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 8 The Black-Scholes model

3. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 3 Notations V(S, t; ,  ; E, T; r) S and t are variables  and  are parameters of the asset E and T are parameters of the contract r is parameter of the currency

4. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 4 Assumption Perfect markets Complete markets No arbitrage. Risk factor dynamics:

5. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 5 Assumptions of BS The underlying price moves continuously Interest rates are known and constant The variance of returns is constant Perfect capital markets no transaction costs short sales are allowed markets operate continuously price taking

6. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 6 V(S,t)

7. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 7 Delta Hedging Form a delta-balanced portfolio: V-  S

8. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 8 No Arbitrage

9. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 9 Black-Scholes-Merton Equation time S T ? 0 payoff

10. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 10 Black-Scholes-Merton Equation Read at home: Dividend-paying stock Currency Commodity Forwards Options on Futures

11. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 11

12. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 12 Digital=Binary options Payoff = 1 if S T >K, and 0 otherwise (Call) stock K Payoff at maturity

13. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 13 Digital=Binary options Payoff = 1 if S T <K, and 0 otherwise (Put) stock K Payoff at maturity

14. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 14 Exchange Option (Margrabe 78) r should be replaced by the difference of yields of the two assets.

15. Zvi WienerFE-Wilmott-IntroQF Ch8 slide 15 Home Assignment Read chapter 8 in Wilmott. Follow Excel files coming with the book.