GENERALIZATION ON OPTIMAL MULTIPLE ...

MATHEMATICAL CONTROL AND RELATED FIELDS Volume 5, Number 4, December 2015

doi:10.3934/mcrf.2015.5.807 pp. 807–826

GENERALIZATION ON OPTIMAL MULTIPLE STOPPING WITH APPLICATION TO SWING OPTIONS WITH RANDOM EXERCISE RIGHTS NUMBER

Noureddine Jilani Ben Naouara and Faouzi Trabelsi D´ epartement de Math´ ematiques Institut Sup´ erieur d’Informatique et de Math´ ematiques de Monastir Avenue de la Korniche, B.P. 223, 5000 Monastir, Tunisia and Unit´ e de Recherche “Multifractales Et Ondelettes” (UR11ES53) Facult´ e des Sciences de Monastir Avenue de l’Environnement - Monastir - 5000 Universit´ e de Monastir, Tunisia

(Communicated by Qing Zhang) Abstract. 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.

1. Introduction. Optimal multiple stopping arises in some fields of applied probability such as insurance and finance. The most important and to our acknowledge, the unique application in finance, is the valuation problem of Swing options in energy market. This problem is concerned with choosing a sequence of stopping times separated by a fixed refracting time δ > 0, in order to maximize the expected value of the sum of the reward process in continuous time. Therefore, the holder of such option, called perpetual swing option is faced with the following optimization problem: Zn (0) := sup E [Yτ1 + . . . + Yτn ] , n ≥ 1. (1) n (τ1 ,...,τn )∈S0,δ

where • {Yt , Ft , t ≥ 0} is a nonnegative process with right-continuous paths, defined on a probability space (Ω, F, P), and adapted to a filtration F = {Ft }t≥0 that satisfies the usual conditions. n • Sσ,δ := {(τ1 , . . . , τn ); τ1 ∈ Sσ ; τi ∈ Sτi−1 +δ , ∀i = 2, . . . , n}, n ≥ 1, is the set of stopping times vector’s, with Sσ := {τ ∈ S; τ ≥ σ} where S is the collection of all F-stopping times with values in [0, +∞], and the constant δ > 0 is a refracting time. 2010 Mathematics Subject Classification. Primary: 93E20, 49J55, 49K45, 49L20; Secondary: 60G40, 60G50, 60J70. Key words and phrases. Optimal stopping, optimal multiple stopping, stopping times, hitting times, strong optimal stopping times strategy, diffusion process, Swing option, dynamic programming.

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NOUREDDINE JILANI BEN NAOUARA AND FAOUZI TRABELSI

In other words, the holder of a perpetual Swing option has the right to exercise the option several times (at a fixed number n ≥ 1 of stopping times) whenever wants, unless the period of successive exercise times is less than the given refracting time δ > 0. However, in some situations in the energy market, a fixed number of exercise times may not be realistic, since swing option may be conditioned by other phenomena, such as (political decisions, wars, catastrophes, explorations, smuggling,. . .). Therefore it is suitable to model the number of right exercises on swing options by a random variable Ξ. Hence, the decision regarding the number of exercise times depends on the information given both by the reward process (Yt )t≥0 and the discrete random variable Ξ. The latest, will be supposed to be independent of the stopping reward process (Yt )t≥0 . We are thus led to study an optimal multiple stopping times problem with a random exercise rights number. This problem is novel and mathematically of a great technical difficulty and we believe that it significantly advances the study of option pricing. For the case where the holder has only one right to mark the reward process under an infinite horizon, the optimal multiple stopping problem (1) becomes the one of pricing a perpetual American option. The reader can consult among others the following references: [7] provides a new characterization of excessive functions for arbitrary one-dimensional regular diffusion processes, using the notion of concavity. They characterize the value function of the optimal stopping time problem as the smallest nonnegative concave majorant of the reward function, and they generalize the results given by [8] for standard Brownian motion. [12] generalizes the results by [7] for a diffusion process that started in bounded and closed subinterval of the state space, to the case of any state space subinterval. When applied, they give the perfect time to monitor a genetic disease. Regarding optimal multiple stopping times problem (1), the main works on this topic deal with the pricing problem of financial instruments with multiple exercise rights of American type. The reader can consult on this topic the following references: [3] in a first attempt at a theory for pricing swing options with multiple exercise rights of the American type, studies the optimal multiple stopping problem with a special reward function in the case of the Black-Scholes model, providing a rather complete solution. [2] formulates and solves optimal multiple stopping times problems for general linear regular diffusions and reward functions, using novel methods for classical optimal stopping times problems. They also provide examples for mean-reverting diffusions. [3] and [2] show that the optimal value can be calculated sequentially. [6] generalizes an optimal multiple stopping with a constant refract period to random refraction times, and they develop the theory and reduce the problem into a sequence of ordinary optimal stopping time problems. The case where stopped reward processes are a non linear function of the underlying process, is novel and first introduced by [19] who studies and formulate a undiscounted nonlinear optimal multiple stopping times problem, where the underlying price process follows a general diffusion on closed and bounded intervals and where the payoff/reward function is bounded, continuous and superadditive. They prove that the problem can be reduced to a sequence of ordinary optimal stopping time problems thanks to an appropriate dynamic programming equation, and as an application they treat the valuation problem of the perpetual American style discretely monitored Asian options. [18] generalizes all results given by [19] to the case of any unbounded closed subinterval of the state space taking the form [ε, ∞),

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where ε > 0 is a given lower bound. [13] generalizes the works by [19] and [18] to any subinterval of the state space by adding a suitable hypothesis to get the second side in the dynamic programming equation, missed in [19] and [18]. The nonlinear optimal multiple stopping times problem is also discussed in [15] where is studied a very specific class of optimal multiple stopping times problems corresponding to the deterministic reward process ψ(t1 , . . . , tn ). Their key point is the construction of a new reward which is a family of random variables. For such a reward, they prove the existence of optimal multiple stopping times for the described class by a constructive method. Here, we outline that the value functions of the recurrent simplified problems can not be obtained explicitly. ! However following [13] for a n X particular reward process ψ(t1 , . . . , tn ) = ϕ Yti , where ϕ is superadditive i=1

satisfying some appropriate hypotheses, we get an explicit solution for the original problem, In this paper, we concentrate mainly on solving optimal multiple stopping with a random exercise rights number, through studying and answering the following key questions: Q1 What is a suitable set of stopping times? Ξ We chose a set Sσ,δ of optimal stopping times with a dimension nm depending on the choice of the enlarged filtration FΞ , where Ξ is a given discrete random variable. Q2 What is the value function of the main problem ? We transform the main problem into a sequence of standard optimal stopping time problems using recurrent equations (dynamic programming) and Ξ canonical projection of the set Sσ,δ . We show that its value function depends on related standard optimal multiple stopping times problems, with a deterministic number of exercises rights at stopping times. Q3 What is the optimal stopping times strategy of the main problem ? The optimal stopping strategy of the main problem is given as a function of the appropriate canonical injection and optimal stopping times strategy (similar to [2]) of a corresponding deterministic exercise rights number problem. Q4 Is there any application to this problem ? As an application, we define and study random exercise rights number monitored perpetual Swing option. 2. Presentation of the problem. The objective of this paper is to study and solve a generalized optimal multiple stopping times problem, of the form: " Ξ # X ZΞ (0) := sup E Yτi , (2) Ξ ~ τ ∈S0,δ

i=1

where 1. (Yt )t≥0 is a nonnegative right-continuous process (could be in a particular a diffusion process) defined on a complete probability space (Ω, F, P) and adapted to a filtration ! F := (Ft )t≥0 satisfying usual conditions. We take S F := σ Ft , and we set lim sup Yt := Y0 and lim sup Yt := Y∞ . 0≤t