1. CF-4 Bank Hapoalim Jun-2001 Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Computational Finance

2. Zvi WienerCF4 slide 2 Plan 1. Monte Carlo Method 2. Variance Reduction Methods 3. Quasi Monte Carlo 4. Permuting QMC sequences 5. Dimension reduction 6. Financial Applications simple and exotic options American type prepayments

3. Zvi WienerCF4 slide 3 Monte Carlo

4. Zvi WienerCF4 slide 4 Monte Carlo

5. Zvi WienerCF4 slide 5 Monte Carlo

6. Zvi WienerCF4 slide 6 Monte Carlo

7. Zvi WienerCF4 slide 7 Speed of convergence Whole circle Upper triangle

8. Zvi WienerCF4 slide 8 Smart Sampling

9. Zvi WienerCF4 slide 9 Spectral Truncation

10. Zvi WienerCF4 slide 10 Variance Reduction Let X(  ) be an option. Let Y be a similar option which is correlated with X but for which we have an analytic formula. Introduce a new random variable

11. Zvi WienerCF4 slide 11 Variance Reduction The variance of the new variable is If 2  cov[X,Y] >  2 var[Y] we have reduced the variance.

12. Zvi WienerCF4 slide 12 Variance Reduction The optimal value of  is Then the variance of the estimator becomes:

13. Zvi WienerCF4 slide 13 Variance Reduction Note that we do not have to use the optimal  * in order to get a significant variance reduction.

14. Zvi WienerCF4 slide 14 Multidimensional Variance Reduction A simple generalization of the method can be used when there are several correlated variables with known expected values. Let Y 1, …, Y n be variables with known means. Denote by  Y the covariance matrix of variables Y and by  XY the n-dimensional vector of covariances between X and Y i.

15. Zvi WienerCF4 slide 15 Multidimensional Variance Reduction Then the optimal projection on the Y plane is given by vector: The resulting minimum variance is where

16. Zvi WienerCF4 slide 16 Variance Reduction Antithetic sampling Moment matching/calibration Control variate Importance sampling Stratification

17. Zvi WienerCF4 slide 17 Monte Carlo in Risk Management Distribution of market factors Simulation of a large number of events P&L for each scenario Order the results VaR = lowest quantile

18. Zvi WienerCF4 slide 18 How to design MC The central point is to model the distribution of relevant risk factors. For example, in pricing you should use the risk-neutral distribution. For risk measurement use true distribution. What should be used for an estimate of frequency of hedge?

19. Zvi WienerCF4 slide 19 Geometrical Brownian Motion

20. Zvi WienerCF4 slide 20 Lognormal process

21. Zvi WienerCF4 slide 21 Euler Scheme

22. Zvi WienerCF4 slide 22 Milstein Scheme

23. Zvi WienerCF4 slide 23 MC for simple options Needs["Statistics`NormalDistribution`"] Clear[MCEuropean, MCEuropeanCall, MCEuropeanPut] nor[mu_,sig_]:=Random[NormalDistribution[mu,sig]];

24. Zvi WienerCF4 slide 24 MC for simple options MCEuropean[s_, T_, r_,  _, n_, exercise_Function]:= Module[{m = N[Log[s]+(r - 0.5  2 )*T], sg=N[  Sqrt[T] ], tbl}, tbl= Table[nor[m, sg], {i, n}]; Exp[-r*T]*Map[exercise, Exp[Join[tbl, 2*m - tbl]]]// {Mean[#], StandardErrorOfSampleMean[#]}& ]

25. Zvi WienerCF4 slide 25 MC for simple options MCEuropeanCall[s_, x_, T_, r_,  _, n_]:= MCEuropean[s, T, r, , n, Max[#-x,0]&] MCEuropeanPut[s_, x_, T_, r_,  _, n_]:= MCEuropean[s, T, r, , n, Max[x-#,0]&]

26. Zvi WienerCF4 slide 26 MC for path dependent options RandomWalk[n_Integer] := FoldList[Plus, 0, Table[Random[] - 1/2, {n}]]; ListPlot[ RandomWalk[500], PlotJoined -> True];

27. Zvi WienerCF4 slide 27 MC for path dependent options The function paths generates a random sample of price paths for the averaging period. It returns a list of numberPaths random paths, each consisting of numberPrices prices over the period from time T1 to time T. The prices at the start of the period are given by the appropriate lognormal distribution for time T1.

28. Zvi WienerCF4 slide 28 MC for path dependent options paths[s_,sigma_,T1_,T_,r_,numberPrices_,numberPaths_]:= Module[{meanAtT1=Log[s]+(r-sigma^2/2)*T1, sigmaAtT1 = sigma*Sqrt[T1], meanPath = 1+ r*(T-T1)/(numberPrices-1), sigmaPath = sigma*Sqrt[(T-T1)/(numberPrices-1)] }, Table[NestList[# nor[meanPath,sigmaPath]&, Exp[nor[meanAtT1,sigmaAtT1]], numberPrices - 1], {i,numberPaths}] ]

29. Zvi WienerCF4 slide 29 MC for Asian options MCAsianCall[s_,x_,sigma_,T1_,T_,r_,numberPrices_,numberPaths_]:= Module[{ t1, t2, t3}, t1 = paths[s,sigma,T1,T,r,numberPrices,numberPaths] ; t2 = Map[Max[0,Mean[#] - x]&, t1]; t3 = Exp[-T*r]*t2; {Mean[t3], StandardErrorOfSampleMean[t3]} ]

30. Zvi WienerCF4 slide 30 Quasi Monte Carlo Van der Corput Halton Haber Sobol Faure Niederreiter Permutations Nets

31. Zvi WienerCF4 slide 31 Quasi Monte Carlo Are efficient in low (1-2) dimensions. Sobol sequences can be used for small dimensions as well. As an alternative one can create a fixed set of well-distributed paths.

32. Zvi WienerCF4 slide 32 Do not use free sequences

33. Zvi WienerCF4 slide 33 Other MC applications Pricing Optimal hedging Impact of dividends Bounds on a basket Prepayments Tranches of MBS

34. Zvi WienerCF4 slide 34 Other MC related topics Use of analytical approximations Richardson extrapolation Ratchets example American properties Bundling Modeling Fat tails

35. CF-4 Bank Hapoalim Jun-2001 Zvi Wiener The Hebrew University of Jerusalem mswiener@mscc.huji.ac.il Value of Value-at-Risk

36. Zvi WienerCF4 slide 36 P&L VaR 1 day 1% probability 1d 1 week 1% probability 1w

37. Zvi WienerCF4 slide 37 -3-2 0.5% 1% 100% VaR A VaR is not sub-additive. Cumulative distribution of assets A and B. 0

38. Zvi WienerCF4 slide 38 Joint distribution of A and B The integral over the gray triangle is 0.99*0.99 + 0.99*0.005*2/2 = 0.98505, which means that there are more than 1% chances to be outside the gray area and VaR A+B > VaR A +VaR B =$2. -3-2 VaR A+B AB -2 -3 Total loss $1 Total loss $2 0.005 0.99 0.005 0.9801 0.00495 0.000025 0.00495

39. Zvi WienerCF4 slide 39 Model Bank’s choice of an optimal system Depends on the available capital Current and potential capital needs Queuing model as a base

40. Zvi WienerCF4 slide 40 Required Capital Let A be total assets C – capital of a bank  - percentage of qualified assets k – capital required for traded assets

41. Zvi WienerCF4 slide 41 Maximal Risk (Assets) The coefficient k varies among systems, but a better (more expensive) system provides more precise risk measurement, thus lower k. Cost of a system is p, paid as a rent (pdt during dt). A max is a function of C and p.

42. Zvi WienerCF4 slide 42 Risky Projects Deposits arrive and are withdrawn randomly. All deposits are of the same size. Invested according to bank’s policy. Can not be used if capital requirements are not satisfied.

43. Zvi WienerCF4 slide 43 Arrival of Risky Projects We assume that risky projects arrive randomly (as a Poisson process with density ). This means that there is a probability dt that during dt one new project arrives.

44. Zvi WienerCF4 slide 44 Arrival of Risky Projects A new project is undertaken if the bank has enough capital (according to the existing risk measuring system). We assume that one can NOT raise capital or change systems quickly.

45. Zvi WienerCF4 slide 45 Termination of Risky Projects We assume that each risky project disappears randomly (as a Poisson process with density  ).

46. Zvi WienerCF4 slide 46 Termination of Risky Projects We assume that each risky project disappears randomly (as a Poisson process with density  ). This means that there is a probability n  dt that during dt one out of n existing projects terminates. With probability (1-n  dt) all existing projects will be active after dt.

47. Zvi WienerCF4 slide 47 Profit We assume that each existing risky project gives a profit of  dt during dt. Thus when there are n active projects the bank has instantaneous profit (  n-p)dt.

48. Zvi WienerCF4 slide 48 States After C and p are chosen, the maximal number of active projects is given by s=A max (C,p).  0  1  2  s-1  s 0 1 2 s-1 s  2  s 

49. Zvi WienerCF4 slide 49 States  0  1  2  s-1  s 0 1 2 s-1 s  2  s  Stable distribution:  0 =  1   1 =  2 2  …  s-1 =  s s 

50. Zvi WienerCF4 slide 50 Probabilities Probability of losing a new project due to capital requirements is equal to the probability of being in state s, i. e.  s. Termination of projects does not have to be Poissonian, only mean and variance matter.

51. Zvi WienerCF4 slide 51 Expected Profit An optimal p (risk measurement system) can be found by maximizing the expected profit stream.

52. Zvi WienerCF4 slide 52 Example Capital requirement as a function of p (price) and q (scaling factor), varies between 1.5% and 8%.

53. Zvi WienerCF4 slide 53 Example q=0.5 q=1 q=3 p A max

54. Zvi WienerCF4 slide 54 Example of a bank Capital $200M Average project is $20K On average 200 new projects arrive each day Average life of a project is 2 years 15% of assets are traded and q=1 spread  =1.25%

55. Zvi WienerCF4 slide 55 Bank’s profit as a function of cost p. C=$200M, arrival rate 200/d, size $20K, average life 2 yr., spread 1.25%, q=1, 15% of assets are traded. rent p Expected profit

56. Zvi WienerCF4 slide 56 rent p Expected profit Bank’s profit as a function of cost p. C=$200M, arrival rate 200/d, size $20K, average life 2 yr., spread 1%, q=1, 5% of assets are traded.

57. Zvi WienerCF4 slide 57 Conclusion Expensive systems are appropriate for banks with low capitalization operating in an unstable environment Cheaper methods (like the standard approach) should be appropriate for banks with high capitalization small trading book operating in a stable environment many small uncorrelated, long living projects

58. Zvi WienerCF4 slide 58 A simple intuitive and flexible model of optimal choice of risk measuring method.