Fast and accurate calculation of American option prices

Abstract

We propose a very efficient numerical method to solve a nonlinear partial differential problem that is encountered in the pricing of American options. In particular, by using the front-fixing approach originally developed in Wu and Kwok (J Financ Eng 6:83–97, 1997) and Nielsen et al. (J Comput Finance 5:69–97, 2002) in conjunction with a suitable change of the time variable, a (nonlinear) partial differential problem is obtained which can be solved very efficiently by means of a finite difference scheme enhanced by repeated Richardson extrapolation. Numerical results are presented showing that the novel algorithm yields excellent results, and performs significantly better than a finite different method with Bermudan approximation.

7 Appendix: A finite difference method with Bermudan approximation

Let us show how to compute the American option price P(S, t) by a finite difference method with Bermudan approximation. In particular, we follow the approach developed in Ballestra and Sgarra (2010).

First of all, let us recall that a Bermudan option is an option that can be exercised only at a finite (discrete) set of times prior or equal to maturity [rather than at any time prior or equal to maturity, see Wilmott (1998)].

In the interval [0, T] let us consider $M_B+1$ equally spaced time levels $t_0$, $t_1$, $\ldots$, $t_{M_B}$, such that $t_i = i \Delta t$, $i=0,1,\ldots ,M_B$, where $\Delta t = \frac{T}{M_B}$, and let us approximate P(S, t) with the price of a Bermudan option that can be exercised only at the dates $t_0$, $t_1$, $\ldots$, $t_{M_B}$. Then, we can compute $P(S,t_{M_B-1})$, $P(S,t_{M_B-2})$, $\ldots$, $P(S,t_{0})$ according to the following recursion procedure: set $P(S,t_{M_B}) = \max (E-S,0)$ and, for $k = M_B-1,M_B-2,\ldots ,0$, first of all solve the following partial differential problem:

$$\begin{aligned}&\displaystyle \frac{ \partial V(S,t)}{\partial t} + \frac{1}{2} \sigma ^2(S,t) \frac{ \partial ^2 V(S,t)}{\partial S^2} + r S \frac{ \partial V(S,t)}{\partial S} -r V(S,t) = 0, \,\,\,\, S > 0, \,\,\,\, t \in [t_k,t_{k+1}),\end{aligned}$$

(53)

$$\begin{aligned}&\displaystyle V(0,t) = E, \,\,\,\,\, \lim _{S \rightarrow + \infty } V(S,t) = 0, \,\,\,\, t \in [t_k,t_{k+1}),\end{aligned}$$

(54)

$$\begin{aligned}&\displaystyle V(S,t_{k+1}) = P(S,t_{k+1}), \,\,\,\,\, S \ge 0, \end{aligned}$$

(55)

and then set

$$\begin{aligned} P(S, t_k) = \max (V(S, t_k),E-S), \,\,\,\,\, S \ge 0. \end{aligned}$$

(56)

Note that, according to (53)–(55), $V(S,t_k)$ represents the price at time $t_k$ of a European option with maturity $t_{k+1}$ and payoff equal to $P(S,t_{k+1})$, $k=0,1,\ldots ,M_B-1$. Moreover, condition (56) is imposed in order to take into account the possibility of early exercise at the dates $t_0$, $t_1$, $\ldots$, $t_{M_B}$ [see, for example, Wilmott (1998)]. Clearly, the option price $P(S, t_k)$ that is obtained according to the above recursion procedure, $k=0,1,\ldots ,M_B$, is an approximation of the American option price that becomes more and more accurate as the number $M_B+1$ of exercise dates increases.

Problems (53)–(55) can be solved by numerical approximation. In particular, as done in Ballestra and Sgarra (2010) [see also Rad et al. (2015)], we apply a standard second-order finite difference scheme. To this aim, the space domain of problems (53)–(55), namely $[0, +\infty )$, is replaced with a bounded one, say $[0, S_{\max }]$, where $S_{\max }$ is chosen such that the $V(S_{\max }, t)$ is of order $10^{-11}$ or smaller. Precisely, consistently with the second of relations (10), and in order to provide a fair comparison with the front-fixing method, in the numerical experiments performed in Sect. 5 we set $S_{\max } = b(0) x_{\max }$. Then, in the interval $[0, S_{\max }]$ let us consider $N_B$ equally spaced points $S_0, S_1, \ldots , S_N$, such that $S_i = i \Delta S$, where $\Delta S = \frac{S_{\max }}{N_B}$. A finite difference approximation of V(S, t) at $S = S_i$, $i=0,1,\ldots , M_B$, and $t = t_k$, $k=0,1,\ldots ,M_B$, is then obtained as follows (for the sake of simplicity, we focus our attention on a single value of $k \in \{0,1,\ldots ,M_B\}$). The space derivatives in (53) are discretized using a standard three-point centered finite difference scheme:

$$\begin{aligned}&\frac{\partial V(S_i,t_k)}{\partial S} \simeq \frac{V(S_{i+1},t_k) - V(S_{i-1},t_k)}{2 \Delta S}, \,\,\,\,\, \frac{\partial ^2 V(S_i,t_k)}{\partial S^2}\nonumber \\&\qquad \simeq \frac{V(S_{i+1},t_k) - 2 V(S_{i},t_k) + V(S_{i-1},t_k)}{\Delta S^2}, \nonumber \\&\qquad i = 1,2,\ldots ,N_B-1, \end{aligned}$$

(57)

and the time derivative is discretized using the Euler implicit scheme:

$$\begin{aligned} \frac{\partial V(S_i,t_k)}{\partial t} \simeq \frac{V(S_{i},t_{k+1}) - V(S_{i},t_k)}{\Delta t}, \,\,\,\,\, i = 1,2,\ldots ,N_B-1. \end{aligned}$$

(58)

Moreover, to impose the boundary conditions (54), the final condition (55) and the Bermudan constraint (56), we set

$$\begin{aligned} V(S_0,t_k) = E, \,\,\,\,\, V(S_{N_B},t_k) = 0, \end{aligned}$$

(59)

$$\begin{aligned} V(S_i,t_{k+1}) = P(S_i,t_{k+1}), \,\,\,\,\, i = 0,1,\ldots ,N_B, \end{aligned}$$

(60)

and

$$\begin{aligned} P(S_i, t_k) = \max (V(S_i, t_k),E-S_i), \,\,\,\,\, i = 0,1,\ldots ,N_B. \end{aligned}$$

(61)

By substituting (57), (58) into (53) and by taking into account (59)–(61) we obtain, for each Bermudan date $t_k$, $k=0,1,$ $\ldots ,M_B-1$, a system of algebraic linear equations that yields the Bermudan option prices $V(S_0,t_k)$, $V(S_1,t_k)$, $\ldots$, $V(S_{N_B},t_k)$. In particular, these systems are tridiagonal and thus they are easily solved by using the well-known Thomas algorithm [see Thomas (1949), and Wilmott (1998) for all the computational details].

Moreover, following a common approach, for $k=0,1,$ $\ldots ,M_B-1$, once the Bermudan option price at time $t_k$ is found, an approximation of the exercise boundary $B(t_k)$ can be obtained as the point at which the payoff function $\max (E-S,0)$ intersects the piecewise linear interpolation of $V(S_0,t_k)$, $V(S_1,t_k)$, $\ldots$, $V(S_{N_B},t_k)$.

Finally, according to Ballestra and Sgarra (2010), a Richardson extrapolation step is used to enhance the accuracy of time discretization. By doing that we obtain a finite difference scheme that, in the presence of smooth solutions, is second-order accurate in both space and time [for further details the reader is referred to Ballestra and Sgarra (2010) and Rad et al. (2015)].