Evaluation of Asian option by using RBF approximation

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1 Boundary Elements and Other Mesh Reduction Methods XXVIII 33 Evaluation of Asian option by using RBF approximation E. Kita, Y. Goto, F. Zhai & K. Shen Graduate School of Information Sciences, Nagoya University, Nagoya, Japan Abstract This paper describes the evaluation of the price of the Asian option by using the radial bases function (RBF) approximation. In the previous study, we described the evaluation method of the European and the American options. In this paper, the Asian option is considered. A governing differential equation is discretized with the Crank-Nicholson scheme and the RBF approximation. The system of equations is solved for the option price. The numerical results are compared with the FDM solutions in order to confirm the validity of the formulation. 1 Introduction Recently, financial derivatives have been widely used and their importance has increased. The importance of the derivative transaction is increasing for the adequate sharing of the financial risk. The option transaction is one of the most important financial derivatives and therefore, several schemes have been presented by many researchers for their pricing [1, 2]. Several financial options have been developed; the European option, American option, Look-Back option, Exotic option and so on. In previous studies, the authors described the pricing of the European and American options [3, 4]. The Asian option will be considered in this paper. In the Asian option, the payoff is performed according to the time-average value of the asset price. The Asian option can be classified into the average rate option, the average strike option and so on. While, in the former, the payoff depends on the difference between the time-average value of the asset of the asset price and the expiration price, in the latter, the payoff depends on the difference between the average value and the asset price on an expiration date. In this paper, we focused on the average strike option. doi:1.2495/be64

2 34 Boundary Elements and Other Mesh Reduction Methods XXVIII The price of the average strike option can be evaluated as the solution of the Black-Scholes differential equation by taking the payoff condition on an expiration date. The Black-Scholes equation is discretized according to the Crank- Nicolson scheme on the time axis and the option price is approximated with Radial Bases Function with unknown parameters at each time step. The initial value of the parameter on the expiration date is determined from the payoff condition. Then, the parameters on the pricing day are evaluated according to the backward algorithm from the expiration date to the pricing date. The numerical solutions are compared with the finite difference solutions. The remaining of the paper is organized as follows. In section 2, the evaluation of the average strike option is formulated. The numerical examples are shown in section 3. Finally, the obtained results are summarized in section 4. 2 Formulation The Asian option is also known as the average option. In the American and European options, the payoff depends on the difference between the asset price S(t) and the expiration price. On the other hand, in the Asian option, the time-average value of the asset price S(t) is estimated first and then, the payoff is exercised according to the difference between the time-average value and the expiration price or the asset price on the expiration date. If the payoff depends on the expiration price, the Asian option is called as the average rate option. If the payoff depends on the asset price on the expiration date, the Asian option is called as the average strike option. In this study, we will consider the European-type average strike option. 2.1 Governing equation and boundary condition First, we will define the time-average value of the asset price S as the function: I = t S(τ)dτ. (1) In the European-type average strike option, the payoff depends on the difference between the time-average value and the asset price on the expiration date. The governing differential equation of the option is given as: V t + S V I σ2 S 2 2 V V + rs rv = (2) S2 S If the function R is defined from the asset price S as R = 1 S t S(τ)dτ = I S, (3) the price V is given as V (S, R, t) =SH(R, t). (4)

3 Boundary Elements and Other Mesh Reduction Methods XXVIII 35 Substituting equations (3) and (4) to equation (2), we have where the operator F is defined as H t + FH =, (5) F = 1 2 σ2 R 2 2 +(1 r)r R2 R. (6) The payoff condition of the average strike option on the expiration date t = T is defined as follows, in the case of European-call type, ( max S 1 T and, in the case of European-put type, ( 1 t max T t ) S(τ)dτ, (7) ) S(τ)dτ S,. (8) where max(a 1,a 2 ) means the bigger one among a 1 and a 2. Now, we consider the pricing of the average strike option in the call-type. Substituting equations (3) and (4) to (7), we have the payoff condition on the expiration date t = T ; SH(R, T )=Smax (1 RT ),, and therefore, H(R, T )=max (1 RT ),. (9) Finally, the governing equation and the boundary condition of the average strike option are given by equation (5) and (9), respectively. 2.2 Solution using RBF Discretizing the equation (5) with Crank-Nicolson Scheme, we have H(t + t) H(t) +(1 θ)fh(t + t)+θfh(t) = (1) t where the parameter θ is taken in the range of θ 1. Defining the parameters H(t) =H m and H(t + t) =H m+1,wehave AH m+1 = BH m (11)

4 36 Boundary Elements and Other Mesh Reduction Methods XXVIII where A =1+(1 θ) tf B =1 θ tf. The derivative price H governed with the equation (5) is approximated with the RBF function as H = λ n φ n (12) where N and λ j denote the total number of data points and the unknown parameters, respectively. Substituting equation (12) to equation (11), we have 2.3 Algorithm A λ m+1 n φ n = B Aφ n λ m+1 n = λ m n φ n Bφ n λ m n (13) The algorithm of the solution procedure is defined as 1. Distribute N data points on R R max and discretize t T with T/M. 2. Solve equation (12) to evaluate H T on the expiration date t = T. 3. Approximate H T by equation (12) to evaluate λ T n on the expiration date t = T. 4. t T t. 5. Solve equation (13) to estimate λ t n. 6. t t t. 7. IF t,gotostep5. 8. Evaluate H from equation (12) and λ n on the date t =. 3 Numerical example First we will adopt the radial bases function: φ(r, R j )= c 2 + R R j 2 (14) where r 2 j = S S j. The parameters are specified in Table 1. The total number of the data points are 11. They are distributed uniformly in the range of R 1.. For comparison with the finite difference solutions, the time-step size is taken as t =.5;the number of the time-step is M = 1.

5 Boundary Elements and Other Mesh Reduction Methods XXVIII 37 Table 1: Parameters for numerical result. Expiration date T =.5 [year] Risk free interest rate r =.1 Volatility σ =.4 Crank-Nicolson method θ =.5 Maximum R R max =1. Time step size t =.5 Number of time step M = 1 Number of stock data points N = 11 Table 2: The condition number to each c. RBF parameter c Condition number For determining the parameter c in the equation (13), we will estimate the condition number of the coefficient matrix Bφ j in equation (12). The results are shown in Table 2. In the case of the parameter c =.4, the numerical results are shown in Fig. 1. In Fig. 1, the abscissa and the ordinate denote R and H, respectively. Figure 1 indicates that the price H dose not well converge to. For improving the computational accuracy, instead of the above RBF (14), we will take the another RBF: φ(r, R j )= 1 c2 + R R j 2 (15) The parameter c in equation (15) is taken as c =.4 and the other parameters are specified in Table 1. The numerical results are shown in Fig. 2. The finite difference solutions are shown in Fig. 3. We notice from Figs. 1 and 2 that the use of the equation (15) obtain good convergence to improve the computational accuracy.

6 38 Boundary Elements and Other Mesh Reduction Methods XXVIII Option Value H t t.25 t R Figure 1: Values of European average strike call option, RBF: Multiquadrics, c =.4. Option Value H t t.25 t R Figure 2: Values of European average strike call option, RBF: Reciprocal Multiquadrics, c =.4. 4 Conclusions This paper described the evaluation of the Asian option by using the RBF approximation. The Asian option can be classified into the average rate option and the average strike options. While, in the former, the payoff depends on the difference between the time-average value of the asset price and the expiration price, in the

7 Boundary Elements and Other Mesh Reduction Methods XXVIII 39 Option Value H t t.25 t R Figure 3: Values of European average strike call option, FDM. latter, the payoff depends on the average value and the asset price on the expiration date. In this paper, we focused on the average strike option. In the average strike option, the introduction of the new function leads to the different governing equation as the European and the American options. The use of the Crank-Nicholson scheme and the RBP approximation transforms the governing differential equation to the system of equations. The system of equations are solved in the backward algorithm from the expiration time t = T to the time t =. First, the multi quadratic RBF was adopted for the analysis. The results show that the convergence of the solution is not good. Next, the reciprocal multi quadratic RBF was applied. The results converged well and agreed well with the finite difference solutions. In the future plan, we are going to apply the formulation to the other options. References [1] G. Courtadon. A more accurate finite difference approximation for valuation of options. Journal of Financial and Quantitative Analysis, Vol. 17, pp , [2] P. Wilmott, J. Dewynne, and S. Howison. Option Pricing: Mathematical Models and Computation. Oxford Financial Press, [3] E. Kita and Y. Goto. Evaluation of the european stock option by using the rbf approximation. In A. Kassab, C. A. Brebbia, E. Divo, and D. Poljak, editors, Boundary Elements XXVII (Orlando, USA, 25), pp , 25.

8 4 Boundary Elements and Other Mesh Reduction Methods XXVIII [4] Y. Goto and E. Kita. Estimation of american option using radial bases function approximation. In V. M. A. Leitao, C. J. S. Alves, and C. A. Duarte, editors, Proceedings of the ECCOMAS Thematic Conference on Meshless Methods (Meshless 25), pp. D41.1 4, 25.