1 Introduction to reducing variance in Monte Carlo simulations


 Cori Hamilton
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1 Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by collectig iid samples from the distributio, X 1,, X ad usig the sample mea X() = 1 X j (1) Lettig σ = V ar(x) deote the variace of the distributio, we coclude that V ar(x()) = σ () The cetral limit theorem asserts that as, the distributio of def Z = σ (X() µ) teds to N(0, 1), the uit ormal distributio Lettig Z deote a N(0, 1) rv, we coclude that for sufficietly large, Z Z i distributio From here we obtai for ay z 0, P ( X() µ > z σ ) P ( Z > z) = P (Z > z) (We ca obtai ay value of P (Z > z) by referrig to tables, etc) For ay α > 0 o matter how small (such as α = 005), lettig z α/ be such that P (Z > z α/ ) = α/, we thus have P ( X() µ > z α/ σ ) α, which implies that the uow mea µ lies withi the iterval X() ± z α/ σ with (approximately) probability 1 α This allows us to costruct cofidece itervals for our estimate: we say that the iterval X() ± z α/ σ is a 100(1 α)% cofidece iterval for the mea µ Typically, we would use (say) α = 005 i which case z α/ = z 005 = 196, ad we thus obtai a 95% cofidece iterval X() ± (196) σ The legth of the cofidece iterval is (196) σ which of course teds to 0 as the sample size gets larger I practice we would ot actually ow the value of σ ; it would be uow (just as µ is) But this is ot really a problem: we istead use a estimate for it, the sample variace s () defied by s () = 1 1 (X j X ) It ca be show that s () σ, with probability 1, as ad that E(s ()) = σ, 1
2 So, i practice we would use s() is place of σ whe costructig our cofidece itervals For example, a 95% cofidece iterval is give by X() ± (196) s() The followig recursios ca be derived; they are useful whe implemetig a simulatio requirig a cofidece iterval: X +1 = X + X +1 X, + 1 ( S+1 = 1 1 ) S + ( + 1)(X +1 X ) 1 Applicatio to Mote Carlo simulatio I Mote Carlo simulatio, istead of collectig the iid data X 1,, X, we simulate it Moreover, we ca choose as large as we wat; = 10, 000 for example, so the cetral limit theorem justificatio for costructig cofidece itervals ca safely be used Thus we ca immediately obtai cofidece itervals for Mote Carlo estimates But simulatio also allows us to be clever: We ca purposely try to iduce egative correlatio amog the variables X 1,, X, or geerate copies that while havig the same mea, have a smaller variace, so that the variace of the estimator i (1) becomes smaller tha σ resultig i a smaller cofidece iterval The idea is to try to get eve better estimates by reducig the ucertaity i our estimate I the ext sectios, we explore ways of doig this 13 Atithetic variates method Let X i deote our copies of X (each has the same distributio hece the same mea µ ad variace σ ) but let us ot assume that they are idepedet Let = m, for some m 1, that is, is eve Note that where X() = 1 m m X j = 1 m Y j = Y (m), (3) m ad we coclude that Y 1 = X 1 + X Y = X 3 + X 4 Y m = X 1 + X, The two estimators Y (m) ad X() for E(X) i (3) are idetical Because they are idetical, we ca ad will use Y (m) i what follows Moreover, E(Y i ) = E(X) = µ (remember we are assumig that the X i all have the same distributio hece the same mea) This meas that for purposes of argumet here we ca view each Y i as the ed
3 copy that we wish to simulate from (istead of the X i ) We let Y = X 1+X deote a geeric Y i The problem of estimatio ca be recast as we are tryig to estimate µ = E(Y ) Computig variaces, V ar(y ) = (1/4)(σ + σ + Cov(X 1, X )) = (1/)(σ + Cov(X 1, X )) I the case whe the X i are iid, Cov(X 1, X ) = 0 ad thus V ar(y ) = σ / yieldig (as we already ow, recall ()) V ar(y (m)) = σ But if Cov(X 1, X ) < 0, the V ar(y ) < (1/)σ yieldig V ar(y (m)) < σ ; variace is reduced So it is i our iterest to somehow create some egative correlatio withi each pair (X 1, X ), (X 3, X 4 ),, but eep the pairs iid so that the Y i are iid (ad thus the CLT still applies); for the V ar(y (m)) will be lowered from what it would be if we simply used iid copies of the X i To motivate how we might create the desired egative correlatio, recall that we ca geerate a expoetially distributed rv X 1 = (1/λ) l (U) with U uiformly distributed o (0, 1) Now istead of usig a ew idepedet uiform to geerate a secod such copy, use 1 U which we well ow is also uiformly distributed o (0, 1); that is, defie X = (1/λ) l (1 U) Clearly X 1 ad X are egatively correlated sice if U icreases, the 1 U decreases ad the fuctio l(y) is a icreasig fuctio of y: X 1 icreases iff U icreases iff 1 U decreases iff X decreases More geerally, for ay distributio F (x) = P (X x) with iverse F 1 (y) we could geerate a egatively correlated pair via X 1 = F 1 (U), X = F 1 (1 U) sice F 1 (y) is a mootoe icreasig fuctio of y The radom variables U ad 1 U have a correlatio coefficiet ρ = 1, they are egatively correlated (to the largest extet), thus the mootoicity preserves the property of egative correlatio; ρ X1,X < 0 (ot ecessarily 1 though) I a geeral Mote Carlo simulatio our X is of the form X = h(u 1,, U ), for some (perhaps very complicated) fuctio h, ad some (perhaps large), that is, we eed iid U i to geerate each copy of X For example, if we are cosiderig X = C = ( 1 S i K) +, the payoff at time T = of a Asia call optio uder the biomial lattice model, the rewritig where the Y i are the iid updow rvs, we have h(u 1, U ) = 1 S i = (1/)S 0 Y 1 [1 + Y ], ( (1/)S 0 (ui{u 1 p} + di{u 1 > p})[1 + (ui{u p} + di{u > p})] K) + This fuctio is mootoe decreasig i U 1 ad U : as either variable icreases, they will exceed the value p ad hece the idicators will ted towards the lower value d as opposed to the higher value u > d Because the vectors (U 1, U ) ad (1 U 1, 1 U ) are idetically distributed, so are the rvs X 1 = h(u 1, U ) ad X = h(1 U 1, 1 U ); i particular they have the same mea E(X) But the mootoicity of h results i egative correlatio betwee them, Cov(X 1, X ) < 0 I geeral, as log as the fuctio h is mootoe (either icreasig or decreasig) i each variable, the it ca be show that X 1 = h(u 1,, U ) ad X = h(1 U 1,, 1 U ) are 3
4 ideed egatively correlated, ad are referred to as atithetic variates Agai, because the vectors (U 1, U, U ) ad (1 U 1, 1 U, 1 U ) have the same distributio, so do X 1 ad X ; i particular they have the same mea E(X) But because of the iduced egative correlatio (whe h is mootoe) the two are themselves egatively correlated copies: Propositio 11 If the fuctio h for geeratig X = h(u 1,, U ) is mootoe i each variable, the X 1 = h(u 1,, U ) ad X = h(1 U 1,, 1 U ) with the U i iid uiform o (0, 1) are i fact egatively correlated; Cov(X 1, X ) < 0 (Equivaletly E(X 1 X ) < E(X 1 )E(X ) = E (X)) Algorithm for usig atithetic variates to estimate µ = E(X), whe X = h(u 1,, U ) is mootoe i the U i : The method of simulatig our pairs is straightforward: 1 Geerate U 1, U Costruct a first pair: Set X 1 = h(u 1,, U ) ad X = h(1 U 1,, 1 U ) Now idepedetly geerate ew iid uiforms to costruct aother pair X 3, X 4 ad so o pair by pair util reachig m (large) desired pairs 3 Use the estimate where m Y (m) = Y j, Y 1 = X 1 + X Y = X 3 + X 4 Y m = X m 1 + X m To costruct our (ew ad better) cofidece iterval: Defie the sample variace as s (m) = 1 m 1 m (Y j Y m ) The the iterval Y (m) ± z α/ s(m) is a 100(1 α)% cofidece iterval for the mea µ As a very simple example recall that we ca estimate π by observig that π/4 = E( 1 U ) Sice h(y) = 1 y is mootoe decreasig i y, we ca use atithetic variates Thus we would use X 1 = 1 U1, X = 1 (1 U 1 ) for our first pair, X 3 = 1 U, X 4 = 1 (1 U ) ad so o 4
5 Remar 11 I a real simulatio applicatio, computig exactly Cov(X 1, X ) whe X 1 ad X are atithetic is ever possible i geeral; after all, we do ot eve ow (i geeral) either E(X) or V ar(x) But this is ot importat sice our objective was oly to reduce the variace, ad we accomplished that 14 Atithetic ormal rvs I may fiace applicatios, the fudametal rvs eeded to costruct a desired output copy X are uit ormals, Z 1, Z, For example, whe usig geometric Browia motio for asset pricig, our payoffs typically ca be writte i the form X = h(z 1,, Z ) Notig that Z is also a uit ormal if Z is, ad that the correlatio coefficiet betwee them is ρ = 1, the followig is the Gaussia aalogue to Propositio 11 Propositio 1 If the fuctio h for geeratig X = h(z 1,, Z ) is mootoe i each variable, the X 1 = h(z 1,, Z ) ad X = h( Z 1,, Z ) with the Z i iid N(0, 1) are i fact egatively correlated; Cov(X 1, X ) < 0 Example with a Asia call optio: Suppose for example you wish to estimate the expected payoff of a Asia call optio (termiatio date T ) averaged over the time poits 0 = t 0 < t 1 < t < < t = T The payoff is the X = C T = ( 1 S(t ) K) + We ext show how to costruct the atithetic pairs 1 Geerate iid N(0, 1) rvs, Z 1,, Z Set Recursively set L i = e σ t i t i 1 Z i +µ(t i t i 1 ), i {1,,, } (4) S(t 1 ) = S(0)L 1 S(t ) = S(t 1 )L = S 0 L 1 L S(t ) = S(t 1 )L = S 0 L 1 L L Set X 1 = ( 1 S(t ) K) + 3 Now reset the L i i (5) by usig Z 1,, Z i place of Z 1,, Z, that is, set L i = e σ t i t i 1 Z i +µ(t i t i 1 ), i {1,,, } (5) 5
6 4 Recursively set S(t 1 ) = S(0)L 1 S(t ) = S(t 1 )L = S 0 L 1 L S(t ) = S(t 1 )L = S 0 L 1 L L Set X = ( 1 S(t ) K) + 5 Set Y = X 1 + X Deotig the above copy by Y 1 = Y, we ca geerate a secod idepedet such copy by startig agai at (1) with a ew (idepedet) Z 1,, Z ad settigy = Y Repeatig this procedure m times yields our desired m iid copies of Y 15 Cotrol variates Suppose we wish to estimate µ = E(X) usig Mote Carlo simulatio (eg, usig X() with iid copies of X) Lettig C be ay other rv, with mea E(C), ad b a costat, ote that the rv Y give by Y = X b(c E(C)), (6) has the same desired mea: E(Y ) = E(X) Thus, if we could simulate iid copies of Y, Y 1, Y, the we could use as our estimate Y () istead of X() Notig further that σ Y = V ar(y ) = σ X + b σ C bσ X,C, (7) we see that by choosig C ad b wisely, it might be possible to reduce variace, that is, to have σy < σ X thus resultig i the lower variace estimator Y () tha the usual X() If X is oegative, the this would amout to choosig b > 0 ad selectig C ad X to have high positive correlatio, but i geeral may possibilities might come ito play Before we ivestigate this further, ote that it would be very helpful if C was already part of the simulatio of X i the sese that wheever we simulated a copy of X, a copy of C ecessarily came out for free alog the way Also we wat C to be such that we exactly ow the value E(C) The idea beig that we do ot wat to have to icrease our wor A example would be X = ([S(t 1 )+S(t )]/ K) + ad C = S(t ) for a Asia call optio This method of itroducig such a C for purpose of reducig variace is the cotrol variates method, ad C E(C) is called the cotrol variate for estimatig E(X) For a give C, we ca view (7) as a fuctio of b, g(b) = σy (b), ad the usig elemetary calculus, set g (b) = bσc σ x,c = 0 ad solve for the miimum b This yields: b = σ X,C σc, (8) σy (b ) = σx(1 ρ X,C), (9) 6
7 where ρ X,C = σ X,C /(σ X σ C ) deotes the correlatio coefficiet Thus by choosig ay C for which σ X,C 0 we ca always reduce variace, ad it is desirable to choose a C that is strogly correlated with X I practice we would ot be able to compute the value of b exactly sice it is uliely that we would ow σ X,C ad maybe ot eve σc But we could estimate it i advace by simulatio: Choose large ad use b b () = (X i X())(C i E(C)) (C i E(C)) (10) I other words we would first (just oce) ru a simulatio (large ) to obtai the estimate b (), ad the use that fixed value throughout our desired Mote Carlo simulatio Examples with GBM 1 Asia call optio: With 0 < t 1 < < t = T, the payoff at time T is X = ( 1 S(t i ) K) + A atural choice for C is the stoc itself at the termial value, C = S(T ) = S(t ) We certaily ca compute E(C) it is part of the simulatio ayhow ad is clearly positively correlated with X Aother choice would be the payoff of a Europea call C = (S(T ) K) +, sice i this case E(C) is ow exactly from the BlacScholes optios pricig formula A eve better choice (more correlatio) would be to use C = (Π S(t i) K) +, the payoff of a geometrically averaged Asia optio Here to, it turs out that the expected payoff E(C) is exactly ow (a formula exists), ad because it icorporates all values of the GBM, it yields a higher correlatio with X It should ot be surprisig that the choice of C might also deped o the strie price K For example, if K is very small compared to S(0), oe would, with high probability, obtai a positive payoff of 1 S(t i ) K, yieldig a high correlatio with (say) ay of the choices of C metioed above, whereas if K is very large compared to S(0) the oe would, with high probability, obtai o payoff at all, thus yieldig a low correlatio with a choice of C = S(T ) (but still a high oe with the geometrically averaged payoff above) There is o obvious best choice that wors with all payoffs; oe must tae ito cosideratio the specific structure of a payoff, ad its parameters I payoffs with multiple assets such as a spread optio, oe ca use a cotrol that uses all (or some) of the assets For example, for X = ( S 1 (T ) S (T ) K) + oe might try C = S 1 (T ) S (T ) 7
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