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# Market Risk Modelling

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Basics of Market risk modelling

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### Market Risk Modelling

1. 1. Market Risk Modelling By A.V. Vedpuriswar July 31, 2009
2. 2. Volatility
3. 3. Basics of volatility <ul><li>Volatility is a huge issue in risk management. </li></ul><ul><li>Volatility is a key parameter in modelling market risk </li></ul><ul><li>The science of volatility measurement has advanced a lot in recent years. </li></ul><ul><li>Here we look at some basic concepts and tools. </li></ul>
4. 4. Estimating Volatility <ul><li>Calculate daily return u 1 = ln S i / S i-1 </li></ul><ul><li>Variance rate per day </li></ul><ul><li>We can simplify this formula by making the following simplifications. </li></ul><ul><li>u i = (S i – S i-1) / S i-1 </li></ul><ul><li>ū = 0 m-1 = m </li></ul><ul><li>If we want to weight </li></ul>
5. 5. Estimating Volatility <ul><ul><li>Exponentially weighted moving average model means weights decrease exponentially as we go back in time. </li></ul></ul><ul><ul><li> n 2 =   2 n-1 + (1 -  ) u 2 n-1 </li></ul></ul><ul><ul><li>=  [  n-2 2 + (1-  )u n-2 2 ] + (1-  )u n-1 2 </li></ul></ul><ul><ul><li> = (1-  )[u n-1 2 +  u n-2 2 ] +  2  n-2 2 </li></ul></ul><ul><ul><li> = (1-  ) [u n-1 2 +  u 2 n-2 +  2 u n-3 2 ] +  3  2 n-3 </li></ul></ul><ul><ul><li>If we apply GARCH model, </li></ul></ul><ul><ul><li> n 2 = Y V L +  u n-1 2 +  2 n-1 </li></ul></ul><ul><ul><li>V L = Long run average variance rate </li></ul></ul><ul><ul><li>Y +  +  = 1. If Y = 0,  = 1-  ,  =  , it becomes exponentially weighted model. </li></ul></ul><ul><ul><li>GARCH incorporates the property of mean reversion. </li></ul></ul>
6. 6. Problem <ul><li>The current estimate of daily volatility is 1.5%. The closing price of an asset yesterday was \$30. The closing price of the asset today is \$30.50. Using the EWMA model, with λ = 0.94, calculate the updated estimate of volatility . </li></ul>
7. 7. Solution <ul><li>h t = λ σ 2 t-1 + ( 1 – λ ) r t-1 2 </li></ul><ul><li>λ = .94 </li></ul><ul><li>r t-1 = ln[(30.50 )/ 30] </li></ul><ul><li>= .0165 </li></ul><ul><li>h t = (.94) (.015) 2 + (1-.94) (.0165) 2 </li></ul><ul><li>Volatility = .01509 = 1.509 % </li></ul>
8. 8. Greeks
9. 9. Introduction <ul><li>Greeks help us to measure the risk associated with derivative positions. </li></ul><ul><li>Greeks also come in handy when we do local valuation of instruments. </li></ul><ul><li>This is useful when we calculate value at risk. </li></ul>
10. 10. Delta <ul><li>Delta is the rate of change in option price with respect to the price of the underlying asset. </li></ul><ul><li>It is the slope of the curve that relates the option price to the underlying asset price. </li></ul><ul><li>A position with delta of zero is called delta neutral. </li></ul><ul><li>Delta keeps changing. </li></ul><ul><li>So the investor’s position may remain delta neutral for only a relatively short period of time. </li></ul><ul><li>The hedge has to be adjusted periodically. </li></ul><ul><li>This is known as rebalancing. </li></ul>
11. 11. Gamma <ul><li>The gamma is the rate of change of the portfolio’s delta with respect to the price of the underlying asset. </li></ul><ul><li>It is the second partial derivative of the portfolio price with respect to the asset price. </li></ul><ul><li>If gamma is small, it means delta is changing slowly. </li></ul><ul><li>So adjustments to keep a portfolio delta neutral can be made only relatively infrequently. </li></ul><ul><li>However, if gamma is large, it means the delta is highly sensitive to the price of the underlying asst. </li></ul><ul><li>It is then quite risky to leave a delta neutral portfolio unchanged for any length of time. </li></ul>
12. 12. Theta <ul><li>Theta of a portfolio is the rate of change of value of the portfolio with respect to change of time. </li></ul><ul><li>Theta is also called the time decay of the portfolio. </li></ul><ul><li>T heta is usually negative for an option. </li></ul><ul><li>As time to maturity decreases with all else remaining the same, the option loses value. </li></ul>
13. 13. Vega <ul><li>The Vega of a portfolio of derivatives is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset. </li></ul><ul><li>High Vega means high sensitivity to small changes in volatility. </li></ul><ul><li>A position in the underlying asset has zero Vega. </li></ul><ul><li>The Vega can be changed by adding options. </li></ul><ul><li>If V is Vega of the portfolio and V T is the Vega of the traded option, a position of –V/ V T in the traded option makes the portfolio Vega neutral. </li></ul><ul><li>If a hedger requires the portfolio to be both gamma and Vega neutral, at least two traded derivatives dependent on the underlying asset must usually be used. </li></ul>
14. 14. Rho <ul><li>Rho of a portfolio of options is the rate of change of value of the portfolio with respect to the interest rate. </li></ul>
15. 15. Problem <ul><li>Suppose an existing short option position is delta neutral and has a gamma of － 6000. Here, gamma is negative because we have sold options. Assume there exists a traded option with a delta of 0.6 and gamma of 1.25. Create a gamma neutral position. </li></ul>
16. 16. Solution <ul><li>To gamma hedge, we must buy 6000/1.25 = 4800 options. </li></ul><ul><li>Then we must sell (4800) (.6) = 2880 shares to maintain a gamma neutral and original delta neutral position. </li></ul>
17. 17. Problem <ul><li>A delta neutral position has a gamma of － 3200. There is an option trading with a delta of 0.5 and gamma of 1.5. How can we generate a gamma neutral position for the existing portfolio while maintaining a delta neutral hedge? </li></ul>
18. 18. Solution <ul><li>Buy 3200/1.5 = 2133 options </li></ul><ul><li>Sell (2133) (.5) = 1067 shares </li></ul>
19. 19. <ul><li>Suppose a portfolio is delta neutral, with gamma = - 5000 and vega = - 8000. A traded option has gamma = .5, vega = 2.0 and delta = 0.6. How do we achieve vega neutrality? </li></ul>Problem
20. 20. <ul><li>To achieve Vega neutrality we can add 4000 options.  Delta increases by (.6) (4000) = 2400 </li></ul><ul><li>So we sell 2400 units of asset to maintain delta neutrality. </li></ul><ul><li>As the same time, Gamma changes from – 5000 to ((.5) (4000) – 5000 = - 3000. </li></ul>
21. 21. <ul><li>Suppose there is a second traded option with gamma = 0.8, vega = 1.2 and delta = 0.5. </li></ul><ul><li>if w 1 and w 2 are the weights in the portfolio, </li></ul><ul><li> - 5000 + .5w 1 + .8w 2 = 0 - 8000 + 2.0w 1 + 1.2w 2 = 0 </li></ul><ul><li>w 1 = 400 w 2 = 6000. </li></ul><ul><li>This makes the portfolio gamma and vega neutral. </li></ul><ul><li>Now let us examine delta neutrality. </li></ul><ul><li>Delta = (400) (.6) + (6000) (.5) = 3240 </li></ul><ul><li>3240 units of the underlying asset will have to be sold to maintain delta neutrality. </li></ul>
22. 22. Value at Risk
23. 23. Introduction <ul><li>Value at Risk (VAR) is probably the most important tool for measuring market risk. </li></ul><ul><li>VAR tells us the maximum loss a portfolio may suffer at a given confidence interval for a specified time horizon. </li></ul><ul><li>If we can be 95% sure that the portfolio will not suffer more than \$ 10 million in a day, we say the 95% VAR is \$ 10 million. </li></ul>
24. 24. <ul><li>Average revenue = \$5.1 million per day </li></ul><ul><li>Total no. of observations = 254. </li></ul><ul><li>Std dev = \$9.2 million </li></ul><ul><li>Confidence level = 95% </li></ul><ul><li>No. of observations < - \$10 million = 11 </li></ul><ul><li>No. of observations < - \$ 9 million = 15 </li></ul>Illustration
25. 25. <ul><li>Find the point such that the no. of observations to the left = (254) (.05) = 12.7 </li></ul><ul><li>(12.7 – 11) /( 15 – 11 ) = 1.7 / 4 ≈ .4 </li></ul><ul><li>So required point = - (10 - .4) = - \$9.6 million </li></ul><ul><li>VAR = E (W) – (-9.6) = 5.1 – (-9.6) = \$14.7 million </li></ul><ul><li>If we assume a normal distribution, </li></ul><ul><li>Z at 95% ( one tailed) confidence interval = 1.645 </li></ul><ul><li>VAR = (1.645) (9.2) = \$ 15.2 million </li></ul>
26. 26. Problem <ul><li>The VAR on a portfolio using a one day horizon is USD 100 million. What is the VAR using a 10 day horizon ? </li></ul>
27. 27. Solution <ul><li>Variance scales in proportion to time. </li></ul><ul><li>So variance gets multiplied by 10 </li></ul><ul><li>And std deviation by √10 </li></ul><ul><li>VAR = 100 √10 = (100) (3.16) = 316 </li></ul><ul><li>( σ N 2 = σ 1 2 + σ 2 2 ….. = N σ 2 ) </li></ul>
28. 28. Problem <ul><li>If the daily VAR is \$12,500, calculate the weekly, monthly, semi annual and annual VAR. Assume 250 days and 50 weeks per year. </li></ul>
29. 29. Solution Weekly VAR ＝ (12,500) (√5) ＝ 27,951 Monthly VAR ＝ ( 12,500) (√20) ＝ 55,902 Semi annual VAR ＝ (12,500) (√125) ＝ 139,754 Annual VAR ＝ (12,500) (√250) ＝ 197,642
30. 30. Variance Covariance Method
31. 31. Problem <ul><li>Suppose we have a portfolio of \$10 million in shares of Microsoft. We want to calculate VAR at 99% confidence interval over a 10 day horizon. The volatility of Microsoft is 2% per day. Calculate VAR. </li></ul>
32. 32. Solution <ul><li>σ = 2% = (.02) (10,000,000) = \$200,000 </li></ul><ul><li>Z (P = .01) = Z (P =.99) = 2.33 </li></ul><ul><li>Daily VAR = (2.33) (200,000) = \$ 466,000 </li></ul><ul><li>10 day VAR = 466,000 √10 = \$ 1,473,621 </li></ul>Ref : Options, futures and other derivatives, By John Hull
33. 33. Problem <ul><li>Consider a portfolio of \$5 million in AT&T shares with a daily volatility of 1%. Calculate the 99% VAR for 10 day horizon. </li></ul>
34. 34. Solution <ul><li>σ = 1% = (.01) (5,000,000) = \$ 50,000 </li></ul><ul><li>Daily VAR = (2.33) (50,000) = \$ 116,500 </li></ul><ul><li>10 day VAR = \$ 111,6500 √10 = \$ 368,405 </li></ul>
35. 35. Problem <ul><li>Now consider a combined portfolio of AT&T and Microsoft shares. Assume the returns on the two shares have a bivariate normal distribution with the correlation of 0.3. What is the portfolio VAR.? </li></ul>
36. 36. Solution <ul><li>σ 2 = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2 ῤ Pw 1 W 2 σ 1 σ 2 </li></ul><ul><li> = (200,000) 2 + (50,000) 2 + (2) (.3) (200,000) (50,000) </li></ul><ul><li>σ = 220,277 </li></ul><ul><li>Daily VAR = (2.33) (220,277) = 513,129 </li></ul><ul><li>10 day VAR = (513,129) √10 = \$1,622,657 </li></ul><ul><li>Effect of diversification = (1,473,621 + 368,406) – (1,622,657) </li></ul><ul><li>= 219,369 </li></ul>
37. 37. Monte Carlo Simulation
38. 38. What is Monte Carlo VAR? <ul><li>The Monte Carlo approach involves generating many price scenarios (usually thousands) to value the assets in a portfolio over a range of possible market conditions. </li></ul><ul><li>The portfolio is then revalued using all of these price scenarios. </li></ul><ul><li>Finally, the portfolio revaluations are ranked to select the required level of confidence for the VAR calculation. </li></ul>
39. 39. Step 1: Generate Scenarios <ul><li>The first step is to generate all the price and rate scenarios necessary for valuing the assets in the relevant portfolio, as well as the required correlations between these assets. </li></ul><ul><li>There are a number of factors that need to be considered when generating the expected prices/rates of the assets: </li></ul><ul><ul><li>Opportunity cost of capital </li></ul></ul><ul><ul><li>Stochastic element </li></ul></ul><ul><ul><li>Probability distribution </li></ul></ul>
40. 40. Opportunity Cost of Capital <ul><li>A rational investor will seek a return at least equivalent to the risk-free rate of interest. </li></ul><ul><li>Therefore, asset prices generated by a Monte Carlo simulation must incorporate the opportunity cost of capital. </li></ul>
41. 41. Stochastic Element <ul><li>A stochastic process is one that evolves randomly over time. </li></ul><ul><li>Stock market and exchange rate fluctuations are examples of stochastic processes. </li></ul><ul><li>The randomness of share prices is related to their volatility. </li></ul><ul><li>The greater the volatility, the more we would expect a share price to deviate from its mean. </li></ul>
42. 42. Probability Distribution <ul><li>Monte Carlo simulations are based on random draws from a variable with the required probability distribution, usually the normal distribution . </li></ul><ul><li>The normal distribution is useful when modeling market risk in many cases. </li></ul><ul><li>But it is the returns on asset prices that are normally distributed, not the asset prices themselves. </li></ul><ul><li>So we must be careful while specifying the distribution. </li></ul>
43. 43. Step 2: Calculate the Value of the Portfolio <ul><li>Once we have all the relevant market price/rate scenarios, the next step is to calculate the portfolio value for each scenario. </li></ul><ul><li>For an options portfolio, depending on the size of the portfolio, it may be more efficient to use the delta approximation rather than a full option pricing model (such as Black-Scholes) for ease of calculation. </li></ul><ul><li>Δoption = Δ(ΔS) </li></ul><ul><li>Thus the change in the value of an option is the product of the delta of the option and the change in the price of the underlying. </li></ul>
44. 44. Other approximations <ul><li>There are also other approximations that use delta, gamma (Γ) and theta (Θ) in valuing the portfolio. </li></ul><ul><li>By using summary statistics, such as delta and gamma, the computational difficulties associated with a full valuation can be reduced. </li></ul><ul><li>Approximations should be periodically tested against a full revaluation for the purpose of validation. </li></ul><ul><li>When deciding between full or partial valuation, there is a trade-off between the computational time and cost versus the accuracy of the result. </li></ul><ul><li>The Black-Scholes valuation is the most precise, but tends to be slower and more costly than the approximating methods. </li></ul>
45. 45. Step 3: Reorder the Results <ul><li>After generating a large enough number of scenarios and calculating the portfolio value for each scenario: </li></ul><ul><ul><li>the results are reordered by the magnitude of the change in the value of the portfolio (Δportfolio) for each scenario </li></ul></ul><ul><ul><li>the relevant VAR is then selected from the reordered list according to the required confidence level </li></ul></ul><ul><li>If 10,000 iterations are run and the VAR at the 95% confidence level is needed, then we would expect the actual loss to exceed the VAR in 5% of cases (500). </li></ul><ul><li>So the 501st worst value on the reordered list is the required VAR. </li></ul><ul><li>Similarly, if 1,000 iterations are run, then the VAR at the 95% confidence level is the 51st highest loss on the reordered list. </li></ul>
46. 46. Formula used typically in Monte Carlo for stock price modelling
47. 47. Advantages of Monte Carlo <ul><li>This method can cope with the risks associated with non-linear positions. </li></ul><ul><li>We can choose data sets individually for each variable. </li></ul><ul><li>This method is flexible enough to allow for missing data periods to be excluded from the VAR calculation. </li></ul><ul><li>We can incorporate factors for which there is no actual historical experience. </li></ul><ul><li>We can estimate volatilities and correlations using different statistical techniques. </li></ul>
48. 48. Problems with Monte Carlo <ul><li>Cost of computing resources c an be quite high. </li></ul><ul><li>Speed can be slow. </li></ul><ul><li>Random Numbers may not be all that random. </li></ul><ul><li>Pseudo random numbers are only a substitute for true random numbers and tend to show clustering effects. </li></ul><ul><li>Quasi-Monte Carlo techniques have been developed to produce quasi-random numbers that are more uniformly spaced. </li></ul>
49. 49. <ul><li>Monte Carlo is based on random draws from a variable with the required probability distribution, often normal distribution. </li></ul><ul><li>As with the variance-covariance approach, the normal distribution assumption can be problematic . </li></ul><ul><li>Monte Carlo can however, be performed with alternative distributions. </li></ul><ul><li>Model risk is the risk of loss arising from the failure of a model to sufficiently match reality, or to otherwise deliver the required results. </li></ul><ul><li>For Monte Carlo simulations, the results (value at risk estimate) depend critically on the models used to value (often complex) financial instruments. </li></ul>
50. 50. Historical Simulation
51. 51. Introduction <ul><li>Historical simulation is one of the three most common approaches used to calculate value at risk. </li></ul><ul><li>Unlike the Monte Carlo approach, it uses the actual historical distribution of returns to simulate the VAR of a portfolio. </li></ul><ul><li>Use of real data, coupled with ease of implementation, has made historical simulation a very popular approach to estimating VAR. </li></ul>
52. 52. Few assumptions <ul><li>Historical simulation avoids the assumption that returns on the assets in a portfolio are normally distributed. </li></ul><ul><li>Instead, it uses actual historical returns on the portfolio assets to construct a distribution of potential future portfolio losses. </li></ul><ul><li>From this distribution, the VAR can be read. </li></ul><ul><li>This approach requires minimal analytics. </li></ul><ul><li>All we need is a sample of the historic returns on the portfolio whose VAR we wish to calculate. </li></ul>
53. 53. Steps <ul><li>Collect data </li></ul><ul><li>Generate scenarios </li></ul><ul><li>Calculate portfolio returns </li></ul><ul><li>Arrange in order. </li></ul>
54. 54. Problem What is VAR (90%) ? % Returns Frequency Cumulative Frequency - 16 1 1 - 14 1 2 - 10 1 3 - 7 2 5 - 5 1 6 - 4 3 9 - 3 1 10 - 1 2 12 0 3 15 1 1 16 2 2 18 4 1 19 6 1 20 7 1 21 8 1 22 9 1 23 11 1 24 12 1 26 14 2 27 18 1 28 21 1 29 23 1 30
55. 55. <ul><li>10% of the observations, i.e, (.10) (30) </li></ul><ul><li>= 3 lie below -7 </li></ul><ul><li>So VAR = -7 </li></ul>Solution
56. 56. Advantages <ul><li>Simple </li></ul><ul><li>No normality assumption </li></ul><ul><li>Non parametric </li></ul>
57. 57. Disadvantages <ul><li>Reliance on the past </li></ul><ul><li>Length of estimation period </li></ul><ul><li>Weighting of data </li></ul><ul><li>Data issues </li></ul>
58. 58. Comparison of different VAR modeling techniques
59. 59. Simulation vs Variance Covariance <ul><li>Simulation approaches are preferred by global banks due to: </li></ul><ul><ul><li>flexibility in dealing with the ever-increasing range of complex instruments in financial markets </li></ul></ul><ul><ul><li>the advent of more efficient computational techniques in recent years </li></ul></ul><ul><ul><li>the falling costs in information technology </li></ul></ul><ul><li>However, the variance-covariance approach might be the most appropriate method for many smaller firms, particularly when : </li></ul><ul><ul><li>they do not have significant options positions </li></ul></ul><ul><ul><li>they prefer to outsource the data requirement component of their risk calculations to a company such as RiskMetrics </li></ul></ul><ul><ul><li>significant savings can often be made by using outsourced volatility and correlation data, compared to internally storing the daily price histories required for simulation techniques </li></ul></ul>
60. 60. Model Validation
61. 61. Basel Committee Standards <ul><li>Banks that prefer to use internal models must meet, on a daily basis, a capital requirement that is the higher of either: </li></ul><ul><ul><li>the previous day's value at risk </li></ul></ul><ul><ul><li>the average of the daily value at risk of the preceding 60 business days multiplied by a minimum factor of three </li></ul></ul><ul><li>VAR must be computed on a daily basis. </li></ul><ul><li>A one-tailed confidence interval of 99% must be used. </li></ul><ul><li>The minimum holding period should be 10 trading days . </li></ul><ul><li>The minimum historical observation period should be one year. </li></ul>
62. 62. <ul><li>Banks should update their data sets at least once every three months. </li></ul><ul><li>Banks can recognize correlations within broad risk categories. </li></ul><ul><li>Provided the relevant supervisory authority is satisfied with the bank's system for measuring correlations , they may also recognize correlations across broad risk factor categories. </li></ul><ul><li>Banks' internal models are required to accurately capture the unique risks associated with options and option-like instruments. </li></ul><ul><li>The Basel Committee has also specified qualitative factors that banks must meet before they are permitted to use internal models. </li></ul>
63. 63. <ul><li>The Basel Committee prescribes an increase in capital requirements if, based on a sample of 250 observations (a one-year observation period), the VAR model underpredicts the number of exceptions (losses exceeding the 99% confidence level). </li></ul><ul><li>For such purposes, three 'zones' have been distinguished by the Committee. </li></ul><ul><li>Green Zone : 0-4 exceptions </li></ul><ul><li>Yellow zone : 5-9 exceptions </li></ul><ul><li>Red zone : 10 or more exceptions </li></ul>
64. 64. Stress Testing
65. 65. Introduction <ul><li>Stress testing involves analysing the effects of exceptional events in the market on a portfolio's value. </li></ul><ul><li>These events may be exceptional, but they are also plausible. </li></ul><ul><li>And their impact can be severe. </li></ul><ul><li>Historical scenarios or hypothetical scenarios can be used. </li></ul>
66. 66. Two approaches to Stress testing <ul><li>Single-factor stress testing (sensitivity testing) involves applying a shift in a specific risk factor to a portfolio in order to assess the sensitivity of the portfolio to changes in that risk factor. </li></ul><ul><li>Multiple-factor stress testing (scenario analysis) involves applying simultaneous moves in multiple risk factors to a portfolio to reflect a risk scenario or event that looks plausible in the near future. </li></ul>
67. 67. Conducting Stress Tests <ul><li>From a computational viewpoint, stress testing can be thought of as a variant of simulation methods. </li></ul><ul><li>It merely uses a different technique to generate scenarios. </li></ul><ul><li>Once scenarios have been developed, the next step is to analyze the effect of each scenario on portfolio value. </li></ul><ul><li>This can sometimes be done in the same way as a simulation to calculate VAR. </li></ul><ul><li>Stress tests can typically be run by inputting the stressed values of the risk factors into existing models and recalculating the portfolio value using the new data. </li></ul>
68. 68. Extreme Value Theory <ul><li>EVT is a branch of statistics dealing with the extreme deviations from the mean of statistical distributions. </li></ul><ul><li>The key aspect of EVT is the extreme value theorem . </li></ul><ul><li>According to EVT, given certain conditions, the distribution of extreme returns in large samples converges to a particular known form, regardless of the initial or parent distribution of the returns. </li></ul>
69. 69. EVT Parameters <ul><li>This distribution is characterized by three parameters – location, scale and shape (tail). </li></ul><ul><li>The tail parameter is the most important as it gives an indication of the heaviness (or fatness) of the tails of the distribution. </li></ul><ul><li>The EVT approach is very useful because the distributions from which return observations are drawn are very often unknown. </li></ul><ul><li>EVT does not make strong assumptions about the shape of this unknown distribution. </li></ul>