AN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM MODEL WITH STOCHASTIC INTEREST RATES

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1 Inernaional Journal of Pure and Applied Mahemaics Volume 76 No , ISSN: (prined version url: hp:// PA ijpam.eu AN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM MODEL WITH STOCHASTIC INTEREST RATES Hsien-Jen Lin Deparmen of Applied Mahemaics Aleheia Universiy Tamsui, New Taipei Ciy, 2513, TAIWAN Absrac: Recenly, a spo maringale measure pricing mehod o derive pricing formulas of quano forward conracs wihin he Heah, Jarrow and Moron (1992 ineres rae model was published. Alhough he spo maringale measure pricing mehod is ineresing, i requires ha one knows somehing abou he dependence beween he discoun facor and he payoff of he derivaive securiy. Thus, more procedures are required for compuaions. This paper proposes anoher simple approach. In paricular, we demonsrae he forward measure pricing mehodology o he valuaion of quano forward conracs wihin he HJM ineres rae model and provide grea ease in deriving closed form soluions. AMS Subjec Classificaion: 91B28, 6G48, 6J75 Key Words: quano forward conracs, Heah-Jarrow-Moron model, forward measure pricing mehod 1. Inroducion Recenly, a spo maringale measure pricing mehod o derive pricing formulas of quano forward conracs wihin he Heah, Jarrow and Moron (1992, hereafer HJM ineres rae model under deerminisic volailiies was pub- Received: January 13, 212 c 212 Academic Publicaions, Ld. url:

2 55 H.-J. Lin lished. Alhough he spo maringale measure pricing mehod is ineresing, i requires ha one knows somehing abou he dependence beween he discoun facor and he payoff of he derivaive securiy. Thus, more procedures are required for compuaions. The idea of erm srucure modelling wih he direc use of forward measures was previously exploied by Hansen (1994, who used he forward inducion o produce an arbirage-free diffusion-ype model for prices of a finie family of zero-coupon bond wih differen mauriies. I is well known ha he forward measure pricing mehodology (Jamshidian, 1987, 1989; and Geman e al., 1995 has been widely used in pricing securiies when ineres raes are sochasic. This paper proposes anoher simple mehod which uses he forward measure pricing mehodology o he valuaion of quano forward conracs wihin he HJM ineres rae model. The echnique provides grea ease in deriving closed form soluions for various derivaives conrac wih European-syle payoffs under sochasic ineres raes. An ouline for his paper is as follows: Secion 2 inroduces he erminology, noaion, and assumpions of he model along he same lines as in Amin and Jarrow (1991. Secion 3 applies he forward measure pricing mehodology o he valuaion of quano forward conracs wihin he HJM ineres rae model, and Secion 4 concludes he paper. 2. The Economy Le W be a n-dimensional sandard Brownian moion given on a filered probabiliy space (Ω,F,P,(F. The filraion (F is assumed o be he righ-coninuous and P-compleed version of he naural filraion of W. We consider a coninuous-ime rading economy wih a rading inerval [,T ] for he fixed horizon dae T. As here will be considerable noaion, for easy reference, we lis i all in one place. f i (,T = he ih foreign marke insananeous forward rae conraced a ime for borrowing and lending a ime T wih T T. f(,t = he domesic marke insananeous forward rae conraced a ime for borrowing and lending a ime T wih T T. B i (,T = he ime price in he ih foreign currency of a foreign zero-coupon bond wih mauriy T and uni face value. B(,T = he ime price in he domesic currency of a domesic zero-coupon bond wih mauriy T and uni face value. Q i ( = he spo exchange rae a ime (denominaed in domesic currency

3 AN EASY METHOD TO PRICE QUANTO FORWARD per uni of he ih foreign currency. Q i = he prescribed exchange rae (denominaed in domesic currency per uni of he ih foreign currency. K i = he srike price denominaed in he ih foreign currency. r i ( = he spo ineres rae a ime in he ih foreign marke. r( = he spo ineres rae a ime in he domesic marke. B i ( = he foreign savings accoun a ime in he ih foreign marke, B i ( = exp[ ri (udu]. B( = he domesic savings accoun a ime, B( = exp[ r(udu]. Z i ( = he price of a given asse a ime, expressed in unis of he ih foreign currency. P = he domesic (spo maringale probabiliy measure. P T = he domesic forward maringale probabiliy measure. Here he superscrip i indicaes ha a given process represens a quaniy relaed o he ih foreign marke. As in Amin and Jarrow (1991, Heah e al. (1992, and Wu (28, he following assumpions and relaions are also used: Assumpion 1. (Domesic Forward Ineres Rae Dynamics df(,t = α(,td+σ(,t dw(, [,T], T [,T ], (1 where α(,t and σ(,t are subjec o some regulariy condiions (see Amin and Jarrow, Using (1, we see ha he dynamics of domesic bond price processes under he marke probabiliy measure P are db(,t = B(,T [(r( α (,T+ 12 ] σ (,T 2 d σ (,T dw(, (2 wherefor any [,T] wehave α (,T = T α(,udu, σ (,T = T σ(,udu (see Heah e al., 1992, for a proof. As usual, he do sands for he Euclidean inner produc in R n. Also, we wrie o denoe he Euclidean norm in R n. Assumpion 2. (The ih Foreign Forward Ineres Rae Dynamics df i (,T = α i (,Td+σ i (,T dw(, [,T], T [,T ], (3 where he drif and volailiies are assumed o saisfy he same condiions as for he corresponding erms in Assumpion 1. Similarly, one can show ha he dynamics of B i (,T under he marke measure P are

4 552 H.-J. Lin db i (,T = B i (,T [( r i ( α i (,T+ 1 σ i (,T 2 ] d σ i (,T dw(, (4 2 where for any [,T] we have α i (,T = T α i (,udu, σ i (,T = T σ i (,udu. Assumpion 3. (The ih Spo Exchange Rae Dynamics dq i ( = Q i ([µ Q i(d+σ Q i( dw(], (5 where µ Q i( and σ Q i( are subjec o some regulariy resricions (see Amin and Jarrow, Assumpion 4. The price of an arbirary foreign asse Z i ha pays no dividends saisfies dz i ( = Z i ([µ Z i(d+σ Z i( dw(], (6 where he drif and volailiies are assumed o saisfy he same condiions as for he corresponding erms in Assumpion 3. I follows from Proposiion 1 in Heah e al. (1992 ha we can readily derive he dynamics of B(,T, B i (,T and Z i ( under he domesic maringale measure P as follows: db(,t = B(,T[r(d σ (,T dw (], (7 [( ] db i (,T = B i (,T r i (+σ Q i( σ i (,T d σ i (,T dw (, (8 and dz i ( = Z i ([(r i ( σ Z i( σ Q i(d+σ Z i( dw (]. (9 3. Pricing Formulas for Quano Forward Conracs This secion demonsraes how o value quano forward conracs in he preceding economy by he forward measure pricing mehod. Recenly, pricing formulas of quano forward conracs wihin he Heah, Jarrow and Moron

5 AN EASY METHOD TO PRICE QUANTO FORWARD (1992 ineres rae model by he spo maringale measure pricing mehod was published (see Wu, 28; Musiela and Rukowski, 25 and he references herein, in which he compuaion of he risk-neural pricing formula, B 1 (E P T (B(TX F, requires ha we know somehing abou he dependence beween he discoun facor B(T and he payoff X of he derivaive securiy. Thus, more procedures are required for compuaions. I is well known ha he forward measure pricing mehodology (Jamshidian, 1987 and 1989; and Geman e al., 1995 has been widely used in pricing securiies when ineres rae are sochasic. Accordingly, we focus on he forward measure pricing mehodology o he valuaion of quano forward conracs wihin he HJM ineres rae model. Definiion 1. A probabiliy measure P T on (Ω,F T equivalen o P wih he Radon-Nikodým densiy given by he formula dp T dp = 1 B(TB(,T P a.s. (1 is called he forward maringale measure for he selemen dae T. When he bond price is governed by (7, in view of (1 and Girsanov s heorem, he Radon-Nikodým densiy of P T wih respec o P equals ( dp T dp = ε T σ (u,t dw (u P a.s., (11 where he member on he righ-hand side of (11 is he Doléans-Dade exponenial, which is given by he following expression ε ( σ (u,t dw (u ( = exp σ (u,t dw (u 1 2 Furhermore, he process W T given by he formula W T ( = W (+ σ (u,t 2 du. σ (u,tdu, [,T], (12 follows a sandard n-dimensional Brownian moion under he forward measure P T. Geman (1989 observed ha he forward price of any financial asse follows a (local maringale under he forward neural probabiliy associaed wih he selemen dae of a forward conrac.

6 554 H.-J. Lin Lemma 1. The forward price a ime for he delivery dae T of an aainable coningen claim X, seling a ime T, equals F X (,T = E PT (X F, [,T], (13 provided ha X is P T inegrable. In paricular, he forward price process F X (,T, [,T], is a maringale under he forward measure P T. The following well known lemma provides a version of he risk neural valuaion formula which is ailored o he sochasic ineres rae framework. Lemma 2. The arbirage price of an aainable coningen claim X which seles a ime T is given by he formula Π (X = B(,TE PT (X F, [,T]. (14 Nex we deal wih he four common quano forward conracs as follows: Theorem 1. A quano forward conrac wih he payoff a expiry equals f 1 (T = Q i (T(Z i (T K i. Then for T, he risk-neural price a ime of he quano forward conrac is f 1 ( = Q i ((Z i ( K i B i (,T. (15 Proof. By applying Lemmas 1, 2, we obain ( f 1 ( = B(,TE PT Q i (T(Z i (T K i F ( Q i (TZ i (T = B(,TE PT B(T,T F ( Q B(,TK i i (TB i (T,T E PT B(T,T F = Q i ((Z i ( K i B i (,T. Theorem 2. A quano forward conrac wih he payoff a expiry equals f 2 (T = Q i (Z i (T K i (such a conrac is also known as a guaraneed exchange rae forawrd conrac (a GER forward conrac for shor. Then for T, he risk-neural price a ime of he quano forward conrac is f 2 ( = Q i (Z i (QA(,T K i B(,T, (16

7 AN EASY METHOD TO PRICE QUANTO FORWARD where and QA(,T = B(,T B i ρ(,t, (17 (,T ρ(,t = { exp T ( ( } σ Z i(u+σ i (u,t σ Q i(u σ i (u,t+σ (u,t du. (18 Proof. I follows immediaely from (8, (9 and (12 ha ( Z i ( d B i (,T = Z i ( [( B i σ (,T Z i(+σ i (,T ( ( dw T ( σ Q i( σ i (,T+σ (,T ] d. (19 In view of he general valuaion (14, i is clear ha we need o evaluae he condiional expecaions ( f 2 ( = B(,TE Qi PT (Z i (T K i F ( Z = B(,T Q i i (T E PT B i (T,T F B(,TK i Qi We know from (19 ha = B(,T Q i I B(,TK i Qi. Z i (T B i (T,T = Z i { ( T B i (,T ρ(,texp dw T (u 1 2 T ( σ Z i(u+σ i (u,t σ Z i(u+σ i (u,t 2 du }. (2 To evaluae I, we inroduce an auxiliary probabiliy measure P R by seing ( dp ( R = ε T σ dp Z i(u+σ i (u,t dw T (u = η T, P T a.s. T By virue of he Bayes rule, i is easily seen ha I equals ( I = Zi ( B i (,T ρ(,te ηt P T η F = Zi ( B i (,T ρ(,te P R (1 F

8 556 H.-J. Lin and hus f 2 ( = Q i (Z i (QA(,T K i B(,T. = Zi ( B i (,T ρ(,t Theorem 3. A quano forward conrac wih he payoff a expiry equals f 3 (T = Q i Z i (T Q i (TK i. Then for T, he risk-neural price a ime of he quano forward conrac is f 3 ( = Q i Z i (QA(,T K i Q i (B i (,T, (21 where QA(,T is given by (17. Proof. f 3 ( = B(,TE PT ( Qi Z i (T Q i (TK i F = B(,T Q i ( E PT Z i (T F B(,TK i ( E PT Q i (T F = B(,T Q i Z i ( ( Q i (TB i (T,T B i (,T ρ(,t B(,TKi E PT B(T,T F = Q i Z i (QA(,T B(,TK iqi (B i (,T B(,T = Q i Z i (QA(,T K i Q i (B i (,T. Theorem 4. A quano forward conrac wih he payoff a expiry equals f 4 (T = Q i (TZ i (T Q i K i. Then for T, he risk-neural price a ime of he quano forward conrac is f 4 ( = Q i (Z i ( K i Qi B(,T. (22 Proof. ( f 4 ( = B(,TE PT Q i (TZ i (T Q i K i F ( Q i (TZ i (T = B(,TE PT B(T,T F B(,T Q i K i = B(,T Qi (Z i ( B(,T B(,T Q i K i = Q i (Z i ( K i Qi B(,T.

9 AN EASY METHOD TO PRICE QUANTO FORWARD Concluding Remarks The forward measure is convenien in calculaing various coningen claim prices under sochasic ineres raes. This sudy proposes a simple mehod which uses he forward measure pricing mehodology o derive he valuaion formulas for quano forward conracs wihin he Heah, Jarrow and Moron (1992 ineres rae model. This echnique provides grea ease in deriving closed form soluions for various derivaives conrac wih European-syle payoffs under sochasic ineres raes. For insance, i has been applied o he valuaion of quano opions under sochasic ineres. References [1] K.I. Amin, R.A. Jarrow, Pricing foreign currency opions under sochasic ineres raes, Journal of Inernaional Money and Finance, 1 (1991, [2] H. Geman, The imporance of he forward neural probabiliy in a sochasic approach of ineres raes, Working Paper, ESSEC (1989. [3] H. Geman, E.K. Nicole, J.C. Roche, Changes of Numéraire, Changes of Probabiliy Measure and Opion Pricing, Journal of Applied Probabiliy, 32 (1995, [4] O.K. Hansen, Theory of an arbirage-free erm srucure, In: Economic and Financial Compuing, Summer (1994, [5] D. Heah, R. Jarrow, A. Moron, Bond pricing and he erm srucure of ineres raes: A new mehodology for coningen claim valuaions, Economerica, 6 (1992, [6] F. Jamshidian, Pricing of coningen claims in he one facor erm srucure model, Working Paper, Merrill Lynch Capial Markes (1987. [7] F. Jamshidian, An exac bond opion formula, Journal of Finance, 44 (1989, [8] M. Musiela, M. Rukowski, Maringale Mehods in Financial Modelling, Springer-Verlag, Berlin Heidelberg (25. [9] T.P. Wu, Pricing and hedging quano forward conracs under HJM model, Journal of he Chinese Saisical Associaion, 46 (28,

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