Pricing american options under proportional transaction costs using a penalty approach and a finite difference scheme

In this paper we propose a penalty method combined with a finite difference scheme for the Hamilton-Jacobi-Bellman (HJB) 
equation 
 arising in pricing American options under proportional transaction costs. 
In this method, the HJB equation is approximated by a nonlinear 
partial differential equation with penalty terms. 
 We prove 
that the viscosity solution to the penalty equation converges to that of 
the original HJB equation when the penalty parameter tends to positive infinity. We then present an upwind finite difference scheme for 
solving the penalty equation and show that the approximate 
solution from the scheme 
converges to 
the viscosity solution of 
the 
penalty equation. A numerical algorithm for solving the discretized nonlinear system 
is proposed and analyzed. Numerical results are presented to demonstrate the accuracy of the method.


1.
Introduction. In a financial market, an option is a contract in which one party (the writer) sells to another party (the holder) the right, but not the obligation, to buy (call option) or sell (put option) a specified amount of an underlying asset such as a stock at a fixed price (the exercise or strike price) on or before a given date (expiry date). There are two major types of options: European options and American options. A European option gives the holder the right to buy (for a call option) or to sell (for a put option) the underlying asset at the strike price on the expiry date, while an American option, in contrast to a European option, allows the holder to exercise it at any time before or on the expiry date.
Options are tradable in a financial market. Thus, how to determine the price of an option is an important topic in financial engineering. In a complete market, Black and Scholes [2] used a no-arbitrage argument to price a European option on a stock under the conditions that the interest rate is constant, the underlying stock price is a geometric Brownian motion and there are no transaction costs on trading the underlying stock and the bond. The Black-Scholes model has been widely used for valuing simple European options. However, in the presence of transaction costs on trading in the bond or/and stock, the Black-Scholes option pricing methodology is no longer valid. In the open literature, there are four main approaches to the problem of option pricing with transaction costs (see, for example, 366 WEN LI AND SONG WANG [3,4,12,13,15,19,25,8]). One of them is the utility based option pricing approach which is an optimal portfolio selection problem to maximize an investor's expect utility of terminal wealth.
The utility based option pricing approach was first proposed in [15] to price European options under proportional transaction costs. It was further developed in [10] in which the authors showed that computing the reservation price of a European option involves solving two singular stochastic optimal control problems and the value functions of these problems are the unique viscosity solutions to a fully nonlinear HJB equation satisfying respectively different boundary conditions. Based on the study in [10], Davis and Zariphopoulou [11] showed that calculating the reservation purchase price of an American option under proportional transaction costs involves solving a combination of a singular stochastic control and an optimal stopping problems. Furthermore, they proved that the value functions of the singular stochastic control problems with an optimal stopping are the unique viscosity solutions of the corresponding fully nonlinear HJB equations. Thus, using utility based method to price a European option and an American option is equivalent to finding the solutions to the HJB equations in [10] and [11], respectively.
Since closed-form solutions of the HJB equations are not available, numerical methods are necessary for approximating the solutions. In the open literature, most authors (see, for example, [9,10,22,32,33,5]) used Markov chain approximation schemes for solving the HJB equations. These schemes are explicit in time and thus computationally very expensive. Recently, we proposed a penalty approach to solve the HJB equations arising in the European option pricing model under proportional transaction costs (cf. [20]). In the approach, we first approximated the HJB equation by a nonlinear PDE with penalty terms which penalize the part of the solution violating the constraints and showed that the viscosity solution to the penalty equation converges to that of the HJB equation based on the power penalty methods proposed in [30,31,16]. We then proposed and analyzed an upwind finite difference method for solving the penalty equations arising from the penalty method [21]. In this paper, we extend the penalty method combined with the finite difference scheme proposed in [20,21] for the HJB equation in the European option pricing model to that arising in pricing American options under proportional transaction costs. In this approach, we first approximate the HJB equation by a quasi-nonlinear parabolic PDE containing three penalty terms with a penalty constant ρ and show that the viscosity solution to the penalty equation converges to that of the original HJB equation as ρ approaches infinity. We will then propose a finite difference scheme for the resulting penalty equation and prove that the solution to the discretized equation system converges to that of the penalty equation.
The rest of this paper is organized as follows. In the next section, we give a brief account of the formulation of the American options valuation problem as HJB equations using the utility maximization theory. In Section 3, we propose a penalty method for the HJB equations and establish a convergence theory for the penalty method. In Section 4, we present a discretization scheme for the penalty problem and show that the solution from the discretization scheme converges to the viscosity solution of the penalized problem. An iterative algorithm and its convergence will be presented in Section 5. In Section 6 we present some numerical results to illustrate the theoretical findings.

2.
The American option pricing model. In this section, we briefly discuss the utility based option pricing model for valuing American call and put options with proportional transaction costs. A more detailed discussion can be found in [11,9].
2.1. American option pricing via utility maximization. Consider a time interval [0, T ] and a continuous-time economy with a risky stock and a risk-less bond. Assume that the price of the stock at time u, denoted as S u , evolves according to the following geometric Brownian motion where µ and σ are respectively constant drift rate and volatility, Z u , representing a single source of uncertainty in the market, is a standard Brownian motion on a filtered probability space denoted as (Ω, F, (F u ) 0≤u≤T , P ). We also assume that the price of the bond, B(u), at time u is determined by the following ordinary differential equation We suppose that the investors in the economy must pay transaction costs when buying or selling the stock and the transaction costs are proportional to the amount transferred from the stock to the bond.
Let β u denote the amount the investors hold in the bond and α u the number of shares of the stock held by the investors at time u ∈ [0, T ], then the evolution equations for β u and α u are where θ ∈ [0, 1) represents the transaction costs in percentage of the traded amount in the stock, and L u and M u denote respectively the cumulative number of shares bought and sold up to u. At time u the liquidated cash value of the stock is S u (α u − θ|α u |) and the investors' wealth, denoted as W u , is given by We now describe the utility based option pricing approach. The idea of the utility based option pricing approach is to consider the optimal portfolio problem of an investor whose objective is to find an admissible trading strategy so that the utility of the terminal wealth is maximized. To use this approach to value reservation purchase price of American call and put options, we need to define the following three different utility maximization problems.
Problem 1 (Utility maximization for an investor without an option). Consider an investor who trades in the underlying stock and the bond. Assume that the investor holds β dollars in the bond and α shares of the stock whose price is S at time t ∈ [0, T ]. The objective of the investor is to maximize the expected utility of wealth at the terminal time T over the set of feasible strategies, i.e., where V 0 (t, α, β, S) denotes the investor's time t maximum expected utility of terminal wealth (also known as value function), E t denotes the expectation operator conditional on the time t information (α, β, S), U (·) is a utility function and Λ 0 (t, α, β, S) is the set of admissible strategies available to the investor, defined as the set of right-continuous, measurable, F-adapted, increasing processes, L u and M u (t ≤ u ≤ T ), such that the following conditions are satisfied: 1. The associated processes (α Lu,Mu , β Lu,Mu , S u ) satisfy (1), (2) and (3) in [t, T ] with the initial state (t, α, β, S).
The choice of the utility function U is non-unique. In this work, we use the following exponential utility function: where γ > 0 is a constant risk aversion parameter. Problem 1 is a utility maximization problem without any options. Let u + := max{0, u}. In what follows, we define two other optimization problems with buying an American call option and an American put option respectively.
Problem 2 (Utility maximization for an investor buying a call option). Consider an investor who trades in the market for the stock and the bond, and in addition, purchases a cash-settled American call option written on the stock with strike price K and expiry date T . If the investor exercises the option at a stopping time τ ∈ [t, T ], then he/she receives the amount of (S τ − K) + dollars and faces the utility maximization problem without an option. Thus, the investor's time τ expected utility of terminal wealth is given by The investor's objective is to choose an admissible trading strategy and an exercise time to maximize (7), i.e., where Λ τ c (t, α t , β t , S t ) denotes the investor's admissible strategies which are defined as the set of right-continuous, measurable, F-adapted, increasing processes, L u and M u (t ≤ u ≤ τ ), such that the following conditions are satisfied: Problem 3 (Utility maximization for an investor buying a put option). Assume that the investor trades in the market for the stock and the bond, and in addition, purchases a cash-settled American put option written on the stock with strike price K and expiry date T . If the investor choose to exercise the option at time τ , he receives the payoff (K − S τ ) + dollars and faces the utility maximization problem without an option. That is, the investor's value function at time τ is defined as The investor's problem is to choose an admissible trading strategy and an exercise time to maximize (9), i.e., where τ ∈ [t, T ] is a stopping time and Λ τ p (t, α t , β t , S t ) is the investor's admissible strategies which are defined as the set of right-continuous, measurable, F-adapted, increasing processes, L u and M u (t ≤ u ≤ τ ), such that the following conditions are satisfied: 1. The associated processes (α Lu,Mu , β Lu,Mu , S u ) satisfy (1) to (3) in [t, τ ] with the initial state (t, α, β, S).
Remark 1. In this model, the investor's wealth is required to be positive at any time u ∈ [0, T ], i.e., there is no bankruptcy in our economy. Item 2 in each of Problems 2-3 represents the no-bankruptcy restriction. These conditions ensure that the investor's wealth is positive at all trading times. Since an American option can be exercised at any time before or on the expiry date, the model allows the investor to pursue trading strategies for which the liquidated cash value of the stock and bond, W u , is negative, as long as the amounts from exercising the option are large enough to cover the negative liquidated value.
Using the above definitions, we now define the reservation purchase prices of American call and put options as follows.
Definition 2.1 (reservation purchase price of an American call option). Consider an investor who starts trading at time t = 0 with holding β dollars in the bond and α shares of the stock whose price is S. Assume that the investor only can buy the option at the initial time t = 0, then the investor's reservation purchase price of an American call option is defined as the amount, P c , such that V 0 (0, α, β, S) = V ax c (0, α, β − P c , S). Definition 2.2 (reservation purchase price of an American put option). Consider an investor who starts trading at time t=0 with holding β dollars in the bond and α shares of the stock whose price is S. Assume that the investor only can buy the option at the initial time t = 0, then the investor's reservation purchase price of an American put option is defined as the amount, P p , such that . Clearly, the definitions are based on the no-arbitrage principle, that is, the expected return for an investor does not depend on whether he/she buys options or not. From the above definitions we see that computing the reservation purchase prices of American call and put options involves three value functions defined in (5), (8) and (10) respectively. By the principle of dynamic programming, Davis and Zarphopoulou [11] have derived HJB equations governing these value functions, as given in the next subsection.
2.2. The HJB equations. It has been shown in [11] that each of the value functions V 0 , V ax c and V ax p satisfies a set of HJB equations. In what follows, we list these equations without proof. A detailed deduction of these HJB equations can be found in [11].
Let L k , k = 1, 2, 3, be the linear differential operators defined respectively by Then, the value function V 0 solves the following HJB equation with the terminal condition where For i = c and p, the value function V ax i is a solution to the HJB equation with the terminal condition where V ex c and V ex p are the value functions defined in (7) and (9), respectively, and (14) and (17) are nonlinear and they do not have in general classical solutions. Using the notion of viscosity solution, Davis and Zarphopoulou [11] showed that the value function V 0 is the unique viscosity solution of (14) satisfying the terminal condition (15) and V ax i , i ∈ c, p are the unique viscosity solutions of (17) satisfying, respectively, the terminal conditions (18)- (20). We will discuss this briefly in the next subsection.

2.3.
Unique solvability of the HJB equations. In this subsection we first introduce the definitions of constrained viscosity solutions and then we will present a brief account of the results on the unique solvability of the HJB problems (14)- (20).
The concept of viscosity solution was first introduced in [6] for handling weak solutions of nonlinear first order PDEs. For a general introduction to the viscosity solution theory, we refer to [7,14]. The concept of constrained viscosity solution was introduced in [24,18] to handle control problems with state constrains.
In order to introduce the notion of viscosity solution, let us consider a fully non-linear second order PDE of the following form: with the terminal condition where Ω ⊆ R n is an open set, W : [0, T ] × Ω → R is an unknown function, and DW and D 2 W denote respectively the gradient and Hessian of W with respect to (t, x 1 , ..., x n ). Let S n denote the set of n × n symmetric matrices, then F is a mapping We assume that F is continuous in all its arguments and satisfies the following degenerate ellipticity condition: Using the above notation, we now state the definition of constrained viscosity solution. 1

Remark 2.
In the above definition, we can replace "local maximum (minimum) point" by "strict local maximum (minimum) point", "global maximum (minimum) point" or "strict global maximum (minimum) point". We can also assume that the extremum of W − ψ has the value zero.
With the definitions of viscosity solutions, we can now state the following theorems: where Ω 0 is defined in (16). Then, V 0 is the unique constrained viscosity solution of (14) and (15). Proof. See the proof of Theorem 3.2 in [20].
We now state the comparison result in the following lemma which will be used in the proof of the unique solvability of (17) and (18). For a proof of this lemma, we refer to [11].
Lemma 2.5. Let i ∈ {c, p} and u i be a bounded upper semi-continuous viscosity subsolution of (17) on [0, T )×Ω ax i , and v i be a lower semi-continuous function which is bounded from below, exhibits sublinear growth and is a viscosity supersolution of (17) (21) and (22) respectively for i = c and p. Then, V ax i is the unique constrained viscosity solutions of (17) and (18). Proof. A proof of the existence of a viscosity solution to (17)- (18) can be found in [11]. We now use Lemma 2.5 to show that the viscosity solution is unique.
For each i ∈ {c, p}, suppose that there are two viscosity solutions, V ax i and W ax i , to (17)- (18). Since V ax i is a subsolution and W ax i is supersolution to (17) 3. The penalty method and its analysis. Penalty methods have been successfully used for option pricing problems without transaction costs and HJB equations in both infinite and finite dimensions (cf., for example, [30,31,16,17,34,35]).
Recently, we propose a penalty method in [20] for the European option pricing problem under proportional transaction costs. The HJB problem (14)- (15) arising from Problem 2.1 is of the form considered in [20]. Using the penalty method, the HJB equation (14)- (15) is approximated by where λ > 1 is a penalty parameter and [v] − := min{v, 0} for any function v. It has been proved in [20] that the viscosity solution of the above penalized PDEs converges to that of the original HJB equation as the penalty parameter λ → ∞.
Thus, in what follows, we will discuss a penalty method for the HJB equations (17)- (18) only. Motivated by the penalty equation (25), we consider the following equation approximating (17)- (18): which penalizes any of the negative parts of V ρ − V ex i , L 2 V ρ and L 3 V ρ . Also note that L 1 V ρ ≥ 0 for any ρ > 1. We next prove that for each i ∈ {c, p} the viscosity solution to (27)- (28) converges to that of (17)-(18) as ρ → ∞. We start this discussion with the following lemma whose proof is the same as that in [7,14].
i , then υ is continuous and υ i ρ → υ as ρ → ∞ on the compact set. Using Lemma 3.1, we are ready to establish a convergence theory for the penalty approach. This is given in the following theorem.
Theorem 3.2. Let i ∈ {c, p} and V i be the unique constrained viscosity solution of (17)- (18). For each ρ > 1, let υ i ρ be the constrained viscosity solution of (27)-(28) for i ∈ {c, p}. Then υ i ρ → V i as ρ → ∞. Proof. We only consider the case that i = c as the proof for i = p is essentially the same as that for i = c.
To prove this theorem, we first need to show that the solution υ c ρ to (27)-(28) is locally bounded uniformly in ρ. Since the proof of this requires a result in the next section, it will be given in Appendix B.
Let υ := υ c and υ := υ c denote respectively the upper and lower weak limits of υ c ρ defined by (29) and (30) respectively. In what follows, we show that υ = υ. First, from the definitions of upper and lower weak limits in (29) and (30) we see that υ ≤ υ.
From this inequality we have when ρ is sufficiently large, since [z] − ≤ 0 for any z. This is a contradiction, since we assumed that υ c ρ is a constrained viscosity solution of (27).
Thus, (36) is proved and we conclude that υ is a supersolution of (17) on [0, T ) × Ω ax c . υ and υ satisfy the terminal condition Since υ c ρ is a constrained viscosity solution of (27) and (28), we have that for all ρ > 1, Hence, the theorem is proved. 4. Discretization and convergence. Several efficient discretization schemes can be found in the open literature for Black-Scholes operators similar to that in (11). Two major ones are upwind finite difference and fitted finite volume schemes (cf., for example, [27,29]). In this section, we will develop numerical schemes for solving the HJB equations in the previous sections. Since the discretization of (25)- (26) has been discussed in [21], in what follows we only consider the discretization of (27)- (28). For brevity, we only show the case that i = c in (27)- (28).
Before presenting the scheme, we first rewrite (27)-(28) as the following equivalent form: Note that the solution domain Ω ax c is an unbounded region. In practice, we are only interested in option prices defined in a finite region Ω R given below: where R α , R α , R β , R β and R S are positive constants. Now, we define a uniform mesh for Ω R with mesh nodes where E, M and P are positive integers, and Let h = max{h 1 , h 2 , h 3 } and Z h and ∂Z h denote respectively the index sets of the internal and boundary mesh nodes defined respectively by Clearly, each grid point (i, j, k) ∈Z h := ∂Z h ∪Z h corresponds to a state (α j , β j , S k ).
Let V n ijk denote the approximation (to be determined) to the solution of (39)-(40) at the mesh node (t n , α i , β j , S k ). Using the finite difference operators we propose a discretization scheme for (39) as follows: for n = N −1, N −2, ..., 0 and (i, j, k) ∈ Z h , where V ex,n ijk denotes an approximation to V ex (t n , α i , β j , S k ), v − = min{v, 0} for any function v, and L ∆ 1 , L ∆ 2 , L ∆ 3 are defined as follows: Clearly, the above are difference operators approximating respectively the continuous ones in (11)- (13). The discretization of the first-order spatial derivatives used above is based on the well-known upwind finite differencing technique (cf., for example, [28]).
Based on (28) and (40), the terminal and the boundary conditions for (41) are defined respectively as for all feasible (n, i, j, k). Before presenting the convergence analysis of the scheme, we first state the following theorem: Theorem 4.1. Let V ex,λ,n ijk be defined in (45) for any feasible (n, i, j, k) and λ > 1.
Then, V ex,λ,n ijk converges to V ex c (t n , α i , β j , S k ) as ∆ := (h, ∆t) → (0, 0) and λ → ∞. Proof. The proof of this theorem is just a combination of the convergence results in [20] and [21] and thus it is omitted.
The following comparison principle for (27) is also needed for the proof of our convergence result. With the above theorem and lemma, we establish the following theorem: The proof is rather lengthy and we will put it in Appendix A.

5.
Decoupling algorithm and its convergence. In this section we consider the implementation of the scheme (41)-(47). Note that (41)-(47) is a coupled non-linear system. In computation, we use the following algorithm to decouple it.
Proof. We will follow the notation in Algorithm A. To prove this theorem, we first show that the sequence {V n,l } ∞ l=0 generated by the iterative method is monotonically increasing, i.e., V n,l ≤ V n,l+1 for l ≥ 1.
From (48) we have since, by (50)-(52), Note that both (48) and (61) have the same boundary conditions, i.e., V n,l ijk = V n,l+1 ijk for any (i, j, k) ∈ ∂Z h . Thus, using the notation in the proof of Lemma 5.1, we may write (48) and (61) as the following respective matrix forms: where A n,l , b n , c n+1 and v n are as defined in (56) with A n,l an M -matrix. Therefore, from the above we have A n,l (V n,l+1 − V n,l ) ≥ 0.
Since A n,l is an M -matrix, we have Therefore, the monotonicity of iteration process is proved. From Lemma 5.1 we have that V n,l is bounded for any l = 0, 1, 2, .... Combining the boundedness and monotonicity we see that V n,l is convergent. Finally, from the construction of (48) and (52) it is obvious that V n,l ijk , m l 0 (n, i, j, k), m l 2 (n, i, j, k) and m l 3 (n, i, j, k) solve (41) when l → ∞. Thus, we have proved the theorem.
6. Numerical results. In this section, we use the scheme (41)-(47) and that in [21] to compute the value functions V ax i (i = c, p) and V 0 respectively, and present the computed reservation purchase prices, P c and P p , of American call and put options with different values of the risk aversion parameter γ in utility function (6).
We now illustrate the performance of the schemes using the following test example: Test Example: Reservation purchase prices of American call and put options with the strike price K = 2.6, expiry date T = 1, the initial price of stock S 0 = 2.6 and various values of α 0 , β 0 and γ. Other parameters are: the interest rate r = 0.05, the drift parameter µ = 0.1, the volatility σ = 0.2 and the proportional transaction cost parameter θ = 0.05.
To solve this problem, we choose R α = R β = 2, R α = 6, R S = 5 and R β = 10. The discretization and penalty parameters are chosen to be E = 41, M = 61, P = 26, N = 25 and ρ = 1000. Comparable mesh and penalty parameters were used for computing V 0 using the scheme in [21]. The problem is solved by a MATLAB code in double precision under the Linux environment. We compute the reservation purchase prices of an American call option for γ = 0.1 and various values of initial holdings in the bond β 0 and share α 0 . The results are reported in Table 1 and Figure 1.
Please note that the price of an American option is higher than or equal to that of its European counterpart because there are more exercising opportunities for an American option than for a European option. It is known that in the economy without transaction costs, the price of an American call option is equivalent to that of its European counterpart. In order to compare the reservation purchase prices of an American call option and its European counterpart involving transaction costs, we also compute the prices of European options, denoted by P b,c , using the scheme in [21]. The results for both American and European options are listed in Table 1.  Table 1. Computed reservation purchase price P c of the American call option for γ = 0.1 and the reservation purchase price P b,c of its European counterpart.
From Table 1 and Figure 1, we see that reservation purchase price of this American call option with transaction costs is the same as that of its European counterpart, coinciding with the case of options without transaction costs. Both of purchase prices of an American and a European call options decrease as the initial holding of the stock α 0 increases. This conforms to the theory of supply and demand that the more stocks an investor holds at t = 0, the less the investor wants to purchase the stocks and therefore the investor will reduce the call option price. From Figure  1 we also see that the reservation purchase prices are independent of β 0 , i.e., the initial holding in the bond does not affect the reservation price of an American call option.
To investigate the influence of the risk aversion parameter γ on the reservation prices, we assume that the investor's initial holding of the share is zero, i.e., α 0 = 0, and compute the reservation purchase prices of the American call option and its European counterpart for γ = 0.01, 0.1, 0.3, and 0.5. The results are listed and depicted in Table 2 and Figure 2 respectively. It is clear from Table 2 and Figure 2 that the reservation purchase prices of an American call option are identical to the corresponding purchase prices of a European call option.
From Table 2 and Figure 2 we observe that the purchase price is a decreasing function of γ. Noting that the investment risk is also a decreasing function of γ, this result is very realistic and true as when an investor is willing to take more risk, he/she is willing to purchase a call option at a higher price, and vice versa. As in the case in Figure 1, the purchase prices are independent of the initial bond holding β.  Table 2. Computed reservation purchase price of the American call option for α 0 = 0 and the reservation purchase price of its European counterpart.
Finally, let us discuss the influence of γ on the reservation purchase price of an American put option and compare the price to that of its European counterpart. Assume that the investor's initial holding in the stock is zero, i.e., α 0 = 0. We compute the reservation purchase prices of an American put option and its European counterpart, denoted as P b,p , for γ = 0.01, 0.1, 0.3 and 0.5. The results are showed in Table 3 and Figure 3. From the table and the figure, we have the following observations: 1. The reservation purchase price of an American put option is greater than that of its European counterpart. This coincide with what we commented earlier in this section.   Table 3. Computed reservation purchase prices of the American put option for α 0 = 0 and the reservation purchase prices of its European counterpart.
Before closing this section, we have the following comments.

Remark 4.
It is worth pointing out that each of the HJB equations in the original problem is defined on an infinite domain and does not have any Dirichlet boundary conditions. However, we solve it numerically on a finite region such as Ω R defined in Section 4 and define an artificial (homogeneous) Dirichlet boundary condition on each of the boundary segments as the exact boundary condition is unknown. The accuracy of the numerical solution depends on the choice of the boundary conditions. From the numerical experiments we have found that the computational errors are essentially located near the boundary of the finite domain Ω R . A theoretical justification for this artificial boundary conditions is given in [23]. Thus, in the  numerical results presented above, we only plot the computed values at the mesh points which are some distances away from the boundary segments.
Remark 5. We have observed from our computational results that the reservation purchase prices of both American call and put options under proportional transaction costs are independent of β 0 . 7. Conclusion. In this paper we have studied the problem of American option pricing with proportional transaction costs. We propose a penalty approach combined with an upwind finite difference scheme to compute the reservation purchase prices of American call and put options. We have proved that both the penalty method and the discretization scheme are convergent. An iterative scheme for the discretized nonlinear system has also been proposed and analyzed. Numerical results show that the reservation purchase price of an American call option is equivalent to that of its European counterpart and the reservation purchase price of an American put option is greater than that of its European counterpart. This means that premature exercise of an American call option is not optimal.
Appendix A: Proof of Theorem 4.3. A standard method for proving the convergence of a finite difference scheme was developed by Barles and Souganidis [1] who showed that any monotone, stable and consistent scheme converges to the exact solution provided that there exists a comparison principle. In what follows, we will show that the scheme (41) is stable, monotone and consistent which, combined with Lemma 4.2, guarantee the convergence of the solution.