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The American put option value for different intensities of exponential distribution. Parameters: σ = 0.2, ρ = 7.5, µ = 0.06, q = −0.01.
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In this paper we study perpetual American call and put options in an exponential L\'evy model. We consider a negative effective discount rate which arises in a number of financial applications including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor. We show that in this...
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... understand the impact of jumps on the value function and on the stopping region we compare various intensities of exponential distribution ρ and various intensities of arrival rate λ, leaving fixed the other parameters, in Figure 5 and Figure 6, respectively. Note that the increase in ρ corresponds to a decrease in the average sizes of the jumps, which, we recall, are downward jumps. ...
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In this paper we study perpetual American call and put options in an exponential L\'evy model. We consider a negative effective discount rate which arises in a number of financial applications including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor. We show that in this...
Citations
... 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]. ...
... 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. ...
... 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]). ...
In this paper we consider the following optimal stopping problem
where the process is a jump-diffusion process, is a family of stopping times while g and are fixed payoff function and discount function, respectively. In a financial market context, if or and is the expectation taken with respect to a martingale measure, describes the price of a perpetual American option with a discount rate depending on the value of the asset process . If 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 . This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of . We also present a put-call symmetry for American options with asset-dependent discounting. In the case when 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 .
... However, when one of these assumptions does not hold, a one-sided policy may not be optimal. For example, De Donno et al. (2020) and Palmowski et al. (2021) explained that the optimal stopping region for perpetual American puts could be two-sided under Lévy models when the discount rate is negative. Christensen et al. (2013) and Mordecki and Eguren (2021) proved a verification theorem for two-sided optimal stopping. ...
This study examines the continuous-time optimal stopping problem with an infinite horizon under Markov processes. Existing research focuses on finding explicit solutions under certain assumptions of the reward function or underlying process; however, these assumptions may either not be fulfilled or be difficult to validate in practice. We developed a continuous-time Markov chain (CTMC) approximation method to find the optimal solution, which applies to general reward functions and underlying Markov processes. We demonstrated that our method can be used to solve the optimal stopping problem with a random delay, in which the delay could be either an independent random variable or a function of the underlying process. We established a theoretical upper bound for the approximation error to facilitate error control. Furthermore, we designed a two-stage scheme to implement our method efficiently. The numerical results show that the proposed method is accurate and rapid under various model specifications.
... Compared to implied volatility, the implied dividend is typically a relatively weak in the case of American options. In addition, other complicating factors, such as a negative interest rate (Hendrik Frankena 2016), may lead to complex-shaped early-exercise regions (e.g., we may encounter different continuation regions) (Donno, Palmowski, and Tumilewicz 2020). ...
... By swapping the strike with the spot price and the interest rate with the dividend yield, an American call value equals the corresponding American put. The relationship is also valid under negative discount rates (Donno, Palmowski, and Tumilewicz 2020). Because with Equation (6) we can get two option prices from one computation, only one function evaluation is required to compute American call and put prices. ...
... However, two continuation regions may arise, when both the interest rate and dividend yield become negative (Battauz, Donno, and Sbuelz 2015;Donno, Palmowski, and Tumilewicz 2020) in the case of American options. Figure 2 presents an American put solution with two early-exercise points, so that the continuation regions are discontinuous. ...
Extracting implied information, like volatility and dividend, from observed option prices is a challenging task when dealing with American options, because of the complex-shaped early-exercise regions and the computational costs to solve the corresponding mathematical problem repeatedly. We will employ a data-driven machine learning approach to estimate the Black-Scholes implied volatility and the dividend yield for American options in a fast and robust way. To determine the implied volatility, the inverse function is approximated by an artificial neural network on the effective computational domain of interest, which decouples the offline (training) and online (prediction) stages and thus eliminates the need for an iterative process. In the case of an unknown dividend yield, we formulate the inverse problem as a calibration problem and determine simultaneously the implied volatility and dividend yield. For this, a generic and robust calibration framework, the Calibration Neural Network (CaNN), is introduced to estimate multiple parameters. It is shown that machine learning can be used as an efficient numerical technique to extract implied information from American options, particularly when considering multiple early-exercise regions due to negative interest rates.
... Therefore, in this case a single continuation region appears. In general, for the negative ω, we can observe a double continuation region; for more details, see De Donno et al. (2020). ...
The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process St is a geometric Lévy process with negative exponential jumps, i.e., St=seζt+σBt−∑i=1NtYi. The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function VAPutω(s) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that VAPutω(s) takes a simplified form.
... The importance of these models has been rapidly developing in the current low-interest environments. We refer the reader to Battauz et al. (2012), Battauz et al. (2015), andDe Donno et al. (2020) for a detailed literature review on the American option problem with a negative discount rate. ...
... Similar to other important applications, Poisson observation models can potentially be used for approximating optimal strategies in the deterministic discrete-time models (see Section 1 of Palmowski et al., 2020 for the accuracy of approximations). As discussed in Section 1.4, these models can also be used to approximate the continuous observation case (De Donno et al., 2020). ...
... One of our main motivations of this study is to derive an efficient numerical approach for the computation of optimal solutions in the continuous observation case (De Donno et al., 2020) that involves the integration of the resolvent measure with respect to the Lévy measure; this is required due to the fact that the process can jump to an interval or jump over it. In our case, on the other hand, the obtained expression is simpler and works for a general spectrally one-sided Lévy process, without the need of integration with respect to the Lévy measure. ...
We consider the Lévy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case, the stopping region that characterizes the optimal stopping time is either a half‐line or an interval. The objective of this paper is to obtain explicit expressions of the stopping and continuation regions and the value function, focusing on spectrally positive and negative cases. To this end, we compute the identities related to the first Poisson arrival time to an interval via the scale function and then apply those identities to the computation of the optimal strategies. We also discuss the convergence of the optimal solutions to those in the continuous observation case as the rate of observation increases to infinity. Numerical experiments are also provided.
... Therefore, in this case a single continuation region appears. In general, for the negative ω, we can observe a double continuation region; for more details, see De Donno et al. (2020). ...
The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as \begin{equation*} V^{\omega}_{\text{A}^{\text{Put}}}(s) = \sup_{\tau\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^\tau \omega(S_w) dw} (K-S_\tau)^{+}], \end{equation*} where is a family of stopping times, is a discount function and is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process is a geometric L\'evy process with negative exponential jumps, i.e. . The asset-dependent discounting is reflected in the function, so this approach is a generalisation of the classic case when is constant. It turns out that under certain conditions on the function, the value function is convex and can be represented in a closed form; see Al-Hadad and Palmowski (2021). We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of such that takes a simplified form.
... Similar type of techniques were applied in Buonaguidi and Muliere [4] to solve a statistical problem: the Bayesian sequential testing of two simple hypothesis. More recently, De Donno et al. [7] found disconnected continuation regions in American put options with negative discount rates in Lévy models. ...
... We obtain explicit expressions for the value functions in (2.4) and (5.1) and apply the normal-reflection condition at the edge of the two-dimensional state space for (X, Y ) to characterise the optimal stopping boundaries as the minimal solutions to the appropriate first-order nonlinear ordinary differential equations. Other optimal stopping problems with exponential positive discounting rates were recently considered by Xia and Zhou [40], Battauz et al. [3]- [4], De Donno et al. [8], and [14] among others. Optimal stopping problems for three-dimensional continuous Markov processes having the running maximum or minimum as well as the running maximum drawdown or drawup as components were recently studied by Peskir [31]- [32], Glover et al. [20], and [16]- [18] among others. ...
We present analytic solutions to some optimal stopping problems for the running minimum of a geometric Brownian motion with exponential positive discounting rates. The proof is based on the reduction of the original problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the minimal solutions of certain first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual dual American lookback options with fixed and floating strikes in the Black-Merton-Scholes model from the point of view of short sellers.
... It comes from the fact that when at time t = 0 the discount rate is negative then it is worth to wait since discounting might increase the value of payoff. This phenomenon has been already observed for fixed negative discounting (see [6,7,8,26,63]) or in the case of American capped options with positive interest rate (see [16,27]). 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. ...
... Our finding supplements [31,38] who extend to the Lévy market the findings by [17]. An analogous result for the negative discount rate case was obtained in [6,7,8,26]. A comprehensive review of the put-call duality for American options is given in [28]. ...
... Theorem 10. Let (A6) hold and assume that the stock price process S t follows (26). ...
In this paper we consider the following optimal stopping problem where the process is a jump-diffusion process, is a family of stopping times and g and are fixed payoff function and discounting function, respectively. In a financial market context, if or and is the expectation taken with respect to a martingale measure, describes the price of a perpetual American option with a discount rate depending on the value of the asset process . If 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 . This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of . We also prove a put-call symmetry for American options with asset-dependent discounting. In the case when is a geometric L\'evy process we give exact expressions using the so-called omega scale functions introduced in Li and Palmowski (2018). We prove that the analysed value function satisfies HJB and we give sufficient conditions for the smooth fit property as well. Finally, we analyse few cases in detail performing extensive numerical analysis.
... We consider the Lévy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case as in De Donno et al. [24], the stopping region that characterizes the optimal stopping time is either a half-line or an interval. The objective of this paper is to obtain explicit expressions of the stopping and continuation regions and the value function, focusing on spectrally positive and negative cases. ...
... The importance of these models has been rapidly developing in the current low-interest environments. We refer the reader to [9,10,24] for a detailed literature review on the American option problem with a negative discount rate. ...
... In this context, Battauz et al. [9,10] considered the Brownian motion case for the analysis of the double continuation region. Recently, De Donno et al. [24] extended the results to the spectrally one-sided Lévy case and multiple stopping (swing option) cases. ...
We consider the L\'evy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case as in De Donno et al. [24], the stopping region that characterizes the optimal stopping time is either a half-line or an interval. The objective of this paper is to obtain explicit expressions of the stopping and continuation regions and the value function, focusing on spectrally positive and negative cases. To this end, we compute the identities related to the first Poisson arrival time to an interval via the scale function and then apply those identities to the computation of the optimal strategies. We also discuss the convergence of the optimal solutions to those in the continuous observation case as the rate of observation increases to infinity. Numerical experiments are also provided.
