Changes of Numeraire for Pricing Futures, Forwards, and Options

Transcription

1 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Mark Schroder Michigan Sae Universiy A change of numeraire argumen is used o derive a general opion pariy, or equivalence, resul relaing American call and pu prices, and o obain new expressions for fuures and forward prices. The general pariy resul unifies and exends a number of exising resuls. The new fuures and forward pricing formulas are ofen simpler o compue in mulifacor models han exising alernaives. We also exend previous work by deriving a general formula relaing exchange opions o ordinary call opions. A number of applicaions o diffusion models, including sochasic volailiy, sochasic ineres rae, and sochasic dividend rae models, and jump-diffusion models are examined. A self-financing porfolio is called a numeraire if securiy prices, measured in unis of his porfolio, admi an equivalen maringale measure. The mos commonly used numeraire is he reinvesed shor-rae process; he corresponding equivalen maringale measure is he risk-neural measure. Geman, El Karoui, and Roche (1995 show ha oher numeraires can simplify many asse pricing problems. In his aricle, we build on heir resuls and, using he reinvesed asse price as he numeraire, unify and exend he lieraure on opion pariy, or equivalence, resuls relaing American call and pu prices for asse and fuures opions. The same numeraire change is used o obain new pricing formulas for fuures and forwards ha are ofen simpler o compue in mulifacor models. Finally, we use a numeraire change o simplify exchange opion pricing, exending a similar resul in Geman, El Karoui, and Roche o dividend-paying asses. The change of numeraire mehod is mos inuiive in he conex of foreign currency derivaive securiies. As discussed by Grabbe (1983, an American call opion o buy 1 DM, wih dollar price process S, for K dollars is equivalen o an American pu opion o sell K dollars, wih DM price process K/S, for a srike price of 1 DM. The dollar price of he call mus herefore equal he produc of he curren exchange rae, S, and he DM price of he pu. The call price is compued using he dollar value of a U.S. I am graeful o Kerry Back (he edior and Cosis Skiadas for heir many helpful suggesions. Thanks also o Peer DeMarzo, Ma Jackson, Naveen Khanna, David Marshall, Rober McDonald, Jim Moser, Phyllis Payee, and an anonymous referee for heir commens. This aricle subsumes Schroder (1992. Address correspondence o Mark Schroder, The Eli Broad Graduae School of Managemen, Deparmen of Finance, Michigan Sae Universiy, 323 Eppley Cener, Eas Lansing, MI , or schrode7@pilo.msu.edu. The Review of Financial Sudies Winer 1999 Vol. 12, No. 5, pp c 1999 The Sociey for Financial Sudies

2 The Review of Financial Sudies/v12n51999 money marke accoun as he numeraire, while he pu price is compued using he dollar value of a German money marke accoun as he numeraire. Corresponding o he change of numeraire is a change in probabiliy measure, from he risk-neural measure for dollar-denominaed asses o he risk-neural measure for DM-denominaed asses. As suggesed in Grabbe (1983, and developed in laer aricles, an analogous relaion applies o any asse opion. A call opion o buy one uni of an asse, wih dollar price process S, for K dollars is he same as a pu opion o sell K dollars, worh K/S unis of asse, for one uni of asse. Muliplying he asse denominaed pu price by he curren asse price convers he price ino dollars. The same numeraire change can be used o obain he ineres pariy heorem which expresses he ime zero dollar forward price, G (T, for ime T delivery of one DM as he spo currency rae imes he raio of wo discoun bond prices: G (T = S B (T /B (T, where B (T is he ime zero dollar price of a discoun bond paying $1 a T, and B (T is he ime zero DM price of a discoun bond paying 1 DM a T. This resul can be exended o forward conracs on any asse. A key issue examined in his aricle is he change of measure ha corresponds o a change of numeraire. Under he risk-neural measure, he drif rae of he reurns of he asse price S is he shor rae minus he dividend rae. In Secion 1 we show ha he drif rae of he reurns of S 1 (he price of dollars in unis of asse under he new measure is he dividend rae minus he shor rae. The reversal of he roles of he shor rae and dividend rae is inuiive because under he new numeraire he asse is riskless while dollars are risky. Example 1 shows ha he change of measure can resul in more suble modificaions and can change boh he inensiy and disribuion of jumps in jump-diffusion models. In Secion 2 we show ha he measure change also alers he drif erms of nonprice sae variables, such as in sochasic volailiy and sochasic ineres rae models. Example 1. Assume ha he shor rae and dividend rae are boh zero, and he asse price follows a Poisson jump process wih inensiy λ under he risk-neural probabiliy measure, Q. A jump ime τ i,i = 1, 2,...,he sock price raio has he Bernoulli disribuion { us(τi, wih Q-probabiliy p S(τ i = ds(τ i, wih Q-probabiliy 1 p, and beween jumps, ds /S = (1 µλ d, τ i < <τ i+1, i =, 1,..., where µ pu + (1 pd is he expeced price raio a jumps, and τ. 1144

3 Changes of Numeraire for Pricing Fuures, Forwards, and Opions A change of numeraire o he underlying asse price is associaed wih he new measure Q, where d Q/dQ = S T /S. A jumps, he value of a dollar measured in unis of he asse saisfies 1 { S(τ i 1 = u 1 S(τ i 1, wih Q-probabiliy puµ 1 d 1 S(τ i 1, wih Q-probabiliy (1 pdµ 1, and beween jumps ds( 1 /S( 1 = (µ 1λ d, τ i < <τ i+1, i =, 1,... The inensiy of he jump process under Q is µλ, which can be obained using he maringale propery of S under Q: Q(τ 1 > = E Q (1 {τ1 >}S T /S = E Q (1 {τ1 >}S /S = e (1 µλ Q(τ 1 >. The disribuions of he reurns of S under Q and S 1 under Q are idenical only in he special case when u = d 1 and µ = 1. We show ha subjec o some common echnical resricions (Assumpions 1 and 2 below, any American call price formula is he same, afer a change of numeraire, o an American pu price formula. This resul is useful for obaining prices, derivaives of prices wih respec o model parameers, and early exercise boundaries for pu opion formulas from he properies of he corresponding call opion formula. Previous aricles derive he correc pu-call equivalence formulas only for some special cases. The geomeric Brownian moion case (see Example 2 below is derived in McDonald and Schroder (199, Bjerksund and Sensland (1993, and, for fuures opions, in Byun and Kim ( Chesney and Gibson (1993 use a change of numeraire o obain a closed-form European formula for sock-index opions when he shor rae is sochasic from Jamshidian and Fein s (199 closedform European formula for opions on asses wih a sochasic payou rae. However, he change of measure is incorrec, in par because i neglecs o make he appropriae modificaion o he drif erm of he sae variable. 1 The jump probabiliies under Q can be verified using he general resuls in he appendix, or from Q ({S(τ 1 = us(τ 1 } {τ 1 } = pue Q (1 { τ1 }S(τ 1 /S(, for any T, and S(τ 1 /S( = ue (1 µλτ 1 on {S(τ 1 = us(τ 1 }. 2 Bjerksund and Sensland (1993 apply a resul in Olsen and Sensland (1991 which demonsraes ha he curren asse price can be facored ou in cerain conrol heory problems where he fuure reward is muliplicaive in he price of an asse. Their resul could be used o derive he pariy resul in a diffusion seing when he reurn volailiy is any funcion of he price, subjec o he price process being sricly posiive (such as he CEV model below. The Olsen and Sensland (1991 resuls can be generalized by allowing he payoff in Proposiion 1 below o depend on a vecor of conrols. See also Kholodnyi and Price (1998, who derive equivalence resuls for geomeric Brownian moion and he binomial model. They use no-arbirage argumens o derive general equivalence resuls in a seing where each opion price is a deerminisic funcion of he curren underlying asse price (for example, Markovian S and deerminisic r and δ. In he foreign currency conex, he equivalence resuls are in erms of he generaors of he domesic and foreign evoluion (or presen value operaors. 1145

4 The Review of Financial Sudies/v12n51999 Example 5 below shows he correc measure change in ha model. Carr and Chesney (1996 derive a formula relaing call and pu prices in a one-facor model in which he volailiy of he underlying price obeys a symmery condiion (see Example 3 below. Baes (1991 derives equivalence formulas for American pu and call opions on fuures for some special cases o es classes of opion pricing models. Example 8 builds on his idea and derives general condiions under which he equivalence formula akes a paricularly simple form: swiching he roles of he curren fuures price and he srike price in he American call opion formula gives he price of an oherwise idenical American pu opion. Secion 1 presens he numeraire change mehod and he general resuls using he reinvesed asse price as he numeraire. Secion 2 presens examples of hese resuls. The Appendix derives he numeraire change for a general jump-diffusion model ha includes all he Secion 2 examples as special cases. 1. The Reinvesed Asse Price as he Numeraire We presen he general change of numeraire argumen before giving he main resuls. Fix a finie ime horizon [, T ]. 3 Le Y denoe some reinvesed asse price process. Tha is, Y is he ime balance of an invesmen sraegy of buying an asse and reinvesing all dividends ino new shares. Le R represen he reinvesed shor rae wih uni iniial invesmen: R = exp( r sds, where r is he shor rae process. If π is he sae price densiy process, hen πy and π R are P-maringales. I follows ha Y/R is a Q-maringale, where dq/dp π T R T. Tha is, when measured in unis of he numeraire R, Y is a maringale wih respec o he risk-neural probabiliy measure Q. Geman, El Karoui, and Roche (1995 show ha we ge he same resul when we replace R wih anoher self-financing porfolio V wih V = 1 (and V/R a Q-maringale. Then Y/V is a Q-maringale, where d Q/dP π T V T (or, equivalenly, d Q/dQ V T /R T. Tha is, when measured in unis of he numeraire V, Y is a Q-maringale. This simple change of numeraire is he basis for all he pricing resuls below. The resuls are very general in ha we allow for incomplee markes and price and sae variable dynamics which are neiher coninuous nor Markovian. The main assumpion is he exisence of a risk-neural measure. Assumpion 1. There exiss a risk-neural measure, Q, such ha every reinvesed price process relaive o he reinvesed shor-rae process is a Q-maringale. 3 We assume hroughou he exisence of a complee probabiliy space (, F, F, P and a filraion F = {F ; } which saisfies he usual condiions. See Proer (1992 for he required condiions and for all he resuls on semimaringale heory needed in his aricle. All processes are assumed o be adaped. 1146

5 Changes of Numeraire for Pricing Fuures, Forwards, and Opions The self-financing porfolio ha serves as he numeraire for our main resuls is he reinvesed asse price process wih uni iniial balance. Le S be a semimaringale represening he price process of an asse wih a proporional dividend payou rae δ. 4 We assume hroughou ha S is sricly posiive. 5 The value of he numeraire porfolio a any ime is S exp( δ sds/s. The probabiliy measure Q ha corresponds o he new numeraire is Q(A = E Q (1 { A } Z T, A F. (1 where Z is defined as he raio of he new and old numeraires: Z e (δ s r s ds S /S, [, T ]. (2 In oher words, he Radon-Nikodym derivaive is d Q/dQ = Z T. All he resuls of his secion hold when S is replaced by a fuures price process F (wih delivery dae D T ifweseδequal or. To jusify his, we consruc a numeraire porfolio wih value F exp( r sds/f a any ime by mainaining a long posiion of exp( r sds/f fuures conracs a and adding or subracing mark-o-marke gains and losses from a money marke accoun, whose ime-zero balance is se o $1 [his sraegy is described in Duffie (1992: chap. 7]. Alernaively, we can use he fac ha F is a Q- maringale and direcly define Z o be F /F. The following proposiion provides a general pricing formula under a change of numeraire o he reinvesed asse price. The consan K will serve as he srike price in he opion pricing applicaions below. The process S represens he price of KS dollars measured in unis of he asse S. Proposiion 1. Define S = KS /S and Q by (1. Then he ime-zero price of an asse wih he F τ -measurable payoff P τ a he sopping ime τ [, T ] is ( E Q e τ rsds P τ = E Q (e τ δsds P τ S τ /K. Furhermore ds = S (r δ d + dm d S = S (δ r d + d M, S = K, 4 I is easy o exend he resuls o discree dividends. In addiion o he proporional dividend rae δ, suppose he asse pays discree cash dividends C i a sopping imes T i, i = 1, 2,...All he resuls are hen generalized by adding he erm log[1 + C i /S(T i ]o δ sds hroughou. The exponenial T i [,] of his addiional erm represens he addiional shares of asse accumulaed by reinvesing he discree dividends ino new shares purchased a he ex-dividend price S. 5 In many applicaions he resuls exend o he case where S has an absorbing boundary a zero. In a diffusion model, for example, consruc a modified sock price process whose diffusion erm is killed he firs ime he price his a small posiive consan. The dominaed convergence heorem can be used o evaluae he limi of he expecaion (in Corollary 1, for example as his consan goes o zero. 1147

6 The Review of Financial Sudies/v12n51999 where M and M and local maringales under Q and Q, respecively. 6 The quadraic variaions of M and M saisfy (dm 2 /S 2 = (d M 2 / S 2 beween jumps. 7 Proof. Assumpion 1 implies ha he price is given by he firs expecaion. Applying he numeraire change, ( E Q e τ ( rsds P τ = E Q Z τ e τ δsds P τ S τ /K = E Q (e τ δsds P τ S τ /K, where he las equaliy is obained using ieraed expecaions and he maringale propery of Z. The equaion for he reurns of S follows because he raio of he reinvesed price process o he reinvesed shor rae process is a Q-maringale. The equaion for he reurns of S follows because he raio of he shor rae price process o he reinvesed price process is a Qmaringale. The equaliy, beween jumps, of he insananeous volailiies of reurns follows from Iô s lemma and from he Girsanov Meyer heorem (a generalizaion of Girsanov s heorem o a non-brownian seing, which implies ha M M is absoluely coninuous in beween jumps. The proposiion shows ha he insananeous reurn variances of S and S are idenical beween jumps. Example 1 illusraes ha a jumps, he squared reurns will generally be differen. The firs applicaion of Proposiion 1 relaes call prices o pu prices under a change of numeraire. Corollary 1. Define S = KS /S and Q by Equaion (1. Then he value of a call opion on S is he same, afer a change of numeraire, as he value of a pu opion on S: ( E Q e τ rsds max[s τ K, ] = E Q (e τ δsds max[s S τ, ], for any sopping ime τ T. 6 A sufficien condiion for M and M o be maringales under Q and Q, respecively, is ha r and δ are bounded processes. 7 The quadraic variaion of any semimaringale Y is denoed by [Y, Y ], and can be decomposed ino is coninuous and jump componens: [Y, Y ] = [Y, Y ] c + s ( Y s 2, where [Y, Y ] c = [Y c, Y c ] and Y c is he pah-by-pah coninuous par of Y. For coninuous Y (or for beween jumps, i is common o wrie (dy 2 insead of d[y, Y ]. The quadraic variaion is invarian o changes in measure. See Proer (1992: chap. II for he formal definiions. 1148

7 Changes of Numeraire for Pricing Fuures, Forwards, and Opions The lef-hand side represens he value of a European call opion expiring a τ wih srike price K and underlying price process S. The righ-hand side represens he value of a European pu opion, also expiring a τ, bu wih a srike price S and underlying price process S. The roles of he shor rae and asse payou rae are reversed in he call and pu price expressions. Corollary 1 also holds for American opions under Assumpion 2 below. Corollary 2. Define S = KS /S and Q by Equaion (1. Then he value of an asse-or-nohing binary opion on S is he same, afer a change of numeraire, as he value of a cash-or-nohing binary opion on S: ( E Q e τ rsds S τ 1 { Sτ K } = S E Q (e τ δsds 1 { S S τ }, for any sopping ime τ T. Anoher inerpreaion is obained if τ min[t, inf{: S K }] and S has coninuous sample pahs. Then he lef-hand side is he value of a barrier, or firs-ouch digial, opion paying K dollars when he asse price S rises o K ; and he righ-hand side is he value of a barrier opion paying S when S falls o S (he evens {S τ K } and {S S τ } are idenical. 8 When δ, Corollary 2 can be obained from Theorem 2 in Geman, El Karoui, and Roche (1995. The resul is derived independenly under he assumpions of geomeric Brownian moion and consan r and δ by Carr (1993, Dufresne, Keirsead, and Ross (1997, and Ingersoll (1997. When r and δ are deerminisic (exenions o he sochasic case are sraighforward, Corollary 2 shows ha any European opion price can be derived from he probabiliies Q(S T K and Q(S S T [see also Theorem 2 in Geman, El Karoui, and Roche (1995]. The nex corollary presens a new fuures price expression. Le F (T denoe he ime-zero fuures price for delivery of asse S a ime T. Wih coninuous marking o marke, he fuures price equals he risk-neural expecaion of he spo price a delivery [see Duffie (1992: chap. 7]: F (T = E Q (S T. (3 When he ineres rae and he payou rae are deerminisic, he fuures price is simply F (T = S exp[ T (r s δ s ds]. When eiher he ineres rae or he payou rae is sochasic, however, a change of measure o Q gives an expression ha is ofen easier o compue and also more clearly emphasizes he role of he cos of carry in fuures pricing. 8 Reiner and Rubinsein (1991 price a variey of binary and one-sided barrier opions assuming he asse price follows geomeric Brownian moion. 1149

8 The Review of Financial Sudies/v12n51999 Corollary 3. The fuures price is he produc of he spo price and he expecaion, under Q, of he exponenial of he cos of carry: ( T F (T = S E Q e (r s δ s ds. In he general diffusion model in he appendix, for example, he Q-expecaion on he righ-hand side doesn depend on he sock price process if he insananeous covariance beween asse reurns and changes in he sae variable is no a funcion of he asse price. Example 5 shows ha he compuaion of he fuures price using Corollary 3 is paricularly simple wih a consan volailiy sock reurn process and an Ornsein Uhlenbeck sae variable driving eiher r or δ. The nex corollary presens a new forward price expression. Le B (T denoe he ime dollar price of a discoun bond paying $1 a T : ( B (T = E Q e T r s ds F, T. (4a Le B (T denoe he ime price, measured in unis of asse, of a discoun bond paying one uni of he asse a T : ( B (T = S 1 E Q e T r s ds S T F = E Q (e T δ s ds F, T. (4b Leing G (T denoe he ime-zero forward price for delivery of asse S a ime T, Duffie (1992: chap. 7 shows ha ( G (T = E Q e T / rsds S T B (T. (5 Corollary 4 follows from Equaions (4b and (5. Corollary 4. The forward price is given by he produc of he spo price and he raio of asse and dollar denominaed discoun bond prices: G (T = S B (T /B (T, where B (T and B (T are defined by Equaion (4. The main advanage of Corollary 4 is in a model where boh he shor rae and payou rae are sochasic. If he shor rae is deerminisic, hen forward and fuures prices are equal and Corollary 3 can be used. If he payou rae is deerminisic, hen Corollary 4 holds rivially. When we se δ r and reinerpre S as a fuures price wih delivery dae T (which implies ha he forward on he fuures conrac is equivalen o a forward on he asse underlying he fuures conrac, hen Corollary 4 provides a 115

9 Changes of Numeraire for Pricing Fuures, Forwards, and Opions simple expression for he raio of forward and fuures prices on he same underlying asse. The final applicaion of Proposiion 1 is o he valuaion of exchange opions. Le S a and S b denoe wo asse prices and δ a and δ b denoe heir corresponding payou raes. Then ds i = S i (r δ i d + dmi, i {a, b}, where M i is a Q-local maringale. Corollary 5 expresses he value of an exchange opion as an ordinary call opion by changing he numeraire o he reinvesed price of asse a. Corollary 5. Define S b = S bsa /Sa and d Q/dQ = e (δa s rsds ST a /Sa. Then he value of an opion o receive one uni of asse b in exchange for one uni of asse a is he same, afer a change of numeraire, as he value of a call opion on S b : ( E Q e τ rsds max[sτ b Sa τ, ] = E Q (e τ δa s ds max[ S τ b Sa, ], for any sopping ime τ T. Furhermore d S b = S b (δa δ b d + d M b, S b = Sb, where M b is a local maringale under Q. The righ-hand side of he firs equaion is he value of an ordinary call opion wih underlying asse process S b, shor rae process δ a, and fixed srike price S a. Corollary 5 exends a similar resul in Geman, El Karoui, and Roche (1995 o dividend-paying asses and American-syle exercise (under Assumpion 2. To apply Proposiion 1 o American opions, we need o assume ha he price of an American opion is he supremum, over all sopping imes τ,of he risk-neural expeced discouned payoff from exercising a τ. Assumpion 2. Le p be he ime zero price of an American opion allowing he holder o exercise and receive, a any sopping ime τ [, T ], he payoff P τ, where P is an adaped process. Then ( p = sup E Q e τ rsds P τ (6 τ [,T ] Karazas (1988 proves Equaion (6 in a complee markes diffusion seing for American opions on asses. When markes are incomplee, his characerizaion is problemaic [see Duffie (1992: chap. 2]. Because of possible ineracion beween he sae price densiy and he choice of exercise policy, he wo-sep procedure of firs deermining he risk-neural T 1151

10 The Review of Financial Sudies/v12n51999 measure and hen compuing Equaion (6 may no be valid. Neverheless, i is common in he lieraure o ignore his ineracion and firs assign a marke price of risk o he relevan sae variables (in effec, deermining he risk-neural measure, hen price opions as in Equaion (6. 2. Examples The examples in his secion are all special cases of he general jumpdiffusion model presened in he appendix. Throughou he remainder of he aricle, I le W [W 1,...,W d ] and W [ W 1,..., W d ] be vecors of d independen sandard Brownian moions under he measures Q and Q, respecively. Example 2. Consan elasiciy of variance (CEV. The risk-neural asse price process is ds = (r δ d + νs ξ dw 1, ξ [ 1, 1], S where ν and ξ are consans, and r and δ are deerminisic. 9 Geomeric Brownian moion corresponds o ξ =. Closed-form soluions for European call and pu opions in his model have been derived by Cox (1975 [see also Schroder (1989]. Under he measure Q, S is also a CEV process: d S S = (δ r d + ν S ξ d W 1, S = K, wih an absorbing boundary a zero (see Foonoe 7, where ν ν(ks ξ and ξ ξ. Using Corollary 1, we obain he pricing formula for he American pu from he formula for he American call by exchanging S and K, exchanging r and δ, and replacing ν wih ν and ξ wih ξ. For he case of geomeric Brownian moion, he equivalence formula is paricularly simple because ν = ν and ξ = ξ. The nex example shows ha he Carr and Chesney (1996 pu-call symmery resul can be obained from Corollary 1. Example 3. Carr and Chesney (1996. Le he risk-neural asse price process saisfy ds = (r δ d + σ(s dw 1 S, where r and δ are deerminisic and σ( f ( log( / yk for some bounded funcion f and fixed y R +. The funcional form of σ saisfies 9 The resuls in his and all he succeeding examples are unchanged if any of he consan parameers are permied o be deerminisic funcions of ime. 1152

11 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Carr and Chesney s symmery condiion which ensures ha σ(s = σ(ŝ,, where Ŝ yk/s. 1 The dynamics of Ŝ under Q are herefore d Ŝ Ŝ = (δ r d + σ(ŝ d W 1, Ŝ = yk/s. When S represens a fuures price process (and δ = r, he reurn disribuions of S and Ŝ are idenical. Applying Corollary 1 and rearranging, we obain E Q ( e τ rsds max[s τ K, ] S K = E Q (e τ δsds max[y Ŝ τ, ]. Ŝ y The numeraor on he lef-hand side is he price of a call opion on S wih srike price K. The numeraor on he righ-hand side is he price of a pu opion on Ŝ wih srike price y, and wih he roles of r and δ swiched. These call and pu opions have he same moneyness in he sense ha Ŝ /y = K/S. For he case of geomeric Brownian moion, where f is a consan funcion, we le y S o reconcile he resul wih Example 2. Example 4. Sochasic volailiy model of Heson (1991. The risk-neural asse price and volailiy processes are ds = (r δ d + ν dw 1 S, dν = (µ κν d + ψ ν (ρdw ρ 2 dw 2, where r and δ are deerminisic, and µ, κ, ψ and ρ [ 1, 1] are consans. Recall ha W 1 and W 2 are independen sandard Brownian moions under Q, and herefore ρw ρ 2 W 2 is sandard Brownian moion wih insananeous correlaion of ρ wih W 1. Under he numeraire change, d S = (δ r d + ν d W 1 S, S = K, dν = (µ [κ ρψ]ν d ψ ν (ρd W ρ 2 d W 2. The numeraire change resuls in a modificaion o he mean reversion parameer of he volailiy process, and reverses he sign of he covariance beween insananeous asse reurns and volailiy changes. 1 Proposiion 1 implies ha he ime insananeous reurn volailiy of Ŝ is σ(s = σ(yk/ŝ. The funcional form of σ implies σ(yk/ŝ = σ(ŝ. 1153

12 The Review of Financial Sudies/v12n51999 The nex example includes a one-dimensional sochasic sae variable driving eiher he shor rae or he dividend rae. Example 5. Sochasic dividend and sochasic shor rae models. Le he risk-neural asse price and one-dimensional sae variable saisfy ds S = (r δ d + σ dw 1, dx = (µ κ X d + ν(ρdw ρ 2 dw 2, where he coefficiens σ, µ, κ, ν, and ρ [ 1, 1] are consans. Under he numeraire change, d S S = (δ r d + σ d W 1, S = K, dx = (µ + ρσν κ X d ν(ρd W ρ 2 d W 2. The sae variable sill is Ornsein Uhlenbeck under Q, bu wih a differen drif parameer. (a Sochasic shor rae. Le δ be deerminisic and r = f (X for some f : R R. This model includes Ornsein Uhlenbeck ( f (x = x, x R and lognormal ( f (x = e x shor rae processes. From Corollary 3, he price of a fuures conrac on S for delivery a T is F (T = S exp ( T δ s ds E Q [ exp ( T This is simpler han evaluaing he sandard Equaion (3: ( T F (T = E Q (S T = S exp δ s ds ] r s ds. [ T E Q (exp r s ds 1 ] 2 σ 2 T + σ WT 1. Corollary 1 implies ha he price of a call opion on S is equal, afer he numeraire change, o he price of a pu opion on S wih a deerminisic shor rae and a sochasic dividend rae. (b Sochasic dividend rae. Le r be deerminisic and δ = f (X.We obain he fuures and forward prices from eiher Corollary 3 or Corollary 4: ( T F (T = G (T = S exp r s ds E Q [ exp ( T δ s ds ]. 1154

13 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Again, his is simpler han evaluaing he sandard Equaion (3: ( T ( [ T F (T = S exp r s ds E Q exp δ s ds 1 ] 2 σ 2 T + σ WT 1. Corollary 1 implies ha a call opion on S is equal, afer he numeraire change, o he price of a pu opion on S wih a deerminisic dividend rae and a sochasic shor rae. Example 5 shows ha opion pricing models for sochasic dividend models can be obained from sochasic ineres rae models and vice versa. The European call opion formula in Jamshidian and Fein s (199 Ornsein Uhlenbeck δ and consan r model can be obained, via some parameer changes, from he European pu opion formula in Rabinovich s (1989 Ornsein Uhlenbeck r and consan δ model. The example also illusraes how Corollary 3 can simplify fuures pricing by reducing a wo-facor problem o a one-facor problem when he volailiy erms of he asse reurns and he sae variable do no depend on he asse price. Example 6. Exchange opions. The risk-neural price processes of asses a and b saisfy ds a S a ds b S b = (r δ a d + σ a dw 1, = (r δ b d + σ b (ρdw ρ 2 dw 2, where he volailiy coefficiens, σ a and σ b, and he insananeous correlaion beween asse reurns, ρ [ 1, 1], are consans. The shor rae, r, and he dividend raes, δ a and δ b, are funcions of he m-dimensional sae variable vecor, X, which saisfies dx = µ(x d + φ(x dw, where µ is m 1, φ is m d, and again W [W 1,...,W d ] is sandard Brownian moion under Q. The price raio S b S bsa /Sa under Q is he consan volailiy process d S b / S b = (δ a δ b d + (σ a ρσ b d W 1 1 ρ 2 σ b d W 2, S b = Sb, where again W [ W 1,..., W d ] is sandard Brownian moion under Q. The drif of X under Q is modified by adding he produc of he volailiy of asse a and he firs column of he sae-variable volailiy marix: dx = [µ(x + σ a φ(x e]d φ(x d W, 1155

14 The Review of Financial Sudies/v12n51999 where e = [1,,...,]. From Corollary 5, he valuaion of an exchange opion can be reduced o he compuaion of an ordinary call opion on S wih shor rae δ a and dividend rae δ b. The nex example is he jump-diffusion model of Meron (1976. Example 7. Meron (1976. As in Example 1, he asse price follows a Poisson jump process wih inensiy λ under he risk-neural probabiliy measure, Q. A jump ime τ i,i = 1, 2,...,he sock price raio is lognormally disribued: log[s(τ i /S(τ i ] (α, γ 2, where (m,vdenoes a normal disribuion wih mean m and variance v. Beween jumps he sock price saisfies ds = [r δ λ(e α+γ 2 /2 1] d S + σ dw 1, τ i < <τ i+1, i =, 1,..., where τ. Using he same calculaion as in Example 1 (or he general resuls in he appendix, he inensiy under Q is equal o he produc of he inensiy under Q and he expeced price raio a jumps: λ = λ exp(α γ 2. The appendix shows ha he disribuion funcions under Q and Q of he sock price raio, denoed by ( and (, respecively, saisfy (dy = (dy exp( α 1 2 γ 2 y. From [ 1 (dy = exp 1 ( ] log(y α 2 dy, 2πγy 2 γ i is sraighforward o show ha he logarihm of he sock price raio under Q is sill normally disribued wih variance γ 2, bu wih mean α + γ 2. The dynamics of S KS /S are herefore log[ S(τ i / S(τ i ] ( α γ 2,γ 2, i = 1, 2,..., d S S = [ ] δ r λ(e α γ 2 /2 1 d + σ d W 1, τ i < <τ i+1, i =, 1,... When he mean sock reurn a jumps is zero, ha is α = γ 2 /2, hen λ = λ and he jump reurns of S under Q and S under Q have he same 1156

15 Changes of Numeraire for Pricing Fuures, Forwards, and Opions disribuions. In his special case, Corollary 1 implies ha simply swiching he roles of he srike price and he curren asse price, and swiching he roles of he shor rae and dividend rae in he American call price formula gives he American pu price. Baes (1991 derives his special case for fuures opions from he parial differenial equaion for he opion price. When applied o he case of fuures opions, several previous examples conain special cases in which he disribuions of he reurns of F under Q and F under Q are idenical. 11 In such cases he equivalence relaionship in Corollary 1 akes a paricularly simple form: he American pu price is obained from he American call price formula by simply swiching he roles of he srike price and curren fuures price. A change of variables can hen be used o relae American call and pu prices on he same underlying fuures price process. Example 8 shows general condiions under which Corollary 1 can be used o relae call and pu prices on he same underlying fuures price process, and, using he ideas of Baes (1991, shows how hese condiions can be esed. I is easy o show ha he same condiions imply ha he geomeric average of he early exercise boundaries of oherwise idenical American calls and pus is equal o he srike price. Example 8. Empirical implicaions for fuures opions. Le F denoe he fuures price for delivery a D, where D T. The fuures price is assumed o follow a jump-diffusion process under Q wih inensiy λ(x and jumps a τ i,i = 1, 2,...,when he fuure price raio has he disribuion Q (F(τ i /F(τ i y = (y, y, and beween jumps, df = λ (X (1 µ d + σ(x dw 1 F τ i < <τ i+1, i =, 1,..., where τ and µ y R + yd (y is he expeced price raio a jumps. The m-dimensional sae variable vecor, X, saisfies dx = µ(x d + φ(x dw, where W [W 1,...,W d ] is sandard Brownian moion under Q, and he coefficiens µ and φ have he appropriae dimensions. The shor rae may also be a funcion of X. We obain an equivalence formula for calls and pus wih he same ime zero underlying price by defining ˆF = F 2 /F (= F F /K. Using he resuls in he appendix, ˆF is a jump-diffusion process under Q wih inensiy 11 See Example 1 when u = d 1 and µ = 1; Example 2 when ξ = ; Examples 4 and 5 when ρ = ; and Example 7 when α = σ 2 /

16 The Review of Financial Sudies/v12n51999 λ ( = λ ( µ and dynamics Q ( ˆF(τ i / ˆF(τ i y 1 = (y, y >, i = 1, 2,..., d ˆF ˆF = λ (X (1 µ 1 d + σ(x d W 1, τ i < <τ i+1, i =, 1,..., ˆF = F, dx = [µ(x + σ(x φ(x e]d φ(x d W, where (dy = (dyyµ 1, and, as earlier, e [1,,...,] and W [ W 1,..., W d ] is sandard Brownian moion under Q. Sufficien condiions for he disribuions of F under Q and ˆF under Q o be idenical are (a φ( e = (he insananeous changes in he sae variables and fuures price are uncorrelaed, and (b λ (no jumps or (y = [y 1, u (du, y >. Noe ha he resricion on he disribuion funcion in (b implies ha µ = 1. When he fuures price raio a jumps has a discree disribuion, as in Example 1, hen he resricion on is equivalen o (y = y 1 (y 1, y > [ (y and (y 1 are he Q-probabiliies of oucomes y and y 1, respecively]. When he jump disribuion funcion is differeniable, as in Example 7, hen he resricion is equivalen o (y = (y 1 y 3, y >. Defining x = K/F, hen Corollary 1 implies he following relaionship beween calls and pus on fuures prices wih he same iniial value: ( E Q e τ rsds max[f τ F x, ] = xe Q (e τ rsds max[f /x ˆF τ, ], (7 for any x >. Under condiions (a and (b, Equaion (7 relaes call and pu prices on he same underlying fuures price process. We can es hese condiions by comparing he relaive prices of American calls and pus. For example, if boh condiions hold, hen oherwise idenical a-he-money calls and pus should be priced he same. Baes (1991 proves Equaion (7, using parial differenial equaion mehods, for he cases of geomeric Brownian moion, Meron s (1976 jumpdiffusion model wih zero-mean jump reurns (α = γ 2 /2 in Example 7 above, and for he case of a diffusion sock price process and an uncorrelaed one-dimensional sae variable represening sochasic volailiy. 1158

17 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Appendix: Price and Sae-Variable Dynamics This appendix derives price and sae-variable processes under a change of numeraire and corresponding change of measure for a general class of diffusion and jump-diffusion processes. Le W [W 1,...,W d ] be a vecor of d independen sandard Brownian moions under he risk-neural measure Q. The asse price, S, and he sae variables, X [X 1,...,X m ], saisfy ds S = (r δ d + σ(s, X dw 1 dx = µ(x d + φ(x dw, where µ is m 1, and φ is m d. To simplify noaion (and wihou loss of generaliy, he differenial of S is defined as a funcion of he differenial of W 1 only, and hus he volailiy process σ is a scalar. The shor rae, r, and payou rae, δ, are given by r = β(x and δ = κ(x, where β and κ are real-valued funcions. I is easy o generalize he model o allow he parameers of X, as well as r and δ, o depend on S also. The Radon Nikodym derivaive [Equaion (2] has he explici soluion [ Z = exp 1 ] σ(s s, X s 2 ds + σ(s s, X s dw 1 s. 2 Define he d-lengh column vecor e = [1,,...,]. By Girsanov s heorem, W = e σ(s s, X s ds W is d-dimensional sandard Brownian moion under Q, where d Q/dQ = Z T. Defining S = KS /S and applying Iô s lemma and Girsanov s heorem, we obain, d S = (δ r d + σ(ks / S, X d W 1 S, S = K, dx = [µ(x + φ(x σ (KS / S, X e] d φ(x d W. The modificaion o he drif, φ(x σ (KS / S, X e, represens he insananeous covariance beween asse reurns and he incremens in he vecor of sae variables. We now inroduce d jump processes, each indexed by i, i = 1,...d. Each jump process is characerized by he double sequence (Tn i, J n i; n = 1, 2,..., where T n represens he ime and Jn i he amoun of he nh jump.12 Le B(R denoe he Borel σ -algebra of subses of he real line. For each se A B(R and i {1,...d}, he couning process N i (A represens he number of jumps wih a magniude in he se A by ime. The jump processes are assumed o be independen of W and are assumed o saisfy [N i ((,, N 1 ((, ] =, a.s., i 1; ha is, he jumps of he firs process do no 12 See Brémaud (1981, for all he needed resuls on poin processes. This discussion borrows heavily from chaper VIII. 1159

18 The Review of Financial Sudies/v12n51999 coincide wih he jumps of he oher processes. 13 The couning measure p i (d dy is defined as p i ((, ] A = N i (A, A B(R, i = 1,...,d. Le λ i (dy denoe he inensiy kernel of pi (d dy, i = 1,...,d; for each A B(R, λ i (A is an F -predicable process. Wrie he inensiy kernel as λ i (dy = λi i (dy, i = 1,...,d, where λ i λ i (R and i (dy = λi (dy/λi on {λ i > }. The process i is a disribuion funcion for each. Loosely speaking, λ i d can be inerpreed as he probabiliy, condiional on F, of a jump in he nex d unis of ime; i (A can be inerpreed as he probabiliy of a jump wih magniude in he se A condiional on F and given ha a jump occurs a. Define he compensaed poin processes q [q 1,...,q d ], where q i (d dy = p i (d dy λ i i (dyd, i = 1,...,d. For any bounded and F -predicable process f (, y, he process M i defined by M i = f (s, yq i (ds dy, >, i = 1,...,d R is a maringale [see Brémaud (1981 for less resricive condiions on f ]. A he jump imes, M i (T i n = Mi (T i n + f i (T i n, J i n, n = 1, 2,..., and beween jumps, dm i = f i (, yλ i i (dyd, T i n 1 < < T i n, n = 1, 2,... R The asse price and he sae variables saisfy ds = (r δ d + σ(s, X dw 1 + g(s, X, yq 1 (d dy, S R dx = µ(x d + φ(x dw + G(X, yq(d dy, R where σ and g are real-valued funcions, µ is m 1, and φ and G are each m d. We allow λ i and i o be funcions of S and X. 13 The Brownian moion W inroduced above is defined on he probabiliy space ( W, F W, P W. The jump processes are defined on he probabiliy space ( p, F p, P p where he filraion is ha generaed by he hisory of he processes: F p = σ (N i s (A; s [, ], A B(R, i {1,...d}. On he produc space, (, F, P ( W p, F W he couning jump processes and W are independen. F p, P W P p, 116

19 Changes of Numeraire for Pricing Fuures, Forwards, and Opions The Radon Nikodym derivaive [Equaion (2] is [ Z = exp 1 ] σ 2 s 2 ds + σ s dw 1 s g 1 (S s, X s, yλ 1 s 1 (dyds R [1 + g(s(t 1 1 n, X (Tn, J 1 n ]1 { Tn 1 }. n 1 The Girsanov Meyer heorem implies W = e σ(s s, X s ds W is d-dimensional sandard Brownian moion under Q. The inensiy kernel of p 1 under Q is characerized by λ 1 = λ 1 [1 + g(s, X, y] 1 (dy, R and 1 (dy = 1 (dy 1 + g(s, X, y R [1 + g(s, X, y] 1 (dy. The following simple heurisic derivaion can be used o obain he inensiy kernel under Q. Suppose here have been exacly n 1 jumps in he asse price before ime. Then λ 1 (dyd = Q ( T 1 n [, + d], J 1 n [y, y + dy] F = Z 1 E Q ( Z+d 1 { T 1 n [,+d], J 1 n [y,y+dy] } F, where he maringale propery of Z and ieraed expecaions have been { used o ge he second equaliy. Now subsiue Z +d = Z [1 + g(s, X, y]on Tn 1 [, + d]} (ignoring smaller-order erms o ge λ ( 1 (dyd = [1 + g(s, X, y]q T 1 n [, + d], J 1 n [y, y + dy] F = [1 + g(s, X, y]λ 1 (dyd. The inensiy kernels of (p 2,...,p d are unalered by he measure change. The compensaed poin processes under Q are herefore q [ q 1,..., q d ], where q 1 (d dy = p 1 (d dy λ 1 1 (dyd and q i = q i, i = 2,...,d. The processes under he numeraire change saisfy d S = (δ r d + σ(ks / S, X d W 1 S g(ks / S, X, y R 1 + g(ks / S, X, y q1 (d dy, 1161

20 The Review of Financial Sudies/v12n51999 where dx = µ (KS / S, X d φ(x d W + G(X, y q(d dy, R µ (KS / S, X µ(x + φ(x σ (KS / S, X e + G(X, ye[ λ 1 1 (dy λ1 1 (dy]. R References Baes, D., 1991, Opion Pricing Under Asymmeric Processes, wih Applicaions o Opions on Deuschemark Fuures, working paper, Universiy of Pennsylvania. Bjerksund, P., and G. Sensland, 1993, American Exchange Opions and a Pu-Call Transformaion: A Noe, Journal of Business Finance and Accouning, 2, Brémaud, P., 1981, Poin Processes and Queues, Springer-Verlag, New York. Byun, S., and I. Kim, 1996, Relaionships Beween American Pus and Calls on Fuures Conracs, working paper, Korea Advanced Insiue of Science and Technology. Carr, P., 1993, Deriving Derivaives of Derivaive Securiies, working paper, Cornell Universiy. Carr, P., and M. Chesney, 1996, American Pu Call Symmery, working paper, Morgan Sanley, Groupe H.E.C. Chesney, M., and R. Gibson, 1993, Sae Space Symmery and Two Facor Opion Pricing Models, in J. Janssen and C. H. Skiadas (eds., Applied Sochasic Models and Daa Analysis, World Scienific Publishing, River Edge, N.J. Cox, J., 1975, Noes on Opion Pricing I: Consan Elasiciy of Variance Diffusions, working paper, Sanford Universiy. Duffie, D., 1992, Dynamic Asse Pricing Theory, Princeon Universiy Press, Princeon, N.J. Dufresne, P., W. Keirsead, and M. Ross, 1997, Maringale Pricing: A Do-I-Yourself Guide o Deriving Black-Scholes, in Equiy Derivaives: Applicaions in Risk Managemen and Invesmen, Risk Books, London. Geman, H., N. El Karoui, and J. Roche, 1995, Changes of Numeraire, Changes of Probabiliy Measure and Opion Pricing, Journal of Applied Probabiliy, 32, Grabbe, J., 1983, The Pricing of Call and Pu Opions on Foreign Exchange, Journal of Inernaional Money and Finance, 2, Heson, S., 1991, A Closed-Form Soluion for Opions wih Sochasic Volailiy, wih Applicaion o Bond and Currency Opions, working paper, Yale Universiy. Ingersoll, J., 1997, Digial Conracs: Simple Tools for Pricing Complex Derivaives, working paper, Yale Universiy. Jamshidian, F., and M. Fein, 199, Closed Form Soluions for Oil Fuures and European Opions in he Gibson Schwarz Model: A Noe, working paper, Merrill Lynch Capial Markes. Karazas, I., 1988, On he Pricing of American Opions, Applied Mahemaics and Opimizaion, 17,

21 Changes of Numeraire for Pricing Fuures, Forwards, and Opions Kholodnyi, V., and J. Price, 1998, Foreign Exchange Symmery, World Scienific Publishing, River Edge, N.J. McDonald, R., and M. Schroder, 199, A Pariy Resul for American Opions, working paper, Norhwesern Universiy. Meron, R., 1976, Opion Pricing when Underlying Sock Reurns are Disconinuous, Journal of Financial Economics, 3, Olsen, T., and G. Sensland, 1991, Invarian Conrols in Sochasic Allocaion Problems, in D. Lund and B. B. Øksendal (eds., Sochasic Models and Opion Values, Elsevier Science, New York. Proer, P., 1992, Sochasic Inegraion and Differenial Equaions, Springer-Verlag, New York. Rabinovich, R., 1989, Pricing Sock and Bond Opions when he Defaul-Free Rae is Sochasic, Journal of Financial and Quaniaive Analysis, 24, Reiner, E., and M. Rubinsein, 1991, Unscrambling he Binary Code, RISK, 4, Schroder, M., 1989, Compuing he Consan Elasiciy of Variance Opion Pricing Formula, Journal of Finance, 44, Schroder, M., 1992, Some Opion Pricing Resuls Obained Using a Change of Numeraire, Ph.D. summer paper, Norhwesern Universiy. 1163