Table of Contents
ISRN Applied Mathematics
Volume 2011 (2011), Article ID 643749, 8 pages
http://dx.doi.org/10.5402/2011/643749
Research Article

Fourier Transform of the Continuous Arithmetic Asian Options PDE

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 43600 Selangor, Malaysia

Received 23 June 2011; Accepted 2 August 2011

Academic Editor: A. Bellouquid

Copyright © 2011 Zieneb Ali Elshegmani and Rokiah Rozita Ahmad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original paper is properly cited.

Abstract

Price of the arithmetic Asian options is not known in a closed-form solution, since arithmetic Asian option PDE is a degenerate partial differential equation in three dimensions. In this work we provide a new method for computing the continuous arithmetic Asian option price by means of partial differential equations. Using Fourier transform and changing some variables of the PDE we get a new direct method for solving the governing PDE without reducing the dimensionality of the PDE as most authors have done. We transform the second-order PDE with nonconstant coefficients to the first order with constant coefficients, which can be solved analytically.

1. Introduction

An Asian option is one whose payoff includes a time average of the underlying asset price. Asian options can be classified by their method of averaging, such as arithmetic or geometric. The geometric average Asian option is easy to price because a closed-form solution is available [1]. So in this work we only focus on arithmetic Asian option, which is the most commonly used, though an exact analytical solution for arithmetic average rate Asian options has not existed. This missing solution is primarily because the arithmetic average of a set of lognormal random variables is not lognormally distributed. There are several approaches to pricing arithmetic Asian options. The first approach is deriving approximations of closed-form solutions. Turnbull and Wakeman [2] and Levy [3] find approximate valuation formula by matching the first several moments of the arithmetic average. Geman and Yor [4] derive an analytical solution for pricing arithmetic Asian option in terms of the inverse Laplace transform. However, this transform is only applicable in some cases. A one-dimensional PDE is derived to price Asian options contingent on dividend paying stocks by Večeř [5]. Vecer and Xu [6] show that the price of arithmetic Asian option satisfies an integrodifferential equation in the case that the underlying asset is driven by special martingale processes, of which the Levy process is a special case. Dewynne and Shaw [7] 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 then they obtained asymptotic solutions for Black-Scholes PDE for Asian options for low-volatility limit which is the big problem on using Laplace transform. Cruz-Báez and González-Rodrígues [8] obtain the same solution of Geman and Yor for arithmetic Asian options using partial differential equations, integral transforms, and mathematica programming, instead Bessel’s processes. Elshegmani et al. [9] derive a modified arithmetic Asian options PDE, together with its analytical solution. In addition, there are several numerical approaches to pricing arithmetic Asian options such as Monte Carlo’s simulation, Binomial Tree, and Finite Element Method.

In this paper, we derive the PDE for continuous arithmetic Asian option, and give a new method for solving this equation using Fourier transform.

2. Derivation of The PDE for Continuous Arithmetic Asian Option

We begin by assuming that the spot price 𝑆𝑡 of the underlying asset of the Asian option satisfies the stochastic differential equation𝑑𝑆𝑡=𝜇𝑆𝑡𝑑𝑡+𝜎𝑆𝑡𝑑𝑊𝑡,(2.1) where 𝑊𝑡 is a standard Brownian motion, 𝜇 and 𝜎 are constants.

We now consider continuous arithmetic Asian option with the average rate defined by the running sum of the underlying asset price𝐴𝑡=𝑡0𝑆𝑢𝑑𝑢,(2.2) or in differential form 𝑑𝐴𝑡=𝑆𝑡𝑑𝑡.(2.3) Suppose an Asian option has pay-off function 𝜑(𝑆𝑇,𝐴𝑇/𝑇) at an expiration date 𝑇. Then, the value of the Arithmetic Asian option at time 𝑡 is𝑉𝑡,𝑆𝑡,𝐴𝑡=𝑒𝑟(𝑇𝑡)𝐸𝜑𝑆𝑇,𝐴𝑇𝑇𝐹𝑡.(2.4) By multidimension of Ito’s Lemma, we have𝑑𝑉=𝜕𝑉𝜕𝑡+𝜇𝑆𝜕𝑉+𝜎𝜕𝑆2𝑆22𝜕2𝑉𝜕𝑆2+𝑆𝜕𝑉𝜕𝐴𝑑𝑡+𝜎𝑆𝜕𝑉𝜕𝑆𝑑𝑊.(2.5) Set up a portfolio of one Asian option and a number Δ of the underlying assets, the value of this portfolio isΠ=𝑉𝑡,𝑆𝑡,𝐴𝑡Δ𝑆,(2.6) and the change in the value of this portfolio is𝑑Π=𝑑𝑉𝑡,𝑆𝑡,𝐴𝑡Δ𝑑𝑆,𝑑Π=𝜕𝑉+𝜎𝜕𝑡2𝑆22𝜕2𝑉𝜕𝑆2+𝑆𝜕𝑉𝜕𝐴𝑑𝑡+𝜇𝑆𝜕𝑉𝜕𝑆Δ𝑑𝑆.(2.7) To get rid of the stochastic term, choose Δ=𝜕𝑉/𝜕𝑆,𝑑Π=𝜕𝑉+𝜎𝜕𝑡2𝑆22𝜕2𝑉𝜕𝑆2+𝑆𝜕𝑉𝜕𝐴𝑑𝑡.(2.8) But we know that the change in the value of a portfolio with risk-less asset is𝑑Π=𝑟Π𝑑𝑡=𝑟𝑉𝑟𝑆𝜕𝑉𝜕𝑆.(2.9) From (2.9) and (2.8), yield its𝜕𝑉+1𝜕𝑡2𝜎2𝑆2𝜕2𝑉𝜕𝑆2+𝑟𝑆𝜕𝑉𝜕𝑆+𝑆𝜕𝑉𝑉𝜕𝐴𝑟𝑉=0,𝑇,𝑆𝑇,𝐴𝑇𝑆=𝜑𝑇,𝐴𝑇𝑇.(2.10) This is the Black-Scholes PDE of the continuous arithmetic Asian options.

There are four different types of the continuous arithmetic Asian options depending on the pay-off function as follows. (1)Arithmetic average fixed strike call option: 𝜑𝑆𝑇,𝐴𝑇𝑇𝐴=max𝑇𝑇𝑘,0.(2.11)(2)Arithmetic average fixed strike put option: 𝜑𝑆𝑇,𝐴𝑇𝑇𝐴=max𝑘𝑇𝑇,0.(2.12)(3)Arithmetic average floating strike call option: 𝜑𝑆𝑇,𝐴𝑇𝑇𝑆=max𝑇𝐴𝑇𝑇,0.(2.13)(4)Arithmetic average floating strike put option: 𝜑𝑆𝑇,𝐴𝑇𝑇𝐴=max𝑇𝑇𝑆𝑇,0.(2.14)

3. Analytical Solution of The PDE

To solve (2.10) we will make the first change of the variables 𝑉(𝑡,𝑆,𝐴)=𝑒𝑟𝑡𝑓(𝑡,𝑆,𝐴),𝜕𝑓+1𝜕𝑡2𝜎2S2𝜕2𝑓𝜕𝑆2+𝑟𝑆𝜕𝑓𝜕𝑆+𝑆𝜕𝑓𝜕𝐴=0.(3.1) Assume further 𝑧=ln𝑠,𝑠>0,𝜏=𝑇𝑡, and taking into account that𝑆2𝜕2𝑉𝜕𝑆2=𝜕2𝑓𝜕𝑧2𝜕𝑓𝜕𝑧,𝑆𝜕𝑓=𝜕𝑆𝜕𝑓𝜕𝑧,(3.2)

then (3.1) becomes:𝜕𝑓+𝜎𝜕𝜏22𝜕2𝑓𝜕𝑧2𝜕𝑓𝜕𝑧+𝑟𝜕𝑓𝜕𝑧+𝑒𝑧𝜕𝑓𝜕𝐴=0,𝜕𝑓+𝜎𝜕𝜏22𝜕2𝑓𝜕𝑧2+𝜎𝑟22𝜕𝑓𝜕𝑧+𝑒𝑧𝜕𝑓𝜕𝐴=0.(3.3) Assume 𝑧=𝑖𝑦 where 𝑖 is a complex number, then we get𝜕𝑓𝜎𝜕𝜏22𝜕2𝑓𝜕𝑦2𝜎𝑖𝑟22𝜕𝑓𝜕𝑦+𝑒𝑖𝑦𝜕𝑓𝜕𝐴=0.(3.4) Note that all the coefficients in the above equation are constant except the coefficient with the term 𝜕𝑓/𝜕𝐴, so we can easily apply the Fourier transform.

Fourier’s transform for a function 𝑓(𝑥) is defined by𝐹{𝑓(𝑥)}=𝑔(𝜔)=𝑓(𝑥)𝑒𝑖𝜔𝑥𝑑𝑥,(3.5) and the inverse Fourier transform is𝐹11{𝑔(𝜔)}=𝑓(𝑥)=2𝜋𝑔(𝜔)𝑒𝑖𝜔𝑥𝑑𝜔.(3.6) Some properties of the Fourier transform that we need in this work as follows:𝐹𝜕𝑛𝑓𝜕𝑥𝑛=(𝑖𝜔)𝑛𝐹𝑒𝑔(𝜔),𝑖𝑎𝑥𝐹[]𝑓(𝑥)=𝑔(𝜔𝑎),𝑓(𝑥𝑎)=𝑒𝑖𝑎𝜔𝐹𝑒𝑔(𝜔),𝑖𝑎𝑥=𝛿(𝜔𝑎),(3.7) where 𝑎 is a constant, and 𝛿(𝜔) is a Dirac delta function.

Applying the Fourier transform in 𝑧 on (3.4), 𝜕𝑔(𝜏,𝜔,𝐴)+𝜎𝜕𝜏2𝜔22𝜎𝑔(𝜏,𝜔,𝐴)+𝜔𝑟22𝑔(𝜏,𝜔,𝐴)+𝜕𝑔(𝜏,𝜔+1,𝐴)𝜕𝐴=0,(3.8)𝜕𝑔(𝜏,𝜔,𝐴)=𝜎𝜕𝜏2𝜔22𝜎+𝜔𝑟22𝑔(𝜏,𝜔,𝐴)+𝜕𝑔(𝜏,𝜔+1,𝐴)𝜕𝐴.(3.9) Assume 𝑔(𝜏,𝜔,𝐴)=𝑒[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))]𝜏(𝜏,𝜔,𝐴).(3.10) Then (3.9) is reduced to𝜕(𝜏,𝜔,𝐴)=𝜕𝜏𝜕(𝜏,𝜔+1,𝐴).𝜕𝐴(3.11) Applying another Fourier transform in 𝜔,𝜕𝜏,𝜔,𝐴=𝜕𝜕𝜏𝜏,𝜔,𝐴𝑒𝜕𝐴𝑖.𝜔(3.12) One solution of the above equation is𝜏,𝜔,𝐴=𝑐𝜏+𝐴𝑒𝑖𝜔.(3.13) From the initial condition, we have0,𝜔,𝐴=𝜑𝜔,𝐴=𝑐𝐴𝑒𝑖,𝜑𝜔𝑐=𝜔,𝐴𝐴𝑒𝑖,𝜔=𝜑𝜔,𝐴(3.14)𝜏,𝜔,𝐴=𝜑𝜔,𝐴𝜏+𝐴𝑒𝑖𝜔.(3.15) Applying the inverse Fourier transform in 𝜔, [],[]𝑒(𝜏,𝜔,𝐴)=𝜑(𝜔,𝐴)𝜏𝛿(𝜔)+𝐴𝛿(𝜔1)(3.16)𝑔(𝜏,𝜔,𝐴)=𝜑(𝜔,𝐴)𝜏𝛿(𝜔)+𝐴𝛿(𝜔1)[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))]𝜏,𝑓(𝜏,𝑖𝑧,𝐴)=𝐹1[],1𝑔(𝜏,𝜔,𝐴)(3.17)𝑓(𝜏,𝑦,𝐴)=2𝜋[](𝜑(𝜔,𝐴)𝜏𝛿(𝜔)+𝐴𝛿(𝜔1))𝑒[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))]𝜏𝑒𝑖𝜔𝑧1𝑑𝜔,(3.18)𝑓(𝜏,𝑧,𝐴)=2𝜋[](𝜑(𝜔,𝐴)𝜏𝛿(𝜔)+𝐴𝛿(𝜔1))𝑒[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))]𝜏𝑒𝜔𝑧1𝑑𝜔,(3.19)𝑓(𝑡,𝑆,𝐴)=2𝜋𝑆𝜔[](𝜑(𝜔,𝐴)(𝑇𝑡)𝛿(𝜔)+𝐴𝛿(𝜔1))𝑒[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))](𝑇𝑡)1𝑑𝜔,(3.20)𝑉(𝑡,𝑆,𝐴)=2𝜋𝑆𝜔[](𝜑(𝜔,𝐴)(𝑇𝑡)𝛿(𝜔)+𝐴𝛿(𝜔1))𝑒[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))𝑟](𝑇𝑡)𝑑𝜔.(3.21) To prove that expression (3.21) is a direct solution for (2.1), we will differentiate, (3.14) with respect to all variables, and then substituting into (2.1),𝜕𝑉𝜎𝜕𝑡=2𝜔22𝜎+𝜔𝑟221𝑟𝑉(𝑡,𝑆,𝐴)𝜑2𝜋(𝜔,𝐴)𝑆𝜔𝛿(𝜔)𝑒[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))𝑟](𝑇𝑡)𝑑𝜔,𝜕𝑉𝜕𝑆=𝜔𝑆𝜔1𝜕𝑉(𝑡,𝑆,𝐴),2𝑉𝜕𝑆2=𝜔(𝜔1)𝑆𝜔2𝑉(𝑡,𝑆,𝐴),𝜕𝑉=1𝜕𝐴2𝜋𝑆𝑖𝜔[](𝜑(𝜔,𝐴)𝛿(𝜔1))𝑒[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))𝑟](𝑇𝑡)𝑑𝜔.(3.22) Substituting (3.22) into (2.10) yields𝜕𝑉+𝜎𝜕𝑡2𝑆22𝜕2𝑉𝜕𝑆2+𝑟𝑆𝜕𝑉𝜕𝑆+𝑆𝜕𝑉𝜎𝜕𝐴𝑟𝑉=2𝜔22𝜎+𝜔𝑟22𝜎𝑟𝑉+2𝑆22𝜔(𝜔1)𝑆𝜔2𝑉+12𝜋𝜑(𝜔,𝐴)𝑆𝜔𝛿(𝜔)𝑒[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))𝑟](𝑇𝑡)𝑑𝜔+𝑟𝑆𝜔𝑆𝜔1𝑉1+𝑆2𝜋𝑆𝜔[](𝜑(𝜔,𝐴)𝛿(𝜔1))𝑒[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))𝑟](𝑇𝑡)𝜎𝑑𝜔𝑟𝑉=2𝜔22𝜎𝑟𝜔+2𝜔2𝜎+𝑟+2𝜔22𝜎2𝜔2+1+𝑟𝜔𝑟2𝜋𝑆𝜔[]𝑒𝜑(𝜔,𝐴)𝑆𝛿(𝜔1)𝛿(𝜔)[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))𝑟](𝑇𝑡)𝑑𝜔=0,(3.23) since we have 𝑆=𝑒𝑖𝑧, 𝑆𝛿(𝜔1)=𝑒𝑖𝑧𝛿(𝜔1)=𝑒𝑖𝑧𝑒𝑖𝑧(𝜔1)=𝑑𝜔𝑒𝑖𝑧(𝜔1+1)=𝑒𝑖𝑧𝜔𝑑𝜔=𝛿(𝜔),(3.24) so𝑆𝑖𝜔[]𝑒𝛿(𝜔)𝛿(𝜔)[(𝜎2𝜔2/2)+𝜔(𝑟(𝜎2/2))𝑟](𝑇𝑡)𝑑𝜔=0.(3.25)

4. Conclusion

The valuation of the arithmetic Asian options with continuous sampling has been an outstanding issue in finance for several decades. Describing the distribution of the integral of lognormals is found to be challenging. In this paper, we have solved the problem with the PDE approach. We show that the governing PDE from the second order can be transformed to a simple PDE from the first order with constant coefficients, which can be easily solved. We have the solution for all types of the continuous arithmetic Asian options only by changing the pay-off function with respect to which one of the options we want to price. Our approach could be extended to the continuous arithmetic Asian options with constant dividend yield.

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