1. Copyright K.Cuthbertson, D. Nitzsche 1 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture VaR: Statistical Issues Version 1/9/2001

2. Copyright K.Cuthbertson, D. Nitzsche 2 TOPICS Monte Carlo Simulation Historic Simulation+Bootstapping Stress Testing

3. Copyright K.Cuthbertson, D. Nitzsche 3 Monte Carlo Simulation

4. Copyright K.Cuthbertson, D. Nitzsche 4 Monte Carlo Simulation: (VaR of Long FX-Call option) Options have a non-linear (convex) payoff structure Distribution of gains/losses NOT normally distributed Assets Held : 1 option on stock-1 Black-Scholes is used to price the option during the MCS. Find the VaR over a 5-day horizon

5. Copyright K.Cuthbertson, D. Nitzsche 5 Distribution  C is “non-normal”. Stock Price SoSo 0 0.4. 1.0 Call premium as S changes “Delta” is the slope of this curve C= Call premium For an equal  S + or - 1 in stock price,the change in the call premia are NOT equal: this gives “non-normality” for  C 0.3 1.0 CoCo

6. Copyright K.Cuthbertson, D. Nitzsche 6 Monte Carlo Simulation-1: Single Asset: 5-day VaR AIM: To generate artificial data on the returns of the underlying asset and then plot a histogram of the change in the call premium. We can then find the 5% lower cut-off point for the value of the call (ie. It’s VaR). Call premium is V (1) and stock price (the underlying) is P (1). Initial call value V 0 (1) = BS(P 0 (1), T 0 …..)

7. Copyright K.Cuthbertson, D. Nitzsche 7 Monte Carlo Simulation-1: Single Asset Statisticians have told us that the (daily) RETURN R 1 on the stock is N (0,  1 ) Hence R 1 = 0 +  1  1  1 -niid(0,1) Note: Daily (proportionate) price change is DEFINED as R = ln (P t+1 /P t ) and therefore (obviously): P t+1 = P t exp(R 1 ) = P t exp(  1  1 ) Over 5-days - use “root-T” rule P t+5 = P t exp (  1  1 )

8. Copyright K.Cuthbertson, D. Nitzsche 8 Monte Carlo Simulation-1: Single Asset 1) “Draw” 1000 values of  1 from a special National Lottery “drum” which has “balls” with mean of zero and s.d. = 1 2) Hence we now have 1000 “new” stock returns R 1 from using: R 1 = 0 +  1  1 This is the end of the MCS. 2a) Now calc 1000 “new” stock prices P (1) t+5 = P (1) 0 exp (  1.  1 ) 3) The 1000 “new artificial” call premia at t+5 from BS: V (1) t+5 = = BS(P (1) t+5, T 0 - 5/365, …..) 4) Plot the 1000 values of  V t+5 = V t+5 - V 0 in a histogram and read off,. the 5% lower cut-off point

9. Copyright K.Cuthbertson, D. Nitzsche 9 Monte Carlo Simulation-1: Alternative to ‘root-T rule Generating “P” over 1-year: (Excel/Gauss) 1) P(t) = P(t-1) [ 1 +  +   t ] where  = annual mean return (eg 0.10),  = annual volatility (eg.0.20) Equivalent to R t =  +   t where  t -niid(0,1) and R t = (P t - P t-1 ) /P t-1

10. Copyright K.Cuthbertson, D. Nitzsche 10 Monte Carlo Simulation-1: Single Asset Generating “P” over 5-days: If we do not use the “root-T” rule then we can generate P t+5 “directly” (in Excel/Gauss) 1) P(t) = P(t-1) [ 1 +   t +   t 1/2  t ] or, 2) P(t) = P(t-1) exp[ (  -  2 /2)  t +  (  t) 1/2  t ] In simple case above we had  = 1-day volatility,  t = 1,  =0 in (2) More general is:  = annual mean return (eg 0.10)  = annual volatility (eg.0.20) and  t = 1/365 ( ie. One day) Use a ‘do-loop’ over 5 ‘periods’

11. Copyright K.Cuthbertson, D. Nitzsche 11 Monte Carlo Simulation-1: Single Asset In Excel you can generate an approximate  t ~N(0,1) variable using  t = [  i RAND(.) i - 6 ] with the sum from 1 to 12. P.S. RAND are independent random variables drawn from a uniform distribution over {0 to 1}

12. Copyright K.Cuthbertson, D. Nitzsche 12 Monte Carlo Simulation-2: Two Assets Assume US resident holds One Call on stock-1 and One call on stock-2 The two underlying “stock prices” are P (1) and P (2) Initial call values V 0 (1) = BS(P 0 (1), T 0 …..) V 0 (2) = BS(P 0 (2), TT 0 …..) Initial value of portfolio of 2-calls V p,0 = V 0 (1) + V 0 (2)

13. Copyright K.Cuthbertson, D. Nitzsche 13 Monte Carlo Simulation-2: Two Assets 1) MCS Generate: Stock-1 return = R 1 and stock-2 return = R 2. Returns drawn from N(  1,  2,  ) R 1 = 0 +  1  1 and R 2 = 0 +  2  2 If  1 and  2 are positively correlated and their correlation coefficient is 0.75 then a positive value for  1 will coincide with a positive value of  2, 75% of the time  = 2x2 covariance matrix of  i,j ‘s of the error terms  and  12 /  1  2

14. Copyright K.Cuthbertson, D. Nitzsche 14 Monte Carlo Simulation-2: Two Assets Do 1000 draws of  1 and  2 and hence obtain 1000 values of R 1, R 2 (which have a correlation of 0.75). Hence our “artificial data” mimics the real data. Monte Carlo Simulation is now over. Use this generated data to determine ‘new’ stock prices after 5 days (here using the ‘root-T’ rule) P (1) t+5 = P (1) 0 exp (  1  1 ) P (2) t+5 = P (2) 0 exp (  2  2 )

15. Copyright K.Cuthbertson, D. Nitzsche 15 Monte Carlo Simulation-2: Two Assets Now “plug in” these values in Black-Scholes to give 1000 values for the call premia V (1) and V (2) and hence the new values for your portfolio V (1) = BS(P t+5 (1), T 0 - 5/365 …..) V (2) = BS(P t+5 (2), TT 0 - 5/365 …..) V p,t+5, = V (1) + V (2) Calculate $  V p = V p,t+5 - V p,0, 1000 times and find 5 th percentile from the histogram.

16. Copyright K.Cuthbertson, D. Nitzsche 16 Monte Carlo Simulation-Approximations 1) Delta or Variance-Covariance method: dV(option) =  ( N i  i ) dS i N=number of options held, dS i are from MCS If on same underlying then dV(option) =  p dS  where  p =  ( N i  i ) 2) Delta-Gamma method dV(option) =  ( N i  i ) dS i + (1/2)  ( N i  i ) (dS i ) 2

17. Copyright K.Cuthbertson, D. Nitzsche 17 Historic Simulation Method + Bootstrapping and Stress Testing

18. Copyright K.Cuthbertson, D. Nitzsche 18 Historic Simulation Method To use say 1000 days of actual historic data to revalue the portfolio (eg. Daily data over last 3 years). Plot the changes in value in a histogram and “read off” the required percentile VaR cut off point. This is a NON PARAMETRIC METHOD since we do not estimate any variances and covariances or assume normality. We merely use the historic returns, so our VaR estimates encapsulate whatever, distribution the returns might embody ( eg.Student’s t) as well as the autocorrelation in individual returns Also, the historic data “contain” the correlations between the returns on the different assets

19. Copyright K.Cuthbertson, D. Nitzsche 19 ‘STYLISED’ HISTORIC SIMULATION (DAILY DATA) Currently hold $1 held in each asset Day =123456…. 1000 R 1 +2+1+4-3-2-1+2 R 2 +1+20-1-5-6-5  V p +3+3+4-4-7-7-5 $100m in each asset Change in value of portfolio in histogram 1% tail then 10 ‘negatives’ in this tail - ‘read off’ the 10th most negative  V p - this is your VaR forecast

20. Copyright K.Cuthbertson, D. Nitzsche 20 Summary:Historic Simulation Method Example: PORTFOLIO OF 10 STOCKS with $10m in each stock and current value V 0. What is YOUR FORECAST OF (daily) VaR of your portfolio ? You have “historic” daily data on each of the 10 stock returns (many of which will be negative), for a total of 1000 days Assume you “write” 10 returns in column-1 = “c1”, 10 returns on “c2” ….., 10 returns on “c1000” etc Simply revalue your FIXED CURRENT holdings V i of each asset and hence of the portfolio, using each of the 10 returns on each of the successive days 1,2,…. 1000. You then get 1,000 “new” values for your portfolio and change in portfolio value. Each of these is a forecast of your daily VaR Plot a histogram of  V and “read off” the 1% lower tail cut off point

21. Copyright K.Cuthbertson, D. Nitzsche 21 Historic Simulation+”Bootstrapping” Problem: You do not have enough data points to estimate the lower tail cut off point (eg. 1000 observations implies only 10 will be in the lower 1% tail) You have “historic” daily data on each of the 10 stock returns (many of which will be negative), for a total of 1000 days Assume you “write” 10 returns in column-1 = “c1”, 10 returns on “c2” ….., 10 returns on “c1000” etc Draw randomly from a uniform distribution with an equal probability of drawing any number between 1 and 1000. If you draw “20” then take the 10-returns in c20 and revalue the portfolio. Call this $-value V (1)

22. Copyright K.Cuthbertson, D. Nitzsche 22 Historic Simulation+”Bootstrapping” Repeat above for 10,000 “trials/runs” (with replacement), obtaining 10,000 possible (alternative) values V (i) - V 0, for the change in value of your portfolio of 10 stocks “With replacement” means that in the 10,000 runs you will “choose” some of the 1000 columns more than once. Plot a histogram of the 10,000 values of  V (i) - some of which will be negative, and read off the “5% cut off” value- This is VaR p (Note: The mean/median of the distribution should be close to V 0 )

23. Copyright K.Cuthbertson, D. Nitzsche 23 BOOTSTRAP A SUB-SET OF DATA More Problems: Is data >3 years ago useful for forecasting tomorrow? Use most recent data - say last 100 days ? 1% tail then only 1 ‘negative’ in this tail, in the actual data ! Actual data no. 99, might be -50% and this has a weight of 1/100 for the next 100 days of the forecast. OK or not? BOOTSTRAP Draw number with equal probability over range 0-100, with replacement. Choose that “day”. Revalue portfolio Repeat 10,000 times

24. Copyright K.Cuthbertson, D. Nitzsche 24 Stress Testing Subject portfolio to ‘extreme’ historic events (eg. 1987 crash) Full Valuation Method: (eg.Black-Scholes for options) Problem: Would correlations in your “scenario” ( in this case for 1987 ) be the same as in the crash of 2005?

25. Copyright K.Cuthbertson, D. Nitzsche LECTURE ENDS HERE

26. Copyright K.Cuthbertson, D. Nitzsche 26 *