BlackScholes Formula Using Long Memory

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Black-Scholes Formula Using Long Memory

• Yaozhong Hu (???)
• University of Kansas
• hu_at_math.ku.edu
• www.math.ku.edu/hu
• 2007?7????

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Black-Scholes Formula Using Long Memory

• Simple example
• Black and Scholes theory
• Fractional Brownian motion
• Arbitrage in Fractal Market

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

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1. Simple Example

• 1.1 A Simple example
• MCDONALDS CORP (MCD)
• Friday, Jul 13-2007, 51.91

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Last Fridays Prices of Mcdonalds Corp
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• 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

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

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Financial Derivatives

• Call
• Put

• European
• American

Many Other options
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• 1.3 How to price an option
• How to fairly price an option?
• If (future) stock price is known then it is easy
• Example

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

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

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

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Simulation of Brownian Motion
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Prices of Yahoo! INC (YHOO)
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• Mathematical Theory
• L. Bachalier 1900
• A. Einstein 1905
• N. Wiener 1923

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2. Black and Scholes Theory

• 2.1 History, Continued
• Bachelier model
• can take negative values!!!
• Black and Scholes Model
• Geometric Brownian motion

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• 2.2. Black-Scholes Model
• Market consists of

Bond Stock
P(t) is Geometric Brownian Motion
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Simulation of Geometric Brownian Motion
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• 2.3 Black and Scholes Formula
• The Price of European call option is given

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• p current price of the stock
• s 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!!!

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• 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,

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

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

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

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3. Fractional Brownian Motion

• 3.1 Long Memory
• Long Memory Joseph Effect
• Self-Similar

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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.
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• 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)

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

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• 3.2 Fractional Brownian Motion
• Let 0 Hurst parameter H is a Gaussian process satisfying

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• 3.3 Properties
• 1. Self-similar has the same property law
  as
• 2. Long-range dependent if H1/2

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• 3. If H 1/2 , standard Brownian motion
• 4. h1/2, Positively correlated
• 5. H
• 6. Not a semi-martingale
• 7. Not Markovian
• 8. Nowhere differentiable

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Daily return, over 1024 trading days
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• Granger, C.W.J.
• Long memory relationships and aggregation
• of dynamic models.
• J. Econometrics, 1980, 227-238.
• The Nobel Memorial Prize, 2003

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4. Arbitrage in Fractal Market

• 4.1 Simple minded Fractal Market
• The market consists of a bond and a stock

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• 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

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• 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)8x314

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

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

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• 4.2 Arbitrage continued
• Arbitrage is a self-financing portfolio such that

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For this model there is arbitrage opportunity!!!
Roger, Shiryaev, Kallianpur
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5. Itô Integral and Itô formula

• 5.1 Why Integration
• Need to sum, product, limit

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• 5.2 Itô Integral
• Ducan, Hu, and Pasik-Ducan

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• 5.3 Itô Formula
• Chain Rule
• Itô formula

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• Conventional Product

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• Wick Product

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• Duncan Hu Pasik-Duncan
• Stochastic calculus for fractional Brownian
  motion.
• SIAM J. Control Optimization. 2000, 582-612

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6. Fractal Market II

• 6.1 Fractal Market with Wick Product
• The market is given by a bond and a stock

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• 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

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• 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.

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7. Fractal Black-Scholes Formula

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Comparison between Classical BS and Fractal BS

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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

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• Optimal Consumption and Portfolio and Stochastic
  Volatility

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• Optimal Consumption and Portfolio and Stochastic
  Volatility continued

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• Biagini, F. Hu, Y. Oksendal, B. and Zhang,
  T.S.
• Stochastic Calculus of Fractional Brownian
  Motion.
• Book, Spring, 200x (7x

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• Hu, Y.
• Integral transformations and anticipative
• calculus for fractional Brownian motions.
• Mem. Amer. Math. Soc. 175 (2005),
• no. 825, viii127 pp.

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Thanks