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## Black-Scholes Formula Using Long Memory

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### Black-Scholes Formula Using Long Memory

Yaozhong Hu (胡耀忠)

University of Kansas

hu@math.ku.edu

www.math.ku.edu/~hu

2007年7月于烟台

Black-Scholes Formula Using Long Memory

- Simple example
- Black and Scholes theory
- Fractional Brownian motion
- 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

1.2 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

1.3 How to price an option

How to fairly price an option?

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

Example

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.

IntroducedBrownian 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

2. Black and Scholes Theory

2.1 History, Continued

Bachelier model

can take negative values!!!

Black and Scholes Model

Geometric Brownian motion

2.3 Black and Scholes Formula

The Price of European call option is given

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 Timesof Wednesday, 15th October 1997

Scholes and Merton “won the Nobel Memorial Prize in Economics Science yesterday for work that enablesinvestors 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

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)

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

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

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

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

Let Z be the total wealth at time t associated with the portfolio (ut, vt):

The portfolio is self-financing if

4.2 Arbitrage continued

Arbitrage is a self-financing portfolio such that

For this model there is arbitrage opportunity!!!

- Roger, Shiryaev, Kallianpur

5.2 Itô Integral

Ducan, Hu, and Pasik-Ducan:

Duncan Hu Pasik-Duncan

Stochastic calculus for fractional Brownian motion.

SIAM J. Control Optimization. 2000, 582-612

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

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.

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

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 anticipative

calculus 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？”

海纳百川，有容乃大,

壁立千仞，无欲则刚！

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