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Numerical Methods for Pricing American Options with Time-Fractional PDE Models.

1. Introduction

Fractional differential equations have wide applications in the fields of physics modelling (see, e.g., a book [1] and the references therein). Based on the fact that the fractional-order derivatives are characterized by the "globalness" and can provide a powerful instrument for the description of memory and hereditary properties of different substances, recently the fractional differential equations are applied to the area of mathematical finance by generalizing the Black-Scholes (B-S) equations to the fractional order. In this subject, there are mainly two types of fractional derivatives involved: a space-fractional derivative and a time-fractional derivative. Regarding a space-fractional derivative, Carr and Wu [2] introduced a finite moment log stable (FMLS) model and showed that the model outperforms other widely used financial models. Cartea and del-Castillo-Negrete [3] successfully connected the FMLS process with the space-fractional derivatives. Later on, Chen et al. [4] derived the explicit closed-form formula for European vanilla options under the FMLS model. As for time-fractional derivative, Wyss [5] derived a closed-form solution for European vanilla options under a time-fractional B-S equation. Kumar et al. [6] further explored the European option pricing under time-fractional B-S equation using the Laplace transform. Later on using tick-to-tick date, Cartea [7] found that the value of European-style options satisfies a fractional partial differential equation with the Caputo time-fractional derivative. Jumarie [8] derived a time-and-space-fractional B-S equation. Liang et al. [9] introduced bifractional B-S models in which they assume that the underlying asset follows a fractional Ito process and the change of option price with time is a fractal transmission system. Chen et al. [10] derived an analytic formula for pricing double barrier options based on a time-fractional B-S equation.

To the best of our knowledge, we are only aware of one paper Chen et al. [11] studying the American option pricing under fractional derivatives. In [11], they studied predictor-corrector finite difference methods for pricing American options under the FMLS model which is a kind of space-fractional derivative model. In this paper we study the pricing of American options with time-fractional model which has essential difference to the space-fractional model. Following the model in [10], we assume that the underlying asset price still follows the classical Brownian motion, but the change in the option price is considered as a fractal transmission system. The price of such American option follows a time-fractional partial differential equation (PDE) with free boundary. The solution of the time-fractional PDE with free boundary is much more challenging than solving the fractional PDEs with fixed boundary for European option pricing in [10]. In this paper we develop two numerical methods, Laplace transform method and finite difference method, to solve the free-boundary problem of time-fractional PDE. The Laplace transform for partial differential equation can result in an ordinary differential equation; that is, it can convert partial differential equations into ordinary differential equations. However the solution of ordinary differential equations is much simpler than that for partial differential equations. This is the intuition of using Laplace transform method for solving the PDEs with free boundary. The Laplace transform method has been successfully applied to the pricing American options with classical (integer) B-S equations (see [12]). In the current work, by applying the Laplace transform to time-fractional PDE with respect to time, constructing the general solutions to the resulted ordinary differential equation (ODE), and then using the (free) boundary conditions, we derive a nonlinear equation for the free boundary (or optimal early exercise boundary) function in Laplace space. This nonlinear equation is solved by the secant method and the approximate free boundary in the Laplace space is obtained. Then the Laplace inversion is applied to get the optimal early exercise boundary in the time space. Finally the approximate value of the American option is obtained. Also motivated by [13, 14] for solving integer-order PDEs with free boundary, we develop a boundary-searching finite difference method to solve the time-fractional PDEs free boundary. The comparison of the Laplace method with the finite difference method is made by several examples. The numerical results show that the Laplace transform method is much more efficient than the finite difference method.

The rest of paper is arranged as follows: In Section 2 we describe the American option pricing models with time-fractional derivatives; in Section 3 we introduce the Laplace transform methods for the problem; in Section 4 we give the boundary-searching finite difference methods for the problem; in Section 5 we provide numerical examples to compare the performance of the Laplace transform method with FDM; the conclusions are made in the final section.

2. American Option Pricing Models with Time-Fractional Derivatives

Assume that the underlying asset price is governed by the constant elasticity of variance (CEV) model (see, e.g., Cox [15]):

[dS.sub.t] = (r-q) [S.sub.t]dt + [delta][S.sup.[beta]+1.sub.t][dW.sub.t], (1)

where r is the risk-free interest rate, q is the dividend yield, and [W.sub.t] is standard Brownian motion. [sigma](S) = [delta][S.sup.[beta]] represents the local volatility function and [beta] can be interpreted as the elasticity of [sigma](S). If [beta] = 0, then SDE (1) becomes the lognormal diffusion model.

Denote by f(S, t) the price of American option with S being underlying and t being the current time. From SDE (1), f(S, t) should satisfy the modified B-S equation

[mathematical expression not reproducible] (2)

However, it is argued that the time derivative [partial derivative]f/[partial derivative]t should be replaced by the fractional derivative [[partial derivative].sup.[alpha]]f/[partial derivative][t.sup.[alpha]] (0 < [alpha] < 1) under the assumption that the change in the option price follows a fractal transmission system (see, e.g., [9, 10]); that is,

[mathematical expression not reproducible] (3)

The meaning of the assumption is that the diffusion of the option price depends on the history of the time to maturity. Moreover it is suggested by [9, 10] that the following right-hand side modified Riemann-Liouville derivative can be used:

[mathematical expression not reproducible] (4)

For convenience, use the coordinate transform [tau] = T - t and denote

V(S, [tau]) = f(S, T - [tau]) = f(S, t). (5)

Then we calculate

[mathematical expression not reproducible] (6)

Recall from [16] that the left-hand side Caputo fractional derivative is defined as

[mathematical expression not reproducible] (7)

Then (6) gives that

[mathematical expression not reproducible] (8)

According to (3), (5), (8) and the facts

[mathematical expression not reproducible] (9)

the valuation of American put option can be formulated as a time-fractional free-boundary problem:

[mathematical expression not reproducible] (10)

V(S, 0) = max (K - S, 0), (11)

V([S.sub.f] ([tau]), [tau]) = K - [S.sub.f] ([tau]), (12)

[partial derivative]V([S.sub.f]([tau]), [tau])/[partial derivative]S = -1, (13)

[mathematical expression not reproducible] (14)

The main purpose of this paper is to solve problem (10)-(14).

3. Laplace Transform Methods

For [lambda] > 0, define the Laplace-Carson transform (LCT) as

[mathematical expression not reproducible] (15)

The LCT is essentially the same as the Laplace transform (LT) and the relationship between LCT and LT is

[L.sub.C] [V(S, [tau])]([lambda]) = [lambda]L [V(S, [tau])]([lambda]). (16)

The reason of using the LCT is to simplify the notations in the later analysis. Using the Laplace transform formula for the Caputo fractional derivative (see (2.253) in [16]),

[mathematical expression not reproducible] (17)

and relationship (16), the LCT for [mathematical expression not reproducible] is found as

[mathematical expression not reproducible] (18)

Taking LCT to (10)-(14), we have

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

The solution of governing equation (19) is given by

[mathematical expression not reproducible] (23)

For the case [beta] = 0, the basis functions [[phi].sub.[lambda]](S) and [[psi].sub.[lambda]](S) have the following forms (the derivation is analogous to that in [17]):

[mathematical expression not reproducible] (24)

By simple calculation, we have

[mathematical expression not reproducible] (25)

For the case [beta] [not equal to] 0, the functions [[phi].sub.[lambda]](S) and [[psi].sub.[lambda]](S) are given by (the derivation is similar to that in [18])

[mathematical expression not reproducible] (26)

where [M.sub.k,m](x) and [W.sub.k,m](x) are the Whittaker functions, [I.sub.v](x) and [K.sub.v](x) are the modified Bessel functions (see, e.g., [18]), and

[mathematical expression not reproducible] (27)

Moreover, it can be calculated that

[mathematical expression not reproducible] (28)

Next, we give a particular solution [u.sub.[lambda]](S) and derive the nonlinear equation for [S.sub.f](X). By matching conditions (20)-(22), we derive that

[mathematical expression not reproducible] (29)

where

[mathematical expression not reproducible] (30)

Furthermore, letting S = [S.sub.f](X) in formula (23) and using conditions (20) and (21), we derive the nonlinear equation for [S.sub.f]([lambda]):

[mathematical expression not reproducible] (31)

where

[mathematical expression not reproducible] (32)

Using the secant method to solve (31), we obtain [S.sub.f]([lambda]) for different values of [lambda]. Finally, the optimal exercise boundary and put option price can be expressed in terms of the Laplace inversion:

[mathematical expression not reproducible] (33)

4. Finite Difference Methods

Suppose [S.sub.max] is a large enough positive number such that V([S.sub.max], [tau]) [approximately equal to] 0 for all [tau] [member of] [0, T]. Define uniform time and space mesh

[mathematical expression not reproducible] (34)

The time-fractional derivative [mathematical expression not reproducible] can be formulated as (see [19])

[mathematical expression not reproducible] (35)

with

[mathematical expression not reproducible] (36)

We approximate the space derivatives by central difference:

[mathematical expression not reproducible] (37)

Inserting (35) and (37) into PDE (10) and omitting the higher term O[([DELTA][tau]).sup.2-[alpha]]) and 0([([DELTA]S).sup.2]), we obtain the FDM scheme (with notation [V.sup.n.sub.j] [approximately equal to] V([S.sub.j], [[tau].sub.n])) as follows:

[c.sub.j-1] [V.sup.n.sub.j-1] + [c.sub.j][V.sup.n.sub.j] + [c.sub.j+1] + [V.sup.n.sub.j+1] = [d.sup.n.sub.j], (38)

where

[mathematical expression not reproducible] (39)

The initial condition is [V.sup.0.sub.j] = max(K - [S.sub.j], 0), and the left-hand side boundary and the boundary conditions are specified by the following algorithm.

Differently from fixed boundary problems, how to determine the moving boundaries [S.sub.f]([tau]) is the key for pricing American option. We design a simple algorithm, namely, boundary-searching method, for solving time-fractional free-boundary problem (10)-(14).

Boundary-Searching Algorithm

Step 1. Let [S.sub.f](0) = K. Since [V.sup.0.sub.j] = max(K - [S.sub.j], 0), we determine J([[tau].sub.0]) by

[mathematical expression not reproducible] (40)

So we know [mathematical expression not reproducible].

Step 2. For n = 1, ..., M

Step 2.1. Search backward for J([[tau].sub.n]).

For i = J([[tau].sub.n-1]) - 1, J([[tau].sub.n-1]) - 2, ..., 1, 0

Solve(38) with boundary conditions [V.sup.n.sub.i] = K - [S.sub.i] and [V.sup.n.sub.N] = 0.

If the solutions [V.sup.n.sub.i+1] and [V.sup.n.sub.i] satisfy ([V.sup.n.sub.i+1] - [V.sup.n.sub.i])/[DELTA]S [less than or equal to] -1, then

J([[tau].sub.n]) = i + 1; Break;

End If

End For

Then we have that [mathematical expression not reproducible]. More accurately searching of [S.sub.f]([[tau].sub.n]) can be carried out in the next substep.

Step 2.2. Let [mathematical expression not reproducible]. Then redefine the boundary conditions for (38) as

[mathematical expression not reproducible] (41)

with Lagrange basis functions

[mathematical expression not reproducible] (42)

Then using the solutions of (38) with the new boundary condition (41), we calculate the approximation

[mathematical expression not reproducible] (43)

We find the appropriate value of 0 [less than or equal to] [theta] [less than or equal to] 1 such that

[partial derivative]/[partial derivative]S V([S.sup.[theta].sub.f] ([[tau].sub.n]), [[tau].sub.n]) [approximately equal to] -1. (44)

Step 3. Output early exercise boundaries [S.sup.[theta].sub.f] ([[tau].sub.n]) and American put option prices [V.sup.n.sub.j] (for j = 0, 1, ..., J([[tau].sub.n]) - 1, [V.sup.n.sub.j] = K - [S.sub.j] and for j = J([[tau].sub.n]), ..., N, [V.sup.n.sub.j] are obtained from FDM (38) with the new boundary condition (41)).

Remark 1. (a) The boundary-searching method is based on the fact that the early exercise boundary [S.sub.f](T) is monotonically decreasing in [tau] and the derivative [partial derivative]V/[partial derivative]S is monotonically increasing with respect to S. (b) Note that the left-hand side of (41) is just the quadratic Lagrange approximation of V([S.sup.[theta].sub.f]([[tau].sub.n]), [[tau].sub.n]). The role of Step 2.2 is to numerically search a value [mathematical expression not reproducible] such that

[partial derivative]/[partial derivative]SV([S.sub.f] ([[tau].sub.n]), [[tau].sub.n]) = -1,

V([S.sub.f] ([[tau].sub.n]), [[tau].sub.n]) = K - [S.sub.f] ([[tau].sub.n]). (45)

In fact, the boundary-searching method still works without Step 2.2; however, Step 2.2 can lead to more accurate results.

5. Numerical Examples

In this section, we implement and compare the Laplace transform method and the boundary-searching finite difference method (FDM). The Laplace inversion is calculated by Gaver-Wynn-Rho (GWR) algorithm in [20]. The mesh parameters are taken as M = 200, N = 800, and [S.sub.max] = 200 for FDM. The volatility at time t = 0 is defined by [[sigma].sub.0] = [delta][S.sup.[beta].sub.0].

Table 1 lists the computational values of American put options. Columns entitled "LTM" and "FDM" represent the Laplace transform method and the boundary-searching FDM, respectively. For [alpha] = 1, the time-fractional derivative reduces to the first-order derivative in time and the problem becomes the classical American option pricing problem. From Table 1, we can see that when [alpha] = 1 the results are consistent with the prices listed in [12]. At time to maturity [tau] = 3, one could observe that the option prices are decreasing as a is decreasing. Moreover, the LTM takes much less CPU time than the FDM as observed from Table 1.

In Figure 1, we plot the early exercise boundaries computed by LTM and FDM for the cases of [alpha] = 0.7 and [alpha] = 0.4. The numbers of mesh nodes are taken as M = 50, N = 200 for FDM. It can be seen that all the boundaries at different obtained by LTM are very close time t obtained by LTM are very close to those by FDM. Figure 2 illustrates the effect of different time-fractional derivative order [alpha] on the early exercise boundaries and option prices. It can be seen from Figure 2(a) that when the value of [alpha] is decreasing, the exercise region will be shrunk in the region of time to maturity [tau] close to 0 and will be enlarged in the region of [tau] close to T. From Figure 2(b) we observe that the time-fractional derivatives have little effect on the option price for the cases of deep-in-the-money ([S.sub.0] [much less than] K) and deep-out-the-money ([S.sub.0] [much greater than] K) and have significant effect near to on-the-money ([S.sub.0] [approximately equal to] K).

6. Conclusions

There have been increasing applications of fractional PDEs in the field of option pricing. In the history most focuses have been on the valuation of options without early exercise features. In current work we study the American option pricing with time-fractional PDEs. The option pricing problem is formulated into a time-fractional PDE with free boundary. Two methods, namely, Laplace transform method (LTM) and finite difference method (FDM), are proposed to solve the time-fractional free-boundary problem. Numerical examples show that the LTM is more efficient than the FDM. The methods in this paper are quite different to the predictor-corrector approach for pricing American options under space-fractional derivatives in [11]. The extension of the methods in this paper to the American option pricing under space-fractional models will be left for the future work. In addition it will be also interesting to extend the methods to the newly established models [21, 22].

http://dx.doi.org/10.1155/2016/5614950

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by the Fundamental Research Funds for the Central Universities (Grant nos. 15CX141110 and JBK1307012) and National Natural Science Foundation of China (Grant no. 11471137).

References

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[3] A. Cartea and D. del-Castillo-Negrete, "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and Its Applications, vol. 374, no. 2, pp. 749-763, 2007.

[4] W. Chen, X. Xu, and S.-P. Zhu, "Analytically pricing European-style options under the modified Black-Scholes equation with a spatial-fractional derivative," Quarterly of Applied Mathematics, vol. 72, no. 3, pp. 597-611, 2014.

[5] W. Wyss, "The fractional Black-Scholes equation," Fractional Calculus & Applied Analysis, vol. 3, no. 1, pp. 51-61, 2000.

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[7] A. Cartea, "Derivatives pricing with marked point processes using tick-by-tick data," Quantitative Finance, vol. 13, no. 1, pp. 111-123, 2013.

[8] G. Jumarie, "Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations," Insurance: Mathematics and Economics, vol. 42, no. 1, pp. 271-287, 2008.

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Zhiqiang Zhou (1) and Xuemei Gao (2)

(1) School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Wenjiang 611130, China

(2) School of Economic Mathematics and School of Finance, Southwestern University of Finance and Economics, Chengdu, Wenjiang 611130, China

Correspondence should be addressed to Zhiqiang Zhou; zqzhou@2014.swufe.edu.cn

Received 26 September 2015; Accepted 22 December 2015

Academic Editor: George Tsiatas

Caption: Figure 1: Early exercise boundaries computed by LTM and FDM.

Caption: Figure 2: Early exercise boundaries (a) and option prices at [tau] = T = 3 (b) with different [alpha] and fixed [[sigma].sub.0] = 0.4, [beta] = -1, r = 0.05, q = 0, and K = 40.
Table 1: Prices of American put option at [tau] = 3.

r = 0.05, q = 0, T = 3, [S.sub.0] = 40, K = 40, [beta] = 0

               [[sigma].sub.0]   [[sigma].sub.0]
                   = 0.1.           = 0.2.

[alpha]         LTM     FDM      LTM      FDM

1.0           1.2189   1.2362   3.4116   3.4792
0.9           1.1771   1.1912   3.2651   3.3157
0.7           1.0959   1.1028   2.9817   3.0071
0.4           0.9778   0.9793   2.5770   2.5829
0.2           0.9005   0.9002   2.3184   2.3191

CPU time (s)   1.65    234.68   1.73    241.34

r = 0.05, q = 0,T = 3, [S.sub.0] = 40, K = 40, [beta] = -1

               [[sigma].sub.0]  [[sigma].sub.0]
                   = 0.1.           = 0.2.

[alpha]         LTM     FDM      LTM     FDM

1.0           1.1877   1.2020   3.3325   3.3834
0.9           1.1485   1.1604   3.1918   3.2297
0.7           1.0722   1.0802   2.9208   2.9400
0.4           0.9609   0.9657   2.5347   2.5397
0.2           0.8877   0.8922   2.2876   2.2898
CPU time (s)   1.53    239.21   1.98     247.22
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Date:Jan 1, 2016
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