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Financial Engineering

This chapter delves into exponential and logarithmic functions, differentiation, Taylor series, random variables, distribution functions, moments, properties, and portfolio analysis in financial engineering. Key concepts in probability, variance, and transformations are explored along with practical examples and calculations related to financial risk management.

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Financial Engineering

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

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

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

  4. Exp(x) x FE-Wilmott-IntroQF Ch4

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

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

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

  8. Differentiation and Taylor series FE-Wilmott-IntroQF Ch4

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

  10. Taylor seriesone variable FE-Wilmott-IntroQF Ch4

  11. Taylor seriestwo variable FE-Wilmott-IntroQF Ch4

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

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

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

  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 FE-Wilmott-IntroQF Ch4

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

  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. FE-Wilmott-IntroQF Ch4

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

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

  20. Main properties FE-Wilmott-IntroQF Ch4

  21. Portfolio of Random Variables FE-Wilmott-IntroQF Ch4

  22. Portfolio of Random Variables FE-Wilmott-IntroQF Ch4

  23. 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

  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%. FE-Wilmott-IntroQF Ch4

  25. 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

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

  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 FE-Wilmott-IntroQF Ch4

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

  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 FE-Wilmott-IntroQF Ch4

  30. FRM-99, Question 21 FE-Wilmott-IntroQF Ch4

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

  32. Uniform Distribution 1 a b FE-Wilmott-IntroQF Ch4

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

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

  35. Normal Distribution FE-Wilmott-IntroQF Ch4

  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 FE-Wilmott-IntroQF Ch4

  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! FE-Wilmott-IntroQF Ch4

  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. FE-Wilmott-IntroQF Ch4

  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. FE-Wilmott-IntroQF Ch4

  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. FE-Wilmott-IntroQF Ch4

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

  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 FE-Wilmott-IntroQF Ch4

  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 FE-Wilmott-IntroQF Ch4

  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 FE-Wilmott-IntroQF Ch4

  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 FE-Wilmott-IntroQF Ch4

  46. FRM-98, Question 10 FE-Wilmott-IntroQF Ch4

  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 FE-Wilmott-IntroQF Ch4

  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 FE-Wilmott-IntroQF Ch4

  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. FE-Wilmott-IntroQF Ch4

  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 FE-Wilmott-IntroQF Ch4

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