New Pricing Formula for Arithmetic Asian Options. Using PDE Approach


 Martin Malone
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1 Applied Mathematical Sciences, Vol. 5, 0, no. 77, New Pricing Formula for Arithmetic Asian Options Using PDE Approach Zieneb Ali Elshegmani, Rokiah Rozita Ahmad and 3 Roza Hazli Zakaria, School of Mathematical Sciences,Faculty of Science and Technology University Kebangsaan Malaysia Bangi Selangor D. Ehsan, Malaysia 3 Department of Economics, Faculty of Economics and Administration University of Malaya Kuala Lumpur, Malaysia Abstract Pricing, and hedging Arithmetic Asian options are difficult, since at present there is no closedform analytical solution exists to price them. This difficulty has led to the development of various methods and models used to price these instruments. In this paper, we propose partial differential equations (PDEs) approach to value the continuous arithmetic Asian options. We provide analytical solution for Rogers and Shi PDE [9] by using changes of some variables and Fourier transform. This is the contribution of this paper since to the best of our knowledge; RogersShi PDE has only been solved numerically. We transform the second order PDE of the arithmetic Asian options to the ODE from the first order, and then we give it s analytical solution.. Introduction Asian options are pathdependent option whose payoff depends on the average value of the underlying assets during a specific set of dates across the life of the option. Because the payoff of the Asian options depends on the average value of the underlying asset, volatility in the average value tends to be smoother and lower than that of the plain vanilla options. Therefore, Asian options, which tend to be less expensive than comparable Plain vanilla put or calls appear very attractive. The two basic forms of
2 380 Z. A. Elshegmani et al averages in Asian options (arithmetic and geometric), both can be structured as calls or puts. A geometric average Asian option is easy to price because a closedform solution is available [], However the most difficult one is arithmetic type, which is the most commonly used, though an exact analytical solution for arithmetic average rate Asian options does not exist. This missing solution is primarily because the arithmetic average of a set of lognormal random variables is not lognormally distributed. Since no general analytical solution for the price of the arithmetic Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options and approach the problem of its valuation some of them: () Monte Carlo simulations. () Binomial trees and lattices. (3) The PDE approach. (4) General numerical methods. (5) Analytical approximations. This work follows the PDE approach. There are many authors have studied the PDE of Asian options, most recently, Geman and Yor [7] use Laplace transform in time of the Asian option price. However, this transform exist for some cases. Rogers and Shi [9] transform the problem of valuing Asian options to the problem of solving a parabolic equation in two variables from the second order. However it is difficult to use numerical methods to solve this PDE, and they derive lowerbound formulas for Asian options by computing the expectation based on some zeromean Gaussian variable. Zhang [] present a theory of continuouslysampled Asian option pricing, and he solves the PDE with perturbation approach. Vecer s approach [0] is based on Asian option as option on a traded account; he provides a onedimensional PDE for Asian options. Francois and Tony [6] derive accurate and fast numerical methods to solve Rogers and Shi PDE. Chen and Lyuu [] develop the lowerbound pricing formulas of Rogers and Shi PDE [9] to include general maturities instead one year. CruzBaez and GonzalezRodrigues [3] obtain the same solution of Geman and Yor for arithmetic Asian options using Partial differential equations, integral transforms, and Mathematica programming, instead Bessel processes. Dewynne and Shaw [4] provide a simplified means of pricing arithmetic Asian options by PDE approach, they derive an analytical formula for the Laplace transform in time of the Asian option, and they obtain asymptotic solutions for BlackScholes PDE for Asian options for lowvolatility limit which is the big problem on using Laplace transform. Elshegmani and Ahmad [5] provide a new method for solving arithmetic Asian option PDE using Mellin transform; they reduce the second order PDE to the first order and the give its final solution. In this work we consider the PDE approach, and give a simple solution for the PDE of Asian option which proposed by Rogers and Shi, using Fourier transform.
3 New pricing formula for arithmetic Asian options Transformation techniques The value of the arithmetic Asian options is characterized by the following PDE with a boundary condition [6]: V V V V σ S rs ( S A) rv = 0, t S S t A V T, S, A = ϕ( S, A), (.) where S is a stock price, r is a constant interest rate, σ is a constant asset volatility, T is t the expiration date, and = Stdt () is the average of the stock price at time t. At t 0 There are four different types of the arithmetic Asian options according to the payoff function: ) Fixed strike call option ) Fixed strike put option 3) Floating strike call option 4) Floating strike put option ϕ S, A = ( A k) ϕ S, A = ( k A) ϕ S, A = ( S A) ϕ S, A = ( A S) where k is a strike price, also named an exercise price. For the case of fixed strike call option with exercise price k, the call option at time t is Inthemoney if At () > k, Atthemoney if At () = k, Outof themoney if A() t < k, and vice versa for fixed strike put option, the put option is Inthemoney if At () < k, Atthemoney if At () = k,
4 3804 Z. A. Elshegmani et al Outofthemoney if At () > k. In this work we propose the following transformation, which allow us to transform the PDE (.) to a simpler parabolic equation in two variables ta k V ( t, S, A) = Sf ( t, ξ), ξ = T, S f f f σξ rξ 0, = t ξ T ξ f T,ξ = ϕξ, (.) for the case of fixed strike call option ϕ ξ ( ξ ) ξ ϕ( ξ) = max ( ξ,0) = ξ, floating strike call strike put ϕ( ξ) = max ( ξ,0 ) = ( ξ ). = max,0 =, fixed strike put option ϕ( ξ) = max ξ,0 = ( ξ), and floating Note that, Eq. (.) has been obtained by Rogers and Shi [9] using different approaches. From the above transformation we can see that, if ξ = 0 then the option is atthemoney, and if ξ > 0, the option is inthemoney. In this work we will price or value only the options which are inthemoney, which are the most important, as options atthemoney or outofthemoney are worthless. Also Rogers and Shi [9] obtain the analytical solution for the case of ξ 0, f t ξ = ( e ) ξe rt rt ( t) rt ( t) (, ). So that, we shall consider only the case whenξ > 0. Under the substitution x = ln ξ, ξ > 0, Eq. (.) becomes x f f e f σ r σ = t x T x f T, x = ϕ( x). 0, (.3)
5 New pricing formula for arithmetic Asian options 3805 We can see that, most of the coefficients in the above equation are constants except one, and this coefficient is exponential function, so we can easily use Fourier transform. Fourier transform for a function f(x) is defined by and the inverse Fourier transform is { } { } iω x F f( x) = g( ω) = f( x) e dx, { } iω x F g( ω) = f( x) = g( ω) e dω, π n f n F ( iω) g( ω), n = x 0 x x iωx ( iω) x ( iω) x F e = e e dx= e dx e =, x> 0, ω and { } where * denotes the convolution operation. Applying Fourier transform in x on Eq. (.3) F f ( x) f ( x) = g ( ω) g ( ω), 0 Letτ = T t ht (, ω) iωσ σ ω iω irω h(, t ω) = 0, t T ( ω ) ht, = ϕ( ω). ( ω) where T is the expiration date, then Eq. (.4) is transformed to h i i = irω τ T h(0, ω) = ϕ( ω) (, τω) ωσ σω ω ( ω ) h(, t ω), The above equation is an ordinary differential equation from the first order with initial condition, its solution is: (.4) (.5)
6 3806 Z. A. Elshegmani et al i i h(, ) exp ωσ ir σ ω ω τ ω = ϕ ω ω τ. T ( (.6) Applying inverse Fourier transform on Eq. (.6) to get the function f (, tx ). iω x iωσ σ ω iω f ( tx, ) = ϕ( ω) e exp irω ( T t) dω. π (.7) T ( We have assumed that x = ln ξ, ξ > 0, x = m ln ξ, this equation has a symmetric graph about ξaxis, so that, we will take x= ln ξ, x 0 and multiples the integration by. iω iωσ σ ω iω f ( t, ξ ) = ϕωξ exp irω ( T t) dω. π (.8) T ( The solution of the arithmetic Asian option PDE then is given by where f (, t ξ) satisfied Eq. (.8). ta k V ( t, S, A) = Sf ( t, ξ), ξ = T, S
7 New pricing formula for arithmetic Asian options 3807 To prove that, our solution (.8) is a direct solution for Eq. (.), derivative Eq. (.8) with respect to all variables, and then substitute them into Eq. (.) f iω iωσ σ ω iω = ϕωξ irω t π T ( iωσ σ ω iω exp irω ( T t) dω, T ( (.9) f iω iωσ σ ω iω = ϕ ( ω) iωξ exp irω ( T t) dω, ξ π (.0) T ( ϕ ω ω iω ξ irω T t dω. (.) f iω iωσ σ ω iω = ( ) exp ( ) ξ π T ( Substituting equations (.9) (.) into Eq. (.) yield f f f σξ rξ = t ξ T ξ iω iωσ σ ω iω σ iω ϕ( ωξ ) irω ( ω iω) riω ξ π T ( ω ) T iωσ σ ω iω exp irω ( T t) dω, T ( = = ω x from the Fourier transform we have F{ ξ } F{ e }
8 3808 Z. A. Elshegmani et al ( ω ) ( ω ) iω iωσ σ ω iω σ iω ϕ( ω) ξ irω ( ω iω) riω π T T iωσ σ ω iω exp irω ( T t) dω 0. = T ( The last equality because = 0. T ( ω ) T ( ω ) iωσ irω σ ω iω σ ( ω iω) riω iω So that, our solution is a direct solution of the arithmetic Asian option PDE. Conclusion We have studied partial differential equation arising in the valuation of arithmetic Asian option. We approach the problem of computing the price of all types of arithmetic Asian options. We consider the PDE of Rogers and Shi for arithmetic Asian options which has been studied by many authors in the literatures, because it is not easy to solve, most of them tried to improve its numerical solution. In this paper we have obtained a direct solution for Rogers and Shi PDE for arithmetic Asian options using means of partial differential equation and Fourier transform. We have shown that, the second order PDE of arithmetic Asian option in three variables can be transformed to ODE from the first order with an initial condition, which can be solved analytically. References [] E. Barucci, S. Polidoro and V. Vespri. Some results on partial differential equations and Asian options. Math. Models Methods Appl. Sci. (3) pp , 00. [] K. Chen and Y. Lyuu. Accurate pricing formula for Asian options, Journal of applied Mathematics and Computation, 88, pp.774, 007.
9 New pricing formula for arithmetic Asian options 3809 [3] D. CruzBaez and J. GonzalezRodrigues. A different approach for pricing Asian options. Journal of applied Mathematics Letters,, pp , 008. [4] J. Dewynne and W. Shaw. Differential equations and asymptotic solutions for arithmetic Asian options: Black Scholes formulae for Asian rate calls. Journal of applied Mathematics, 9, pp , 008. [5] Z. A. Elshegmani and R. R. Ahmad. Analytical solution for an arithmetic Asian option using Mellin transform. International Journal of Mathematical Analysis, 5, pp , 0. [6] D. Francois and L. Tony. Efficient pricing of Asian options by the PDE approach, Journal of Computational Finance, 8, pp.5564, 004. [7] H. Geman and M. Yor. Bessel process, Asian options and perpetuities, Mathematical Finance 3(4), pp , 993. [8] G. H. Meyer. On pricing American and Asian options with PDE Methods, Acta Math. Univ. Comenianae, vol. LXX,, pp , 00. [9] L. C. G. Rogers and Z. Shi. The value of an Asian option, Journal of Applied Probability, 3, pp , 995. [0] J. Vecer. A new PDE approach for pricing arithmetic average Asian option, Journal of Computational Finance, 4, pp. 053, 00. [] J. Zhang. Theory of continuouslysampled Asian option pricing, (working paper)city University of Hong Kong, 000. Received: May, 0
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