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Financial Risk Management

Financial Risk Management . Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook. Chapter 4 Quantitative Analysis Monte Carlo Methods. Following P. Jorion 2001 Financial Risk Manager Handbook. Monte Carlo. Monte Carlo Simulation. Simulating Markov Process. The Wiener process.

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Financial Risk Management

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  1. Financial Risk Management Zvi Wiener Following P. Jorion,Financial Risk Manager Handbook FRM

  2. Chapter 4Quantitative AnalysisMonte Carlo Methods Following P. Jorion 2001 Financial Risk Manager Handbook FRM

  3. Monte Carlo Zvi Wiener

  4. Monte Carlo Simulation Zvi Wiener

  5. Simulating Markov Process The Wiener process The Generalized Wiener process The Ito process Zvi Wiener

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

  7. The Geometric Brownian Motion Zvi Wiener

  8. value time Zvi Wiener

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

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

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

  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 Zvi Wiener

  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 Zvi Wiener

  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 Zvi Wiener

  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 Zvi Wiener

  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 Zvi Wiener

  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 Zvi Wiener

  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 Zvi Wiener

  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 Zvi Wiener

  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. Zvi Wiener

  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. Zvi Wiener

  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. Zvi Wiener

  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. Zvi Wiener

  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. Zvi Wiener

  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. Zvi Wiener

  26. Risk-neutral approach Standard approach assumes some risk aversion and utility function. Risk neutral approach - change probabilities in order to get Zvi Wiener

  27. Accuracy Sampling variability Antithetic Variable Technique Control Variable Technique Quasi-Random Sequences Very difficult to use for American types. Zvi Wiener

  28. Monte Carlo Zvi Wiener

  29. Monte Carlo Zvi Wiener

  30. Monte Carlo Zvi Wiener

  31. Monte Carlo Zvi Wiener

  32. Speed of convergence Whole circle Upper triangle Zvi Wiener

  33. Smart Sampling Zvi Wiener

  34. Spectral Truncation Zvi Wiener

  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. Zvi Wiener

  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 Zvi Wiener

  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? Zvi Wiener

  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. Zvi Wiener

  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. Zvi Wiener

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

  41. Multiple Sources of Risk Correlation matrix R Cholesky decomposition R=A AT, where A is a lower triangular matrix with zeros in the upper left corner. Then  = A  Example: Zvi Wiener

  42. Cholesky Decomposition Zvi Wiener

  43. FRM-99, Question 29 Covariance matrix: Let =A AT, where A is lower triangular, be a Cholesky decomposition. Then the four elements in the upper left hand corner of A, a11,a12,a21, a22, are respectively: A. 3%, 0%, 4%, 2% B. 3%, 4%, 0%, 2% C. 3%, 0%, 2%, 1% D. 2%, 0%, 3%, 1% Zvi Wiener

  44. FRM-99, Question 29 Covariance matrix: Let =A AT, where A is lower triangular, be a Cholesky decomposition. Then the four elements in the upper left hand corner of A, a11,a12,a21, a22, are respectively: A. 3%, 0%, 4%, 2% B. 3%, 4%, 0%, 2% C. 3%, 0%, 2%, 1% D. 2%, 0%, 3%, 1% Zvi Wiener

  45. FRM-99, Question 29 Zvi Wiener

  46. FRM-99, Question 29 Zvi Wiener

  47. FRM-99, Question 29 Given the following covariance matrix: 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 Zvi Wiener

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