1. Mafinrisk Market Risk Course
15/04/2017
Mafinrisk Market Risk Course
Value at Risk Models: simulation approaches
Session 8
Andrea Sironi

2. Mafinrisk - Simulation Approaches
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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6. 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%
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12. Historical simulations vs parametric approach
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13. Historical simulations vs parametric approach
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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
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16. Mafinrisk - Simulation Approaches
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:
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18. Mafinrisk - Simulation Approaches
Hybrid approach
VaR (99%) “prudent”: 1.09%
VaR (99%) realistic: linear interpolation
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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
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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 euro.
Hp. 1: Rf=3%, Volatility MIB 30 = 20%
Hp. 2: normal distribution with mean 0.15% and standard deviation standard 1.5%
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25. Mafinrisk - Simulation Approaches
Monte Carlo Simulations
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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
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29. Monte Carlo Simulations
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31. Monte Carlo Simulations
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32. Monte Carlo Simulations
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34. Monte Carlo Simulations
Final result
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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
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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
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37. Mafinrisk - Simulation Approaches
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 stock market crash, 1992 currency crisis, 1994 bond markets collapse, 2000 equity markets, etc.)
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38. Mafinrisk - Simulation Approaches
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
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39. Mafinrisk - Simulation Approaches
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. Mafinrisk - Simulation Approaches
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. Mafinrisk - Simulation Approaches
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
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42. Mafinrisk - Simulation Approaches
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
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43. Mafinrisk - Simulation Approaches
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.
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44. Mafinrisk - Simulation Approaches
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 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