Black-Scholes Formula Using Long Memory. Yaozhong Hu ( 胡耀忠 ) University of Kansas email@example.com 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.
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Yaozhong Hu (胡耀忠)
University of Kansas
6. New Fractal market
7. Fractal Black and Scholes formula
8. Stochastic volatility and others
1.1 A Simple example
MCDONALD’S CORP (MCD)
Friday, Jul 13-2007, $51.91
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
Right to buy one share of MCD at the end of one year with $60
Many Other options
How to fairly price an option?
If (future) stock price is known then it is easy
Math model of market
Probability distribution of future stock price
Stochastic Differential Equations
Théorie de la spéculation
Ann. Sci. École Norm. Sup. 1900, 21-86.
Robert Brown (1828)
An British botanist observed that
pollen grains suspended in water perform a continual swarming motion
L. Bachalier 1900
A. Einstein 1905
N. Wiener 1923
2.1 History, Continued
can take negative values!!!
Black and Scholes Model
Geometric Brownian motion
Market consists of
P(t) is Geometric Brownian Motion
The Price of European call option is given
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!!!
Scholes and Merton “won the Nobel Memorial Prize in Economics Science yesterday for work that enablesinvestors to price accurately their bets on the future,
Itô stochastic calculus
a mathematical tool from probability
3.1 Long Memory
Long Memory = Joseph Effect
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. …”
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)
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
Let 0 < H < 1. Fractional Brownian motion with Hurst parameter H is a Gaussian process satisfying
1. Self-similar: has the same property law as
2. Long-range dependent if H>1/2
4. h>1/2, Positively correlated
5. H<1/2, Negatively correlated
6. Not a semi-martingale
7. Not Markovian
8. Nowhere differentiable
Long memory relationships and aggregation
of dynamic models.
J. Econometrics, 1980, 227-238.
The Nobel Memorial Prize, 2003
4.1 Simple minded Fractal Market
The market consists of a bond and a stock
Arbitrage in a market is an investment strategy
which allows an investor,
who starts with nothing,
to get some wealth
without risking anything
Example: 5 shares of GE
8 shares of Sun
If GE goes down $2/share
if Sun goes up $3/share
then wealth change
At time instant t
ut the total shares in bond
vt the total shares in stock
The portfolio is self-financing if
Arbitrage is a self-financing portfolio such that
5.1 Why Integration
Need to sum, product, limit
Ducan, Hu, and Pasik-Ducan:
Stochastic calculus for fractional Brownian motion.
SIAM J. Control Optimization. 2000, 582-612
6.1 Fractal Market with Wick Product
The market is given by a bond and a stock
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
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.
Markets of two securities:
Theorem: Let (XT-K)+ be a European call option settled at time T. Then the risk-minimizing hedging price is
Stochastic Calculus of Fractional Brownian Motion.
Book, Spring, 200x (7≤x<∞)
Integral transformations and anticipative
calculus for fractional Brownian motions.
Mem. Amer. Math. Soc. 175 (2005),
no. 825, viii+127 pp.