2 Control variates. λe λti λe e λt i where R(t) = t Y 1 Y N(t) is the time from the last event to t. L t = e λr(t) e e λt(t) Exercises

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1 96 ChapterVI. Variance Reduction Methods stochastic volatility ISExSoren5.9 Example.5 (compound poisson processes) Let X(t) = Y + + Y N(t) where {N(t)},Y, Y,... are independent, {N(t)} is Poisson(λ) with epochs T, T + T,... and the Y i have common density f(y). If we take P to again make {X(t)} compound Poisson with parameters λ, f, then PrVRIS5.a OutPr7.0a OutPr8.0aa L t = e λr(t) N(t) f(y i ) e e λt(t) f(y i ) λe λti λe e λt i where R(t) = t Y Y N(t) is the time from the last event to t. Exercises. (TP) Consider Monte Carlo integration with g 0 and assume that g ch for some c > and some density. The hit-or-miss estimator is I`Uch(Y ) g(y ) where U, Y are generated independent as uniform(0, ), resp. from the density h. Show that its expectation is z = R g as desired, but that the variance is the variance of the importance sampling estimator using sampling from h.. (TP) Let F be N`µ, σ and consider a likelihood ratio (df /df)(x) proportional to e ax bx. Identify F..3 (TP) Show in Example.6 that the likelihood ratio is exp (µ µ)b T (µ µ) σ T/ where T is the maturity. OutPr.0a.4 (A) Consider the stochastic volatility setting in Exercise??. with K = 50. Do importance sampling using a change of drift 0 = µ µ for {B T }, choosing µ to satisfy S 0 exp (r 0.5 /)T + E µ B T K (cf. Example.6; the implementation here simply ignores the period with the higher volatility). For the likelihood, see Exercise??.3. What do you expect from the same idea when K = 00 or 50? Control variates The idea is to look for a r.v. W which has a strong correlation (positive or negative) with Z and a known mean w, generate (Z, W ),..., (Z R, W R ) rather than Z,...,Z R and combine the empirical means ẑ, ŵ to an estimator with lower variance than ẑ. The naive method is to choose some arbitrary constant α and consider the estimator ẑ + α(ŵ w). The point is that since w is known, we are in position to just add a term with mean zero so that the mean of the new estimator still is z. The variance is where σz + α σw + ασ ZW, (.) S6.c σ Z = Var Z, σ W = Var W, σ ZW = Cov(Z, W).

2 . Control variates 97 In general, nothing can be said about how (.) compares to the variance σz of the CMC estimator ẑ (though sometimes a naive choice like α = works to produce a lower variance). However, it is easily seen that (.) is minimized for α = σzw /σ W, and that the minimum value is σz ( ρ ) where ρ = Corr(Z, W) = σ ZW σ Z σw (.) S6.d One then simply estimates the optimal α via the empirical covariance matrix, where s Z = s = α = s ZW s W (Z i ẑ), s W =, (W i ŵ), s ZW = (Z i ẑ)(w i ŵ), and uses the estimator ẑ CV = ẑ + α(ŵ w) which has the same asymptotic properties as ẑ + α(ŵ w); in the particular, the asymptotic variance is σ Z ( ρ )/n, and a confidence interval is constructed by replacing σ Z, ρ by their empirical values s Z, s4 ZW /s Z s W. The procedure reduces the variance by a factor ρ. Thus, one needs to look for a control variate W with ρ as close to as possible. The exact value of ρ will be difficult to asses apriori, so that in practice one would just try to make W and Z as dependent as possible (in some vague sense). It is, however, an appealing feature that even if one is not very succesful, the resulting variance is never increased. There is also an interesting relation to standard regression analysis. In fact, the calculation of ẑ CV amounts to use a regression of Z upon W, fit a regression line by least squares and calculate the level of the line at the known value w of EW; see Fig. II.4.. This is seen as follows: the assumption underlying the regression is EZ i = m + βw i = m + β(w i ŵ) (.3) 0.8f (m = m + βŵ), with least squares estimates m = ẑ, β = R (Z i ẑ)(w i ŵ) R (W i ŵ) = α. Thus, the level of the fitted regression line at w is m + β(w ŵ) which (replacing the w i by the W i ) is the same as ẑ CV.

3 98 ChapterVI. Variance Reduction Methods z ẑ CV w w FIGURE VI. fig:5. For this reason, often the term regression adjusted control variates is used. The similarity is, however, formal: regression analysis via least squares is based upon the assumption of linear dependence (and preferably normal errors) whereas nothing like this is needed for regression adjusted control variates (one may, however view the method as inference in the limiting bivariate normal distribution of (ẑ, ŵ)). The literature pays quite a lot of attention to control variates without regression adjustment (i.e., α is assigned some arbitrary value), but to our mind, it is difficult to imagine situations where one would prefer this to regression adjustment. Ex7.b OutEx6.0a Example. Consider again the Monte Carlo integration problem in Example??. A suitable control for Z = g(u) is then W = f(u) with where f is close to g (to get ρ close to ) and w = Ef(U) = 0 f(x)dx is analytically available. Example. A famous example of control variates occur in Asian options where the key step in estimating the price is evaluating the expected value of [s 0 A K] + where A = p eb it/p /p is the average of a discretely sampled Brownian motion {B t }, with drift say µ and variance σ (s 0 > 0, K, T are constants). The idea is that whereas the distribution of A is untractable, so is not the case for the case for the geometric average A = ( p e B it/p ) /p = p e (p i+)yi/p

4 . Control variates 99 where Y i = B it/p B (i )T/p. Namely, since the Y i are i.i.d. N ( µt/p, σ T/p ), we have that log A is normal with mean and variance multiple correlation θ = µt p (p i + ), resp. ω = σ T p 3 (p i + ). Thus, we can take W = [s 0 A K] + as control variate since the expectation log(k/s 0) (s 0 e z /ω K) πω e (z θ) dz can be evaluated explicitly (we omit the elementary but tedious details as well as reduction of θ, ω above). An extension of the control variate method is multiple controls where the single control W above is replaced by a (row) vector W = (W... W p ), with the means w,..., w p explicitly available. Denote by W r = (W r... W rp ) the copy of W generated together with Z r and let W = ( W,...,W p ) be the average over r =,...,R. The multiple control estimator is then ẑ + α i [W i w i ] = ẑ + [W w]α (.4) OutVR9.0c (representing the α i as a column vector α). Writing the covariance matrix of the (Z r, W r ) as the variance of (.4) is ( ) σ Σ ZW, Σ W Z Σ W W σ R( +α T Σ W W α+σ ZW α ) = ( σ + R α i α j σ WiW j + i,j= α i σ ZWi ). (.5) OutVR9.0d The minimization w.r.t. α is straightforward: equating the partial derivatives w.r.t. the α i to 0 yields a system of linear equations with solution α = Σ W W Σ W Z, and (.5) then becomes ( σ Σ ZW Σ W W R Σ ) W Z = R ( ρ ZW )σ where ρ ZW = Σ ZWΣ W W Σ W Z/σ is the multiple squared correlation coefficient between Z and W, commonly interpreted as the fraction of the variance of Z that can be explained by linear dependence with the W i.

5 00 ChapterVI. Variance Reduction Methods Markov process In practice, the unknowns Σ ZW, Σ W W have to be replaced by estimates, that is, the empirical values S ZW = S W W = (Z r ẑ)(w r W), r= (W r W) T (W r W), r= so that the multiple control estimator and its variance estimate are ẑ S W W S W Z, resp. ( s S ZW S W W R S W Z). Again, the calculations of the method are formally similar to regression, this time the multiple regression model EZ r = z + β i w ri = z + β i (w ri w i ) = z + Tβ (.6) 0.8fxyz where T has rith element w ri w i (here w i = R w ri/r). The least squares estimator of β = ( β... β p ) T is OutPr7.0b β = ( T T T ) T T Z = S W W S W Z and the residual variance is Z ẑ T β ( = Z r ẑ R p R p Exercises r=. β i (w ri w i )). (A) Consider an European call option with a maturity of T = 3 years, strike price K, underlying asset price process {S t} 0 t T (S 0 = 00)and risk-free interest rate 4%. It is assumed that {S t} 0 t T evolves like geometric Brownian motion with stochastic volatility σ(t), such that {σ(t)} is Markov with states 0.5 (the baseline volatility) and 0.75, and switching intensities λ + =, λ = 3 for transitions , resp It is easy to see that a risk-neutral measure can be described by dx t = (r σ(t) /)dt + σ(t)db t where X t = log S t log S 0. Give simulation estimates of the option price for K = 50,00, 50 using the control variates: where Y t is given by S T, Y T, Y T, e Y T, e rt [S 0e Y T K] + dy t = (r 0.5 /)dt + 0.5dB t (with the same driving Brownian motion). Use both single and multiple controls and report on the explanatory power of the control variates in form of the multiple correlation coefficient corresponding to various subsets of the controls.

6 3. Antithetic sampling 0 3 Antithetic sampling Here one generates Z,...,Z n not as i.i.d. but as pairwise dependent and as negatively correlated as possible. That is, one takes n = m and generates m i.i.d. random pairs (Z, Z ), (Z 3, Z 4 ),..., (Z n, Z n ) such that the marginal distribution of Z i is the same (as for the CMC method) for all i (even and uneven) but Z j and Z j may be dependent. The estimator is ẑ Anth = (Z + + Z n )/n with variance n σ Anth = m Var ( ) Z + Z = 4m (σ CMC+σ CMC+σ CMCρ) = n σ CMC(+ρ) where ρ = Corr(Z, Z ). Thus, ρ should be negative for obtaining variance reduction, and preferably as close to - as possible for the method to be efficient. ExVRAnth9.a Example 3. In Monte Carlo integration (considering dimension d = for simplicity), a standard choice is to take Z = g(u), Z = g( U) when estimating z = 0 g. If g is monotone, Chebycheff s covariance inequality then yields ρ 0. If g(x) = x, Z + Z = so the antithetic estimator has zero variance. For g(x) = x, one gets ρ = E[ U ( U) ] (EU ) VarU = In general for g(x) = x n, ρ E[ U n ( U) n] + (EU n ) VarU n so that the variance reduction vanishes as n. /5 + /3 / /3 /5 /3 = 7 8. /4 n + /(n + ) /(n + ) /(n + ) We know of no realistic example where the variance reduction obtained by antithetic sampling is dramatic. For some further non-standard discussion, see Evans & Swartz [6]. n 3a Conditional Monte Carlo Here Z CMC is replaced by Z Cond = E[Z CMC W] for some r.v. W (more generally, one could consider E[Z CMC G] for some σ field G). Clearly, EZ Cond = EZ CMC = z. Since σ CMC = Var(Z CMC ) = Var(E[Z CMC W]) + E(Var[Z CMC W]) = σ Cond + E(Var[Z CMC W]) σ Cond, stating that Corr(f (X), f (X)) 0 if X is a r.v. and f, f non decreasing functions