Utility-based valuation and hedging of basis risk with partial information

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1 Uiliy-based valuaion and hedging of basis risk wih parial informaion Michael Monoyios Mahemaical Insiue, Universiy of Oxford May 20, 2010 Absrac We analyse he valuaion and hedging of a claim on a non-raded asse using a correlaed raded asse under a parial informaion scenario, when he asse drifs are unknown consans. Using a Kalman filer and a Gaussian prior disribuion for he unknown parameers, a full informaion model wih random drifs is obained. This is subjeced o exponenial indifference valuaion. An expression for he opimal hedging sraegy is derived. An asympoic expansion for small values of risk aversion is obained via PDE mehods, following on from payoff decomposiions and a price represenaion equaion. Analyic and semi-analyic formulae for he erms in he expansion are obained when he minimal enropy measure coincides wih he minimal maringale measure. Simulaion experimens are carried ou which indicae ha he filering procedure can be beneficial in hedging, bu someimes needs o be augmened wih he increased opion premium, ha akes ino accoun parameer uncerainy, in order o be effecive. Empirical examples are presened which conform o hese conclusions. 1 Inroducion This aricle analyses he opimal valuaion and hedging of a coningen claim in an incomplee marke under a parial informaion scenario. The hedger does no know he values of he asses expeced reurns, which are filered from price observaions, and rading sraegies are required o be adaped o he asse price filraion. We assume volailiies and correlaion are known consans, so we assume approximaely coninuous price monioring. We make his approximaion as drif parameer uncerainy is much more severe han covariance uncerainy, as discussed by Rogers [27] in an opimal invesmen conex, and by Monoyios [22] in an opimal hedging problem. Parial informaion problems have usually been sudied in he conex of opimal invesmen (Rogers [27], Lakner [19], Brendle [8], and Björk, Davis and Landén [7]). Nagai and Peng [24] rea risk-sensiive conrol, while Pham [25] and Xiong and Zhou [28] sudy mean-variance porfolio problems. Uiliy-based hedging of claims under parial informaion has received lile aenion, hough some menion of parial informaion pricing was made in Dufresne and Hugonnier [9]. The incomplee marke is a basis risk model in which a claim on a non-radeable asse Y is hedged using a correlaed raded sock S. This has been sudied by many auhors in he compleely observable case, ofen when asse prices follow log-brownian moions wih consan parameers (Davis [10], Henderson [15], Monoyios [21, 22]) or when some parameers depend on he non-raded asse only (Musiela and Zariphopoulou [23] and, in a muli-dimensional case, Imkeller and co-auhors [1, 2, 3]). In he scalar versions of hese models, an explici nonlinear expecaion represenaion for he exponenial uiliy-based price is available. In our approach, significan differences arise. We begin wih lognormal processes for S, Y, bu wih unknown consan drifs, hence considered as random variables wih some prior disribuion a ime zero. Afer filering he drifs from price observaions, he resuling full informaion model wrien in he observaion filraion has sochasic drifs which depend on 1

2 boh asse prices. This is a deparure from he models in he lieraure and he explici resuls of he full informaion case are no longer available. Preliminary ideas on using filering mehods o deal wih drif parameer uncerainy in he basis risk model were oulined by Monoyios [22]. Tha analysis used wo separae onedimensional filers for each asse drif, and he indifference valuaion and hedging program was no carried ou. Here, we use a wo-dimensional filer, hus inroducing coupling beween he opimal filers conneced wih he drifs of he asses. The filered drif of S can depend on Y and vice versa. The raher specialised case in [22] is no, in general, obained in he full wo-dimensional analysis. We carry ou he uiliy-based valuaion and hedging program in he derived compleely observable model wih random drifs. We derive an opimal hedging formula (Theorem 1), giving he hedge raio in erms of derivaives of he indifference price wih respec o boh asse prices. This is a modificaion of he full informaion resul (which conains derivaives wih respec o he non-raded facors only). The exra erm reflecs addiional risk induced by drif parameer uncerainy. Using he PDE saisfied by he indifference price we derive some payoff decomposiions (Lemma 1, Corollary 1) and a price represenaion equaion (Corollary 2) wrien under he minimal enropy maringale measure (MEMM) Q E. Mania and Schweizer [20] (see also Becherer [5] and Kallsen and Rheinländer [17]) obain similar (bu less explici) resuls using backward sochasic differenial equaion mehods in a coninuous semimaringale model. Our resuls are explici, in ha he erms appearing in he relaions are idenified in erms of he indifference price and is derivaives, given he more concree seing of our model. These resuls are used o derive an asympoic expansion (Theorem 2) of he indifference price, o linear order in he risk aversion, wih he zeroh order erm being he marginal price (he zero risk aversion limi of he indifference price). Small risk aversion resuls (of a slighly differen form) for indifference valuaion have been obained by Kramkov and Sirbu [18], bu for a uiliy funcion defined on he posiive real line, so no direcly applicable here. The asympoic expansion relies on idenifying he MEMM Q E. In general, his involves he soluion of a sochasic conrol problem. To obain more explici formulae and carry ou numerical experimens, we specialise in Secion 5 o he case where Q E = Q M, he minimal maringale measure. This corresponds o he case where he prior variance of S is less han or equal o ha of Y. We show (Theorem 3) ha, even wih sochasic risk premia generaed by he Kalman filer, he disribuion of he erminal value of he non-raded asse is lognormal, wih a ime-dependen variance and mean ha is a funcion of boh asse prices and ime. From his we derive new analyic formulae for mos of he erms in he price expansion, involving BS-syle formulae wih a random dividend yield ha is a funcion of ime and he asse prices, and a ime-dependen volailiy. One erm in he firs order correcion is no obainable in closed form. This is he Q M -expecaion of he quadraic variaion of he gains process of he marginal hedging sraegy (he opimal sraegy in he zero risk aversion limi), and so is compued via simulaion. We invesigae numerically wheher uiliy-based valuaion and hedging, when coupled wih learning based on filering he asse drifs, can ouperform oher echniques. We es agains (i) he opimal price and hedge in he absence of filering, implemened using he full informaion resuls wih he iniial values of he asse drifs (so no updaed by filering), (ii) he perfec correlaion Black-Scholes (BS)-syle price and hedge, and (iii) is correlaion-weighed modificaion, proposed by Hulley and McWaler [16], in which he laer wo mehods do no require knowledge of he drif parameers. As a calibraion, we also use he genuine full informaion hedge, using he rue asse drifs, once again fixed hrough he hedging inerval. We simulae many asse price hisories and compue he disribuion of erminal hedging errors, wih he hedging programs saring a he same iniial wealh. We also compue he effec of using he appropriae opion premium (allied o he paricular hedging mehod) as he iniial wealh. The laer compuaion is designed o show he combined effec of valuaion as well as hedging on he final profi and loss. The resuls indicae ha filering he asse price drifs can ofen improve he erminal profi and loss disribuion, bu here are insances when he prior is well suied o producing 2

3 good resuls wihou filering, or when he correlaion-weighed BS hedge is also a good hedge, because is approximaion of zero drif for he underlying asse under he minimal measure is a foruious one. The parial informaion mehod almos always gives an improvemen when one combines he filered hedge wih he appropriae valuaion a ime zero for he claim. This can be raced o he fac ha he parameer uncerainy induces a higher effecive volailiy, so he agen charges a higher premium for he claim han wihou parameer uncerainy. Finally, we show some examples wih real daa, hedging a baske of socks wih index fuures, and hese conform o our earlier conclusions. The res of he paper is organised as follows. In Secion 2 we se up he model and use he Kalman-Bucy filer o conver he parial informaion model o a full informaion model wih random drifs, given in Proposiion 1. In Secion 3 we carry ou exponenial indifference valuaion in he derived full informaion model, presen he opimal hedging formula (Theorem 1), and discuss he required smoohness of he dual value funcion, required for he validiy of our resuls. In Secion 4 we derive he payoff decomposiions and price represenaion resuls leading o he asympoic expansion (Theorem 2) for he indifference price. Secion 5 derives analyic formulae for all bu one erm in he price expansion in he case when Q E = Q M. Numerical experimens are carried ou in Secion 6, and some empirical examples in Secion 7. Secion 8 concludes. 2 The model The seing is a probabiliy space (Ω, F, P ), equipped wih a filraion F := (F ) 0 T carrying a wo-dimensional Brownian moion (B S, Z S ). Define a Brownian moion B Y correlaed wih B S according o B Y := ρb S + 1 ρ 2 Z S, wih ρ [ 1, 1] a known consan. A raded sock price S := (S ) 0 T and a non-raded asse price Y := (Y ) 0 T follow ds = σ S S (λ S d + db S ), dy = σ Y Y (λ Y d + db Y ), where σ S > 0 and σ Y > 0 are known consans. The risk premia λ S, λ Y are F 0 -measurable random variables, so would be known consans if a financial agen had access o he filraion F. The new feaure in his aricle is ha an agen will be required o use sraegies adaped o he observaion filraion generaed by he asse prices, so λ S, λ Y will be unknown consans whose values will be filered from price observaions. For simpliciy, he ineres rae is aken o be zero. In aking σ S, σ Y, ρ as known, we imply ha hey could be inferred from quadraic and cross-variaions, so his is an approximaion of coninuous asse price monioring. We make his assumpion o focus on he more severe problem of drif uncerainy, hough i would be ineresing o sudy similar problems wih discree monioring of asse prices, and unknown volailiies and correlaion as well as uncerain drifs. A European coningen claim pays he non-negaive random variable C(Y T ) a ime T, where C( ) is a bounded coninuous funcion. As is well-known, if he correlaion is perfec hen he claim can be perfecly replicaed by a BS-syle hedge ha does no require knowledge of he asse price drifs, and so is robus wih respec o drif parameer uncerainy. In he compleely observable incomplee case exponenial uiliy-based valuaion has been sudied by a number of auhors. In [21, 22] i is shown ha opimal valuaion combined wih hedging is beneficial compared wih he BS approach, in erms of he erminal hedging error disribuion produced by selling he claim a he appropriae price (he indifference price or he BS price) and invesing he proceeds in he corresponding hedging porfolio. If one focuses exclusively on he hedging sraegy, so begins wih he same iniial wealh for boh sraegies, Hulley and McWaler [16] provide evidence ha he he improved performance is no always guaraneed, signifying ha uiliy indifference mehods rely o some exen on heir iniial valuaion of he derivaive o be effecive. We shall see examples of his in Secion 6. 3

4 We focus here on he issue of drif parameer uncerainy, and he fac ha indifference pricing requires perfec knowledge of λ S, λ Y, which are virually impossible o esimae accuraely. This can ruin he effeciveness of indifference mehods, as shown in [22]. We relax he assumpion ha he agen knows he values of λ S, λ Y, and invesigae if opimal valuaion and hedging is made feasible in his case by filering he asse price drifs. The reurn parameers λ S, λ Y are hence modelled as random variables wih some prior disribuion. Define he processes ξ S, ξ Y by ξ S := 1 ds u σ S = λ S + B S, ξ Y := 1 0 S u σ Y S 0 0 dy u Y u = λ Y + B Y, 0 T. Using he Iô formula hese may be expressed as deerminisic funcions of he asse prices, given by ξ S ξ S (, S ) = 1 ( ) σ S log S σs, ξ Y ξ Y (, Y ) = 1 ( ) σ Y log Y σy. (1) For breviy of noaion we shall ofen wrie ζ ζ(, S, Y ) for any process ζ ha is a funcion of ime and curren asse prices, whenever no confusion arises. We regard he wo-dimensional process ( ) ξ S Ξ :=, 0 T, ξ Y as an observaion process in a Kalman-Bucy filering framework, corresponding o noisy observaions of he signal process (in our case, an unknown consan) Λ, defined by ( ) λ S Λ :=. Define he observaion filraion F := ( F ) 0 T by λ Y F = σ(ξ S u, ξ Y u ; 0 u ), 0 T. We assume a Gaussian prior disribuion, given by wih ( λ S Λ 0 = 0 λ Y 0 Law(Λ F 0 ) = N(Λ 0, V 0 ), (2) ) ( v S, V 0 = 0 c 0 c 0 v0 Y Y 0 ), c 0 = ρ min(v S 0, v Y 0 ), (3) for given consans λ S 0, λ Y 0, v S 0, v Y 0, which may be deermined as described below. A moivaion for his prior is he idea ha an agen migh use daa before ime zero o make a poin esimae of Λ, and could hen use he disribuion of he esimaor as he prior. Wih hisorical daa for ξ S (respecively, ξ Y ) over a ime inerval inerval S (respecively, Y ), hen an unbiased esimaor of Λ is Gaussian according o (2) and (3) wih λ i 0 equal o he poin esimae of λ i, and v i 0 = 1/ i, for i = S, Y. Hence, we shall suppose ha Λ, considered as a random variable, is bivariae normal according o (2) and (3). This disribuion will be updaed via subsequen observaions of ξ S, ξ Y over he hedging inerval [0, T ]. 2.1 Two-dimensional Kalman-Bucy filer We are firmly wihin he realm of a wo-dimensional Kalman filering problem. In [22], Monoyios oulined preliminary ideas connecing he parial informaion basis risk model o a Kalman filering problem, and wo one-dimensional Kalman-Bucy filers were used o obain he filered random drifs. Here, we properly rea he filering problem wih a wo-dimensional Kalman filer. 4

5 Remark 1. In principle, one could model he unknown risk premia λ S, λ Y as processes following linear SDEs, wihou leaving he Kalman filering framework, as Brendle [8] does in he conex of an opimal porfolio problem. We do no pursue his here, as we are seeking maximally explici formulae for he indifference price and opimal hedge. I is no cerain a his poin ha his is feasible in he linear SDE framework. This is lef for fuure research. where The observaion and signal SDEs are D = dξ = Λd + DdB, dλ = ( 1 0 ρ 1 ρ 2 ( ) 0, 0 ) ( ) B S, B = Z S. The opimal filer Λ := E[Λ F ], 0 T, is herefore he wo condiional expecaions λ i := E[λ i F ], 0 T, i = S, Y. The condiional variances and covariance are defined by v i := E [(λ i λ ] i ) 2 F, 0 T, i = S, Y, [ c := E (λ S λ S )(λ Y λ Y ) F ], 0 T, and he covariance marix will be denoed by ( v S V := c c v Y ), 0 T. (4) As usual wih a Kalman filer, his will be a deerminisic funcion of ime. For ρ 2 1, define he funcions m := min(v S, v Y ), M := max(v S, v Y ), b := M ρ 2 m 1 ρ 2, 0 T, (5) and noe ha b = m = M when he asse variances v S = v Y are equal. The Kalman-Bucy filer convers he parial informaion model o a compleely observable model as given below. Proposiion 1. On he filered probabiliy space (Ω, F T, F, P ) we have a full informaion model wih asse price dynamics given by ds = σ S S ( λ S d + d B S ), dy = σ Y Y ( λ Y d + d B Y ), (6) where B S, B Y are correlaed (P, F)-Brownian moions wih correlaion ρ, and λ S, λ Y are Fadaped processes, given in erms of he asse prices and ime as follows. For i, j {S, Y }, if m 0 = v0 i < v j 0 = M 0, hen λ i = λi 0 + m 0 ξ i 1 + m 0, λj = λj 0 + b 0ξ j 1 + b 0 ρ The covariance marix V in (4) is given by ( λ i 0 + b 0 ξ i 1 + b 0 λi 0 + m 0 ξ i ), 1 + m 0 0 T, i, j {S, Y }. (7) v i = m, v j = M = ρ 2 m + (1 ρ 2 )b, c = ρm, 0 T, i, j {S, Y }, (8) wih m, M, b defined in (5), and m, b given by m = m m 0, b = b 0, 0 T. (9) 1 + b 0 In he case ha m 0 = v0 S = v0 Y for all [0, T ]. = M 0, hen (7) (9) sill hold, wih b 0 = m 0, and hence b = m 5

6 Proof. By he Kalman-Bucy filer (for example, Theorem V.9.2 in Fleming and Rishel [12]), he process Λ saisfies he SDE d Λ = V ( DD T ) 1 (dξ Λ d) =: V ( DD T ) 1 dn, Λ0 = Λ 0, (10) where N is he innovaions process, defined by N := Ξ 0 Λ u du, 0 T, (11) and is a wo-dimensional correlaed F-Brownian moion, given by ( ) BS N =, B S, B Y = ρ, 0 T. (12) Using (11), (12) and he price dynamics in he form ( ) ( S σ S ) S d = Y σ Y dξ, Y gives he dynamics (6) of S, Y in he observaion filraion F. The covariance marix V saisfies he Riccai equaion dv d = V ( DD T ) 1 V, B Y wih V 0 given in (3). Then F := V 1 saisfies he Lyapunov equaion df d = ( DD T ) 1. Lenghy (bu sraighforward) calculaions confirm ha he Lyapunov equaion is solved by (8) and (9). Using hese formulae in he filering equaion (10) we find ha for i, j {S, Y }, if m 0 = v0 i < v j 0 = M 0, hen d λ i = m d B i = m (dξ i λ i d), λi 0 = λ i 0, d( λ j ρ λ i ) = b (d B j ρd B i ) = b [d(ξ j ρξ i ) ( λ j ρ λ i )d], λj 0 = λj 0. and in he case ha m 0 = v S 0 = v Y 0 = M 0, hese SDEs are valid wih b = m. Solving hese SDEs we obain (7). Remark 2. Wrien explicily, he dependence of he random risk premia on he asse prices is according o λ S λ S (, S ), λy λ Y (, S, Y ), if v0 S < v0 Y, λ S λ S (, S ), λy λ Y (, Y ), if v0 S = v0 Y, (13) λ S λ S (, S, Y ), λy λ Y (, Y ), if v0 S > v0 Y, saisfying he SDEs d λ S = m d B S, d λ Y ρd λ S = b (d B Y ρd B S ), if v0 S < v0 Y, d λ S = m d B S, d λ Y = m d B Y, if v0 S = v0 Y, d λ Y = m d B Y, d λ S ρd λ Y = b (d B S ρd B Y ), if v0 S > v0 Y. When boh asse drifs have equal prior variance, he opimal filers decouple, somewha similarly o he siuaion ha arises when one uses wo one-dimensional Kalman filers on each asse, as done in Monoyios [22] (hough one can have differen prior variances for each asse wih wo one-dimensional filers). In general, he asse price wih smaller prior variance eners he formulae for boh random risk premia. Wih he inerpreaion ha he prior disribuion is se using pas daa for he asse prices, hen he longer hisorical daase influences he opimal filers for boh asses, and he shorer daase only gives informaion on is respecive asse price drif. The inuiion behind his resul is ha esimaion of he drif of a geomeric Brownian moion depends only on he lengh of he ime inerval for which i is observed. (14) 6

7 3 Exponenial valuaion and hedging wih random drifs On he sochasic basis (Ω, F, F, P ), we consider exponenial indifference valuaion and hedging of he claim. An agen rades he sock wih F-adaped sraegy θ = (θ ) 0 T, an S-inegrable process represening he number of shares held in he porfolio. Denoe he porfolio wealh process by X = (X ) 0 T. For [0, T ], given X = x, he wealh evoluion is given by T T X u = x + θ u ds u = x + σ S π u ( λ S udu + d B u S ), u T, where π := θs. Denoe by Θ (respecively, Π) he se of admissible θ (respecively, π), defined shorly. The se of equivalen maringale measures is denoed by P e := {Q P S is a local (Q, F)-maringale}. Denoe by H(Q, P ) he relaive enropy beween Q P e and P : [ ] dq dq H(Q, P ) := E log, (if finie, else H(Q, P ) := ). dp dp The se of measures wih finie relaive enropy is denoed by P e,f := {Q P e H(Q, P ) < }, and we assume his se is nonempy. The se of admissible sraegies is defined in a similar manner o Becherer [4] and Mania and Schweizer [20], as Θ := {θ (θ S) is a (Q, F)-maringale for all Q P e,f }. For measures Q P e,f denoe he likelihood raio process by a (P, F)-maringale Γ Q : Γ Q := dq dp, 0 T. (15) bf We assume he agen has an exponenial uiliy funcion U(x) := exp( αx), x R, α > 0, wih risk aversion parameer α. The primal value funcion is he maximal expeced uiliy from erminal wealh from rading, wih he addiional random erminal endowmen of a shor posiion in he claim: u C (, x, s, y) := sup E,x,s,y [U(X T C(Y T ))], (16) π Π where E,x,s,y denoes expecaion given (X, S, Y ) = (x, s, y), for [0, T ]. Denoe by u 0 he value funcion when no claim is sold. The indifference selling price a ime T is p(, S, Y ), where he funcion p : [0, T ] R 2 + is defined as usual by u C (, x + p(, s, y), s, y) = u 0 (, x, s, y). As always wih exponenial uiliy, we anicipae ha he indifference price is independen of iniial wealh. Denoe he opimal sraegy for (16) by π C, and he opimal sraegy wih no claim by π 0. The opimal hedging sraegy π (H) is defined by π (H) := π C π 0. (17) 7

8 Well-known dualiy heory for he problem (16) (see Delbaen e al [11], for example) implies ha he primal value funcion has he represenaion u C (, x, s, y) = exp ( αx H C (, s, y) ), (18) where he funcion H C originaes from he dual problem o (16), and is defined by [ ( ) ] H C (, s, y) := inf E Q Γ Q T,s,y log αc(y T ), (19) Q P e,f where E Q,s,y denoes Q-expecaion given (S, Y ) = (s, y). Denoe by H 0 he funcion in (19) when no claim is presen, or equivalenly when C 0. This is he value funcion corresponding o he problem of minimising he relaive enropy beween Q P e,f and P, so ha H 0 (0,, ) = H(Q E, P ). To emphasise he link wih Q E, we shall someimes wrie H 0 H E. Applying he definiion of he he indifference price and using he separable form (18) of he value funcion leads o he well-known enropic represenaion 3.1 Opimal hedging heorem Γ Q p(, s, y) = 1 α (HC (, s, y) H 0 (, s, y)). (20) The resul below is a represenaion for he opimal hedging sraegy in erms of derivaives of he indifference price. Theorem 1. Suppose he indifference pricing funcion p : [0, T ] R + is of class C 1,2 ([0, T ] R + ). Then he opimal hedge for a shor posiion in he claim is o hold θ (H) shares of S a [0, T ], given by ( ) θ (H) p = s (, S, Y ) + ρ σy Y p σ S S y (, S, Y ), 0 T. Remark 3. The required regulariy of he indifference price for he validiy of he heorem is esablished in Secion 3.3. The addiional erm p s (, S, Y ), compared wih oher papers [1, 15, 21, 23] in which he drif parameers do no depend on he raded sock price, reflecs he addiional risk induced by parameer uncerainy. I is easy o see ha he formula sill holds if he volailiies and correlaion are also funcions of ime and curren asse prices. Proof. The HJB equaion associaed wih he primal he value funcion (16) is u C + max π A X,S,Y u C = 0, where A X,S,Y denoes he generaor of (X, S, Y ) under P. Performing he maximisaion in he HJB equaion gives he opimal feedback conrol π C (, s, y) in erms of derivaives of he value funcion. Then using he separable form (18) of he value funcion, we obain he opimal sraegy as π C = π C (, S, Y ), where π C (, s, y) = λ S σ S α 1 ) (sh Cs + ρ σy α σ S yhc y. A similar formula holds for he opimal sraegy π 0 in he case when no claim is presen, wih H C replaced by H 0. Applying he definiion (17) of he opimal hedging sraegy, we obain π (H) (, s, y) = 1 ] [s(h Cs H 0s ) + ρ σy α σ S y(hc y Hy) 0. The resul now follows from he enropic represenaion (20) of he indifference price. 8

9 3.2 The dual sochasic conrol problem We consider he dual problem in (19) from a sochasic conrol perspecive, o derive a PDE for H C (, s, y). We discuss he exisence of a sufficienly smooh soluion o his PDE, so ha he value funcion and indifference price are smooh enough for he hedging heorem o be valid. From he resuling PDE saisfied by he indifference price we shall derive, in he nex secion, payoff decomposiions and an indifference price represenaion equaion, leading o an asympoic expansion for he indifference price, for small values of risk aversion. Paramerise he measures Q P e,f via F-adaped processes ψ in he P -maringale Γ Q of (15), according o Γ Q = E( λ S B S ψ ẐS ), 0 T, where ẐS is a (P, F)-Brownian moion orhogonal o B S. By he Girsanov Theorem we have a wo-dimensional (Q, F)-Brownian moion ( B S,Q, ẐS,Q ) defined by B S,Q := B S + 0 λ S udu, Ẑ S,Q := ẐS + 0 ψ u du, 0 T, wih ψ = 0 corresponding o he minimal maringale measure Q M, so ha ẐS is also a (Q M, F)- Brownian moion. Then, for Q P e,f, we have [ ] [ ] E Q,s,y log ΓQ T Γ Q = E Q 1 T [ ],s,y ( λ S 2 u) 2 + ψu 2 du <, (21) where he inegrabiliy condiion on he righ hand side is associaed wih he finie enropy condiion and ensures ha he sochasic inegrals on he lef hand side have zero expecaion. Le Ψ denoe he se of inegrands ψ such ha (21) is saisfied. Then H C in (19) is he value funcion of he sochasic conrol problem H C (, s, y) := inf ψ Ψ EQ,s,y where, under Q P e,f, he sae variables S, Y follow [ 1 T [ ] ( λ S 2 u) 2 + ψu 2 du αc(y T ) ], (22) ds = σ S S,Q S d[ B, dy = σ Y Y ( λ Y ρ λ S 1 ρ 2 ψ )d + d ] Y,Q B, (23) and B Y,Q is a (Q, F)-Brownian moion given by B Y,Q = ρ B S,Q + 1 ρ 2ẐS,Q. The risk premia λ S, λ Y are funcions of he asse prices according o (13). The HJB equaion for H C is H C +A QM S,Y HC ( λ S ) 2 +min ψ [ 1 2 ψ2 1 ρ 2 σ Y yψh C y ] = 0, H C (T, s, y) = αc(y), (24) where A QM S,Y is he generaor of (S, Y ) under QM. The funcion H 0 corresponding o finding he minimal enropy measure Q E saisfies (24) wih erminal condiion H 0 (T, s, y) = 0. The opimal feedback conrol in (24) is ψ C, given by ψ C (, s, y) = 1 ρ 2 σ Y yh C y (, s, y). (25) and in paricular, he problem wih no claim gives he inegrand ψ E ψ 0 in he densiy of he minimal enropy measure: dq E dp = E( λ S B S ψ E ẐS ) T, where he process ψ E is given by ψ E = ψ E (, S, Y ), 0 T, wih ψ E (, s, y) = 1 ρ 2 σ Y yh 0 y(, s, y). (26) 9

10 Insering he opimal feedback conrol (25) back ino he Bellman equaion yields he PDE H C + A QM S,Y HC ( λ S ) (1 ρ2 )(σ Y yh C y ) 2 = 0, H C (T, s, y) = αc(y). Using he fac ha H 0 saisfies he same PDE wih zero erminal condiion, along wih he enropic represenaion (20) of he indifference price, we subrac he PDE for H 0 from ha for H C o yield ha he indifference price saisfies p + A QM S,Y p α(1 ρ2 )(σ Y yp y ) ρ2 σ Y yp y ψ E (, s, y) = 0, p(t, s, y) = C(y), where we have used (26). Using he fac ha he drif of Y under Q E is given by (23) wih ψ = ψ E, we recas he above PDE ino he form p + A QE S,Y p α(1 ρ2 )(σ Y yp y ) 2 = 0, p(t, s, y) = C(y). (27) For α = 0 his PDE becomes linear and he indifference price becomes he he so-called marginal uiliy-based price p E, given as an expecaion of he payoff under Q E, as is well-known: p E (, s, y) := lim p(, s, y) = E Q E α 0,s,yC(Y T ). Remark 4 (The case Q E = Q M ). In he case when v S 0 v Y 0, hen by (13), λ S loses all dependence on he non-raded asse price, he infimum in he dual problem (22) for C = 0 is achieved by ψ E = 0, and he MEMM coincides wih he minimal maringale measure, Q E = Q M. Then H 0 loses dependence on he non-raded asse price and (26) gives ψ E = 0, as i should. The indifference pricing PDE (27) hen becomes racable, as he unknown funcion ψ E (, s, y) in he generaor A QE S,Y is se o zero. We shall herefore focus in Secion 5 on he case Q E = Q M o obain more explici resuls. 3.3 Regulariy of he value funcion In his secion we discuss he required smoohness of he dual value funcion H C, and hence of he indifference price, required for he validiy of Theorem 1. We ouline how smoohness can be esablished in he case when v S 0 v Y 0. The argumens are idenical (bu wih modified formulae in places) when v S > v Y 0. The exisence of sufficienly smooh soluions o semi-linear PDEs of he ype (24) has been considered by Pham [26] and Benh and Karlsen [6], and similar echniques could in principle be used o esablish ha H C is indeed a classical soluion o (24). We do no pursue his here, bu insead follow Davis [10] and make he ransformaions S := 1 σ S log S, Y := 1 σ Y log Y, f(y) := C(exp(σ Y y)). The funcion H C expressed in he new variables is J, defined by he sochasic conrol problem [ ] 1 T [ J(, s, y) := inf ψ EQ L 2 (, S u ) + ψu 2 ] du + f(yt ) 2 S = s, Y = y, (28) where he funcion L(, s) is given by ransforming he formula for λ S from Proposiion 1 ino he new variables: ( λ S L(, s) = m 0 S ) m 0 2 σs + s. The dynamics of he sae variables in (28) are ds = 1 2 σs S,Q d + d B, dy = [a(, S, Y ) 1 ρ 2 Y,Q ψ ]d + d B, 10

11 where a(, s, y) is equal o λ Y ρ λ S 1 2 σy, ransformed o he new variables: [ λ Y a(, s, y) = b 0 ρλ S 0 (Y 0 ρs 0 ) + 1 ] b 0 2 (σy ρσ S ) + y ρs 1 2 σy, wih b 0 = m 0 (and hence b = m ) in he case where v0 S = v0 Y. Then a is Lipschiz in s, y for all [0, T ]. The poin of making his ransformaion is ha (28) is a sandard form of sochasic conrol problem, whose soluion can be shown o be given by a classical soluion of he HJB equaion J +a(, s, y)j y 1 2 σs J s J ss+ρj sy J yy L2 (, s)+min ψ [ 1 2 ψ2 ] 1 ρ 2 ψj y = 0, (29) wih J(T, s, y) = f(y). The salien feaure of his equaion is ha i is uniformly ellipic or parabolic 1 for ρ 2 1, and he proof of Lemma 1 and Theorem 2 in Davis [10] can be adaped o show ha he value funcion J is he unique classical soluion of (29), and hence ha he funcion H C (, s, y) is a classical soluion of (24), and hus he primal value funcion u C is smooh enough for he proof of Theorem 1 o be valid. 4 Payoff decomposiions and asympoic expansions 4.1 Payoff decomposiions and price represenaion We shall obain an asympoic represenaion of he indifference price valid for small values of risk aversion, following from payoff decomposiions and a price represenaion equaion. We work under he MEMM Q E, wih asse price dynamics given by (23) wih ψ = ψ E. Define he local (Q E, F)-maringale L by L := 1 ρ 2 σ Y 0 Y u p y (u, S u, Y u )dẑs,qe u, 0 T, (30) where ẐS,QE is a (Q E, F)-Brownian moion orhogonal o he Brownian moion B S,QE driving he sock under Q E. Noe ha from (25), (26), he enropic represenaion (20) of he indifference price and he inegrabiliy condiion (21), ha L is a (Q E, F)-maringale. We have he following decomposiion of he claim payoff. Lemma 1 (Payoff decomposiion). The claim payoff admis he decomposiion T C(Y T ) = p(, S, Y ) + θ u (H) ds u + L T L 1 2 α( L T L ), 0 T, (31) where θ (H) is he opimal hedging sraegy for he claim, given in Theorem 1. Remark 5. Mania and Schweizer [20] have obained a similar resul in a backward sochasic differenial equaion represenaion (see equaion (4.5) in Theorem 13 of [20]), in a more absrac model. The new feaure of Lemma 1 is ha he inegrand θ (H) and he Q E -maringale L are given in erms of derivaives of he indifference price. Naurally, we have been able o obain his because we are considering a more explici model, allowing he Iô formula and PDE mehods o be exploied. Remark 6. Lemma 1 is similar in spiri o Theorem 7 in Musiela and Zariphopoulou [23], bu our hedging sraegy depends, of course, on derivaives of he indifference price wih respec o boh sae variables (S, Y ), raher han he single variable Y of [23]. 1 In oher words, wriing he second order erms as a 11 J ss + 2a 12 J sy + a 22 J yy, we have a 2 12 a 11a

12 Proof of Lemma 1. This is esablished by compuing he differenial dp(, S, Y ) under Q E and using he PDE saisfied by p, o give dp(, S, Y ) = 1 2 α(1 ρ2 )[σ Y Y p y (, S, Y )] 2 d + p s (, S, Y )ds + σ Y Y,QE Y p y (, S, Y )d B, where B Y,QE is a (Q E, F)-Brownian moion driving he non-raded asse price. We have B Y,QE = ρ B S,QE + 1 ρ 2ẐS,QE, as well as ds = σ S S,QE S d B. Using hese relaions, along wih he opimal hedging formula of Theorem 1 and he definiion (30) of L, convers (32) o dp(, S, Y ) = 1 2 αd L + θ (H) ds + dl, and he resul follows by inegraing from o T. (32) Define he process L E as he α 0 limi of L, given by replacing he indifference price p wih he marginal price p E in he definiion (30) of L: L E := 1 ρ 2 σ Y 0 Y u p E y (u, S u, Y u )dẑs,qe u, 0 T. (33) We have he immediae corollary below, from seing α = 0 in Lemma 1, he Föllmer-Schweizer- Sondermann [13, 14] decomposiion of he payoff under Q E in our specific model. Corollary 1 (Föllmer-Schweizer-Sondermann decomposiion). The claim payoff admis he decomposiion T C(Y T ) = p E (, S, Y ) + θu E ds u + L E T L E, 0 T, (34) where p E is he marginal uiliy-based price of he claim, θ E is he marginal hedging sraegy for he claim, given by Theorem 1 wih p E in place of he indifference price, and L E is he process in (33). The following corollary of Lemma 1 follows by aking condiional expecaions of (31) under Q E, given (S, Y ) = (s, y). Corollary 2 (Indifference price represenaion). The indifference pricing funcion p : [0, T ] R 2 + has he represenaion p(, s, y) = p E (, s, y) + 1 2,s,y[ L αeqe T L ]. (35) Remark 7. A more absrac form of his resul appears in Mania and Schweizer [20] (see he las equaion before Theorem 18 in heir paper). Once again, he new feaure here is he explici idenificaion of he Q E -maringale L wih he derivaive of he indifference price according o (30). 4.2 Asympoic expansions Denoe by v(, s, y) := var QE,s,y[C(Y T )], he condiional variance of he claim payoff under Q E. Define he gains process G E associaed wih he marginal hedging sraegy by G E := 0 θ E u ds u, 0 T. The asympoic expansion for he indifference price o firs order in α is hen given by he following heorem. 12

13 Theorem 2. The indifference pricing funcion p(, s, y) has he asympoic expansion Proof. Wrie p(, s, y) = p E (, s, y) + 1 ( ) 2 α v(, s, y) E,s,y[ G QE E T G E ] + O(α 2 ). p(, s, y) = p E (, s, y) + αp (1) (, s, y) + O(α 2 ). Inser his expansion ino he price represenaion equaion (35), and use he definiion (30) of L, o obain αp (1) (, s, y)+o(α 2 ) = 1 2 α(1 ρ2 )(σ Y ) 2 E QE,s,y T Yu 2 Equaing erms of order α and using he definiion (33) of L E, we obain ( p E y (u, S u, Y u ) + αp (1) y (u, S u, Y u ) + O(α 2 )) 2 du. p (1) (, s, y) = 1 2 EQE,s,y[ L E T L E ]. (36) Now use he Föllmer-Schweizer-Sondermann decomposiion (34) o compue v(, s, y) = var QE,s,y[C(Y T )] = E,s,y[(C(Y QE T ) p E (, s, y)) 2 ] = E QE,s,y [ (G E T G E + L E L E ) 2]. The processes G E, L E are orhogonal Q E -maringales, so his becomes v(, s, y) = E QE,s,y[ G E T G E + L E T L E ], and insering his ino (36) gives he resul. Remark 8. Conras he expansion in Theorem 2 wih he corresponding resul in he full informaion case, which is obained from a Taylor expansion of a non-linear expecaion represenaion of he indifference price, and which is of he form (see [21], for insance) p FI (, y) = p E,FI (, y) α(1 ρ2 )v FI (, y) + O(α 2 ), (full informaion expansion) (37) (where he superscrip FI denoes full informaion). In his case, Q E = Q M, and he indifference price has no dependence on s. As a resul, L E,FI = ((1 ρ 2 )/ρ 2 ) G E,FI, and he variance of he payoff in he full informaion case is given by v FI (, y) := var QE,y [C(Y T )] [ = E QE,y L E,FI T L E,FI + G E,FI T G E,FI ] = 1 [ ρ 2 EQE,y G E,FI T G E,FI ]. The firs order erm in he price expansion of he heorem reduces o 1 ( 2 α v FI [ (, y) E QE,y G E,FI T G E,FI ] ) = 1 2 α(1 ρ2 )v FI (, y), in accordance wih (37). This expansion can be wrien in analyic form, so numerical compuaion for non-zero risk aversion is considerably easier in he compleely observable case han in he parial informaion model. 13

14 5 Analyic formulae when Q E = Q M From now on we specialise o he case when v0 S v0 Y, so ha λ S on he non-raded asse price, and Q E = Q M, as in Remark 4. For convenience, define he process ν by λ S (, S ) does no depend ν := λ Y ρ λ S, 0 T, wih ν 0 = λ Y 0 ρλ S 0. The dynamics of Y under Q M are dy = σ Y Y,QM Y ( ν d + d B ), Y,QM wih B a Q M -Brownian moion. The dynamics of ν under Q M follow from ransforming (14) from P o Q M, giving d ν = 1 ρ 2 b dẑs, where ẐS is a Q M -Brownian moion (and also a P -Brownian moion) perpendicular o he Brownian moion B S,QM driving he sock, relaed o B Y,QM by B Y,QM = ρ B S,QM + 1 ρ 2ẐS. (38) From Proposiion 1, when v0 S v0 Y, ν is a funcion of he curren asse prices, ν ν(, S, Y ), given by ν(, s, y) = ν 0 + b 0 (ξ Y (, y) ρξ S (, s)), (39) 1 + b 0 wih b 0 = m 0 in he case ha v S 0 = v S 0, and where ξ S (, s) and ξ Y (, y) are he funcions in (1) evaluaed a S = s and Y = y. The following resul shows ha log Y T is normal under Q M, wih a mean which depends on he curren asse prices, and a wih a ime-dependen variance. Theorem 3. Suppose v S 0 v Y 0. Under Q M, condiional on S = s, Y = y, wih log Y T N ( µ(, s, y), Σ 2 () ), µ(, s, y) = log y + σ Y ν(, s, y)(t ) 1 ( σ Y ) 2 (T ), 2 Σ 2 () = [ 1 + (1 ρ 2 )b (T ) ] (σ Y ) 2 (T ), (40) where ν(, s, y) is given in (39) and b is given in (9). (When v0 S = v0 Y wih b = m, for 0 T.) he same formulae hold Proof. We use he SDEs for Y and ν under Q M. Applying he Iô formula o log Y under Q M, we obain T log Y T = log Y + σ Y ν u du 1 T 2 (σy ) 2 (T ) + σ Y From he dynamics of ν under Q M we have ν u = ν + u 1 ρ 2 b r dẑs r, u T. Hence, afer changing he order of inegraion in a double inegral, we have T ν u du = ν (T ) + T 1 ρ 2 b u (T u)dẑs u. Y,QM d B u, 0 T. (41) 14

15 Insering his ino (41) and using (38) gives log Y T = log Y + σ Y [ ν (T ) + 1 ρ 2 T 1 2 (σy ) 2 (T ), 0 T. ] T (1 + b u (T u)) dẑs S,QM u + ρ d B u The sochasic inegrals are orhogonal Gaussian processes wih zero condiional expecaion given (S, Y ) = (s, y), from which he disribuion of log Y T follows. Using his disribuion we obain BS-syle formulae for he marginal price wih a dividend rae ha depends on he asse prices, and wih a ime-dependen volailiy. We also ge an analyic formula for he condiional variance v(, s, y) in Theorem 2. Denoe by µ j he j h momen of he payoff under Q M : [ µ j (, s, y) := E QM,s,y C j (Y T ) ], T, j N. The marginal pricing funcion p E (, s, y) = E QM,s,yC(Y T ) is given by p E (, s, y) = µ 1 (, s, y), and he variance v(, s, y) := var QM,s,y[C(Y T )] is given by v(, s, y) = µ 2 (, s, y) µ 2 1(, s, y). (42) For a pu opion of srike K, he firs and second momens are given by he following formulae. Lemma 2. Wih µ(, s, y) and Σ() as in Theorem 3, define q q(, s, y) by µ(, s, y) = log y q(, s, y) 1 2 Σ2 (). The marginal price a ime [0, T ] of a pu opion wih payoff (K Y T ) + is p E (, S, Y ), given by p E (, s, y) = KΦ( d 1 (, s, y) + Σ()) y exp( q(, s, y))φ( d 1 (, s, y)), [ 1 ( y ) d 1 (, s, y) := log q(, s, y) + 1 ] Σ() K 2 Σ2 (), where Φ( ) denoes he sandard cumulaive normal disribuion funcion. The second momen of he payoff under Q M is given by µ 2 (, s, y) = K 2 Φ( d 1 (, s, y) + Σ()) 2Ky exp( q(, s, y))φ( d 1 (, s, y)) + y 2 exp(σ 2 () 2q(, s, y))φ( d 1 (, s, y) Σ()). To implemen he opimal hedging sraegy we need he derivaives wih respec o s and y of he indifference price. Approximaing he indifference price by he asympoic expansion of Theorem 2, we obain he derivaives of p E, and of he variance v. The calculaions are similar (bu no idenical o) hose in [21, 22], bu some nice cancellaions occur o give formulae wih a similar flavour o hose in [21, 22]. Lemma 3. The marginal hedging sraegy for a shor pu posiion is θ E (, s, y) = ρ σy σ S y s exp( q(, s, y))φ( d 1(, s, y)). 15

16 The derivaives of he variance in (42) saisfy v s (, s, y) + ρ σy σ S y s v y(, s, y) = 2ρ σy y [ σ S s e q(,s,y) KΦ( d 1 (, s, y)) y exp(σ 2 () q(, s, y))φ( d 1 (, s, y) Σ()) ] 2p E (, s, y)θ E (, s, y). The final erm we need o implemen he opimal hedge wih he asympoic expansion of Theorem 2 is he quadraic variaion erm involving G E. Using he explici formula for he marginal price, his erm is given as follows. Lemma 4. The expecaion under Q M of he quadraic variaion G E for a pu opion saisfies E,s,y[ G QM E T G E ] = ( ρσ Y ) T 2 E Q M,s,y [Y u exp( q(u, S u, Y u ))Φ( d 1 (u, S u, Y u ))] 2 du. This expression will be evaluaed numerically by Mone-Carlo simulaion, as will is derivaives wih respec o s and y. 6 Numerical resuls and experimens We conduced exensive numerical invesigaions ino he parial informaion valuaion and hedging scheme. An iniial gauge of he effec of parameer uncerainy on he valuaion and hedging of he claim is given in Table 2, showing pu opion prices and hedging sraegies a ime zero from he parial informaion model and he full informaion model, for varying correlaion and for risk aversions α = 0 and α = We se he means of he prior equal o he rue risk premia. These and oher parameers are shown in Table 1. The opion prices are higher in he parial informaion case as he agen incorporaes he risk from parameer uncerainy ino a higher effecive volailiy, given by (40). This effec does no ranslae o he hedges, in ha he agen generally akes a bigger shor posiion wih full informaion, excep for high correlaion and nonzero risk aversion. This is a firs indicaion ha he bigges effec of parameer uncerainy on uiliy-based mehods is incorporaed ino he valuaion of he claim, as opposed o he hedge. This will be borne ou by simulaion resuls which follow, and is no alogeher surprising. Uiliy-based valuaion insiss on achieving unchanged uiliy only when selling he opion a he indifference price, and no necessarily by rading judiciously wih an opion premium ha is a odds wih he agen s objecive. Noe he dependence of he resuls on he Q M -drif of Y, given by σ Y ν in he parial informaion case, and by he corresponding quaniy wih he rue risk premia in he full informaion case. The negaive of his drif acs as a dividend yield in he opion pricing formulae, he pu opion premia increase as his drif decreases, and he hedge raios become larger in absolue value. This is refleced in he lower half of Table 2. We hen conduced simulaion experimens on he hedging of he claim. Using a given prior disribuion λ i 0, v0, i i = S, Y, eiher chosen or esimaed randomly from simulaed daa over [ i, 0], i = S, Y (and seing λ i 0 o he poin esimaes of he risk premia, wih v0 i = 1/ i ), we hedged a shor posiion in a pu opion of srike K over [0, T ], using some fixed rebalancing inerval δ (we used δ = 1/252, one rading day in all resuls we repor). The prior disribuion was updaed over [0, T ] and he claim was hedged using he opimal sraegy of Theorem 1 wih he indifference price approximaed by he asympoic expansion of Theorem 2, or by is α 0 limi, he marginal price. We generaed a erminal hedging error over he simulaed pahs and repeaed his over many price pahs o produce a hedging error disribuion, and compared his disribuion wih ha produced from alernaive sraegies, namely: The uiliy-based hedge which does no incorporae he learning from filering. This uses he indifference hedging formulae of he compleely observable incomplee model wih 16

17 Table 1: Parial and full informaion (FI) pu opion prices and hedge raios a ime zero, for risk aversion α = 0 (indexed wih superscrip E) and α = 0.01 from he firs order in α expansions (indexed wih superscrip 1). The parameers in he upper half of he able are as in Table 1, and in he lower half of he Table we have λ Y = λ Y 0 = The BS price is 9.95 and he BS hedge is ρ p E p 1 p E,FI p 1,FI θ E θ 1 θ E,FI θ 1,FI Table 2: Parameers for prices and hedge raios in Table 2 S 0 Y 0 K T λ S σ S λ Y σ Y λ S 0 λ Y 0 v S 0 v Y year he asse drifs se o heir iniial values λ S 0, λ Y 0, and kep fixed hroughou he hedging ime-frame. To order α he indifference price a [0, T ] is given by (37), where we ake he drif of Y under Q M o be σ Y (λ Y 0 ρλ S 0 ). The opimal hedging sraegy is given by Theorem 1 (wih, of course, no derivaive wih respec o s). The BS-syle hedge which assumes ha S is a perfec proxy for Y, given by θ BS = σy Y σ S S y BS(, Y ; σ Y ), 0 T. (43) where BS(, y; σ Y ) denoes he BS formula wih underlying asse price y and volailiy σ Y. A varian of he BS hedge proposed by Hulley and McWaler [16], which muliplies he hedge in (43) by he correlaion ρ. In effec, his approximaes he Q M -drif of Y by zero. We also carried ou he ess using he full informaion hedge, wih he drif of Y under Q M aken o be is rue value σ Y (λ Y ρλ S ) =: σ Y ν. This sraegy is no available in realiy o he agen, bu we include i as a calibraion. I is no guaraneed o produce he bes resuls, as any finie sample of daa may no reflec he rue drifs of he asses. The iniial wealh was se o he ime zero BS price for all he sraegies. We also compued he hedging error when saring wih he appropriae opion premium corresponding o he hedging program, o assess he effec of valuaion as well as he benefis or oherwise of he hedging sraegy. Firs, we presen resuls in which we used he marginal hedging sraegies (wih or wihou filering) for he uiliy-based mehods. In his case analyic formulae were available for all quaniies involved, and wih he ensuing fas compuaion we carried ou exensive simulaions over a range of scenarios, a represenaive sample of which we repor. Table 3 shows summary saisics for he hedging error disribuions when he prior risk premia were se equal o he rue risk premia, λ i 0 = λ i, for i = S, Y. Of course, in his case, he 17

18 full informaion resuls will be idenical o hose wihou filering. We use his as a base case and hen vary he prior o illusrae how he benefis or oherwise of he parial informaion approach vary wih he prior. The hedge wih filering gives a higher mean, median and expeced uiliy han he oher sraegies. The sandard deviaion is higher han ha of he unfilered hedge and he correlaionweighed BS hedge. The BS hedge is he wors performer, and is massively improved by weighing i by he correlaion. Wih he parameers used in he upper half of he able, he rue drif of Y under he minimal measure is posiive (we have ν := λ Y ρλ S = 0.05). Under hese condiions he uiliy-based hedges end o generae a posiive erminal wealh by under-hedging (ha is, aking a small shor posiion), as he opion ends o end up ou of he money. The correlaion-weighed BS hedge in effec approximaes he Q M -drif of Y drif by zero, so lower han he rue value. I hen ends o over-hedge (ha is, ake a larger a shor posiion han he uiliy-based hedges) and hus under-perform. Conversely, if we change he risk premium of Y o λ Y = 0.325, hen he Q M -drif of Y becomes negaive, and we obain he resuls in he lower half of Table 3 (over a fresh se of simulaed pahs). Here, he correlaion-weighed BS hedge ouperforms he uiliy-based hedge wihou learning, bu once again he filering has improved he performance so ha i urned ou o be he bes sraegy in erms of mean, median and expeced uiliy. If we incorporae he effec of valuaion, and sar he hedging programs a he indifference price implied by he hedging mehod, hen he filering procedure massively ouperforms he oher mehods. We found his o be rue in virually all our simulaion resuls. These iniial resuls indicae ha he filering procedure can improve he performance of he uiliy-based mehod wihou filering, regardless of iniial wealh, and regardless of he fac he he unfilered hedge used he rue values of he risk premia. The laer poin reflecs he fac ha asse daa over any finie ime period may well no reflec he rue values of he drifs (he noorious difficuly of drif parameer uncerainy) and he updaing of filering has couneraced his o some exen. The qualiy of sraegies ha do no use filering appear o be relaed o he perceived Q M -drif of Y relaive o is rue value. This is indicaed by he resuls in Table 4. In he upper half of he able, he agen who does no incorporae filering perceives he Q M -drif of Y as greaer han he rue value, hough boh are posiive. The non-filered sraegy underhedges, and when he opion is unlikely o be exercised, his is a successful policy (when all hedges begin wih common iniial wealh). The improvemen in hedging offered by filering is now less pronounced (he mean hedging error wih filering is larger han wihou, bu he median is no). A cavea o his is ha if agens use heir respecive opion premia as he iniial wealh, hen he agen who does no incorporae learning will suffer. Similarly, he correlaionweighed BS hedge approximaes he Q M -drif of Y by zero, and does no perform so well in his case. Overall, he filering procedure appears o be of benefi. This is suppored by he resuls in he lower half of he able. Now he agen who does no filer perceives he Q M -drif of Y as negaive, and he correlaion-weighed BS hedge ouperforms he uiliy-based hedge wihou filering, bu if we incorporae filering, hen he uiliy-based mehod has an improved performance. Occasionally, we found ha he filering procedure was no beneficial, usually when he prior gave an exremely poor esimae of ν := λ Y ρλ S. The filering hen appears o be of limied use in improving he hedge, and relies on is increased valuaion of he claim o be a all effecive. To his end, he upper half of Table 5 shows resuls in he case ha he risk premium of Y is iniially badly over-esimaed and he risk premium of S is badly under-esimaed. In his scenario, ν = 0.375, while he iniial value of his quaniy in he prior is ν 0 = The non-filering agen perceives he pu as much less risky han in realiy, and his hedging sraegy produces losses. Bu he prior is so poor ha he filering procedure fails o updae i drasically enough o change his percepion, and he filered hedge is also poor. The filering mehod performs beer han he non-filered hedge if he agens incorporae heir iniial valuaions of he opion ino he iniial wealh, bu in his case boh sraegies do worse han he correlaionweighed BS hedge. If we improve he prior a lile, we ge he resuls shown in he lower half 18

19 Table 3: Hedging error saisics as a fracion of he iniial wealh (he BS price a ime zero), over 40, 000 asse price pahs, wih he risk premia of he prior se o be equal o he rue risk premia. The uiliy-based hedges used he marginal (α = 0) hedging sraegies. Parameers in he upper half of he able are as in Table 1, δ = 1/252, and ρ = The ime zero opion premia are p E 0 = 11.93, p E,NF 0 = p E,FI = 9.39 (NF denoing no filering), p BS 0 = Figures in parenheses show resuls obained if he opion premium corresponding o he hedging mehod was used as he iniial wealh (so sandard deviaions and he BS-relaed saisics are unchanged), again as a percenage of he ime zero BS price. The las column of he able is he expeced uiliy of he erminal wealh wih random endowmen. In he lower half of he Table we se λ Y 0 = λ Y = 0.325, wih all oher parameers as before. In his case he iniial uiliy-based opion premia are p E 0 = 12.96, p E,NF 0 = p E,FI = Mean SD Median EU(X T C(Y T )) Filered hedge (0.2637) (0.4337) ( ) Non-filered hedge ( ) (0.1654) ( ) BS hedge ρ.bs hedge Full informaion hedge ( ) (0.1654) ( ) Wih λ Y = λ Y 0 = Filered hedge (0.2628) (0.4316) ( ) Non-filered hedge (0.0026) (0.1703) ( ) BS hedge ρ.bs hedge Full informaion hedge (0.0026) (0.1703) ( ) Table 4: Hedging error saisics wih λ Y 0 = (upper able) and λ Y 0 = (lower able). The uiliy-based hedges used α = 0, and all oher parameers as in Tables 1 and 3. The ime zero opion premia are for he upper able are p E 0 = 10.94, p E,NF 0 = 8.34, p E,FI = 9.39, wih he BS price and he full informaion price as in Table 3. For he lower able we have p E 0 = 12.96, p E,NF 0 = Wih λ Y 0 = Mean SD Median EU(X T C(Y T )) Filered hedge (0.1652) (0.3476) ( ) Non-filered hedge ( ) (0.0895) ( ) BS hedge ρ.bs hedge Full informaion hedge ( ) (0.1632) ( ) Wih λ Y 0 = Filered hedge (0.3593) (0.5224) ( ) Non-filered hedge (0.0966) (0.2456) ( ) BS hedge ρ.bs hedge Full informaion hedge ( ) (0.1645) ( ) 19