Application of Fast N-Body Algorithm to Option Pricing under CGMY Model

The N-body problem is an active research topic in physics for which there are two major algorithms for efficient computation, the fast multipole method and treecode, but these algorithms are not popular in financial engineering. In this article, we apply a fast N-body algorithm called the Cartesian treecode to the computation of the integral operator of integro-partial differential equations to compute option prices under the CGMY model, a generalization of a jump-diffusion model. We present numerical examples to illustrate the accuracy and effectiveness of the method and thereby demonstrate its suitability for application in financial engineering.

evaluation of PIDEs computationally expensive. An efficient approach for computing the integral term is to use the FFT [7] [8] [9] [10], but this requires two FFT operations per time-step and direct use of FFT requires either a uniform grid or an additional interpolation scheme [11]. Also, an efficient scheme which transforms the PIDEs to pseudo-differential equations and applies FFT to the equations has been suggested [12], but this also requires two FFT operations per time-step for path-dependent option pricing.
The present paper introduces an application of the Cartesian treecode [13] which has been used as an efficient algorithm for the N-body problem. The N-body problem, which simulates the interactions between individual particles in a multi-particle system, is an active research topic in physics. However, direct computation for this problem is computationally expensive and there are two major efficient algorithms: the fast multipole method (FMM, [14] [15] [16] [17]) and treecode [13] [18]. FMM is relatively complicated and the performance is not sufficient for the actual computation. On the other hand, although the computational burden of treecode is greater, its implementation is straightforward and easier to use in practice.
Little study has been done on the application of these fast N-body algorithms to financial engineering. One application of FMM called the fast Gauss transform (FGT, [16]) has been implemented under Merton's jump-diffusion model [19] to solve the corresponding PIDE [9]. However, its computational efficiency was observed to be poor because the method requires more grid points than the FFT method to achieve similar accuracy. Recently, the improved fast Gauss transform (IFGT, [20]) has been applied and numerical experiments have shown that this method is more efficient than the FFT method [21]; however, IFGT can be applied only to Merton's jump-diffusion models.
In the present paper, we apply another efficient approach, called the Cartesian treecode [13], to the CGMY model ( [11] [22]). To our best knowledge, this is the first paper applying treecode to option pricing modeling. Using numerical examples, we examine its computational accuracy and ease of implementation for the computation of the integral operators of the PIDE. The use of the treecode makes the computation of option pricing faster than the original finite difference method without losing accuracy. Furthermore, we examine whether treecode is also applicable in the field of financial engineering.

Cartesian Treecode
In the N-body problem studied in physic, the main goal is to compute the summation of pairwise interactions over the particles in a given system (e.g., gravitational forces of stars in a galaxy or the Coulomb forces of atoms in a molecule).
Herein, we consider N particles located at 1 2 , , , N x x x  . Then given coefficients 1 2 , , , N C C C  and potential

( )
, K x y which describes some relation between particles at positions x and y, we want to compute However, naive computation of the direct summation results in an ( ) 2 O N computational cost. The Cartesian treecode [13] is a type of treecode used to evaluate the screened Coulomb (Yukawa) potential efficiently, which involves potential ( ) The intuitive idea behind treecode is that one imagines wanting to compute we can approximate the force for each cluster as that of a star whose mass is equal to the total sum of the stars in the cluster and is located at the center of the cluster (called the cluster point). Treecode applies the Barnes-Hut algorithm [18] to build a hierarchy of such clusters and employs an efficient recursive computation of (far-field) Taylor coefficients to compute particle-cluster approximations of the screened Coulomb potential, reducing the computational cost from A more concrete explanation of the Cartesian treecode is as follows (see [13] for details). First, particles ( ) 1 2 , , , n x x x  are divided into a hierarchy of clusters c's by the Barnes-Hut algorithm, Second, truncated expansion of the Taylor series of ( ) For each cluster c, the Taylor expansion is applied if : can be calculated independently and computation of ( ) c i F x is more efficient (notice that in order to compute (1), the summation is required at each i x ). Furthermore, the upshot of this Cartesian treecode is that the Taylor coefficients Generally, FMM is a more sophisticated model for N-body problems in the sense that it also uses near-field expansions, which reduces the computational cost from

Application to Solving a PIDE
We apply the Cartesian treecode to option pricing under the CGMY model with diffusion coefficients ([11] [22]). We assume the underlying stock price S to follow a jump process so that the European option price ( ) , V S t satisfies the following PIDE (see [2] for details of the model and the derivation of the PIDE): where K is the strike price. Substituting ( ) The major drawback of this approach is that we need to compute the integral operator given in this PIDE at each time-step; however, use of the Cartesian treecode makes this efficient. Under the CGMY model, where we employ the following approximation [25] for the last term: Then the PIDE is solved numerically via a simple Crank-Nicholson scheme: ( ) ( ) The boundary conditions are given by x r Use of an iterative scheme [11] is also possible as a means of maintaining second-order accuracy with respect to time, but for simplicity ( )

Recursive Computation of
The Taylor coefficients can be evaluated as follows. Let On the other hand, and differentiation with respect to z k times (

Numerical Examples
We conducted the following numerical experiments, which were executed in

Conclusion
In this article, we describe applying one of the fast N-body algorithms used in physics, the Cartesian treecode [13], high efficiency compared to the original finite difference method without losing accuracy. In addition, the Cartesian treecode can be directly applied to a nonuniform grid, and the use of transformation maps [10] to increase accuracy is also possible. In this paper, we consider a one-dimensional case, but the treecode method is applicable to up to three-dimensional cases, and therefore we can apply the code to a two-dimensional PIDE for pricing basket and spread options.
On the other hand, because the treecode method is only applicable to up to three-dimensional cases, the limitation of this approach is that we cannot compute the price of a multi-asset option which involves more than three assets. Application to path-dependent option pricing is also interesting and left for future research.