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### Fundamentals of Stochastic Calculus and Quantitative Finance: Insights from Wilmott ###

This document provides a comprehensive overview of key concepts in stochastic calculus as introduced in Chapter 7 of Paul Wilmott's "Introduces Quantitative Finance." It covers essential topics such as coin tossing, the Markov property, martingale property, quadratic variation, Brownian motion, and stochastic differential equations. Additionally, it explores models of financial dynamics including geometric Brownian motion, mean-reverting processes, and popular methods for simulating yields. These concepts are foundational for understanding financial modeling and analysis in quantitative finance. ###

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### Fundamentals of Stochastic Calculus and Quantitative Finance: Insights from Wilmott ###

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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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