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Options valuation by using radial basis function approximation
ARTICLE IN PRESS
Engineering Analysis with Boundary Elements 31 (2007) 836–843 www.elsevier.com/locate/enganabound
Options valuation by using radial basis function approximation Yumi Goto, Zhai Fei, Shen Kan, Eisuke Kita Graduate School of Information Science, Nagoya University, Nagoya 464-8601, Japan Received 30 August 2006; accepted 6 February 2007 Available online 11 April 2007
Abstract This paper describes the valuation scheme of European, barrier, and Asian options of single asset by using radial basis function approximation. The option prices are governed with Black–Scholes equation. The equation is discretized with Crank–Nicolson scheme and then, the option price is approximated with the radial basis functions with unknown parameters. In the European and the barrier options, the prices are governed with Black–Scholes equation. The governing option of the Asian option, however, is different from them of the others. In that case, one has to adopt the other radial basis functions than that for the original Black–Scholes equation. Finally, numerical results are compared with theoretical and finite difference solutions in order to confirm the validity of the present formulation. r 2007 Elsevier Ltd. All rights reserved. Keywords: Option Contract; European option; Barrier option; Asian option; Radial basis function
1. Introduction Recently, financial derivatives are widely dealt and the importance is expanded. The importance of the derivative transaction is increasing for the adequate sharing of the financial risk. Therefore, it is considered that the option contract is one of the most important financial derivatives. The options can be classified into a European and an American options and so on. The European option price can be determined by analytically solving the so-called Black–Sholes equation. Since the other option price cannot be determined analytically, many researchers have presented the numerical schemes [1–3]. The existing schemes are mainly classified into binomial tree, finite differential and Monte Carlo methods. In the finite difference method, the governing difference equation is discretized with finite difference approximation and to be solved. If an option is constructed with many assets, huge computer memory is necessary in the finite difference method. The binomial tree method also has similar features as the finite difference method. Therefore, if the option is constructed with many assets, the Monte Carlo method is more effective. In the Corresponding author. Tel./fax: +81 52 789 3521.
E-mail address: [email protected] (E. Kita). 0955-7997/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2007.02.001
Monte Carlo method, the history of the asset price fluctuation is generated randomly and the option price is determined as their expected value. The computational accuracy of the Monte Carlo method depends on the quality of the algorithm to generate random number. On the other hand, this paper describes the evaluation scheme of the option price by using radial basis function approximation. Since the present scheme has the similar algorithm as the finite difference method, the computational accuracy does not depend on the random number generator and a large computer memory is necessary if the option is constructed with a lot of assets. However, when designing a option for private person or small business firm, the options are often constructed with small number of assets. In the cases, the present method as well as the finite difference method is effective. Besides, the present method does not need structured grid or mesh, unlike the finite difference method. The application of the radial basis function approximation to the option valuation was already presented by Hon and his co-workers [4,5], Boztosum [6] and Fasshauer [7]. While European and American options were considered in their studies, this paper focuses on the barrier and Asian options as well as European option. The Asian option price is governed with Black–Scholes equation [8]. However, the
ARTICLE IN PRESS Y. Goto et al. / Engineering Analysis with Boundary Elements 31 (2007) 836–843
use of the function transformation often changes the governing equation from the original Black–Scholes equation. In that case, we should adopt the different radial basis function than that for the governing equation of European options. In the present formulation, the governing equation is discretized according to the Crank–Nicolson scheme on the time interval and the option price is approximated with radial basis function with unknown parameters at each time step. The initial values of the parameters are determined from the strike condition on the expiration date. Then, the parameters at the date of purchase are evaluated according to the backward algorithm from the expiration date to the date of purchase. The numerical solutions are compared with the analytical ones. In the numerical examples, the results are compared with the solutions with binomial tree and finite differential methods. The remaining of this paper is organized as follows. In Section 2, the option contracts are described. In Section 3, the valuation scheme for the European option of one asset is explained and the numerical solutions are compared with the theoretical ones. The valuation schemes of the barrier and Asian options are explained in Sections 4 and 5, respectively. Finally, the conclusions are summarized in Section 6. 2. Option contract An option is a contract between a buyer and a seller. The option contract is the right, but not the obligation, to buy (for a call option) or sell (for a put option) a specific amount of a given stock, commodity, currency, index, or debt, at a specified price (the strike price) during a specified period of time. For simplicity, this article will discuss only options connected to listed stocks. 2.1. European, American and exotic options Note that there are two basic types of options, the American and the European. An American (or Americanstyle) option is an option contract that can be exercised at any time between the date of purchase and the expiration date. A European (or European-style) option is an option contract that can only be exercised on the expiration date. An exotic option is an option contract which has more complex features than European and American options. An exotic option is an option contract that can be exercised according to the average value of the asset price during a specified period of time and their maximum and minimum prices. In this paper, we will focus on the European, the barrier and the Asian options. 2.2. Barrier option Barrier options can be classified into knock-out and knock-in options. Considering the barrier price K, the
837
knock-out option can be exercised unless the asset price S reaches the barrier K during the day of purchase and expiration day. The knock-in option can be exercised if the asset price S overtakes the barrier K. The knock-out options can be classified into ‘‘up-andout’’ and ‘‘down-and-out’’. The up-and-out option can be exercised unless the asset price S reaches the barrier K from beneath the barrier and the down-and-out option can be done unless the asset price reaches the barrier from above the barrier. Besides, the knock-in options can be classified into ‘‘up-and-in’’ and ‘‘down-and-in’’. The up-and-in option can be exercised if the asset price reaches the barrier from beneath the barrier and the down-and-in option can be done if the asset price reaches the barrier from above the barrier. The valuation algorithms of the options are almost similar and therefore, the down-and-out option is described in Section 4. 2.3. Asian option An Asian option, which is also called an average option, is an option whose payoff is linked to the average value of an asset before the expiration date. There are two basic forms; an average rate option or an average strike option. An average rate option or average price option is a cashsettled option whose payoff is based on the difference between the average value of the asset during the period from the day of purchase and the expiration date and a strike price E. An average strike option is a cash settled whose payoff is based on the difference between the average value of the asset during the period and the asset price at the expiration date. The European-type average strike option is described in Section 5. 3. European option 3.1. Governing equation and strike condition Now, we consider the option constructed with only one asset. When the asset price at the date t is indicated with S, the option price V ðS; tÞ is governing with Black–Scholes equation [3,8]: qV ðS; tÞ 1 2 2 q2 V ðS; tÞ qV ðS; tÞ þ s S rV ¼ 0, þ rS 2 qt 2 qS qS
(1)
where the volatility s is considered to be constant between the date of purchase and the expiration date. Defining the differential operator 1 q2 q F 1 ¼ s2 S2 2 þ rS r, 2 qS qS
(2)
we can rewrite Eq. (1) to qV ðS; tÞ þ F 1 V ðS; tÞ ¼ 0. qt
(3)
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For European call option, the strike condition is given as V ðS; TÞ ¼ maxðSðTÞ E; 0Þ,
(4)
Substituting Eq. (11) into Eq. (8), we have N X
ljtþDt H 1 fj ¼
N X
ltj G 1 fj .
(12)
and, for European put option, the strike condition is as
j¼1
V ðS; TÞ ¼ maxðE SðTÞ; 0Þ,
Holding the above equation for the collocation points, we have
(5)
where E denotes the strike price and the function maxðS E; 0Þ gives the bigger one between S E and 0.
j¼1
(13)
Ax ¼ b, where
3.2. Discretization
A ¼ ½G 1 fj ,
Discretizing equation (3) with Crank–Nicolson scheme, we have V ðS; t þ DtÞ V ðtÞ þ ð1 yÞF 1 V ðS; t þ DtÞ Dt þ yF 1 V ðS; tÞ ¼ 0,
ð6Þ
where the parameter y is specified in the range of 0pyp1. Rearranging the above equation for V ðtÞ and V ðt þ DtÞ, we have ½1 þ ð1 yÞDtF 1 V ðS; t þ DtÞ ¼ ½1 yDtF 1 V ðS; tÞ
(7)
x ¼ fltj g b¼
and
H 1 fj . ½ltþDt j
The coefficient matrix A and vector H 1 fj can be calculated analytically. The parameter l is determined from the numerical result at the previous time-step t þ Dt. Therefore, Eq. (13) is solved for ltj . Eq. (13) is solved iteratively from the expiration date t ¼ T to the date of purchase t ¼ 0. Once we can get the parameter l0j at the date t ¼ 0, the option price at the date of purchase is estimated from l0j by Eq. (11).
and then, H 1 V tþDt ¼ G 1 V t ,
(8)
where
The algorithm is as follows.
V ðS; tÞ ¼ V t , V ðS; t þ DtÞ ¼ V tþDt , H 1 ¼ 1 þ ð1 yÞDtF 1 , G 1 ¼ 1 yDtF 1 . In this study, the option price V t is approximated with the multi-quadric radial basis function (MQ-RBF); qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðS; S j Þ fj ¼ c2 þ kS Sj k2 , (9) or the reciprocal multi-quadric radial basis function (RMQ-RBF); 1 fðS; S j Þ fj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , c2 þ kS Sj k2
(10)
where Sj is the asset price at the collocation point j for approximating the option price V . The parameter c is the support radius of the radial basis function. It is taken in this study as c ¼ 1. The radial basis function approximation of the option price V t is given as Vt ’
3.3. Algorithm
N X
ltj fj ,
(11)
j¼1
where N and lj denote the total number of the collocation points at the date t and the unknown parameters, respectively.
1. N collocation points are taken uniformly in the range of the asset price 0pSpSmax . 2. The time interval 0ptpT is discretized with the timestep T=M between the date of purchase t ¼ 0 and the strike date t ¼ T where M denotes the number of time steps. 3. The option price V T at the expiration date t ¼ T is calculated from the strike condition (4) or (5). 4. The parameter lTj on the expiration date T is calculated from Eq. (11) on V T . 5. t T Dt. 6. Eq. (12) is solved for ltj . 7. t t Dt. 8. If t40, the process goes to step 6. 9. Substituting l0j into Eq. (11) leads to V 0 . 3.4. Numerical example A European put option is considered here. The MQRBF, which is shown in Eq. (9), is adopted first. The parameters are listed in Table 1. One discussed the effect of the total number of the collocation points N and the timestep size Dt to the computational accuracy. First we will discuss the effect of the total number of the collocation points N to the computational accuracy. The time-step size is specified as Dt ¼ T=M ¼ 0:005. The computational error e and the condition number of the coefficient matrix are shown in Table 2. The error is
ARTICLE IN PRESS Y. Goto et al. / Engineering Analysis with Boundary Elements 31 (2007) 836–843 Table 1 Parameters for numerical result
Table 3 Parameters for numerical result T ¼ 0:5 (year) E ¼ 10:0 r ¼ 0:05 s ¼ 0:2 y ¼ 0:5 Smax ¼ 30 c¼1
Expiration date Exercise price Risk free interest rate Volatility Crank–Nicolson method Maximum stock value RBF parameter
Table 2 The condition number and the error for N N
Condition number
e
5
61 121 151
0.0210884 0.000289184 0.000505691
4:64 10 6:45 108 1:99 1010
0.0002895
ε
839
0.000289
T ¼ 0:5 (year) E ¼ 10:0 r ¼ 0:05 s ¼ 0:2 y ¼ 0:5 Smax ¼ 30 M ¼ 100 Dt ¼ 0:005 N ¼ 121 c ¼ 1:0
Expiration date Exercise price Risk free interest rate Volatility Crank–Nicolson method Maximum stock value Number of time step Time-step size Number of stock data points RBF parameter
Table 4 Results of European put option by using MQ-RBF Stock S
V Anal:
V RBF
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0
9.7531 7.7531 5.7531 3.75318 1.79871 0.44197 0.04834 0.00277 0.00010 0.00000
9.7531 7.7531 5.7531 3.75318 1.79823 0.44055 0.04780 0.00271 0.00010 0.00008
0.0002885
0.02
0.01 Δt
0.00625
0.005
Fig. 1. Variation of the relative error with Dt.
defined as e¼
N 1X jV ðS j ; tÞRBF V ðS j ; tÞAnal: j, N j¼1
(14)
where V ðS j ; tÞRBF and V ðS j ; tÞAnal: denote the numerical solution by the present method and the theoretical solution, respectively. We notice that the computational error decreases and the condition number increases according to the increase of the total number of the collocation points. The total number of the collocation points strongly affects the computational accuracy. If the condition number of the coefficient matrix is relatively low, it is good to take large number of the collocation points. So, we will take N ¼ 121. In case of N ¼ 121, the effect of the time-step size Dt to the computational accuracy is shown in Fig. 1. The abscissa and the ordinate indicate the computational error and the time-step size, respectively. We notice that the error is very huge for Dt40:02 and that the computational error is getting less at Dt ¼ 0:005. Numerical results by using the MQ-RBF are compared with the theoretical ones in Table 4. The parameters are
Table 5 Results of European put option by using RMQ-RBF Stock S
V Anal:
V RBF
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0
9.7531 7.7531 5.7531 3.75318 1.79871 0.44197 0.04834 0.00277 0.00010 0.00000
9.7531 7.7531 5.7531 3.75318 1.79823 0.44055 0.04780 0.00271 0.00010 0.00008
specified as shown in Table 3. Numerical results agree well with the theoretical ones (Table 4). Finally, the RMQ-RBF (10) is adopted, instead of the MQ-RBF. One takes the same parameters as them for the MQ-RBF (Table 3). Numerical results are shown in Table 5, as well as theoretical ones. We notice that the results by using RMQ-RBF agree with them by MQ-RBF. 4. Barrier option 4.1. Governing equation and strike condition We shall consider the down-and-out option of the expiration price E and the barrier K. The option becomes
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840
invalid if the asset price S reaches the barrier E from above the barrier during the day of purchase and the expiration date. Even unless the asset price S reaches the barrier E, i.e., S4K, the option is European call option. The price of the down-and-out option is governed with qV þ F 1 V ¼ 0 ðS4KÞ, qt V ¼ 0 ðSpKÞ.
ð15Þ ð16Þ
The strike condition on the expiration day is given as V ðS; TÞ ¼ maxðSðTÞ E; 0Þ.
(17)
Table 6 Parameters for numerical analysis T ¼ 0:5 (year) E ¼ 10:0 r ¼ 0:05 s ¼ 0:2 y ¼ 0:5 H ¼ 9:0 Smax ¼ 30 Dt ¼ 0:05 M ¼ 100 N ¼ 121 c ¼ 1:0
Expiration date Exercise price Risk free interest rate Volatility Crank–Nicolson method Barrier Maximum stock value Time-step size Number of time step Number of stock data points RBF parameter
If S reaches K, the option is invalid. Therefore, ðS ¼ KÞ.
Therefore, the payoff X is as ( maxðS E; 0Þ ðS4KÞ; X¼ 0 ðSpKÞ:
(18)
(19)
4.2. Discretization and algorithm While the strike condition in the barrier option is different from the European option, the differential equation (15) is identical. Therefore, the discretized equation derived from Eq. (15) is the same as that in the European option; i.e., Eq. (12). The algorithm is also similar to that in Section 3.3 except for the strike condition. The valuation algorithm of the down-and-out option is as follows. 1. N collocation points are taken uniformly in the range of the asset price 0pSpS max . 2. The time interval 0ptpT is discretized with the timestep T=M between the date of purchase t ¼ 0 and the strike date t ¼ T. M denotes the number of time steps. 3. The option price V T at the expiration date t ¼ T is calculated from the strike condition (17). 4. The parameter lTj on the expiration date T is calculated from Eq. (11) on V T . 5. t T Dt. 6. Eq. (12) is solved for ltj . 7. t t Dt. 8. If t40, the process goes to step 6. 9. Substituting l0j into Eq. (11) leads to V 0 .
Table 7 Results of down-and-out option Stock S
V Anal:
V MQ
V RMQ
1 3 5 7 9 11 13 15 17 19
0.0000 0.0000 0.0000 0.0000 0.0000 1.3998 3.2591 5.2475 7.2469 9.2469
0.0000 0.0000 0.0000 0.0000 0.0000 1.3985 3.2589 5.2474 7.2469 9.2466
0.0000 0.0000 0.0000 0.0000 0.0000 1.3985 3.2589 5.2474 7.2465 9.2383
10
MQ– RBF RMQ– RBF Analytical
8 Option Value V
V ðK; tÞ ¼ 0
6 4 2 0 0
2
4
6
8 10 Stock S
12
14
16
18
Fig. 2. Values of European down-and-out call option.
5. Asian option 5.1. Governing equation and strike condition
4.3. Numerical example We will adopt the MQ-RBF and the RMQ-RBF. The parameters are specified as shown in Table 6. Numerical results are shown in Table 7 and Fig. 2, as well as theoretical solutions. The abscissa and the ordinate denote the option price V and the asset price S, respectively. We notice that the results by using MQRBF agree well with the theoretical ones and that the results by using RMQ-RBF are very slightly different from the theoretical ones.
We shall consider that the payoff of an Asian option depends on an average rate of an asset. The average rate of an asset S is given as Z 1 t SðtÞ dt. t 0 Introducing the function Z t I¼ SðtÞ dt, 0
(20)
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we have the partial differential equation governing the Asian option price V 2
qV qV 1 2 2 q V qV þS þ sS rV ¼ 0. þ rS qt qI 2 qS qS 2 Using the function Z 1 t I R¼ SðtÞ dt ¼ S 0 S
(21)
(22)
(23)
where H denotes the option price. Substituting Eqs. (22) and (23) to (21), we have qH þ F 2H ¼ 0 qt
(24)
and then, 1 q2 q . F 2 ¼ s2 R2 2 þ ð1 rÞR 2 qR qR
(25)
In case of an call option, the payoff on the expiration date t ¼ T is given as Z 1 t max S SðtÞ dt; 0 T 0 and in case of an put option, Z t 1 max SðtÞ dt S; 0 . T 0
(26)
5.2. Discretization and algorithm According to the same formulation as the European option, we have the following equation from Eq. (24): N X
ltþDt H 2 fj ¼ j
N X
j¼1
ltj G2 fj ,
(27)
j¼1
where
x ¼ fltj g
and
The coefficient matrix A and vector H 1 fj can be calculated analytically. The parameter l is determined from the numerical result at the previous time-step t þ Dt. Therefore, Eq. (28) is solved for ltj . The algorithm is similar to the European option except for the discretized equation and the strike condition. The valuation algorithm of the present method is as follows. 1. N collocation points are taken uniformly in the range of the asset price 0pSpS max . 2. The time interval 0ptpT is discretized with the timestep T=M between the date of purchase t ¼ 0 and the strike date t ¼ T. M denotes the number of time steps. 3. The option price V T at the expiration date t ¼ T is calculated from the strike condition (26). 4. The parameter lTj on the expiration date T is calculated from Eq. (11) on V T . 5. t T Dt. 6. Eq. (27) is solved for ltj . 7. t t Dt. 8. If t40, the process goes to step 6. 9. Substituting l0j into Eq. (11) leads to V 0 .
First, we adopt the MQ-RBF: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðR; Rj Þ ¼ c2 þ kR Rj k2 .
G 2 ¼ 1 yDtF 2 . Holding the above equation for the collocation points, we have (28)
(29)
The parameters are specified in Table 8. The parameter R is taken within 0:0pRp1:0 and the number of data points is N ¼ 101. The time step is Dt ¼ 0:0005 ðM ¼ 1000Þ, which is determined for comparison with finite difference method. The parameter c in the radial basis function (29) is determined the condition number of the matrix G 2 fj in Eq. (27). Table 9 indicates the effect of the parameter c to the condition number. From this results, we will take the parameter c ¼ 0:04. Numerical results are shown in Fig. 3. Besides, finite difference solutions are shown in Fig. 5. Comparing the figures shows that the results by using MQ-RBF are different from them by finite difference solutions at t ¼ 0:25 and 0:5. For improving the computational accuracy, instead of the radial basis function (29), we adopt the RMQ-RBF: 1 fðR; Rj Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . c2 þ kR Rj k2
H 2 ¼ 1 þ ð1 yÞDtF 2 ,
Ax ¼ b,
A ¼ ½G 2 fj ,
5.3. Numerical example
In this paper, we will consider only a call option. By taking Eqs. (22) and (23), we have the payoff on the expiration date t ¼ T as follows: R SHðR; TÞ ¼ S max 1 ; 0 . T Finally we have the strike condition for Eq. (24) as R HðR; TÞ ¼ max 1 ; 0 . T
where
H 2 fj . b ¼ ½ltþDt j
leads to V ðS; R; tÞ ¼ SHðR; tÞ,
841
(30)
Fig. 4 shows the results by using the radial basis function (30) with c ¼ 0:04. The other parameters are shown in Table 8. The results by explicit-scheme finite difference method are shown in Fig. 5. Comparing Fig. 3 with 4
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1
Table 8 Parameters for numerical result
0.8 Option Value H
T ¼ 0:5 (year) r ¼ 0:1 s ¼ 0:4 y ¼ 0:5 Rmax ¼ 1:0 Dt ¼ 0:0005 M ¼ 1000 N ¼ 101
Expiration date Risk free interest rate Volatility Crank–Nicolson method Maximum R Time-step size Number of time step Number of stock data points
t=0 t=0.25 t=0.5
0.6 0.4 0.2 0 0
Table 9 The condition number to each c
0.4
0.6
0.8
1
R
RBF parameter c
Condition number
0.03 0.035 0.04 0.045 0.05
2:47 107 1:09 108 4:86 108 2:18 109 9:87 109
Fig. 4. Values of European average strike call option, reciprocal multiquadric RBF, c ¼ 0:04.
1
t=0 t = 0.25 t = 0.5
0.8 Option Value H
1
t=0 t=0.25 t=0.5
0.8 Option Value H
0.2
0.6 0.4 0.2
0.6 0 0.4
0
0.2
0.4
0.6
0.8
1
R 0.2
Fig. 5. Values of European average strike call option, FDM.
0 0
0.2
0.4
0.6
0.8
1
R Fig. 3. Values of European average strike call option, multi-quadric RBF, c ¼ 0:04.
indicates that the use of new radial basis function (30) improves the numerical results. Finally, we will discuss the effect of the parameter c to the accuracy of the numerical results by using MQ-RBF and RMQ-RBF. The accuracy of the numerical results is estimated by the difference from the finite difference solutions. The error estimator is defined as e¼
N 1X jV ðS j ; tÞRBF V ðS j ; tÞFDM j, N j¼1
where V ðS j ; tÞRBF and V ðS j ; tÞFDM denote the solution by the present method and finite solution, respectively. Fig. 6 shows the effect of the parameter computational error. In case of MQ-RBF,
(31) numerical difference c to the numerical
solutions can be obtained at 0:005pcp0:04. In case of RMQ-RBF, numerical solutions can be obtained at cp0:06. Besides, the error estimator in MQ-RBF is much greater than that in RMQ-RBF. Therefore, we can say that the RMQ-RBF is more effective than the MQ-RBF. 6. Conclusion This paper describes the valuation of the European, the barrier, and the Asian options by using radial basis function approximation. In Section 3, the valuation scheme for European option of one-asset is explained and the numerical results are compared with a theoretical solution. The results agreed well with the theoretical solution. Therefore, the formulation was extended to the other options such as the barrier, and the Asian options. In Sections 4 and 5, the valuation schemes for the barrier and the Asian options were explained. Since the barrier option is governed with the same differential equation as the European option, it could be solved by using the same radial basis functions as that for the European option.
ARTICLE IN PRESS Y. Goto et al. / Engineering Analysis with Boundary Elements 31 (2007) 836–843
0.06
843
should be applied to the Asian option. In the future, we would like to discuss the relationship between the radial basis function and the differential operator of the governing equation.
MQ–RBF RMQ–RBF
Error
0.04
References 0.02
0.02
0.04 Parameter c
0.06
0.08
Fig. 6. Effect of parameter c to computational error.
However, in the Asian option, the different radial basis functions were necessary because their governing equation were different from that for the European and the barrier options. As shown by numerical results, while the MQ-RBF and the RMQ-RBF give the good computational accuracy for the European and the barrier options, the RMQ-RBF
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Engineering Analysis with Boundary Elements 31 (2007) 836–843 www.elsevier.com/locate/enganabound
Options valuation by using radial basis function approximation Yumi Goto, Zhai Fei, Shen Kan, Eisuke Kita Graduate School of Information Science, Nagoya University, Nagoya 464-8601, Japan Received 30 August 2006; accepted 6 February 2007 Available online 11 April 2007
Abstract This paper describes the valuation scheme of European, barrier, and Asian options of single asset by using radial basis function approximation. The option prices are governed with Black–Scholes equation. The equation is discretized with Crank–Nicolson scheme and then, the option price is approximated with the radial basis functions with unknown parameters. In the European and the barrier options, the prices are governed with Black–Scholes equation. The governing option of the Asian option, however, is different from them of the others. In that case, one has to adopt the other radial basis functions than that for the original Black–Scholes equation. Finally, numerical results are compared with theoretical and finite difference solutions in order to confirm the validity of the present formulation. r 2007 Elsevier Ltd. All rights reserved. Keywords: Option Contract; European option; Barrier option; Asian option; Radial basis function
1. Introduction Recently, financial derivatives are widely dealt and the importance is expanded. The importance of the derivative transaction is increasing for the adequate sharing of the financial risk. Therefore, it is considered that the option contract is one of the most important financial derivatives. The options can be classified into a European and an American options and so on. The European option price can be determined by analytically solving the so-called Black–Sholes equation. Since the other option price cannot be determined analytically, many researchers have presented the numerical schemes [1–3]. The existing schemes are mainly classified into binomial tree, finite differential and Monte Carlo methods. In the finite difference method, the governing difference equation is discretized with finite difference approximation and to be solved. If an option is constructed with many assets, huge computer memory is necessary in the finite difference method. The binomial tree method also has similar features as the finite difference method. Therefore, if the option is constructed with many assets, the Monte Carlo method is more effective. In the Corresponding author. Tel./fax: +81 52 789 3521.
E-mail address: [email protected] (E. Kita). 0955-7997/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2007.02.001
Monte Carlo method, the history of the asset price fluctuation is generated randomly and the option price is determined as their expected value. The computational accuracy of the Monte Carlo method depends on the quality of the algorithm to generate random number. On the other hand, this paper describes the evaluation scheme of the option price by using radial basis function approximation. Since the present scheme has the similar algorithm as the finite difference method, the computational accuracy does not depend on the random number generator and a large computer memory is necessary if the option is constructed with a lot of assets. However, when designing a option for private person or small business firm, the options are often constructed with small number of assets. In the cases, the present method as well as the finite difference method is effective. Besides, the present method does not need structured grid or mesh, unlike the finite difference method. The application of the radial basis function approximation to the option valuation was already presented by Hon and his co-workers [4,5], Boztosum [6] and Fasshauer [7]. While European and American options were considered in their studies, this paper focuses on the barrier and Asian options as well as European option. The Asian option price is governed with Black–Scholes equation [8]. However, the
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use of the function transformation often changes the governing equation from the original Black–Scholes equation. In that case, we should adopt the different radial basis function than that for the governing equation of European options. In the present formulation, the governing equation is discretized according to the Crank–Nicolson scheme on the time interval and the option price is approximated with radial basis function with unknown parameters at each time step. The initial values of the parameters are determined from the strike condition on the expiration date. Then, the parameters at the date of purchase are evaluated according to the backward algorithm from the expiration date to the date of purchase. The numerical solutions are compared with the analytical ones. In the numerical examples, the results are compared with the solutions with binomial tree and finite differential methods. The remaining of this paper is organized as follows. In Section 2, the option contracts are described. In Section 3, the valuation scheme for the European option of one asset is explained and the numerical solutions are compared with the theoretical ones. The valuation schemes of the barrier and Asian options are explained in Sections 4 and 5, respectively. Finally, the conclusions are summarized in Section 6. 2. Option contract An option is a contract between a buyer and a seller. The option contract is the right, but not the obligation, to buy (for a call option) or sell (for a put option) a specific amount of a given stock, commodity, currency, index, or debt, at a specified price (the strike price) during a specified period of time. For simplicity, this article will discuss only options connected to listed stocks. 2.1. European, American and exotic options Note that there are two basic types of options, the American and the European. An American (or Americanstyle) option is an option contract that can be exercised at any time between the date of purchase and the expiration date. A European (or European-style) option is an option contract that can only be exercised on the expiration date. An exotic option is an option contract which has more complex features than European and American options. An exotic option is an option contract that can be exercised according to the average value of the asset price during a specified period of time and their maximum and minimum prices. In this paper, we will focus on the European, the barrier and the Asian options. 2.2. Barrier option Barrier options can be classified into knock-out and knock-in options. Considering the barrier price K, the
837
knock-out option can be exercised unless the asset price S reaches the barrier K during the day of purchase and expiration day. The knock-in option can be exercised if the asset price S overtakes the barrier K. The knock-out options can be classified into ‘‘up-andout’’ and ‘‘down-and-out’’. The up-and-out option can be exercised unless the asset price S reaches the barrier K from beneath the barrier and the down-and-out option can be done unless the asset price reaches the barrier from above the barrier. Besides, the knock-in options can be classified into ‘‘up-and-in’’ and ‘‘down-and-in’’. The up-and-in option can be exercised if the asset price reaches the barrier from beneath the barrier and the down-and-in option can be done if the asset price reaches the barrier from above the barrier. The valuation algorithms of the options are almost similar and therefore, the down-and-out option is described in Section 4. 2.3. Asian option An Asian option, which is also called an average option, is an option whose payoff is linked to the average value of an asset before the expiration date. There are two basic forms; an average rate option or an average strike option. An average rate option or average price option is a cashsettled option whose payoff is based on the difference between the average value of the asset during the period from the day of purchase and the expiration date and a strike price E. An average strike option is a cash settled whose payoff is based on the difference between the average value of the asset during the period and the asset price at the expiration date. The European-type average strike option is described in Section 5. 3. European option 3.1. Governing equation and strike condition Now, we consider the option constructed with only one asset. When the asset price at the date t is indicated with S, the option price V ðS; tÞ is governing with Black–Scholes equation [3,8]: qV ðS; tÞ 1 2 2 q2 V ðS; tÞ qV ðS; tÞ þ s S rV ¼ 0, þ rS 2 qt 2 qS qS
(1)
where the volatility s is considered to be constant between the date of purchase and the expiration date. Defining the differential operator 1 q2 q F 1 ¼ s2 S2 2 þ rS r, 2 qS qS
(2)
we can rewrite Eq. (1) to qV ðS; tÞ þ F 1 V ðS; tÞ ¼ 0. qt
(3)
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For European call option, the strike condition is given as V ðS; TÞ ¼ maxðSðTÞ E; 0Þ,
(4)
Substituting Eq. (11) into Eq. (8), we have N X
ljtþDt H 1 fj ¼
N X
ltj G 1 fj .
(12)
and, for European put option, the strike condition is as
j¼1
V ðS; TÞ ¼ maxðE SðTÞ; 0Þ,
Holding the above equation for the collocation points, we have
(5)
where E denotes the strike price and the function maxðS E; 0Þ gives the bigger one between S E and 0.
j¼1
(13)
Ax ¼ b, where
3.2. Discretization
A ¼ ½G 1 fj ,
Discretizing equation (3) with Crank–Nicolson scheme, we have V ðS; t þ DtÞ V ðtÞ þ ð1 yÞF 1 V ðS; t þ DtÞ Dt þ yF 1 V ðS; tÞ ¼ 0,
ð6Þ
where the parameter y is specified in the range of 0pyp1. Rearranging the above equation for V ðtÞ and V ðt þ DtÞ, we have ½1 þ ð1 yÞDtF 1 V ðS; t þ DtÞ ¼ ½1 yDtF 1 V ðS; tÞ
(7)
x ¼ fltj g b¼
and
H 1 fj . ½ltþDt j
The coefficient matrix A and vector H 1 fj can be calculated analytically. The parameter l is determined from the numerical result at the previous time-step t þ Dt. Therefore, Eq. (13) is solved for ltj . Eq. (13) is solved iteratively from the expiration date t ¼ T to the date of purchase t ¼ 0. Once we can get the parameter l0j at the date t ¼ 0, the option price at the date of purchase is estimated from l0j by Eq. (11).
and then, H 1 V tþDt ¼ G 1 V t ,
(8)
where
The algorithm is as follows.
V ðS; tÞ ¼ V t , V ðS; t þ DtÞ ¼ V tþDt , H 1 ¼ 1 þ ð1 yÞDtF 1 , G 1 ¼ 1 yDtF 1 . In this study, the option price V t is approximated with the multi-quadric radial basis function (MQ-RBF); qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðS; S j Þ fj ¼ c2 þ kS Sj k2 , (9) or the reciprocal multi-quadric radial basis function (RMQ-RBF); 1 fðS; S j Þ fj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , c2 þ kS Sj k2
(10)
where Sj is the asset price at the collocation point j for approximating the option price V . The parameter c is the support radius of the radial basis function. It is taken in this study as c ¼ 1. The radial basis function approximation of the option price V t is given as Vt ’
3.3. Algorithm
N X
ltj fj ,
(11)
j¼1
where N and lj denote the total number of the collocation points at the date t and the unknown parameters, respectively.
1. N collocation points are taken uniformly in the range of the asset price 0pSpSmax . 2. The time interval 0ptpT is discretized with the timestep T=M between the date of purchase t ¼ 0 and the strike date t ¼ T where M denotes the number of time steps. 3. The option price V T at the expiration date t ¼ T is calculated from the strike condition (4) or (5). 4. The parameter lTj on the expiration date T is calculated from Eq. (11) on V T . 5. t T Dt. 6. Eq. (12) is solved for ltj . 7. t t Dt. 8. If t40, the process goes to step 6. 9. Substituting l0j into Eq. (11) leads to V 0 . 3.4. Numerical example A European put option is considered here. The MQRBF, which is shown in Eq. (9), is adopted first. The parameters are listed in Table 1. One discussed the effect of the total number of the collocation points N and the timestep size Dt to the computational accuracy. First we will discuss the effect of the total number of the collocation points N to the computational accuracy. The time-step size is specified as Dt ¼ T=M ¼ 0:005. The computational error e and the condition number of the coefficient matrix are shown in Table 2. The error is
ARTICLE IN PRESS Y. Goto et al. / Engineering Analysis with Boundary Elements 31 (2007) 836–843 Table 1 Parameters for numerical result
Table 3 Parameters for numerical result T ¼ 0:5 (year) E ¼ 10:0 r ¼ 0:05 s ¼ 0:2 y ¼ 0:5 Smax ¼ 30 c¼1
Expiration date Exercise price Risk free interest rate Volatility Crank–Nicolson method Maximum stock value RBF parameter
Table 2 The condition number and the error for N N
Condition number
e
5
61 121 151
0.0210884 0.000289184 0.000505691
4:64 10 6:45 108 1:99 1010
0.0002895
ε
839
0.000289
T ¼ 0:5 (year) E ¼ 10:0 r ¼ 0:05 s ¼ 0:2 y ¼ 0:5 Smax ¼ 30 M ¼ 100 Dt ¼ 0:005 N ¼ 121 c ¼ 1:0
Expiration date Exercise price Risk free interest rate Volatility Crank–Nicolson method Maximum stock value Number of time step Time-step size Number of stock data points RBF parameter
Table 4 Results of European put option by using MQ-RBF Stock S
V Anal:
V RBF
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0
9.7531 7.7531 5.7531 3.75318 1.79871 0.44197 0.04834 0.00277 0.00010 0.00000
9.7531 7.7531 5.7531 3.75318 1.79823 0.44055 0.04780 0.00271 0.00010 0.00008
0.0002885
0.02
0.01 Δt
0.00625
0.005
Fig. 1. Variation of the relative error with Dt.
defined as e¼
N 1X jV ðS j ; tÞRBF V ðS j ; tÞAnal: j, N j¼1
(14)
where V ðS j ; tÞRBF and V ðS j ; tÞAnal: denote the numerical solution by the present method and the theoretical solution, respectively. We notice that the computational error decreases and the condition number increases according to the increase of the total number of the collocation points. The total number of the collocation points strongly affects the computational accuracy. If the condition number of the coefficient matrix is relatively low, it is good to take large number of the collocation points. So, we will take N ¼ 121. In case of N ¼ 121, the effect of the time-step size Dt to the computational accuracy is shown in Fig. 1. The abscissa and the ordinate indicate the computational error and the time-step size, respectively. We notice that the error is very huge for Dt40:02 and that the computational error is getting less at Dt ¼ 0:005. Numerical results by using the MQ-RBF are compared with the theoretical ones in Table 4. The parameters are
Table 5 Results of European put option by using RMQ-RBF Stock S
V Anal:
V RBF
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0
9.7531 7.7531 5.7531 3.75318 1.79871 0.44197 0.04834 0.00277 0.00010 0.00000
9.7531 7.7531 5.7531 3.75318 1.79823 0.44055 0.04780 0.00271 0.00010 0.00008
specified as shown in Table 3. Numerical results agree well with the theoretical ones (Table 4). Finally, the RMQ-RBF (10) is adopted, instead of the MQ-RBF. One takes the same parameters as them for the MQ-RBF (Table 3). Numerical results are shown in Table 5, as well as theoretical ones. We notice that the results by using RMQ-RBF agree with them by MQ-RBF. 4. Barrier option 4.1. Governing equation and strike condition We shall consider the down-and-out option of the expiration price E and the barrier K. The option becomes
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840
invalid if the asset price S reaches the barrier E from above the barrier during the day of purchase and the expiration date. Even unless the asset price S reaches the barrier E, i.e., S4K, the option is European call option. The price of the down-and-out option is governed with qV þ F 1 V ¼ 0 ðS4KÞ, qt V ¼ 0 ðSpKÞ.
ð15Þ ð16Þ
The strike condition on the expiration day is given as V ðS; TÞ ¼ maxðSðTÞ E; 0Þ.
(17)
Table 6 Parameters for numerical analysis T ¼ 0:5 (year) E ¼ 10:0 r ¼ 0:05 s ¼ 0:2 y ¼ 0:5 H ¼ 9:0 Smax ¼ 30 Dt ¼ 0:05 M ¼ 100 N ¼ 121 c ¼ 1:0
Expiration date Exercise price Risk free interest rate Volatility Crank–Nicolson method Barrier Maximum stock value Time-step size Number of time step Number of stock data points RBF parameter
If S reaches K, the option is invalid. Therefore, ðS ¼ KÞ.
Therefore, the payoff X is as ( maxðS E; 0Þ ðS4KÞ; X¼ 0 ðSpKÞ:
(18)
(19)
4.2. Discretization and algorithm While the strike condition in the barrier option is different from the European option, the differential equation (15) is identical. Therefore, the discretized equation derived from Eq. (15) is the same as that in the European option; i.e., Eq. (12). The algorithm is also similar to that in Section 3.3 except for the strike condition. The valuation algorithm of the down-and-out option is as follows. 1. N collocation points are taken uniformly in the range of the asset price 0pSpS max . 2. The time interval 0ptpT is discretized with the timestep T=M between the date of purchase t ¼ 0 and the strike date t ¼ T. M denotes the number of time steps. 3. The option price V T at the expiration date t ¼ T is calculated from the strike condition (17). 4. The parameter lTj on the expiration date T is calculated from Eq. (11) on V T . 5. t T Dt. 6. Eq. (12) is solved for ltj . 7. t t Dt. 8. If t40, the process goes to step 6. 9. Substituting l0j into Eq. (11) leads to V 0 .
Table 7 Results of down-and-out option Stock S
V Anal:
V MQ
V RMQ
1 3 5 7 9 11 13 15 17 19
0.0000 0.0000 0.0000 0.0000 0.0000 1.3998 3.2591 5.2475 7.2469 9.2469
0.0000 0.0000 0.0000 0.0000 0.0000 1.3985 3.2589 5.2474 7.2469 9.2466
0.0000 0.0000 0.0000 0.0000 0.0000 1.3985 3.2589 5.2474 7.2465 9.2383
10
MQ– RBF RMQ– RBF Analytical
8 Option Value V
V ðK; tÞ ¼ 0
6 4 2 0 0
2
4
6
8 10 Stock S
12
14
16
18
Fig. 2. Values of European down-and-out call option.
5. Asian option 5.1. Governing equation and strike condition
4.3. Numerical example We will adopt the MQ-RBF and the RMQ-RBF. The parameters are specified as shown in Table 6. Numerical results are shown in Table 7 and Fig. 2, as well as theoretical solutions. The abscissa and the ordinate denote the option price V and the asset price S, respectively. We notice that the results by using MQRBF agree well with the theoretical ones and that the results by using RMQ-RBF are very slightly different from the theoretical ones.
We shall consider that the payoff of an Asian option depends on an average rate of an asset. The average rate of an asset S is given as Z 1 t SðtÞ dt. t 0 Introducing the function Z t I¼ SðtÞ dt, 0
(20)
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we have the partial differential equation governing the Asian option price V 2
qV qV 1 2 2 q V qV þS þ sS rV ¼ 0. þ rS qt qI 2 qS qS 2 Using the function Z 1 t I R¼ SðtÞ dt ¼ S 0 S
(21)
(22)
(23)
where H denotes the option price. Substituting Eqs. (22) and (23) to (21), we have qH þ F 2H ¼ 0 qt
(24)
and then, 1 q2 q . F 2 ¼ s2 R2 2 þ ð1 rÞR 2 qR qR
(25)
In case of an call option, the payoff on the expiration date t ¼ T is given as Z 1 t max S SðtÞ dt; 0 T 0 and in case of an put option, Z t 1 max SðtÞ dt S; 0 . T 0
(26)
5.2. Discretization and algorithm According to the same formulation as the European option, we have the following equation from Eq. (24): N X
ltþDt H 2 fj ¼ j
N X
j¼1
ltj G2 fj ,
(27)
j¼1
where
x ¼ fltj g
and
The coefficient matrix A and vector H 1 fj can be calculated analytically. The parameter l is determined from the numerical result at the previous time-step t þ Dt. Therefore, Eq. (28) is solved for ltj . The algorithm is similar to the European option except for the discretized equation and the strike condition. The valuation algorithm of the present method is as follows. 1. N collocation points are taken uniformly in the range of the asset price 0pSpS max . 2. The time interval 0ptpT is discretized with the timestep T=M between the date of purchase t ¼ 0 and the strike date t ¼ T. M denotes the number of time steps. 3. The option price V T at the expiration date t ¼ T is calculated from the strike condition (26). 4. The parameter lTj on the expiration date T is calculated from Eq. (11) on V T . 5. t T Dt. 6. Eq. (27) is solved for ltj . 7. t t Dt. 8. If t40, the process goes to step 6. 9. Substituting l0j into Eq. (11) leads to V 0 .
First, we adopt the MQ-RBF: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðR; Rj Þ ¼ c2 þ kR Rj k2 .
G 2 ¼ 1 yDtF 2 . Holding the above equation for the collocation points, we have (28)
(29)
The parameters are specified in Table 8. The parameter R is taken within 0:0pRp1:0 and the number of data points is N ¼ 101. The time step is Dt ¼ 0:0005 ðM ¼ 1000Þ, which is determined for comparison with finite difference method. The parameter c in the radial basis function (29) is determined the condition number of the matrix G 2 fj in Eq. (27). Table 9 indicates the effect of the parameter c to the condition number. From this results, we will take the parameter c ¼ 0:04. Numerical results are shown in Fig. 3. Besides, finite difference solutions are shown in Fig. 5. Comparing the figures shows that the results by using MQ-RBF are different from them by finite difference solutions at t ¼ 0:25 and 0:5. For improving the computational accuracy, instead of the radial basis function (29), we adopt the RMQ-RBF: 1 fðR; Rj Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . c2 þ kR Rj k2
H 2 ¼ 1 þ ð1 yÞDtF 2 ,
Ax ¼ b,
A ¼ ½G 2 fj ,
5.3. Numerical example
In this paper, we will consider only a call option. By taking Eqs. (22) and (23), we have the payoff on the expiration date t ¼ T as follows: R SHðR; TÞ ¼ S max 1 ; 0 . T Finally we have the strike condition for Eq. (24) as R HðR; TÞ ¼ max 1 ; 0 . T
where
H 2 fj . b ¼ ½ltþDt j
leads to V ðS; R; tÞ ¼ SHðR; tÞ,
841
(30)
Fig. 4 shows the results by using the radial basis function (30) with c ¼ 0:04. The other parameters are shown in Table 8. The results by explicit-scheme finite difference method are shown in Fig. 5. Comparing Fig. 3 with 4
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1
Table 8 Parameters for numerical result
0.8 Option Value H
T ¼ 0:5 (year) r ¼ 0:1 s ¼ 0:4 y ¼ 0:5 Rmax ¼ 1:0 Dt ¼ 0:0005 M ¼ 1000 N ¼ 101
Expiration date Risk free interest rate Volatility Crank–Nicolson method Maximum R Time-step size Number of time step Number of stock data points
t=0 t=0.25 t=0.5
0.6 0.4 0.2 0 0
Table 9 The condition number to each c
0.4
0.6
0.8
1
R
RBF parameter c
Condition number
0.03 0.035 0.04 0.045 0.05
2:47 107 1:09 108 4:86 108 2:18 109 9:87 109
Fig. 4. Values of European average strike call option, reciprocal multiquadric RBF, c ¼ 0:04.
1
t=0 t = 0.25 t = 0.5
0.8 Option Value H
1
t=0 t=0.25 t=0.5
0.8 Option Value H
0.2
0.6 0.4 0.2
0.6 0 0.4
0
0.2
0.4
0.6
0.8
1
R 0.2
Fig. 5. Values of European average strike call option, FDM.
0 0
0.2
0.4
0.6
0.8
1
R Fig. 3. Values of European average strike call option, multi-quadric RBF, c ¼ 0:04.
indicates that the use of new radial basis function (30) improves the numerical results. Finally, we will discuss the effect of the parameter c to the accuracy of the numerical results by using MQ-RBF and RMQ-RBF. The accuracy of the numerical results is estimated by the difference from the finite difference solutions. The error estimator is defined as e¼
N 1X jV ðS j ; tÞRBF V ðS j ; tÞFDM j, N j¼1
where V ðS j ; tÞRBF and V ðS j ; tÞFDM denote the solution by the present method and finite solution, respectively. Fig. 6 shows the effect of the parameter computational error. In case of MQ-RBF,
(31) numerical difference c to the numerical
solutions can be obtained at 0:005pcp0:04. In case of RMQ-RBF, numerical solutions can be obtained at cp0:06. Besides, the error estimator in MQ-RBF is much greater than that in RMQ-RBF. Therefore, we can say that the RMQ-RBF is more effective than the MQ-RBF. 6. Conclusion This paper describes the valuation of the European, the barrier, and the Asian options by using radial basis function approximation. In Section 3, the valuation scheme for European option of one-asset is explained and the numerical results are compared with a theoretical solution. The results agreed well with the theoretical solution. Therefore, the formulation was extended to the other options such as the barrier, and the Asian options. In Sections 4 and 5, the valuation schemes for the barrier and the Asian options were explained. Since the barrier option is governed with the same differential equation as the European option, it could be solved by using the same radial basis functions as that for the European option.
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0.06
843
should be applied to the Asian option. In the future, we would like to discuss the relationship between the radial basis function and the differential operator of the governing equation.
MQ–RBF RMQ–RBF
Error
0.04
References 0.02
0.02
0.04 Parameter c
0.06
0.08
Fig. 6. Effect of parameter c to computational error.
However, in the Asian option, the different radial basis functions were necessary because their governing equation were different from that for the European and the barrier options. As shown by numerical results, while the MQ-RBF and the RMQ-RBF give the good computational accuracy for the European and the barrier options, the RMQ-RBF
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