1 Estimating sensitivities

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Vivien Dawson
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1 Copyright c 27 by Karl Sigma 1 Estimatig sesitivities Whe estimatig the Greeks, such as the, the geeral problem ivolves a radom variable Y = Y (α) (such as a discouted payoff) that depeds o a parameter α of iterest (such as iitial price S, or volitility σ, etc.). I additio to estimatig the expected value K(α) def = E(Y (α)) (this might be, for example, the price of a optio), we wish to estimate the sesitivity of E(Y ) with respect to α, that is, the derivative of E(Y ) with respect to α, K (α) = = lim h K(α + h) K(α). h 1.1 Sample-path approach If the mappig α Y (α) is ice eough, we ca iterchage the order of takig expected value ad derivative, [ dy = E. (1) Uder this sceario, K (α) itself is a expected value so we ca estimate it by stadard Mote Carlo: Simulate iid copies of dy ad take the empirical average. To dispese with the otio that such a iterchage as i (1) is always possible (o, it is ot!) oe merely eed cosider the of a digital optio with payoff Y = Y (S ) = e rt I{S(T ) > K}, where S(T ) = S e X(T ) = S e σb(t )+(r σ2/2)t. (The risk-eutral probability is beig used for pricig purposes.) I this case, dy = sice the idicator is a piecewise costat fuctio of S, thus E[ dy =. But E(Y ) = e rt P (S(T ) > K) is a ice smooth fuctio of S >, with a o-zero derivative. (Yes, the sample paths of Y are ot differetiable (or cotiuous eve) at the value of S for which S(T ) = K, but P (S(T ) = K) =, so this poit ca be igored.) O the other had, a Europea call optio, with payoff Y = e rt (S(T ) K) + satisfies dy = e rt e X(T ) I{S(T ) > K} which ca be re-writte as dy = e rt S(T ) S I{S(T ) > K}, ad ideed it ca be proved that (1) holds. The basic coditio eeded to esure that the iterchage is legitimate is uiform itegrability of the rvs {h 1 (Y (α + h) Y (α))} as h. We preset a sufficiet coditio for this ext. Propositio 1.1 Suppose that Y (α) is 1. wp1, differetiable at the poit α, ad satisfies 2. there exists a iterval I = (α ɛ, α + ɛ), (some ɛ > ), ad a o-egative rv B with E(B) < such that wp1 The (1) holds. Y (α 1 ) Y (α 2 ) α 1 α 2 B, α 1, α 2 I. 1

2 Proof : For sufficietly small h, Y (α + h) Y (α ) hb, wp1, thus Y (α + h) Y (α ) h B, wp1, ad the result follows by the domiated covergece theorem (lettig h ). For the Europea call optio, we have Y (S + h) Y (S ) e rt he X(T ), so the above propositio applies with B = e rt e X(T ). For the digital call optio, Y (S ) is ot differetiable at the poit S where S e X(T ) = K; it is ot eve cotiuous there. A geeral rule is that if the mappig α Y (α) wp1 is cotiuous at all poits, ad differetiable except at most a fiite umber of poits, the the iterchage will be valid. Eve if oe ca justify the iterchage (1), it may ot be possible to explicitly compute the derivative dy, thus rederig the sample-path approach impractical. So clearly other methods are eeded for estimatig K(α). 1.2 Score fuctio approach I our ext approach, we expad the expected value i E(Y ) as a itegral, ad the brig the derivative iside. To illustrate the basics of this method, let us first cosider computig the of a optio with a termiatio date T, such as a Europea call with payoff (S(T ) K) +, where S(t) = S e X(t), t is GBM, with X(t) = σb(t) + (r σ 2 /2)t a risk-eutral versio of BM (for pricig purposes). The discouted payoff is Y = e rt (S(T ) K) +, ad the price is give by E(Y ) = e rt E((S(T ) K) + ) = (x K) + dx, (2) where = f S (x) is the desity fuctio of the logormal rv S(T ) (which depeds o S ). Notig that the cdf F (x) = P (S(T ) x) = P (Z l(x)), where Z is a uit ormal ad l(x) = l (x/s ) (r σ 2 /2)T σ T we ca differetiate F (x) wrt x to obtai the desity i closed form where θ is the uit ormal desity, = 1 xσ T θ(l(x)), θ(x) = 1 2π e x2 /2, x R., To compute the, we differetiate (2) with respect to S, but we differetiate the itegral expressio as opposed to usig the expected value expressio: = e rt d (x K) + dx = e rt (x K) + d dx. (3) The mai poit is that ot oly ca we ca iterchage the order of itegral ad derivative, but the payoff fuctio (x K) + is a costat with respect to S ; oly the desity depeds o S. (Justificatio for the iterchage is easily established sice the itegrad is o-egative 2

3 ad is a very smooth fuctio of S.) The itegral i (3) is ot obviously a expected value, but we ca make it so: Lettig = d, we ca express (3) as a expected value by simply dividig by ad the multiplyig by the factor i the itegrad = e rt (x K) + dx = e rt E [(S(T ) K) +. (4) Thus we ca use Mote Carlo simulatio: K) + f(s(t )) simulate (large) iid copies of X = (S(T ), X 1,..., X, ad take the empirical average as our estimate: e rt 1 X i. We have expressed the as a weighted expected value of the payoff, (S(T ) K) +, i which f(s(t the weight )) is determied by what is called the score fuctio. The form of the fial expressio of (4) remids us of the likelihood ratio i importace samplig, ad because of the similarity, this method is sometimes called the likelihood ratio method, eve though we are ot chagig measure here; we are still usig. To use this method i practice, however, we must compute. Because of the expoetial form of θ(x), it is easily see that = l(x) l(x), ad l(x) = (S σ T ) 1. Thus yieldig = l(x) S σ T, f(s(t )) l(s(t )) = S σ T. But if we use a uit ormal Z to costruct S(T ) via S(T ) = S e σ T Z+(r σ 2 /2)T, the (check!) l(s(t )) = Z, ad we fially arrive at = Z S σ T. (5) Thus from (4) we arrive at: = e rt E [(S(T ) K) + Z S σ. (6) T 3

4 Algorithm for estimatig the, 1. Geerate iid uit ormals, Z 1,..., Z. 2. Defie ad 3. Use the estimate of a Europea call optio: Y i = S e σ T Z i +(r σ 2 /2)T, X i = (Y i K) + Z i S σ, i = 1, 2,...,. T e rt 1 X i Score method for computig the of other optios: The expressio i (6) exteds immediately to ay optio with a payoff give by a fuctio h(s(t )) such as the digital optio h(s(t )) = I{S(T ) > K}; the of ay such optio the has the form = e rt h(x) [ dx = e rt E h(s(t )) Z S σ T. (7) We ca also hadle some path-depedet optios such as a Asia call. We outlie this here. For fixed time poits < t 1 < t 2 < < t k = T, the payoff is give by (S K) + where S = 1 k S(t i ). k I this case, the payoff fuctio h is a fuctio of k variables, x = (x 1,..., x k ), h(s(t 1 ),..., S(t k )) = (S K) +. We thus obtai our via ( 1 h(x) = h(x 1, x 2,..., x k ) = k = e rt k +; x i K) h(x) dx, (8) where deotes the joit desity of (S(t 1 ),..., S(t k )). Lettig Z 1,..., Z k deote iid uit ormals we ca recursively costruct S(t 1 ) = S e σ t 1 Z 1 +(r σ 2 /2)t 1 (9) S(t 2 ) = S(t 1 )e σ t 2 t 1 Z 2 +(r σ 2 /2)(t 2 t 1 ) (1). (11) S(t k ) = S(t k 1 )e σ t k t k 1 Z k +(r σ 2 /2)(t k t k 1 ). (12) The joit desity ca thus be writte as the product of coditioal logormal desities: = f 1 (x 1 S )f 2 (x 2 x 1 ) f k (x k x k 1 ); 4

5 f 1 (x 1 S ) is the desity of S(t 1 ) ad it depeds o S. f 2 (x 2 x 1 ) is the coditioal desity of S(t 2 ) give that S(t 1 ) = x 1, so it is the desity of x 1 e σ t 2 t 1 Z 2 +(r σ 2 /2)(t 2 t 1 ), ad so o with f k (x k x k 1 ) the coditioal desity of S(t k ) give that S(t k 1 ) = x k 1 ; it has the desity of x k 1 e σ t k t k 1 Z k +(r σ 2 /2)(t k t k 1 ). Oly f 1 depeds o S, ot the others, so ad we coclude (from (5)), that = f 1 (x 1 S )f 2 (x 2 x 1 ) f k (x k x k 1 ) Fially, from (8) this yields the as f(s(t 1 ),..., S(t k )) f((s(t 1 ),..., S(t k ))) = Z 1 S σ t 1. f = 1 (x 1 S ) f 1 (x 1 S ), ( = e rt E (S K) + Z 1 S σ ). (13) t 1 Algorithm for estimatig the, Begi Loop: of a Asia call optio: For j = 1,..., : Geerate k iid uit ormals Z 1,..., Z k. Set X j = (S K) + Z 1 S σ t 1. Ed Loop. Now use the estimate e rt 1 X i Score method for computig the V ega of optios: Let = d dσ, where is the desity of S(T ). The similar to computig the, we ca compute the V ega of a optio with payoff h(s(t )) via dσ = e rt h(x) dx = e rt E [h(s(t )). (14) Oce agai, the poit is that oly the desity depeds o σ. Oe ca easily check that f(s(t )) = Z2 1 Z T, σ 5

6 whe we use a uit ormal Z to costruct S(T ). Usig this, oe ca the show that the score fuctio for estimatig the Vega of a Asia call (payoff = (S K) + ) is give by f((s(t 1 ),..., S(t k ))) f((s(t 1 ),..., S(t k ))) = k Z 2 i 1 σ Z i ti t i 1, where we are usig the iid ormals, Z 1,..., Z k to costruct (S(t 1 ),..., S(t k )). 1.3 Forward-differece, ad cetral differece approach I the followig, istead of tryig to estimate K (α) itself, we are cotet with estimatig a approximatio of it. I the forward-differece approach we choose a small h > ad cosider usig as our approximatio [ K (α) h 1 (K(α + h) K(α)) = E h 1 (Y (α + h) Y (α)). I the cetral-differece approach we choose a small h > ad cosider usig as our approximatio [ K (α) (2h) 1 (K(α + h) K(α h)) = E (2h) 1 (Y (α + h) Y (α h)). I both cases, the approximatio eds up expressed as a expected value so ca be estimated usig Mote Carlo simulatio: Geerate iid copies of h 1 (Y (α + h) Y (α)) ad average, or geerate iid copies of (2h) 1 (Y (α + h) Y (α h)) ad average. It is importat to ote that sice i practice we will wat to estimate K(α) too, ot just K (α), the cetral-differece approach seems o the face of it to offer o advatages over the fiite-differece approach. But actually it does have a advatage as ca be see by comparig the error ivolved. Assumig that K(α) has at least two derivatives, a Taylor s series expasio yields K(α + h) = K(α) + K (α)h + K (α)h 2 /2 + o(h 2 ) (15) K(α h) = K(α) K (α)h + K (α)h 2 /2 + o(h 2 ). (16) Usig the first equatio by itself yields h 1 (K(α + h) K(α)) K (α) = K (α) h/2 + o(h), (17) while subtractig the secod from the first yields (2h) 1 (K(α + h) K(α h)) K (α) = o(h). (18) The poit is that the cetral-differece estimator rids us of the K (α)h/2 term, leavig us oly with a o(h) term as error; o(h)/h, as h. I ay case, the bias of our estimates (the error), is uder cotrol; choosig h sufficietly small will brig us as close as we wish to our desired aswer K (α). We have said othig yet about how we would geerate a copy of (say) h 1 (Y (α+h) Y (α)). We could, for example, geerate Y (α), ad the idepedetly geerate Y (α+h). But it might be possible to geerate both Y (α) ad Y (α + h) usig commo radom umbers. For example, cosider the digital call payoff Y = e rt I{S(T ) > K}. Sice S(T ) = S e X(T ), we ca geerate a copy of X(T ) ad use it for both S e X(T ) ad (S + h)e X(T ), ad hece for Y (S ) ad Y (S + h); they become positively correlated. This has the ice effect of reducig the variace of h 1 (Y (α + h) Y (α)) from what it would be if we geerated both idepedetly, but also saves us from havig to do extra simulatios. Not all applicatios allow such a ice couplig, but it is good to take advatage of it if it ca be doe. 6

7 1.3.1 Variace of the estimates I geeral, whereas the bias of our estimate (the error) seems uder cotrol (either K (α) h/2+ o(h) or just o(h)) thus motivatig us to choose a very small h >, the variace might ot be cooperative as h gets small. To see this, observe that V ar(h 1 (Y (α + h) Y (α))) = h 2 [V ar(y (α + h) Y (α)). (19) The h 2 term wars us that the variace might blow up as h gets small. For example, if we were to geerate Y (α) ad Y (α + h) idepedetly, the (assumig cotiuity as h ) V ar(y (α+h) Y (α)) 2V ar(y (α)) ad ideed h 2 [V ar(y (α+h) Y (α)). If we were able to use commo radom umbers, the it ca be show that typically, the improvemet is of the form V ar(y (α + h) Y (α)) = O(h) that is, V ar(y (α + h) Y (α)) Ch for some costat C > as h. But oce agai the h 2 domiates, ad it could still happe that h 2 [V ar(y (α + h) Y (α)). Sice i the ed, we will be cosiderig a estimate of the form Ỹ (α) = 1 X i, where the X i are iid copies of h 1 (Y (α + h) Y (α)), it would seem prudet to cosider the value of as well as the choice of h >, whe carryig out our simulatio. We would like to kow What happes to V ar(ỹ (α)) as ad h? Notig that V ar(ỹ (α)) = 1 h 2 (V ar(y (α + h) Y (α))), let us cosider the case whe we have (at worst) simulated Y (α) ad Y (α + h) idepedetly, so that as metioed above V ar(y (α + h) Y (α)) 2V ar(y (α)). Thus we ca assume there is a costat C > such that V ar(ỹ (α)) C 1 h 2 as ad h. (So we ca write V ar(ỹ (α)) = O( 1 h 2 ).) We thus see that i order to prevet the variace from blowig up but istead tedig to as ad h, we eed to have h 2. The poit is that we do ot wat h to be so very small relative to the sample size. 7

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