{'fraction_non_alphanumeric': 0.06306366940368825, 'fraction_numerical': 0.015850047112666577, 'mean_word_length': 3.829676580529823, 'pattern_counts': {'":': 0, '<': 16, '<?xml version=': 0, '>': 37, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Analysis of the optimal exercise boundary of American put option with delivery lags

We show that an American put option with delivery lags can be decomposed as a European put option and another American-style derivative. The latter is an option for which the investor receives the Greek Theta of the corresponding European option as the running payoff, and decides an optimal stopping time to terminate the contract. Based on the this decomposition, we further show that the associated optimal exercise boundary exists, and is a strictly increasing and smooth curve. We also analyze its asymptotic behavior for both large maturity and small time lag using the free-boundary method.


Introduction
With a few exceptions, models of optimal stopping time problems assume that the player is able to terminate the underlying stochastic dynamics immediately after the decision to stop, or to bring a new project online without any delays after the decision to invest. In fact, both stopping stochastic dynamics and initiating a new project take time. In this paper, we consider a general class of optimal stopping problems, where there exists a time lag between the player's decision time and the time that the payoff is delivered.
As an example, we study American put options with delivery lags in details. In practice, there may exist a time lag between the time that the option holder decides to excise the option and the time that the payoff is delivered. Such delivery lags may be specified in financial contracts, where the decision to exercise must be made before the exercise takes place. They are called make-your-mindup options (see Chapter 6 of [13] and Chapter 9 of [18]). For example, the option holder must give a notice period before she exercises, and she cannot change her mind. On the other hand, even for standard American derivatives, the option holder may not be able to exercise immediately, when there exist liquidation constraints in financial markets.
The purpose of this paper is to study the effects of delivery lags in the model of American put options. Our paper makes three specific contributions.
We first solve a general optimal stopping problem with a given time lag to deliver the payoff. To this end, we use the reflected backward stochastic differential equation (BSDE) method and, therefore, no Markovian assumption is required. This is in contrast to the existing literature (see [1], [3], [15] and [16] with more references therein), where the Markovian property is required. We refer to [9] and [10] for an introduction of BSDE and reflected BSDE. To solve the problem, we introduce a new obstacle process, which is the projection (conditional expectation) of the payoff of the original optimal stopping problem with delivery lags. We then transform the original problem to a standard optimal stopping problem (without delivery lags) with this new obstacle process as the modified payoff (see Lemma 2).
Due to the existence of the projection operator, if the original payoff is nonlinear (as the payoff of the American put option herein), the nonlinearity will propagate via the conditional expectation, resulting in a more complicated nonlinear function as the modified payoff. In our case, it is the corresponding European put option price. This makes the analysis of the associated optimal exercise boundary much more challenging. In the existing literature of optimal stopping with delivery lags (see [1] and [15] for example), however, the authors assume that original payoff is linear, so the modified payoff is also linear as a consequence of the linearity of the conditional expectation. Hence, the treatments of the optimal stopping problems with and without delivery lags are essentially the same in their models.
Our second contribution is an early exercise premium decomposition formula for American put options with delivery lags. This helps us overcome the difficulty of handling the European option price as the modified payoff. We show that an American put option with delivery lags can be decomposed as a European put option and another American-style derivative. The latter is an option for which the investor receives the Greek Theta of the corresponding European option as the running payoff, and decides an optimal stopping time to terminate the contract (see Lemma 6). The decomposition formula (12) in Lemma 6 can also be regarded as a counterpart of the early exercise premium representation of standard American options, and is crucial to the analysis of the associated optimal exercise boundary.
Using the free-boundary method, we then give a detailed analysis of the associated optimal exercise boundary. An essential difficulty herein is the nonmonotonicity of the value function with respect to the stock price (a similar phenomenon also appears in [6]). As a result, it is not even clear ex ante whether the optimal exercise boundary exists or not. This is in contrast to standard American options, for which the value function, subtracted by the payoff, is monotonic with respect to the stock price, so the stopping and continuation regions can be easily separated.
As the third contribution, we prove that the optimal exercise boundary exists and is a strictly increasing and smooth curve, with its end point closely related to the zero crossing point of the Greek Theta of the corresponding European option (see Theorem 13). Intuitively, when Theta is positive, the running payoff of the new American-style derivative is also positive, so the investor will hold the option to receive the positive Theta continuously. In the contrary, when Theta is negative, one may think that the investor would then exercise the option to stop her losses. However, we show that when Theta is negative but not too small, the investor may still hold the option and wait for Theta to rally at a later time to recover her previous losses. We further quantify such negative values of Theta by identifying the asymptotic line of the optimal exercise boundary, which turns out to be the optimal exercise boundary of the corresponding perpetual option (see Theorem 16). We also prove the convergence of the optimal exercise boundary as the time lag tends to zero. As expected, it will converge to the optimal exercise boundary of standard American options (see Theorem 18).
The paper is organized as follows. In section 2, we solve a general optimal stopping problem with delivery lags via the reflected BSDE method. In section 3, we introduce the model of American put options with delivery lags, together with their early exercise premium decomposition and their perpetual version. We then give a detailed analysis of the associated optimal exercise boundary via the free-boundary method in section 4, and conclude the paper with the properties of the Greek Theta in the appendix.

Optimal stopping with delivery lags
In this section, we introduce a general optimal stopping problem with delivery lags, which includes American put options in the next section as a special case.
Let W be a one-dimensional Brownian motion on a probability space (Ω, F , P). Denote by F = {F t } t≥0 the augmented filtration generated by W . Let T > 0 represent the fixed maturity, and δ ∈ [0, T ) represent the time lag. For t ∈ [0, T ], we introduce the admissible set T ], and {τ 0 ≤ s} ∈ F s for any s ∈ [t, T ]}. The player chooses an optimal stopping time τ 0, * ∈ R 0 t in order to maximize the following objective functional with the terminal data ξ, the running payoff f , the discount factor R, and the payoff S as the given data, and the stopping time τ 0 ∈ R 0 t .
If the maximum value exists, then τ 0, * is called the optimal stopping time, and the maximum value y δ t is called the value process, where Note that δ = 0 corresponds to classical optimal stopping problems (see, for example, [7], [14] and [17] among others). For δ > 0, if the player decides to stop at some stopping time τ 0 , then the payoff will be delivered at τ 0 + δ rather than τ 0 , so there is a time lag of the delivery of the payoff. For this reason, the problem (1) is referred to as the optimal stopping problem with delivery lags.
We also observe that (1) is trivial for t ∈ (T − δ, T ] since, in this situation, Y δ t (τ 0 ) is independent of choice of τ 0 , and we may simply choose the optimal stopping time as the maturity T . Thus, we focus on the case t ∈ [0, T − δ] throughout the paper.
(ii) The running payoff f is an F-adapted process, and is H 2 -square integrable, i.e. E[ T 0 |f t | 2 dt] < ∞; (iii) The discount factor R satisfies dR t = −r t R t dt with R 0 = 1, and r is a uniformly bounded F-adapted process.
(iv) The payoff S is a continuous F-adapted process, and is uniformly square Lemma 2 Suppose that Assumption 1 holds. Let the process Y δ t , t ∈ [0, T − δ], be given as Then, the following assertions hold: where Y δ is uniformly square integrable, Z δ is H 2 -square integrable, and K δ is an increasing and continuous process with K 0 = 0 and K T being square integrable.
(ii) The value process of (1) is given as , and the optimal stopping time is given as Proof. (i) It is standard to check that the processŶ δ t , t ∈ [0, T −δ), is continuous and uniformly square integrable. Hence, it follows from sections 5 and 6 in [9] that the reflected BSDE (4) admits a unique solution (Y δ , Z δ , K δ ).
(ii) The proof is adapted from Proposition 2.3 in [9]. We provide its details for the reader's convenience. We first show that Y δ t ≥ y δ t . Applying Itô's formula to R t Y δ t , we obtain that For any stopping time τ 0 ∈ R 0 t , it follows from (5) that and on {τ 0 < T − δ}, where we have used (3) in both equations. It follows that Taking supremum over τ 0 ∈ R 0 t gives Y δ t ≥ y δ t . Next, we prove the reserve inequality using τ 0, * . Note that the Skorohod condition in (4) implies that the measure dK δ s is carried by the set {s : Y δ s = Y δ s }. Hence, dK δ s = 0 for s ∈ [t, τ 0, * ]. It then follows from (5) that from which we conclude that Y δ t ≤ y δ t .

American put option with delivery lags
As an application of the optimal stopping with delivery lags, we consider a concrete model of American put option with delivery lags in this section.

Assumption 2
The terminal data ξ = (K − X T ) + , the running payoff f t = 0, the interest rate r t = r and the payoff S t = (K − X t ) + , where X represents the stock price and follows and the strike price K, the interest rate r, the dividend rate q < r, and the volatility σ are all positive constants.
Under the above assumption, the optimal stopping problem (1) reduces to for t ∈ [0, T − δ], so y δ is the value process of the American put option with time lag δ, if the probability measure P is interpreted as the risk-neutral measure.
which is the time t value of the corresponding European put option with maturity t + δ. Denote by P (·, ·) the value function of the European put option with maturity T . Then, we haveŶ δ t = P (T − δ, X t ). Moreover, Lemma 2 (ii) implies that the value process y δ is given as where Y δ solves the reflected BSDE (4). Due to Assumption 2, the Feynman-Kac formula in section 9 of [9] further yields that with Ω T −δ = [0, T − δ ) × R + , and the operator L given as the Black-Scholes differential operator Note that if δ = 0, P (T − δ, X) = (K − X) + , and variational inequality (8) reduces to the standard variational inequality for American put options (see, for example, [7], [14] and [17]).
Furthermore, since variational inequality (8) is with smooth coefficients and obstacle, the solution V δ (·, ·) to (8) admits the following regularities. The proof follows along the similar arguments used in Chapter 1 of [11] (or more recently [19] and [20]) and is, thus, omitted.
Proposition 3 Suppose that Assumption 2 holds. Then, the viscosity solution V δ (·, ·) is also the unique bounded strong solution to variational inequality (8), Herein, W 2,1 p,loc (Ω T −δ ) is the set of all functions whose restriction on any

Remark 4
In the existing literature of optimal stopping with delivery lags (see [1] and [15] for example), they assume that the payoff S is a linear function of the underlying asset X. A consequence of this assumption is that the processŶ δ is also linear in X, which follows from the linearity of the conditional expectation, and the obstacle in the variational inequality is therefore also a linear function. In our case, the payoff S is a piecewise linear function of the underlying asset X (with a kink point at K), and this kink point propagates via the conditional expectation, resulting in a nonlinear obstacle P (T − δ, ·). This differentiates our problem from the existing optimal stopping problems with delivery lags, and makes the analysis of the corresponding optimal exercise boundary much more challenging.

Comparison with standard American put option
We first make a comparison between the American put options with and without delivery lags. Intuitively, the existence of delivery lags results in the loss of the opportunities to exercise the option and, therefore, the option value is smaller than the value of the standard American put option. Moreover, the longer the lag period is, the smaller the option value should be. The following proposition 5 confirms the above intuition.
Proposition 5 Suppose that Assumption 2 holds. Then, the American put option's value function V δ (·, ·) with time lag δ is decreasing with respect to δ for δ ∈ [0, T ), and moreover, Proof. To prove the monotone property of V δ (·, ·) with respect to δ, we first establish that To It follows from the comparison principle (see [11] or [19]) for variational inequality (8) On the other hand, it follows from (7) that from which we obtain that V δ2 (t, X) ≥ V δ1 (t, X), for (t, X) ∈ Ω T −δ1 , using the comparison principle for (8) in the domain Ω T −δ1 .
To prove the second inequality in (9), we use the early exercise premium representation of the standard American put option (see [13] for its proof): where e is the early exercise premium given as with X 0 (s) being the corresponding optimal exercise boundary, and p(t, X; s, ξ) being the transition density/the fundamental solution of the operator L.
which in turn yields that using the comparison principle for (8) in the domain Ω T −δ .

An early exercise premium decomposition formula
We derive a decomposition formula for the American put option with delivery lags, which can be regarded as a counterpart of the early exercise premium representation (11) for the standard case. Such a decomposition formula is crucial to the analysis of the optimal exercise boundary in sections 3.3 and 4. We show that the American put option with delivery lags can be decomposed as the European put option P (T − δ, X) and another American-style derivative with the running payoff Θ δ (X), where Θ δ (·) is the Greek Theta of the European option: Θ δ (X) = −∂ t P (T − δ, X).
Lemma 6 Suppose that Assumption 2 holds. Then, for t ∈ [0, T − δ], the value of the American put option with time lag δ can be decomposed as where U δ (t, X t ) = ess sup The optimal stopping time for (13) is τ 0, * as given in Lemma 2.
Finally, it is immediate to check that τ 0, * is the optimal stopping time for (13) using the definition of U δ t .

Remark 7
One of the advantages of the optimal stopping formulation (13) is that it does not have final payoff and only has running payoff, and this will facilitate our analysis of the associated optimal exercise boundary. In the rest of the paper, we shall focus our analysis on the optimal stopping problem (13) and its associated optimal exercise boundary, which will in turn solve the original optimal stopping problem (6).
The Feynman-Kac formula in Section 9 of [9] yields that U δ (t, X) is the unique (viscosity) solution of the variational inequality Introduce the transformation 1 Then (14) reduces to where N T −δ = (0, T − δ ] × R, and For the latter use, we present some basic properties of the Greek Θ(X), or equivalently, the function θ(x) with x = ln X − ln K, whose proof is given in Appendix A.1.

Thanks to Proposition 3 and transformation
Proposition 9 Suppose that Assumption 2 holds. Then, the viscosity solution u(·, ·) is also the unique bounded strong solution to variational inequality (16),

The perpetual case and its optimal exercise boundary
We consider the perpetual version of the optimal stopping problem (13), whose solution admits explicit expressions (cf. (22) and (23) below). The perpetual problem is also closely related to the asymptotic analysis of the optimal exercise boundary in section 4. Suppose that Assumption 2 holds. For any F-stopping time τ 0 ≥ t, we consider Using the similar arguments as in section 3.2, we obtain that U δ ∞ (X) = u ∞ (x), where x = ln X − ln K, and u ∞ (·) is the unique bounded strong solution to the stationary variational inequality with u ∞ ∈ W 2 p,loc (R) for p ≥ 1, and (u ∞ ) ′ ∈ C(R). In the domain {u ∞ (x) = 0}, the player will exercise the option. Since − L0 = 0 ≥ θ(x) by (18), and {θ(x) ≤ 0} = {x ≤ X} by Proposition 8, it follows that We can then define the optimal exercise boundary X as 2 The continuity of u ∞ (·) implies that u ∞ (x) = 0 for x ≤ X and, therefore, the player will exercise the option in (−∞, X]. Moreover, it follows from (19) and (20) that X ≤ X. The next proposition relates variational inequality (18) to a free-boundary problem, which in turn provides the explicit expressions for u ∞ (·) and X.
Proposition 10 For x > X, it holds that u ∞ (x) > 0. Moreover, (u ∞ (·), X) is the unique bounded solution to the free-boundary problem and satisfies X > X > −∞. Proof.
Step 1. We prove that (u ∞ (·), X) satisfies the free-boundary problem (21). To this end, we first show what u ∞ (x) > 0 for x > X. Since u ∞ (x) > 0 for x > X, we only need to show that u ∞ > 0 on (X, X]. If not, let x 1 , x 2 ∈ [ X, X ] be such that Using variational inequality (18) and Proposition 8, we obtain that The comparison principle then implies that u ∞ (x) ≤ 0 for x ∈ (x 1 , x 2 ), which is a contradiction.
Step 2. we prove that (u ∞ (·), X) is actually the unique solution to (21). To this end, we first show that if (u ∞,1 (·), X 1 ) is any solution solving (21), then it is necessary that X 1 < X. If not, by (21) and Proposition 8, we have The strong comparison principle (see [8]) then implies that u ∞,1 (x) > 0 for x > X 1 .
Next we compare u ∞,1 (x) with an auxiliary function with λ + and λ − being, respectively, the positive and negative characteristic roots of L: It is clear that In turn, Hence, the comparison principle implies that u ∞,1 (x) ≥ w(x) for x ∈ (X 1 , X 1 + 1). In turn, (u ∞,1 ) ′ (X 1 ) ≥ w ′ (X 1 ) > 0, which contradicts the smooth-pasting condition (u ∞,1 ) ′ (X 1 ) = 0. Now we show that (u ∞ (·), X) is the unique solution to (21). If not, let (u ∞, 1 , X 1 ) be another solution of the free-boundary problem (21). Without loss of generality, we may assume that X 1 < X < X. It is immediate to check that where we have used the fact θ(x) < 0 for any x ≤ X < X. The comparison principle then implies that u ∞, 1 (x) ≤ u ∞ (x) and, in particular, On the other hand, applying Taylor's expansion to u ∞,1 (x) yields which further implies that u ∞,1 (x) > 0 if x is close enough to X 1 . Thus, we obtain a contradiction.
Step 3. We prove that X > X > −∞. Since we have already showed that X < X in Step 2, it is sufficient to prove that X > −∞.
In fact, using the free-boundary formulation (21), we further obtain that its solution must have the form where the constants C is to be determined, λ − is the negative root of the characteristic equation for L, and p(x) = p(T − δ, x) is the price of the European put option (cf. (35) with t = T − δ).
In order to fix the constant C and the optimal exercise boundary X, we make use of the boundary and smooth-pasting conditions in (21), and obtain that where d 1 and d 2 are the same as d 1 and d 2 in (34) except that x is replaced by X (see Appendix A.1 for the notations). Thus, we obtain that and X is the zero crossing point of the algebra equation Next, we prove that the zero crossing point of l(x) = 0 exists and is unique. It is clear that, when x → −∞, Hence, l(x) is negative provided x is small enough. On the other hand, by (37) and (38), we have as x → +∞, and therefore, Hence, l(x) is positive provided x is large enough. Thus, we deduce that there exists at least one zero crossing point of l(x) = 0. Thanks to the uniqueness of the solution to the free-boundary problem (21), we know that the zero crossing point of the algebra equation (23) is also unique, from which we conclude that X > −∞.

Analysis of the optimal exercise boundary
We study the optimal exercise boundary of the American put option with time lag δ under Assumption 2. We first demonstrate our main results through Figures 1 and 2 as below.    (15)). Figure 2 illustrates that the whole region Ω T −δ is divided by a curve X δ (t) into two parts. In the left region, the investor will exercise the option (with time lag δ), and in the right region the investor will hold the option. Hence, X δ (t) is called the optimal exercise boundary. We will prove that it is a strictly increasing and smooth function, with Ke X as its asymptotic line when the maturity T goes to infinity, and Ke X as the end point at time T − δ. If we denote by x(·) the optimal exercise boundaries under the coordinates (τ, x), as shown in Figure 1, then we have the relationship (24)

The optimal exercise boundary
Due to Remark 7, we will mainly work with variational inequality (16) for u(·, ·) in the following. Recall N T −δ = (0, T − δ ] × R. Define the exercise domain ER and the continuation domain CR as Lemma 11 Let X and X be given in Proposition 8 and (20), respectively. Then, it holds that Proof. If (τ, x) ∈ ER, then u(τ, x) = 0 and variational inequality (16) reduces to In order to prove that ER ⊇ { x ≤ X }, we compare u(·, ·) and u ∞ (·), the latter of which is the solution to variational inequality (18). Note that The comparison principle for variational inequality (16) in the domain N T −δ then implies that u(τ, x) ≤ u ∞ (x). But if x ≤ X, according to the free-boundary problem (21), u ∞ (x) = 0. In turn, u(τ, x) = 0. This proves that {x ≤ X } ⊆ ER.
Intuitively, when θ(x) is positive (i.e. x > X), the running payoff in (13) is also positive, so the investor will hold the option. In the contrary, when θ(x) is non-positive (i.e. x ≤ X), one may think that the investor would then exercise the option to stop her losses. However, the above lemma shows that for x ≤ X, the investor may still hold the option, and wait for the running payoff to rally at a later time to recover her previous losses.
Proof. The definition of x 1 (τ ) implies that x(τ ) ≤ x 1 (τ ) and u(τ, x) > 0 for x ≥ x 1 (τ ). Moreover, it follows from Lemma 11 that The continuity of u implies that u(τ * , x(τ * )) = u(τ * , x 1 (τ * )) = 0. Let x * be a maximum point of u(τ * , ·) in the interval [x(τ * ), x 1 (τ * )]. Suppose that u(τ * , x * ) > 0; otherwise u(τ * , x) ≡ 0 in the interval [x(τ * ), x 1 (τ * )], which contradicts the definition of x * (τ ). Since On the other hand, by the continuity of u, there exits a neighborhood of (τ * , x * ) such that u > 0, so ∂ τ u − Lu = θ. In turn, where we have used the transformation (15) and the decomposition (12) in the first two equalities, and (10) in the last inequality, we further get This is a contradiction. Thus, we must have x(τ * ) = x 1 (τ * ). From the above lemma, we deduce that the exercise region and the continuation region are equivalent to We state the first main result of this section, which is about the monotone and regularity properties of the optimal exercise boundary x(τ ).

Remark 14
Under the original coordinates (t, X), the optimal exercise boundary X δ (t) then satisfies the following properties: (i) Monotonicity: X δ (t) is strictly increasing in t; (ii) Position: X δ (t) is with the end point

Asymptotic behavior for large maturity
We study the asymptotic behavior of the optimal exercise boundary x(τ ) and the value function u(τ, x) as τ → ∞, and as the second main result in this section, we show that they converge to their stationary counterparts X and u ∞ (x), respectively. To this end, we consider the following auxiliary optimal stopping time problem perturbed by rǫ.
for any F-stopping time τ 0 ≥ t and any ǫ ≥ 0. This will help us to achieve the lower bound and, therefore, the asymptotic behavior of the optimal exercise boundary x(τ ). Following along the similar arguments used in section 3.3, we obtain that u ǫ ∞ (x) = U ǫ ∞ (X), where x = ln X − ln K, and u ǫ (·) is the unique strong solution to the stationary variational inequality with u ǫ ∞ ∈ W 2 p,loc (R) for p ≥ 1, and (u ǫ ∞ ) ′ ∈ C(R). In contrast to variational inequality (18), it is not clear how to reduce variational inequality (28) to a free-boundary problem, and to obtain its explicit solution. Nevertheless, we are able to derive a local version of the free-boundary problem with ǫ > 0 small enough, which is sufficient to obtain the asymptotic behavior of the optimal exercise boundary later on.
. Then X ≤ X ǫ < X, and u ǫ ∞ (x) > 0 for any x ∈ (X ǫ , X), where X and X are given in (20) and Proposition 8, respectively. Moreover, X ǫ → X as ǫ → 0 + . See Figure 4 below. Proof. Note that the running payoff in the optimal stopping problem (27) satisfies Since u ∞ (x) > 0 for x > X, and X > X by Proposition 10, it holds that u ∞ (X) > 0. Let ǫ > 0 be small enough such that ǫ < u ∞ (X). Using the inequality u ǫ In turn, the definition of X ǫ and the continuity of u ǫ ∞ (·) imply that X ǫ < X. Repeating the similar arguments used in Proposition 10, we obtain that u ǫ ∞ (x) > 0 for x ∈ (X ǫ , X). Furthermore, the inequality u ∞ (x) ≥ u ǫ ∞ (x) and Proposition 10 imply that In turn, the definition of X ǫ implies that X ǫ ≥ X.
Next, we prove that X ǫ → X as ǫ → 0 + . In fact, from the definition of X ǫ and the continuity of u ǫ ∞ , we know that u ǫ ∞ (X ǫ ) = 0. Using the inequality On the other hand, applying Taylor's expansion to u ∞ (x) yields and is continuous, we deduce that if ǫ is small enough, then X ǫ ≤ X + ǫ/κ. Recalling X ǫ ≥ X, we conclude that X ǫ → X as ǫ → 0 + .
Finally, we send t → ∞ in the above inequalities, and conclude the convergence of x(2t) to X by Proposition 15.
Remark 17 Under the original coordinates (t, X), it follows from the relationship (24) and Theorem 16 that X δ (t) → Ke X as T → ∞, so Ke X is the asymptotic line of the optimal exercise boundary X δ (t). Theorem 16 also establishes the connection between the optimal stopping problems (13) and (17): U δ (t, X) → U δ ∞ (X) uniformly in X ∈ R + as T → ∞. Moreover, it follows from the decomposition formula (12) that the value function of the American put option with time lag δ has the long maturity limit:

Asymptotic behavior for small time lag
Our third main result in this section is about the convergence of the optimal exercise boundary when the time lag δ → 0. Denote by X 0 (t) the optimal exercise boundary of the corresponding standard American put option. It is well known that X 0 (t) is a strictly increasing and smooth function with X 0 (T ) = K. We refer to [13] for its proof.
From Proposition 5, V δ converges to V 0 in C(Ω T ) as δ → 0. Hence, the above estimate (31) implies that V δ also converges weakly to V 0 in W 2,1 p (N n T ) and, therefore, In turn, weakly in L p (N n T ). Now suppose that X δ (t) does not converge to X 0 (t). Then there exist t 0 ∈ [0, T ) and a sequence {X δm } ∞ m=1 such that when δ m → 0, X δm (t 0 ) does not converge to X 0 (t 0 ).
Since X 0 (t) is continuous and strictly increasing with X 0 (T ) = K, we may assume there exists ǫ > 0 and an integer M such that X 0 (t 0 ) + 2ǫ < min{X δm (t 0 ), K} for any m ≥ M . See Figure 5 below. Other cases can be treated in a similar way.
By the continuity and strictly increasing property of both X 0 (t) and X δ (t), we can find η > 0 such that the compact set [t 0 , t 0 + η] × [X 0 (t 0 ) + ǫ, X 0 (t 0 ) + 2ǫ] is in the exercise region of V δm and the continuation region of V 0 . Therefore, in this compact set, V δm (t, X) = P (T − δ m , X), V 0 (t, X) > K − X, and where we use the notation Θ(T − δ m , ·) to emphasize its dependence on T − δ m . However, from Proposition 8, it is immediate to check that lim δm→0 Θ(T − δ m , X) = qX − rK < 0, for X < K, which is a contradiction to (32).
• Ke X • K Figure 5: Non-convergence of the free boundaries X δ (t) to X 0 (t) as δ → 0.
A The proof of Proposition 8 (i) We first show that the function θ(x) (or equivalently, Θ(X) with X = Ke x ) has the following explicit form To this end, let x = ln X − ln K and p(t, x) = P (t, X). It is well known that p(t, x) has the following explicit expression (see [13] for example) where d t 1 and d 2 t are the same as d 1 and d 2 in (34) except that δ is replaced T − t: Differentiating p(t, x) against t yields that where we have used the fact that Thus we have proved (33).
To prove Proposition 8, we use the following two elementary inequalities: For d ≥ 0, We first show that there exists X such that θ(X) = 0. Note that this is equivalent to show that θ(X)/N ′ (−d 2 ) = 0, where d 2 is the same as d 2 in (34) except that x is replaced by X.
Next, we show that θ(x) N ′ (−d2) is strictly increasing in x, so its zero crossing point X is unique. Indeed, note that Let h(d) := N (−d)d N ′ (−d) . Then, we calculate its derivative against d as It is obvious that h ′ (d) > 0 when d ≤ 0. For d > 0, by using the inequalities (37) and (38), we obtain that In turn, h(d 2 ) ≤ h(d 1 ), which yields that by noting that h(d 2 ) < lim d→∞ h(d) = 1 and r > q.