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 4, see www.wiley.co.uk/wilmottwww.wiley.co.uk/wilmott Math

3. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 3 e Natural logarithm 2.718281828459045235360287471352662497757… e x = Exp(x) e 0 = 1 e 1 = e

4. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 4 x Exp(x)

5. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 5 Ln Logarithm with base e. e ln(x) = x, or ln(e x ) = x Determined for x>0 only!

6. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 6 Ln x Ln(x)

7. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 7 Differentiation and Taylor series x f(x)

8. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 8 Differentiation and Taylor series

9. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 9 Differentiation and Taylor series x x+  x

10. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 10 Taylor series one variable

11. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 11 Taylor series two variable

12. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 12 Differential Equations Ordinary Partial Boundary conditions Initial Conditions

13. FE-W http://pluto.mscc.huji.ac.il/~mswiener/zvi.html EMBAF Chapter 2 Quantitative Analysis Fundamentals of Probability Following P. Jorion 2001 Financial Risk Manager Handbook

14. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 14 Random Variables Values, probabilities. Distribution function, cumulative probability. Example: a die with 6 faces.

15. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 15 Random Variables Distribution function of a random variable X F(x) = P(X  x) - the probability of x or less. If X is discrete then If X is continuous then Note that

16. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 16 Random Variables Probability density function of a random variable X has the following properties

17. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 17 Independent variables Credit exposure in a swap depends on two random variables: default and exposure. If the two variables are independent one can construct the distribution of the credit loss easily.

18. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 18 Moments Mean = Average = Expected value Variance

19. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 19 Its meaning... Skewness (non-symmetry) Kurtosis (fat tails)

20. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 20 Main properties

21. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 21 Portfolio of Random Variables

22. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 22 Portfolio of Random Variables

23. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 23 Product of Random Variables Credit loss derives from the product of the probability of default and the loss given default. When X 1 and X 2 are independent

24. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 24 Transformation of Random Variables Consider a zero coupon bond If r=6% and T=10 years, V = $55.84, we wish to estimate the probability that the bond price falls below $50. This corresponds to the yield 7.178%.

25. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 25 The probability of this event can be derived from the distribution of yields. Assume that yields change are normally distributed with mean zero and volatility 0.8%. Then the probability of this change is 7.06% Example

26. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 26 Quantile Quantile (loss/profit x with probability c) 50% quantile is called median Very useful in VaR definition.

27. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 27 FRM-99, Question 11 X and Y are random variables each of which follows a standard normal distribution with cov(X,Y)=0.4. What is the variance of (5X+2Y)? A. 11.0 B. 29.0 C. 29.4 D. 37.0

28. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 28 FRM-99, Question 11

29. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 29 FRM-99, Question 21 The covariance between A and B is 5. The correlation between A and B is 0.5. If the variance of A is 12, what is the variance of B? A. 10.00 B. 2.89 C. 8.33 D. 14.40

30. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 30 FRM-99, Question 21

31. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 31 Uniform Distribution Uniform distribution defined over a range of values a  x  b.

32. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 32 Uniform Distribution abab 1

33. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 33 Normal Distribution Is defined by its mean and variance. Cumulative is denoted by N(x).

34. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 34 Normal Distribution 66% of events lie between -1 and 1 95% of events lie between -2 and 2

35. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 35 Normal Distribution

36. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 36 Normal Distribution symmetric around the mean mean = median skewness = 0 kurtosis = 3 linear combination of normal is normal 99.99 99.90 99 97.72 97.5 95 90 84.13 50 3.715 3.09 2.326 2.000 1.96 1.645 1.282 1 0

37. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 37 Lognormal Distribution The normal distribution is often used for rate of return. Y is lognormally distributed if X=lnY is normally distributed. No negative values!

38. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 38 Lognormal Distribution If r is the expected value of the lognormal variable X, the mean of the associated normal variable is r-0.5  2.

39. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 39 Student t Distribution Arises in hypothesis testing, as it describes the distribution of the ratio of the estimated coefficient to its standard error. k - degrees of freedom.

40. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 40 Student t Distribution As k increases t-distribution tends to the normal one. This distribution is symmetrical with mean zero and variance (k>2) The t-distribution is fatter than the normal one.

41. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 41 Binomial Distribution Discrete random variable with density function: For large n it can be approximated by a normal.

42. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 42 FRM-99, Question 13 What is the kurtosis of a normal distribution? A. 0 B. can not be determined, since it depends on the variance of the particular normal distribution. C. 2 D. 3

43. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 43 FRM-99, Question 16 If a distribution with the same variance as a normal distribution has kurtosis greater than 3, which of the following is TRUE? A. It has fatter tails than normal distribution B. It has thinner tails than normal distribution C. It has the same tail fatness as normal D. can not be determined from the information provided

44. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 44 FRM-99, Question 5 Which of the following statements best characterizes the relationship between normal and lognormal distributions? A. The lognormal distribution is logarithm of the normal distribution. B. If ln(X) is lognormally distributed, then X is normally distributed. C. If X is lognormally distributed, then ln(X) is normally distributed. D. The two distributions have nothing in common

45. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 45 FRM-98, Question 10 For a lognormal variable x, we know that ln(x) has a normal distribution with a mean of zero and a standard deviation of 0.2, what is the expected value of x? A. 0.98 B. 1.00 C. 1.02 D. 1.20

46. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 46 FRM-98, Question 10

47. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 47 FRM-98, Question 16 Which of the following statements are true? I. The sum of normal variables is also normal II. The product of normal variables is normal III. The sum of lognormal variables is lognormal IV. The product of lognormal variables is lognormal A. I and II B. II and III C. III and IV D. I and IV

48. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 48 FRM-99, Question 22 Which of the following exhibits positively skewed distribution? I. Normal distribution II. Lognormal distribution III. The returns of being short a put option IV. The returns of being long a call option A. II only B. III only C. II and IV only D. I, III and IV only

49. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 49 FRM-99, Question 22 C. The lognormal distribution has a long right tail, since the left tail is cut off at zero. Long positions in options have limited downsize, but large potential upside, hence a positive skewness.

50. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 50 FRM-99, Question 3 It is often said that distributions of returns from financial instruments are leptokurtotic. For such distributions, which of the following comparisons with a normal distribution of the same mean and variance MUST hold? A. The skew of the leptokurtotic distribution is greater B. The kurtosis of the leptokurtotic distribution is greater C. The skew of the leptokurtotic distribution is smaller D. The kurtosis of the leptokurtotic distribution is smaller

51. Zvi WienerFE-Wilmott-IntroQF Ch4 slide 51 Home Assignment Read chapters 4, 5 in Wilmott. Read and understand the xls files!! Build a module for pricing of the Max, Min and Mixture programs (BRIRA). Analyze the program offered by BH. Build a module for pricing of this program. Describe in terms of options the client’s position in the program offered by FIBI.