product has the basic characteristics of a bond except that the issuer

has the right to redeem it at maturity by repayment of its nominal value or delivery of

previously fixed number of specified shares. In general, structured products can be


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categorized into with and without coupon payments. Products with coupon payments

are called as ‘reverse convertibles’, while those without coupon payments are named as

‘discount certificates’.

2.5.2 Corridor Products

The payout of a corridor structured product depends on whether the underlying at

maturity is quoted within a certain range.

2.5.3 Guarantee Products

A guarantee product is similar to that of a corridor product. The only difference is that

in a guarantee product, fixed minimum repayments are guaranteed to the investor. So, if

price of the underlying falls below the reference value, then the investor will always get

the guaranteed amount.

2.5.4 Turbo Products

The payout of a turbo product is doubled if the underlying is quoted within a certain

price range at maturity. This is called turbo effect. But there are three possibilities at

maturity. If, for example, L and K are lower and upper reference prices, then at

maturity, if

1. St fixing ≤ L, the product is redeemed in shares;

2. L < St fixing < K, a cash settlement with s(2 St fixing - L) occurs;

3. K ≤ St fixing, the maximum amount s (2K - L) will be paid.

2.6 Products with exotic option components

2.6.1 Barrier Products

Barrier products are the most common type of structured products, where the embedded

option is a barrier one. The redemption of a barrier product depends upon whether the

underlying reaches a certain fixed price barrier during its lifetime. In a knock in product,

if the underlying reaches or crosses a fixed pre specified lower price barrier, then the

stocks are delivered at maturity. In such a case the product behaves like a classic

product. A knock in pays the maximum amount if the underlying is always above this

barrier regardless of the St Fixing . In the case of a knock out product, if the underlying


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reaches or crosses the pre specified upper price barrier, then the issuer loses his choice

of redemption and the products behaves like a regular bond in this case. In a partial time

knock in product, the barrier criterion is tested only within a certain time interval,

generally a few months immediately before maturity.

2.6.2 Rainbow Products

The rainbow products have more than one underlying. In a rainbow product, the issuer

has the right to choose between the specified underlying on redemption.

2.7 Structure of structured products

Commodity linked notes like other types of structured products provide partial or 100%

capital protection depending upon the investor’s specific needs. A typical commodity

linked bond provides 100% capital protection at maturity independent of the

performance of the underlying commodity or commodity index. The structure of a

simple commodity linked note can be sketched as

This figure shows that when an investor buys a structured product (equity, commodity

or any other asset as an underlying), he/ she actually has bought a package which

consists of a bond and an option or a swap (forward or future) and the fee on top of it.

The payoff of a structured product (note) is equal to the par amount of the note plus a

commodity / equity etc linked coupon. The payoff is either

1.

Zero, if the underlying has depreciated from the initial agreed upon strike level

BondCommodity

linked Note Principal

PayoffOption/ Swap


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2. Or the participation rate times the percentage change in the underlying

commodity/ equity times the par amount of the note7

In order to understand how a structured note works, we consider a simple example,

where an investor wants to invest 100 DKK over five years with full capital protection

and exposure to the S & P Commodity index. This means that the investor will get at

least his/ her 100 DKK at the end of five years no matter if the index depreciates or

appreciates. The investment bank will apply the appropriate interest rate (treasury

bonds, LIBOR or REPO rate) and find out the present value of 100 DKK (future value).

For example, if the five years interest rate was 5 %. Then the present value of this five

years zero coupon bond will be 78.35 DKK (100 * 1.05-5) today. It means that the

structure provider (investment bank) will have 100-78.35 = 21.65 DKK to purchase an

option or a futures/ forwards contract. Now let’s consider that a five – years S & P call

option costs 23.65 and 2 is the administration and margin costs, then the investor will

benefit from 83. 87 % ((21.65 – 2) / 23.65) participation in the S & P index’s upside.

.

2.7.1 The Bond Component

The bond component of a structured product is the most important part of it. It is also

the major part of any structured product. The bond component ensures that the investor

will receive the agreed amount of his/ her investment at maturity. The agreed amount

can be a 100 % of the invested capital or it can also be partial protection depending

upon the product. Structured products in general have the characteristics of a zero

coupon bond but it can also have coupon payments (annual or a semi- annual). The

main advantage of a zero coupon bond is that the investor gets all his investment back at

the same time instead of coupon payments at the end each period (annual or semi-

annual). The risk free interest rate applied to a zero coupon bond ensures that the present value of the investment will grow continuously until maturity. The risk free

interest rate is normally taken from the government bonds (the rate at which the state

borrows money).

If an amount A is invested for n years at an interest rate R per annum and if R is

compounded once per annum i.e. m=1, then the terminal value of the investment will be

7 Lehman Brothers ( equity – Linked Notes)


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(1 + )

If → +∞ , we compound more and more frequently, then we obtain the well knowncompounding frequency interest rates and the future value or the terminal value will be given as

In the same way the present value of a future amount can be written as

−

The pricing of a zero coupon bond or any other fixed income security can be derived if

we know the zero coupon (ZC) yield curve. The term structure of ZC rates (also knownas ZC yield curve) is the curve relating maturities t (time horizons) with the

corresponding ZC interest rate R(t ).

= ∑ (1+())

=1 = ∑ =1 () (1)

Here

•

B(t) means the discount factor at time t ( the prices of zero coupon rates with face

value of 1)

• R(t) is the zero coupon rate derived from B(t) and

• F (t) is the known cash flow or also called the principal amount.

When a structure note/ bond have the features of a coupon bond, then it can be

considered as the portfolio of zero coupon bonds. The price of such a bond can be

written as the present value of the sum of all cash flows (coupon payments) for each period plus the principal amount and can by the following expression

= ∑ . + = ∑ . (1 + )− + (1 + )− (2)

Where,

• CFt is the cash flow in time t


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2.7.2 The Option Component

The option component is the 2nd part of a structured product. Option component

provides the chances of payoff. Options are of two types i.e. a call and a put. A call

option provides its holder the right to buy an asset at a certain date on a pre specified price while the put option gives its holder the right to sell an asset at a pre specified

price on a certain date. Call options are normally embedded in structured products

because it is easy to earn on something whose price is increasing rather than decreasing

as in case of a put option. Option component is also the risky part of any structured,

because the payoff depends on the performance of the underlying. If the option

embedded in a structured product expires out of the money (i.e. the strike price of the

call option is higher than the corresponding price of the underlying) than it will not beexercised and the holder will get no profit but instead loose the money to buy that

option but he will still receive his invested capital. If on the other hand, an option

expires in the money, then it will be exercised and the holder will earn profit along with

the guaranteed capital. The expression at the money means that when the strike price of

the call option is equal to the price of the underlying.

There are two possibilities to exercise an option. In a European style option, the holder

can exercise his right to buy or sell an asset only at the maturity of the option. In an

American style, the holder can exercise his right to buy or sell the underlying before the

maturity of the option too.

2.7.3 Swaps

Commodity linked structured products can also be found with swaps in their structures.

Forwards are an example of swap and commodity swaps are in fact a series of forward

contracts on a commodity with different maturity dates and the same delivery prices. 8

The commodity linked products as mentioned by Schwartz9

8 John C Hull: P-173 Option, Futures and Other derivatives

issued for example by

Sunshine company during 80’s or by the Mexican Government during 1979 backed by

silver and oil respectively are examples of the products with forward type component in

the structured product, where the company promised to pay either the face value or

market value of the underlying commodity.

9 Eduardo S. Schawartz: The Pricing of Commodity Linked bonds


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2.7.4 Participation Rate

Participation rate determines that how much the product will participate in the

performance of the underlying. It can be defined as the exposure of a product to

movements in the price of its underlying. A participation rate of 100 % means that theinvestor would receive the return that will be exactly equal to the rise in the price of the

underlying. For example if the underlying has increased by 25% at maturity, then the

investor will also receive 25% return. But if it is low as mentioned in the example on

page 9 i.e. 87.83 % then the investor will get DKK 21(87.83% * 25).

Participation rate depends on the value of the option embedded in the structured

product, the administration and other issuing costs and the present value of the bond

component of the product. The participation rate depends on many factors. For example

if the issuing costs of the product are low then the participation rate can be higher.

Similarly, if the value of embedded option is high/ low then the participation rate can

lower/ high. Participation rate is generally not set prior to the expiry of the issuance

period and it appears as estimate in the prospectus. The participation rate can be

calculated by the following relation

= ∗ 100 (3)

Participation rate is also named as Gearing. The above equation also shows that there

other factor which determines the participation rate. For example, the interest rate used

to calculate the present value of the bond component, the life time of the product and

volatility of the underlying asset. For example a low interest rate will result in high

present value of the bond component and can reduce the participation rate and vice

versa. Similarly volatility of the underlying asset can also affect participation rate. If the

volatility of the underlying asset is lower, consequently the option will have lower value

and ultimately a higher participation rate. Cheaper options embedded in the structured

products also result in high participation rate. For example, exotic options are generally

cheaper than plain vanilla options. Therefore, now a day exotic options are generally

embedded in the structured product which increase the participation rate and can result

in higher payoffs at the end.


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3 Understanding Options

Options are classified into plain vanilla and exotic options. Plain vanilla options are

standard options while the exotic options are complex in nature. The complex options

have low prices as compared to the standard options. Therefore, exotic options are

generally embedded in structured products, which also make the products interesting for

investor’s point of view. Some examples of exotic options are barrier, chooser, look

back, Asian, Himalayan and basket options. Their detail will be discussed later on. If

the price of an embedded option/ options is lower, then the participation rate will be

higher and more payoff for the investor.

3.1 Exotic Options

An option whose characteristics, including strike price calculations/ determinations,

payoff characteristics, premium payment terms or activation/expiration mechanisms

vary from standard call and put options or where the underlying asset involves

combined or multiple underlying assets are called exotic options (Das2001, p718).

Exotic options are also called thirds generation risk management products. Although it

is hard to classify all the options, but they can be roughly divided into five to six

categories.10

3.2 Path dependent options

In path dependent options the final payoff depends on particular path that asset prices

follow over their life rather than asset’s value at expiration. The path of the underlying

determines payoff and structure of the options. Path dependent options are further

divided into weak and strong path dependent options. In strong path dependency, the

payoff depends on some property of the asset price path along with the value of theunderlying at present moment of time and some other extra variable (Wilmot 2007,

p252). Examples of strong path dependent options are

3.2.1 Asian options

Asian options are examples of strong path dependent options. In Asian options the

payoff is determined by comparing the strike/ spot price of the underlying with the

10 Das divided exotic options into five classes while Wilmott into six categories.


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average value of strike/ spot price during a specific period of time. They are strongly

path dependent because their value prior to expiry depends on the path taken and not

just where they have reached. Asian options were originated from Tokyo office of the

bankers trust in 1987. Asian options are normally cheaper than the plain vanilla options

because averages are less volatile and therefore less risky. Average can be calculated by

means of arithmetic, or geometric average of the prices. In Asian options

• There is a specific period over which the prices are taken. End of the averaging

interval can be shorter than or equal to the options expiration date, the starting

value can be any time before. In particular, after an average option is traded, the

beginning of the averaging period typically lies in the past, so that parts of the

values contributing to the average are already known.

• The market generally uses discrete sampling, like daily fixing.

• Weighting different weights may be assigned to the prices to account for a non-

linear, i.e. skewed, price distribution

• The wide range of variations covers also the possible right for early execution.

Asian options are popular in risk management for currencies and commodities because

they provide protection against rapid price movements or manipulation in thinly tradedunderlying at maturity, i.e. reduction of significance of the closing price through

averaging. These options reduce hedging costs because they are cheaper than standard

options. Average Price Options can be used to hedge a stream of (received) payments

(e.g. a USD average call can be bought to hedge the ongoing EUR revenues of a US

based company). Different amounts of the payments can be reflected in flexible

weights, i.e. the prices related to higher payments are assigned a higher weight than

those related to smaller cash flows when calculating the average. With Average Strike

Options the strike price can be set at the average of the underlying price which is a

helpful structure in volatile or hardly predictable markets.

An average price call pays (AT – K) +, where AT denotes the geometric or arithmetic

average price of the underlying. The geometric average of the underlying can becalculated as

= ∏ =0 (4)


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And the arithmetic or the simple average can be calculated as

= 1 ∑ =0 (5)

Week path dependency means that the option depends only on the underlying price and

the time. Barrier options are examples of week path dependent options. The payoff in

these options depends on if the underlying hits a pre specified value at some time before

expiry.

3.2.2 Lookback options

In these options the purchaser has the right at expiration to set the strike price of the

options at the most favorable price for the asset that has occurred during a specified

time. In a lookback call option, the buyer can choose to buy the underlying asset at the

lowest price that has occurred over a specified period, typically the life of the option.

Details about lookback options can be found in Fx options and structured products by

Uwe Wystrup.

3.2.3 Ladder options

The strike price in these options is periodically reset based on the underlying evolutionof the asset price. A ladder option can be identical to lookback when the amount of reset

is set to infinity.

3.2.4 Barrier options

Barrier options are weekly path dependent options. Das also classified them as limit

dependent options because their payoff depends on the realized asset path via its level.

Certain aspects of the contract are triggered if the asset price becomes too high or toolow. For example, an up- and – out call option pays off the usual max (S-K, 0) at expiry

unless at any time previously the underlying asset has traded at a value Su or higher. It

means if the asset reaches this level then it is said to ‘knock out’ and become worthless.

The option can also be “knocked in” instead of “Knock out”, where the payoff is

received only if the level is reached (Wilmott 2007, P288). Barrier options can be

divided into two types (out option and in option) i.e. up- and – out, down- and- out, up-

and- in and down- and- in.


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• The ‘out option’ pays off only if a specified level is not reached. Otherwise the

option is said to have knocked out and becomes worthless.

• The ‘in option’ pays off as long as the level is reached before expiry. If the

barrier is reached then it is said to have knocked in. In options contracts starts

worthless and only become active when the predetermined barrier is reached.

If the barrier is set above the initial asset value then it is said to have an ‘up option’ and

if the barrier is set below the initial asset value then it is said to have ‘down option’

Barrier options generally are of American style. It means that the barrier level is active

during the entire duration of the option: any time between today and maturity the spot

hits the barrier, the option becomes worthless. If the barrier level is only active atmaturity the barrier option is of European style and can in fact be replicated by a

vertical spread and a digital option.

Apart from a lower or an upper barrier, double barrier options are also available. Double

barrier options have both upper and lower barrier. In double ‘out’ option the contract

becomes worthless if either of the barriers is reached. In a double ‘in’ option one of the

barriers must be reached before expiry, otherwise the option expires worthless.

In some cases a so called rebate is paid if for example in an ‘out’ option the barrier level

is reached. The rebate may be paid as soon as the barrier is triggered or not until expiry.

The above mentioned barrier options are standard in nature. The barrier options can also

have exotic type features for example resetting of barrier, outside barrier options, soft

barriers and Parisian options. Detail discussion can be found in Wilmott 2007, p300.

3.3 Time dependent options

In time dependent options the buyer has the right to nominate a specific characteristic of

the option as a function of time for example the expiration of the option. Preference or

chooser option is an example of time dependent option. In a chooser option, at a

predetermined date (normally after commencement and before expiry) the buyer can

choose if the contract should be a call or a put option. Bermudan options are also

example of time dependent options, where early exercise of the option is possible on

certain dates or periods.


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3.4 Multifactor options

In multi factor options, the payoff depends on the relationship between multiple assets.

It means there is second source of randomness such as a second underlying asset.

Compound, basket, exchange, quanto, rainbow are the examples of multifactor options.

In Compound options (options on options), the holder has the right but not an obligation

to buy or sell another predetermined options at a pre agreed time. The compound

options can be a call on a call, a put on a put, a call on a put and put on a call.

Compound options have two strike prices and two exercise dates. For examples in a call

on a call option, on the first exercise date T1, the holder of the compound option is

entitled to pay the first strike price K1, and receive a call option. This call option gives

him the right to buy the underlying asset for the second strike price K2 on the secondexercise date. The compound option will be exercised on the first exercise date only if

the value of the option on the date is greater than the first strike price.

In basket options the payoff is based on the cumulative performance of the underlying

assets and in exchange options the holder has the right to exchange one asset for

another. The underlying assets can be individual stocks or stock indices, currencies or

commodities etc. If the payoff is determined on performance of maximum or minimum

of two more underlying assets, then these option are named as Rainbow option. A

quanto option can be any cash-settled option, whose payoff is converted into another

currency at maturity than that of the underlying asset at a pre-specified rate, called the

quanto factor. There can be quanto plain vanilla, quanto barriers, quanto forward starts,

quanto corridors, etc.

3.5 Payoff modified options

These options entail adjustment to the linear and smooth payoffs that are associated

with conventional options (Das 2001, p723). Examples include

• Digital options: Digital options have discontinuous payouts irrespective of the

normal options whose payoffs are smooth. In a normal option, if it is further in

the money the higher the payout to the purchaser. While in digital option the

payout is normally fixed provided if certain conditioned are met. For examples,

in the typical structure of digital option, if the strike price is reached the payouts


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are fixed predetermined amounts no matter how much the option is in the

money.

• Contingent premium: A contingent premium option is basically a European

option. The premium will be paid to the writer if the contingent premium option

finishes "in the money". Otherwise, if the option expires "at the money" then, no

premium will be paid. It means no premium is paid in the beginning of the

contract and is due at expiration of the options only if it expires in the money. In

other words, the contingent premium structure is a combination of a

conventional option and a digital option.

• Power options: A power option is a derivative with payoff depending on the asset

price at expiry raised to some power α , where α is higher than 1.


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4 Option Pricing Theory

This section will discuss the underlying concepts in Black & Scholes option pricing

theory. For example, the assumption of model, stochastic process, Markove property,

generalized Wienner process, geometric Brownian motion, Ito’s lemma, risk neutral

evaluation and finally the Black & Scholes option pricing formulae are discussed.

4.1 Assumptions

The famous Black and Scholes (B&S) model has several underlying assumptions like

other option pricing models. The understanding of these assumptions will help to

analyze the advantages and the drawbacks of the model. The underlying assumptions

are discussed below

1. The markets are efficient i.e. the markets are assumed to be liquid. There is price

continuity. The markets are fair and provide all information to all the players. It

means no transaction costs in buying or selling stock or options.

2. The underlying is perfectly divisible and short selling is allowed. A seller who

does not own a security will simply accept the price of the security from a buyer

and will agree to settle with buyer on the same future date by paying him an

amount equal to the price of the security on that date

3. There are no costs of carrying to the commodity (evaporation, obsolescence,

insurance etc) and that the commodity is held for speculative purposes like a

stock.

4. The commodity price, firm value and the interest rate follow continuous time

diffusion processes. It means that the interest rate is known and it is constant

(risk free) through time (Schwartz 1982). In other words there exists a risk free

security that returns $1 at time T when 1e-r(T-t) is invested at time T.

5. The stock/commodity price follows a random walk in continuous time

(geometric Brownian motion) with a variance rate proportional to the square of

the price. Thus the distribution of the possible stock/ commodity price at the end

of any infinite interval is log normal and the variance rate of the return on the

stock/ commodity is constant

6. The model deals with European style options only that can be exercised at

maturity only.


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4.2 Stochastic Process

We know that the prices of commodities, stocks and interest rates etc change over time

in the financial markets. If the change in value is uncertain over time i.e. if the change in

price of a commodity, equity or currency exchange is unpredictable over time then this

kind of price behavior is called a stochastic process. In other words any variable whose

value changes over time in an uncertain way is said to follow a stochastic process (Hull

2008, p 259) and it can be discrete time and continuous time stochastic process. In a

discrete time process the value of a variable is assumed to change at fixed time intervals

of time, while changes can take place at any time in a continuous time stochastic

process.

4.2.1 Properties of a stochastic process

4.2.2 The Markov Property

A stochastic process is said to have the Markov property, when only the present value

of a variable is relevant to predict its future value (Hull 2008, p 259) i.e. the process has

no memory beyond where it is now. It means that the past history of that variable and

pattern of changes in value would be irrelevant to predict future prices. So it means that

to predict the future price of a commodity bundle, the only relevant price will be the

today’s price and it will be independent of its price during the last week or year.

4.2.3 Wiener Process

Wiener process is a particular type of Markov process which has a mean change of zero

and a variance rate of 1.0 per year. Wiener process is also called Brownian motion

(named after a Scottish botanist Robert Brown). Brownian motion has been used in physics

to describe the motion of the particle that is subject to a large number of small molecularshocks. It is among the simplest type of continuous stochastic process. In mathematical finance,

this concept was first used by Louis Bachelier during the 1900 in his PHD thesis, where he

presented the stochastic analysis of stock and option markets

A variable Z follows a Wiener process if it has the following two properties (Hull 2008,

p261)

1.

The change ∆Z during a small period of time ∆t will be


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∆ = √ ∆ (6)

Here, ε has a standardized normal distribution with mean zero and standard deviation of

1, that is; N (0, 1).

2. The values of ∆ Z for any two different short interval of time ∆t are independent.

It means that Z has independent increments and∆Z 1 is independent of ∆Z 2 if ∆t1

does not overlap with ∆t2.

The first property shows that ∆ Z has a normal distribution with i.e.

Mean of ∆ Z = Z (t) – Z (0) = 0

Standard deviation of ∆ Z = √ ∆

And variance of ∆ Z = ∆ t

The Wiener process is both the Markov and Martingale process (zero drift stochastic

process). By martingale process, it means that the expected value at any future time is

equal to its value today. Martingale property is an important part of the risk neutral

evaluation.

4.2.4 Generalized Wiener Process

It is clear from the Wiener process that if we choose it as a model then the stock/

commodity price can take negative values at any point in time with a probability of 0.5

and it will have a constant zero mean and it is not an ideal model to price stock prices.

So we have to consider a better model called Generalized Wiener Process. The basic

Wiener Process also states that∆Z has a zero drift rate and a variance rate of 1.0. Zerodrift means that the expected value of Z at any future time is equal to its current value

and the variance rate of 1.0 means that the change in a time interval of length T equals

T. Here we consider a discrete time random walk

X0 = x, Xi = Xi-1 + a ∆t + b √∆t εi where ε i ~ N (0, 1)

And the increments are given by ∆ Xi = a ∆t + b √∆t εi


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Here ‘a’ is the constant drift rate and ‘b’ is the volatility rate. When we take smaller and

smaller time steps ∆t, then the above equation can be written as

() = + + () (7)

The above equation is called the Generalized Wiener Process. The above equation can

be written in the differential form as follows

= + (8)

So for a stock price we can conclude that

•

Its expected percentage change in a short period of time remains constant, notit’s expected absolute change in a short period of time.

• The size of the future stock price movements is proportional to the level of the

stock price

4.2.4 Geometric Brownian motion

Generalized wiener process fails to capture a key aspect of the stock / commodity prices

i.e. the percentage return required by the investors is independent of the stock price. Itmeans that the investor will demand the same return, does not matter if the stock price is

DKK 10 or DKK 100. So, the assumption of constant expected drift rate needs to be

replaced by the assumption that expected return (i.e. expected drift divided by the stock

price) is constant. So, if P is the price of a commodity bundle at time t, then the

expected drift rate in P should be assumed to be µP for some constant parameter µ. It

means that the expected increase in P (in a short period of time ∆t) is µP ∆t. The

parameter µ is the expected rate of return on the price, expressed in decimal form.

If the volatility of the commodity price is zero, then the model implies that

∆P = µ P∆t And as ∆t approaches to zero then

dP = µP dt or = (9)

Integrating the equation (7) between 0 and time T, we get


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= 0 (10)

PT and P0 are the commodity prices at time T and time 0 respectively. This equation

shows that when variance rate is zero, the commodity price grows at a continuously

compounded rate of µ per unit of time. But in practice, commodity prices or stock prices

exhibit volatility. Here it can be assumed that the variation in the percentage return in

short period of time ∆t is the same regardless of the commodity price. It means that the

investor is just as uncertain of the percentage return when the price is DKK 100 as when

it is DKK 10. It means that the standard deviation of the change in a short period of time

∆t should be proportional to the commodity price and it can be written as follows

= + Or

= + P > 0 (11)

This equation is called the Geometric Brownian motion. The important feature of this

equation is that the commodity or stock price will never become negative.

4.2.6 Ito’s Lemma

Ito’s lemma is the most important rule of stochastic calculus. It was discovered by the

mathematician K. Ito in 1951. According to Ito’s lemma the ordinary rules of calculus

do not apply to the stochastic processes For example, consider a function F(Z) = Z 2

where Z is a Brownian motion. Then according to ordinary calculus, dF (Z) = 2ZdZ.

But this is not true for stochastic processes. In order to drive rules for stochastic

calculus, we have to apply Taylor expansion i.e.

= + 12 2 …….. (12)

So from the F (Z) = Z2 we have = 2Z and

2 = 2 substituting these values in the

above Taylor expression we get = 2 +

Now we consider the Geometric Brownian motion equation i.e.

=

+

(13)


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Here (, ) = (, ) =

Now consider the function () = log

Then () = 1 2(logP) = 12

And by Ito’s lemma = + 122(,) 2 and substituting the values weobtain = 1 1222 ∗ 1 (14)

And substituting equation (13) in equation (14), we get

= + 122

Or = 122 + (15)

Since and constant, this equation shows that F= log P follows a generalized Wiener process. It has a constant drift rate of 122 and constant variance rate of 2. Thechange in ln P between time 0 and some future time T is therefore normally distributed

with mean 122 and variance 2. It means that

ln ~∅0 + 2 , 2 (16)

Equation (16) shows that is normally distributed. A variable has a lognormaldistribution if the natural logarithm of variable is normally distributed. This model

implies that that the price of a commodity bundle/ stock price at time T, given the price

today, are log normally distributed. The standard deviation of the stock price is√. Itis proportional to the square root of how far ahead we are looking (Hull 2007, p 271).

4.2.7 Risk Neutral Valuation

Risk- neutral valuation is an important concept in option pricing and particularly while

deriving the famous Black- Scholes option pricing equation. According to this principal

we can assume that the world is risk neutral when pricing options. It means that present

value of any cash flow can be obtained by discounting its expected future value at risk


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free interest rate. So, in order to calculate the payoff of a derivative at particular time by

risk neutral valuation we assume that expected future return from the underlying asset is

the risk free interest rate “r”, that is, = . Then we have to calculate the expected payoff from the derivative and finally discount this expected payoff at the risk free

interest rate. In a risk- neutral world all individual investors are indifferent to risk and

they require no compensation (premium) for risk and the expected return on all

securities is the risk free interest rate. The solutions obtained in the risk- neutral world

also hold in the real world (risk- averse world) because in the risk-averse assumption

the expected growth rate in the stock price changes and discount rate that must be used

for any payoffs from the derivative changes. It happens that these two changes always

offset each other exactly (Hull 2007, p290).

5 The Black- Scholes Equation (BS)

The Black- Scholes model was derived by Fischer Black and Myron Scholes in the

early 1970s for the pricing of stock options. Robert Merton also participated in the

creation of this novel model. Therefore, sometimes, this model is also named as the

Black- Scholes- Merton model. This model was perhaps the biggest breakthrough in the

field of option pricing and rapidly got its acceptance among the financial engineers. Thesuccess of financial engineering in the last 30 years is highly because of this model. In

1997, Myron Scholes and Robert Merton because of creating this model were also

awarded the Nobel Prize for economics.

The BS model shows that the value of an option depends on two factors i.e. the stock

price and the time to expiry/ maturity provided that the ideal market conditions

discussed earlier hold. Therefore, it is possible to create a hedge position by having a

long position in one option and a short position by some amount ∆ in the underlying ,

whose value will not depend on the stock price. If this portfolio is hedged continuously

then the portfolio of these two will be risk free and the expected return will be the risk

free interest rate. The riskless portfolio can be created because stock price and the

derivative price are both affected by the same underlying source of uncertainty i.e.

stock/ commodity price movements. In any short period of time, the price of the

derivative is perfectly correlated with the price of the underlying (Hull 2007, p285).


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In order to find the value “f” of a derivative written on an asset by BS model, the stock/

commodity price P follows geometric Brownian motion given in equation (11). In

general, the value f will be a function of many parameters in the contract i.e.

f (P, t, σ , μ ,K,T, r )

Here

• P and T are variables

• σ and μ are associated with the underlying asset’s price

• Strike price K and time to maturity T depends on the specific details of the

contract.

• And interest rate r depends on the currency in which the asset is quoted.

We assume here that value of an option is a function of time t and current price P of the

underlying and drop the other parameters. Therefore we can write as follows

f (P, t )

To begin with we assume that we know the value f of the option and

• Form a portfolio, which we hold for a period of length dt , by taking a long position

in the option and a short position of a quantity Δ in the underlying.

• Determine the quantity Δ so that our portfolio is risk-free over a time period of

length dt .

• By a no arbitrage argument, the rate of return of this risk-free portfolio must be

equal to the risk-free interest rate r . What comes out of this restriction is an equation

whose solution is the option price f .

Suppose we know the value of the option f (P, t) at time t and Π denote the value of a

portfolio consisting of a long position in the option and a short position in a quantity Δ,

delta, of the underlying

Π() = (, ) Δ · (17)

In equation (12), the term f (P, t ) is the option part of the portfolio and Δ · P is the shortasset position (negative sign).


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During the next time period of length dt the change in the portfolio value is given by

Π = (, ) Δ · (18)

(Δ is fixed during the period t , t + dt ). The portfolio Π is self financing, replicating, andhedging strategy. It replicates a risk free investment. There is no stochastic term,

therefore it is hedged.

From Ito’s lemma we can compute df and we get

(, ) = + + 1222

(19)

Substituting equation (19) into equation (18), we get

d Π = + + 1222

- Δ · dP

Re-arranging the above equation,

Π = + 1222

+ ∆ (20)

In the right hand side of equation (20), the expression containing the term dt is the

deterministic term while the with dP is random. The random term is the risk factor in

the portfolio. The risk in the portfolio can be removed if

∆ = 0 (21)

I.e. if, for the small period of time from t to t + dt , we choose the quantity Δ as

= ∆ (22)

Therefore, equation (17) becomes

Π() = (, ) (23)

Then the randomness reduces to zero. The reduction in randomness is called hedging.

The perfect elimination of risk by exploiting correlation between two instruments


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(option and the underlying in our portfolio) is called Delta hedging (Wilmott, p142).

Delta hedging is an example of perfect hedging. It means if we are allowed at any point

in time t to continuously re-balance our portfolio by choosing the quantity Δ in the

underlying (i.e. if we are entitled to continuous trading), then we have constructed a

portfolio which is risk-less and with dynamics given by

Π = + 1222

(24)

If there are no arbitrage opportunities in our market, then it must hold that our risk-less

portfolio should yield the risk-free interest rate r , i.e.

Π = Π (25)

Substituting the values of d Π and Π from equation 24, and 23 in equation 25 we get

+ 1222

=

After re-arranging the above equation we get

+

1222

+ = 0 (26)

Equation (26) is the famous BS equation. The price of any option which depends on P

and t must satisfy the BS equation otherwise it cannot be price of the derivative without

creating arbitrage opportunities for the traders. BS equation is a linear parabolic partial

differential equation. By the term linear means, if we have two solutions of the equation

then the sum of the two solutions is itself a solution. Parabolic means it has a secondorder derivative with respect to one variable P and a first order derivative with respect

to the variable t . Therefore equations of this type are called heat or diffusion equations

of mechanics (Wilmott 2007, p158).

The BS equation shows that value of a stock option when expressed in terms of the

value of the underlying, does not depend on drift rate or expected return µ. This is

dropped out when the dP term is eliminated from the portfolio. It is also clear from this


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equation that the variables appear in this equation (stock price, time, volatility and risk-

free interest rate) are all independent of risk preferences.

Another point to be noted from equation (26) is that we can not see whether this

equation is valuing a call or a put option. It means we will have to specify the final

conditions and specify the option value f as a function of the underlying at expiry date

T. For example, if we have a call option, then we know that f(P,T) = max( P- K, 0) and

for a put option the final condition will be f(P,T) = max(K- S, 0)

5.1 Options on dividend paying stock

Let q be the amount of continuous and constant dividend yield. Then equation (26) can

be re-written as

+

1222

+ ( ) = 0 (27)

5.2 Commodity Options

Options are called commodity options when the underlying is a commodity or

commodities. Commodity options differ from the security options in the way that it

cannot be exercised before the future fixed (expiry) date. Therefore, in a European

option rather than an American style option11

+

1222

+ ( + ) = 0 (28)

, the holder of the commodity option can

choose whether or not he wants to buy the commodity at the specified price. Then we

have to adjust the general BS equation (26). Commodities have the cost of carry. That

is, the commodities cannot be held without storage cost. So, if we assume that q is the

storage cost associated with the commodity, which means that if we simply hold the

commodity, it will lose value even if the price remains fixed. It means that for each unit

of commodity held an amount qP dt will be required during the short period of time dt

to finance the holding (like a negative dividend). Therefore, equation (26) can be

modified as follows (Wilmott, p148)

11 F. Black, The Pricing of Commodity Contracts, Journal of financial Economics, 3 (March 1976)


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The BS model can also be derived in many other ways instead of the classical risk-

neutral valuation. For example the Martingale approach, the binomial model and capital

asset pricing model (CAPM)

5.3 Options on many underlying

Options with many underlying assets are called basket options, options on baskets or

rainbow options. The basket can be any weighted sum of the underlying assets as long

as the weights are all positive. The value of these options depends on price, time to

maturity and additional variable i.e. the correlation between assets in the basket. The

payoff from the basket depend on the degree of correlation among the underlying assets

i.e. if the underlying assets are highly correlated with each other, then the option’s

payoff will be high and vice versa. The basic option pricing with one underlying can be

extended to more than one underlying too. First of all the idea of lognormal random

walk needed to be extended for multiple assets. I.e. The geometric Brownian motion of

an asset price (equation 11) can be easily extended to multiple assets. Therefore,

equation 11 can be written as

= + (29)

Here is the price of ith asset, i= 1, 2, 3…..,d and and are the drift and volatilityof the assets and dZ is the Wiener process for the respective asset. dZi can be still

considered to be as a random number drawn from normal distribution with a mean of

zero and a standard deviation of dt0,05 i.e .

[] = 0 2 =

In baskets options, the assets are also correlated with each other, therefore the log

normal random walks are also considered to be correlated with each other. That is

[] =

Here is the correlation coefficient between ith and jth random walks. The symmetricmatrix with ρij as the entry in the ith row and jth column is called the correlation matrix.

For example, if we have a basket option with three underlying assets i.e. n = 3 and thecorrelation matrix can be written as


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1 12 1321 1 2331 32 1

Here,

= 1

=

. The correlation matrix is positive definite.

We can apply multi dimensional ito’s lemma to manipulate functions of many random

variables. So, if we a have a function of variables Pn where n= 1, 2, 3 ….. And time t,

then = + 12∑=1 ∑ =1

+ ∑

=1 (30)

The pricing model for basket options following BS model can be derived in the same

way as for the single asset, i.e. by setting up a portfolio consisting of one basket option

and short a number∆ o f each o f th e asset p r ice P in th e basket. Employ the

multidimensional Ito’s Lemma, take Δi = ∂V/ ∂Pi to eliminate the risk, and set the

return of the portfolio equal to the risk-free rate. We are able to arrive at the multi

dimensional version of the Black and Scholes equation (Wilmott 2007, p277)

+ 12∑=1 ∑ =1

+ ∑

=1 = 0 (31)

5.4 Black- Scholes Pricing Formulas

Consider now the case of a call option with maturity T and strike price K written on a

stock / commodity paying no dividends. Assume that we stand at time t , that the current

price of the underlying is P0, that the interest rate is r and that the volatility of the stock

is σ . Here P follows geometric Brownian motion described by equation (11).

Then we know that

= −(−)[max( ) , 0] (32)

Recalling, we know that P follows a lognormal distribution with mean

122 2 and

[] = (

−)

(33)


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Then value of the European call option can written as

= 0 (1) −(−)(2) (34)

Equation (34) can also be written as

= −(−)[0 (1)(−) (2)] (35)

Where N(.) is the cumulative probability distribution function for standardized normal

distribution and

1=

lnK +r+ (T−t)

σ√ T−t

And 2 = 1 √

N(d2) is the probability that the option will be exercised (P > K) in a risk- neutral world

so that KN(d2) i.e. strike price times the probability that the strike price will be paid.

The expression 0 (1)(−) is the expected value in risk- neutral of a variablewhich is equal to PT if PT > K and to zero otherwise. We can also say that the expected

value of the call option at maturity will be 0 (1)(−) (2).The value of a European put option P on a non dividend paying underlying can be

written as

= 0 (1) + −(−)(2) (36)

The value of European call and put option on dividend paying stocks can be written as

= 0 −(−)(1) −(−)(2) (37)

= 0 −(−)(1) + −(−)(2) (38)

5.5 Upper and Lower bounds for the call option

Structured products have in general embedded call options and therefore it is important

to evaluate whether the prices of call option follows the no-arbitrage argument. We

know that call price has boundary conditions (upper and lower bounds). Boundary


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conditions tell us how the solution must behave for all time at certain values of the asset

(Wilmott 2007, P159). If an option is above the upper bound or below the lower bound,

then there exist arbitrage opportunities for investors. So from equation (37), the lower

bound for a call option can be written as

+ −(−)(2) ≥ 0 −(−)(1) Or

≥ 0 −(−)(1) −(−)(2)

And the upper bound for call option is given by

≤ 0

It means that the option value is always less than the corresponding underlying asset or

the underlying asset is always worth more than the corresponding call option.

Therefore, stock price is an upper bound to the option price. Similarly, for a put option,

the option can never be worth more than the strike price i.e. it can not be worth more

than the present value of K today (Hull 2008, p206). If it is not true, then the arbitrageur

could make riskless profit by writing the option and investing the proceeds of the sale at

the risk free interest rate.

≤ ≤ exp ()

5.6 Forward Contract

A forward contract is an agreement to buy or sell the asset P at a future date T (delivery

date) for a fixed price K (delivery price). The payoff from a long position of forward

contract is therefore

(,) = (39)

The delivery price K is typically delivered so that at initiation the contract has value

zero. By no arbitrage argument we have

= 0 (40)


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A forward contact sometimes also called a cash forward sale. There are some

disadvantages associated with the forward contracts i.e.

• Default risk, particularly if the prices are either high or low by the delivery date,

which negate the main value of a forward contract- price certainty

• The only way to legally terminate a contract was by mutual agreement, which would

be unlikely when the market price was significantly different from the delivery

price;

• There was no easy way to resell the contract, because it had customized terms that

specifically suited the seller and buyer—hence, forward contracts were highly

illiquid.

Eventually, organized exchanges developed that solved these basic problems. To lower

the risk of default, the exchanges required that money to be deposited with a 3rd party to

ensure the performance of the contract. The exchanges also standardized the contracts

by stipulating the types of contracts that they would sell, including its terms.

Standardized contracts were easier to sell or to offset with another contract that

eliminated the liability of the original contract. Standard specifications include the

amount of the commodity, the grade, and delivery dates. These standard forwardcontracts were called futures, and the exchanges developed listings for these contracts

that greatly increased their liquidity.

5.7 Futures contracts

A futures contract is an agreement between two parties to buy or sell an asset at a

certain time in the future for a certain price, the current futures price of the asset.

Futures contracts are similar to forward contracts, but the principal difference lies in theway payments are being made. In a forward contract, the whole gain or loss is realized

at the end of the life of the contract. But in case of the futures contract, the gain or loss

is realized day by day through a mechanism known as marking to market .

In mark to market mechanism, the contract is settled every day and simultaneously a

new contract, with the same maturity, is written according to the current futures price of

the underlying. Any profit or loss during the day is recorded in the account of the

contract holder. At any point in time, the value of the futures contract itself is therefore


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zero (i.e. it is a zero sum game, which means that short side’s loss or gain is the long

side’s gain or loss) , but we have also a capital gain which comes from the previous

days cash settlements. The futures price of an asset, for a given maturity, varies from

day to day, but at maturity it must be the same as the price of the underlying.

In general, at any point in time t the value of the futures contract is zero, but we have a

capital gain given by

Capital gain = change in futures prices = Ft − F 0

At the expiry of the contract, the total cash flows will amount to

(S T − F T −1) + · · · + (F 2 − F 1) + (F 1 − F 0) = S T − F 0

In order to find out the futures price F(P,T) for a futures contract with expiry T on an

asset P, we assume that the price follows Geometric Brownian motion given in equation

(8). We create a portfolio Π consisting of one long futures contract and short Δ of the

underlying. The portfolio value is given by

Π = −ΔP (41)

We know that the value of the futures contract is zero. But, when computing the

portfolio change during the next time period we need to take into account the cash

settlement which is given by the change in the futures price, i.e.

Π = – Δ (42)

Applying Ito’s Lemma, we finally get

d Π = + + 1222

- Δ · dP (43)

Assuming thatΔ = and to eliminate the risk, the following condition must hold

Π = Π (44)

Substituting equations 41and 43 into 44, we get


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+

+

1

222

22 Δ · = r(∆P)dt

+1

2

22

+ r∆Pdt Δ +

= 0

+

1222

+ = 0 (45)

The important point to note in equation (45) is that there are only three terms instead of four as

in BS equation. The final condition is

(,) =

i.e. The futures price and the underlying must have the same value at maturity.

eval prod struct

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