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 11 Multi-Asset Options

3. Zvi WienerFE-Wilmott-IntroQF Ch11 slide 3 Multidimensional Lognormal Random Walk

4. Zvi WienerFE-Wilmott-IntroQF Ch11 slide 4 Correlation Matrix Note that  ii = 1 and  ji =  ij. Positive definite: for any vector y we have y T  y  0.

5. Zvi WienerFE-Wilmott-IntroQF Ch11 slide 5 Measuring Correlation Cov(X,Y) = E( (E(X)-X) (E(Y)-Y) ) Let R a (t), R b (t) be returns on assets a and b on day t. Then Problems: time delay in observations, missing data, time delay in response (CPI), correlation does not mean dependence (copula).

6. Zvi WienerFE-Wilmott-IntroQF Ch11 slide 6 Correlation Can be a bad measure of dependence! Consider X ~ N(0, 1) and X 2. Are they dependent? What is the correlation? Cointegartion: two time series are cointegrated if their linear combination has constant mean and standard deviation. Bonds of emerging markets …

7. Zvi WienerFE-Wilmott-IntroQF Ch11 slide 7 Options on Many Underlyings Exchanging one asset for another Max(Q a S a -Q b S b, 0 ) Note that: Max(x, y) = y + Max(x-y, 0) Min(x, y) = x – Max(x-y, 0) Similarity reduction = Application of Numeraire

8. Zvi WienerFE-Wilmott-IntroQF Ch11 slide 8 Options on Many Underlyings Analytical solution Reasonable approximation Numerical Approach trees Monte Carlo (bad for American style) finite difference

9. Zvi WienerFE-Wilmott-IntroQF Ch11 slide 9 Home Assignment Read chapter 11 in Wilmott. Follow Excel files coming with the book. Read and analyze (do not price) the structured notes of LEUMI and HAPOALIM – submit to Oz before the end of February.