A Comparison Study of ADI and LOD Methods on Option Pricing Models

This paper will focus on pricing options in marketing with two basic assets with risk and one basic asset without risk. In so doing, the Black-Scholes model and the European options which is applicable at the due date were used. By investigating the European option to find the proper price, it is necessary to solve an equation with partial derivatives which has two spatial variables. The finite differences will be used for such equations. Finite differences for one dimensional equations commonly ends in a three diagonal set which will be solved by calculation costs O(n) in which n is the number of discrete points. But here, since the problems are two dimensional, the Alternating Direction Implicit (ADI) and Locally One-Dimensional (LOD) are used to reduce the calculation costs. The open cost is at the level of discrete points and this is the advantage of these methods. Moreover, these methods enjoy acceptable stability. Though ADI and LOD are equal and easy in calculations, evaluating these methods in pricing the option indicates that the ADI method is sensitive to discontinuity or non-derivation which is the common property of income function; therefore, this thesis proposes the LOD method.


Introduction
Imagine than an option contract for two underlying assets with the present prices of x and y and the future prices of X 1 and X 1 at the maturity time of T has been signed. If the volatility of the first and second assets are shown by 1 2 , σ σ and the ρ is the correlation coefficient between the prices of the two underlying assets and r is interest rate, then ( ) , , u u x y t = is the three-variate u function. One of the variables is t and the other two variables are the spatial variables. By To use numerical methods, a limited range will be considered. Therefore, Equation (1) will be considered on a finite range where M and L are selected so large that the error in the u price become negligible [1].
The third condition is Neumann boundary condition which is not used in the present thesis. A combination of the first and the second condition might also be used [2].

Weight Methods for the Two Dimensional Black-Scholes Equation
The two dimensional Black-Scholes equation can be discretized as the one dimensional Black-Scholes equation. To simplify the computations, the discrete operator L is defined as follows [3]. )   2  2  2  2  1,  ,  1,  , 1  ,  , 1  1  2  2  2   1, 1  , 1  1,  ,  1 The explicit method is as follows And the implicit method is as follows

− = ∆
The dimensions of this set equals the numbers of points in the net. Since in the two dimensional mode the number of points is twice the number of points in the one dimensional mode; therefore, we have a large set of , x y N N dimensions which is too expensive to solve. The same can be found in Crank Nicolson method for 0 1 θ ≤ ≤ Therefore these methods are not recommended. ADI and LOD methods will be proposed in the next section. These methods are both unconditionally stable and have low computational costs. In these methods, by decomposing the L operator to two x and y operators at each step, a number of three diagonal sets with x N or y N dimension will be solved to be able to go to the next step [4].

ADI Method for the Two-Dimensional Black-Scholes Model
To obtain the Alternating Direction Implicit (ADI), the time derivation will be estimated as, In fact, we added and subtracted one intermediate step 1 2 n t + and divided the time derivation into two parts. Now, by the using the LOD of L, we try to find the following equation to solve the equation.
We will explain the way of obtaining the above mentioned equation in the following. To get from n − to ( ) 1 n − + in ADI method, we first solve , and every one of them is a three diagonal set from the x N dimension. By this action, the values of u will be obtained in to obtain u in ( ) 1 n + phase, we solve   . in the first time step, the derivations should be implicitly estimated in relation to x and the derivations will be explicitly estimated in relation to y variable. In other words, we write ( ) In the second time step, i.e. in we will act in reverse. In other words, we implicitly estimate the derivation in relation to y and explicitly in relation to x. Therefore, we have 1  1  1  1  1  1  1  1  2  2  2  2  2  2  ,  ,  1,  ,  1,  , 1  ,  , 1  2  2  1  2  2  2   1  1  1  1  2  2  2  2  1, 1 1, 1 1, 1 1, 1 1 2 2 In the above equations, the derivation of 2 u x y ∂ ∂ ∂ is explicitly written and y in the last sentence is implicitly written. Now, if we multiply the two sides of the In the present equations, the coefficient if we transfer the values in the n − step to the right side and define ( ) then, we get to the below equation The sentence end Considering the (9), we follow the same procedure for the second half-step.
First, we define which is a three diagonal set. The sentence  As it can be seen, in the first half-step of y N , the three diagonal set will be solved from the dimension of x N and in the second half-step of x N , the three diagonal set will be solved from the dimension of Therefore, the calculation cost of the method to get from the n step to the 1 n + step is from the rank of y N x N , that is, the rank of the number of the network points.

LOD Method
The LOD method, similar to ADI method is divided into two steps in each time step. The first stage estimations are implicit and in relation to x and the second step is explicit and in relation to y. the Black-Scholes model is rewritten as [5], in which the derivations for x and y are written separatel y.
We will solve the half-step Now the two following sets are made. which are called the first phase and the second phase sets respectively. Therefore, the algorithm of LOD method has two phases. In the first phase, the Equation (12) will be fragmented and summarized. The coefficients of i α , i β , and i γ are defined as The only difference between ADI and LOD is in the right hand side value of And the right side values are defined as

Meassuring the Numerical Method Error
If x is the exact value of a numerical quantity and x is an approximate quantity.
Absolute error is defined as  x is considered as x. the error must be investigated in a limited area of the present prices. This area is called G [1].
The number of ( ) , i j which are in this neighborhood are defined as We also use the root mean square error (RMSE) on a specific region. The RMSE is defined as where N is the number of points on the gray region show in Figure 1.

All Cash or None
First, the option of two cash assets or none will be considered. We assume that by having two assets x, y the income of option is as follows [3]. 0 unless where X 1 , X 2 are the prices of x, y.
The function figure is as follows.
The following values will be used for the numerical simulation of the parameters.  Figure 2 with the mentioned parameters, the exact answer is For this example, the errors of both ADI and LOD methods with different time and location step length are presented in Table 1 and Table 2.

Result
As shown in Table 1 u , and source terms, f and g, generated from the ADI and LOD methods. We used time step size, 0.5 t ∆ = , and space step size    The source term in the ADI method exhibits oscillation around 2 y X = which is from the y-derivatives in the source term. On the other hand, for the LOD method, we don't have the y-derivatives in the source term and solution 1 2 u is monotone around 2 y X = . Therefore, for the ADI we have an LOD solution at the first step. After one complete time step, the result with the ADI shows nonsmooth numerical solution. However, the LOD method results in a smooth numerical solution [1].