For these techniques to be efficient, we need to use. M then we introduce techniques to reduce Var[Y ] while. Var[Y ]

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Cordelia Wilson
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1 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 261 Variance reduction a Idea: Since the Monte Carlo Error is E[Y ] 1 M M j=1 Var[Y ] Y (ωj) Cα M then we introduce techniques to reduce Var[Y ] while keeping E[Y ] unchanged. For these techniques to be efficient, we need to use particular features of Y... a See for instance Monte Carlo methods in financial engineering, by P. Glasserman

2 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 262 Control Variates Suppose that we want to compute E[Y ] and instead of sampling just Yj we also sample an auxiliary r.v., Xj for which we know E[X]. Then, for a given β, we consider the unbiased estimator VM = 1 M M j=1 Yj β 1 M M j=1 (Xj E[X]) Questions: Is there a way to choose optimally β to minimize Var[VM]? Does the strategy reduce the computational effort?

3 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 263 Now we compute Var[VM](β) = 1 M Var[Y βx] = 1 M {Var[Y ]+β2 Var[X] 2βCov[Y,X]} and we minimize over β, yielding β = Cov[Y,X] Var[X] and Var[VM](β )= 1 M Var[Y ] ( 1 (Cov[Y,X])2 Var[X]Var[Y ] ) < 1 M Var[Y ]

4 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 264 Obs: -As long as X is correlated with Y the above procedure reduces the variance... - In practice, we can approximate β by using sample covariances and variances (at least for large M)

5 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 265 Does control variates pay in terms of computational work? Assume that the work to generate the pair (Xj,Yj) is twice the work to generate Yj. Method # samples Computat. Work ( MC C ) 2 ( ɛ Var[Y ] C ) 2 ɛ Var[Y ] ( MC + Cont.Var. C ) 2 ( ɛ Var[V (β )] 2 C ) 2 ɛ Var[V (β )] We see that the strategy only pays if Var[Y ] > 2Var[V (β )] i.e. 1 > 2(1 (ρxy ) 2 )iff(ρxy ) 2 > 1/2.

6 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 266 Obs: As said before, we need to use some knowledge of Y to find a suff. correlated X!!

7 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 267 Antithetic Variates Let Y = g(x) ands.t.x has a symmetric distribution around its mean. Assume E[X] =0. Then X and X are identically distributed, yielding E[g(X)] = E[g( X)] and E[Y ]=E [ ] g(x)+g( X). 2 We then use the unbiased estimator VM = 1 M M j=1 g(xj)+g( Xj) 2

8 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 268 Againwemayaskifitpaystodosointermsof computational work... We have Var[VM] = Var[g(X)+g( X)] 4M Then, assuming that computing the pair (g(x),g( X)) takes double the work for g(x), we need. 2Var[V1] Var[Y ] i.e. Var[g(X)+g( X)] 2Var[g(X)] or just Cov[g(X),g( X)] < 0.

9 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 269 Obs: If g is linear then Var[g(X)+g( X)] = 0 so we expect the method to work for functions that are close to linear On the other hand, for g(x) =X 2, Cov[g(X),g( X)] = Var[g(X)] > 0 and it is worse to apply antithetic variates than to use standard Monte Carlo!

10 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 270 Importance Sampling Idea: Change the probability measure to reduce the variance!! Let ρx,ρz be pdfs s.t. ρ X(x) ˆρZ(x) <C. Then E[Y ]=E[g(X)] = g(x)ρx(x)dx R = g(x) ρ X(x) ˆρZ(x)dx R ˆρZ(x) }{{} =ĝ(x) =E[ĝ(Z)] However, E[(g(X)) 2 ]ande[(ĝ(z)) 2 ] may be different!!

11 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 271 Example: Let g>0 and choose ˆρ = Cgρ, with the normalizing constant ( C = R gρ ) 1 Then E[(ĝ(Z)) 2 ]= R g 2 ρ Cgρ ρ = ( R gρ ) 2 = E[ĝ(Z)] 2 and we have a zero variance estimator!!

12 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 272 This approach is not practical because we need to know C = ( gρ) 1 R which is equivalent to solve the original problem! It indicates though that ˆρ has to follow the product gρ as much as possible. Think of the case where g is nonzero for x A. How should you choose ˆρ?

13 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 273 Numerical Example, Variance reduction: Ex 5.13 Look at J. Carlsson s implementation uppg5_13.m Uses antithetic variates and control variates.

14 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 274 Consider the computation of a call option on an index Z, πt = e r(t t) E[max(Z(T ) K, 0)], (37) where Z is the average of d stocks, Z(t) 1 d d i=1 Si(t) and dsi(t) = rsi(t)dt + σisi(t)dwi(t), i = 1,...,d with volatilities σi 0.2 (2 + sin(i)) i =1,...,d.

15 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 275 The correlation between Wiener processes is given by E[dWi(t)dWi (t)] = exp( 2 i i /d))dt 1 i, i d. The goal of this exercise is to experiment with two different variance reduction techniques, namely the antithetic variates and the control variates. From now on we take d = 10, r =0.04 and T =0.5 inthe example above.

16 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 276 For the application of control variates to (37) use the geometric average Ẑ(t) { d i=1 Si(t)} 1 d, compute ˆπt = e r(t t) E[max(Ẑ(T ) K, 0)] exactly (hint: see which SDE Ẑ follows and find a way to

17 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 277 apply Black-Scholes formula). Then approximate πt ˆπt + t) M e r(t M j=1 { max(z(w (T,ω j)) K, 0) max(ẑ(w (T,ω j)) K, 0) }.

18 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 278

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22 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 282

23 SPRING 2008, CSC KTH - Numerical methods for SDEs, Szepessy, Tempone 283 Can you improve the use of Control Variates for this example?

IEOR E4703: Monte-Carlo Simulation

IEOR E4703: Monte-Carlo Simulation Simulation Efficiency and an Introduction to Variance Reduction Methods Martin Haugh Department of Industrial Engineering and Operations Research Columbia University