Black-Scholes Formula Using Long Memory

1. 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

2. Black-Scholes Formula Using Long Memory • Simple example • Black and Scholes theory • Fractional Brownian motion • Arbitrage in Fractal Market

3. 5. Itô integral and Itô formula 6. New Fractal market 7. Fractal Black and Scholes formula 8. Stochastic volatility and others

5. 1. Simple Example 1.1 A Simple example MCDONALD’S CORP (MCD) Friday, Jul 13-2007, $51.91

6. Last Friday’s Prices of Mcdonald’s Corp

7. 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

8. Example (call option) Right to buy one share of MCD at the end of one year with $60

9. Financial Derivatives • European • American • Call • Put Many Other options

10. 1.3 How to price an option How to fairly price an option? If (future) stock price is known then it is easy Example

11. Future stock price is unknown Math model of market Probability distribution of future stock price Stochastic Differential Equations

12. 1.4 History Louis Bachelier Théorie de la spéculation Ann. Sci. École Norm. Sup. 1900, 21-86. IntroducedBrownian motion Solved problem

13. 1.5 History of Brownian Motion Robert Brown (1828) An British botanist observed that pollen grains suspended in water perform a continual swarming motion

14. Simulation of Brownian Motion

15. Prices of Yahoo! INC (YHOO)

16. Mathematical Theory L. Bachalier 1900 A. Einstein 1905 N. Wiener 1923

17. 2. Black and Scholes Theory 2.1 History, Continued Bachelier model can take negative values!!! Black and Scholes Model Geometric Brownian motion

18. 2.2. Black-Scholes Model Market consists of Bond: Stock: P(t) is Geometric Brownian Motion

19. Simulation of Geometric Brownian Motion

20. 2.3 Black and Scholes Formula The Price of European call option is given

21. 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!!!

22. 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,

23. 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.”

24. 2.4 Main Idea and Tool Itô stochastic calculus a mathematical tool from probability stochastic analysis

25. 2.5 Extension of Black and Scholes Models • Jump Diffusions • Markov Processes • Semimartignales • Long memory processes

26. 3. Fractional Brownian Motion 3.1 Long Memory Long Memory = Joseph Effect Self-Similar

27. 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. …”

28. 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)

30. 大江东去 一江春水向东流 大江北上?

35. Time-series record of the Nile River minimum water levels from 662-1284 AD

37. 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

38. 3.2 Fractional Brownian Motion Let 0 < H < 1. Fractional Brownian motion with Hurst parameter H is a Gaussian process satisfying

39. 3.3 Properties 1. Self-similar: has the same property law as 2. Long-range dependent if H>1/2

40. 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

42. Daily return, over 1024 trading days

43. Granger, C.W.J. Long memory relationships and aggregation of dynamic models. J. Econometrics, 1980, 227-238. The Nobel Memorial Prize, 2003

44. 4. Arbitrage in Fractal Market 4.1 Simple minded Fractal Market The market consists of a bond and a stock

45. 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

46. 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

47. A portfolio (ut, vt) At time instant t ut the total shares in bond vt the total shares in stock

48. Let Z be the total wealth at time t associated with the portfolio (ut, vt): The portfolio is self-financing if

49. 4.2 Arbitrage continued Arbitrage is a self-financing portfolio such that

50. For this model there is arbitrage opportunity!!! • Roger, Shiryaev, Kallianpur