Black-Scholes Formula Using Long Memory

Yaozhong Hu ( 胡耀忠 )University of Kansas

hu@math.ku.eduwww.math.ku.edu/~hu

2007 年 7 月于烟台

Black-Scholes Formula Using Long Memory

1. Simple example

2. Black and Scholes theory

3. Fractional Brownian motion

4. Arbitrage in Fractal Market

5. Itô integral and Itô formula

6. New Fractal market

7. Fractal Black and Scholes formula

8. Stochastic volatility and others

1. Simple Example

1.1 A Simple example

MCDONALD’S CORP (MCD)

Friday, Jul 13-2007, $51.91

Last Friday’s Prices of Mcdonald’s Corp

1.2 Option

Buy the stock at the current time

Alternatively, buy an option

Option = right (not obligation) to buy (sell) a share of the stock with a specific price K at (or before) a specific future time T

T = Expiration date

K = strike price

Example (call option)

Right to buy one share of MCD at the end of one year with $60

Stock Price Option Payoff

$70 $10

$65 $5

$28 $0

$2 $0

– European

– American

Financial Derivatives

– Call

– Put

Many Other options

1.3 How to price an option

How to fairly price an option?

If (future) stock price is known then it is easy

Example

Strike price K Future Price P(T) Should Pay

Up to inflation

$60 $65 $5

$60 $50 $0

Future stock price is unknown

Math model of market

Probability distribution of future stock price

Stochastic Differential Equations

1.4 History

Louis Bachelier

Théorie de la spéculation

Ann. Sci. École Norm. Sup. 1900, 21-86.

Introduced Brownian motion

Solved problem

1.5 History of Brownian Motion

Robert Brown (1828)

An British botanist observed that

pollen grains suspended in water perform a continual swarming motion

Simulation of Brownian Motion

Prices of Yahoo! INC (YHOO)

–Mathematical Theory

L. Bachalier 1900

A. Einstein 1905

N. Wiener 1923

2. Black and Scholes Theory

2.1 History, Continued

Bachelier model

can take negative values!!!

Black and Scholes Model

Geometric Brownian motion

2.2. Black-Scholes Model

Market consists of

)]()[()(

)()(

tdWbdttPtdP

dttrAtdA

Bond:

Stock:

P(t) is Geometric Brownian Motion

Simulation of Geometric Brownian Motion

2.3 Black and Scholes Formula

The Price of European call option is given

)()( rTKepBS

duezwherez

u

2

2

2

1)(

Tr

K

p

T 2log

1 2

p = current price of the stock

σ = the volatility of stock price

r = interest rate of the bond

T = expiration time

K = Strike price

It is independent of the mean return of the stock price!!!

New York Times of Wednesday, 15th October 1997

Scholes and Merton “won the Nobel Memorial Prize in Economics Science yesterday for work that enables investors to price accurately their bets on the future,

a break through that has helped power the explosive growth in financial markets since the 1970’s and plays a profound role in the economics of everyday life.”

2.4 Main Idea and Tool

Itô stochastic calculus

a mathematical tool from probability

stochastic analysis

2.5 Extension of Black and Scholes Models

– Jump Diffusions

– Markov Processes

– Semimartignales

– Long memory processes

3. Fractional Brownian Motion

3.1 Long Memory

Long Memory = Joseph Effect

Self-Similar

Holy Bible, Genesis (41, 29-30)

Joseph said to the Pharaoh

“… God has shown Pharaoh what he is about to do. Seven yeas of great abundance are coming throughout the land of Egypt, but seven years of famine will follow them. Then all the abundance in Egypt will be forgotten, and the famine will ravage the land. …”

Hurst H.E. spent a lifetime studying the

Nile and the problems related to water storage.

He invented a new statistical method

rescaled range analysis (R/S analysis)

Yearly minimal water levels of the Nile

River for the years 622-1281 (measured

at the Roda Gauge near Cairo)

大江东去

一江春水向东流

大江北上 ?

Time-series record of the Nile River

minimum water levels from 662-1284 AD

Hurst, H. E.

1. Long-term storage capacity of reservoirs. Trans. Am. Soc. Civil Engineers, 116 (1995), 770-799

2. Methods of using long-term storage in reservoirs.

Proc. Inst. Civil Engin. 1955, 519-577.

3. Hurst, H. E.; Black, K.P. and Simaika, Y.M.Long-Term Storage: An Experimental Study. 1965

3.2 Fractional Brownian Motion

Let 0 < H < 1. Fractional Brownian motion with Hurst parameter H is a Gaussian process satisfying

2/])([)(

,0)(222 HHHH

s

H

t

H

t

ststBBE

BE

3.3 Properties

1. Self-similar: has the same property law as

2. Long-range dependent if H>1/2

HtB

0n

11

)(

),,cov()(

nrthen

BBBnr Hn

Hn

H

Ht

H B

3. If H = 1/2 , standard Brownian motion

4. h>1/2, Positively correlated

5. H<1/2, Negatively correlated

6. Not a semi-martingale

7. Not Markovian

8. Nowhere differentiable

Company Hurst Parameter Est. Error

Alcoa 0.5363 0.0157

Applied Mat. 0.5307 0.0105

Boeing 0.5358 0.0077

Capital One 0.5177 0.0104

GE 0.5129 0.0096

General Mills 0.5183 0.0095

IBM Corp. 0.5202 0.0088

3M Corp. 0.5145 0.0081

Merck 0.5162 0.0075

Wal-Mart 0.5103 0.0078

Intel 0.5212 0.0078

Daily return, over 1024 trading days

Granger, C.W.J.

Long memory relationships and aggregation

of dynamic models.

J. Econometrics, 1980, 227-238.

The Nobel Memorial Prize, 2003

4. Arbitrage in Fractal Market

4.1 Simple minded Fractal Market

The market consists of a bond and a stock

)()()()(

)()(

tBdttbtPtdP

dttrAtdAH

4.2 There is Arbitrage

Opportunity

Arbitrage in a market is an investment strategy

which allows an investor,

who starts with nothing,

to get some wealth

without risking anything

Mathematical Meaning of Arbitrage

Example: 5 shares of GE

8 shares of Sun

If GE goes down $2/share

if Sun goes up $3/share

then wealth change

5x(-2)+8x3=14

A portfolio (ut, vt)

At time instant t

ut the total shares in bond

vt the total shares in stock

Let Z be the total wealth at time t

associated with the portfolio (ut, vt):

The portfolio is self-financing if

)()( tPvtAuZttt

market fromout and inmoney No

)()( tPvtdAudZttt

4.2 Arbitrage continued

Arbitrage is a self-financing portfolio such that

0)0 P(Z)(

0Z )(

0 )(

T

T

0

iii

ii

Zi

For this model there is For this model there is arbitrage opportunity!!! arbitrage opportunity!!!

Roger, Shiryaev, KallianpurRoger, Shiryaev, Kallianpur

5. Itô Integral and Itô formula

5.1 Why IntegrationNeed to sum, product, limit

0 be shouldmean The on.perturbati random is )(

change of ratemean is )(,)()(

system aldifferenti stochastic model to

)))(()((2

1lim)))(((

2

1lim

)))(()((2

1lim)(

exmplefor

)))(()((2

1lim

))((lim))((lim)(

.

.

t

.

1

1

1

1

111

11

1

11

H

tt

t

H

ttt

Ht

Htkk

Ht

Htk

Ht

Htkk

Ht

Ht

Htkk

Ht

Htk

Ht

Htk

R

Ht

Wx

xbWxxbx

BBtFtFBBtF

BBtFtFBtF

BBtftf

BBtFBBtFBtF

kkkk

kk

kk

kkkk

5.2 Itô IntegralDucan, Hu, and Pasik-Ducan:

R

1 )()(lim)( Ht

Hti

Ht ii

BBtfdBtf

5.3 Itô FormulaChain Rule

Itô formula

ttt dggfgdf )(')(

dtBfHtdBBfBdf Ht

HHt

Ht

Ht )('')(')( 12

Conventional Product

nmnm xxx

Wick Product

2

2

2

0)!()!(!

!!1

1

xxx

xxx

Example

xxx knmnm

kknkmk

nmnm k

Duncan Hu Pasik-Duncan

Stochastic calculus for fractional Brownian motion.

SIAM J. Control Optimization. 2000, 582-612

6. Fractal Market II

6.1 Fractal Market with Wick ProductThe market is given by a bond and a stock

)()()()(

)()(

tdBdttbtPtdP

dttrAtdAH

6.2 Arbitrage

A portfolio (ut, vt): ut the total investment in bond and vt the total investment in stock.

Let Zt be the total wealth at time t associated with the portfolio (u,v):

The portfolio is self-financing if

)()( tPvtAuZttt

)()( tdPvtdAudZttt

Hu and Oksendal

No Arbitrage Opportunity in the market!

Fractional white noise calculus and

applications to finance.

Infinite Dimensional Analysis, Quantum

Probability and Related Topics, 2003, 1-32.

)()( rTKepFBS

duezwherez

u

2

2

2

1)(

2

log1 22 H

H

TrT

K

p

T

7. Fractal Black-Scholes Formula

)()(

rTKepBS

ScholesandBlackclassicaltocompare

Tr

K

p

T 2log

1 2

)(

)()()( and logarithm natural thedenotes log Here

,)(

21

loglog ,

)(2

1)(

)),(()()(

))(),(

22

22

~2

2

1

),[

22)(

)(2

)(

22

1)(1

tA

tXtvtZu(t)

xezh

where

ttxyyate

tTyk

where

tXktXetv

bygivenistvt(uθ(t)portfoliogreplicatiningcorrespondthe

H

HH

TTz

c

R

Hdzzh

tT

zy

HH

tT

Stock

Value

Current

Price

Strike Price

Price +20 BS FBS hurst s b

AA 1.45

35.89 34.0035.4

52.10 1.90 0.51 0.011

0.000051

AMAT 1.95

14.46 12.5014.4

51.99 2.00 0.53 0.010

40.00005

7

COF 0 62.02 60.2258.9

91.99 2.04 0.53 0.009

0.000057

GE

5.42

37.61 35.7341.2

51.99 2.09 0.54 0.018

20.00006

2

GIS 0 44.62 42.7641.9

51.99 1.16 0.51 0.008

20.00004

8

Comparison between Classical BS and Fractal BS

8. Stochastic volatility and others

Markets of two securities:

Theorem: Let (XT-K)+ be a European call option settled at time T. Then the risk-minimizing hedging price is

Httttt

ttttttt

dBdtYmdYxfYf

t

dWXdtXdXdtrdv

)( and for )(

and above as are )0,(W and , r, where

,

t

1` )( TTP vKXEv

Optimal Consumption and Portfolio and

Stochastic Volatility

v(t)X(t)u(t)A(t)Z(t)

). (u(t),v(t)θ(t)

dBdtYmdYandx fYf

t

dWtXdtXdXdtrdv

Htttt

tttttt

process wealth The

portfolio a choosecan investor an suppose

)( for )(

and above as are )0,(W and , r,

,

t

t

Optimal Consumption and Portfolio and Stochastic Volatility continued

SPDE) gfound(usinbeen hassolution explicit the

portfolio andn consumptio optimal thefind

Define

constant.given be 01 and000Let

if c respect to with financing-self is say that We

2

0

1

21

(t)) (u*(t),v*θ*(t) C*(t)

(T))(Zγ

D(t)dtc

γ

DE))),c(J(θ(

})\{,(-γ , T , DD

c(t)dtv(t)dX(t)-u(t)dA(t)dZ(t)

γc,θz

T γ

Biagini, F.; Hu, Y. Oksendal, B. and Zhang, T.S.

Stochastic Calculus of Fractional Brownian Motion.

Book, Spring, 200x (7≤x<∞)

Hu, Y.

Integral transformations and anticipativecalculus for fractional Brownian motions.

Mem. Amer. Math. Soc. 175 (2005),no. 825, viii+127 pp.

卫公孙朝问于子贡曰：“仲尼焉学？”

子贡曰：“文武之道，未坠于地，在人。贤者识其大者，不贤者识其小者，莫不有文武之道焉。夫子焉不学？而亦何常师之有？”

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