Carol Alexander ICMA Centre, University of Reading. Aanand Venkatramanan ICMA Centre, University of Reading

Transcription

1 Analyic Approximaions for Spread Opions Carol Alexander ICMA Cenre, Universiy of Reading Aanand Venkaramanan ICMA Cenre, Universiy of Reading 15h Augus 2007 ICMA Cenre Discussion Papers in Finance DP Copyrigh 2007 Alexander and Venkaramanan. All righs reserved.! "#$%&'*+*, -#$%&',*+.#/// # "7" 1

2 ABSTRACT Even in he simple case ha wo price processes follow correlaed geomeric Brownian moions wih consan volailiy no analyic formula for he price of a sandard European spread opion has been derived, excep when he srike is zero in which case he opion becomes an exchange opion. This paper expresses he price of a spread opion as he price of a compound exchange opion and hence derives a new analyic approximaion for is price and hedge raios. This approximaion has several advanages over exising analyic approximaions, which have limied validiy and an indeerminacy ha renders hem of lile pracical use. Simulaions quanify he accuracy of our approach and demonsrae he indeerminacy and inaccuracy of oher analyic approximaions. The American spread opion price is idenical o he European opion price when he wo price processes have idenical drifs, and oherwise we derive an expression for he early exercise premium. A pracical illusraion of he model calibraion uses marke daa on American crack spread opions. 1 JEL Code: C02, C29, G12 Keywords: Asse pricing, Spread opions, Exchange opions, American Opions Carol Alexander Chair of Risk Managemen and Direcor of Research, ICMA Cenre, Universiy of Reading, Reading, RG6 6BA, UK. c.alexander@icmacenre.rdg.ac.uk Aanand Venkaramanan PhD Suden, ICMA Cenre, Universiy of Reading, Reading, RG6 6BA, UK. a.venkaramanan@icmacenre.rdg.ac.uk 1 We would like o hank Prof. Thorsen Schmid of he Dep. of Mahemaics, Leipzig Universiy for very useful commens on a earlier draf of his paper.

3 1. INTRODUCTION A spread opion is an opion whose pay-off depends on he price spread beween wo correlaed underlying asses. If he asse prices are S 1 and S 2 he payoff o a spread opion of srike K is [ωs 1 S 2 K, 0} where ω = 1 for a call and ω = 1 for a pu. Early work on spread opion pricing by Ravindran [1993], Shimko [1994], Kirk [1996] assumed each forward price process is a geomeric Brownian moion wih consan volailiy and ha hese processes have a consan nonzero correlaion: we label his he 2GBM framework for shor. The 2GBM assumpion allows a simple analyic approximaion for he spread opion price by reducing he dimension of he uncerainy from wo o one. However, as we shall explain below, hese approximaions suffer from an indeerminacy ha renders hem pracically useless. The 2GBM framework is racable bu i capures neiher he implied volailiy smiles ha are derived from marke prices of univariae opions nor he implied correlaion smile ha is eviden from marke prices of spread opions. In fac correlaion frowns raher han smiles are a prominen feaure in spread opion markes. This is because he pay-off o a spread opion decreases wih correlaion. Hence if marke prices of ou-of-he-money call and pu spread opions are higher han he sandard 2GBM model prices wih consan correlaion he implied correlaions ha are backed ou from he 2GBM model wih have he appearance of a frown. Alexander and Scourse [2004] derive approximae analyic prices of European spread opions on fuures or forward conracs ha display boh volailiy smiles and a correlaion frown. They assume he asse prices have a bivariae lognormal mixure disribuion and hence obain prices as a weighed sum of four differen 2GBM spread opion prices, each of which may be obained using an analyic approximaion such as ha of Kirk [1996]. However, mos spread opions are raded on asses ha pay dividends or have carry coss. For insance spread opions on equiy indices and opions on commodiy spreads are common. And in mos cases he opions are American, as is he case for he crack spread opions ha we consider laer in his paper. Numerical approaches o pricing and hedging spread opions ha are boh realisic and racable include Carr and Madan [1999] and Dempser and Hong [2000] who advocae models ha capure volailiy skews on he wo asses by inroducing sochasic volailiy o he price processes. And he addiion of price jumps can explain he implied correlaion frown, as in he spark spread opion pricing model of Carmona and Durrleman [2003a]. However pricing and hedging in his framework necessiaes compuaionally inensive numerical resoluion mehods such as he fas Fourier ransform. Oher models provide only a range for spread opion prices, as in Durrleman [2001] and Carmona and Durrleman [2005], who provide upper and lower bounds ha can be very narrow for cerain parameer values. For a deailed survey of hese models and a comparison of heir performances, he reader is referred o Carmona and Durrleman [2003b]. Recenly Li e al. [2006] define a bivariae normal process for he underlying asses and express he price as an expecaion of he ransformed payoff. Even reaining he simpliciy of he 2GBM framework an exac analyic price for a spread opion wih non-zero srike is elusive. In his paper we express he price as ha a compound exchange opion and hus derive a new analyic approximaion for he price and hedge raios of a spread opion. Our approximaion always provides a unique and close approximaion o he exac price, i is easy o calibrae and hedge raios are simple o compue. By conras, oher analyic approximaions are only valid for spread opions of cerain srikes and he calibraed opion price is no Copyrigh 2007 Alexander and Venkaramanan 1

4 unique. The ouline of his paper is as follows: Secion 2 provides he background o our work, beginning wih a summary of he exchange opion pricing formula of Margrabe [1978] since his is cenral o our model. We also provide a derivaion of he approximaion saed in Kirk [1996] since his is no available in he lieraure, and we exend he approximaion o allow for non-zero dividends or carry coss. Secion 3 derives he compound exchange opion represenaion, he analyic approximaion o he price and hedge raios of spread opions and remarks on he model calibraion. A simulaion exercise demonsraes he accuracy of our approximaion compared wih ha of Kirk [1996]. Secion 4 exends he framework o accommodae early exercise and here an empirical demonsraion of he superioriy of our analyic approximaion is based on he pricing and hedging performance for American crack spread opions raded on he New York Mercanile Exchange NYMEX during The final secion summarises our resuls and concludes. 2. BACKGROUND Here we assume ha he wo underlying asse prices follow correlaed geomeric Brownian moions wih consan volailiies and consan correlaion. We presen Margrabe s formula for he price a European exchange opion and he approximae pricing formulae for spread opions wih non-zero srike ha are in common use Margrabe s Exchange Opion Pricing Formula When he srike of he spread opion is zero he opion is called an exchange opion, since he buyer has he opion o exchange one underlying asse for he oher. The fac ha he sirke is zero allows one o reduce he pricing problem o a single dimension, using one of he asses as numeraire. If S 1, and S 2, are he spo prices of wo asses a ime hen he payoff o an exchange opion a he expiry dae T is given by [S 1,T S 2,T ] +. Bu his is equivalen o an ordinary call opion on x = S 1, /S 2, wih uni srike. Assume ha he risk-neural price dynamics are governed by wo correlaed geomeric Brownian moions wih consan volailiies given by: ds i, = r q i S i, d + σ i S i, dw i, i = 1, 2 1 where, W 1, and W 2, are Wiener processes under risk neural measure, r is he assumed consan risk-free ineres rae and q 1 and q 2 are he assumed consan dividend yields of he wo asses. The volailiies σ 1 and σ 2 are also assumed o be consan as is he reurns correlaion: dw 1,, dw 2, = ρd Using risk-neural valuaion he price of an exchange opion is given by P = E Q { e rt [S 1,T S 2,T ] +} = e rt E Q {S 2,T max {x T 1, 0}} where x follows he process dx = q 2 q 1 x d + σx dw Copyrigh 2007 Alexander and Venkaramanan 2

5 wih dw = ρ dw 1, + 1 ρ 2 dw 2, σ = σ1 2 + σ2 2 2ρσ 1 σ 2 Since he boh he asses grow a he risk-free rae, he relaive drif of S 1 wih respec o S 2 due o r is zero. Bu as he dividend yields of he asses may be differen, x drifs a he rae of q 2 q 1. Margrabe [1978] shows ha under hese assumpions he price P of an exchange opion is given by P = S 1, e q 1T Φd 1 S 2, e q 2T Φd 2 2 where Φ denoes he sandard normal disribuion funcion and d 1 = ln S1, S 2, + q 2 q 1 + 1σ2 T 2 σ T d 2 = d 1 σ T σ = σ1 2 + σ2 2 2ρσ 1 σ Analyic Approximaions o Spread Opion Prices and Hedge Raios In his secion we review analyic approximaions o spread opion prices and hedge raios in he lieraure raher han he numerical approximaion mehods described in he inroducion. Analyic approximaions are preferred over numerical echniques like fas-fourier ransform, PDE mehods and rees for heir compuaional ease and availabiliy of closed form formulae for hedge raios. Kirk [1996] presens an approximae formula for pricing European spread opions on fuures or forwards. The mehod exends ha of Margrabes o non-zero bu very small srike values. When K S 2, he displaced diffusion process S 2, + K can be assumed o be approximaely log-normal. Then, he raio beween S 1, and S 2, + Ke rt is also approximaely log-normal and can be expressed as a geomeric Brownian moion process. We ouline he main seps of he derivaion of he formula in appendix A. Rewrie he pay-off o he European spread opion as: [ ] + [ωs 1,T S 2,T K] + S1,T = K + S 2,T ω 1 K + S 2,T = K + S 2,T [ωz T 1] + where ω = 1 for a call and ω = 1 for a pu, Z = S 1, Y and Y = S 2, + Ke rt. The price f a ime for a spread opion on S 1 and S 2 wih srike K, mauriy T and payoff [ωs 1 S 2 K] + is given by: { f = E Q Y e rt max {ωz T 1, 0} } = ω S 1, e q 1T Φ ωd 1Z Ke rt + S 2, e r r+ q 2 T Φ ωd 2Z 3 Copyrigh 2007 Alexander and Venkaramanan 3

6 where d 1Z = ln Z + r r + q 2 q σ2 T σ T d 2Z = d 1Z σ T 2 σ = σ1 2 + S2, S2, σ2 2 2ρσ 1 σ 2 Y Y A slighly modified represenaion of Kirk s formula is P = P Ke rt + S 2, = ω Z e q 1T Φωd 1Z e r r+ q 2T Φωd 2Z 4 This represenaion reduces he dimension of he pricing problem from wo o one, which is useful when we exend he formula o price American spread opions in secion 4. The price hedge raios for Kirk s approximaion are sraighforward o derive from equaion 3. Le z x denoes he dela of y wih respec o x and Γ z xy denoe he gamma of z wih respec o x and y. The wo delas and pure gammas are quie similar o ha of Black-Scholes : f S 1 = ωe q 1T Φωd 1Z f S 2 = ωe q 1T Φωd 2Z Γ f S 1 S 1 = e q 1T φd 2Z S 1, σ T Γ f S 2 S 2 = e q 1T φd 2Z Ke rt + S 2, σ T The cross gamma, i.e., he second order derivaive of price wih respec o boh he underlying asses is given by Γ f S 1 S 2 = e q 1T φd 1Z Ke rt + S 2, σ T = e r r+ q 2T φd 2Z S 1, σ T Under he 2GBM assumpion oher price approximaions can derived ha are similar o Kirk s approximaion in ha hey reduce he dimension of he uncerainy from wo o one 2. For insance le S = S 1, e q 1T S 2, e q 2T and choose an arbirary M >> max {S, σ }. Then an analyic spread opion price approximaion based on an approximae lognormal disribuion for M + S is: f = M + S Φd 1M M + Ke rt Φd 2M where d 1M = ln M+S M+K + r q σ2 T σ T d 2M = d 1M σ T σ = σ1 2 + / σ2 2 2ρσ 1 σ 2 M + S 2 see Eydeland and Wolyniec [2003] Copyrigh 2007 Alexander and Venkaramanan 4

7 To avoid arbirage a spread opion mus be priced consisenly wih he prices of opions on S 1 and S 2. This implies seing σ i in 1 equal o he implied volailiy of S i for i = 1, 2. Then he implied correlaion is calibraed by equaing he model and marke prices of he spread opion. Alhough he 2GBM model assumes consan volailiy he marke implied volailiies are no consan wih respec o srike. So he srikes K 1 and K 2 a which he implied volailiies σ 1 and σ 2 are calculaed have a significan influence on he resuls. In Kirk s approximaion here is an indeerminacy ha arises from he choice of srike for he single asse implied volailiies and he calibraed value of he implied correlaion of a spread opion wih srike K will no be independen of his choice. The problem wih spread opion price approximaions such as Kirk s is ha he implied volailiy and correlaion parameers are ill-defined. There are infiniely many pairs S 1,, S 2, for which S 1, S 2, = K and hence very many possible choices of K 1 and K 2. Similarly here are infiniely many combinaions of marke implied σ 1, σ 2, and ρ ha yield he same σ in equaion 3. Hence he consrucion does no lead o a unique price for he opion. To calibrae he model some convenion needs o be applied. We have ried using he single asse s a-he-money ATM forward volailiy o calibrae spread opions of all srikes, and several oher convenions. None of hese gave reasonable resuls. This poins o anoher major drawback of he approximaions: for large values of K, he log-normaliy approximaion does no hold and neiher do he assumpions of consan drif and volailiy parameers. Hence he formulae have limied validiy. To our knowledge we do no know of any approximaion oher han ours ha is free of a srike convenion. 3. SPREAD OPTIONS AS COMPOUND EXCHANGE OPTIONS In his secion we derive a represenaion of he price of a spread opion as he sum of he prices of wo exchange opions. These exchange opions are: o exchange a call on one asse wih a call on he oher asse, and o exchange a pu on one asse wih a pu on he oher asse. We reain he 2GBM assumpion since our purpose is o compare our approximaion wih Kirk s approximaion. Our approximaion arises because we assume ha he call and pu opions in he exchange opions have consan volailiy. The accuracy of he approximae prices is quanified by simulaion The Compound Exchange Opion Represenaion Le Θ, F, F 0, Q be a filered probabiliy space, where Θ is he se of all possible evens θ such ha S 1,, S 2, [0,, F 0 is he filraion produced by he sigma algebra of he price pair S 1,, S 2, 0 and Q is a bivariae risk neural probabiliy measure and L = {θ Θ : ω S 1,T S 2,T K 0} A = {θ Θ : S 1,T mk 0} B = {θ Θ : S 2,T m 1K 0} The payoff o he spread opion of srike K a ime T can be wrien 1 L ω [S 1,T S 2,T K] = 1 L ω [S 1,T mk] [S 2,T m 1K] = 1 L ω 1 A [S 1,T mk] 1 B [S 2,T m 1K] A [S 1,T mk] 1 1 B [S 2,T m 1K] Copyrigh 2007 Alexander and Venkaramanan 5

8 where m is any posiive real number. Since a European opion price a ime depends only on he erminal price densiies, we have f = e rt E Q {ω1 L [S 1,T S 2,T K]} { = e rt E Q ω1 L 1 A [S 1,T mk] 1 B [S 2,T m 1K] } A [S 1,T mk] 1 1 A [S 2,T m 1K] = e rt E Q {ω 1 L A [S 1,T mk] 1 L B [S 2,T m 1K] } L B [m 1K S 2,T ] 1 1 L A [mk S 1,T ] { [ = e rt E Q ω [S 1,T mk] + [S 2,T m 1K] +] } + { + e rt E Q [ω [m 1K S 2,T ] + [mk S 1,T ] + ] +} = e rt E Q { [ω [U1,T U 2,T ]] +} + E Q { [ω [V2,T V 1,T ]] +} 5 where U 1,T, V 1,T are pay-offs o European call and pu opions on asse 1 wih srike mk and U 2,T, V 2,T on asse 2 wih srike m 1K respecively. This shows ha a spread opion is exacly equivalen a compound exchange opion CEO on wo call opions, wih prices U 1, and U 2, and wo pu opions wih prices V 1,T and V 2,T. We now describe he processes of he wo call and pu opions U i, and V i,, i = 1, 2. From 1: Tha is where Similarly where du i, = U i, d + U i, ds i, + 1 S i, 2 2 U i, ds 2 Si, 2 i, = Ui, U i, + rs i, + σ 2 i S 2 2 U i, U i, i, d + σ S i, Si, 2 i S i, dw i, S i, = ru i, d + σ i S i, Ui, dw i, 6 du i, = ru i, d + ξ i U i, dw i, 7 ξ i = σ i S i, U i, U i, S i, dv i, = rv i, d + η i V i, dw i, 8 η i = σ i S i, V i, V i, S i, 3.2. A New Analyic Approximaion In his secion we make he approximaion ha ξ i and η i are consan hroughou [, T] wih { } F Ui, ξ i = σ i E Q X i,s S i, { } F η i = σ i E Q Y i,s V i, 9 S i, Copyrigh 2007 Alexander and Venkaramanan 6

9 where X i, = S i, U i,, Y i, = S i, V i,, and s [, T]. Under he 2GBM assumpion for he spread opion s underlying prices he exchange opion price disribuions will no be lognormal. However hey will be approximaely lognormal if he opion remains deep in-he-money ITM or deep ou-of-he-money OTM unil expiry. An inuiive explanaion of his is ha he he price of a deep ITM is a linear funcion of he relaive price of he wo underlying asses and under he 2GBM assumpion he relaive price disribuion is lognormal. The price of a deep OTM exchange opion is approximaely zero. For a mahemaical jusificaion of his assumpion, noe ha { X i, and } Y i, are maringales { and } under he equivalen maringale measure Q we have X i, = E Q Xi,s F and Yi, = E Q Yi,s F. Hence σ Xi = ξi 2 + σi 2 2ξ i σ i = ξ i σ i So σ Xi = σ i X i, Ui, 1, and σ Yi = σ i Yi, Vi, Hence as σ Xi 0, he ne quadraic variaion T dx i,s, dx i,s = T σ 2 Xi ds 0 and X i,s X i, for s [, T]. The exchange opions ha are used o consruc our pricing formula do no need o have raded prices. We calibrae he model o he marke prices of spread opions only. All four exchange opions are deermined by he single parameer m and we may choose his o minimise he volailiies of X i, and Y i, and, as a resul, minimize he approximaion error. Therefore we express ξ i and η i as in equaion 9 and assume hem o be consan. We now apply Margrabe s formula o derive an analyic price for he spread opion as he price of a compound exchange opion on wo calls and pus. The risk neural price of he spread opion a ime is given by: f = e rt E Q { [ω U1,T U 2,T ] +} + e rt E Q { [ω V2,T V 1,T ] +} 11 so is price may be obained using Margrabe s formula: where f = e rt ω [U 1, Φω d 1U U 2, Φω d 2U V 1, Φ ω d 1V V 2, Φ ω d 2V ] 12 d 1A = ln A1, A 2, + q 2 q A σ2 T σ A T d 2A = d 1A σ A T 13 and σ U = σ V = ξ1 2 + ξ2 2 2ρξ 1 ξ 2 η1 2 + η2 2 2ρη 1 η 2 The prices of opions U i and V i are given by: U i, = S i, e q i T Φ d 1i K i e rt Φ d 2i V i, = K i e rt Φ d 2i S i, e q i T Φ d 1i 14 Copyrigh 2007 Alexander and Venkaramanan 7

10 where d 1i = ln Si, K i + r q i σ2 i T σ i T d 2i = d 1i σ i T where K 1 = mk and K 2 = m 1K. Hence he four vanilla opions ha are used in he model calibraion are deermined by he parameer m which is calibraed m o minimize he volailiies of X i, and Y i,. Under he assumpion of complee markes here exis a leas wo opion price pairs {U 1,, U 2, } and {V 1,, V 2, } such ha 12 holds. Noe ha in pracice he calibraed value of m will depend on he srike and mauriy of he spread opion. In equaion 12 here are wo erms on he righ hand side, one represening he discouned expeced pay-off o he exchange opion wih pay-off [U 1,T U 2,T ] + and he oher represening he discouned expeced pay-off o he exchange opion wih pay-off [V 2,T V 1,T ] +. To see his, noe ha for a call spread opion: f = e rt Φ d 1U U 1, Φ d 2U U 2, + e rt V 2, Φ d 2V Φ d 1V V 1, Φ d 2U Φ d 1V 15 Φd where Φ d 2U is he risk neural probabiliy ha U 1,T > U 2,T and U 1U 1, is he condiional expecaion of U 1,T given U 1,T > U 2,T. Similarly, Φ d 1V is he risk neural probabiliy ha V 2,T > V 1,T Φd 2U Φ d and V 2V is he condiional expecaion of V 2, Φ d 1V 2,T given V 2,T > V 1,T Approximae Price Hedge Raios The dela and gamma hedge raios for our analyic approximaion are sraighforward o derive by differeniaing he model price wih respec o each underlying. Again, le z x denoes he dela of z wih respec o x and Γ z xy denoe he gamma of z wih respec o x and y: f S i = f U i U i S i + f V i V i S i Γ f S i S i = Γ f U i U i S i 2 + Γ U i S i f U i + Γ f V i V i S i 2 + Γ V i S i f V i Γ f S 1 S 2 = Γ f S 2 S 1 = Γ f U 1 U 2 U 1 S 1 U 2 S 2 + Γ f V 1 V 2 V 1 S 1 V 2 S 2 16 The CEO model Greeks given by equaions 16 are beer approximaions han hose derived from Kirk s formula for he reasons discussed in he previous secion. So oher approximaion mehods can lead o subsanial hedging errors as well as inaccurae pricing Hedging Volailiy and Correlaion Risks Spread opions may be dela-gamma hedged by aking posiions in he underlying asses and opions on hese. Bu hedging volailiy and correlaion is more complicaed. In his nex secion we derive an expression for he spread opion price sensiiviy o correlaion. Copyrigh 2007 Alexander and Venkaramanan 8

11 We remark ha oher analyic approximaions yield correlaion sensiiviies ha are proporional o he opion vega because hey are all based on a volailiy of he form: σ = ω 1 σ ω 2 σ 2 2 2ω 3 ρσ 1 σ 2 where all he erms on he righ hand side are consan. Hence he sensiviy of volailiy o correlaion is consan, and his implies ha he opion price s correlaion sensiiviy is jus a consan imes he opion vega. Also he model implied correlaion is no clearly defined. Correlaion is merely calibraed as a free parameer independen of he spread opion srike and he underlying volailiies. Thus marke senimens such as he correlaion frown are no capured by he model. Bu hen i is meaningless o hedge he spread opion correlaion based on he calibraed values of model parameers. Moreover, volailiy hedging is complicaed by he fac ha one is likely o hedge he volailiies wih he wrong opions if he srike convenion is no chosen correcly. Theoreically here are infiniely many possible srikes for he wo vanilla opions and he srikes are chosen wihou relaing hem o vega risks. Hence he hedging errors accrued from incorrec vega hedging along wih every oher unhedged risk are aribued o correlaion risk. Clearly hese models fail o quanify correlaion risks accuraely and his is likely o have a serious effec on he P&L of he hedging porfolio. We now srucure he CEO model so ha he implied correlaion ρ is direcly relaed o m, he only independen and herefore cenral parameer. The vanilla opion implied volailiies and he exchange opion volailiies are hen also deermined by m. This consrucion provides a closed form formula for he sensiiviy of he spread opion price o correlaion. In oher words he correlaion smile or frown is endogenous o he model. In he following we wrie he spread opion price as f = f U 1, U 2, σ U, V 1, V 2, σ V where: σ U = ξ1 2 + ξ2 2 2ρξ 1 ξ 2 σ V = η1 2 + η2 2 2ρη 1 η 2 Here σ U and σ V are he volailiies of he exchange opions on calls and pus respecively. The sensiiviy of he opion price wih respec o correlaion is hus: d f dρ = d f dm dm dρ where d f dm = f du 1 U 1 dm + f du 2 U 2 dm + f dσ U σ U dm + f dv 1 V 1 dm + f dv 2 V 2 dm + f dσ V σ V dm The above equaion shows ha he vegas of he spread opion affec he correlaion sensiiviy. By conras wih oher analyic approximaions, in he CEO model he volailiy and correlaion hedge raios may be independen of each oher. We se dσ U = 0 and dσ V = 0. In oher words, we dρ dρ choose m and ρ so ha he volailiy of he spread opion is invarian o changes in correlaion. We call his volailiy he correlaion invarian volailiy CIV. Copyrigh 2007 Alexander and Venkaramanan 9

12 The oal derivaive of σ U is: dσ U = σ U ξ 1 dξ 1 + σ U ξ 2 dξ 2 + σ U ρ dρ Hence dσ U dρ = σ U dξ 1 dm ξ 1 dm dρ + σ U dξ 2 dm ξ 2 dm dρ + σ U ρ = A dm dρ ξ 1ξ 2 σ U 17 where 3 A = 1 dξ1 σ U dm ξ 1 ρξ 2 + dξ 2 dm ξ 2 ρξ 1 Similarly dσ V dρ = B dm dρ η 1η 2 σ V 18 where B = 1 dη1 σ V dm η 1 ρη 2 + dη 2 dm η 2 ρη 1 Now equaion 17 implies ha dm dρ = 1 A ξ 1 ξ 2 σ U 1 = ξ 1 ξ 2 dξ1 ξ dm 1 ρξ 2 + dξ 2 ξ dm 2 ρξ 1 1 = ξ 1 ξ 2 dξ1 ξ dm 1 + dξ 2 ξ dm 2 ρ dξ1 ξ dm 2 dξ 2 ξ dm 1 = g ξ 1, ξ 2, ρ 1 19 Similarly: dm dρ 1 = η 1 η 2 dη1 η dm 1 + dη 2 η dm 2 ρ dη1 η dm 2 dη 2 η dm 1 = g η 1, η 2, ρ 1 20 We hen have d f dm = K U 1 U 2 dv 1 dv 2 f U1 + f U2 + f V1 + f V2 K 1 K 2 dk 1 dk The firs order derivaives of ξ i and η i wih respec o m can be calculaed from heir respecive implied volailiies σ 1 and σ 2 eiher numerically or by assuming a quadraic or cubic spline funcion of heir srikes. σ i = a i K 2 i + b i K i + c i where i = 1, 2 and a i, b i, and c i are some consans ha can be esimaed using curve fiing mehods. Copyrigh 2007 Alexander and Venkaramanan 10

13 and d f dρ = d f dm dm dρ = Kg ξ 1, ξ 2, ρ 1 du 1 du 2 f U1 + f U2 + Kg η 1, η 2, ρ 1 dv 1 dv 2 f V1 + f V2 dk 1 dk 2 dk 1 dk Calibraion Our calibraion problem reduces o calibraing a single parameer m for each spread opion by equaing marke prices of he spread opions o 12. Then he single asse opions srikes are deermined because K 1 = mk and K 2 = m 1K. And he implied correlaion beween he opions is also deermined: since he opions follow he same Wiener processes as he underlying prices, heir implied correlaion is he same. Le f M be he marke price of he spread opion and f be he price of a spread opion given by equaion 12. Then he calibraion problem reduces o he following opimisaion problem: such ha, a a given ieraion j: 1. m j saisfies he equaion : g ξ 1, ξ 2, ρ j g η 1, η 2, ρ j = 0 2. dm dρ 1 = j gξ 1,ξ 2,ρ j 3. σ Xi + σ Yi is a minimum where g x, y, z = 1 xy x m x + y m y z min f M f m, ρ 23 x m y y m x. The above problem can be solved using a one-dimensional gradien mehod. The firs order differenial of f wih respec o ρ is given by d f f dρ = Kg ξ 1, ξ 2, ρ 1 du 1 + f du 2 + f dv 1 + f dv 2 24 U 1 dk 1 U 2 dk 2 V 1 dk 1 V 2 dk Comparison wih Kirk s Approximaion In his secion we calibrae our model o simulaed spread opion prices and compare he calibraion errors wih hose derived from Kirk s approximaion. We have used prices S 1 = 65 and S 2 = 50, and spread opion srikes ranging beween 9.5 and 27.5 wih a sep size of 1.5 and mauriy 30 days. The spread opion prices were simulaed using quadraic local volailiy and local correlaion funcions ha are assumed o be dependen only on he price levels of he underlying asses and no on ime. The a-he-money volailiies were boh 30% and he a-he-money forward correlaion was Figure 1 compares he implied correlaions calibraed from he compound exchange opion formula wih hose obained from Kirk s approximaion. The resuls illusrae he poor performance Copyrigh 2007 Alexander and Venkaramanan 11

14 of Kirk s approximaion for high srike values. Using Kirk s approximaion he roo mean square percenage calibraion error RMSE, i.e. where each error is expressed as a percenage of he opion price, was 9% using he srike convenion and 9.3% using he consan ATM volailiy o deermine σ 1 and σ 2. By conras he exchange opion model s pricing errors are exremely small he RMSE was 0.53% and he implied correlaion values in figure 1 show greaer sabiliy. FIGURE 1: Implied Correlaions from Kirk s and CEO Approximaions Kirk 1 implied volailiies are calculaed using K 1 = S 1,0 K 2 and K 2 = S 2,0 + K and Kirk 2 uses ATM consan volailiy Correlaion CEO Kirk 1 Kirk Srike This simulaion exercise illusraes he main problem wih oher price approximaions for spread opions. When we apply a convenion for fixing he srikes of he implied volailiies σ 1, σ 2, ake he implied volailiies from he single asse opion prices and hen calibrae he implied correlaion o he spread opion price we obain oally unrealisic resuls excep for opions wih very low srikes. For high srike opions he model s lognormaliy assumpion is simply no valid PRICING AND HEDGING AMERICAN SPREAD OPTIONS In his secion derive he early exercise premium for a spread opion and exend boh Kirk s approximaion and our approximaion o pricing American spread opions. The price of American 4 Aemps o use he exchange opion srike convenion wih K 1 = mk, TK and K 2 = mk, T 1K led o even greaer pricing errors. A possible quick fix could be o change he srike convenion so ha i can be differen for each spread opion, bu his is very ad hoc. Copyrigh 2007 Alexander and Venkaramanan 12

15 syle opions on single underlying asses is mainly deermined by he ype of he underlying asse, he prevailing discoun rae, and he presence of any dividend yield. The opion o exercise early suggess ha hese opions are more expensive han heir European counerpars bu here are many insances when i is no opimal o exercise an opion early. American calls on nondividend paying socks and calls or pus on forward conracs are wo examples where i is never opimal o exercise he opion early see James [2003]. Since no raded opions are perpeual he expiry dae forces he price of American opions o converge o he price of heir European counerpars. Before expiry, he prices of American calls and pus are always greaer han or equal o he corresponding European calls and pus The Early Exercise Premium of a Spread Opion In he free boundary pricing mehods of McKean [1965], Kim [1990], Carr e al. [1989], Jacka [1991], and ohers he price of an American opion wih payoff [ωs K] + on one underlying asse wih price process 1 is given by: PS, = P E S, T, ω + ω ω T T qs e qs Φ ω d 1 S, B, s ds rke rs Φ ω d 2 S, B, s ds 25 where ω = 1 for a call and -1 for a pu and B is he early exercise boundary. In he case of muliple underlying asses he behaviour of American opions is similar o ha of single asse American opions, wih some noable excepions. Rubinsein [1991] was he firs o noe ha an American exchange opion is equivalen o a sandard opion in a modified ye equivalen financial marke. The problem pricing reduces o ha of pricing a plain vanilla opion by aking one of he asses as he numeraire insead of he money marke accoun wih he corresponding equivalen maringale measure. Then he prices of such claims can be found using he early exercise premium EEP represenaion see Deemple [2005]. We can express he price of an American spread opion as a sum of is European counerpar and an early exercise premium. Consider he simple case of an American exchange opion on wo asses. Le S 1, and S 2, be he prices of wo asses a ime given by equaion 1 and x = S 1, S 2,, as in secion 2.1. Then he payoff o an exchange opion a mauriy T is S 2,T [x T 1, 0}. Le P E and P A be he prices of a European and American exchange opion respecively. The EEP represenaion gives where, P A x = P E x + T T d 1 x, B, T, q 1, q 2, σ x = ln q 1 x e q 1s Φd 1 x, B s, s, q 1, q 2, σ x ds q 2 e q 2s Φd 2 x, B s, s, q 1, q 2, σ x ds 26 x B + q 2 q σ2 x T σ x T d 2 x, B, T, q 1, q 2, σ x = d 1 x, B, T, q 1, q 2, σ x σ x T Copyrigh 2007 Alexander and Venkaramanan 13

16 and σ x is as defined in secion 2.1. This shows ha for he early exercise premium o be posiive we require q 1 > 0. 5 Now consider he case when he wo underlying asses are fuures conracs. Since fuures do no have dividends, he above equaion implies P A = P E. Hence whils an American opion on a single fuures conrac may be worh more han he corresponding European opion his is no necessarily he case for American opions on muliple asses. Broadie and Deemple [1997] provide a deailed discussion of pricing American opions on wo asses saing properies of he exercise region and giving a recursive inegral equaion which is saisfied by he early exercise boundary. A presen here are no efficien mehods available o calculae he early exercise boundary in he wo asse case. However in he following secions we reduce he dimension of he problem o one 4.2. Exension of Kirk s Formula In secion 2.1 he random variable Z was approximaely log normal for small K values and his allowed one o express he price of a European pu spread as ha of an ordinary European pu. By he same consrucion we can use Z o express he price of an American pu spread as an ordinary American pu on Z wih srike 1. The inrinsic value of he opion a ime is given by, [Ke rt S 1, + S 2, ] + = [Y S 1, ] + The above resembles he payoff of an exchange opion wrien on Y and S 1,, and boh processes are observable in he marke. Recalling equaions 1 and 32 we have, P A Z = P Z + T T q 1Z e q 1 s Φ d 1 Z, B s, s, q 1, q 2, σds q 2e q 2 s Φ d 2 Z, B s, s, q 1, q 2, σds 27 where q 1 = q 1 and q 2 = r r + q 2. A he early exercise boundary, i.e., when Z = B, he price given by equaion 27 equals 1 B. 1 B = P B + T T q 1B e q 1 s Φ d 1 B, B s, s, q 1, q 2, σds q 2e q 2 s Φ d 2 B, B s, s, q 1, q 2, σds 28 This is he value mach condiion. Moreover, a B he slope of he price curve of equaion 27 is ha of 1 B. This is called as he high conac condiion and i can be obained by differeniaing equaion 28 wih respec o B, giving: P B, 1, T B 1 = 5 If q 1 = 0 and q 2 > 0 hen P A < P E. T q B 2e q 2 s Φ d 2 B, B s, s, q1, q2, σds T q B 1B e q 1 s Φ d 1 B, B s, s, q1, q2, σds Copyrigh 2007 Alexander and Venkaramanan 14

17 4.3. Exension of Compound Exchange Opion Formula In secion 3 we showed how a European spread opion price is equivalen o he price of an exchange opion on wo deep in-he-money call opions. We may choose m o be sufficienly small so ha he opion price processes closely imiae ha of underlying asses and hence carry coss or dividends on he underlying asses can aler he prices of hese in-he-money opions considerably. Therefore any change in price of he underlying asses due o dividends or carry coss mus be accouned for when pricing a spread opion as a compound exchange opion. Consider he price process of each underlying asse and he corresponding call opions. The soluions o heir sochasic differenial equaions a ime are given by, S i,t = S i, e r q i 1 2 σ2 i T +σ i dw i U i,t = U i, e r q i 1 2 ξ2 i T +ξ i dw i Dividing S i,t by U i,t and using he approximaion ξ i σ i, o eliminae he sochasic erm q i = 1 Si,T Ui,T ln ln + q i T S i, 0 U i, 0 We now rewrie equaions 7 and 8 as: du i, = r q i U i, d + ξ i U i, dw i dv i, = r q i V i, d + ξ i V i, dw i 29 where q 1 and q 2 are he equivalen dividend yields of he opions. I should be noed ha even hough we shall be pricing American spread opions, he wo call opions wih prices U 1 and U 2 remain European syle opions. Alhough he exchange opion may be exercised before mauriy, he call opions may be exercised only a expiry. Since he compound exchange opion replicaes he cash flow of a spread opion, when exercised hey will yield he same payoff. Since mos of he rades are cash seled his is adequae. Even in commodiy markes where he opions are exercised by he physical delivery of goods, his adjusmen can be jusified as he underlying fuure conracs expiry dae is he same as or laer han ha of he spread opion. Le us now resric our analysis o he case ha here are no dividend yields or carry coss, such as when he underlying asses are fuure conracs. We now price an American spread opion as an American compound exchange opion using he early exercise premium represenaion given by equaion 26. Define maringale processes X = U 1, U 2, and Y = V 2,. Then American spread opion V 1, price is given by f A = f E X, ω + ω T q1x e q 1 s Φω d 1 X, B s, s, q1, q2, σ X ds T T + f E Y, ω + ω T q2e q 2 s Φω d 2 X, B s, s, q1, q2, σ X ds q 1Y e q 1 s Φω d 1 Y, B s, s, q 1, q 2, σ Y ds q2e q 2 s Φω d 2 Y, B s, s, q1, q2, σ Y ds and hence American spread opions on fuures or non dividend paying socks are worh he same as heir European counerpars. 30 Copyrigh 2007 Alexander and Venkaramanan 15

18 4.4. Empiricial Resuls We now es he pricing performance of he exchange opion approximaion using 1:1 American crack spread opion daa raded a NYMEX beween Sepember 2005 and May The crack spread opions are on gasoline - crude oil and are raded on he price differenial beween he fuures conracs of WTI ligh swee crude oil and gasoline. Opion daa for American syle conracs on each of hese individual fuures conracs were also obained for he same ime period along wih he fuures prices. The size of all he fuures conracs is 1000 bbls. Figures 2 and 3 depic he implied volailiy skews in gasoline and crude oil on several of he days during he sample period. These pronounced negaive implied volailiy skews indicae ha a suiable pricing model should exhibi a posiive skew in implied correlaion as a funcion of he spread opion srike. FIGURE 2: Implied Volailiy of Gasoline Mar Mar Mar Mar Mar Mar Mar Mar Mar Srike We compare he resuls of Kirk s approximaion wih he exchange opion approximaion by calibraing each model o he marke prices of he gasoline - crude oil crack spread over consecuive rading daes saring from 1s March 2006 o 15h March 2006, hese being days of paricularly high rading volumes. From figure 4 we can clearly see ha Kirk s approximaion gives an error ha increases drasically for high srike values, as was also he case in our simulaion resuls. On he oher hand he compound exchange opion model errors were found o be close o zero for all srikes on all daes. Figure 5 shows ha he implied correlaions ha are calibraed from he compound exchange opion approximaion exhibi a realisic, posiively sloped skew on each day of he sample. However, he implied correlaions compued from Kirk s approximaion were found o be equal o 0.99 for all srikes and on every day. Copyrigh 2007 Alexander and Venkaramanan 16

19 FIGURE 3: Implied Volailiy of Crude Oil Implied Volailiy Mar Mar Mar Mar Mar Mar Mar Mar Mar Srike FIGURE 4: Kirks and CEO Pricing Errors Kirks Error 1 Error EO Model Error 0 05 Mar Mar Mar Mar Mar Daes Srikes 14 Copyrigh 2007 Alexander and Venkaramanan 17

20 FIGURE 5: Implied Correlaion Skews of CEO Model Mar Mar Mar Mar Mar Srike Figures 6 and 7 compare he wo delas and gammas of each model, calibraed on 1s March 2006 and depiced as a funcion of he spread opion srike. The same feaures are eviden on all oher days in he sample: a every srike he exchange opion dela is much smaller han he dela ha is obained hrough Kirk s formula. Similar remarks apply o he gamma hedges, paricularly for he gamma hedge on crude oil. We conclude ha he use of Kirk s approximaion may lead o significan over hedging. 5. CONCLUSION This paper highlighs cerain difficulies wih pricing and hedging spread opions based on approximaions such as ha of Kirk [1996]. There are wo subsanial problems: he approximaion is only valid for spread opions wih low srikes and an arbirary srike convenion is necessary o deermine he implied volailiies in he calibraion. Thus he approximae prices and hedge raios only apply o spread opions wih very low srikes and even hese have quesionable accuracy, since heir values depend on he ad hoc choice of srike convenion. We have esed several srike convenions for fixing he implied volailiies of he single asse opions bu in each case heir marke prices are inconsisen wih he marke prices of spread opions, excep for spread opions wih very low srikes. Moreoever, for he crack spread opion daa all choices of srike convenion yielded almos consan correlaions ha were very close o 1, which is unrealisic. By conras, he compound exchange opion approximaion provides accurae prices a all srikes and realisic values for implied correlaion. Oher advanages of he compound exchange opion approximaion are he ease of calibraion and he simple compuaion of he opion s price sensi- Copyrigh 2007 Alexander and Venkaramanan 18

21 FIGURE 6: Dela wih respec o Gasoline lef and Crude Oil righ CEO Kirks Dela CEO Kirk Srikes Srikes FIGURE 7: Gamma wih respec o Gasoline lef and Crude Oil righ 0.07 CEO Kirks CEO Kirks Srikes Srikes Copyrigh 2007 Alexander and Venkaramanan 19

22 iviies. We have found empirically ha he compound exchange opion approach specifies delas and gammas ha are much smaller han he delas and gammas from Kirk s approximaion and we hus have reason o suppose ha he use of similar approximaions will lead o subsanial over hedging of spread opion posiions. REFERENCES C. Alexander and A. Scourse. Bivariae normal mixure spread opion valuaion. Quaniaive Finance, 46: , M. Broadie and J. Deemple. The valuaion of american opions on muliple asses. Mahemaical Finance, 73: , R. Carmona and V. Durrleman. Pricing and hedging spread opions in a log-normal model. Technical repor, Technical Repor, 2003a. R. Carmona and V. Durrleman. Pricing and hedging spread opions. Siam Rev, 454: , 2003b. R. Carmona and V. Durrleman. Generalizing he black-scholes formula o mulivariae coningen claims. Journal Of Compuaional Finance, 92:43, P. Carr and D. Madan. Opion valuaion using he fas fourier ransform. Journal of Compuaional Finance, 24:61 73, P. Carr, R.A. Jarrow, and R. Myneni. Alernaive Characerizaions of American Pu Opions. Cornell Universiy, Johnson Graduae School of Managemen, M. Dempser and S.S.G. Hong. Spread opion valuaion and he fas fourier ransform. Mahemaical Finance, Bachelier Congress, Geman, H., Madan, D., Pliska, S.R. and Vors, T., Eds., Springer Verlag, Berlin, 1: , J. Deemple. American-syle Derivaives: Valuaion and Compuaion. Chapman & Hall/CRC, V. Durrleman. Implied correlaion and spread opions. Technical repor, Technical Repor, Deparmen Of Operaions, Alexander Eydeland and Krzyszof Wolyniec. Energy and power risk managemen: new developmens in modeling, pricing and hedging. John Wiley & Sons, S.D. Jacka. Opimal sopping and he american pu. Mah. Finance, 11:14, P. James. Opion heory. Wiley New York, I.N. Kim. The analyic valuaion of american opions. Review of Financial Sudies, E. Kirk. Correlaion in energy markes. Managing Energy Price Risk, Minqiang Li, Shijie Deng, and Jieyun Zhou. Closed-form approximaions for spread opion prices and greeks. SSRN elibrary, W. Margrabe. The value of an opion o exchange one asse for anoher. The Journal of Finance, 33 1: , Copyrigh 2007 Alexander and Venkaramanan 20

23 H.P. McKean. Appendix: A free boundary problem for he hea equaion arising from a problem in mahemaical economics. Indusrial Managemen Review, 62:32 39, K. Ravindran. Low-fa spreads. RISK, 610:56 57, M. Rubinsein. One for anoher. Risk, 410:30 32, D. Shimko. Opions on fuures spreads: hedging, speculaion and valuaion. Journal of Fuures Markes, 142: , Copyrigh 2007 Alexander and Venkaramanan 21

24 A APPENDIX: DERIVATION OF KIRK S APPROXIMATION In his appendix we derive he approximae pricing formula presened in Kirk [1996]. The derivaion has no been documened in he lieraure, and neiher were dividends included in he formula. The payoff o a spread opion is given by [ωs 1,T S 2,T K] + = K + S 2,T [ωz T 1] + where Z = S 1, /Y and Y = S 2, + Ke rt. By Io s lemma: dz = Z ds 1, + Z dy Z ds S 1, Y T 2 S1, 2 1, dz ds 1, = dy 2 dy + ds 1, dy Z S 1, Y Y S 1, Y 2 Z Y 2 Y Z S 1, Y ds 1, dy 31 We have dy = ds 2, + Kre rt d and for K S 2 dy Y = S 2, Y r q 2 d + σ 2 dw 2, = r q 2 d + σ 2 dw 2, 32 where σ 2 = S2, Y σ 2, r = S2, Y r, and q 2 = S2, Y q 2 are assumed o be consan. Hence 31 can be rewrien: dz Z = r r q 1 q 2 d + σ 2 2 σ 1 σ 2 ρ d + σ 1 dw 1, σ 2 dw 2, Le W 3, be a Brownian moion ha is uncorrelaed wih W 2, and such ha Then, dw 1, = ρdw 2, + 1 ρ 2 dw 3, dz Z = r r q 1 q 2 d + σ 2 2 σ 1 σ 2 ρ d + ρσ 1 σ 2 dw 2, + σ 1 1 ρ2 dw 3, Define dw2, = dw 2, σ 2 d. Using Girsanov s heorem, le P be he new probabiliy measure under which boh W2, and W 3, are maringales. The Radon-Nikodym derivaive wih respec o he risk-neural probabiliy Q is hen given by: We now have dp dq = e 1 2 σ2 2 T+ σ 2 W 2, dz Z = r r q 1 q 2 d + σ 1 ρ σ 2 dw 2, + 1 ρ 2 σ 1 dw 3, = r r q 1 q 2 d + σ dw, say Copyrigh 2007 Alexander and Venkaramanan 22

25 The sandard deviaion of W is given by σ = = σ 1 ρ σ ρ 2 σ1 2 2 σ1 2 + S2, S2, σ2 2 2ρσ 1 σ 2 Y Y 33 since W 2, and W 3, are independen Weiner processes. Noe ha Z is approximaely log-normal and is also observable in he marke. Hence he spread opion can be priced by reaing i as a plain vanilla opion defined on an observable asse whose price process is described by Z and wih a srike K = 1. Therefore he price P a ime for an opion on S 1, and S 2, wih srike K, mauriy T and payoff [ωs 1 S 2 K] + is given by: P = ω S 1, e q 1T Φ ωd 1 Ke rt + S 2, e r r+ q 2 T Φ ωd 2 34 where ω = 1 for a call and ω = 1 for a pu, d 1 = ln Z + r r + q 2 q σ2 T σ T d 2 = d 1 σ T Copyrigh 2007 Alexander and Venkaramanan 23