Convergence rate of free boundary of numerical scheme for American option

Based on the optimal estimate of convergence rate $O(\Delta x)$ of the value function of 
an explicit finite difference scheme for the American put option problem in [6], an $O(\sqrt{\Delta x})$ rate of 
 convergence of the free boundary resulting from a general compatible 
 numerical scheme to the true free boundary is proven. 
 A new criterion for the 
 compatibility of a generic numerical scheme to the PDE problem is presented. 
 A numerical example is also included.

inequality, where the free boundary is the optimal exercise boundary. As there is no closed form solution for American option pricing, numerical calculations are the only available approaches to pricing the options, as well as to seeking the optimal exercise boundaries.
By PDE methods, Liang, Hu, Jiang and Bian obtained in [10] the explicit finite difference scheme (EFDS) convergence rate estimate O((∆x) 2/3 ) for the value function of the American put option. The rate estimate is improved to O(∆x) in [6] by Hu, Liang and Jiang, which is optimal. In [6], the convergence rate of the free boundary with a modified definition is also considered. However, the proposed scheme is difficult to implement because the definition of the modified free boundary includes an apriori unknown constant. To overcome a technical difficulty, in [11] a new definition of a slightly shifted free boundary for a more general model with jump is given and it is also proved that this slightly shifted free boundary is actually within an error of √ ∆x to the actual free boundary. In this paper, we use the optimal convergence rate for the value function to study the convergence rate of the free boundary. We consider a generic finite difference scheme and propose a new compatibility criterion. We shall establish an error estimates of √ ∆x of the numerical free boundary to the actual free boundary. More properties of the numerical free boundaries are also obtained.
This paper is organized as follows: In Section 2, we introduce the model, the EFDS, and a criterion for compatibility of numerical schemes. In Section 3, some properties of the scheme are collected and proved. In Section 4 we prove a nondegeneracy result: the second order derivative of the true solution, as well as the numerical solution, experience a jump across the free boundary. In Section 5, we use the non-degeneracy and error estimate to prove our main result: the distance between the free boundary of the numerical solution and the true free boundary is O( √ ∆x). A numerical example is given in Section 6.
2. Modeling and numerical scheme. In the Black-Scholes theory, the underlying asset price {S t } t 0 is assumed to be a geometric Brownian motion (cf. [5,7]): S t = S 0 er t+σWt wherer is the growth rate of the underlying asset, σ > 0 measures the volatility, and {W t } t 0 is the standard Brownian motion. Assume that the interest rate is a constant r > 0, strike price is K and expiration date is T . It is well-known (c.f. [15,Ch. 7]) that the no arbitrage price of the American put option at time t with asset price S is the solution V (S, t) of the variational inequality where It is well-known that no closed-form solutions are available for the variational inequality (2.1) (e.g. see [4]). In 1979, Cox and Rubinstein (see [3]) was the first to propose the Binomial Tree Method (BTM), a numerical scheme derived directly from the Black-Scholes theory by replacing the Brownian motion by (discrete) random walk; this is one of the most popular approaches for numerical evaluation and theoretical investigation of many mathematical finance problems, including the American put option.
Under the dimensionless variables where g + = max{g, 0} and Here we remark that the solution satisfies u 0 so min L[u], u − g + = 0. Denote by h the time mesh step and k the space mesh size, we claim that a generic EFDS for (2.2) can be written as where B is defined by where a 0 , a + , a − are constants depending on specific discretization. We illustrate this with the following examples: Example 1. In [6], the following discrization is used: This corresponds to .

XINFU CHEN, BEI HU, JIN LIANG AND YAJING ZHANG
For the compatibility of an EFDS, we propose the following new criterion. First note that the linear equation Clearly, these three special solutions determine completely and characterize fully the corresponding operator L. We denote by That is, To obtain expected exponents (λ, ν, γ), it is necessary and sufficient to take (2.10) We say that the discretization scheme ( Clearly, such definition of compatibility can be extended to higher order and operators in multi-space dimension.
In the current situation, the CFL stability condition can be written as If we use discretization (2.7) we have If we use discritization (2.5) or (2.6), we can show that We define the Modified Binomial Tree Model (MBTM) by setting (λ, µ, γ) = (1, µ, µ) and a 0 = 0, i.e.
Clearly, the difference between our MBTM and the classical Cox-Rubinstein's BTM is the same as the difference between continuously compounded interest rate and simply compounded interest rate.
Remark 2.1. It has been proved that BTM is equivalent to EFDS with h = 1 2 k 2 ; for European option, see Xu, Qian and Jiang [14]; for American option, see Qian, Xu, Jiang and Bian [13]. Jiang and Dai [8,9] studied American options on BTM in a partial differential equation framework, and proved the uniform convergence of the BTM for American options and their approximated free boundaries in the sense of viscosity solutions (without convergence rate).
In the sequel, we shall assume that B is given by (2.4) with (a 0 , a + , a − ) given by (2.10). We always assume that the stability condition (2.12) and the compatibility condition (2.11) are satisfied.
3. Preliminary. It is well-known that the variational problem (2.2) is well-posed. We quote the following for readers' convenience: Lemma 3.1. The variational inequality (2.2) admits a unique solution and the solution satisfies u t 0. In addition, define the optimal exercise boundary x = s(t) by Then u(x, t) > g + (x) when x > s(t) and u(x, t) = g(x) when x s(t). Moreover, s(·) ∈ C ∞ ((0, ∞)), s < 0 < s and These results can be found in [1] and [2]; see also [12] and [7, §6.5].
Before prove our convergence result, we state a comparison principle.
The proof follows by a mathematical induction on n for t = nh.
Proposition 1 (Convergence Theorem). Let u h be solution of (2.3) where B is given by (2.4) and (2.10). Assume (2.12) and Then the error to the solution of problem

4)
where C 0 is a positive constant which depends only on the given data.
Proof. Our finite difference operator is slight different from the one in [6]. Here we only need to show that our solution u h is O(k) distance away from the solutionũ h of the finite difference solution given in [6]: whereB is given by (2.4) and (2.5). We denote the constants by (ã 0 ,ã ± ). Our assumption on (a 0 , a ± ) implies that For any bounded and Lipschitz in x continuous function φ, Using a 0 + a + + a − = e −γh and a 0 +ã + +ã − = e −γh we obtain

Now we show thatũ h + Ck is an upper solution of problem (2.3). One can show that ũ
Proof. The monotone increasing property of u in t follows from the comparison principle first for 0 < δ h, and then step-by-step for all δ > 0. Similarly, one can show that u is decreasing in x. To prove the second inequality in (3.6), we first note that min It is also clear that, It then follows by comparison principle that w u h . We now define the approximate free boundary x = s h (t).
Proof. The monotonicity of s h follows from the monotonicity of u h in t. For boundedness, we argue as follows. Since u h t 0 and u h x 0, the limit exists, and is the solution of the infinite horizon problem: 4. Non-degeneracy property. At the free boundary x = s(t), u − g = 0 and (u − g) x = 0 (indeed also (u − g) t = 0). The following shows that u − g grows quadratically as x moves to the right from the free boundary; we call it the nondegeneracy property.

5.
Convergence rate of the free boundary. Here we prove the convergence rate of the approximate free boundary x = s h (t) to the actual free boundary x = s(t).
Theorem 1 (Main Theorem). Assume (2.7) and (3.3). Then the difference of the approximate free boundary s h (t) and the real one s(t) has the estimate: for some positive constant C 1 which does not depend on h and k.
Proof. Fix t > 0. If s h (t) s(t), then by Proposition 1, definition of s h , and Proposition 4, we obtain This implies that s h (t) s(t) + C 0 /μ √ k. On the other hand, if s h (t) < s(t) − k, then, as s h (t + h) s h (t), we can write s(t) = s h (t + h) + y + ϑk with ϑ ∈ [0, 1) and y = nk for some integer n 1. By Proposition 1, definition of s, and Proposition 5 we have √ k. Combining both cases, we obtain the assertion of the theorem. 6. A numerical example. We give a numerical example for Example 3. Here we choose µ = 0.5 and consider the solution in time interval [0, 0.1]. Calculation interval is [−5, 5]. As an American put option has no closed form solution, we treat the approximation solution for h = h * = ∆t = 1.5 × 10 −7 , as "exact solution" and S h * (t) to be a benchmark for comparing to the others.   We found that the errors are smaller than h 1/4 . According to the theoretical result, the error is bounded by O(h * 1/4 ), and this is mostly attributed to the singularity of the solution near (x, t) = (0, 0). On the scheme, the first step is already of size h away from 0.
The corresponding figures of approximated free boundary with different "h"s vs time t are shown in the Figure 1.