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.

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 [

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

We begin by assuming that the spot price

We now consider continuous arithmetic Asian option with the average rate defined by the running sum of the underlying asset price

There are four different types of the continuous arithmetic Asian options depending on the pay-off function as follows.

Arithmetic average fixed strike call option:

Arithmetic average fixed strike put option:

Arithmetic average floating strike call option:

Arithmetic average floating strike put option:

To solve (

then (

Fourier’s transform for a function

Applying the Fourier transform in

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.