Financial Engineering

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

Math Following Paul Wilmott, Introduces Quantitative Finance Chapter 4, see www.wiley.co.uk/wilmott http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

e • Natural logarithm • 2.718281828459045235360287471352662497757… • ex = Exp(x) • e0 = 1 • e1 = e FE-Wilmott-IntroQF Ch4

Exp(x) x FE-Wilmott-IntroQF Ch4

Ln • Logarithm with base e. • eln(x) = x, or ln(ex) = x • Determined for x>0 only! FE-Wilmott-IntroQF Ch4

Ln Ln(x) x FE-Wilmott-IntroQF Ch4

Differentiation and Taylor series f(x) x FE-Wilmott-IntroQF Ch4

Differentiation and Taylor series FE-Wilmott-IntroQF Ch4

Differentiation and Taylor series x+x x FE-Wilmott-IntroQF Ch4

Taylor seriesone variable FE-Wilmott-IntroQF Ch4

Taylor seriestwo variable FE-Wilmott-IntroQF Ch4

Differential Equations • Ordinary • Partial • Boundary conditions • Initial Conditions FE-Wilmott-IntroQF Ch4

Chapter 2Quantitative AnalysisFundamentals of Probability Following P. Jorion 2001 Financial Risk Manager Handbook http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

Random Variables • Values, probabilities. • Distribution function, cumulative probability. • Example: a die with 6 faces. FE-Wilmott-IntroQF Ch4

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 FE-Wilmott-IntroQF Ch4

Random Variables • Probability density function of a random variable X has the following properties FE-Wilmott-IntroQF Ch4

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. FE-Wilmott-IntroQF Ch4

Moments • Mean = Average = Expected value Variance FE-Wilmott-IntroQF Ch4

Its meaning ... Skewness (non-symmetry) Kurtosis (fat tails) FE-Wilmott-IntroQF Ch4

Main properties FE-Wilmott-IntroQF Ch4

Portfolio of Random Variables FE-Wilmott-IntroQF Ch4

Product of Random Variables • Credit loss derives from the product of the probability of default and the loss given default. When X1 and X2 are independent FE-Wilmott-IntroQF Ch4

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%. FE-Wilmott-IntroQF Ch4

Example • 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% FE-Wilmott-IntroQF Ch4

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

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 FE-Wilmott-IntroQF Ch4

FRM-99, Question 11 FE-Wilmott-IntroQF Ch4

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 FE-Wilmott-IntroQF Ch4

Uniform Distribution • Uniform distribution defined over a range of values axb. FE-Wilmott-IntroQF Ch4

Uniform Distribution 1 a b FE-Wilmott-IntroQF Ch4

Normal Distribution • Is defined by its mean and variance. Cumulative is denoted by N(x). FE-Wilmott-IntroQF Ch4

66% of events lie between -1 and 1 95% of events lie between -2 and 2 Normal Distribution FE-Wilmott-IntroQF Ch4

Normal Distribution FE-Wilmott-IntroQF Ch4

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 FE-Wilmott-IntroQF Ch4

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! FE-Wilmott-IntroQF Ch4

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. FE-Wilmott-IntroQF Ch4

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. FE-Wilmott-IntroQF Ch4

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. FE-Wilmott-IntroQF Ch4

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

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 FE-Wilmott-IntroQF Ch4

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 FE-Wilmott-IntroQF Ch4

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 FE-Wilmott-IntroQF Ch4

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 FE-Wilmott-IntroQF Ch4

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 FE-Wilmott-IntroQF Ch4

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 FE-Wilmott-IntroQF Ch4

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. FE-Wilmott-IntroQF Ch4

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 FE-Wilmott-IntroQF Ch4

Financial Engineering