Pricing American options time-capped by a drawdown event in a L\'evy market

TL;DR

This paper derives an explicit formula for pricing perpetual American put options with a drawdown cap in a Le9vy market, using martingale and fluctuation theory, and supports it with numerical analysis.

Contribution

It introduces a novel explicit pricing formula for American options with drawdown constraints in Le9vy markets, extending existing models.

Findings

Explicit pricing formula derived for the option.

Optimal stopping rule identified as first crossing below a threshold.

Numerical analysis confirms theoretical results.

Abstract

This paper presents a derivation of the explicit price for the perpetual American put option time-capped by the first drawdown epoch beyond a predefined level. We consider the market in which an asset price is described by geometric L\'evy process with downward exponential jumps. We show that the optimal stopping rule is the first time when the asset price gets below a special value. The proof relies on martingale arguments and the fluctuation theory of L\'evy processes. We also provide a numerical analysis.

Equations

$$S_t=e^{X_t},$$

$$X_t=x+\mu t+\sigma B_t-\sum _{k=1}^{N_t}{U}_k,$$

$${\overline{S}}_t=e^{\overline{x}}\vee \underset{0\le u\le t}{\sup }{S}_u$$

$${\tau }_D=\inf \left\{t\ge 0:\frac{{\overline{S}}_t}{S_t}\ge e^c\right\}$$

$$V(x,\overline{x})=\underset{\tau \in \mathcal{T}}{\sup }{\mathbb{E}}_{x,\overline{x}}\left[e^{-r\tau \wedge {\tau }_D}(K-S_{\tau \wedge {\tau }_D}{)}^{+}\right],$$

$$V(x,\overline{x})={\mathbb{E}}_{x,\overline{x}}[e^{-r{\tau }^{\ast }\wedge {\tau }_D}(K-S_{{\tau }^{\ast }\wedge {\tau }_D}{)}^{+}].$$

$${\tau }^{\ast }=\inf \left\{t\ge 0:X_t\le a^{\ast }\right\},$$

$${\tau }^{\ast }=\inf \{t\ge 0:X_t\le b(\overline{X_t})\}$$

$$D=\{(x,\overline{x})\in {\mathbb{R}}^2:V(x,\overline{x})=(K-e^x{)}^{+}\}.$$

$$Z_t=(X_t,{\overline{X}}_t)$$

$${\tau }^{\ast }=\inf \{t\ge 0:X_t\in D\},$$

$$V(y,\overline{x})-V(x,\overline{x})$$

$$\le {\mathbb{E}}_{y,\overline{x}}[e^{-r{\tau }_y\wedge {\tau }_D(y,\overline{x})}(K-e^{X_{{\tau }_y\wedge {\tau }_D(y,\overline{x})}}{)}^{+}]-{\mathbb{E}}_{x,\overline{x}}[e^{-r{\tau }_y\wedge {\tau }_D(y,\overline{x})}(K-e^{X_{{\tau }_y\wedge {\tau }_D(y,\overline{x})}}{)}^{+}]$$

$$=\mathbb{E}[e^{-r{\tau }_y\wedge {\tau }_D(y,\overline{x})}(K-e^{y+X_{{\tau }_y\wedge {\tau }_D(y,\overline{x})}}{)}^{+}]-\mathbb{E}[e^{-r{\tau }_y\wedge {\tau }_D(y,\overline{x})}(K-e^{x+X_{{\tau }_y\wedge {\tau }_D(y,\overline{x})}}{)}^{+}]$$

$$\le \mathbb{E}[e^{-r{\tau }_y\wedge {\tau }_D(y,\overline{x})}(K-e^{y+X_{{\tau }_y\wedge {\tau }_D(y,\overline{x})}})]-\mathbb{E}[e^{-r{\tau }_y\wedge {\tau }_D(y,\overline{x})}(K-e^{x+X_{{\tau }_y\wedge {\tau }_D(y,\overline{x})}})]$$

$$=(e^x-e^y)\mathbb{E}[e^{-r{\tau }_y\wedge {\tau }_D(y,\overline{x})+X_{{\tau }_y\wedge {\tau }_D(y,\overline{x})}}]=e^x-e^y.$$

$$V(y,\overline{x})-V(x,\overline{x})\le e^x-e^y=(K-e^y)-(K-e^x)$$

$$V(y,\overline{x})\le (K-e^y)\le (K-e^y{)}^{+}.$$

$$V(y,\overline{x})\ge (K-e^y{)}^{+}.$$

$$\vartheta =\inf \{t>0:X_t=b-\epsilon \}$$

$${\mathbb{E}}_{x,\overline{x}}[e^{-r\vartheta \wedge {\tau }_D(x,\overline{x})}(K-e^{X_{\vartheta \wedge {\tau }_D(x,\overline{x})}}{)}^{+}]\ge {\mathbb{E}}_{x,\overline{x}}[e^{-r{\tau }_b\wedge {\tau }_D(x,\overline{x})}(K-e^{X_{{\tau }_b\wedge {\tau }_D(x,\overline{x})}}{)}^{+}],$$

$$\mathcal{L}$$

$$+$$

$$\frac{\partial }{\partial \overline{x}}f(x,\overline{x})=0\text{for}x=\overline{x}.$$

$$(\mathcal{L}\hat{V}-r\hat{V})(x,\overline{x})$$

$$\hat{V}(x,\overline{x})$$

$$\displaystyle\hat{V}(x,\overline{x})\big{|}_{x=b(\overline{x})}$$

$$\displaystyle\frac{\partial}{\partial x}\hat{V}(x,\overline{x})\big{|}_{x=b(\overline{x})}$$

$$e^{-rt}\hat{V}(X_t,{\overline{X}}_t)=\hat{V}(x,\overline{x})+\sigma {\int }_0^t{e}^{-ru}\frac{\partial }{\partial x}\hat{V}(X_u,{\overline{X}}_u)d{B}_u+{\int }_0^t{e}^{-ru}\frac{\partial }{\partial \overline{x}}\hat{V}(X_u,{\overline{X}}_u)d{\overline{X}}_u$$

$$+{\int }_0^t{e}^{-ru}\left(\mathcal{L}\hat{V}(X_u,{\overline{X}}_u)-r\hat{V}(X_u,{\overline{X}}_u)\right)du$$

$$\displaystyle\quad+\frac{1}{2}\int_{0}^{t}\left(\frac{\partial}{\partial x}\hat{V}(x,\overline{x})\big{|}_{x=b(\overline{x})}-\frac{\partial}{\partial x}(K-e^{x})^{+}\big{|}_{x=b(\overline{x})}\right)\operatorname{d}\!L(s),$$

$$e^{-rt}$$

$$+{\mathbb{E}}_{x,\overline{x}}{\int }_0^t{e}^{-ru}\left(\mathcal{L}\hat{V}(X_u,{\overline{X}}_u)-r\hat{V}(X_u,{\overline{X}}_u)\right)du\le \hat{V}(x,\overline{x})$$

$$\hat{V}(x,\overline{x})={\mathbb{E}}_{x,\overline{x}}[e^{-r{\tau }^{\ast }\wedge {\tau }_D}(K-S_{{\tau }^{\ast }\wedge {\tau }_D}{)}^{+}]$$

Full text

Keywords. Black-Scholes model $\star$ pricing $\star$ American option $\star$ optimal stopping $\star$ cap $\star$ Lévy process

1. 1 Introduction

2. 1.1 Main results

3. 1.2 Literature overview

4. 1.3 Organisation of the paper

5. 2 Proof of main result

6. 2.1 Basic facts and verification lemma

7. 2.2 Value function $V_{a}$

8. 2.3 Optimal stopping threshold $a^{*}$

9. 2.4 Proof of Theorem 1

10. 3 Numerical analysis

1. Introduction

1.1. Main results

In mathematical finance, there has been growing interest in extending classical American options to include more complex financial products. In this paper, we consider the perpetual American put option time-capped by the first drawdown epoch beyond a predefined level. Such instrument should appeal to investors due to their reduced liability and lower cost compared to standard options. Popularity of this type of financial instrument strongly depends though on understanding their pricing, hedging, and optimal exercise policies. The desire to understand these features was our motivation for this paper.

This type of financial instrument compliments the capped options involving a cap to the asset price, where the option is terminated if the asset value exceeds a specified threshold. One of the simplest examples of capped options was introduced in 1991 by the Chicago Board of Options Exchange European options written on the S&P 100 and S&P 500 with a cap on their payoff function (see [3]). For other papers related to this type of cap, see [40, 41] and references therein.

This type of financial instrument is also strongly related to the deterministic cap (hence American option with finite maturity) and with other time-caps like the first passage time over given level of the asset price.

In this paper, consider the asset price $S_{t}$ in the Lévy-type market, that is, we assume that under the risk-neutral measure,

[TABLE]

with

[TABLE]

where $x=X_{0}=\log s$ and $\sigma\geq 0$. In equation (1), $B_{t}$ denotes a Brownian motion, $N_{t}$ is a homogeneous Poisson process with intensity $\lambda$ and $\{U_{k}\}_{\{k\in\mathbb{N}\}}$ is a sequence of independent identically distributed exponential random variables with mean $\rho^{-1}$. We assume that $B_{t}$, $N_{t}$ and $\{U_{k}\}_{\{k\in\mathbb{N}\}}$ are mutually independent and that all processes considered live in a common filtered probability space $(\Omega,{\mathcal{F}},\{\mathcal{F}_{t}\}_{\{t\geq 0\}},{\mathbb{Q}})$ with natural filtration $\{{\mathcal{F}}_{t}\}_{\{t\geq 0\}}$ of $X_{t}$ satisfying the usual conditions. We allow $\lambda=0$, which corresponds to the standard Black-Scholes model. For simplicity, we assume that no dividends are paid to the holders of the underlying asset.

Although geometric Brownian motion hence the Black-Scholes market widely used in everyday applications is a special case of our model, we add the possibility of the negative jumps in the asset price to better fit the evolution of the stock price process to real data. In recent years, the empirical study of financial data reveals that the distribution of the log-return of stock price exhibits features which cannot be captured by the normal distribution such as heavy tails and asymmetry. With a view to replicate these features more effectively and to reproduce a wide variety of implied volatility skews and smiles, there has been a general shift in the literature to model a risky asset with a geometric Lévy process analyzed in this paper rather than a geometric Brownian motion, see [31] for details. We still keep in mind that there is no uniqe martingale measure in this set-up, so one can choose, for example, one minimizing the entropy; see e.g. [5, Chap. 10.5] for details and discussion.

By

[TABLE]

we denote the running maximum of the asset price where $e^{\overline{x}}$ is the historical maximum of the underlying asset price prior to the beginning of the contract, where $d\vee f=\max\{d,f\}$. Similarly, we will denote $d\wedge f=\min\{d,f\}$. For the fixed threshold $c>0$ let

[TABLE]

be the first time when the (relative) drawdown is greater than $e^{c}$. In the main result of this paper we identify the closed-form formula for the price of the American put option with the random maturity determined to be a drawdown event given by

[TABLE]

for a family of stopping times $\mathcal{T}$ and fixed strike price $K>0$. Above, the subindex $x,\overline{x}$ attached to the expectation ${\mathbb{E}}_{x,\overline{x}}$ underlines the dependence of the mean on the initial asset price $S_{0}=e^{x}$ and the historical (observed) supremum $\overline{S}_{0}=e^{\overline{x}}$. We will skip the subindex when $x=\overline{x}=0$. The expectation is taken with respect of measure ${\mathbb{Q}}$. When $x=0$, we will skip this index. We restrict the domain of $V$ to $\mathcal{D}:=\{(x,\overline{x})\in\mathbb{R}^{2}:x\leq\overline{x}\}$. In the main result, we also find the optimal exercise rule, that is, a stopping rule $\tau^{*}$ such that

[TABLE]

Our main result is the following theorem.

Theorem 1**.**

The optimal stopping barrier is the first downward asset price time

[TABLE]

where $a^{*}$ is the unique solution of equation (40). Moreover, we have $V(x,\overline{x})=V_{a^{*}}(x,\overline{x})$ for the function $V_{a^{*}}(x,\overline{x})$ identified in Theorem 2.

1.2. Literature overview

Random termination appears in the mathematical finance literature, for example, as the default time of a company (see, e.g., [2, p. 27] or [42]) or as an asset-price-independent time cap following an exponential or Erlang distribution (see [4, 1]. In some cases, the time cap is chosen to be unobservable; see [10].

Time-cap is closely related to cancellable options, which are terminated early when a specific event occurs. In fact, the price of the stopped option consists of the value of the cancellable option and the discounted payoff (under the risk-neutral measure) at the event time, provided the event happens before maturity. Typically, this event is described as the first or last time when the underlying asset price reaches a specific threshold; see, e.g., [12, 11] and references therein. Similar early termination features can be found in game options (like, e.g., Israeli options) where the seller has the right to terminate the contract early, subject to a fixed penalty paid to the buyer; see the seminal paper [15] and subsequent works such as [21, 40, 41].

There are other studies addressing derivatives that closely resemble those analyzed in this paper. Egloff, Farkas and Leippold in [8] price American options with stochastic stopping time constraints, where the exercise of the option is restricted to specific conditions being met. To some extent, our motivation for considering a time-capped option is very similar: we want to introduce an option that remains valid until is either terminated by the buyer or triggered by an event described above, whichever occurs first. However, the difference in pricing is crucial. In [8] the buyer can only exercise the option when a pre-specified condition, associated with the performance of the underlying asset, is satisfied and when it is possible to transform the constrained pricing problem into an unconstrained optimal stopping problem. For the time-capped American option considered in this paper, this idea cannot be realized.

Several authors have also explored the concept of time-capping in American options. In [33], the random cap is a first hitting time of a fixed barrier by the underlying asset price. Finally, there are other related works. For example, [37, 36, 22] investigate Russian options that terminate when the stock price hits its running maximum for the last time, as well as American options that terminate when the stock price reaches a prespecified level for the last time.

We choose the drawdown event in pricing our American option due to its importance for the financial markets aiming at minimizing potential losses for the seller. The interest in the drawdown process has arisen after recent financial crises. It has also been used as dynamic risk measure or measure of relative regret, for example, the time-adjusted measure of performance known as the Calmar ratio. Thus, the drawdown process has become very important among practitioners.

The list of papers addressing contracts that incorporate drawdown or drawup feature is quite long; see, e.g. [6, 13, 18, 26, 27, 29, 30, 32, 34, 35, 38, 39] and references therein.

In [28] we analyzed the same problem, but only for the Black-Scholes market. The arguments given in this article requires much more complex analysis, though.

1.3. Organisation of the paper

This paper is organized as follows. In the next section, we give the proof of the main result. The proof is based on choosing a stopping rule and calculating the option value based on it. Next, a verification step via HJB system ensures that the chosen rule is indeed optimal and the achieved price is fair. In Section 3 we present a numerical analysis.

2. Proof of main result

2.1. Basic facts and verification lemma

The proof of the main result is based on the following key observations. Their proofs are the same as in [28] but we add them here for the completeness of the arguments.

Proposition 1**.**

The optimal stopping time $\tau^{*}$ is of the following form:

[TABLE]

for some function $b$.

Proof.

Let

[TABLE]

Note that

[TABLE]

is a Feller process. By [24, Thm. 2.7, p. 40 and (2.2.80), p. 49] it follows that

[TABLE]

that is, $D$ a stopping region, where the option should be exercised immediately. Suppose there exist $(x,\overline{x})\in D$. We additionally assume that $x<\log K$, otherwise the immediate payout is zero. Let $\tau_{y}$ and $\tau_{x}$ be the optimal stopping rule for starting point $(y,\overline{x})$ and $(x,\overline{x})$ respectively. Observe that if $y<x<\log(K)$, then for a chosen $c$, we have $\tau_{D}(y,\overline{x})\leq\tau_{D}(x,\overline{x})$, where $\tau_{D}(x,\overline{x})$ denotes $\tau_{D}$ for the starting point $(x,\overline{x})$. Further, we have:

[TABLE]

Therefore we get

[TABLE]

and further

[TABLE]

On the other hand the payoff function of the option cannot be higher than its value function, therefore

[TABLE]

This gives $(y,\overline{x})\in D$. This leads to the conclusion that for a certain $\overline{x}$ the optimal stopping region can be achieved by the pair $(X_{t},\overline{X_{t}})$ when $X_{t}$ drops down to some value $b(\overline{X_{t}})$ before it reaches its past maximum.

The question remains whether the optimal stopping region can also be reached from below, when both $X_{t}$ and $\overline{X_{t}}$ hit a certain level for the first time. We will show that this scenario is not possible. Indeed, assume a contrario that there exists a threshold $b$ such that there exist $x$ and $\overline{x}<b$ satisfying $(x,\overline{x})\notin D$ and $(b,b)\in D$. Let us take a positive $\varepsilon$ such that $b-\varepsilon>\overline{x}$. Let $\vartheta$ be the first time that process $X_{t}$ reaches the level $b-\varepsilon$ from below, i.e.

[TABLE]

and let $\tau_{b}$ be the first time when process $X_{t}$ reaches the level $b$ from below. Clearly, we have $\vartheta<\tau_{b}$ and $e^{X_{\vartheta}}<e^{X_{\tau_{b}}}.$ Therefore

[TABLE]

which contradicts the assumption and completes the proof.

∎

We will also need the following verification lemma. By

[TABLE]

we denote the generator of the Markov process $(X_{t},\overline{X}_{t})$ and the domain of this generator includes the functions $f\in\mathcal{C}^{2}_{0}(\mathbb{R})$ such that

[TABLE]

Lemma 1**.**

Let $\hat{V}(x,\overline{x}):\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a function defined on $\mathcal{D}:=\{(x,\overline{x})\in\mathbb{R}^{2}:x\leq\overline{x}\}$. Assume that $\hat{V}(x,\overline{x})\in\mathcal{C}^{2}_{0}(\mathbb{R})$ and that it fulfills condition (2). Assume that for some function $b$,

[TABLE]

[TABLE]

[TABLE]

Then $\hat{V}(x,\overline{x})\geq V(x,\overline{x})$.

Remark 1**.**

Conditions (7) and (8) are the so-called smooth paste conditions of the value function. Note that the condition (7) is mainly required to write (8) which is used in the proof of Lemma 1.

Proof.

Due to the assumed smoothness of $\hat{V}$, the smooth paste conditions (7) - (8) and an appropriate version Itô’s theorem (see [7, p. 208]) we have

[TABLE]

where $L$ is a local time of the process $X-b(\overline{X})$ at 0. Observe that $\operatorname{d}\!\overline{X}_{u}={\mathbb{I}}\{X_{u}=\overline{X}_{u}\}\operatorname{d}\!X_{u}$.

Now, requirement (2) guarantees that the integral over $\operatorname{d}\!\overline{X}_{u}$ is zero. Similarly, the smooth-paste conditions make sure that the integral over the local time also vanishes. Finally, relations (3) and (4) lead to the conclusion that the integral over $\operatorname{d}\!u$ is non-positive. If we take the expectation of both sides, we get the following result

[TABLE]

since the integral over Brownian motion is a zero-mean local martingale. Hence, according to the assumptions made, the process $e^{-rt\wedge\tau_{D}}\hat{V}(X_{t\wedge\tau_{D}},\overline{X}_{t\wedge\tau_{D}})=e^{-rt\wedge\tau_{D}}\hat{V}(Z_{t\wedge\tau_{D}})$ is a supermartingale and $\hat{V}(x,\overline{x})$ is a superharmonic function that dominates the payout. Now, from [24, (2.2.80), p. 49] we additionally know that $\hat{V}$ is also lower semi-continuous. It allows us to use [24, Thm. 2.7, p. 40] and claim that it is the optimal solution to the considered stopping problem. Hence $\hat{V}(x,\overline{x})\geq V(x,\overline{x})$. ∎

Remark 2**.**

By Proposition 1 we know that the optimal stopping rule $\tau^{*}$ is of the one-sided form. In the next step, we postulate that the optimal stopping boundary is even more specific, namely, that

• •

$b(\overline{x})=a^{*}$ for some optimal $a^{*}$ when $\overline{x}<a^{*}+c$;

• •

$b(\overline{x})=\overline{x}-c$ when $a^{*}+c<\overline{x}<\log(K)+c$.

We will calculate the value function

[TABLE]

for this postulated stopping rule $\tau^{*}$ and we will show that all the assumptions of Lemma 1 are satisfied for $\hat{V}$. Hence in this case $\hat{V}(x,\overline{x})\geq V(x,\overline{x})$ by Lemma 1 and $\hat{V}(x,\overline{x})\leq V(x,\overline{x})$ due to the fact that we choose a specific stopping rule. Thus, $\hat{V}(x,\overline{x})=V(x,\overline{x})$ is a true value function and $\tau^{*}$ is the optimal stopping rule.

From general stopping theory applied to the Markov process $(S_{t\wedge\tau_{D}},\overline{S}_{t\wedge\tau_{D}})$ (see [24, Thm. 2.7, p. 40 and (2.2.80), p. 49]) it follows that the stopping region is the set when the value function meets the payout function and hence it is unique which is due to the existence of the value function $V(x,\overline{x})$. In other words, our stopping region is unique as well.

2.2. Value function $V_{a}$

According to Remark 2 we will first identify the value function

[TABLE]

where

[TABLE]

Then we choose $a$ in such a way to satisfy all the conditions of the verification Lemma 1 are satisfied for $\hat{V}(x,\overline{x})=V_{a}(x,\overline{x})$.

To realize the first goal, we introduce the so-called scale functions. We define a Laplace exponent of process $X_{t}$ as

[TABLE]

We assumed that ${\mathbb{Q}}$ is a risk-neutral measure and hence $e^{-rt}S_{t}$ is a ${\mathbb{Q}}$-local martingale which is equivalent to

[TABLE]

or that

[TABLE]

For $r\geq 0$ the so-called scale function is defined as a continuous function $W^{(r)}:[0,\infty)\rightarrow[0,\infty)$ such that:

[TABLE]

which gives

[TABLE]

see for details [23]. The exponents $\gamma_{i}$ have the following form:

[TABLE]

where

[TABLE]

Additionally, coefficients $C_{i}$ are given by:

[TABLE]

With the first scale function we associate the second one given by

[TABLE]

The value function $V_{a}$ for the stopping rule (9) is given in the next theorem.

Theorem 2**.**

The following holds.

For $x>a$ and $\overline{x}-x<c$:

• (i)

If $\overline{x}<a+c$ then we have

[TABLE]

where $V_{1}$, $V_{2}$, $V_{3}$, $V_{4}$, $V_{5}$, $V_{6}$, $V_{7}$ are given in (18), (13), (19), (14), (25), (28), (30), respectively.

• (ii)

If $a+c<\overline{x}<\log(K)+c$ then we have

[TABLE]

where $V_{10}$, $V_{11}$, $V_{12}$, $V_{13}$ are given in (31), (32), (33), (34), respectively.

• (iii)

If $\overline{x}>\log(K)+c$ then we have

[TABLE]

where $V_{14}$, $V_{15}$, $V_{16}$ are given in (36), (37), (38), respectively.

For $x\leq a$ or $\overline{x}-x\geq c$:

• (iv)

[TABLE]

Proof.

Assume first that $\overline{x}<a+c$. Then

[TABLE]

From [20, eq. (2.3)] we have that

[TABLE]

and

[TABLE]

From [17] we know that

[TABLE]

is a martingale. To identify other terms, we introduce the following change of measure:

[TABLE]

Moreover, we know that $(X,\mathbb{P})$ is also a spectrally negative Lévy process and its Laplace exponent is given by:

[TABLE]

Therefore, under $(X,{\mathbb{P}})$ the process $X$ has the same form (1) but with new parameters:

[TABLE]

Let us now define $W^{(0)}$ scale function for $(X,\mathbb{P})$. We will denote it by $W^{\mathbb{P}}$. That is,

[TABLE]

By (15) observe that

[TABLE]

This gives

[TABLE]

and that

[TABLE]

where $\tilde{C}_{i}=C_{i}$ and $\tilde{\gamma}_{i}=\gamma_{i}-1$ for $i=1,2,3$. Additionally, we have

[TABLE]

Now sing [20, eq. (2.4)], we get

[TABLE]

and

[TABLE]

Moving on to $V_{5}$, observe that

[TABLE]

Let us introduce the following notations to deal with the last terms

[TABLE]

We will also need

[TABLE]

By

[TABLE]

we denote the Lévy measure of the Lévy process $X_{t}$. Now, let us define the following two events:

[TABLE]

The first one is associated with drawdown exceeding the threshold with a Poissonian jump, the latter is related to the hitting the threshold by creeping. From [20, eq. (3.10, 3.11)] we have

[TABLE]

and

[TABLE]

We can represent events $A_{o}$ and $A_{c}$ as follows

[TABLE]

[TABLE]

For the first event, $A_{o}^{5}$, the first condition $\underline{X}_{\tau_{D}}\geq u$ disappears as the jump sizes are unbounded, allowing $\underline{X}_{\tau_{D}}$ to be arbitrarily small. Additionally, observe that no matter if we take $u=-\infty$ or $u=a$, we get $(a+c-u)\wedge c=c$ and $a+c\vee(u+c)=a+c$. As a result, we get $\frac{W^{(r)}((x-u)\wedge a)}{W^{(r)}(a)}=1$. This gives

[TABLE]

The first integral equals

[TABLE]

and the double integral from the square bracket is

[TABLE]

Observe that $\sum_{i=1}^{3}\frac{C_{i}}{\gamma_{i}+\rho}=0$ since $C_{i}=\frac{2}{\sigma^{2}}\frac{\gamma_{i}+\rho}{(\gamma_{i}-\gamma_{j})(\gamma_{i}-\gamma_{k})}$ for $i,j,k\in\{1,2,3\},\ i\neq j,i\neq k,j\neq k$. Additionally,

[TABLE]

because

[TABLE]

Therefore, we can simplify $\Gamma_{\mathbb{Q}}$ in the following way

[TABLE]

To handle the expected value on ${\mathbb{P}}$ measure, we need to introduce the similar notations as for ${\mathbb{Q}}$ measure:

[TABLE]

[TABLE]

and

[TABLE]

Finally, let

[TABLE]

By combining everything together, we get

[TABLE]

Now, let us consider an event, when the first drawdown of the process $X_{t}$ starting from $a+c$ occurs after the process hits level $\log(K)+c$. We can then split the time until drawdown into two sub-intervals: from 0 to $\tau_{\log(K)+c}^{+}$ and from $\tau_{\log(K)+c}^{+}$ to $\tau_{D}$. As time before reaching $\log(K)+c$ is independent from time to first drawdown starting from $\log(K)+c$, we can write the following relation:

[TABLE]

Let us first handle the denominator. Again, we want to specify events $A_{o}$ and $A_{c}$ for $V_{8}$:

[TABLE]

and

[TABLE]

With these, we get the formula for $V_{8}$ as follows

[TABLE]

as the first integral is equal to $1$ and the double integral is the same as one derived for in $V_{5}$ in (22). On the other hand, from formula (3.3) from [20] we have:

[TABLE]

and therefore

[TABLE]

Similarly, using the same argument for ${\mathbb{P}}$ counterparts and applying equality (17), we get

[TABLE]

Now, observe that for $V_{9}$, event $A_{o}$ is the same as for $V_{8}$:

[TABLE]

and only the lower bound of $\underline{X}_{\tau_{D}}$ changes for $A_{c}$ in the following way

[TABLE]

Similarly to $V_{8}$, we get

[TABLE]

Finally, by (26)

[TABLE]

For the last component of $V$, i.e. $V_{7}$, we only need the event $A_{o}$. It is impossible to get a non-zero payout from the option for the stock price starting from $Ke^{c}$, if the drawdown does not occur by a Poissonian jump. Hence in this case

[TABLE]

and then

[TABLE]

Furthermore, note that

[TABLE]

Similarly:

[TABLE]

Finally, we have

[TABLE]

This completes the derivation of value function $V$ for $\overline{x}<a+c$.

Let us now consider the case where $a+c\leq\overline{x}<\log(K)+c$. In this case we have

[TABLE]

Observe that $V_{7}$ appears in both cases when $\overline{x}$ is smaller or greater than $a+c$.

Calculations of $V_{10}$ are similar to calculations of $V_{3}$:

[TABLE]

Now, similarly to $V_{2}$ and $V_{4}$, we have:

[TABLE]

The term $V_{12}$ can be identified in a similar way as it was done for $V_{5}$. We introduce

[TABLE]

and

[TABLE]

Then

[TABLE]

Following the analysis of $V_{6}$ changing the starting point of $X_{t}$ from $\overline{x}$ to $a+c$ we can write

[TABLE]

Now, let us consider the case when $\overline{x}\geq\log(K)+c$. Then

[TABLE]

Observe that in this case

[TABLE]

To calculate this expected value, we will use the following Gerber-Shiu measure

[TABLE]

for $x,z\in[0,a]$ and $y\geq 0$. By [16, Thm. 5.5] we have

[TABLE]

Above, the Gerber-Shiu measure is associated with the first downward crossing of [math]. The proof of [16, Thm. 5.5] is given for the classical Cramer-Lundberg risk process, but it remains true without any changes for our Lévy process $X_{t}$ given in (1). We are, on the other hand, interested in the first time when the process starting from $x$ hits the level $\overline{x}-c$. Using the stationarity and independence of increments of Lévy processes, we can shift our starting point from $x$ to $x+c-\overline{x}$ and thus we can search for the first time when $X_{t}$ becomes negative. Additionally, to make the condition $\{X_{\tau_{\overline{x}-c}^{-}}<\log(K)\}$ true, we allow the undershoot $y$ of our shifted process to take values from set $(\overline{x}-\log(K)-c,\ \infty)$. Thus, we get that

[TABLE]

Calculation of this double integral can be separated into three smaller single integrals. Firstly,

[TABLE]

Next,

[TABLE]

Analogically, we have

[TABLE]

because $W^{(r)}(x)=0$ for $x<0$. Combining everything together, we get

[TABLE]

In order to calculate ${\mathbb{E}_{x,\hat{x}}}\left[e^{-r\tau_{D}}\left(K-e^{X_{\tau_{D}}}\right)^{+}{\mathbb{I}}\{\tau_{D}<\tau_{\overline{x}}^{+}\}\right]$ we also need to consider the following equality

[TABLE]

Finally, observe that

[TABLE]

Moving on to the penultimate expected value in (35), observe that

[TABLE]

as in (13) and (14).

Now, similarly to (29), we consider the event

[TABLE]

Thus, we can split the last expected value into

[TABLE]

We start by calculating the expected value under ${\mathbb{Q}}$

[TABLE]

Redoing the calculation on the ${\mathbb{P}}$ measure and combining the results together we get

[TABLE]

∎

2.3. Optimal stopping threshold $a^{*}$

To find the optimal level $a$ we choose $a^{*}$ satisfying condition (8), that is

[TABLE]

Theorem 3**.**

There exists a unique optimal level $a^{*}$ satisfying (39) and it solves the following equation

[TABLE]

where $\eta^{\mathbb{Q}}$, $\Delta^{\mathbb{Q}}$, $\Gamma_{\mathbb{Q}}$ are given in (20), (21), (24), respectively.

Proof.

We start the proof from observing that

[TABLE]

Furthermore,

[TABLE]

and

[TABLE]

Note that $W^{(r)}(0)=0$, $W^{(r)^{\prime}}(0)=\frac{2}{\sigma^{2}}$, $Z^{(r)}(0)=1$. Now, by setting $x=a$, we get

[TABLE]

and that

[TABLE]

Combining everything together we have

[TABLE]

In order to fulfill condition (39) (and (8)) we search for $a^{*}$ satisfying

[TABLE]

Now, using (27), we can rewrite the left-hand side of above equation as follows:

[TABLE]

This gives the equation (40). Using the intermediate value theorem we can prove that the solution of equation (41) always exists. Indeed, observe that by taking $a^{*}\rightarrow-\infty$, the last expression above becomes $rK\frac{W^{(r)}(c)^{2}}{W^{(r)^{\prime}}(c)}$, which is smaller than [math]. On the other hand, with $a^{*}=\log(K)$, we get

[TABLE]

Note that $Z^{(r)}(0)=1$ for the function $Z^{(r)}(x)$ defined in (12) and hence $\sum_{i=1}^{3}\frac{rC_{i}}{\gamma_{1}}=1$. The function $Z^{(r)}(x)$ is then a weighted average of three exponential functions with $C_{2},C_{3},\gamma_{2},\gamma_{3}<0$. Therefore, using fact that $\gamma_{1}=1$ (see (11)), we have $e^{c}-Z^{(r)}(c)>0$ and hence above expression is strictly positive for $a^{*}=\log(K)$. Thus indeed solution $a^{*}$ always exists. ∎

2.4. Proof of Theorem 1

According to Remark 2 it is sufficient to show that $\hat{V}(x,\overline{x})=V_{a^{*}}(x,\overline{x})$ is in the domain of the infinitesimal generator $\mathcal{L}$, that is, that $\hat{V}(x,\overline{x})\in\mathcal{C}^{2}_{0}(\mathbb{R})$ and that the boundary condition (2) is satisfied, that is, that

[TABLE]

Furthermore, we have to verify that all the conditions given in the verification Lemma 1 are satisfied.

The fact that $\hat{V}(x,\overline{x})\in\mathcal{C}^{2}(\mathbb{R})$ follows from Theorem 2 and from the form of the scale functions $W^{(r)}(x)$ and $Z^{(r)}(x)$ given in (10) and (12). The fact that $\hat{V}(x,\overline{x})$ disappear as $x,\overline{x}$ tend to infinity follows directly from the definition of $\hat{V}(x,\overline{x})=V_{a^{*}}(x,\overline{x})$. We will now verify that the condition (2) holds true.

We start from the case $\overline{x}<a^{*}+c$ for which we have

[TABLE]

Note that

[TABLE]

and that

[TABLE]

Therefore

[TABLE]

Additionally, we have

[TABLE]

and that

[TABLE]

Finally, putting all terms together,

[TABLE]

When we set $x=\overline{x}$, all terms cancel out and the condition (2) is fulfilled.

We consider now the case of $a^{*}+c<\overline{x}<\log(K)+c$. In this case we have

[TABLE]

Further,

[TABLE]

and

[TABLE]

Moreover, we have

[TABLE]

and

[TABLE]

Furthermore,

[TABLE]

This gives

[TABLE]

Thus condition (2) is satisfied.

We analyse now the remaining case of $\log(K)+c<\overline{x}$. In this case we have

[TABLE]

Further,

[TABLE]

Using (23) we get

[TABLE]

Additionally, we have

[TABLE]

and

[TABLE]

This gives

[TABLE]

which completes the proof of condition (2).

In the final step, we will verify that all the conditions of the verification Lemma 1 are satisfied.

First, observe that the scale functions $W^{(r)}(x)$ and $Z^{(r)}(x)$ defined as in (10) and (12) fulfill condition (3). Additionally, $e^{x}$ also satisfies this condition, since $\gamma_{1}=1$. Now, if $x>a^{*}$, then for all three cases specified in Theorem 2, function $V_{a^{*}}(x,\overline{x})$ is just a combination of $W^{(r)}(x)$, $Z^{(r)}(x)$ and $e^{x}$. Thus, condition (3) holds.

Now, from the fourth case specified in Theorem 2 we can see that $(\mathcal{L}V_{a^{*}}-rV_{a^{*}})(x,\overline{x})=-rK$ when $x\leq a^{*}$. Therefore, condition (4) is satisfied as well. The form of function $V_{a^{*}}(x,\overline{x})$ defined in Theorem 2 makes equation (5) directly to be satisfied for $x\leq a^{*}$.

Let us first handle the smooth paste conditions before moving on to the fourth condition of the HJB system. Note that equality (8) is true due to our choice of $a^{*}$ solving (41). Furthermore, note that $V_{1}(a^{*},\overline{x})=K-e^{a^{*}}$ and $V_{2}(a^{*},\overline{x})=0$, which implies that condition (7) is satisfied.

Finally, to prove (6), we need to ensure that $V_{a^{*}}(x,\overline{x})$ dominates over the gain function.

Proposition 1 implies that the stopping region is reached by $X_{t}$ process from above. On the other hand, from conditions (7) and (8) we know that $\tau_{a^{*}}^{-}$ is the first time when the candidate for the value function equals its payoff. Let us consider the pair $(x,\overline{x})=(a^{*}+c,a^{*}+c)$. Note that in this case, we have $V_{1}(a^{*}+c,a^{*}+c)=0$, $V_{2}(a^{*}+c,a^{*}+c)=1$, $V_{3}(a^{*}+c)=0$ and $V_{4}(a^{*}+c)=1$. Therefore, from (41) we get that $V_{a^{*}}(a^{*}+c,a^{*}+c)=KZ^{(r)}(c)-e^{a^{*}+c}$. On the other hand, the immediate payout if equal to $K-e^{a^{*}+c}$. By (12) it immediately follows that $Z(x)>1$ for all $x>0$ which makes condition (6) satisfied.

$\square$

3. Numerical analysis

In this section, we analyze several properties of options capped by drawdown. In Figure 1, we present the smooth-pasting of the payoff and value functions, defined by equations (7) and (8). The term ”Projection”, mentioned in the legend, refers to the form of the function $V$, defined on the continuation region, applied to the stopping region. The chart confirms that the smooth-paste condition indeed holds.

Next, Figure 2 illustrates how the option price depends on the initial values $X_{0}=x$ and $\overline{X}_{0}=\overline{x}$. Half of the chart is set to zero due to domain limitation $x\leq\overline{x}$. The plot reveals an interesting behavior of the function, explained below, when both $x$ and $\overline{x}$ are sufficiently high. Figure 3 shows the same function, zoomed into the lower-right corner.

The first notable feature is the non-differentiability of the function at $\overline{x}=\log(K)+c$, when $x\in(\log(K),\log(K)+c)$. The explanation is straightforward: if $\overline{x}<\log(K)+c$, a non-zero payout can still be achieved via diffusion, whereas for $\overline{x}>\log(K)+c$, it can only result from a Poissonian jump. Consequently, the structure of the value function changes, leading to non-differentiability.

Additionally, we observe that for $\overline{x}>\log(K)+c$, the value function is zero when $x<\overline{x}-c$, which is expected, as the option is out of the money and immediately stopped by the drawdown. However, for $x>\overline{x}-c$, the value function increases with respect to $x$, which might be surprising, since for $\overline{x}<\log(K)+c$ it behaves in the opposite manner. This can be explained by the fact that the higher the value of $x$, the longer it takes for the drawdown to occur. As a result, the probability of observing a downward jump that causes $x$ to fall below $\log(K)$ increases.

Next, we perform a sensitivity analysis of both the stopping barrier and the option price with respect to the volatility $\sigma$, the risk-free rate $r$, the jump size parameter $\rho$, and the jump frequency parameter $\lambda$.

In Figure 4, we examine the optimal barrier $e^{a^{*}}$ of the underlying asset price process. It is evident that the barrier increases with higher values of $r$ and lower values of $\sigma$, indicating that an increase in the drift parameter of $X_{t}$ results in an upward shift of the barrier.

Figure 5 presents the sensitivities of the option price with respect to the risk-free rate and volatility. It is clear that the highest price is obtained for the lowest values of $r$ and $\sigma$. However, it is also evident that for a high value of $r=0.5$, the value function increases with $\sigma$. This suggests that the impact of volatility on the option price depends on the values of the other parameters.

A similar analysis is conducted for $\rho$ and $\lambda$. Figure 6 shows that the optimal stopping barrier decreases with increasing $\lambda$ and decreasing $\rho$. This is intuitive, as a smaller $\rho$ implies a larger mean jump size. Likewise, the option price, presented in Figure 7, is highest for the largest value of $\lambda$ and the smallest value of $\rho$, which is sensible, since in such a scenario the likelihood of the process $X_{t}$ dropping significantly below $\log(K)$ increases. Figures 8 and 9 present side views of the chart from Figure 7. There, we observe that the option price grows exponentially as $\rho$ decreases and linearly as $\lambda$ increases. This behavior can be explained by the fact that while $\lambda$ controls the frequency of jumps, $\rho$ exponentially influences the jump sizes, affecting the payout after a jump occurs.