arxiv:1407.5091v1 [q-fin.pr] 18 Jul 014 An exact and explicit forula for pricing Asian options with regie switching Leunglung Chan and Song-Ping Zhu July, 014 Abstract This paper studies the pricing of European-style Asian options when the price dynaics of the underlying risky asset are assued to follow a Markovodulated geoetric Brownian otion; that is, the appreciation rate and the volatility of the underlying risky asset depend on unobservable states of the econoy described by a continuous-tie hidden Markov process. We derive the exact, explicit and closed-for solutions for European-style Asian options in a two-state regie switching odel. Key words: Option pricing; Markov-odulated geoetric Brownian otion; Regie switching; Asian options. 1 Introduction The pricing, hedging and risk anageent of contingent clais has becoe a popular topic because contingent clais are now widely used to transfer risk in financial School of Matheatics and Statistics, University of New South Wales, Sydney, NSW, 05, Australia, Eail: leung.chan@unsw.edu.au School of Matheatics and Applied Statistics, University of Wollongong, Wollongong, NSW 5, Australia, Eail: spz@uow.edu.au 1
arkets. The pioneering work of Black and Scholes (1973) and Merton (1973) laid the foundations of the field and stiulated iportant research in option pricing theory. The Black-Scholes-Merton forula has been widely adopted by traders, analysts and investors. The contingent clais traded in the arket not only include vanilla European options but also exotic options such as Asian options. Asian options are path-dependent options whose payoff depends on an average of the stock prices over a certain tie period. Asian options are used for hedging purpose. Traders ay be interested to hedge against the average price of a coodity over a period. The use of Asian options ay avoid price anipulation near the end of the period. Most Asian options are European style because an Asian option with the Aerican early exercise feature ay be redeeed as early as the beginning of the averaging period and thus lose hedging purpose fro averaging. Since no general closed-for solution for the price of the Asian option based on arithetic averaging is known, a variety of ethods have been developed to study this proble. Many papers studied Asian options including Kena and Vorst (1990), Turnbull and Wakean (1991), Ritchken et al. (1993), Gean and Yor (1993), Curran (1994), Rogers and Shi (1995), Zhang (1995), Boyle et al. (1997) and Fu et al. (1999). Despite its popularity the Black-Scholes-Merton odel fails in various ways, such as the fact that iplied volatility is not constant. During the past few decades any extensions to the Black-Scholes-Merton odel have been introduced in the literature to provide ore realistic descriptions for asset price dynaics. In particular, any odels have been introduced to explain the epirical behavior of the iplied volatility sile and sirk. Such odels include the stochastic volatility odels, jupdiffusion odels, odels driven by Lévy processes and regie switching odels. Maybe the siplest way to introduce additional randoness into the standard Black-Scholes-Merton odel is to let the volatility and rate of return be functions of a finite state Markov chain. There has been considerable interest in applications of regie switching odels driven by a Markov chain to various financial probles. Many papers in a regie switching fraework include Elliott and van der Hoek (1997), Guo (001), Elliott et al. (001), Buffington and Elliott (00a,b), Elliott et al. (003) and Elliott et al. (005). In addition, Boyle and Dravia (007) used the nuerical ethod to solve the syste of coupled partial differential equations for the price of exotic options under regie switching. Siu et al. (008) priced credit default swaps under a Markov-odulated Merton structural odel. Yuen and Yang (009) proposed a recobined trinoial tree to price siple options and barrier options in a jup-diffusion odel with regie switching. Yuen and Yang (010)
used a trinoial tree ethod to price Asian options and equity-indexed annuities with regie switching. Zhu et al. (01) derived a closed-for solution for European options with a two-state regie switching odel. However, there is no closed-for solution to Asian options under a regie switching odel. In this paper, we study the pricing of European-style Asian options when the price dynaics of the underlying risky asset are assued to follow a Markov-odulated geoetric Brownian otion. The Markov-odulated odel provides a ore realistic way to describe and explain the arket environent. It has been entioned in Yao et al. (003) that it is of practical iportance to allow the arket paraeters to respond to the oveents of the general arket levels since the trend of general arket levels is a key factor which governs the price oveents of individual risky assets. We derive an analytical solution for Asian option by eans of the hootopy analysis ethod (HAM). HAM was initially suggested by Ortega and Rheinboldt (1970) and has been successfully used to solve a nuber of heat transfer probles, see Liao (1997), Liao and Zhu (1999), and fluid-flow probles, see Liao and Zhu (1996), Liao and Capo (00). Zhu (006) used HAM to obtain an analytic pricing forula for Aerican options in the Black-Scholes odel. Gounden and O Hara (010) extended the work of Zhu to pricing Aerican-style Asian options of floating strike type in the Black-Scholes odel. Leung (013) used HAM to derive an analytic forula for lookback options under stochastic volatility. This paper is organized as follows. Section describes the asset price dynaics under the Markov-odulated geoetric Brownian otion. Section 3 forulates the partial differential equation syste for the price of a floating-strike Asian option. Section 4 derives an exact, closed-for solution for the floating-strike Asian option. Section 5 discusses a syetry between fixed-strike and floating-strike Asian options. The final section contains a conclusion. Asset Price Dynaics Consider a coplete probability space (Ω, F, P), where P is a real-world probability easure. Let T denote the tie index set [0,T] of the odel. Write {W t } t T for a standard Brownian otion on (Ω,F,P). Suppose the states of an econoy are odelled by a finite state continuous-tie Markov chain {X t } t T on (Ω,F,P). Without loss of generality, we can identify the state space of {X t } t T with a finite set of unit 3
vectors X := {e 1,e,...,e N }, where e i = (0,...,1,...,0) R N. We suppose that {X t } t T and {W t } t T are independent. Let à be the generator [a ij] i,j=1,,...,n of the Markov chain. Fro Elliott et al. (1994), we have the following seiartingale representation theore for {X t } t T : X t = X 0 + t 0 ÃX s ds+m t, (.1) where {M t } t T is an R N -valued artingale increent process with respect to the filtration generated by {X t } t T. We consider a financial odel with two priary traded assets, naely a oney arket account B and a risky asset or stock S. Suppose the arket is frictionless; the borrowing and lending interest rates are the sae; the investors are price-takers. The instantaneous arket interest rate {r(t,x t )} t T of the bank account is given by: r t := r(t,x t ) =< r,x t >, (.) where r := (r 1,r,...,r N ) with r i > 0 for each i = 1,,...,N and <, > denotes the inner product in R N. In this case, the dynaics of the price process {B t } t T for the bank account are described by: db t = r t B t dt, B 0 = 1. (.3) Suppose the stock appreciation rate {µ t } t T and the volatility {σ t } t T of S depend on {X t } t T and are described by: µ t := µ(t,x t ) =< µ,x t >, σ t := σ(t,x t ) =< σ,x t >, (.4) where µ := (µ 1,µ,...,µ N ), σ := (σ 1,σ,...,σ N ) with σ i > 0 for each i = 1,,...,N and <, > denotes the inner product in R N. We assue that the price dynaics of the underlying risky asset S are governed by the Markov-odulated geoetric Brownian otion : ds t = µ t S t dt+σ t S t dw t, S 0 = s 0. (.5) 4
3 Asian option We now turn to the pricing of an Asian option of floating strike type in a regie switching odel. We consider continuously sapled arithetic average. The average fro continuous sapling is given by 1 T T 0 S u du. The price of Asian options based on an arithetic averaging is not known in closed for even in the Black-Scholes-Merton odel. We assue that Q is a risk-neutral easure and the price dynaics of the risky stock under Q are governed by Now define a process ds t = r t S t dt+σ t S t d W t, S 0 = s 0. (3.1) A t = t 0 S u du. (3.) We consider a floating-strike Asian put option whose payoff at tie T is V(T) = ( ) + 1 T A T S T. (3.3) The price at ties t prior to the expiration tie T of this put is given by the riskneutral pricing forula V(t,s,a,x) = E Q [e T t r udu V(T) S t = s,a t = a,x t = x] [ ( ) + = E Q e T r t udu 1 T A T S T S t = s,a t = a,x t = x]. (3.4) WriteV = (V(t,s,a,e 1 ),...,V(t,s,a,e N )).Applying thefeynan-kacforulatothe above equation, then V(t, s, a, x) satisfies the syste of partial differential equations V t +r ts V s +s V a + 1 V σ t s s r tv + V,Ãx = 0, (3.5) 5
and the boundary conditions V(T,s,a,x) = ( a T S ) +, T (3.6) V(t,s,0,x) = 0. (3.7) The price of an Asian option can be found by solving above syste of partial differential equations (PDE) in two space diensions. In the case of the Black-Scholes- Merton odel, Ingersoll (1987) observed that the three-diensional PDE for a floating strike Asian option can be reduced to a two-diensional PDE. However, by using S t A t as the state variable, and hence a nontraded variable as the nuéraire, artingale pricing techniques cannot be exploited. On the other hand, we will adopt S t as the nuéraire and by the change of easure reduce the three-diensional proble (3.4) to a two-diensional proble (for instance see Peskir and Shiryaev (006)). Write {F X t } t T and {F W t } t T for the P-augentation of the natural filtrations generated by {X t } t T and {W t } t T, respectively. For each t T, we define G t as the σ-algebra F X t F W t. We introduce a second process and define Y t = A t S t (3.8) dq dq G T = e T rudus T 0 (3.9) S 0 so that W Q t = W t t σ 0 udu is a standard Brownian otion with respect to {G t } t T under Q and using S t as the nuéraire, the valuation proble (3.4) becoes [( + ] 1 V(t,y,x) = E Q T Y T 1) Y t = y,x t = x. (3.10) Here dy t = ( 1 r t Y t ) dt+σt Y t dŵq t, (3.11) where ŴQ t = W Q t is a standard Brownian otion with respect to {G t } t T under Q. Then via the Feynan-Kac forula, the price of the Asian option V(t,y,x) satisfies the following syste of PDEs: V t +(1 r ty) V y + 1 V σ t y y + V,Ãx = 0, (3.1) 6
and the boundary conditions V(T,y,x) = ( y T 1)+, (3.13) V(t,0,x) = 0. (3.14) For each t T and i = 1,,...,N, let V i = V(t,y,e i ) and V := (V 1,V,...,V N ). We have that V satisfies the syste of coupled PDEs and the boundary conditions for each i = 1,,...,N. V i t +(1 r iy) V i y + 1 σ iy V i y + V,Ãe i = 0, (3.15) V(T,y,e i ) = ( y T 1) +, (3.16) V(t,0,e i ) = 0, (3.17) 4 A closed-for forula In this section, we restrict ourselves to a special case with the nuber of regies N being in order to siplify our discussion. By eans of the hootopy analysis ethod, we derive a closed-for solution for a floating strike Asian option under a regie switching odel. To solve the syste of PDEs effectively, we shall introduce the transforations z = ln(y) and τ i = (T t) σ i,i = 1,. Then the syste of equations (3.15)-(3.17) becoes L 1 V 1 (τ 1,z) = λ 1 ( V1 (τ 1,z) V (τ,z) ) ez σ 1 V 1 (0,z) = 0 li z V 1 (τ 1,z) = 0 V 1 z (4.1) where L 1 = τ 1 z (1+γ 1) z 7 (4.)
and L V (τ,z) = λ ( V (τ,z) V 1 (τ 1,z) ) ez σ V (0,z) = 0 li z V (τ,z) = 0 V z (4.3) where L = τ z (1+γ ) z, (4.4) λ i = a ii σ i and γ i = r i,i = 1,. The hootopy analysis ethod is adopted to solve σi V i,i = 1, fro equations (4.1) and (4.3). Now we introduce an ebedding paraeter p [0,1] and construct unknown functions V i (τ i,z,p),i = 1, that satisfy the following differential systes: (1 p)l 1 [ V 1 (τ 1,z,p) V 0 V 1 (0,z,p) = (1 p) V 0 1 (0,z) li z V1 (τ 1,z,p) = 0 { } 1 (τ 1,z)] = p A 1 [ V 1 (τ 1,z,p), V (τ,z,p)] (4.5) { } (1 p)l [ V (τ,z,p) V 0 (τ,z)] = p A [ V 1 (τ 1,z,p), V (τ,z,p)] V (0,z,p) = (1 p) V 0(0,z) li z V (τ,z,p) = 0 (4.6) Here L i,i = 1, is a differential operator defined as L i = τ i z (1+γ i) z and A i,i = 1, are functionals defined as (4.7) A 1 [ V 1 (τ 1,z,p), V (τ,z,p)] = L 1 [ V 1 (τ 1,z,p)] λ 1 ( V 1 (τ 1,z,p) V (τ,z,p))+ ez σ 1 V 1 z (τ 1,z,p), (4.8) A [ V 1 (τ 1,z,p), V (τ,z,p)] = L [ V (τ,z,p)] λ ( V (τ,z,p) V 1 (τ 1,z,p))+ ez σ 8 V z (τ,z,p). (4.9)
With p = 1, we have L 1 [ V 1 (τ 1,z,1)] = λ 1 ( V 1 (τ 1,z,1) V (τ,z,1)) ez σ1 V 1 (0,z,1) = 0 li z V1 (τ 1,z,1) = 0 L [ V (τ,z,1)] = λ ( V (τ,z,1) V 1 (τ 1,z,1)) ez σ V (0,z,1) = 0 li z V (τ,z,1) = 0 V 1 z (τ 1,z,1) V z (τ,z,1) (4.10) (4.11) Coparing with equations (4.1) and (4.3), it is obvious that V i (τ i,z,1),i = 1, are equal to our searched solutions V i (τ i,z),i = 1,. Now we set p = 0, equations (4.5) and (4.6) becoe L 1 [ V 1 (τ 1,z,0)] = L 1 [ V 1 0 (τ 1,z)] V 1 (0,z,0) = V 1 0(0,z) li z V1 (τ 1,z,0) = 0 L [ V (τ,z,0)] = L [ V 0 (τ,z)] V (0,z,0) = V 0(0,z) li z V (τ,z,0) = 0 (4.1) (4.13) Clearly V i (τ i,z,0),i = 1, will be equal to V i 0(τ i,z) as long as the initial guess 0 satisfies the liiting condition li z V i (τ i,z) = 0. The liiting condition is the only requireent for V i 0(τ i,z). However if in addition L i [ V i 0(τ i,z)] = 0 will speed up a convergence rate of the solution series. Forthechoiceof V 0 i (τ i,z) anycontinuous functionsatisfying theliiting condition can be used. We choose the corresponding European option value as the initial guess with two apparent erits given as in Zhu (006): 1. the boundary condition as z is autoatically satisfied;. L i [ V 0 i (τ i,z)] will becoe 0; which we expect will foster a faster convergence of the series. 9
The closed-for solution of a European put option with a two-state regie switching is given in Zhu, et al. (01): V 0 i (t,s) = Ke ri(t t) + 1 4π { [ e X i(ρ) e X i(ρ) SKe 1 (r i+a 1 +a 1 + σ 1 +σ )(T t) 8 0 ( 1) i 1 ˆf 1 (ρ)(a 1 +a 1 ) M(ρ)(ρ 4 + 1 16 )(σ 1 σ ) (ρ 1 )sin(ˆf (ρ)+θ(ρ) Ŷi(ρ)) (ρ + 1 ] )cos(ˆf (ρ)+θ(ρ) Ŷi(ρ)) [ (ρ 1 )sin(ˆf (ρ)+θ(ρ)+ŷi(ρ)) (ρ + 1 ]} )cos(ˆf (ρ)+θ(ρ)+ŷi(ρ)) + ˆf { 1 (ρ) e [ X i(ρ) sin(ˆf (ρ)+θ(ρ) Ŷi(ρ))+cos(ˆf (ρ)+θ(ρ) Ŷi(ρ)) ] M(ρ) [ ]} e X i(ρ) sin(ˆf (ρ)+θ(ρ)+ŷi(ρ))+cos(ˆf (ρ)+θ(ρ)+ŷi(ρ)) + ˆf 1 (ρ) ρ 4 + 1 16 + e X i(ρ) for i = 1,, where { [ e X i(ρ) (ρ 1 )sin(ˆf (ρ) Ŷi(ρ)) (ρ + 1 ] )cos(ˆf (ρ) Ŷi(ρ)) [ (ρ 1 )sin(ˆf (ρ)+ŷi(ρ)) (ρ + 1 )cos(ˆf (ρ)+ŷi(ρ)) τ = σ 1 σ 4 (T t), α = (a 1 a 1 ), µ = 4a 1a 1 σ1 σ (σ1 σ ), { [( 1 M(ρ) = 4 +α) ρ 4 +µ ] +4ρ 4 ( 1 }1 4 4 +α), θ(ρ) = 1 [ ρ ( 1 +α) ] tan 1 4 ( 1, 4 +α) ρ 4 +µ ]} dρ, (4.14) and X i (ρ) = ( 1) i 1 M(ρ)τ cosθ(ρ), Ŷ i (ρ) = ( 1) i 1 M(ρ)τ sinθ(ρ) ˆf 1 (ρ) = e ρ ln( S K )+r i(t t), ˆf (ρ) = ρ 4 (σ 1 +σ )(T t) ρ ln( S K )+r i(t t). 10
To find the values of V i (τ i,z,1),i = 1,, we can expand the functions V i (τ i,z,p) as a Taylor s series expansion of p where { V1 (τ 1,z,p) = =0 V (τ,z,p) = =0 V 1 (τ 1,z) p! V (τ,z)! p (4.15) { V 1 (τ 1,z) = V p 1 (τ 1,z,p) p=0 V (τ,z) = V (4.16) p (τ,z,p) p=0 To find V 1 (τ 1,z) and V (τ,z) in equation (4.15), we put (4.15) into equations (4.5) and (4.6) respectively and obtain the following recursive relations: L 1 ( V 1 ) = λ 1( V 1 1 V 1 (0,z) = 0 li z V 1 (τ 1,z) = 0 V 1 ) ez σ 1 V 1 1 z = 1,,..., (4.17) L ( V ) = λ ( V 1 V (0,z) = 0 li z V (τ,z) = 0 V 1 1 ) ez σ V 1 z = 1,,..., (4.18) We introduce the following transforation: V i (τ i,z) = e [(1+γ i) z +1 4 τ i(1+γ i ) ]ˆV i (τ i,z). (4.19) We can rewrite equations (4.17) and (4.18) in the for of standard nonhoogeneous diffusion equations ˆV 1 τ 1 ˆV 1 z ˆV 1 (0,z) = 0 li z ˆV 1 (τ 1,z) = 0 = e [ [(1+γ 1) z +1 4 τ 1(1+γ 1 ) ] λ 1 ( V 1 1 1 V ) ez σ1 V 1 ] 1 z (4.0) ˆV τ ˆV z ˆV (0,z) = 0 li z ˆV (τ,z) = 0 = e [(1+γ ) z +1 4 τ (1+γ ) ] [ λ ( V 1 V 1 1 ) ez σ V 1 ] z (4.1) 11
The syste of PDEs (4.0) and (4.1) has a well-known closed-for solution respectively (see Carslaw and Jaeger (1959)): ˆV 1 (τ 1,z) = τ1 e [(1+γ 1) ξ +1 4 τ 1(1+γ 1 ) ] 0 0 [ λ 1 ( V 1 1 1 V ) eξ σ1 V 1 1 ] G1 (τ 1 u,z,ξ)dξdu, ξ (4.) ˆV (τ,z) = τ e [(1+γ ) ξ +1 4 τ (1+γ ) ] 0 0 [ λ ( V 1 1 V 1 ) eξ σ V 1 ] G (τ u,z,ξ)dξdu, ξ (4.3) where { [ ] [ ] 1 (z ξ) (z +ξ) G 1 (τ 1,z,ξ) = exp +exp πτ 1 4τ 1 4τ 1 + (1 γ 1 ) [ (1 γ1 ) πτ 1 exp (1 γ ] 1)(z +ξ) 4 ( z +ξ erfc 1 τ 1 (1 γ 1) )} τ 1, (4.4) { [ ] [ ] 1 (z ξ) (z +ξ) G (τ,z,ξ) = exp +exp πτ 4τ 4τ + (1 γ ) [ (1 γ ) πτ exp (1 γ ] )(z +ξ) 4 ( z +ξ erfc 1 τ (1 γ ) )} τ, (4.5) and erfc(.) denotes the copleentary error function. 1
5 A syetry between fixed-strike and floatingstrike Asian options There are any known syetry results in financial option pricing. Henderson and Wojakowski (00) derived an equivalence of European floating strike Asian calls (or put) and fixed strike Asian puts(or call) in the Black-Scholes-Merton odel. Eberlein and Papapantoleon(005) established a syetry relationship between floating strike and fixed strike Asian options for assets driven by general Lévy processes using a change of nuéraire and the characteristic triplet of the dual process. We also have a syetry between fixed-strike and floating-strike Asian options in the regie switching odel. In the fraework of the regie switching odel, the syetry results for Asian options with respect to G t are stated in the following proposition. Proposition 5.1: Let C f (P f ) denote the floating-strike Asian call (put) option and C x (P x )thefixedstrikeasiancall(put)option; basedonaritheticaveraging, written at tie t = 0 with expiration date T. Then the following syetry results hold: C f (S 0,1,r,0,0,T) = P x (S 0,S 0,0,r,0,T), (5.1) C x (K,S 0,r,0,0,T) = P f (S 0, K S 0,0,r,0,T). (5.) The proof of Proposition 5.1 is siilar to the proof of Theore 1 in Henderson and Wojakowski (00). Hence, we do not repeat it here. 6 Conclusion We consider the pricing of a floating-strike Asian option in a two-state regie switching odel. A closed-for analytical pricing forula for the floating strike Asian option is derived by the eans of the hootopy analysis ethod. 13
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