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In this paper, we extend the results of Carmona and Touzi [6] for an optimal multiple stopping problem to a market where the price process is allowed to jump. We also generalize the problem of valuation swing options to the context of a Lévy market. We prove the existence of multiple exercise policies under an additional condition on Snell envelops. This condition emerges naturally in the case of Lévy processes. Then, we give a constructive solution for perpetual put swing options when the price process has no negative jumps. We use the Monte Carlo approximation method based on Malliavin calculus in order to solve the finite horizon case. Numerical results are given in the last two sections. We illustrate the theoretical results of the perpetual case and give the numerical solution for the finite horizon case.

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... For instance, Carmona and Touzi [9] formulate the valuation of a swing put option as optimal multiple stopping problem, with constant refraction periods, under the geometric Brownian motion model. In a related study, Zeghal and Mnif [27] value a perpetual American swing put when the underlying Lévy price process has no negative jumps. They provide mathematical characterization and numerical solutions to the associated optimal multiple stopping problem. ...

... It involves the evaluation of the expectation E x e −αδ v (k−1) (X δ ) while the distribution of the random variable X δ is most commonly not explicit or even unknown. As is used in [27], Monte Carlo simulation is the most straightforward approach. However, it is far from being practical unless k is a very small number. ...

... Hence, the single optimal stopping problem (2.6) is well defined. To ensure the existence of an optimal stopping time, we may adapt the proof of Proposition 3.2 in [27] to the setting with possibly negative discount rate α and call-like payoff. More precisely, by Lemma 2.1 and Corollary 2.1 we know that U (k) (s) is globally Lipschitz in s ∈ R + , which implies that, by the proof of Proposition 3.2 in [27], the expected jump of e −ατ v (k) (X τ ), at any predictable time τ , is zero, namely, ...

This paper studies a class of optimal multiple stopping problems driven by
L\'evy processes. Our model allows for 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. Moreover, successive exercise opportunities are
separated by i.i.d. random refraction times. Under a wide class of two-sided
L\'evy models with a general random refraction time, we rigorously show that
the optimal strategy to exercise successive call options is uniquely
characterized by a sequence of up-crossing times. The corresponding optimal
thresholds are determined explicitly in the single stopping case and
recursively in the multiple stopping case.

... A wide array of financial applications can be formulated as optimal multiple stopping problems. These include energy delivery contracts such as swing options [12,13,51], derivatives liquidation [25,36,37], real option analysis [15,17,19,44], as well as employee stock options [24,38,39] potentially with additional reload and shout options [18]. In many of these applications, consecutive stopping times are separated by a constant or random period. ...

... Some recent applications of spectrally negative Lévy processes include the pricing of perpetual American and exotic options [1,6], optimal dividend problems [7,33,43], and capital reinforcement timing [21]. For related optimal multiple stopping problems under spectrally negative models, we mention [51] for a swing put option with constant refraction times, and [50] with a more general payoff function without refraction times. For models with more general processes, Leung et al. [41] study a refracted optimal multiple stopping problem driven by a two-sided Lévy process with general random refraction times, and Christensen and Lempa [16] consider a similar problem driven by a general Markov process with exponential refraction times. ...

... This is a critical condition so that the solution of the problem is nontrivial (see, e.g., [13,41,51]). In fact, we can slightly weaken the condition for the case α < 0 (where exp(−αt)K grows to infinity) to accommodate the case ψ(1) = α given ψ ′ (1) < 0 (see [41] for a proof). ...

We study an optimal multiple stopping problem driven by a spectrally negative Levy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Levy process at a constant horizon and hence the solutions in general cannot be attained analytically. Motivated by the maturity randomization (Canadization) technique by Carr (1998), we approximate the refraction times by i.i.d. Erlang random variables. In addition, fitting random jumps to phase-type distributions, our method involves repeated integrations with respect to the resolvent measure written in terms of the scale function of the underlying Levy process. We derive a recursive algorithm to compute the value function in closed form, and sequentially determine the optimal exercise thresholds. A series of numerical examples are provided to compare our analytic formula to results from Monte Carlo simulation.

... For instance, Carmona and Touzi [9] formulate the valuation of a swing put option as optimal multiple stopping problem, with constant refraction periods, under the geometric Brownian motion model. In a related study, Zeghal and Mnif [27] value a perpetual American swing put when the underlying Lévy price process has no negative jumps. They provide mathematical characterization and numerical solutions to the associated optimal multiple stopping problem. ...

... It involves the evaluation of the expectation E x e −αδ v (k−1) (X δ ) while the distribution of the random variable X δ is most commonly not explicit or even unknown. As is used in [27], Monte Carlo simulation is the most straightforward approach. However, it is far from being practical unless k is a very small number. ...

... Hence, the single optimal stopping problem (2.6) is well defined. To ensure the existence of an optimal stopping time, we may adapt the proof of Proposition 3.2 in [27] to the setting with possibly negative discount rate α and call-like payoff. More precisely, by Lemma 2.1 and Corollary 2.1 we know that U (k) (s) is globally Lipschitz in s ∈ R + , which implies that, by the proof of Proposition 3.2 in [27], the expected jump of e −ατ v (k) (X τ ), at any predictable time τ , is zero, namely, ...

This paper studies a class of optimal multiple stopping problems driven by Lévy processes. Our model allows for 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. Moreover, successive exercise opportunities are separated by i.i.d. random refraction times. Under a wide class of two-sided Lévy models with a general random refraction time, we rigorously show that the optimal strategy to exercise successive call options is uniquely characterized by a sequence of up-crossing times. The corresponding optimal thresholds are determined explicitly in the single stopping case and recursively in the multiple stopping case.

... For details we refer to El Karoui [6], when the reward process is non-negative, right continuous, F-adapted and left continuous in expectation and its supremum is bounded in L p , p > 1, Karatzas and Shreve [9] in the continuous setting and Peskir and Shiryaev [13] in the Markovian context. Carmona and Touzi [2] introduced the problem of optimal multiple stopping time where the underlying process is continuous. They characterized the optimal multiple stopping time as the solution of a sequence of ordinary stopping time problems. ...

... In this section, we shall prove that Z () 0 can be computed by solving inductively single optimal stopping problems sequentially. This result is proved in [2] under the assumption that the process X is continuous a.s.. As it is proved by El Karoui [6, Theorem 2.18, p.115], the existence of the optimal stopping strategy for a right continuous, non-negative and F-adapted process X requires assumption (2) in addition to the left continuity in expectation of the process X, i.e. for all τ ∈ S, (τ n ) n≥0 an increasing sequence of stopping times such that τ n ↑ τ , E [X τn ] → E [X τ ]. ...

... We have also that E Y (−i+1) τ * i = sup τ ∈S τ * i−1 +δ E X (−i+1) τ and the stopped supermartingale {Y (−i+1) t∧τ * i , τ * i−1 + δ ≤ t ≤ T } is a martingale. By Theorem 4.10 we generalize Theorem 1 of [2] ...

In their paper [2], Carmona and Touzi have studied an optimal multiple stopping time problem in a market where the price process is continuous. In this paper, we generalize their results when the price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. Then we relate our problem to a sequence of ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman Variational Inequality.

... For instance, Carmona and Touzi [3] formulated the valuation of a swing put option as optimal multiple stopping problem, with constant refraction periods, under the Black-Scholes model. In a related work, Zeghal and Mnif [24] priced a perpetual American swing put option under spectrally positive exponential Lévy models. Later, Leung et al. [15] considered a stock loan with multiple repayments as an optimal multiple stopping problem with negative discounting rate and general i.i.d. ...

... positive refraction periods. Among them, [3] and [24] subsequently established the optimality of threshold type strategies in exercising a perpetual American swing put option, 2 and demonstrated these monotonicity of the optimal thresholds using sub-gradients techniques under two special models. By using mathematical inductions and the supermartingale property of value functions, [15] proved similar results for a multiple-exercising call option under a general Lévy model with arbitrary negative jumps and Phase-type positive jumps. ...

... [24, Proposition 3.1] proved the optimality of threshold type strategy using monotonicity and convexity of the value function, which, is not fully legitimate, because the reward functions in the multiple optimal stopping problems can also be curved, convex functions, leaving arguments based on a put payoff invalid. ...

In the spirit of [Surya07'], we develop an average problem approach to prove the optimality of threshold type strategies for optimal stopping of L\'evy models with a continuous additive functional (CAF) discounting. Under spectrally negative models, we specialize this in terms of conditions on the reward function and random discounting, where we present two examples of local time and occupation time discounting. We then apply this approach to recursive optimal stopping problems, and present simpler and neater proofs for a number of important results on qualitative properties of the optimal thresholds, which are only known under a few special cases.

... In the first, swing options are priced using an extension of the binomial tree algorithm, leading to the so-called forest tree (Lari-Lavassani et al. (2001), Jaillet et al. (2004). In the second, Monte Carlo methods are used, in which conditional expectations are computed using either regression techniques (Barrera-Esteve et al. (2006)) or Malliavin calculus (Mnif and Zhegal (2006), Carmona and Touzi (2008)). In particular, Carmona and Touzi (2008) propose a Monte Carlo approach to the problem of pricing American put options, in a finite time horizon, with multiple exercise rights in the case of geometric Brownian motion. ...

... Energy expenditure increases sharply with higher daily temperature variation, and consequently, price varies. Although these spikes of power consumption are infrequent, they have a large financial impact, and therefore many authors propose to price swing options in a model with jumps (see Mnif and Zhegal (2006), Wilhelm and Winter (2008)). Mnif and Zhegal (2006) extend the results of Carmona and Touzi (2008) to a market with jumps. ...

... Although these spikes of power consumption are infrequent, they have a large financial impact, and therefore many authors propose to price swing options in a model with jumps (see Mnif and Zhegal (2006), Wilhelm and Winter (2008)). Mnif and Zhegal (2006) extend the results of Carmona and Touzi (2008) to a market with jumps. In fact, the multiple stopping time problem for swing options can be reduced to a cascade of single stopping time problems in a Lévy market where jumps are permitted. ...

We consider the problem of pricing swing options with multiple exercise rights in Lévy-driven models. We propose an efficient Wiener–Hopf factorization method that solves multiple parabolic partial integro-differential equations associated with the pricing problem. We compare the proposed method with a finite difference algorithm. Both proposed deterministic methods are related to the dynamic programming principle and lead to the solution of a multiple optimal stopping problem. Numerical examples illustrate the efficiency and the precision of the proposed methods.

... ey used the theory of the Snell envelope to determine the optimal exercise boundaries. Zeghal and Mnif in [9] extended the latter method to Lévy processes for valuing swing options. Ben Latifa et al. in [10] valued swing options with the jump in price process. ...

... (v) Use the least-squares regression method on N opt l paths to estimate P l . (vi) Estimate Y 0 by (9). else (a) Generate N init l sample paths according to the EM scheme in (11), and use these paths for both levels l and l − 1 (coarsen level l to the level l − 1), we refer the interested readers to [21] for more details. ...

In this study, we propose a novel approach for the valuation of swing options. Swing options are a kind of American options with multiple exercise rights traded in energy markets. Longstaff and Schwartz have suggested a regression-based Monte Carlo method known as the least-squares Monte Carlo (LSMC) method to value American options. In this work, first we introduce the LSMC method for the pricing of swing options. Then, to achieve a desired accuracy for the price estimation, we combine the idea of LSMC with multilevel Monte Carlo (MLMC) method. Finally, to illustrate the proper behavior of this combination, we conduct numerical results based on the Black–Scholes model. Numerical results illustrate the efficiency of the proposed approach.

... For instance, Carmona and Touzi [15] formulate the valuation of a swing put option as an optimal multiple stopping problem, with constant refraction periods, under the geometric Brownian motion model. In a related study, Zeghal and Mnif [53] value a perpetual American swing put when the underlying Lévy price process has no negative jumps. Leung et al. [41] consider call swing options, under the assumption of a negative effective discount rate, and, as in the case of Xia and Zhou [52], they derive single continuation regions. ...

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 case a double continuation region arises and we identify the two critical prices. We also generalize this result to multiple stopping problems of swing type, that is, when successive exercise opportunities are separated by i.i.d. random refraction times. We conduct extensive numerical analysis for the Black-Scholes model and the jump-diffusion model with exponentially distributed jumps.

... Such reduction principle is well-known in discrete time (see Haggstrom, 1967; Cairoli and Dalang, 1996). In continuous time it was studied by Carmona and Touzi (2008) and Zeghal and Mnif (2006) under stronger conditions than the ones imposed in the present paper. The main technical difficulty related to Theorem 2.2 is that the value process of the multiple stopping problem typically exhibits discontinuities from the right, (although the cash-flow Z is right-continuous). ...

In this paper we study the pricing problem of multiple exercise options in continuous time on a finite time horizon. For the corresponding multiple stopping problem, we prove, under quite general assumptions, the existence of the Snell envelope, a reduction principle as nested single stopping problems, and a Doob-Meyer-type decomposition for the Snell envelope. The main technical difficulty arises from the fact that the price process of a multiple exercise option typically exhibits discontinuities from the right-hand side, even if the payoff process of the option is right-continuous. We also derive a dual minimization problem for the price of the multiple exercise option in terms of martingales and processes of bounded variation. Moreover, we explain how the primal and dual pricing formulas can be applied to compute confidence intervals on the option price via Monte Carlo methods, and we present a numerical example.

... To overcome these challenges, a variety of approaches have been suggested, including other partial differential equation methods ([13,31]and[eqf12-007]), stochastic dynamic programming ([2,17]), the aforementioned binomial and trinomial trees ([21]and[eqf12-017]), Monte Carlo simulation methods ([27,4,20,16]and[eqf13-006]), approximate techniques ([14,23,29]and[eqf13-024]) and optimal quantization ([1]). Also see[16,19,24,32]for analytic pricing of swing contracts under certain exotic models for the underlying commodity and[28]for game-theoretic aspects of swing options. Beyond the basic contracts, the modern practitioner/research trend is to value swing options using simulation tools which can achieve excellent computational complexity properties with respect to underlying models. ...

Swing options are the main type of volumetric contracts in commodity markets. A swing contract gives the holder the right (but not the obligation) to adjust volume of received commodity at her discretion. Unlike paper assets, trading in physical commodities often takes place over time and therefore involves volume as a second key state variable. Often consumption rates of the commodity are unpredictable and make fixed delivery amounts uneconomic. To mitigate such volume risk, a swing contract gives the buyer the opportunity to manage fluctuating commodity demand levels in exchange for a fixed upfront fee. By exercising her swing up/down rights, the buyer can dynamically match supply and demand levels while hedging her costs. Swing options are widely offered by market makers and used extensively by major energy companies, especially in electricity and fossil fuel markets. Contracts with swing features are available both as stand-alone financial tools, and can also be found embedded within structured physical transactions.

... They focus on the Black-Scholes dynamics. Zeghal and Mnif [12] extend that method to Lévy processes. Unlike the models in which swing actions are only allowed at discrete times, Dahlgren [1] proposes a continuous time model to price the commodity-based swing options. ...

Swing options give contract holders the right to modify amounts of future delivery of certain commodities, such as electricity or gas. In this paper, we assume that these options can be exercised at any time before the end of the contract, and more than once. However, a recovery time between any two consecutive exercise dates is incorporated as a constraint to avoid continuous exercise. We introduce an efficient way of pricing these swing options, based on the Fourier cosine expansion method, which is especially suitable when the underlying is modeled by a Lévy process.

... Products falling in this class are those with mature markets, including crude and heating oil, for which storability (and a global market structure) cushions short-term and local fluctuations in supply and demand. The study of optimal multiple stopping and swing option pricing in Lévy-driven markets has received considerable interest in recent years, for example Zhegal & Mnif (2006) (2015); Latifa et al. (2015); Donno et al. (2017). Suppose that L(t), t ≥ is a Lévy process with characteristic exponent or Lévy symbol ψ L (ξ) := ln E[exp(iξL(1))]. ...

Swing options are a type of exotic financial derivative which generalize American options to allow for multiple early-exercise actions during the contract period. These contracts are widely traded in commodity and energy markets, but are often difficult to value using standard techniques due to their complexity and strong path-dependency. There are numerous interesting varieties of swing options, which differ in terms of their intermediate cash flows, and the constraints (both local and global) which they impose on early-exercise (swing) decisions. We introduce an efficient and general purpose transform-based method for pricing discrete and continuously monitored swing options under exponential Lévy models, which applies to contracts with fixed rights clauses, as well as recovery time delays between exercise. The approach combines dynamic programming with an efficient method for calculating the continuation value between monitoring dates, and applies generally to multiple early-exercise contracts, providing a unified framework for pricing a large class of exotic derivatives. Efficiency and accuracy of the method is supported by a series of numerical experiments which further provide benchmark prices for future research.

... Proof: the proof of this theorem is based on mathematical induction and can be found in Carmona and Touzi [23] for the Black-Scholes framework and in Zeghal and Mnif [108] for a general jump-diffusion model. ...

Although electricity is considered to be a commodity, its price behavior is remarkably different from most other commodities or assets on the market. Since power can hardly be stored physically, the storage-based methodology, which is widely used for valuing commodity derivatives, is unsuitable for electricity. Therefore, new approaches are required to understand and reproduce its price dynamics. Concurrently, the demand for derivative instruments has grown and new types of contracts for energy markets have been introduced. Swing options, in particular, have attracted an increasing interest, offering more flexibility and reducing exposure to strong price fluctuations. In this thesis, we propose a mean-reverting model with seasonality and double exponential jumps. It is able to accurately reproduce the behavior and main peculiarities of electricity’s spot prices. With this model, we can characterize the swing option value as a solution to a partial integro-differential complementarity problem, which we solve numerically. In the last part of the thesis, we present a more complex type of swing options, in which we also include variable electricity volumes in the contract. This formulation leads to a two-dimensional Hamilton-Jacobi-Bellman (HJB) equation. By applying the method of characteristics, this problem is simplified to a sequence of one dimensional HJB equations, which are solved numerically by using a similar approach as before.

... They also provided examples for mean-reverting diffusions. (Zeghal and Mnif, 2006) extended the results of (Carmona and Touzi, 2008) for an optimal multiple stopping problem to a market where the price process is allowed to jump. They also generalize the problem of valuation swing options to the context of a Lévy market. ...

We study and formulate an undiscounted non-linear optimal multiple stopping problem, with an application to the valuation of the perpetual American-style discretely monitored Asian options. When the reward process is continuous, we follow a vector-valued approach. Under the right-continuity of this process, the problem can be reduced to a sequence of ordinary optimal stopping problems. In the Markovian case, we characterise the value function of the problem in terms of excessive functions. Finally, in case of a regular diffusion, we provide an optimal sequence of stopping times. The results are illustrated by some examples, where the value function of the problem is given explicitly.

... For the case n ≥ 2, problem (1) is an optimal multiple stopping one. For the main works on this topic, dealing with the pricing problem of financial instruments with multiple exercise rights of American type or the hedging problem in continuous time model using a finite number of portfolio rebalancing, the reader can consult (Trabelsi and Trad, 2002;Martini and Patry, 1999;Carmona and Touzi, 2008;Carmona and Dayanik, 2008;Zeghal and Mnif, 2006;Ross and Zhu, 2008;Bender, 2011a;Kobylanski et al., 2009) (see Trabelsi and Zoghlami, 2012 for details). We have also the following recent references dealing with optimal multiple stopping: Bender (2011b) studies the pricing problem of multiple exercise options in continuous time on a finite time horizon. ...

In this paper we formulate and solve a class of undiscounted non-linear optimal multiple stopping problems, where the underlying price process follows a general linear regular diffusion on an unbounded and closed subinterval of the state space and where the payoff/reward function is bounded, continuous and superadditive. We use and adapt general theory of optimal stopping for diffusion and we illustrate the developed optimal exercise strategies by the example of valuation of perpetual American-style fixed strike discretely random monitoring Asian put options on any unbounded closed interval of the form [ε,∞), where ε > 0 is a given lower bound.

... To overcome these challenges, a variety of approaches have been suggested, including other partial differential equation methods ([13,31]and[eqf12-007]), stochastic dynamic programming ([2,17]), the aforementioned binomial and trinomial trees ([21]and[eqf12-017]), Monte Carlo simulation methods ([27,4,20,16]and[eqf13-006]), approximate techniques ([14,23,29]and[eqf13-024]) and optimal quantization ([1]). Also see[16,19,24,32]for analytic pricing of swing contracts under certain exotic models for the underlying commodity and[28]for game-theoretic aspects of swing options. Beyond the basic contracts, the modern practitioner/research trend is to value swing options using simulation tools which can achieve excellent computational complexity properties with respect to underlying models. ...

This paper deals with the pricing of spread options on the difference between correlated log-normal underlying assets. We introduce a new pricing paradigm based on a set of precise lower bounds. We also derive closed form formulae for the Greeks and other sensitivities of the prices. In doing so we prove that the price of a spread option is a decreasing function of the correlation parameter, and we analyze the notion of implied correlation. We use numerical experiments to provide an extensive analysis of the performance of these new pricing and hedging algorithms, and we compare the results with those of the existing methods.

... Optimal multiple stopping problems with deterministic refraction periods have been studied quite extensively over the recent years. In particular, theory of Snell envelopes is understood well in both discrete and continuous time, see [9,10,1,5,6,25,33,36]. In the recent study [13], the authors develop a general approach for optimal multiple stopping with random refraction periods. ...

We study optimal multiple stopping of strong Markov processes with random
refraction periods. The refraction periods are assumed to be exponentially
distributed with a common rate and independent of the underlying dynamics. Our
main tool is using the resolvent operator. In the first part, we reduce in?nite
stopping problems to ordinary ones in a general strong Markov setting. This
leads to explicit solutions for wide classes of such problems. Starting from
this result, we analyze problems with fi?nitely many exercise rights and
explain solution methods for some classes of problems with underlying L?evy and
diff?usion processes, where the optimal characteristics of the problems can be
identi?fied more explicitly. We illustrate the main results with explicit
examples.

This paper is a survey article on mathematical theories and techniques used in the study of swing options. In financial terms, swing options can be regarded as multiple-strikeAmerican or Bermudan options with specific constraints on the exerciseability.We focus on two categories of approaches: martingale and Markovian methods. Martingale methods build on purely probabilistic properties of the models whereas Markovian methods draw on the interplay between stochastic control and partial differential equations. We also review other techniques available in the literature.

The issue of making a decision several times and thereby earning a reward is the focus of this chapter. It considers the problem of multiply stopping a general one-dimensional diffusion process with fairly general reward functions at each decision time. A key aspect of the problem is the requirement that succeeding decisions be delayed by at least the length of time of a refraction period following a preceding decision. Using a conditioning argument, the multiple-stopping problem can be solved using an iterative set of single-stopping problems for which several solution approaches are known. The refraction period adds an interesting twist to the problem. A tractable solution method is developed for those processes whose distributions are known. This work is motivated by the recent paper [Carmona and Dayanik (Math Oper Res 32:446–460, 2008)].

We study an optimal control problem related to swing option pricing in a
general non-Markovian setting in continuous time. As a main result we show that
the value process solves a first-order non-linear backward stochastic partial
differential equation. Based on this result we can characterize the set of
optimal controls and derive a dual minimization problem.

This paper develops the theory of optimal multiple stopping times expected value problems by stating, proving, and applying a dynamic programming principle for the case in which both the reward process and the number of stopping times are stochastic. This case comes up in practice when valuing swing options, which are somewhat common in commodity trading. We believe our results significantly advance the study of option pricing. © 2015, American Institute of Mathematical Sciences. All rights reserved.

This paper develops the theory of optimal multiple stopping times
expected value problems by stating, proving, and applying a dynamic programming
principle for the case in which both the reward process and the number
of stopping times are stochastic. This case comes up in practice when valuing
swing options, which are somewhat common in commodity trading. We believe
our results significantly advance the study of option pricing.

In their paper, Carmona and Touzi [8] studied an optimal multiple stopping time problem in a market where the price process is continuous. In this article, we generalize their results when the price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. We relate our problem to a sequence of ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequality.

Le travail présenté dans cette thèse a été motivé par des problématiques posées par l'évaluation de contrats échangés sur le marché du gaz: les contrats de stockage et d'approvisionnement en gaz. Ceux-ci incorporent de l'optionalité et des contraintes, ce qui rend leur évaluation difficile dans un contexte de prix de matières premières aléatoires. L'évaluation de ces contrats mène à des problèmes de contrôle stochastique complexes: switching optimal ou contrôle impulsionnel et contrôle stochastique en grande dimension. La première partie de cette thèse est une revue relativement exhaustive de la littérature, mettant en perspective les différentes approches d'évaluation existantes. Dans une deuxième partie, nous considérons une méthode numérique de résolution de problèmes de contrôle impulsionnel basée sur leur représentation comme solution d'EDSRs à sauts contraints. Nous proposons une approximation à temps discret utilisant une pénalisation pour traiter la contrainte et donnons un taux de convergence de l'erreur introduite. Combinée avec des techniques Monte Carlo, cette méthode a été testée numériquement sur trois problèmes: gestion optimale de biomasse, évaluation d'options Swing et de contrats de stockage gaz. Dans une troisième partie, nous proposons une méthode pour l'évaluation d'options dont le payoff dépend de moyennes mobiles de prix sous-jacents. Elle utilise sur une approximation à dimension finie de la dynamique des processus de moyenne mobile, basée sur un développement en série de Laguerre tronquée. Les résultats numériques fournis incluent des exemples de contrats Swing gaziers à prix d'exercice indexés sur moyennes mobiles de prix pétroliers.

In this article the problem of the American option valuation in a jump diffusion setting is tackeld. The perpetual case is first considered. Without possible discontinuities (i.e. with negative jumps in the call case), known results concerning the currency option value as well the exercise boundary are obtained with a martingale approach. With possible discontinuities of the underlying process at the exercise boundary (i.e. with positive jumps in the call case) original results are derived by relying on first passage time and overshoot associated with a Lévy process. For finite life American currency calls, the formula given by Bates (1991) or Zhang (1995), is rederived in the context of a negative jump size. It is basically an extension of the Barone-Adesi and Whaley (1987) approach. This formula is tested. It is shown that Bates (1996) model generates good results when the process is continuous at the exercise boundary. However, with possible discontinuities results generated by Bates' (1996) model are less accurate.

In the energy markets, in particular the electricity and natural gas markets, many con- tracts incorporate e xibility-of-delivery options, known as ìswingî or ìtake-or-payî op- tions. Subject to daily as well as periodic constraints, these contracts permit the option holder to repeatedly exercise the right to receive greater or smaller amounts of energy. We extract market information from forward prices and volatilities and build a pricing framework for swing options based on a one-factor mean-reverting stochastic process for energy prices which explicitly incorporates seasonal effects. We present a numerical scheme for the valuation of swing options calibrated for the case of natural gas.

Motivated by the analysis of financial instruments with multiple exercise rights of American type and mean revert- ing underlyers, we formulate and solve the optimal multiple-stopping problem for a general linear regular diusion process and a general reward function. Instead of relying on specific properties of geometric Brownian motion and call and put option payos like in most of the existing literature, we use general theory of optimal stopping for diusions, and we illustrate the resulting optimal exercise policies by concrete examples and constructive recipes.

Given a multi-dimensional Markov diffusion X, the Malliavin integration by parts formula provides a family of representations of the conditional expectation E[g(X
2)|X1]. The different representations are determined by some localizing functions. We discuss the problem of variance reduction within this family. We characterize an exponential function as the unique integrated mean-square-error minimizer among the class of separable localizing functions. For general localizing functions, we prove existence and uniqueness of the optimal localizing function in a suitable Sobolev space. We also provide a PDE characterization of the optimal solution which allows to draw the following observation : the separable exponential function does not minimize the integrated mean square error, except for the trivial one-dimensional case. We provide an application to a portfolio allocation problem, by use of the dynamic programming principle.

Recent work by Nualart and Schoutens (2000), where a kind of chaotic property for Lévy processes has been proved, has enabled us to develop a Malliavin calculus for Lévy processes. For simple Lévy processes some useful formulas for computing Malliavin derivatives are deduced. Applications for option hedging in a jump-diffusion model are given.

This paper presents an original probabilistic method for the numerical computations of Greeks (i.e. price sensitivities) in finance. Our approach is based on the {\it integration-by-parts} formula, which lies at the core of the theory of variational stochastic calculus, as developed in the Malliavin calculus. The Greeks formulae, both with respect to initial conditions and for smooth perturbations of the local volatility, are provided for general discontinuous path-dependent payoff functionals of multidimensional diffusion processes. We illustrate the results by applying the formula to exotic European options in the framework of the Black and Scholes model. Our method is compared to the Monte Carlo finite difference approach and turns out to be very efficient in the case of discontinuous payoff functionals.

In this paper we give the closed form solution of some optimal stopping problems for processes derived from a diffusion with jumps. Within the possible applications, the results can be interpreted as pricing perpetual American Options under diffusion-jump information.

Elementary proofs of classical theorems on pricing perpetual call and put options in the standard Black-Scholes model are given. The method presented does not rely on stochastic calculus and is also applied to give prices and optimal stopping rules for perpetual call options when the stock is driven by a Levy process with no positive jumps, and for perpetual put options for stocks driven by a Levy process with no negative jumps 1 This work was partially written at the Laboratoire de Statistique et Probabilites de l'Universite Paul Sabatier, Toulouse, and beneted from helpful discussion with Walter Moreira. Elementary Proofs on Optimal Stopping Ernesto Mordecki y Facultad de Ciencias. Centro de Matematica. Igua 4225. CP 11400. Montevideo. Uruguay. December 29, 2000 1 Introduction 1.1 Consider a model of a nancial market with two assets, a savings account B = fB t g t0 and a stock S = fS t g t0 . The evolution of B is deterministic, with B t = B 0 e rt ; B 0 = 1; r >...

The connection between optimal stopping of random systems and the theory of the Snell envelop is well understood, and its application to the pricing of American contingent claims is well known. Motivated by the pricing of swing contracts (whose recall components can be viewed as contingent claims with multiple exercises of American type) we investigate the mathematical generalization of these results to the case of possible multiple stopping. We prove existence of the multiple exercise policies in a fairly general set-up. We then concentrate on the Black–Scholes model for which we give a constructive solution in the perpetual case, and an approximation procedure in the finite horizon case. The last two sections of the paper are devoted to numerical results. We illustrate the theoretical results of the perpetual case, and in the finite horizon case, we introduce numerical approximation algorithms based on ideas of the Malliavin calculus.

Les deux premiers chapitres de la thèse présentent la construction et l’utilisation d’un arbre qui discrétise le “modèle à seuil“ introduit par Geman et Roncoroni (2006) pour les prix spots d’électricité. Ce modèle incorpore à la fois un retour à la moyenne et des sauts ; la direction de ces derniers dépend du prix du sous-jacent au moment du saut. L’arbre est utilisé dans un premier temps pour la valorisation d’options de type européen, par une méthode d’induction récursive, et dans un contexte de paramètres constants ou déterministes. Des simulations de Monte Carlo confirment la précision de la méthodologie introduite, laquelle est beaucoup plus rapide en temps de calcul. Le grid est ensuite étendu à un contexte multi-niveau pour la valorisation d’options swing (à la fois de type stockage et approvisionnement), avec des paramètres du modèle déterministes ; là encore, des simulations de Monte Carlo confirment la validité des résultats numériques. Le troisième chapitre de la thèse analyse l’impact des rendements des prix des matières premières sur les rendements des actions de compagnies qui les produisent (pétrole, cuivre et blé sont étudiés). Les résultats montrent que les prix des actions sont effectivement influencées par ceux des matières premières et que cette influence a augmenté au cours des dernières années.

We suggest a discrete-time approximation for decoupled forward–backward stochastic differential equations. The Lp norm of the error is shown to be of the order of the time step. Given a simulation-based estimator of the conditional expectation operator, we then suggest a backward simulation scheme, and we study the induced Lp error. This estimate is more investigated in the context of the Malliavin approach for the approximation of conditional expectations. Extensions to the reflected case are also considered. Author Keywords: Monte-Carlo methods for (reflected) forward–backward SDEs; Malliavin calculus; Regression estimation oui

Calcul du prix et des sensibilité d'une option amé par une mé de Monte Carlo

- P.-L Lions
- H Regnier

P.-L. Lions and H. Regnier, Calcul du prix et des sensibilité d'une option amé par une mé de Monte Carlo, Preprint (2001).

Calcul du prix et des sensibilités d'une option américaine par une méthode de Monte Carlo

- P.-L Lions
- H Regnier