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

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

3. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 3 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

4. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 4 Coin Tossing

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

6. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 6 The Martingale Property Some technical conditions are required as well.

7. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 7 Quadratic Variation For example of a fair coin toss it is = i

8. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 8 Brownian Motion

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

10. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 10 Stochastic Integration

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

12. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 12 Stochastic Differential Equations

13. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 13 Simulating Markov Process The Wiener process The Generalized Wiener process The Ito process

14. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 14 time value

15. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 15 Ito’s Lemma dtdX dt00 dX0dt

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

17. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 17 Arithmetic BM dS =  dt +  dX   time S

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

19. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 19 The Geometric Brownian Motion

20. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 20 The Geometric Brownian Motion

21. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 21 Geometric BM dS =  Sdt +  SdX time S

22. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 22 The Geometric Brownian Motion

23. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 23 Mean-Reverting Processes

24. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 24 Mean-Reverting Processes

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

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

27. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 27 Mean Reverting Process dS =  (  -S)dt +  S  dX time S 

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

29. Zvi WienerFE-Wilmott-IntroQF Ch7 slide 29 Home Assignment Read chapter 7 in Wilmott. Follow Excel files coming with the book.