Mafinrisk Market Risk Course

1. MafinriskMarket Risk Course Value at Risk Models: simulation approaches Session 8 Andrea Sironi

2. Agenda • Common features of simulation approaches • Historical simulations • The hybrid approach • Monte Carlo simulations • Stress testing Mafinrisk - Simulation Approaches

3. Simulation Approaches • Problems of the parametric approach • Non-normal distribution of market factors’ returns: higher kurtosis (fat tails) + skewness • Serial correlation of market factors’ returns • Non linear positions (bonds, options, etc.) • Simulation approaches • Historical & Monte Carlo simulations Mafinrisk - Simulation Approaches

4. Simulation Approaches • Full valuation approaches • Every position is repriced for each scenario • No use of sensitivity coefficients (delta, duration, beta, etc.) • No normal distribution assumption • Historical simulations: every position is revalued at the historical conditions (returns) • Monte Carlo simulation: random generation of a large number of scenarios • Logic of the distribution percentile Mafinrisk - Simulation Approaches

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7. Historical simulations • Four phases • Selection of an historical sample of market factors’ returns (e.g. 100 days) • Revaluation of the portfolio for each of the historical values of the market factor • Reconstruction of the empirical frequency distribution of the portofolio market values • Identification of the desired distribution percentile, corresponding to the desired confidence level Mafinrisk - Simulation Approaches

8. Historical simulations • 1. Revalue the position/ portfolio based on historical conditions • 2. Rank P&L • 3. Cut the distribution at the desidered percentile level • Ex. 99% VaR for a long USD position  5.42% • Ex. 95% VaR for a short USD position  5.91% Mafinrisk - Simulation Approaches

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12. Historical simulations vs parametric approach Mafinrisk - Simulation Approaches

13. Historical simulations vs parametric approach Mafinrisk - Simulation Approaches

14. Historical simulations • Advantages • Easy to understand and communicate • No explicit underlying assumption concerning the functional form of the returns distribution • No need to estimate the variance-covariance matrix • Allows to capture the risk profile of portfolios with non linear and non monotonic sensitivity to market factors returns Mafinrisk - Simulation Approaches

15. Historical simulations • Disadvantages • Assumption of stability of the distribution of market factors’ returns • Computationally hard because of full valuation • Size of the historical sample, particularly when time horizon > 1 day • Bad definition of the distributions tails • Risk of overweighting or underweighting the extreme events in the historical sample • Increasing the size of the historical sample there is the risk of deviating from the distribution stationarity assumption Mafinrisk - Simulation Approaches

16. Hybrid approach Boudoukh, Richardson e Whitelaw (1998) • Attempt to combine the advantages of the parametric approach (decreasing weights through exponentially weighted moving averages) e those of historical simulations (no normal distribution assumption) • Long historical series but more weight to recent data • Weight attributed to each individual historical return: Mafinrisk - Simulation Approaches

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18. Hybrid approach • VaR (99%) “prudent”: 1.09% • VaR (99%) realistic: linear interpolation Mafinrisk - Simulation Approaches

19. Monte Carlo Simulations • Problem lack of data: generate new data  Monte Carlo • Originally used for pricing complex derivatives (i.e. exotic options) for which no closed analytic solution was possible  expected value of the payoff present value • Simulate the market factor path n times (respecting arbitrage constraints) and compute the payoff in each simulated scenario  average of these values = expected value Mafinrisk - Simulation Approaches

20. Monte Carlo Simulations • Risk Management: 5 steps • Identify the distribution – f(x) – that best proxy the actual market factor returns distribution • Simulate the market factor evolution n times • Calculate the position market value in each scenario • Build the empirical probability distribution of the changes of the position’s market value • Cut the empirical distribution at the desired confidence level Mafinrisk - Simulation Approaches

21. Monte Carlo Simulations • Pricing • The stochastic process that governs the evolution of the market factor is generally known • The problem concerns the valuation • Risk Management • The problem concerns the choice of the distribution from which to extract the market factor returns Mafinrisk - Simulation Approaches

22. Monte Carlo Simulations • The third step is based on the use of a random generator and the uses a uniform distribution. It can be decomposed into 4 sub-steps • Extract a value U from a uniform distribution [0,1] • Calculate the value x of this function f(x) corresponding to the extracted U value • Determine the inverse of the cumulative function of the original sample distribution • Repeat the previous steps a large number of times Mafinrisk - Simulation Approaches

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24. Monte Carlo Simulations • Example: a bank has bought an at the money call option on the MIB 30 stock index with a maturity of 1 year and a market value of 9.413 euro. • Hp. 1: Rf=3%, Volatility MIB 30 = 20% • Hp. 2: normal distribution with mean 0.15% and standard deviation standard 1.5% Mafinrisk - Simulation Approaches

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26. Monte Carlo Simulations • What about a portfolio which is sentitive to more than just one market factor? • MC simulations MC do not capture, as historical simulations, the correlation structure • We need to introduce a method to simulate the different market factors taking into account their correlations Mafinrisk - Simulation Approaches

27. Monte Carlo Simulations • 5 steps • Estimate variance-covariance matrix • Decompose the original matrix into two symmetric matrices, A and AT “Cholesky decomposition” • Generate scenarios for the different market factors multiplying matrix AT, which reflects the historical correlations of market factors returns, for a vector z of random numers • Calculate the market value change corresponding to each of the simulated scenarios • Calculate VaR cutting the empirical probability distribution at the desired confidence level Mafinrisk - Simulation Approaches

28. Monte Carlo Simulations • Example: 2 positions • Buy a call on MIB 30 (same data as before) • Sell an at the money call on DAX with ne year maturity • Hp. 3) The DAX returns distribution is normal with mean 0.18% and standard deviation 1.24% • Hp. 4) The returns correlation between the two market indices is 0.75 Mafinrisk - Simulation Approaches

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34. Monte Carlo Simulations Final result Mafinrisk - Simulation Approaches

35. Monte Carlo Simulations • Advantages of Monte Carlo simulations • Full valuation: no problems with non linear or non monotonic portfolios • Flexibility: possibility to use any probability distribution functional form • Simulating not only final values but also path: possibility to evaluate the risk profile of path dependent options Mafinrisk - Simulation Approaches

36. Monte Carlo Simulations • Limits of Monte Carlo simulations • Need to estimate market factors’ returns correlations  stability problem • Computationally intensive • Large number of scenarios  one tends to estimate VaR based on values which are not really extremes 10,000 simulations, VaR 99% = 100th worst change Mafinrisk - Simulation Approaches

37. Stress testing • Estimate the effects, in terms of potential losses, of extreme events • The portfolio market value is revalued at the market conditions of very pessimistic scenarios • Similar to a simulation model  based on revaluing the portfolio at simulated conditions • The extreme scenarios can be based on: • statistical techniques (e.g. 10 times standard deviation) • subjective assumptions (e.g. 10% fall of the stock market, 1% parallel shift of the yield curve, etc.) • major historical events (e.g. 1987 stock market crash, 1992 currency crisis, 1994 bond markets collapse, 2000 equity markets, etc.) Mafinrisk - Simulation Approaches

38. Stress testing Derivative Policy Group (1995) Recommendations • 100 basis points parallel shift, upwards or downwards, of the yield curve • 25 b.p. change in the yield curve slope • 10% change in the stock market indices • 6% changes in the FX rates • 20% change in volatility Mafinrisk - Simulation Approaches

39. Stress testing • Not a real VaR model  discretionality • They allow to overcome the restrictive assumptions of VaR models • They allow to simulate the impact of liquidity crisis • They allow to capture the effects of crisis episodes during which significant increases in correlations between different market factors tend to occur • They can be built on specific tailor made assumptions, based on the size, composition and sensitivity of the individual portfolio Mafinrisk - Simulation Approaches

40. Questions & Exercises 1. Which of the following statements concerning Monte Carlo simulations is correct? • Monte Carlo simulations, unlike the parametric approach, have the advantage of preserving the structure of correlations among market factor returns • Monte Carlo simulations have the advantage of not requiring any assumption on the shape of the of the probability distributions of market factor returns • Monte Carlo simulations allow to estimate the VaR of a portfolio, with the desired confidence level, using the percentile technique • Monte Carlo simulations allow to estimate the VaR of a portfolio, with the desired confidence level, using a multiple of standard deviation of market factor returns Mafinrisk - Simulation Approaches

41. Questions & Exercises 2. A European bank computes the VaR associated to its overall position in US dollars, based on parametric VaR and historical simulation. The two results are different (€100,000 and €102,000, respectively) regardless of the fact that they are based on the same data series and the same confidence level. Consider the following statements: I. The distribution of the percent changes in the euro/dollar exchange rate is not normal II. The distribution of the percent changes in the euro/dollar exchange rate is asymmetrical III. The distribution of the percent changes in the euro/dollar exchange rates has a greater kurtosis than the normal distribution Which ones would you agree with? A) Only I B) I, II and III C) Only I and II D) Only I and III Mafinrisk - Simulation Approaches

42. Questions & Exercises 3. Read the following statements on Monte Carlo simulations: I. Monte Carlo simulations are more accurate than the parametric approach when the value of the bank’s portfolio is a linear function of the risk factors, and the risk factor returns are normally distributed. II. Monte Carlo simulations are quicker than the parametric approach. III. Monte Carlo simulations can be made more precise through the delta/gamma approach. IV. Monte Carlo simulations require the assumption that risk factor returns are uncorrelated with each other, since otherwise the Cholesky decomposition could not be computed. Which one(s) would you agree with? A) Only II. B) Only III. C) I and IV. D) None of them Mafinrisk - Simulation Approaches

43. Questions & Exercises 4. Consider the following statements: “historical simulations… i) …are totally distribution-free, meaning that users do not have to make hypotheses on the shape of the probability distribution of market factor returns”; ii) …are stationary, meaning that the variance of market factor returns is supposed to be constant”; iii) …are equivalent to parametric models (including models where volatilities are exponentially-weighted) if the probability of past factor returns is close to normal”; iv) …are extremely demanding in terms of past data, especially if VaR is based on a long holding period”. Which ones would you agree with? A) all; B) ii and iv; C) i and iii; D) only iv. Mafinrisk - Simulation Approaches

44. Questions & Exercises 5. Following a brief period of sharp changes in market prices, a bank using historical simulations to estimate VaR decides to switch to a model based on hybrid simulations, adopting a decay factor  of 0.95. Which of the following is true? A) The new model is likely to lead to an increase in VaR, which can be mitigated by setting  at 0.98; B) The new model is likely to lead to an decrease in VaR, which can be mitigated by setting  at 0.98; C) The new model is likely to lead to an increase in VaR, which can be mitigated by setting  at 0.90; D) The new model is likely to lead to an decrease in VaR, which can be mitigated by setting  at 0.90. Mafinrisk - Simulation Approaches