Dual Valuation and Hedging of Bermudan Options

Transcription

1 SIAM J. FINANCIAL MAH. Vol. 1, pp c 2010 Sociey for Indusrial and Applied Mahemaics Dual Valuaion and Hedging of Bermudan Opions L. C. G. Rogers Absrac. Some years ago, a differen characerizaion of he value of a Bermudan opion was discovered which can be hough of as he viewpoin of he seller of he opion, in conras o he convenional characerizaion which ook he viewpoin of he buyer. Since hen, here has been a lo of ineres in finding numerical mehods which exploi his dual characerizaion. his paper presens a pure dual algorihm for pricing and hedging Bermudan opions. Key words. opimal sopping, dual valuaion, Bermudan opion, hedging AMS subjec classificaions. 62L15, 60G40, 91G40 DOI / Inroducion. his paper derives an algorihm for valuing and hedging a Bermudan 1 opion from a pure dual sandpoin. Apar from various changes of names, pricing a Bermudan opion is he same as solving an opimal sopping problem, which is arguably he simples possible sochasic opimal conrol problem. For as long as derivaives have been priced wihin he Black Scholes paradigm, he radiional value-funcion approach, and he associaed Bellman equaions, have been widely used in he aemp o price Bermudan opions; he whole area has been a mahemaical playground because of he scarciy of closed-form soluions and he consequen need for approximaions, esimaes, and asympoics o come up wih prices. During he las cenury, he value-funcion approach was, in effec, he only mehod available, bu in recen years anoher quie differen dual approach has been discovered; see Rogers [3] and Haugh and Kogan [2]. he main resul is ha if he reward process 2 is denoed Z, hen he value Y0 of he opimal sopping problem can be alernaively expressed as (1.1) Y0 =sup E[Z τ ]= min E [ sup (Z M ) ], τ M M 0 0 where is he se of sopping imes bounded by, he ime horizon for he problem, and M 0 is he se of uniformly inegrable maringales vanishing a zero. he minimum is aained, by he maringale M of he Doob Meyer decomposiion of he Snell envelope process of Z, and in ha case (1.2) sup (Z M )=Y0 almos surely. 0 Received by he ediors Sepember 25, 2009; acceped for publicaion (in revised form) May 1, 2010; published elecronically July 15, hp:// Saisical Laboraory, Universiy of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (L.C.G.Rogers@ saslab.cam.ac.uk). 1 While we shall discuss only discree-ime problems in his paper, he mehodology could be applied o he pricing of American opions afer discreizing he ime. 2 Some minor regulariy condiion needs o be imposed on Z; iissufficienhasup 0 Z L p for some p>1. 604

2 DUAL VALUAION AND HEDGING OF BERMUDAN OPIONS 605 he radiional approach via he value funcion and he Bellman equaions akes he viewpoin of he buyer of he opion, who seeks o choose he bes sopping ime a which o exercise; he firs expression for Y0 in (1.1) embodies his. he dual approach is he soluion of he problem from he viewpoin of he seller of he opion, who seeks a hedging maringale Y0 + M, whose value a all imes will be a leas he value of he reward process; and he resul (1.2) shows ha for he perfec choice of M = M his does indeed happen. Since he dual approach was discovered, here have been various aemps o apply i in pracice, wih mixed success; choosing a good maringale is a leas as difficul as choosing a good sopping ime! he early paper of Andersen and Broadie [1] usesanumericalapproxi- maion o he value funcion o sugges a good maringale o use and in his way obains quie igh bounds on boh sides for a number of es examples. However, a a concepual level, i has been an ousanding issue o derive a pure dual mehod which solves he opimal sopping problem wihou he need o calculae a value funcion using only he dual characerizaion of (1.1). Le us amplify he disincion. Pure primal mehods are well undersood; indeed, virually all soluions of he opimal sopping problem (and all soluions prior o he discovery of he dual characerizaion in he early 21s cenury) are of his ype. here are hybrid mehods, such as ha of Andersen and Broadie, bu where is he pure dual mehod? I is he purpose of his shor noe o demonsrae how he soluion may be derived by purely dual mehods akin o he backward recursion of dynamic programming. he key observaion, which is obvious from (1.1), is ha he value of he opimal sopping problem is lef unalered if Z is replaced by Z M, wherem M he algorihm. In his secion, we specify he algorihm by which he given Bermudan opion is o be hedged. We are given a reward process (Z ) =0,..., adaped o he filraion (F ) =0,...,, and he aim is o find some maringale M M 0 such ha (1.1) holds. If(1.1) holds, hen by he earlier resul of [3], [2] we also have(1.2). he consrucion is based on wo very simple observaions: 1. he value of he sopping problem for Z is he same as he value of he sopping problem for Z + N, wheren is any maringale in M Adding a consan o Z adds a consan o he value. he proof of he following resul shows how o solve he problem recursively, by consrucing a sequence of maringales which do an ever beer job of hedging. he idea is ha he pahwise maximum mus become a consan random variable (see (1.2)); while i is no obvious how we shall achieve his in one go, we can easily see how o ensure ha he final value of Z is a consan by subracing a maringale which is equal o Z a ime. he inducive proof consrucs maringales which are consan on some inerval [ k, ] for ever bigger k. Proposiion 1. here exis a sequence of consans a j and a sequence of maringales N (j) M 0, j =1,..., +1, such ha (2.1) max j<i Z(j) i = a j, where (2.2) Z (j) Z N (j).

3 606 L. C. G. ROGERS Proof. he proof proceeds by inducion on j. o sar he inducion, we consider he maringale M (1) = E [Z ] and se N (1) = M (1) E[M (1) ], which is clearly in M 0 and equally clearly achieves (2.1)forj =1,wiha 1 = E[M (1) ]=E[Z ]. Now suppose ha (2.1) isrueforj k, and consider he nonnegaive maringale (2.3) M (k+1) = E [ {Z (k) k a k} + ]. his maringale is consan in [ k, ]; we shall subrac i from Z (k) o form he process Z (k) M (k+1). wo cases hen need o be considered. 1. If Z (k) k >a (k) k,henwehave Z k = a (k) k,and Z Z (k) for all > k, sincem (k+1) is nonnegaive. hus = a k. max k 2. If Z (k) k a k,henm (k+1) iszeroin[ k, ], and by he inducive hypohesis max k Eiher way, he conclusion is he same, namely, ha (2.4) max k We now define = max k< Z(k) = a k. = a k. (2.5) (2.6) N (k+1) = N (k) + M (k+1) E[M (k+1) ], a k+1 = a k + E[M (k+1) ], so ha Z (k+1) Z N (k+1) = Z (k) + E[M (k+1) ] saisfies (2.1), aking j = k +1. Remarks. (i) Applying he proposiion in he case j = +1, welearn ha (2.7) max 0 i Z( +1) i max 0 i { ( +1) } Zi N i = a +1. Hence he value of he opimal sopping problem wih reward process Z ( +1) is equal o a +1, and by he firs observaion, his is acually he value of he opimal sopping problem for he original reward process Z.

4 DUAL VALUAION AND HEDGING OF BERMUDAN OPIONS 607 (ii) he recursive consrucion generaes an increasing sequence a 1 a 2 a +1, increasing he value of he problem as well as he hedging maringale. (iii) Observe ha by adding (2.5) and(2.6) welearnha (2.8) N (k) + a k =. From (2.3) and(2.2) we deduce ha M (k+1) = E [ {Z (k) k a k} + ] = E [ {Z k N (k) k a k} + ] { } + = E Z k k, and aking = k leads o he conclusion (2.9) M (k+1) k = { Z k k} +. he saemen and proof of Proposiion 1 is pure dual ; here is no menion of he value of he sopping problem; we alked only abou he hedging maringales M (k). o ie hings ogeher, we shall now show how he consrucs from he proof of Proposiion 1 relae o he more familiar value process of he opimal sopping problem, (2.10) Y sup E [ Z τ ]. τ,τ is iner- As is well known, Y is he Snell envelope process of he reward process Z, andy preed as he bes ha can be done if by ime he process has no been sopped. Proposiion 2. For all k =0, 1,..., we have (2.11) k+1 k = Y k. Proof. he proof is by inducion on k. Clearly he saemen is rue if k =0,forboh sides of (2.11) areequaloz. Suppose now ha he saemen is rue for all k<n,and

5 608 L. C. G. ROGERS consider n+1 n = M (n+1) n + n n = M (n+1) n + E n[y n+1 ] by inducive hypohesis { } + n = Z n + E n[y n+1] using (2.9) n = {Z n E n [Y n+1]} + + E n [Y n+1] by inducive hypohesis =max { Z n,e n [Y n+1] } = Y n, as required. Remarks. (i) From (2.6) and Proposiion 2 we see ha a k = E[Y k+1 ], which explains why he sequence a k is increasing and why a +1 is he value of he problem. (ii) If we resric our aenion o problems where here is some underlying Markovian srucure, i is no hard o see ha he maringales M (k) consruced are characerized 3 as M (k+1) k = ϕ k (X k ) for all k =0, 1,...,. Inryingouse(2.9) o deermine he funcions ϕ k recursively, we are faced wih he wo seps, condiional expecaion and poinwise maximizaion, which are cenral o he sandard dynamic programming approach. hus i seems unlikely ha in problems wih Markovian srucure he pure dual approach presened here will generae numerical mehodologies ha differ significanly from exising mehodologies for such examples. REFERENCES [1] L. Andersen and M. Broadie, A primal-dual simulaion algorihm for pricing mulidimensional American opions, Managemen Sci., 50 (2004), pp [2] M. B. Haugh and L. Kogan, Pricing American opions: A dualiy approach, Oper. Res., 52 (2004), pp [3] L. C. G. Rogers, Mone Carlo valuaion of American opions, Mah. Finance, 12 (2002), pp We know ha M (1) has his form, and from (2.9) and he Markovian srucure i is obvious by inducion ha his form persiss.