1. http://pluto.huji.ac.il/~mswiener/zvi.html 972-2-588-3049 FRM Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook Financial Risk Management

2. http://pluto.huji.ac.il/~mswiener/zvi.html 972-2-588-3049 FRM Chapter 4 Quantitative Analysis Monte Carlo Methods Following P. Jorion 2001 Financial Risk Manager Handbook

3. Ch. 4, HandbookZvi Wiener slide 3 Monte Carlo

4. Ch. 4, HandbookZvi Wiener slide 4 Monte Carlo Simulation

5. Ch. 4, HandbookZvi Wiener slide 5 Simulating Markov Process The Wiener process The Generalized Wiener process The Ito process

6. Ch. 4, HandbookZvi Wiener slide 6 The Geometric Brownian Motion Used for stock prices, exchange rates.  is the expected price appreciation:  =  total - q. S follows a lognormal distribution.

7. Ch. 4, HandbookZvi Wiener slide 7 The Geometric Brownian Motion

8. Ch. 4, HandbookZvi Wiener slide 8 time value

9. Ch. 4, HandbookZvi Wiener slide 9 Simulating Yields GBM processes are widely used for stock prices and currencies (not interest rates). A typical model of interest rates dynamics: Speed of mean reversion Long term mean

10. Ch. 4, HandbookZvi Wiener slide 10 Simulating Yields  = 0 - Vasicek model, changes are normally distr.  = 1 - lognormal model, RiskMetrics.  = 0.5 - Cox, Ingersoll, Ross model (CIR).

11. Ch. 4, HandbookZvi Wiener slide 11 Other models Ho-Lee term-structure model HJM (Heath, Jarrow, Morton) is based on forward rates - no-arbitrage type. Hull-White model:

12. Ch. 4, HandbookZvi Wiener slide 12 FRM-99, Question 18 If S and Q follow a geometric Brownian Motion which of the following is true? A. Log(S+Q) is normally distributed B. S*Q is lognormally distributed C. S*Q is normally distributed D. S + Q is normally distributed

13. Ch. 4, HandbookZvi Wiener slide 13 FRM-99, Question 19 Considering a one-factor CIR term structure model and the Vasicek model: I. Drift coefficients are different II. Both include mean reversion III. Coefficients of the stochastic term, dz, are different. IV. CIR is a jump-diffusion model. A. All of the above is true B. I and III are true C. II, III, and IV are true D. II and III are true

14. Ch. 4, HandbookZvi Wiener slide 14 FRM-99, Question 19 Considering a one-factor CIR term structure model and the Vasicek model: I. Drift coefficients are different II. Both include mean reversion III. Coefficients of the stochastic term, dz, are different. IV. CIR is a jump-diffusion model. A. All of the above is true B. I and III are true C. II, III, and IV are true D. II and III are true

15. Ch. 4, HandbookZvi Wiener slide 15 FRM-99, Question 25 The Vasicek modle defines a risk-neutral process for r which is dr=a(b-r)dt +  dz, where a, b, and  are constants, and r represents the rate of interest. From the equation we conclude that the model is a: A. Monte Carlo type model B. Single factor term structure model C. Two-factor term structure model D. Decision tree model

16. Ch. 4, HandbookZvi Wiener slide 16 FRM-99, Question 26 The term a(b-r) in the equation dr=a(b-r)dt +  dz, represents which term? A. Gamma B. Stochastic C. Mean reversion D. Vega

17. Ch. 4, HandbookZvi Wiener slide 17 FRM-99, Question 30 For which of the following currencies would it be most appropriate to choose a lognormal interest rate model over a normal model? A. USD B. JPY C. DEM D. GBP

18. Ch. 4, HandbookZvi Wiener slide 18 FRM-99, Question 30 For which of the following currencies would it be most appropriate to choose a lognormal interest rate model over a normal model? A. USD B. JPY C. DEM D. GBP

19. Ch. 4, HandbookZvi Wiener slide 19 FRM-98, Question 23 Which of the following interest rate term structure models tends to capture the mean reversion of interest rates? A. dr=a*(b-r)*dt +  *dz B. dr=a*dt +  *dz C. dr=a*r*dt +  *dz D. dr=a*(r-b)*dt +  *dz Bad question

20. Ch. 4, HandbookZvi Wiener slide 20 FRM-98, Question 24 Which of the following is a shortcoming of modeling a bond option by applying Black- Scholes formula to bond prices? A. It fails to capture convexity in a bond. B. It fails to capture the pull-to-par effect. C. It fails to maintain the put-call parity. D. It works for zero-coupon bond options only.

21. Ch. 4, HandbookZvi Wiener slide 21 FRM-00, Question 118 Which group of term structure models do the Ho-Lee, Hull-White and Heath, Jarrow, Morton models belong to? A. No-arbitrage models. B. Two-factor models. C. Log normal models. D. Deterministic models.

22. Ch. 4, HandbookZvi Wiener slide 22 FRM-00, Question 118 Which group of term structure models do the Ho-Lee, Hull-White and Heath, Jarrow, Morton models belong to? A. No-arbitrage models. B. Two-factor models. C. Log normal models. D. Deterministic models.

23. Ch. 4, HandbookZvi Wiener slide 23 FRM-00, Question 119 A plausible stochastic process for the short- term rate is often considered to be one where the rate is pulled back to some long-run average level. Which one of the following term structure models does NOT include this? A. The Vasicek model. B. The Ho-Lee model. C. The Hull-White model. D. The Cox-Ingersoll-Ross model.

24. Ch. 4, HandbookZvi Wiener slide 24 FRM-00, Question 119 A plausible stochastic process for the short- term rate is often considered to be one where the rate is pulled back to some long-run average level. Which one of the following term structure models does NOT include this? A. The Vasicek model. B. The Ho-Lee model. C. The Hull-White model. D. The Cox-Ingersoll-Ross model.

25. Ch. 4, HandbookZvi Wiener slide 25 Simulations for VaR Choose a stochastic process Generate a pseudo-sequence of variables Generate prices from these variables Calculate the value of the portfolio Repeat steps above many times Calculate VaR from the resulting distribution of values.

26. Ch. 4, HandbookZvi Wiener slide 26 Risk-neutral approach Standard approach assumes some risk aversion and utility function. Risk neutral approach - change probabilities in order to get

27. Ch. 4, HandbookZvi Wiener slide 27 Accuracy Sampling variability Antithetic Variable Technique Control Variable Technique Quasi-Random Sequences Very difficult to use for American types.

28. Ch. 4, HandbookZvi Wiener slide 28 Monte Carlo

29. Ch. 4, HandbookZvi Wiener slide 29 Monte Carlo

30. Ch. 4, HandbookZvi Wiener slide 30 Monte Carlo

31. Ch. 4, HandbookZvi Wiener slide 31 Monte Carlo

32. Ch. 4, HandbookZvi Wiener slide 32 Speed of convergence Whole circle Upper triangle

33. Ch. 4, HandbookZvi Wiener slide 33 Smart Sampling

34. Ch. 4, HandbookZvi Wiener slide 34 Spectral Truncation

35. Ch. 4, HandbookZvi Wiener slide 35 Regular Grid An alternative to MC is using a regular grid to approximate the integral. Advantages: The speed of convergence is error~1/N. All areas are covered more uniformly. There is no need to generate random numbers. Disadvantages: One can’t improve it a little bit. It is more difficult to use it with a measure.

36. Ch. 4, HandbookZvi Wiener slide 36 FRM-99, Question 8 VaR of a portfolio was estimated with 1,000 independent log-normally distributed runs. The standard deviation of the results was $100,000. It was then decided to re-run the VaR calculation with 10,000 independent samples. The standard deviation of the result: A. about 10,000 USD B. about 30,000 USD C. about 100,000 USD D. can not be determined from this information

37. Ch. 4, HandbookZvi Wiener slide 37 FRM-98, Question 34 The value of an Asian option on the short rate. The Asian option gives the holder an amount equal to the average value of the short rate over the period to expiration less the strike rate. With a one-factor binomial model of interest rates what method you will recommend using?

38. Ch. 4, HandbookZvi Wiener slide 38 FRM-98, Question 34 A. The backward induction method, since it is the fastest? B. The simulation method, using path averages, since the option is path dependent. C. The simulation method, using path averages, since the option is path independent. D. Either the backward induction or the simulation method since both methods give the same value.

39. Ch. 4, HandbookZvi Wiener slide 39 FRM-97, Question 17 The measurement error in VaR, due to sampling variation should be greater with: A. more observations and a high confidence level (e.g. 99%). B. fewer observations and a high confidence level. C. more observations and a low confidence level. (e.g. 95%). D. more observations and a low confidence level.

40. Ch. 4, HandbookZvi Wiener slide 40 Multiple Sources of Risk GBM model with j=1,…,N independent risk factors correlated risk factors

41. Ch. 4, HandbookZvi Wiener slide 41 Multiple Sources of Risk Correlation matrix R Cholesky decomposition R=A A T, where A is a lower triangular matrix with zeros in the upper left corner. Then  = A Example:

42. Ch. 4, HandbookZvi Wiener slide 42 Cholesky Decomposition

43. Ch. 4, HandbookZvi Wiener slide 43 FRM-99, Question 29 Covariance matrix: Let  =A A T, where A is lower triangular, be a Cholesky decomposition. Then the four elements in the upper left hand corner of A, a 11, a 12, a 21, a 22, are respectively: A. 3%, 0%, 4%, 2% B. 3%, 4%, 0%, 2% C. 3%, 0%, 2%, 1% D. 2%, 0%, 3%, 1%

44. Ch. 4, HandbookZvi Wiener slide 44 FRM-99, Question 29 Covariance matrix: Let  =A A T, where A is lower triangular, be a Cholesky decomposition. Then the four elements in the upper left hand corner of A, a 11, a 12, a 21, a 22, are respectively: A. 3%, 0%, 4%, 2% B. 3%, 4%, 0%, 2% C. 3%, 0%, 2%, 1% D. 2%, 0%, 3%, 1%

45. Ch. 4, HandbookZvi Wiener slide 45 FRM-99, Question 29

46. Ch. 4, HandbookZvi Wiener slide 46 FRM-99, Question 29

47. Ch. 4, HandbookZvi Wiener slide 47 FRM-99, Question 29 Given the following covariance matrix: A. Log(S+Q) is normally distributed B. S*Q is lognormally distributed C. S*Q is normally distributed D. S + Q is normally distributed