Perpetual American Options with Asset-Dependent Discounting

Abstract

In this paper we consider the following optimal stopping problem

$$\begin{aligned} V^{\omega }_{\text {A}}(s) = \sup _{\tau \in {\mathcal {T}}} {\mathbb {E}}_{s}[e^{-\int _0^\tau \omega (S_w) dw} g(S_\tau )], \end{aligned}$$

where the process $S_t$ is a jump-diffusion process, ${\mathcal {T}}$ is a family of stopping times while g and $\omega$ are fixed payoff function and discount function, respectively. In a financial market context, if $g(s)=(K-s)^+$ or $g(s)=(s-K)^+$ and ${\mathbb {E}}$ is the expectation taken with respect to a martingale measure, $V^{\omega }_{\text {A}}(s)$ describes the price of a perpetual American option with a discount rate depending on the value of the asset process $S_t$. If $\omega$ is a constant, the above problem produces the standard case of pricing perpetual American options. In the first part of this paper we find sufficient conditions for the convexity of the value function $V^{\omega }_{\text {A}}(s)$. This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of $V^{\omega }_{\text {A}}(s)$. We also present a put-call symmetry for American options with asset-dependent discounting. In the case when $S_t$ is a spectrally negative geometric Lévy process we give exact expressions using the so-called omega scale functions introduced in [30]. We show that the analysed value function satisfies the HJB equation and we give sufficient conditions for the smooth fit property as well. Finally, we present a few examples for which we obtain the analytical form of the value function ${V}^{\omega }_{\text {A}}(s)$.

1 Introduction

In this paper the uncertainty associated with the stock price $S_t$ is described by a jump-diffusion process defined on a complete filtered risk-neutral probability space $(\Omega , {\mathcal {F}}, \{{\mathcal {F}}_t\}_{t\ge 0}, {\mathbb {P}})$, where ${\mathcal {F}}_t$ is a natural filtration of $S_t$ satisfying the usual conditions and ${\mathbb {P}}$ is a risk-neutral measure under which the discounted (with respect to a risk-free interest rate) asset price process $S_t$ is a local martingale. We point out that, as noted in [15], introducing jumps into the model, implies lost of completeness of the market which results in the lack of uniqueness of the equivalent martingale measure. Our main goal is the analysis of the following optimal stopping problem

$$\begin{aligned} V^{\omega }_{\text {A}}(s):= \sup _{\tau \in {\mathcal {T}}} {\mathbb {E}}_{s}\left[ e^{-\int _0^\tau \omega (S_w) dw} g(S_\tau )\right] , \end{aligned}$$

(1)

where ${\mathcal {T}}$ is a family of ${\mathcal {F}}_t$-stopping times while g and $\omega$ are fixed payoff function and discount function, respectively. Above ${\mathbb {E}}_s$ denotes the expectation with respect to ${\mathbb {P}}$ when $S_0=s$. We assume that the function g is convex. We allow in this paper for $\omega$ to have negative values as well. In the case when $g(s)=(K-s)^+$ or $g(s)=(s-K)^+$ and ${\mathbb {P}}$ is a martingale measure, this function can be interpreted as the value function of a perpetual American option ¹with asset-dependent²discounting $\omega$ and the payoff function g. In the case of the general theory of stochastic processes, multiplying by the discount factor $e^{-\int _0^\tau \omega (S_w) dw}$ corresponds to killing of a generator of $S_t$ by potential $\omega$.

This problem extends the classical theory of option pricing, where the deterministic discount rate is considered, that is if $\omega (s)=r$, then we obtain the standard form

$$\begin{aligned} V_{\text {A}}(s):= \sup _{\tau \in {\mathcal {T}}}{\mathbb {E}}_{s}\left[ e^{-r\tau } g(S_\tau )\right] \end{aligned}$$

of the perpetual American option’s value function with constant discount rate r.

The main objective of this paper is to find a closed expression of (1) and identify the optimal stopping rule $\tau ^*$ for which the supremum is attained. To do this, we start from presenting in Theorem 1 an inheritance of convexity property from the payoff function to the value function. This corresponds to preserving the convexity by the solution to a certain obstacle problem.

Using this observation and the classical optimal stopping theory presented e.g. in [34] one can identify the optimal stopping region as an interval $[l^*,u^*]$, that is, $\tau ^*=\inf \{t\ge 0: S_t\in [l^*,u^*]\}$. Hence, in general, one can obtain in this case a double continuation region.

Later we focus on the case when $S_t$ is a geometric spectrally negative Lévy process, that is,

$$\begin{aligned} S_t:=e^{X_t} \end{aligned}$$

for a spectrally negative Lévy process $X_t$. In this case, using the fluctuation theory of Lévy processes, we identify value function (1) in terms of the omega scale functions introduced in [30].

For optimal stopping problem (1) we give sufficient conditions under which we can formalise the classical approach here as well. In particular, in Theorem 4 we show that if the value function $V^{\omega }_{\text {A}}(s)$ is smooth enough then it is the unique solution to a certain Hamiltonian-Jacobi-Bellman (HJB) system. Moreover, in the case of geometric Lévy process of the asset price $S_t$, we demonstrate that the regularity of 1 for (0, 1) and $(1,+\infty$) gives the smooth fit property at the ends of the stopping region. We want to underline here that proving the above mentioned regularity of the value function in the case of the jump-diffusion processes (which allows to formulate the HJB equation) in general is very difficult task (see [16] for some deep results related with it). Nevertheless, it is possible in our case thanks to Theorem 4 and Remark 6. Having the convexity of the value function and hence knowing that the value function is defined in terms of the omega scale functions, we can prove the appropriate smoothness condition using the fluctuation theory of Lévy processes. Only then one can apply the HJB equation.

Further, even solving the HJB equation does not give in the straightforward way the form of the the stopping region (except, of course, the fact that it is the set where the value function equals the payoff function). This is the reason why we do not follow this path, but rely on our approach allowing us to define precisely the stopping region.

In Theorem 5 we show the put-call symmetry which holds in our setting as well.

These theoretical results allow us to find the price of the perpetual American option with asset-dependent discounting for some particular cases. We take for example a put option, that is $g(s)=(K-s)^{+}$, and a geometric Brownian motion for the asset price $S_t$. We model $S_t$ also by the geometric Lévy process with exponentially distributed downward jumps. We analyse various discount functions $\omega$. In Sect. 3 we provide a few examples for which we obtain the analytical form of the value function.

The discount rate changing in time or a random discount rate are widely used in pricing derivatives in financial markets. They have proved to be valuable and flexible tools to identify the value of various options. Usually, either the interest rate is independent from the asset price or this dependence is introduced via taking a correlation between gaussian components of these two processes. Our aim is completely different. We want to understand an extreme case when we have strong, functional dependence between the interest rate and the asset price. In particular, we take a close look at the American put option with the discount function $\omega$ having the opposite monotonicity to the payoff function. At first sight, such a case seems to be counter-intuitive, because, for the put option, if the asset price is in higher region one can expect that the interest rate will be lower and the opposite effect one expects for smaller range of asset’s prices. This dependence somehow balances the discount function with the payoff function. However, we can think of an investor who have a strong confidence in the movement of the asset price and wishes to make an extra profit when he is right and suffers a greater loss when he is wrong. This concept resembles an idea that stands behind barrier options, i.e. if an investor believes that it is unlikely that the asset price will hit a given level, he can add a knock-out provision with the barrier set at the support level, so he can reduce the price of an option. By including the barrier provision, he can eliminate paying for those scenarios he feels are unlikely. In our approach, we work in two ways by reducing the premium thanks to incidents being improbable from the investor’s perspective and increasing it for scenarios that are more likely to happen for him. Such a description of the analysed option adequately describes a financial instrument tailored to the risky investor. Let us add that for example up-and-out put option analysed by [31] is a particular case of our option.

One can look at optimisation problem (1) from a wider perspective though. The killing by potential $\omega$ has been known widely in physics and other applied sciences. Then (1) can be seen as a certain functional describing gain or energy and the goal is to optimise it by choosing some optimal stopping time. We focus here on financial applications only and therefore throughout this paper the price is calculated under the martingale measure. It means more formally that there exists a risk-free interest rate r such that the discounted price process is a local martingale under ${\mathbb {P}}$. Note that the discount function $\omega$ may equal the risk-free interest rate r but in our case usually is different.

Our research methodology is based on combining the theory of partial differential equations with the fluctuation theory of Lévy processes.

To prove the convexity we start from Theorem 9 which tells us about the convexity of

$$\begin{aligned} V^{\omega }_{\text {E}}(s, t):= {\mathbb {E}}_{s, t}\left[ e^{-\int _{t}^{T}\omega (S_w) d w} g(S_T)\right] \end{aligned}$$

(2)

for fixed time horizon T, where ${\mathbb {E}}_{s,t}$ is the expectation ${\mathbb {E}}$ with respect to ${\mathbb {P}}$ when $S_t=s$. Namely, the value function $V^{\omega }_{\text {E}}(s,t)$ given in (2) can be presented as the unique viscosity solution to a certain Cauchy problem for some second-order operator related to the generator of the process $S_t$. In fact, applying similar arguments like in [35, Proposition 5.3] and [22, Lemma 3.1], one can show that, under some additional assumptions, this solution can be treated as the classical one. Then we can formulate the sufficient locally convexity preserving conditions for the infinitesimal preservation of convexity at some point. This characterisation is given in terms of a differential inequality on the coefficients of the considered operator. It also allows to prove the convexity of $V^{\omega }_{\text {E}}(s, t)$. Then, in Theorem 1 we apply the dynamic programming principle (see [21]) in order to generalise the convexity property of $V^{\omega }_{\text {E}}(s, t)$ to the value function $V^{\omega }_{\text {A}}(s)$.

Later we focus on the American put option with the value function

$$\begin{aligned} V^{\omega }_{\text {A}^{\text {Put}}}(s):= \sup _{\tau \in {\mathcal {T}}}{\mathbb {E}}_{s}\left[ e^{-\int _{0}^{\tau }\omega (S_w) dw} (K-S_{\tau })^{+}\right] , \end{aligned}$$

where the payoff function $g(s)=(K-s)^+$ for some strike price $K>0$. Using the convexity property mentioned above we can conclude that the optimal stopping rule is defined as the first entrance of the process $S_t$ to the interval [l, u], that is,

$$\begin{aligned} \tau _{l,u}:=\inf \{t\ge 0: S_t\in [l,u]\}. \end{aligned}$$

(3)

In the next step, one has to identify

$$\begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u):={\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau _{l,u}} \omega (S_w) dw} (K-S_{\tau _{l,u}})^{+}\right] \end{aligned}$$

(4)

and take maximum over levels l and u to identify the optimal stopping rule $\tau ^*$ and to find the value function $V^{\omega }_{\text {A}^{\text {Put}}}(s)$. This is done for the geometric spectrally negative Lévy process $S_t=e^{X_t}$ where $X_t$ is a spectrally negative Lévy process starting at $X_0=\log S_0=\log s$. We recall that spectrally negative Lévy processes do not have positive jumps. Hence, in particular, our analysis could be applied for the Black-Scholes market where $X_t$ is a Brownian motion with a drift. To execute this plan we express $v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)$ in terms of the laws of the first passage times and then we use the fluctuation theory developed in [30]. In the whole analysis the use of the change of measure technique developed in [33] is crucial as well.

Optimal levels $l^*$ and $u^*$ of the stopping region $[l^*,u^*]$ and the price $V^{\omega }_{\text {A}^{\text {Put}}}(s)$ of the American put option could be found by application of the appropriate HJB equation.

Finally, to find the price of the American call option we present the put-call symmetry in our set-up.

We analyse in detail the Black-Scholes model and the case when a logarithm of the asset price is a geometric linear drift minus compound Poisson process with exponentially distributed jumps and we take various discount functions $\omega$. The first example shows changes of the American options’ prices in a gaussian and continuous market while the latter is to model the market including downward shocks in the assets’ behaviour. In this paper we present some specific examples for these two cases. In [1] we analyse even more examples using and citing the results of this work (in arxiv version).

Our paper seems to be the first one analysing the optimal problem of the form (1) in this generality for jump-diffusion processes. For the classical diffusion processes Lamberton in [28] proved that the value function given in (1) is continuous and can be characterised as the unique solution to a variational inequality in the sense of distributions. Another crucial paper for our considerations is [8] which introduced discounting via a positive continuous additive functional of the process $S_t$ and used the approach of Itô and McKean [25, Sec. 4.6] to characterise the value function. Note that $t\rightarrow \int _0^t \omega (S_w)dw$ is indeed additive functional. A similar problem was also considered in [32].

The pricing technique developed in this paper can be applied to a wide range of securities and financial contracts where the discounting in the above vein is affected by the underlying asset price process. Apart from the above mentioned examples of options, one can consider for example Executive Stock Options (EOSs) in which the executive may exercise ESOs prematurely and leave the firm if an interesting opportunity arises or for diversification or liquidity reasons. Hence this policy can be determined by publicly available information such as stock prices. As Carr and Linetsky [11] noted this option corresponds either to $\omega (s) =\lambda _f +\lambda _e\mathbbm {1}_{\{s>K\}}$ or $\omega (s) =\lambda _f +\lambda _e\mathbbm {1}_{\{\log s>\log K\}}$, where $\lambda _f$ is a constant intensity of early exercise or forfeiture due to the exogenous voluntary or involuntary employment termination and $\lambda _e$ is the constant intensity of the early exercise due to the executive’s exogenous desire for liquidity or diversification.

In this paper we also prove that in this general setting of asset-dependent discounting, one can express the price of the call option in terms of the price of the put option. It is called the put-call symmetry (or put-call parity). Our finding supplements [20, 24] who extend to the Lévy market the findings by [10]. An analogous result for the negative discount rate case was obtained in [4,5,6, 17]. A comprehensive review of the put-call duality for American options is given in [18]. We also refer to [19, Section 7] and other references therein for a general survey on the American options in the jump-diffusion model.

The paper is organised as follows. In Sect. 2 we introduce basic notations and assumptions that we use throughout the paper and we give the main results of this paper. In Sect. 3 we present a few examples for which we obtain the analytical form of the value function, i.e. for the Black-Scholes market with negative $\omega$ function and for the market with prices being modeled by the geometric Lévy process with zero volatility and downward exponential jumps and $\omega$ being a linear function. Section 4 contains proofs of all relevant theorems. We put into Appendix proofs of auxiliary lemmas.

2 Main Results

2.1 Jump-Diffusion Process

In this paper we assume a jump-diffusion financial market defined formally as follows. On the basic probability space we define a couple $(B_t, v)$ adapted to the filtration ${\mathcal {F}}_t$, where $B_t$ is a standard Brownian motion and $v=v(dt, dz)$ is an independent of $B_t$ homogeneous Poisson random measure on ${\mathbb {R}}_0^{+} \times {\mathbb {R}}$ for ${\mathbb {R}}_0^{+}=[0,+\infty )$. Then the stock price process $S_t$ solves the following stochastic differential equation

$$\begin{aligned} dS_t = \mu (S_{t-}, t) dt + \sigma (S_{t-}, t)dB_t + \int _{\mathbb {R}} \gamma (S_{t-}, t, z) {\tilde{v}}(dt, dz), \end{aligned}$$

(5)

where

• ${\tilde{v}}(dt, dz) = (v-q)(dt, dz)$ is a compensated jump martingale random measure of v,

• v is a homogenous Poisson random measure defined on ${\mathbb {R}}_0^{+}\times {\mathbb {R}}$ with intensity measure

  $$\begin{aligned} q(dt, dz) = dt\; m(dz). \end{aligned}$$

If additionally, the jump-diffusion process has finite activity of jumps, i.e. when

$$\begin{aligned} \lambda :=\int _{{\mathbb {R}}}m(dz)<\infty , \end{aligned}$$

then $N_t=v([0,t]\times {\mathbb {R}})$ is a Poisson process and m can be represented as

$$\begin{aligned} m(dz) = \lambda {\mathbb {P}}\left( e^{Y_i}-1\in dz\right) , \end{aligned}$$

where $\{Y_i\}_{i\in {\mathbb {N}}}$ are i.i.d. random variables independent of $N_t$ with distribution $\mu _Y$. Note that $B_t$ and $N_t$ are independent of each other as well. When additionally $\mu (s,t)=\mu s$, $\sigma (s,t)=\sigma s$ and $\gamma (s,t,z)=sz$, then the asset price process $S_t$ is the geometric Lévy process, that is,

$$\begin{aligned} S_t= e^{X_t}, \end{aligned}$$

(6)

where $X_t$ is a Lévy process starting at $x=\log s$ with a triple $(\zeta , \sigma , \Pi )$ for

$$\begin{aligned} \zeta :=\mu -\frac{\sigma ^2}{2},\quad \Pi (dx):=\lambda \mu _Y (dx). \end{aligned}$$

(7)

This observation follows straightforward from Itô’s rule.

2.2 Assumptions

Before we present the main results of this paper, we state now the assumptions on the model parameters used later on. We denote ${\mathbb {R}}^{+}:=(0,+\infty )$. If we talk about convexity and concavity we mean it in a weak sense allowing these functions to be constants within some regions.

Assumptions (A)

• (A1) The drift parameter $\mu$: ${\mathbb {R}}^{+}\times {\mathbb {R}}_0^{+}\rightarrow {\mathbb {R}}$ and the diffusion parameter $\sigma$: ${\mathbb {R}}^{+}\times {\mathbb {R}}_0^{+}\rightarrow {\mathbb {R}}$ are continuous functions, while the jump size $\gamma$: ${\mathbb {R}}^{+}\times {\mathbb {R}}_0^{+}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is measurable and for each fixed $z\in {\mathbb {R}}$, the function $(s, t) \rightarrow \gamma (s, t, z)$ is continuous.

• (A2) There exists a constant $C>0$ such that

  $$\begin{aligned} \mu ^2(s, t) + \sigma ^2(s, t) + \gamma ^2(s, t, z) \le C s^2 \end{aligned}$$

  for all $(s, t, z) \in {\mathbb {R}}^{+}\times {\mathbb {R}}_0^{+}\times {\mathbb {R}}$.

• (A3) There exists a constant $C>0$ such that

  $$\begin{aligned} |\mu (s_2, t)-\mu (s_1, t)|+ |\sigma (s_2, t)-\sigma (s_1, t)|+ |\gamma (s_2, t, z) - \gamma (s_1, t, z)|\le C|s_2-s_1|\end{aligned}$$

  for all $(s, t, z) \in {\mathbb {R}}^{+}\times {\mathbb {R}}_0^{+}\times {\mathbb {R}}$.

• (A4) There exists a constant $C>-1$ such that

  $$\begin{aligned} \gamma (s, t, z) > Cs \end{aligned}$$

  for all $(s, t, z) \in {\mathbb {R}}^{+}\times {\mathbb {R}}_0^{+}\times {\mathbb {R}}$.

• (A5) $g(s)\in C_{\text {pol}}({\mathbb {R}}^{+})$, where $C_{\text {pol}}({\mathbb {R}}^{+})$ denotes the set of functions of at most polynomial growth.

• (A6) $V^{\omega }_{\text {A}}(s)<\infty$.

Assumptions (A1), (A2), (A3) guarantee that there exists a unique solution to (5). Moreover, (A2) and (A4) imply that

$$\begin{aligned} {\mathbb {P}}(S_t\le 0 \text { for some }t\in {\mathbb {R}}_0^{+})=0 \end{aligned}$$

which is a natural assumption since the process $S_t$ describes the stock price dynamic and its value has to be positive. Additionally, to make the problem of pricing well-defined we assume (A6). We do not provide here necessary conditions for $\omega$ and g that guarantee (A6), we only focus on finiteness of $V^{\omega }_{\text {A}}(s)$. However, it can be easily shown that $\omega \ge 0$ and boundedness of function g are the sufficient conditions for (A6) to hold true.

Remark 1

Note that assumptions (A1)–(A4) are all satisfied for the geometric Lévy process.

2.3 Convexity of the Value Function

Our first main result concerns the convexity of the value function $V^{\omega }_{\text {A}}(s)$.

Theorem 1

Let Assumptions (A) hold. Assume that the payoff function g is convex, $\omega$ is concave such that $\omega \in C^2({\mathbb {R}}^+)$, the stock price process $S_t$ follows (5) and the following inequalities are satisfied

$$\begin{aligned}{} & {} \frac{\partial ^2\gamma (s,t,z)}{\partial s^2} \gamma (s,t,z)\ge 0, \end{aligned}$$

(8)

$$\begin{aligned}{} & {} \left( \frac{\partial ^2\mu (s, t)}{\partial s^2} - 2\frac{d\omega (s)}{ds}\right) \frac{\partial V^{\omega }_{\text {E}}(s, t)}{\partial s} - \frac{d^2\omega (s)}{ds^2}V^{\omega }_{\text {E}}(s, t) \ge 0, \end{aligned}$$

(9)

where $V^{\omega }_{\text {E}}(s, t)$ is defined in (2). Then the value function $V^{\omega }_{\text {A}}(s)$ is convex as a function of s.

Remark 2

We give now sufficient conditions in terms of model parameters for (9) to be satisfied. If $S_t$ is the geometric Lévy process (hence $\mu (s,t)=\mu s$, $\sigma (s,t)=\sigma s$ and $\gamma (s,t,z)=sz$) then (8) is satisfied. Let additionally $g(s)=(K-s)^+$. Then our optimal stopping problem is equivalent to pricing the perpetual American put option with functional discounting. If $\omega$ is non-decreasing function then the function $s\rightarrow V^{\omega }_{\text {E}}(s, t)$ is non-increasing. Moreover, concavity of $\omega$ imply that second term in (9) is nonnegative. Combining all these conditions we can conclude that (9) is satisfied. Concluding, if $\omega$ is concave and non-decreasing, then the value function of the perpetual American put option in geometric Lévy market is convex as a function of the initial asset price s. In Sect. 3 we provide an example which shows that convexity can be not preserved if the above mentioned conditions do not hold.

Remark 3

We have engineered above assumptions to handle mainly the put option (not a call option). The call option can be then handled via put-call symmetry proved in Theorem 5. Hence there is no need to provide sufficient conditions for both cases.

2.4 American Put Option and the Optimal Exercise Time for Spectrally Negative Geometric Lévy Process

Assume now the particular case of (1) with the payoff function

$$\begin{aligned} g(s) = (K-s)^{+}, \end{aligned}$$

that is, the value function $V^{\omega }_{\text {A}}(s)$ describes the price of a perpetual American put option. The value function for this special choice of payoff function is denoted by

$$\begin{aligned} V^{\omega }_{\text {A}^{\text {Put}}}(s):= \sup _{\tau \in {\mathcal {T}}}{\mathbb {E}}_{s}\left[ e^{-\int _{0}^{\tau }\omega (S_w) dw} (K-S_{\tau })\right] . \end{aligned}$$

(10)

Note that above we used the fact that the option will not be realised when it equals zero, hence the plus in the payoff function could be skipped.

From now on we will focus only on the asset price modeled by a spectrally negative geometric Lévy process defined in (6), that is when

$$\begin{aligned} S_t=e^{X_t}, \end{aligned}$$

where $X_t$ is a spectrally negative Lévy process with $X_0 = x=\log s$ and hence $S_0 = s$. This means that $X_t$ does not have positive jumps which corresponds to the inclusion of the support of Lévy measure m on the negative half-line. This is very common assumption which is justified by some financial crashes; see e.g. [2, 3, 12]. One can easily observe that the dual case of spectrally positive Lévy process $X_t$ can be also handled in a similar way. We decided to skip this analysis and focus only on a more natural, from a practical perspective, spectrally negative scenario.

From [34, Chapter III]³ it follows that the optimal stopping rule is of the form

$$\begin{aligned} \tau ^*=\inf \{t\ge 0: V^{\omega }_{\text {A}^{\text {Put}}}(S_t)=(K-S_t)\}. \end{aligned}$$

From Theorem 1 we know that $V^{\omega }_{\text {A}^{\text {Put}}}(s)$ is convex. Moreover, from the definition of the value function it follows that $V^{\omega }_{\text {A}^{\text {Put}}}(s)\ge (K-s)$. Having both these facts in mind, together with linearity of the payoff function, it follows that $V^{\omega }_{\text {A}^{\text {Put}}}(s)$ and g can cross each other in at most two points. This observation leads straightforward to the conclusion about the form of the stopping region. We recall that in (3) and (4) we introduced the entrance time $\tau _{l, u} = \inf \{t\ge 0: S_t \in [l, u]\}$ into the interval [l, u] and the corresponding value function $v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u) = {\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau _{l, u}}\omega (S_w)dw} (K-S_{\tau _{l, u}})\right]$, respectively.

Theorem 2

Let assumptions of Theorem 1 hold. Then, the value function defined in (10) is equal to

$$\begin{aligned} V^{\omega }_{\text {A}^{\text {Put}}}(s) = v^{\omega }_{\text {A}^{\text {Put}}}(s, l^{*}, u^{*}), \end{aligned}$$

where

$$\begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, l^{*}, u^{*}):= \sup _{0\le l\le u\le K}v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u). \end{aligned}$$

The optimal stopping rule is $\tau _{l^*, u^*}$, where $l^*, u^*$ realise the supremum above.

Remark 4

Another characterisation of critical points $l^*$ and $u^*$ via smooth fit property is given in Theorem 4.

Theorem 2 indicates that the optimal stopping rule in our problem is the first time when the process $S_t$ enters the interval $[l^*, u^*]$ for some $l^*\le u^*$. In the case when $l^*=u^*$ the interval becomes a point which is possible as well. In some cases the above observation allows to identify the value function in a much more transparent way. Finally, note that if the discount function $\omega$ is nonnegative, then it is never optimal to wait to exercise the option for small asset prices, that is, always $l^*=0$ and the stopping region is one-sided.

We express the value function in terms of some special functions, called omega scale functions; see [30] for details. To introduce these functions let us define first the Laplace exponent via

$$\begin{aligned} \psi (\theta ):= \frac{1}{t}\log {\mathbb {E}}[e^{\theta X_t}\mid X_0=0], \end{aligned}$$

which is finite at least for $\theta \ge 0$ due to downward jumps. This function is strictly convex, differentiable, equals zero at zero and tends to infinity at infinity. Hence there exists its right inverse $\Phi (q)$ for $q\ge 0$.

The key functions for the fluctuation theory are the scale functions; see [14]. The first scale function $W^{(q)}(x)$ is the unique right continuous function disappearing on the negative half-line whose Laplace transform is

$$\begin{aligned} \int _0^{\infty } e^{-\theta x} W^{(q)}(x) dx = \frac{1}{\psi (\theta )-q} \end{aligned}$$

(11)

for $\theta >\Phi (q)$.

For any measurable function $\xi$ we define the $\xi$-scale functions $\{{\mathcal {W}}^{(\xi )}(x), x\in {\mathbb {R}}\}$, $\{{\mathcal {Z}}^{(\xi )}(x), x\in {\mathbb {R}}\}$ and $\{{\mathcal {H}}^{(\xi )}(x), x\in {\mathbb {R}}\}$ as the unique solutions to the following renewal-type equations

$$\begin{aligned} {\mathcal {W}}^{(\xi )}(x)&= W(x) + \int _0^{x} W(x - y)\xi (y){\mathcal {W}}^{(\xi )}(y)dy, \end{aligned}$$

(12)

$$\begin{aligned} {\mathcal {Z}}^{(\xi )}(x)&= 1 + \int _0^{x} W(x - y)\xi (y){\mathcal {Z}}^{(\xi )}(y)dy, \end{aligned}$$

(13)

$$\begin{aligned} {\mathcal {H}}^{(\xi )}(x)&= e^{\Phi (c)x} + \int _0^x W^{(c)}(x - z)(\xi (z) - c){\mathcal {H}}^{(\xi )}(z) dz, \end{aligned}$$

(14)

where $W(x)=W^{(0)}(x)$ is a classical zero scale function and in equation (14) it is additionally assumed that $\xi (x)=c$ for all $x\le 0$ and some constant $c\in {\mathbb {R}}$. We also define the function $\{{\mathcal {W}}^{(\xi )}(x,z), (x,z)\in {\mathbb {R}}^2\}$ solving the following equation

$$\begin{aligned} {\mathcal {W}}^{(\xi )}(x,z)&= W(x-z) + \int _z^{x} W(x - y)\xi (y){\mathcal {W}}^{(\xi )}(y,z)dy. \end{aligned}$$

(15)

We introduce the following $S_t$ counterparts of the scale functions (12), (13), (14) and (15)

$$\begin{aligned} {\mathscr {W}}^{(\xi )}(s)&:= {\mathcal {W}}^{(\xi \circ \textrm{exp})}(\log s), \end{aligned}$$

(16)

$$\begin{aligned} {\mathscr {Z}}^{(\xi )}(s)&:= {\mathcal {Z}}^{(\xi \circ \textrm{exp})}(\log s), \end{aligned}$$

(17)

$$\begin{aligned} {\mathscr {H}}^{(\xi )}(s)&:= {\mathcal {H}}^{(\xi \circ \textrm{exp})}(\log s), \end{aligned}$$

(18)

$$\begin{aligned} {\mathscr {W}}^{(\xi )}(s,z)&:= {\mathcal {W}}^{(\xi \circ \textrm{exp})}(\log s,z), \end{aligned}$$

(19)

where $\xi \circ \textrm{exp} (x):= \xi (e^x)$.

For $\alpha$ for which the Laplace exponent is well-defined we can define a new probability measure ${\mathbb {P}}^{(\alpha )}$ via

$$\begin{aligned} \frac{d{\mathbb {P}}^{(\alpha )}_{s}}{d{\mathbb {P}}_{s}}\Bigg \vert _{{\mathcal {F}}_t} = e^{\alpha (X_{t}-\log s)-\psi (\alpha )t}. \end{aligned}$$

(20)

By [33] and [27, Cor. 3.10], under ${\mathbb {P}}^{(\alpha )}$, the process $X_t$ is again spectrally negative Lévy process with the Laplace exponent of the form

$$\begin{aligned} \psi ^{(\alpha )}(\theta ):=\psi (\theta +\alpha )-\psi (\alpha ). \end{aligned}$$

(21)

For this new probability measure ${\mathbb {P}}^{(\alpha )}$ we can define $\xi$-scale functions which are denoted by adding a subscript $\alpha$ to the regular counterparts, hence we have ${\mathscr {W}}^{(\xi )}_{\alpha }(s)$, ${\mathscr {Z}}^{(\xi )}_{\alpha }(s)$, ${\mathscr {H}}^{(\xi )}_{\alpha }(s)$ and ${\mathscr {W}}^{(\xi )}_{\alpha }(s, z)$.

We additionally define the following functions

$$\begin{aligned}&\omega _u(s):=\omega (su) \quad \text {and}\quad \omega _u^\alpha (s):=\omega _u(s)- \psi (\alpha ). \end{aligned}$$

The main result is given in terms of the resolvent density at z of $X_t$ starting at $\log s-\log u$ killed by the potential $\omega _u$ and on exiting from positive half-line given by

$$\begin{aligned} r(s,u, z):= {\mathscr {W}}^{(\omega _u)}(\log s-\log u) c_{{\mathscr {W}}^{(\omega )}/{\mathscr {W}}^{(\omega )}}(z)- {\mathscr {W}}^{(\omega _u)}(\log s-\log u, z), \end{aligned}$$

(22)

where

$$\begin{aligned} c_{{\mathscr {W}}^{(\omega )}/{\mathscr {W}}^{(\omega )}}(z):= \lim _{y\rightarrow \infty }\frac{{\mathscr {W}}^{(\omega )}(\log y, z)}{{\mathscr {W}}^{(\omega )}(\log y)}. \end{aligned}$$

Theorem 3

Assume that the stock price process $S_t$ is described by (6) with $X_t$ being the spectrally negative Lévy process and $\omega$ is a measurable, bounded from below, concave and non-decreasing function such that

$$\begin{aligned} \omega (s)=c \text { for all }s\in (0, 1] \text { and some constant }c\in {\mathbb {R}}. \end{aligned}$$

(23)

Then

$$\begin{aligned} \begin{aligned}&v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u) = \frac{{\mathscr {H}}^{(\omega )}(s)}{{\mathscr {H}}^{(\omega )}(l)}(K-l)\mathbbm {1}_{\{s<l\}} + (K-s)\mathbbm {1}_{\{s\in [l, u]\}} \\&\quad + \Bigg \{ \int _0^{\infty } \int _0^{\infty } \frac{{\mathscr {H}}^{(\omega _u)}((\frac{u}{e^y})\wedge l)}{{\mathscr {H}}^{(\omega _u)}(l)} (K-e^{\log l \vee (\log u-y)}) r(s,u, z)\Pi (-z-dy) dz \\&\quad + (K-u) \left( \lim _{\alpha \rightarrow \infty } \left( \frac{s}{u}\right) ^{\alpha }\left( {\mathscr {Z}}^{(\omega _u^\alpha )}_{\alpha }\left( \frac{s}{u}\right) - c_{{\mathscr {Z}}^{(\omega ^\alpha )}_{\alpha }/{\mathscr {W}}^{(\omega ^\alpha )}_{\alpha }}{\mathscr {W}}^{(\omega _u^\alpha )}_{\alpha }\left( \frac{s}{u}\right) \right) \right) \Bigg \} \mathbbm {1}_{\{s>u\}}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} c_{{\mathscr {Z}}^{(\omega ^\alpha )}_{\alpha }/{\mathscr {W}}^{(\omega ^\alpha )}_{\alpha }}&= \lim _{z\rightarrow \infty } \frac{{\mathscr {Z}}^{(\omega ^\alpha )}_{\alpha }(z)}{{\mathscr {W}}^{(\omega ^\alpha )}_{\alpha }(z)} \end{aligned} \end{aligned}$$

and r(s, u, z) is given in (22).

If $l=0$ then assumption (23) is superfluous and

$$\begin{aligned} \begin{aligned}&v^{\omega }_{\text {A}^{\text {Put}}}(s, 0, u) = (K-s)\mathbbm {1}_{\{s\in [0, u]\}} \\ {}&\quad + \Bigg \{ \int _0^{\infty } \int _0^{\infty } (K-e^{\log u-y}) r(s,u, z)\Pi (-z-dy) dz \\&\quad + (K-u) \left( \lim _{\alpha \rightarrow \infty } \left( \frac{s}{u}\right) ^{\alpha }\left( {\mathscr {Z}}^{(\omega _u^\alpha )}_{\alpha }\left( \frac{s}{u}\right) - c_{{\mathscr {Z}}^{(\omega ^\alpha )}_{\alpha }/{\mathscr {W}}^{(\omega ^\alpha )}_{\alpha }}{\mathscr {W}}^{(\omega _u^\alpha )}_{\alpha }\left( \frac{s}{u}\right) \right) \right) \Bigg \} \mathbbm {1}_{\{s>u\}}. \end{aligned} \end{aligned}$$

Remark 5

For the general case when $l>0$ the assumption (23) is a technical one and it is a consequence of the assumption made in [30, Thm. 2.5] which is used in the proof. We believe that this assumption is superfluous though.

2.5 HJB, Smooth and Continuous Fit Properties

The classical approach via HJB system is possible in our set-up as well. More precisely, as before in (6) we have

$$\begin{aligned} S_t=e^{X_t} \end{aligned}$$

for the Lévy process $X_t$ with the triple ($\zeta , \sigma , \Pi )$. We start from the observation that using [37, Thm. 31.5, Chap. 6] and Itô’s formula one can conclude that the process $S_t$ is a Markov process with an infinitesimal generator

$$\begin{aligned} {\mathcal {A}}f(s) = A^C f(s) + A^J f(s), \end{aligned}$$

where $A^C$ is the linear second-order differential operator of the form

$$\begin{aligned} A^C f(s) = \frac{\sigma ^2 s^2}{2} f^{\prime \prime }(s) + \left( \zeta +\frac{\sigma ^2}{2}\right) s f^\prime (s) \end{aligned}$$

and $A^J$ is the integral operator given by

$$\begin{aligned} A^J f(s) = \int _{(-\infty , 0)} \left( f(s e^z) - f(s) - s|z|f^\prime (s)\mathbbm {1}_{\{|z|\le 1\}} \right) \Pi (dz). \end{aligned}$$

The domain $D({\mathcal {A}})$ of this generator consists of the functions belonging to $C^{2}({\mathbb {R}}^+)$ if $\sigma >0$ and $C^{1}({\mathbb {R}}^+)$ if $\sigma =0$. In this paper we prove that $V^{\omega }_{\text {A}}(s)$ satisfies the HJB equation given below with appropriate smooth fit conditions. We recall that 1 is regular for (0, 1) and for the process $S_t$ if ${\mathbb {P}}_1(\tau _{(0, 1)}=0)=1$ for $\tau _{(0, 1)}=\inf \{t> 0: S_t\in (0, 1)\}$. Similarly, we can define regularity for $(1, +\infty )$. Note that regularity of $S_t$ at 1 corresponds to regularity of $X_t$ at 0 for the negative or positive half-line.

Theorem 4

Assume that the asset price is a spectrally negative geometric Lévy process (6). Let $\omega$ be a bounded from below and concave function with the opposite monotonicity to the payoff function g. Assume that $V^{\omega }_{\text {A}}(s)\in D({\mathcal {A}})$ and $g(s)\in C^1({\mathbb {R}}^+)$. Then $V^{\omega }_{\text {A}}(s)$ solves uniquely the following HJB system

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {A}} V^{\omega }_{\text {A}}(s) - \omega (s) V^{\omega }_{\text {A}}(s)= 0, \quad &{} \quad s\notin [l^*,u^*], \\ V^{\omega }_{\text {A}}(s) = g(s), \quad &{}\quad s\in [l^*,u^*]. \end{array}\right. } \end{aligned}$$

(24)

Moreover, if 1 is regular for (0, 1) and for the process $S_t$ then there is a smooth fit at the right end of the stopping region

$$\begin{aligned} (V^{\omega }_{\text {A}})^\prime (u^*)=g^\prime (u^*). \end{aligned}$$

Similarly, if 1 is regular for $(1, +\infty )$ and for the process $S_t$ then there is a smooth fit at the left end of the stopping region

$$\begin{aligned} (V^{\omega }_{\text {A}})^\prime (l^*)=g^\prime (l^*). \end{aligned}$$

Remark 6

Let us consider the American put option. Then from Theorems 2 and 3, we can conclude that smoothness of the value function $V^{\omega }_{\text {A}^{\text {Put}}}(s)$ corresponds to the smoothness of the $\xi$-scale functions for $\omega$, $\omega _u$ and $\omega _u^\alpha$. From the definition of these functions given in (12), (13) and (14) it follows that the smoothness of the latter functions is equivalent to the smoothness of the first scale function observed under measures ${\mathbb {P}}$ and ${\mathbb {P}}^{(\alpha )}$. By [27, Lem. 8.4] the smoothness of the first scale function does not change under the exponential change of measure (20). Thus from [14, Lem. 2.4, Thms 3.10 and 3.11] if follows that

• if $\sigma >0$ then $V^{\omega }_{\text {A}^{\text {Put}}}(s)\in C^{2}({\mathbb {R}}^+)$;

• if $\sigma =0$ and the jump measure $\Pi$ is absolutely continuous or $\int _{-1}^0|x|\Pi (dx)=+\infty$, then $V^{\omega }_{\text {A}^{\text {Put}}}(s)\in C^{1}({\mathbb {R}}^+)$.

Moreover, by [2, Prop. 7], 1 is regular for both (0, 1) and $(1,+\infty )$ if $\sigma >0$. Hence in this case we have $V^{\omega }_{\text {A}}(s)\in D({\mathcal {A}})$ and HJB system (24) with the smooth fit property could be used without any additional assumptions as long as $\sigma >0$. If one has single continuation region $[u^*, +\infty )$ and $\sigma =0$ then by [2, Prop. 7] to get the smooth fit condition at $u^*$ it is sufficient to assume that the drift $\zeta$ of the process $X_t$ is strictly negative.

2.6 Put-Call Symmetry

The put-call parity allows to calculate the American call option price having the put one. We formulate this relation again for $S_t$ being a general geometric Lévy process defined in (6), that is, $S_t=e^{X_t}$ for $X_t$ being a general spectrally negative Lévy process having triple

$$\begin{aligned} (\zeta , \sigma , \Pi ) \end{aligned}$$

for $\zeta$ and $\Pi$ defined in (7) and starting position $X_0=\log S_0=s$. Apart from the function

$$\begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, K, \zeta , \sigma , \Pi , l, u):= {\mathbb {E}}_{s}[e^{-\int _0^{\tau _{l, u}}\omega (S_w)dw} (K-S_{\tau _{l, u}})^{+}] \end{aligned}$$

defined in (4) we denote

$$\begin{aligned} v^{\omega }_{\text {A}^{\text {Call}}}(s, K, \zeta , \sigma , \Pi , l, u):= {\mathbb {E}}_{s}[e^{-\int _0^{\tau _{l, u}}\omega (S_w)dw} (S_{\tau _{l, u}} - K)^{+}]. \end{aligned}$$

Theorem 5

Assume that $\psi (1)=\log {\mathbb {E}}e^{X_1}=\log {\mathbb {E}} S_1$ is finite. Let $l \le u \le K$. Then we have

$$\begin{aligned} v^{\omega }_{\text {A}^{\text {Call}}}(s, K, \zeta , \sigma , \Pi , l, u) = v^{\vartheta ^{(1)}}_{\text {A}^{\text {Put}}}\left( K, s, -\zeta , \sigma , {\hat{\Pi }}, \frac{s}{u}K, \frac{s}{l}K\right) , \end{aligned}$$

(25)

where

$$\begin{aligned} {\hat{\Pi }}(dx)&:= e^{-x}\Pi (-dx), \nonumber \\ \vartheta ^{(1)}(\cdot )&:= \omega \left( \frac{1}{\cdot }sK\right) - \psi (1). \end{aligned}$$

(26)

Moreover, if assumptions of Theorem 1 hold for the function $\vartheta ^{(1)}$ then the American call option admits a double continuation region with optimal stopping boundaries $l^*_c$ and $u^*_c$ such that

$$\begin{aligned} l^*_cu^*=l^*u_c^*=sK, \end{aligned}$$

(27)

where $l^*$ and $u^*$ are the stopping boundaries for the put option.

Remark 7

Note that the value function of the American call option is expressed in terms of the American put option calculated for the Lévy process ${\hat{X}}_t$ being dual to $X_t$ process observed under the measure ${\mathbb {P}}^{(1)}$. In particular, the jumps of the process ${\hat{X}}_t$ have the opposite direction to the jumps of the process $X_t$ for which the put option is priced. In general, determining the conditions for $\omega$ such that $\vartheta ^{(1)}$ satisfies all the assumptions of Theorem 1 seems to be impossible and then we can only work on a case-by-case basis.

2.7 Black-Scholes Model

We can give more detailed analysis in the case of Black-Scholes model in which the stock price process $S_t=e^{X_t}$, where

$$\begin{aligned} X_t = \log s + \zeta t + \sigma B_t \end{aligned}$$

(28)

with $\zeta = \mu -\frac{\sigma ^2}{2}$, while $\mu \in {\mathbb {R}}$ and $\sigma >0$ are the parameters called drift and volatility, respectively. Under the martingale measure we have $\mu = r$, where r is a risk-free interest rate.

Theorem 6

Assume that $\omega$ is a bounded from below, concave and non-decreasing function. For Black-Scholes model (6) with $X_t$ given in (28) the function $v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)$ defined in (4) is given by

$$\begin{aligned} \begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)&= \frac{h(s)}{h(l)}(K-l)\mathbbm {1}_{\{s<l\}} + (K-s)\mathbbm {1}_{\{s\in [l, u]\}} \\ {}&\quad + \frac{h(s)}{h(u)}(K-u)\mathbbm {1}_{\{s>u\}}, \end{aligned} \end{aligned}$$

where h(s) is a solution to

$$\begin{aligned} \frac{\sigma ^2 s^2}{2}h^{\prime \prime }(s) + \mu s h^\prime (s) - \omega (s)h(s) = 0 \end{aligned}$$

(29)

which satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} h(s) = K-s, \quad s \in [l^{*}, u^{*}],\\ \displaystyle \lim _{s\rightarrow \infty }h(s) = \text {const.}\\ \end{array}\right. } \end{aligned}$$

(30)

Remark 8

The optimal boundaries $l^*$ and $u^*$ can be found from the smooth fit property given in Theorem 4.

2.8 Exponential Crashes Market

We can construct more explicit equation for the value function for the case of Black-Scholes model with additional downward exponential jumps, that is, as in (6), $S_t=e^{X_t}$ for

$$\begin{aligned} X_t = \log s + \zeta t + \sigma B_t - \sum _{i=1}^{N_t} Y_i, \end{aligned}$$

(31)

where $\zeta = \mu -\frac{\sigma ^2}{2}$, $N_t$ is the Poisson process with intensity $\lambda >0$ independent of Brownian motion $B_t$ and $\{Y_i\}_{i\in {\mathbb {N}}}$ are i.i.d. random variables independent of $B_t$ and $N_t$ having exponential distribution with mean $1/\varphi >0$. Moreover, under the martingale measure we obtain that $\mu = r + \frac{\lambda }{\varphi +1}$ with r being a risk-free interest rate. The Laplace exponent of $X_t$ starting at 0 is as follows

$$\begin{aligned} \psi (\theta )=\zeta \theta +\frac{\sigma ^2}{2}\theta ^2 - \frac{\lambda \theta }{\varphi +\theta }. \end{aligned}$$

(32)

For this model the price of American put option is easier to determine.

Theorem 7

Assume that $\omega$ is a nonnegative, concave and non-decreasing function. For geometric Lévy model (6) with $X_t$ given in (31) we have $l^*=0$. Furthermore,

• (i) if $\sigma =0$ then

  $$\begin{aligned} \begin{aligned} V^{\omega }_{\text {A}^{\text {Put}}}(s)&:=\sup _{u>0}v^{\omega }_{\text {A}^{\text {Put}}}(s, 0, u) \\ {}&= \sup _{u>0}\bigg \{\left( K - \frac{u\varphi }{\varphi +1}\right) \left( {\mathscr {Z}}^{(\omega _u)}\left( \frac{s}{u}\right) - c_{{\mathscr {Z}}^{(\omega )}/{\mathscr {W}}^{(\omega )}}{\mathscr {W}}^{(\omega _u)}\left( \frac{s}{u}\right) \right) \bigg \}, \end{aligned} \end{aligned}$$

  (33)

  where ${\mathscr {W}}^{(\omega _u)}\left( \frac{s}{u}\right)$ and ${\mathscr {Z}}^{(\omega _u)}\left( \frac{s}{u}\right)$ are given in (16) and (17), respectively, and

  $$\begin{aligned} \begin{aligned} c_{{\mathscr {Z}}^{(\omega )}/{\mathscr {W}}^{(\omega )}}&:= \lim _{z\rightarrow \infty } \frac{{\mathscr {Z}}^{(\omega )}(z)}{{\mathscr {W}}^{(\omega )}(z)}. \end{aligned} \end{aligned}$$

  (34)

  The optimal boundary $u^*$ is determined by the continuous fit condition

  $$\begin{aligned} V^{\omega }_{\text {A}^{\text {Put}}}(u^*) = K-u^{*}. \end{aligned}$$

  (35)

• (ii) If $\sigma >0$ then

  $$\begin{aligned} V^{\omega }_{\text {A}^{\text {Put}}}(s):= & {} \sup _{u>0}v^{\omega }_{\text {A}^{\text {Put}}}(s, 0, u) \nonumber \\= & {} \sup _{u>0}\bigg \{\left( K - \frac{u \varphi }{\varphi +1}\right) \left( {\mathscr {Z}}^{(\omega _u)}\left( \frac{s}{u}\right) - c_{{\mathscr {Z}}^{(\omega )}/{\mathscr {W}}^{(\omega )}} {\mathscr {W}}^{(\omega _u)}\left( \frac{s}{u}\right) \right) \nonumber \\{} & {} \quad + (K-u) \left( \lim _{\alpha \rightarrow \infty } \left( \frac{s}{u}\right) ^{\alpha }\left( {\mathscr {Z}}^{(\omega _u^\alpha )}_{\alpha }\left( \frac{s}{u}\right) - c_{{\mathscr {Z}}^{(\omega ^\alpha )}_{\alpha }/{\mathscr {W}}^{(\omega ^\alpha )}_{\alpha }} {\mathscr {W}}^{(\omega _u^\alpha )}_{\alpha }\left( \frac{s}{u}\right) \right) \right) \bigg \},\qquad \quad \end{aligned}$$

  (36)

  where ${\mathscr {W}}^{(\omega _u^{\alpha })}_{\alpha }\left( \frac{s}{u}\right)$ and ${\mathscr {Z}}^{(\omega _u^{\alpha })}_{\alpha }\left( \frac{s}{u}\right)$ are the scale functions (16) and (17) taken under measure ${\mathbb {P}}^{(\alpha )}$ and

  $$\begin{aligned} \begin{aligned} c_{{{\mathscr {Z}}^{(\omega ^{\alpha })}_{\alpha }}/{{\mathscr {W}}^{(\omega ^{\alpha })}_{\alpha }}}&:= \lim _{z\rightarrow \infty } \frac{{\mathscr {Z}}^{(\omega ^{\alpha })}_{\alpha }(z)}{{\mathscr {W}}^{(\omega ^{\alpha })}_{\alpha }(z)}. \end{aligned} \end{aligned}$$

  The optimal boundary $u^*$ is determined by the smooth fit condition

  $$\begin{aligned} (V^{\omega }_{\text {A}^{\text {Put}}})'(u^*) = -1. \end{aligned}$$

From (21) (see also [33, Prop. 5.6]) we simply note that the Laplace exponent of $X_t$ taken under ${\mathbb {P}}^{(\alpha )}$ is of the same form as (32), i.e.

$$\begin{aligned} \psi ^{(\alpha )}(\theta )=\zeta ^{(\alpha )}\theta +\frac{{\sigma ^{(\alpha )}}^2}{2}\theta ^2 - \frac{\lambda ^{(\alpha )}\theta }{\varphi ^{(\alpha )}+\theta }, \end{aligned}$$

(37)

where $\zeta ^{(\alpha )} = \zeta + \sigma ^2\alpha$, ${\sigma ^{(\alpha )}} = \sigma$, $\lambda ^{(\alpha )} = \frac{\lambda \varphi }{\varphi +\alpha }$ and $\varphi ^{(\alpha )} = \varphi + \alpha$. Hence, finding the scale functions under ${\mathbb {P}}$ and ${\mathbb {P}}^{(\alpha )}$ works in the same manner. To do so, we recall that in (16) and (17) we introduced them via regular $\xi$-scale functions, that is ${\mathscr {W}}^{(\xi )}(s) = {\mathcal {W}}^{(\xi \circ \textrm{exp})}(x)$ and ${\mathscr {Z}}^{(\xi )}(s) = {\mathcal {Z}}^{(\xi \circ \textrm{exp})}(x)$ for $x=\log s$. Therefore, to identify a closed form of (33) and (36) it suffices to find $\xi$-scale functions ${\mathcal {W}}^{(\xi )}(x)$ and ${\mathcal {Z}}^{(\xi )}(x)$ for a given generic function $\xi$. We recall that both $\xi$-scale functions are given as the solutions of renewal equations (12) and (13) formulated in terms of the classical scale function W(x). From the definition of the first scale function given in (11) with $q=0$ and from (32) with $\sigma >0$ we derive

$$\begin{aligned} W(x)=\sum _{i=1}^3 \Upsilon _ie^{\gamma _i x}, \end{aligned}$$

where $\gamma _i$ solves $\psi (\gamma _i)=0$ and $\Upsilon _i:=\frac{1}{\varphi ^\prime (\gamma _i)}$. In turn, if $\sigma =0$ in (32) then

$$\begin{aligned} W(x)=\sum _{i=1}^2 \Upsilon _i e^{\gamma _i x} \end{aligned}$$

with $\gamma _1 = 0$, $\gamma _2 = \frac{\lambda - \varphi \mu }{\mu }$, $\Upsilon _1 = -\frac{\varphi }{\lambda -\varphi \mu }$ and $\Upsilon _2 = \frac{\lambda }{\mu (\lambda -\varphi \mu )}$. Next theorem provides the ordinary differential equations whose solutions are ${\mathcal {W}}^{(\xi )}(x)$ and ${\mathcal {Z}}^{(\xi )}(x)$. We use this result in the next section where we provide specific examples.

Theorem 8

We assume that the function $\xi$ is continuously differentiable. For geometric Lévy model (6) with $X_t$ given in (31) we have

• (i) If $\sigma =0$ then the function ${\mathcal {W}}^{(\xi )}(x)$ solves

  $$\begin{aligned} \begin{aligned} {{\mathcal {W}}^{(\xi )}}''(x)&= \left( (\Upsilon _1+\Upsilon _2)\xi (x)+\gamma _2\right) {{\mathcal {W}}^{(\xi )}}'(x) \\ {}&\quad + \left( (\Upsilon _1+\Upsilon _2)\xi '(x)-\gamma _2\Upsilon _1\xi (x)\right) {{\mathcal {W}}^{(\xi )}}(x) \end{aligned} \end{aligned}$$

  (38)

  with

  $$\begin{aligned} {\left\{ \begin{array}{ll} {{\mathcal {W}}^{(\xi )}}(0) = \Upsilon _1 + \Upsilon _2, \\ {{\mathcal {W}}^{(\xi )}}'(0) = (\Upsilon _1+\Upsilon _2)^2\xi (0) + \Upsilon _2\gamma _2. \end{array}\right. } \end{aligned}$$

  (39)

  Moreover, the function ${\mathcal {Z}}^{(\xi )}(x)$ solves the same equation (38) with

  $$\begin{aligned} {\left\{ \begin{array}{ll} {{\mathcal {Z}}^{(\xi )}}(0) = 1, \\ {{\mathcal {Z}}^{(\xi )}}'(0) = (\Upsilon _1+\Upsilon _2)\xi (0). \end{array}\right. } \end{aligned}$$

  (40)

• (ii) If $\sigma >0$ then the function ${\mathcal {W}}^{(\xi )}(x)$ solves

  $$\begin{aligned} \begin{aligned} {{\mathcal {W}}^{(\xi )}}'''(x)&= \left( \gamma _2+\gamma _3\right) {{\mathcal {W}}^{(\xi )}}''(x) \\ {}&\quad + \left( \Upsilon _2(\gamma _2-\gamma _3)\xi (x)-\gamma _2\gamma _3-\gamma _3\Upsilon _1\xi (x)\right) {{\mathcal {W}}^{(\xi )}}'(x) \\ {}&\quad + \left( \Upsilon _2(\gamma _2-\gamma _3)\xi '(x)+\gamma _2\gamma _3 \Upsilon _1 \xi (x)-\gamma _3 \Upsilon _1\xi '(x)\right) {{\mathcal {W}}^{(\xi )}}(x) \end{aligned} \end{aligned}$$

  (41)

  with

  $$\begin{aligned} {\left\{ \begin{array}{ll} {{\mathcal {W}}^{(\xi )}}(0) = 0, \\ {{\mathcal {W}}^{(\xi )}}'(0) = \Upsilon _2\gamma _2 + \Upsilon _3\gamma _3, \\ {{\mathcal {W}}^{(\xi )}}''(0) = \Upsilon _2 {\gamma _2}^2 + \Upsilon _3{\gamma _3}^2. \end{array}\right. } \end{aligned}$$

  Moreover, the function ${\mathcal {Z}}^{(\xi )}(x)$ solves the same equation (41) with

  $$\begin{aligned} {\left\{ \begin{array}{ll} {{\mathcal {Z}}^{(\xi )}}(0) = 1, \\ {{\mathcal {Z}}^{(\xi )}}'(0) = 0, \\ {{\mathcal {Z}}^{(\xi )}}''(0) = \Upsilon _2(\gamma _2-\gamma _3) \xi (0) - \gamma _3 \Upsilon _1 \xi (0). \end{array}\right. } \end{aligned}$$

Remark 9

Note that in our case $l^*=0$ and from Theorem 8 it follows that assumption (23) is not required.

3 Examples

In this section we present the analytical form of value function (10) for the particular $\omega$ and for the Black-Scholes model and Black-Scholes model with zero volatility and downward exponential jumps. In the first scenario, we take into account only the case of negative $\omega$, while in the second example we focus on the positive $\omega$.

3.1 Black-Scholes Model Revisited

Let

$$\begin{aligned} \omega (s)= -\frac{C}{s+1} - D, \end{aligned}$$

(42)

where C and D are some positive constants. This is the case of the negative effecting discounting which is allowed in our considerations as we already mentioned in Introduction. This type of discount rate arises for example in the case of stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor; see [17] for more details and examples. Discount function (42) has an additional feature of tending to different constants for small and large values of the asset prices. Applying Theorem 6, we obtain where h is a solution to

$$\begin{aligned} \frac{\sigma ^2 s^2}{2}h^{\prime \prime }(s) + \mu s h^\prime (s) - \left( -\frac{C}{s+1} - D\right) h(s) = 0 \end{aligned}$$

(43)

which satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} h(s) = g(s), \quad s \in [l^{*}, u^{*}],\\ \displaystyle \lim _{s\rightarrow \infty }h(s) = \text {const}.\\ \end{array}\right. } \end{aligned}$$

(44)

Firstly, we solve the above equation and then we look for boundaries $l^{*}$ and $u^{*}$ such that $v^{\omega }_{\text {A}^{\text {Put}}}(s, l^{*}, u^{*}) = \displaystyle \sup _{0\le l\le u\le K} v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)$. We find them using the smooth fit conditions. The general solution to (43) is given by

$$\begin{aligned} \begin{aligned} h(s) = K_1 s^{d_1}{}_2F_1(a_1, b_1, c_1; -s) + K_2 s^{d_2} {}_2F_1(a_2, b_2, c_2; -s), \end{aligned} \end{aligned}$$

(45)

where ${_2F_1}(\cdot , \cdot , \cdot ; \cdot )$ is the Gaussian hypergeometric function, $L:= \frac{1}{2} - \frac{\mu }{\sigma ^2}$, $M:= \sqrt{L^2 - \frac{2D}{\sigma ^2}}$, $G:= \sqrt{L^2 - \frac{2(C+D)}{\sigma ^2}}$, while $a_i:= (-1)^{i+1}(M-G)$, $b_i:= (-1)^i(M+G)$, $c_i:= 1+2(-1)^i G$, $d_i:= (-1)^iG+L$ for $i = 1, 2$ and $K_1$, $K_2$ are some constants.

Using formula (45) and the boundary conditions given in (44) we can identify the form of value function (10). Since we consider the negative $\omega$ we obtain a double continuation region. We take one of the summand from (45) for $s\in (0, l^{*})$ and the second one for $s\in (u^{*}, +\infty )$. This choice is made in a such a way that on the given interval we impose to have a greater function of these two. Hence we derive

$$\begin{aligned} V^{\omega }_{\text {A}^{\text {Put}}}(s) = {\left\{ \begin{array}{ll} K_2 s^{d_2} {}_2F_1(a_2, b_2, c_2; -s), &{}s\in (0, l^{*}), \\ K-s, &{}s\in [l^{*}, u^{*}], \\ K_1 s^{d_1} {}_2F_1(a_1, b_1, c_1; -s), &{}s\in (u^{*}, +\infty ). \end{array}\right. } \end{aligned}$$

Using the smooth and continuous fit properties we can find $K_1$ and $K_2$ and show that $l^{*}$ and $u^{*}$ solve the following equation

$$\begin{aligned} 1 + {}_2F_1(a_i, b_i, c_i; -s) K_i D_i + s^{d_i} P_i = 0, \end{aligned}$$

(46)

where

$$\begin{aligned} K_i&:= (K-s) \frac{s^{-d_i}}{{}_2F_1(a_i, b_i, c_i; -s)}, \\ D_i&:= d_i s^{d_i - 1}, \\ P_i&:= -\frac{a_i b_i {}_2F_1(a_i+1, b_i+1, c_i+1; -s)}{c_i} \end{aligned}$$

for $i=1, 2$. We numerically calculate the roots of (46) for $i=1, 2$ and we assign the smaller result to $l^{*}$ and the greater one to $u^{*}$.

Let us assume the given set of parameters $C = 0.001$, $D=0.01$, $K=20$, $\mu =5\%$ and $\sigma =20\%$. The above numerical procedure produces in this case $l^{*}\approx 7.23$ and $u^{*}\approx 8.34$. Figure 1 presents the value function that then arises.

Fig. 1

The value and payoff functions for the given set of parameters: $C = 0.001$, $D=0.01$, $K=20$, $\mu =5\%$ and $\sigma =20\%$

Remark 10

Let us note that $\lim _{s\rightarrow 0^{+}} V^{\omega }_{\text {A}^{\text {Put}}}(s) = \infty$ which means that the price of the option is unlimited even for an arbitrarily low stock price. This is obviously a consequence of the fact that for $s\rightarrow 0^{+}$ the discount function is strictly negative.

3.2 Exponential Crashes Market Revisited

Let

$$\begin{aligned} \omega (s)= Cs, \end{aligned}$$

where C is some positive constant. Note that this discount function is nonegative. Hence from Theorem 7 it follows that $l^*=0$. Let

$$\begin{aligned}&\eta (x) := \omega (e^x) = \omega (s) \quad \text {and}\quad \eta _u(x):=\eta (x+\log u). \end{aligned}$$

(47)

From Theorem 7 (see equation (33)) with $\sigma = 0$ and using (16) and (17) we can conclude that

$$\begin{aligned} V^{\omega }_{\text {A}^{\text {Put}}}(s)=\sup _{u>0} v^{\omega }_{\text {A}^{\text {Put}}}(s, 0, u), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, 0, u)&= \left( K - \frac{u\varphi }{\varphi +1}\right) \biggl ({\mathcal {Z}}^{(\eta _u)}\left( x-\log u\right) \\ {}&\quad - c_{{\mathcal {Z}}^{(\eta )}/{\mathcal {W}}^{(\eta )}} {\mathcal {W}}^{(\eta _u)}\left( x-\log u\right) \biggr ) \end{aligned} \end{aligned}$$

(48)

and from (34)

$$\begin{aligned} \begin{aligned} c_{{\mathscr {Z}}^{(\omega )}/{\mathscr {W}}^{(\omega )}}&= c_{{\mathcal {Z}}^{(\eta )}/{\mathcal {W}}^{(\eta )}}:=\lim _{z\rightarrow \infty } \frac{{\mathcal {Z}}^{(\eta )}(z)}{{\mathcal {W}}^{(\eta )}(z)}. \end{aligned} \end{aligned}$$

From Theorem 8 it follows that ${{\mathcal {W}}^{(\eta )}}(x)$ solves the following ordinary differential equation

$$\begin{aligned} {{\mathcal {W}}^{(\eta )}}''(x) = (Ae^x+B){{\mathcal {W}}^{(\eta )}}'(x) + D e^x {\mathcal {W}}^{(\eta )}(x), \end{aligned}$$

(49)

with $A:= \frac{C}{\mu }$, $B:= \frac{\lambda -\varphi \mu }{\mu }$ and $D:= C\frac{1 + \varphi }{\mu }$. The above equation is also satisfied by ${{\mathcal {Z}}^{(\eta )}}(x)$.

From (39) and (40) we conclude that

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\mathcal {W}}^{(\eta )}}(0) = \frac{1}{\mu }, \\ {{\mathcal {W}}^{(\eta )}}'(0) = \frac{C+\lambda }{\mu ^2}. \end{array}\right. } \end{aligned}$$

(50)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\mathcal {Z}}^{(\eta )}}(0) = 1, \\ {{\mathcal {Z}}^{(\eta )}}'(0) = \frac{C}{\mu }. \end{array}\right. } \end{aligned}$$

(51)

Using substitutions $t = Ae^x$ and $F(t) = {{\mathcal {W}}^{(\eta )}}(x)$ and applying them to (49) we obtain the Kummer’s equation of the form

$$\begin{aligned} t F''(t) + (b - t) F'(t) - a F(t) = 0, \end{aligned}$$

(52)

where $b = 1 - B$ and $a = \frac{D}{A}$.

If b is not an integer, then the general solution to (52) has the form

$$\begin{aligned} f(x) = K_1^W {_1 F_1(a_1, b_1; t)} + K_2^W t^{1-b} {_1 F_1(a_2, b_2; t)}, \end{aligned}$$

where $_1 F_1(\cdot , \cdot ; \cdot )$ is the Kummer confluent function, $a_1 = a$, $b_1 = b$, $a_2 = a-b+1$ and $b_2 = 2-b$, while $K_1^W$ and $K_2^W$ are the constants that can be found based on the initial conditions given in (50). The same expression can be found for ${{\mathcal {Z}}^{(\eta )}}(x)$ with constants $K_1^Z$ and $K_2^Z$ satisfying (51). Having the both forms of ${{\mathcal {W}}^{(\eta )}}(x)$ and ${{\mathcal {Z}}^{(\eta )}}(x)$ and by shifting these scale functions by $\log u$ we can produce figures of ${\mathcal {W}}^{(\eta _u)}(x - \log u)$ and ${\mathcal {Z}}^{(\eta _u)}(x - \log u)$.

The asymptotic behaviour of ${_1F_1}\left( a, b; t\right)$ for $t\rightarrow \infty$ is as follows

$$\begin{aligned} {_1 F_1(a, b; t)} = \frac{\Gamma (b)}{\Gamma (a)} e^t t^{a-b}\left[ 1+O\left( \frac{1}{t}\right) \right] . \end{aligned}$$

(53)

Based on (53) we calculate the constant $c_{{\mathcal {Z}}^{(\eta )}/{\mathcal {W}}^{(\eta )}}$. It has the following form

$$\begin{aligned} c_{{\mathcal {Z}}^{(\eta )}/{\mathcal {W}}^{(\eta )}} = \frac{K_1^Z \frac{\Gamma (b_1)}{\Gamma (a_1)}A^{a_1-b_1} + K_2^Z \frac{\Gamma (b_2)}{\Gamma (a_2)} A^{a_2-b_2}}{K_1^W \frac{\Gamma (b_1)}{\Gamma (a_1)}A^{a_1-b_1} + K_2^W \frac{\Gamma (b_2)}{\Gamma (a_2)} A^{a_2-b_2}}. \end{aligned}$$

Finally, using the continuous fit condition (35) we can find the optimal $u^{*}$.

Let us assume that $C = 0.1$, $K = 20$, $r = 5\%$, $\lambda = 6$, $\varphi = 2$. The continuous fit property produces $u^{*}\approx 4.56$. Figure 2 shows the obtained value function.

Fig. 2

The value and payoff functions for the given set of parameters: $C = 0.1$, $K = 20$, $r = 5\%$, $\lambda = 6$, $\varphi = 2$

4 Proofs

All the details of the proofs given in this section are available at the arxiv version.

Before we prove main Theorem 1 we show the convexity of European option price $V^{\omega }_{\text {E}}(s,t)$ defined in (2) as a function of s. It is done in Theorems 9 and 10. Later, in the proof of Theorem 1, we use a variant of the maximum principle.

Let us introduce a set $E\subset {\mathbb {R}}\times [0,T]$. We use the following notations

• $C_{\alpha }(E)$ is the set of locally Hölder($\alpha$) functions with $\alpha \in (0, 1)$,

• $C_{\text {pol}}(E)$ is the set of functions of at most polynomial growth in s,

• $C^{p, q}(E)$ is the set of functions for which all the derivatives $\frac{\partial ^k}{\partial s^k}\left( \frac{\partial ^l f(s,t)}{\partial t^l}\right)$ with $k +2\,l\le p$ and $0\le l\le q$ exist in the interior of E and have continuous extensions to E,

• $C^{p, q}_{\alpha }(E)$ and $C^{p, q}_{\text {pol}}(E)$ are the sets of functions $f\in C^{p, q}(E)$ for which all the derivatives $\frac{\partial ^k}{\partial s^k}\left( \frac{\partial ^l f(s,t)}{\partial t^l}\right)$ with $k+2\,l\le p$ and $0\le l\le q$ belong to $C_{\text {pol}}(E)$ and $C_{\alpha }(E)$, respectively. Similarly, $C^{p}_{\alpha }(E)$ is the sets of functions $f\in C^{p}(E)$ for which all the derivatives $\frac{\partial ^k f(s,t)}{\partial s^k}$ with $k\le p$ belong to $C_{\alpha }(E)$.

We need the following conditions in the proofs.

Assumptions (B)

There exist constants $C > 0$ and $\alpha \in (0, 1)$ such that

• (B1) $\mu (s,t)\in C^{2,1}_{\alpha }({\mathbb {R}}^{+}\times [0, T])$;

• (B2) $\sigma ^2(s,t)\ge Cs^2$ for all $(s,t)\in {\mathbb {R}}^{+}\times [0,T]$;

• (B3) $\sigma (s,t)\in C^{2,1}_{\alpha }({\mathbb {R}}^{+}\times [0,T])$;

• (B4) $\gamma (s,t,z)\in C^{2,1}_{\alpha }({\mathbb {R}}^{+}\times [0,T])$ with the Hölder continuity being uniform in z;

• (B5) $|\omega (s)|\le C$ for all $s\in {\mathbb {R}}^{+}$;

• (B6) $\omega (s)\in C^{2}_{\alpha }({\mathbb {R}}^{+})$;

• (B7) g(s) is Lipschitz continuous;

• (B8) $g(s)\in C^4_{\alpha }({\mathbb {R}}^{+})$.

Assumptions (C)

There exist a constant $C > 0$ such that

• (C1) $|\frac{\partial \mu (s,t)}{\partial t}|\le Cs$, $|\frac{\partial ^2\mu (s,t)}{\partial s^2}|\le \frac{C}{s}$ for all $(s,t)\in {\mathbb {R}}^{+}\times [0,T]$;

• (C2) $|\frac{\partial \sigma (s,t)}{\partial t}|\le Cs$, $|\frac{\partial ^2\sigma (s,t)}{\partial s^2}|\le \frac{C}{s}$ for all $(s,t)\in {\mathbb {R}}^{+}\times [0,T]$;

• (C3) $|\frac{\partial \gamma (s,t,z)}{\partial t}|\le Cs$, $|\frac{\partial ^2\gamma (s,t,z)}{\partial s^2}|\le \frac{C}{s}$ for all $(s,t,z)\in {\mathbb {R}}^{+}\times [0,T]\times {\mathbb {R}}$;

• (C4) $|\frac{d\omega (s)}{ds}|\le \frac{C}{s}$, $|\frac{d^2\omega (s)}{ds^2}|\le \frac{C}{s^2}$ for all $s\in {\mathbb {R}}^{+}$;

• (C5) $g(s)\in C^3_{\text {pol}}({\mathbb {R}}^{+})$.

Theorem 9

Let all assumptions of Theorem 1 be satisfied. We assume additionally that conditions (B) and (C) hold true. Then $V^{\omega }_{\text {E}}(s, t)$ is convex with respect to s at all times $t\in [0,T]$.

Proof

The first part of the proof proceeds in a similar way as the proof of [22, Prop. 4.1].

Let

$$\begin{aligned} {\mathcal {L}}V^{\omega }_{\text {E}}(s, t) = -\frac{\partial V^{\omega }_{\text {E}}(s, t)}{\partial t} - A_t^C V^{\omega }_{\text {E}}(s, t) - A^J_t V^{\omega }_{\text {E}}(s, t) +\omega (s)V^{\omega }_{\text {E}}(s, t), \end{aligned}$$

where $A_t$ is the linear second-order differential operator of the form

$$\begin{aligned} A_t^C V^{\omega }_{\text {E}}(s, t) = \beta (s, t) \frac{\partial ^2 V^{\omega }_{\text {E}}(s, t)}{\partial s^2} + \mu (s,t) \frac{\partial V^{\omega }_{\text {E}}(s, t)}{\partial s} \end{aligned}$$

with $\beta (s, t) = \frac{\sigma ^2(s, t)}{2}$ and $A^J_t$ is the integro-differential operator given by

$$\begin{aligned} A^J_t V^{\omega }_{\text {E}}(s, t) = \int _{{\mathbb {R}}} \left( V^{\omega }_{\text {E}}(s+\gamma (s,t,z), t) - V^{\omega }_{\text {E}}(s,t) - \gamma (s,t,z)\frac{\partial V^{\omega }_{\text {E}}(s,t)}{\partial s}\right) m(dz). \end{aligned}$$

Lemma 1

Let Assumptions (A) and (B) hold and assume that the stock price process $S_t$ follows (5). Then $V^{\omega }_{\text {E}}(s,t)\in C^{4,1}_{\alpha }({\mathbb {R}}^{+}\times [0,T])\cap C_{\text {pol}}({\mathbb {R}}^{+}\times [0,T])$ and it is the solution to the Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}} V^{\omega }_{\text {E}}(s, t) = 0, \quad &{} (s, t)\in {\mathbb {R}}^{+}\times [0,T), \\ V^{\omega }_{\text {E}}(s, T) = g(s), \quad &{} s\in {\mathbb {R}}^{+}. \end{array}\right. } \end{aligned}$$

(54)

Lemma 2

Let Assumptions (A), (B) and (C) hold and assume that the stock price process $S_t$ follows (5). Then there exist constants $n>0$ and $K>0$ such that the value function $V^{\omega }_{\text {E}}(s, t)$ satisfies

$$\begin{aligned} |\frac{\partial ^2 V^{\omega }_{\text {E}}(s, t)}{\partial s^2}|\le K(s^{-n} + s^{n}) \end{aligned}$$

for all $(s, t) \in {\mathbb {R}}^{+}\times [0,T]$.

Proofs of both above lemmas are given in Appendix. We introduce the function $u^{\omega }:{\mathbb {R}}^{+}\times [0,T]\rightarrow {\mathbb {R}}^{+}$ of the form

$$\begin{aligned} u^{\omega }(s,t):= V^{\omega }_{\text {E}}(s, T-t) \end{aligned}$$

and we prove convexity of $u^{\omega }(s,t)$ with respect to s. Note that it is equivalent to the convexity of the value function $V^{\omega }_{\text {E}}(s, t)$ in s. Furthermore, based on Lemma 1, the function $u^{\omega }(s,t)$ solves the Cauchy problem of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u^{\omega }(s,t)}{\partial t}= \hat{{\mathcal {L}}} u^{\omega }(s,t), \quad &{} (s, t)\in {\mathbb {R}}^{+}\times (0,T], \\ u^{\omega }(s, 0) = g(s), \quad &{} s\in {\mathbb {R}}^{+}, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \hat{{\mathcal {L}}} u^{\omega }(s,t)&= \beta (s, t) \frac{\partial ^2 u^{\omega }(s,t)}{\partial s^2} + \mu (s,t) \frac{\partial u^{\omega }(s,t)}{\partial s} - \omega (s)u^{\omega }(s,t) \\ {}&+ \int _{{\mathbb {R}}} \left( u^{\omega }(s+\gamma (s,t,z), t) - u^{\omega }(s,t) - \gamma (s,t,z)\frac{\partial u^{\omega }(s,t)}{\partial s}\right) m(dz) \end{aligned}$$

with $\beta (s, t) = \frac{\sigma ^2(s, t)}{2}$. Observe that by Lemma 2 there exist constants $n>0$ and $K>0$ such that

$$\begin{aligned} \Big |\frac{\partial ^2 u^{\omega }(s, t)}{\partial s^2}\Big |\le K (s^{-n} + s^n) \end{aligned}$$

(55)

for all $(s, t)\in {\mathbb {R}}^{+}\times [0,T]$.

Let us now define a convex function $\kappa : {\mathbb {R}}^{+} \rightarrow {\mathbb {R}}^{+}$ of the form

$$\begin{aligned} \kappa (s):= s^{n+3} + s^{-n+1} \end{aligned}$$

The assumptions that we put on the coefficients $\mu$, $\sigma$ and $\gamma$ and function $\omega$ and their derivatives imply that each component of $\frac{d^2(\hat{{\mathcal {L}}}\kappa (s))}{ds^2}$ grows at most like $s^{n+1}$ for large s and like $s^{-n-1}$ for small s. The same behaviour characterises $\frac{d^2\kappa (s)}{ds^2}$. In addition, we define the function $\vartheta :{\mathbb {R}}^{+}\times [0,T]\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} \vartheta (s, t):= \left( \frac{\partial ^2\mu (s, t)}{\partial s^2} - 2 \frac{d\omega (s)}{ds}\right) \frac{d\kappa (s)}{ds} - \frac{d^2\omega (s)}{ds^2}\kappa (s) \end{aligned}$$

which also behaves like $\frac{d^2(\hat{{\mathcal {L}}}\kappa (s))}{ds^2}$ at 0 and $+\infty$. Hence we claim that there exist a positive constant C such that

$$\begin{aligned} C\frac{d^2\kappa (s)}{ds^2} - \frac{d^2(\hat{{\mathcal {L}}}\kappa (s))}{ds^2} > -\vartheta (s, t). \end{aligned}$$

(56)

In the second part of the proof, we define the auxiliary function

$$\begin{aligned} u^{\omega }_{\varepsilon }(s, t):= u^{\omega }(s, t) + \varepsilon e^{Ct} \kappa (s) \end{aligned}$$

(57)

for some $\varepsilon >0$.

We carry out a proof by contradiction. Let us then assume that $u^{\omega }_{\varepsilon }(s,t)$ is not convex. For this purpose, we denote by $\Lambda$ the set of points for which $u^{\omega }_{\varepsilon }(s, t)$ is not convex, i.e.

$$\begin{aligned} \Lambda := \{(s, t)\in {\mathbb {R}}^{+}\times [0,T]:\frac{\partial ^2 u^{\omega }_{\varepsilon }(s, t)}{\partial s^2}<0\} \end{aligned}$$

and we assume that the set $\Lambda$ is not empty. From Lemma 2 we know that $u^{\omega }(s, t)$ satisfies (55). Due to this fact and using (57) we claim that there exist a positive constant R such that $\Lambda \subseteq [R^{-1}, R] \times [0,T]$. This is a direct consequence of such a choice of $u^{\omega }_{\varepsilon }(s, t)$ in (57) so that $\frac{d^2 \kappa (s)}{ds^2}$ grows faster than $\frac{\partial ^2 u^{\omega }(s, t)}{\partial s^2}$ for both large and small values of s. Consequently, the set $\Lambda$ is a bounded set. Since the closure of a bounded set is also bounded, we conclude that the closure of $\Lambda$, i.e. $\text {cl}(\Lambda )$, is compact. Due to the fact that a compact set always contains its infimum we can define

$$\begin{aligned} t_0:= \inf \{t\ge 0: (s, t) \in \text {cl}(\Lambda )\text { for some } s \in \mathbb {R^{+}}\}. \end{aligned}$$

From the initial condition, i.e. $u^{\omega }(s, 0) = g(s)$ and convexity of g we have

$$\begin{aligned} \frac{d^2 u^{\omega }_{\varepsilon }(s, 0)}{d s^2} = \frac{d^2 (g(s)+\varepsilon \kappa (s))}{d s^2}\ge \varepsilon \frac{d^2 \kappa (s)}{ds^2}>0 \end{aligned}$$

for all $s\in {\mathbb {R}}^{+}$. Hence we can conclude that $t_0>0$. Moreover, at the point when the infimum is attained, i.e. $(s_0, t_0)$ for some $s_0\in {\mathbb {R}}^{+}$

$$\begin{aligned} \frac{\partial ^2 u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial s^2} = 0. \end{aligned}$$

This is a consequence of the continuity of the function $\frac{\partial ^2 u^{\omega }_{\varepsilon }(s, t)}{\partial s^2}$ in s. In addition, for $t\in [0, t_0)$ we have $\frac{\partial ^2 u^{\omega }_{\varepsilon }(s_0, t)}{\partial s^2}> 0$ and thus, by applying the symmetry of second derivatives at $t= t_0$, we derive

$$\begin{aligned} \frac{\partial ^2}{\partial s^2} \left( \frac{\partial u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial t}\right) = \frac{\partial }{\partial t}\left( \frac{\partial ^2 u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial s^2}\right) \le 0. \end{aligned}$$

(58)

Since $\frac{\partial ^2 u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial s^2} = 0$ and $\frac{\partial ^2 u^{\omega }_{\varepsilon }(s, t_0)}{\partial s^2}$ has a local minimum at $s=s_0$, we have $\frac{\partial ^3 u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial s^3} = 0$ and $\frac{\partial ^4 u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial s^4} \ge 0$. Thus

$$\begin{aligned} \begin{aligned} \frac{\partial ^2(\hat{{\mathcal {L}}}u^{\omega }_{\varepsilon }(s_0, t_0))}{\partial s^2}&\ge \frac{\partial ^2 \mu (s_0, t_0)}{\partial s^2} \frac{\partial u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial s} - \frac{d^2\omega (s_0)}{ds^2}u^{\omega }_{\varepsilon }(s_0, t_0) \\&\quad - 2\frac{d\omega (s_0)}{ds}\frac{\partial u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial s} \\ {}&\quad + \int _{{\mathbb {R}}} \Bigg (\frac{\partial u^{\omega }_{\varepsilon }(s_0+\gamma (s_0,t_0,z), t_0)}{\partial s}\frac{\partial ^2 \gamma (s_0,t_0,z)}{\partial s^2} \\ {}&\quad - \frac{\partial u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial s} \frac{\partial ^2\gamma (s_0,t_0,z)}{\partial s^2}\Bigg )m(dz). \end{aligned} \end{aligned}$$

Since $u^{\omega }_{\varepsilon }(s, t_0)$ is convex in s and $\frac{\partial ^2 u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial s^2} = 0$, applying (8) we can conclude that the integral part of the above expression is nonnegative. Moreover, (9) imply that

$$\begin{aligned} \frac{\partial ^2(\hat{{\mathcal {L}}}u^{\omega }_{\varepsilon }(s_0, t_0))}{\partial s^2} \ge \varepsilon e^{C t_0} \vartheta (s_0, t_0). \end{aligned}$$

(59)

Combining (56) with (58) and (59) at $(s_0, t_0)$ we derive that

$$\begin{aligned} \begin{aligned}&\frac{\partial ^2}{\partial s^2}\left( \frac{\partial u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial t} - \hat{{\mathcal {L}}}u^{\omega }_{\varepsilon }(s_0, t_0)\right) = \varepsilon e^{Ct_0}\frac{d^2}{d s^2}(C\kappa (s_0) - \hat{{\mathcal {L}}}\kappa (s_0)) \\&\qquad > -\varepsilon e^{Ct_0}\vartheta (s_0, t_0) \ge \frac{\partial ^2}{\partial s^2}\left( \frac{\partial u^{\omega }_{\varepsilon }(s_0, t_0)}{\partial t} - \hat{{\mathcal {L}}}u^{\omega }_{\varepsilon }(s_0, t_0)\right) \end{aligned} \end{aligned}$$

which is a contradiction. This confirms that the set $\Lambda$ is empty, and thus $u^{\omega }_{\varepsilon }(s, t)$ is a convex function. Finally, letting $\varepsilon \rightarrow 0$ we conclude that $u^{\omega }(s, t)$ is convex in s for all $t\in [0,T]$. $\square$

Using the same arguments like in the proof of [22, Thm. 4.1], we can resign from Assumptions (B) and (C) in Theorem 9, that is, the following theorem holds true.

Theorem 10

Let assumptions of Theorem 1 hold true. Then $V^{\omega }_{\text {E}}(s, t)$ is convex with respect to s at all times $t\in [0,T]$.

Proof of Theorem 1

As noted in [22, Sec. 7], under conditions (A1)–(A4), for each $p\ge 1$ there exists a constant C such that the stock price process given in (5) satisfies

$$\begin{aligned} {\mathbb {E}}_s\left[ \sup _{0\le t\le T} |S_t|^p\right] \le C(1+s^p). \end{aligned}$$

Together with (A5) and (A6) it implies that the value function given by

$$\begin{aligned} V^{\omega }_{\text {A}_T}(s, t):= \sup _{\tau \in {\mathcal {T}}^T_t} {\mathbb {E}}_{s, t}[e^{-\int _t^{\tau } \omega (S_w) dw} g(S_\tau )] \end{aligned}$$

is well-defined, where ${\mathcal {T}}^T_t$ is the family of ${\mathcal {F}}_t$-stopping times with values in [t, T] for fixed maturity $T>0$. Moreover, we denote

$$\begin{aligned} V^{\omega }_{\text {A}_T}(s):= V^{\omega }_{\text {A}_T}(s, 0). \end{aligned}$$

Let us define now a Bermudan option with the value function of the form

$$\begin{aligned} V^{\omega }_{\text {B}_\Xi }(s, t):= \sup _{\tau \in {\mathcal {T}}_{\Xi }} {\mathbb {E}}_{s, t}[e^{-\int _t^\tau \omega (S_w) dw} g(S_\tau )], \end{aligned}$$

where ${\mathcal {T}}_{\Xi }$ is the set of stopping times with values in $\text {B}_\Xi = \left\{ \frac{n}{2^\Xi }(T-t)+t: n = 0, 1, ..., 2^\Xi \right\} ,$ where $\Xi$ is some positive integer number. To simplify the notation, we denote

$$\begin{aligned} V^{\omega }_{\text {B}_\Xi }(s):= V^{\omega }_{\text {B}_\Xi }(s, 0). \end{aligned}$$

In the next step we use the dynamic programming principle as formulated in [21]. In fact, the dynamic programming principle holds in our more general than in [21] set-up since bivariate process $(S_t, \int _0^t \omega (S_w)dw)$ seen at epochs from $\text {B}_\Xi$ is a discrete-time Markov process and decision space is compact (we have binary decisions: either to stop or to continue); see e.g. [36, Chap. 4.5] and [34, Thm. 1.7, p. 14]. Following again [21], we can conclude that $V^{\omega }_{\text {B}_\Xi }(s, t)$ inherits the property of convexity from its European equivalent $V^{\omega }_{\text {E}}(s, t)$. Next, we generalise this result to the American case $V^{\omega }_{\text {A}}(s)$. As the possible exercise times of the Bermudan option get denser, the value function $V^{\omega }_{\text {B}_\Xi }(s, t)$ converges to $V^{\omega }_{\text {A}_T}(s, t)$. To receive our claim we take the maturity T tending to infinity. $\square$

Proof of Theorem 3

Denoting

$$\begin{aligned} \tau ^{+}_a:=\inf \{t> 0: S_t\ge a\}, \qquad \tau ^{-}_a:=\inf \{t> 0: S_t\le a\} \end{aligned}$$

and keeping in mind that $S_t=e^{X_t}$, from [30] we can conclude that

$$\begin{aligned}&{\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau _{a}^{+}} \omega (S_w)\,dw}; \tau _{a}^{+}<\infty \right] =\frac{{\mathscr {H}}^{(\omega )}(s)}{{\mathscr {H}}^{(\omega )}(a)}, \nonumber \\&\quad {\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau _1^{-}} \omega (S_w)\,dw}; \tau _1^{-}<\infty \right] = {\mathscr {Z}}^{(\omega )}(s) - c_{{\mathscr {Z}}^{(\omega )}/{\mathscr {W}}^{(\omega )}} {\mathscr {W}}^{(\omega )}(s), \end{aligned}$$

(60)

where $\omega (s) = \omega (e^x) = \eta (x)$ and the functions ${\mathscr {Z}}^{(\omega )}(s)$, ${\mathscr {W}}^{(\omega )}(s)$, ${\mathscr {H}}^{(\omega )}(s)$ were defined in (16), (17) and (18).

We consider three possible cases of a position of the initial state $S_0=s$ of the process $S_t$.

• 1. $s<l$: As the process $S_t$ is spectrally negative and starts below the interval [l, u], it can enter this interval only in a continuous way and hence $\tau _{l, u} = \tau ^{+}_l$ and $S_{\tau _{l, u}} = l$. Thus from (60)

  $$\begin{aligned} \begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)&= \frac{{\mathscr {H}}^{(\omega )}(s)}{{\mathscr {H}}^{(\omega )}(l)}(K-l). \end{aligned} \end{aligned}$$

• 2. $s\in [l, u]$: If the process $S_t$ starts inside the interval [l, u] which is an optimal stopping region, we decide to exercise our option immediately, i.e. $\tau _{l, u} = 0$. Therefore, we have $v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u) = K-s.$

• 3. $s>u$: There are three possible cases of entering the interval [l, u] by the process $S_t$ when it starts above u: either $S_t$ enters [l, u] continuously going downward or it jumps from $(u, +\infty )$ to (l, u) or $S_t$ jumps from the interval $(u, +\infty )$ to the interval (0, l) and then, later, enters [l, u] continuously.

We can distinguish these cases in the following way

$$\begin{aligned} \begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)&={\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau _{l, u}}\omega (S_w)dw}(K-S_{\tau _{l, u}}); \tau ^{-}_{u}<\tau ^{-}_{l}\right] \\ {}&\quad + {\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau _{l, u}}\omega (S_w)dw} (K-S_{\tau _{l, u}}); \tau ^{-}_{u} = \tau ^{-}_{l}\right] . \end{aligned} \end{aligned}$$

(61)

To analyse the first component in (61), note that

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau _{l, u}}\omega (S_w)dw} (K-S_{\tau _{l, u}}); \tau ^{-}_{u}<\tau ^{-}_{l}\right] \\ {}&\quad = \int _{(0, \log u-\log l)}(K-e^{\log u-y}){\mathbb {E}}\left[ e^{-\int _0^{\sigma ^{-}_0}\eta _u(X_w)dw}; -X_{\sigma ^{-}_0}\in dy\mid X_0=x-\log u\right] \\ {}&\qquad + (K-u) {\mathbb {E}}\left[ e^{-\int _0^{\sigma ^{-}_0}\eta _u(X_w)dw}; X_{\sigma ^{-}_0}=0\mid X_0=x-\log u\right] . \end{aligned}\nonumber \\ \end{aligned}$$

(62)

From the compensation formula for Lévy processes given in [27, Thm. 4.4] we have

$$\begin{aligned} {\mathbb {E}}\left[ e^{-\int _0^{\sigma ^{-}_0}\eta _u(X_w)dw}; -X_{\sigma ^{-}_0}\in dy\mid X_0=\log s -\log u\right] = \int _0^{\infty }r(s,u,z) \Pi (-z-dy)dz \nonumber \\ \end{aligned}$$

(63)

for r(s, u, z) given in (22). To find ${\mathbb {E}}\left[ e^{-\int _0^{\sigma ^{-}_0}\eta _u(X_w)dw}; X_{\sigma ^{-}_0}=0\mid X_0=x-\log u\right]$, we consider ${\mathbb {E}}\left[ e^{-\int _0^{\sigma ^{-}_0}\eta _u(X_w)dw + \alpha X_{\sigma ^{-}_0}}; \sigma _0^{-}<\infty \mid X_0=x-\log u\right]$ for some $\alpha >0$. Note that using the change of measure given in (20) and [30, Cor. 2.1.] we know that

$$\begin{aligned}{} & {} {\mathbb {E}}^{(\alpha )}\left[ e^{-\int _0^{\sigma ^{-}_0}\eta _u^\alpha (X_w)dw}; \sigma _0^{-}<\infty \mid X_0=x-\log u\right] \\ {}{} & {} \qquad = {\mathcal {Z}}^{(\eta _u^\alpha )}_{\alpha }(x-\log u) - c_{{\mathcal {Z}}^{(\eta ^\alpha )}_{\alpha }/{\mathcal {W}}^{(\eta ^\alpha )}_{\alpha }}{\mathcal {W}}^{(\eta _u^\alpha )}_{\alpha }(x-\log u). \end{aligned}$$

Therefore we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\left[ e^{-\int _0^{\sigma ^{-}_0}\eta _u(X_w)dw};X_{\sigma ^{-}_0} = 0\mid X_0=x-\log u\right] \\ {}&\quad = \lim _{\alpha \rightarrow \infty } e^{\alpha (x-\log u)}\left( {\mathcal {Z}}^{(\eta ^\alpha _u)}_{\alpha }(x-\log u) - c_{{\mathcal {Z}}^{(\eta ^\alpha )}_{\alpha }/{\mathcal {W}}^{(\eta ^\alpha )}_{\alpha }}{\mathcal {W}}^{(\eta ^\alpha _u)}_{\alpha }(x-\log u)\right) . \end{aligned} \end{aligned}$$

(64)

Furthermore, the second component of (61) is equal to

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau _{l, u}}\omega (S_w)dw} (K-S_{\tau _{l, u}}); \tau ^{-}_{u} = \tau ^{-}_{l}\right] \\ {}&\quad = {\mathbb {E}}\left[ e^{-\int _0^{\tau _{l, u}}\eta (X_w)dw} (K-e^{X_{\tau _{l, u}}}); \sigma ^{-}_{\log u} = \sigma ^{-}_{\log l}\mid X_0=x\right] \\ {}&\quad = \int _{\log u-\log l}^\infty \frac{{\mathcal {H}}^{(\eta _u)}(\log u-y)}{{\mathcal {H}}^{(\eta _u)}(\log l)}(K-l){\mathbb {E}}\\&\qquad \times \left[ e^{-\int _0^{\sigma ^{-}_0} \eta _u(X_w)dw}; -X_{\sigma ^{-}_0}\in dy\mid X_0=x-\log u\right] . \end{aligned} \end{aligned}$$

(65)

If $l^*=0$ then we can proceed as before except that we do not need identity (60) and hence assumption (23) is indeed superfluous. $\square$

Proof of Theorem 4

From the facts that $V^{\omega }_{\text {A}}(s)\in D({\mathcal {A}})$ and that the Lévy process $X_t$ is right-continuous and left-continuous over stopping times, using classical arguments, we can conclude that $V^{\omega }_{\text {A}}(s)$ solves uniquely equation (24); see [34, Thm. 2.4, p. 37] and [16] for details. We are left with the proof of the smoothness at the boundary of stopping set that can be handled in the same way as it is done in [29] (see also [16] and [17]). $\square$

Proof of Theorem 5

We recall that

$$\begin{aligned} V^{\omega }_{\text {A}^{\text {Call}}}(s, K, \zeta , \sigma , \Pi , l, u)&= {\mathbb {E}}_{s}[e^{-\int _0^{\tau _{l, u}}\omega (S_w)dw} (S_{\tau _{l, u}} - K)^{+}] \\&= {\mathbb {E}}[e^{-\int _0^{\tau _{l, u}}\eta (X_w)dw} (e^{X_{\tau _{l, u}}} - K)^{+}\mid X_0=x], \end{aligned}$$

where $x=\log S_0=\log s$. By our assumption for general Lévy process $X_t$ we can define a new measure ${\mathbb {P}}^{(1)}_1$ via $\frac{d {\mathbb {P}}^{(1)}_1}{d{\mathbb {P}}_1}\Bigg \vert _{{\mathcal {F}}_t} = e^{X_{t} - \psi (1)t}$. Then

$$\begin{aligned}{} & {} {\mathbb {E}}\left[ e^{-\int _0^{\tau _{l, u}}\eta (X_w)dw} (e^{X_{\tau _{l, u}}} - K)^{+}\mid X_0=x\right] \nonumber \\{} & {} \quad = {\mathbb {E}}^{(1)}\left[ e^{-\int _0^{\tau _{\frac{s}{u}K, \frac{s}{l}K}}(\omega (\frac{1}{{\hat{S}}_w} sK) -\psi (1)) dw}(s - e^{{\hat{X}}_{\tau _{\frac{s}{u}K, \frac{s}{l}K}}})^{+}\mid {\hat{X}}_0=\log K\right] , \end{aligned}$$

(66)

where ${\hat{S}}_t = e^{{\hat{X}}_t}$ and ${\hat{X}}_t = -X_t$ is the dual process to $X_t$ and from [17, 24, 33] it follows that under ${\mathbb {P}}^{(1)}$ it is again Lévy process with the triple $(-\zeta , \sigma , {\hat{\Pi }})$ for ${\hat{\Pi }}$ defined in (26). This completes the proof of identity (25). The rest of the proof can be done along classical lines. $\square$

Proof of Theorem 6

We prove that for the function h satisfying (29) we have

$$\begin{aligned} {\mathbb {E}}_s\left[ \frac{h(S_{\tau _{l, u}})}{h(s)} e^{-\int _0^{\tau _{l, u}}\omega (S_w)dw}\right] = 1. \end{aligned}$$

(67)

Since process $S_t$ is continuous in Black-Scholes model, $S_{\tau _{l, u}}$ equals either to l or u depending on the initial state of $S_t$. We can distinguish three possible scenarios

• 1. $s<l$: As the process $S_t$ is a continuous process and starts below the interval [l, u], then $\tau _{l, u} = \tau ^{+}_l$ and $S_{\tau _{l, u}} = l$. Thus, we get

  $$\begin{aligned} \begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)&= {\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau ^{+}_l}\omega (S_w)dw}; S_{\tau ^{+}_l} = l\right] (K-l) \\ {}&= \frac{h(s)}{h(l)}(K-l). \end{aligned} \end{aligned}$$

  (68)

• 2. $s\in [l, u]$: If the process $S_t$ starts inside the interval [l, u] which is the optimal stopping region, we decide to exercise our option immediately, i.e. $\tau _{l, u} = 0$. Therefore, we have

  $$\begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u) = K-s. \end{aligned}$$

  (69)

• 3. $s>u$: Similarly to the case when $s<l$, the process $S_t$ can enter [l, u] only via u and thus $\tau _{l, u} = \tau ^{-}_u$ and $S_{\tau _{l, u}} = u$. Therefore,

  $$\begin{aligned} \begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)&= {\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau ^{-}_u}\omega (S_w)dw}; S_{\tau ^{-}_u} = u\right] (K-u) \\ {}&= \frac{h(s)}{h(u)}(K-u). \end{aligned} \end{aligned}$$

  (70)

Identities (68), (69) and (70) give the first part of the assertion of the theorem. Note that boundary condition (30) follows straightforward from the definition of the value function of the American put option. We are left with the proof of (67). This follows from [33, Prop. 3.2]. $\square$

Proof of Theorem 7

From Theorem 1 and Remark 2 it follows that the optimal exercise time is the first entrance to the interval $[l^*, u^*]$ and by Theorem 2 the value function $V^{\omega }_{\text {A}^{\text {Put}}}(s)$ is equal to the maximum over l and u of $v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)$ defined in (4). We recall the observation that if the discount function $\omega$ is nonnegative, then it is never optimal to wait to exercise option for small asset prices, that is, always $l^*=0$ in this case and the stopping region is one-sided. We find now function $v^{\omega }_{\text {A}^{\text {Put}}}(s, l, u)$ in the case of (i) and (ii).

If $\sigma =0$, by the lack of memory of exponential random variable, using similar analysis like in the proof of Theorem 3, we have

$$\begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, 0, u)&={\mathbb {E}}(K-e^{\log u- Y})^+ {\mathbb {E}}_{s}\left[ e^{-\int _0^{\tau _{u}^-}\omega (S_w)dw}; \tau ^{-}_{u}<\infty \right] \\&= {\mathbb {E}}(K-e^{\log u-Y})^+\left( {\mathscr {Z}}^{(\omega _u)}\left( \frac{s}{u}\right) - c_{{\mathscr {Z}}^{(\omega )}/{\mathscr {W}}^{(\omega )}} {\mathscr {W}}^{(\omega _u)}\left( \frac{s}{u}\right) \right) . \end{aligned}$$

Observing that

$$\begin{aligned} {\mathbb {E}}(K-e^{\log u-Y})^+= K-\frac{u \varphi }{\varphi +1} \end{aligned}$$

completes the proof of part (i). If $\sigma >0$ then

$$\begin{aligned} v^{\omega }_{\text {A}^{\text {Put}}}(s, 0, u)&= {\mathbb {E}}(K-e^{\log u-Y})^+{\mathbb {E}}\\&\quad \ \times \left[ e^{-\int _0^{\sigma _{0}^-}\eta _u(X_w)dw}; \sigma ^{-}_{0}<\infty , X_{\sigma ^{-}_{0}}<0\mid X_0=x-\log u\right] \\&\quad \ +(K-u){\mathbb {E}}\left[ e^{-\int _0^{\sigma _{0}^-}\eta _u(X_w)dw}; \sigma ^{-}_{0}<\infty , X_{\sigma ^{-}_{0}}=0\mid X_0=x-\log u\right] . \end{aligned}$$

The first increment can be analysed like in the case of $\sigma =0$. The expression for the second component follows from (64). Finally, the smooth fit condition follows straightforward from Theorem 4. $\square$

Proof of Theorem 8

Assume first that $\sigma =0$. Then

$$\begin{aligned} W(x)= \Upsilon _1 e^{\gamma _1 x} + \Upsilon _2 e^{\gamma _2 x} \end{aligned}$$

(71)

with $\gamma _1=0$. To produce ordinary differential equation for ${\mathcal {W}}^{(\xi )}(x)$ we start from equation (12). Putting (71) there and taking the derivative of both sides in the next step gives

$$\begin{aligned} {{\mathcal {W}}^{(\xi )}}'(x) = ((\Upsilon _1+\Upsilon _2)\xi (x)+\gamma _2){\mathcal {W}}^{(\xi )}(x) - \gamma _2 \Upsilon _1 - \gamma _2 \Upsilon _1\int _0^{x}\xi (y){\mathcal {W}}^{(\xi )}(y)dy. \end{aligned}$$

We take the derivative of both sides again to get equation (38). From (12) and (71) we derive both initial conditions (39). Similar analysis can be done for the ${\mathcal {Z}}^{(\xi )}(x)$ scale function producing equation (38) and its initial conditions. This completes the proof of the case (i). The case (ii) when $\sigma >0$ can be proved in a similar way. $\square$

Appendix

Proof of Lemma 1

Firstly, we define the function $f:{\mathbb {R}}^{+}\rightarrow {\mathbb {R}}$ of the form

$$\begin{aligned} f(s) = {\left\{ \begin{array}{ll} -\frac{1}{s}, \quad &{}s\in (0, 1],\\ s, \quad &{}s\in [2, +\infty ) \end{array}\right. } \end{aligned}$$

such that $f(s)\in C^{2}({\mathbb {R}}^{+})$ and $f'(s)>0$ for all $s\in {\mathbb {R}}^{+}$. Taking $Y_t = f(S_t)$ and applying Itô’s lemma on (5), we obtain

$$\begin{aligned} dY_t = {\tilde{\mu }}(Y_{t-}, t) dt + {\tilde{\sigma }}(Y_{t-}, t)dB_t + \int _{{\mathbb {R}}} {\tilde{\gamma }}(Y_{t-}, t, z){\tilde{v}}(dt, dz), \end{aligned}$$

where

$$\begin{aligned}&\begin{aligned} {{\tilde{\mu }}(y, t)}&= \mu (f^{-1}(y), t) f'(f^{-1}(y)) + \frac{\sigma ^2(f^{-1}(y), t)}{2} f''(f^{-1}(y))\\&\qquad + \int _{{\mathbb {R}}} \left( {\tilde{\gamma }}(y, t, z) - f'(f^{-1}(y))\gamma (f^{-1}(y), t, z)\right) m(dz), \end{aligned}\\&\begin{aligned} {{\tilde{\sigma }}(y, t)}&= f'(f^{-1}(y))\sigma (f^{-1}(y), t), \end{aligned}\\&\begin{aligned} {{\tilde{\gamma }}(y, t, z)}&= f(f^{-1}(y) + \gamma (f^{-1}(y), t, z)) - y. \end{aligned} \end{aligned}$$

We define also the functions ${\tilde{\omega }}(y):= \omega (f^{-1}(y))$ and ${\tilde{g}}(y):= g(f^{-1}(y)).$ We can now verify that the functions ${\tilde{\mu }}(y, t)$, ${\tilde{\sigma }}(y, t)$, ${\tilde{\gamma }}(y, t, z)$ and ${\tilde{g}}(y)$ satisfy conditions $(2.2)-(2.5)$ from [35, Section 2]. Let

$$\begin{aligned} v(y, t): = V^{\omega }_{\text {E}}(f^{-1}(y), t). \end{aligned}$$

From [35, Theorem 3.1] it follows that v(y, t) is a viscosity solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \tilde{{\mathcal {L}}} v(y, t) = {\tilde{f}}(y,t), \quad &{} (y, t)\in {\mathbb {R}}\times [0,T), \\ v(y, T) = {\tilde{g}}(y), \quad &{} y\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$

(72)

where

$$\begin{aligned} \tilde{{\mathcal {L}}} v(y, t) = - \frac{\partial v(y, t)}{\partial t} - \frac{{\tilde{\sigma }}^2(y, t)}{2} \frac{\partial ^2 v(y,t)}{\partial y^2} - {\hat{\mu }}(y,t) \frac{\partial v(y, t)}{\partial y} + {\tilde{\omega }}(y)v(y,t) \end{aligned}$$

with ${\hat{\mu }}(y,t) = {\tilde{\mu }}(y,t) - \int _{{\mathbb {R}}}{\tilde{\gamma }}(y,t,z)m(dz)$ and ${\tilde{f}}(y,t) = - \int _{{\mathbb {R}}}\big (v(y+{\tilde{\gamma }}(y,t,z), t) - v(y,t)\big )m(dz).$ In addition, using [35, Prop. 3.3] yields that $v(y,t)\in C({\mathbb {R}}\times [0,T])$ and it satisfies

$$\begin{aligned} |v(y_2,t_2) - v(y_1,t_1)|\le C((1 + |y_2|)|t_2 - t_1|^{\frac{1}{2}} + |y_2 - y_1|) \end{aligned}$$

(73)

for some $C>0$ and for all $t_1, t_2 \in [0,T]$ and $y_1, y_2\in {\mathbb {R}}$. Based on (73) and assumptions put on $\gamma$ we can conclude that ${\tilde{f}}(y,t)\in C_{\alpha }({\mathbb {R}}\times [0,T])\cap C_{\text {pol}}({\mathbb {R}}\times [0,T])$. Then applying [26, Thm. A.14] give us the existence of a unique classical solution w(y, t) to (72) such that $w(y,t)\in C^{2,1}({\mathbb {R}}\times [0, T)) \cap C_{\text {pol}}({\mathbb {R}}\times [0,T])$. In view of the fact that w(y, t) is continuous, we can observe that ${\tilde{f}}(y,t)$ is Lipschitz continuous in y, uniformly in t. Hence from [35, Lem. 3.1] we know that that w(y, t) is also Lipschitz continuous in y, uniformly in t. From the uniqueness result given in [35, Thm. 4.1] we can deduce that $v(y,t) = w(y,t)$. Applying [26, Thm. A.18.] we find that $v(y,t)\in C^{4,1}_{\alpha }({\mathbb {R}}\times [0,T])$. Changing back to the original coordinates, it follows that $V^{\omega }_{\text {E}}(s,t)\in C^{4,1}_{\alpha }({\mathbb {R}}^{+}\times [0,T])\cap C_{\text {pol}}({\mathbb {R}}^{+}\times [0,T])$ and it satisfies (54). $\square$

Proof of Lemma 2

The proof follows in the same way as the proof of Lemma 1. However, this time we apply [26, Thm. A.20] which guarantees the existence of a unique classical solution w(y, t) to (72) satisfying $w(y, t) \in C^{2,1}_{\text {pol}}({\mathbb {R}}\times [0,T])$. Hence, coming back to the original coordinates, we have that $V^{\omega }_{\text {E}}(s, t) \in C^{2,1}_{\text {pol}}({\mathbb {R}}^{+}\times [0,T])$. Therefore, there exist constants $n>0$ and $K>0$ such that $|\frac{\partial ^2 V^{\omega }_{\text {E}}(s, t)}{\partial s^2}|\le K(s^{-n}+s^{n})$ for all $(s, t)\in {\mathbb {R}}^{+}\times [0,T]$. This completes the proof. $\square$