Market Risk Modelling By A.V. Vedpuriswar

1. Market Risk ModellingBy A.V. Vedpuriswar July 31, 2009

2. Volatility

3. Basics of volatility Volatility is a huge issue in risk management. Volatility is a key parameter in modelling market risk The science of volatility measurement has advanced a lot in recent years. Here we look at some basic concepts and tools.

4. Estimating Volatility • Calculate daily return u1 = ln Si / Si-1 • Variance rate per day • We can simplify this formula by making the following simplifications. ui = (Si – Si-1) / Si-1 ū = 0 m-1 = m If we want to weight 3

5. Estimating Volatility • Exponentially weighted moving average model means weights decrease exponentially as we go back in time. n2 = 2n-1+ (1 - ) u2n-1 =  [n-22 + (1- )un-22] + (1- )un-12 = (1- )[un-12 + un-22] + 2n-22 = (1-) [un-12 + u2n-2 + 2un-32 ] + 3 2n-3 • If we apply GARCH model, n2 = Y VL + un-12 + 2n-1 VL = Long run average variance rate Y +  +  = 1. If Y = 0,  = 1-, = , it becomes exponentially weighted model. • GARCH incorporates the property of mean reversion. 4

6. Problem • 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 . 5

7. Solution ht = λσ2t-1 + ( 1 – λ) rt-12 • λ = .94 • rt-1 = ln[(30.50 )/ 30] = .0165 • ht = (.94) (.015)2 + (1-.94) (.0165)2 • Volatility = .01509 = 1.509 % 6

8. Greeks

9. Introduction • Greeks help us to measure the risk associated with derivative positions. • Greeks also come in handy when we do local valuation of instruments. • This is useful when we calculate value at risk.

10. Delta • Delta is the rate of change in option price with respect to the price of the underlying asset. • It is the slope of the curve that relates the option price to the underlying asset price. • A position with delta of zero is called delta neutral. • Delta keeps changing. • So the investor’s position may remain delta neutral for only a relatively short period of time. • The hedge has to be adjusted periodically. • This is known as rebalancing. 9

11. Gamma • The gamma is the rate of change of the portfolio’s delta with respect to the price of the underlying asset. • It is the second partial derivative of the portfolio price with respect to the asset price. • If gamma is small, it means delta is changing slowly. • So adjustments to keep a portfolio delta neutral can be made only relatively infrequently. • However, if gamma is large, it means the delta is highly sensitive to the price of the underlying asst. • It is then quite risky to leave a delta neutral portfolio unchanged for any length of time. 10

12. Theta • Theta of a portfolio is the rate of change of value of the portfolio with respect to change of time. • Theta is also called the time decay of the portfolio. • Theta is usually negative for an option. • As time to maturity decreases with all else remaining the same, the option loses value. 11

13. Vega • 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. • High Vega means high sensitivity to small changes in volatility. • A position in the underlying asset has zero Vega. • The Vega can be changed by adding options. • If V is Vega of the portfolio and VT is the Vega of the traded option, a position of –V/ VT in the traded option makes the portfolio Vega neutral. • 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. 12

14. Rho • Rho of a portfolio of options is the rate of change of value of the portfolio with respect to the interest rate. 13

15. Problem • 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. 14

16. Solution • To gamma hedge, we must buy 6000/1.25 = 4800 options. • Then we must sell (4800) (.6) = 2880 shares to maintain a gamma neutral and original delta neutral position. 15

17. Problem • 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? 16

18. Solution • Buy 3200/1.5 = 2133 options • Sell (2133) (.5) = 1067 shares 17

19. Problem • 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? 18

20. To achieve Vega neutrality we can add 4000 options.  Delta increases by (.6) (4000) = 2400 • So we sell 2400 units of asset to maintain delta neutrality. • As the same time, Gamma changes from – 5000 to ((.5) (4000) – 5000 = - 3000.

21. Suppose there is a second traded option with gamma = 0.8, vega = 1.2 and delta = 0.5. • if w1 and w2 are the weights in the portfolio, • - 5000 + .5w1 + .8w2 = 0 - 8000 + 2.0w1 + 1.2w2 = 0 • w1 = 400 w2 = 6000. • This makes the portfolio gamma and vega neutral. • Now let us examine delta neutrality. • Delta = (400) (.6) + (6000) (.5) = 3240 • 3240 units of the underlying asset will have to be sold to maintain delta neutrality. 20

22. Value at Risk

23. Introduction • Value at Risk (VAR) is probably the most important tool for measuring market risk. • VAR tells us the maximum loss a portfolio may suffer at a given confidence interval for a specified time horizon. • 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.

24. Illustration • Average revenue = $5.1 million per day • Total no. of observations = 254. • Std dev = $9.2 million • Confidence level = 95% • No. of observations < - $10 million = 11 • No. of observations < - $ 9 million = 15 23

25. Find the point such that the no. of observations to the left = (254) (.05) = 12.7 (12.7 – 11) /( 15 – 11 ) = 1.7 / 4 ≈ .4 So required point = - (10 - .4) = - $9.6 million VAR = E (W) – (-9.6) = 5.1 – (-9.6) = $14.7 million If we assume a normal distribution, Z at 95% ( one tailed) confidence interval = 1.645 VAR = (1.645) (9.2) = $ 15.2 million

26. Problem • The VAR on a portfolio using a one day horizon is USD 100 million. What is the VAR using a 10 day horizon ? 25

27. Solution • Variance scales in proportion to time. • So variance gets multiplied by 10 • And std deviation by √10 • VAR = 100 √10 = (100) (3.16) = 316 • (σN2 = σ12 + σ22 ….. = Nσ2) 26

28. Problem • If the daily VAR is $12,500, calculate the weekly, monthly, semi annual and annual VAR. Assume 250 days and 50 weeks per year. 27

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 28

30. Variance Covariance Method

31. Problem • 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. 30

32. Solution • σ = 2% = (.02) (10,000,000) = $200,000 • Z (P = .01) = Z (P =.99) = 2.33 • Daily VAR = (2.33) (200,000) = $ 466,000 • 10 day VAR = 466,000 √10 = $ 1,473,621 Ref : Options, futures and other derivatives, By John Hull 31

33. Problem • Consider a portfolio of $5 million in AT&T shares with a daily volatility of 1%. Calculate the 99% VAR for 10 day horizon. 32

34. Solution • σ = 1% = (.01) (5,000,000) = $ 50,000 • Daily VAR = (2.33) (50,000) = $ 116,500 • 10 day VAR = $ 111,6500 √10 = $ 368,405 33

35. Problem • 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.? 34

36. Solution • σ2 = w12σ12 + w22 σ22 + 2 ῤPw1 W2σ1σ2 • = (200,000)2 + (50,000)2 + (2) (.3) (200,000) (50,000) • σ = 220,277 • Daily VAR = (2.33) (220,277) = 513,129 • 10 day VAR = (513,129) √10 = $1,622,657 • Effect of diversification = (1,473,621 + 368,406) – (1,622,657) = 219,369 35

37. Monte Carlo Simulation

38. What is Monte Carlo VAR? • 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. • The portfolio is then revalued using all of these price scenarios. • Finally, the portfolio revaluations are ranked to select the required level of confidence for the VAR calculation.

39. Step 1: Generate Scenarios • 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. • There are a number of factors that need to be considered when generating the expected prices/rates of the assets: • Opportunity cost of capital • Stochastic element • Probability distribution

40. Opportunity Cost of Capital • A rational investor will seek a return at least equivalent to the risk-free rate of interest. • Therefore, asset prices generated by a Monte Carlo simulation must incorporate the opportunity cost of capital.

41. Stochastic Element • A stochastic process is one that evolves randomly over time. • Stock market and exchange rate fluctuations are examples of stochastic processes. • The randomness of share prices is related to their volatility. • The greater the volatility, the more we would expect a share price to deviate from its mean.

42. Probability Distribution • Monte Carlo simulations are based on random draws from a variable with the required probability distribution, usually the normal distribution. • The normal distribution is useful when modeling market risk in many cases. • But it is the returns on asset prices that are normally distributed, not the asset prices themselves. • So we must be careful while specifying the distribution.

43. Step 2: Calculate the Value of the Portfolio • Once we have all the relevant market price/rate scenarios, the next step is to calculate the portfolio value for each scenario. • 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. • Δoption = Δ(ΔS) • 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.

44. Other approximations • There are also other approximations that use delta, gamma (Γ) and theta (Θ) in valuing the portfolio. • By using summary statistics, such as delta and gamma, the computational difficulties associated with a full valuation can be reduced. • Approximations should be periodically tested against a full revaluation for the purpose of validation. • When deciding between full or partial valuation, there is a trade-off between the computational time and cost versus the accuracy of the result. • The Black-Scholes valuation is the most precise, but tends to be slower and more costly than the approximating methods.

45. Step 3: Reorder the Results • After generating a large enough number of scenarios and calculating the portfolio value for each scenario: • the results are reordered by the magnitude of the change in the value of the portfolio (Δportfolio) for each scenario • the relevant VAR is then selected from the reordered list according to the required confidence level • 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). • So the 501st worst value on the reordered list is the required VAR. • Similarly, if 1,000 iterations are run, then the VAR at the 95% confidence level is the 51st highest loss on the reordered list.

46. Formula used typically in Monte Carlo for stock price modelling

47. Advantages of Monte Carlo • This method can cope with the risks associated with non-linear positions. • We can choose data sets individually for each variable. • This method is flexible enough to allow for missing data periods to be excluded from the VAR calculation. • We can incorporate factors for which there is no actual historical experience. • We can estimate volatilities and correlations using different statistical techniques.

48. Problems with Monte Carlo • Cost of computing resources can be quite high. • Speed can be slow. • RandomNumbers may not be all that random. • Pseudo random numbers are only a substitute for true random numbers and tend to show clustering effects. • Quasi-Monte Carlo techniques have been developed to produce quasi-random numbers that are more uniformly spaced.

49. Monte Carlo is based on random draws from a variable with the required probability distribution, often normal distribution. • As with the variance-covariance approach, the normal distribution assumption can be problematic . • Monte Carlo can however, be performed with alternative distributions. • 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. • For Monte Carlo simulations, the results (value at risk estimate) depend critically on the models used to value (often complex) financial instruments.

50. Historical Simulation