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Financial Engineering

Financial Engineering. Zvi Wiener mswiener@mscc.huji.ac.il 02-588-3049. Elementary Stochastic Calculus. Following Paul Wilmott, Introduces Quantitative Finance Chapter 7. Coin Tossing. R i = -1 or 1 with probability 50% E[R i ] = 0 E[R i 2 ] = 1 E[R i R j ] = 0 Define. Coin Tossing.

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Financial Engineering

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  1. Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

  2. Elementary Stochastic Calculus Following Paul Wilmott, Introduces Quantitative Finance Chapter 7 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

  3. Coin Tossing • Ri = -1 or 1 with probability 50% • E[Ri] = 0 • E[Ri2] = 1 • E[Ri Rj] = 0 • Define FE-Wilmott-IntroQF Ch7

  4. Coin Tossing FE-Wilmott-IntroQF Ch7

  5. Markov Property • No memory except of the current state. • Transition matrix defines the whole dynamic. FE-Wilmott-IntroQF Ch7

  6. The Martingale Property • Some technical conditions are required as well. FE-Wilmott-IntroQF Ch7

  7. Quadratic Variation • For example of a fair coin toss it is = i FE-Wilmott-IntroQF Ch7

  8. Brownian Motion FE-Wilmott-IntroQF Ch7

  9. Brownian Motion • Finiteness – does not diverge • Continuity • Markov • Martingale • Quadratic variation is t • Normality: X(ti) – X(ti-1) ~ N(0, ti-ti-1) FE-Wilmott-IntroQF Ch7

  10. Stochastic Integration FE-Wilmott-IntroQF Ch7

  11. Stochastic Differential Equations dX has 0 mean and standard deviation FE-Wilmott-IntroQF Ch7

  12. Stochastic Differential Equations FE-Wilmott-IntroQF Ch7

  13. Simulating Markov Process • The Wiener process The Generalized Wiener process The Ito process FE-Wilmott-IntroQF Ch7

  14. value time FE-Wilmott-IntroQF Ch7

  15. Ito’s Lemma • dt dX • dt 0 0 • dX 0 dt FE-Wilmott-IntroQF Ch7

  16. Arithmetic Brownian Motion • At time 0 we know that S(t) is distributed normally with mean S(0)+t and variance 2t. FE-Wilmott-IntroQF Ch7

  17. S   time Arithmetic BMdS =  dt +  dX FE-Wilmott-IntroQF Ch7

  18. The Geometric Brownian Motion Used for stock prices, exchange rates.  is the expected price appreciation:  = total - q. S follows a lognormal distribution. FE-Wilmott-IntroQF Ch7

  19. The Geometric Brownian Motion FE-Wilmott-IntroQF Ch7

  20. The Geometric Brownian Motion FE-Wilmott-IntroQF Ch7

  21. S time Geometric BMdS = Sdt + SdX FE-Wilmott-IntroQF Ch7

  22. The Geometric Brownian Motion FE-Wilmott-IntroQF Ch7

  23. Mean-Reverting Processes FE-Wilmott-IntroQF Ch7

  24. Mean-Reverting Processes FE-Wilmott-IntroQF Ch7

  25. Speed of mean reversion Long term mean Simulating Yields • GBM processes are widely used for stock prices and currencies (not interest rates). A typical model of interest rates dynamics: FE-Wilmott-IntroQF Ch7

  26. Simulating Yields •  = 0 - Vasicek model, changes are normally distr. •  = 1 - lognormal model, RiskMetrics. •  = 0.5 - Cox, Ingersoll, Ross model (CIR). FE-Wilmott-IntroQF Ch7

  27. Mean Reverting ProcessdS = (-S)dt + SdX S  time FE-Wilmott-IntroQF Ch7

  28. Other models • Ho-Lee term-structure model • HJM (Heath, Jarrow, Morton) is based on forward rates - no-arbitrage type. • Hull-White model: FE-Wilmott-IntroQF Ch7

  29. Home Assignment • Read chapter 7 in Wilmott. • Follow Excel files coming with the book. FE-Wilmott-IntroQF Ch7

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