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American and Game Contingent Claims with Asymmetric Information and Reflected BSDE

Esmaeeli, Neda | 2016

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 49049 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Zohouri Zangeneh, Bijan; Foroush-Bastani, Ali
  7. Abstract:
  8. In the past decades, an extensive mathematical theory has been developed for the problems of derivative pricing. One of the salient features of this theory is its assumption of a common information flow on which the portfolio decisions of all economic agents are based. In this thesis, we attempt to widen the scope of the pricing financial derivatives by studying two important classes of contingent claims in a financial market with two types of investors on different information levels. Following the well-known link between optimal stopping problems and reflected backward stochastic differential equations (RBSDE) in El Karoui et al. [31], we also investigate the problem of information asymmetry for these equations. A European contingent claim is a contract on a financial market whose payoff depends on the market state at maturity or exercise time. In contrast to their European counterparts, American contingent claims (ACC), such as American call or put options, can be exercised at any time before maturity. Ignoring interest rates, it is well known that the value of the process of an American contingent claim is related to the Snell envelope of the payoff process, i.e. the smallest supermartingale dominating it. The optimal exercise time is given by the hitting time of the payoff process by the Snell envelope. This key observation links optimal stopping problems to reflected backward stochastic differential equations (RBSDE), i.e. BSDE constrained to stay above a given barrier which in the case of the ACC is given by the payoff function. RBSDE in continuous time, the variant related to ACC, were first investigated in El Karoui et al. [31]. In this context the solution process is kept above the reflecting barrier by means of an additional process. As in the classical Skorokhod problem, this process is non-decreasing. The support of the associated positive random measure is included in the set of times at which the solution process touches the barrier. In the first part, we consider American contingent claims (ACC) in a scenario in which the buyer has better information than the seller. While the decisions of the latter are based on the public information flow F = (Ft)t2[0;T ], the buyer possesses additional information modeled by some random variable G which is already available initially. So his information evolution is described by the enlarged filtration G =(Gt)t2[0;T ] with Gt = Ft _ (G). We study the effect of this additional information on the value and the optimal exercise time of an American contingent claim. The situation is similar to an insider’s optimal investment problem in the simplest possible model, where he aims to maximize expected utility from the terminal value of his portfolio, and his investment decisions are based on the associated larger flow of information. Building on results about initial enlargements of filtrations by Jacod [73], we reduce the problem to a standard optimal stopping problem on an enlarged probability space in case G possesses conditional laws with respect to the smaller filtration that are smooth enough (density hypothesis). Under the density hypothesis, we write the value function of an American contingent claim obtained with additional information as the value function of a modified American contingent claim on the enlarged space. To define it as the product of the underlying probability space and the (real) space of possible values of G, we give a factorization of Gstopping times in terms of parametrized Fstopping times. This is a rational choice, since the initial enlargement is related to a measure change on this product space; see for instance Jacod [73] or Amendinger et al. [6]. In the second part, we define a corresponding RBSDE on the product space associated to the initial enlargement of the filtration. BSDE for (initially or progressively) enlarged filtrations have been studied by Eyraud-Loisel [41] or Kharroubi et al. [85]. The approach used in [41] is based on measure changes, which is one, but not the main, tool for our approach. Our treatment of the RBSDE is based on Ito calculus and the canonical decomposition of semimartingales in G. Extending results in El Karoui et al. [31], we rewrite the value function of the American contingent claim with asymmetric information in terms of the solution of the RBSDE on the product space. This provides a solution of the RBSDE with respect to the larger filtration. Possessing additional information, the buyer has a larger value of the expected payoff than the seller. We study the advantage of the buyer in terms of the solutions of two different RBSDE. In the third part, we investigate the problem of the information asymmetry for game contingent claims. A game contingent claim (GCC) is a generalization of an American contingent claim which enables the seller to terminate the contract before maturity, but at the expense of a penalty. Kifer [87] showed that in a setting of a complete market model, a game contingent claim has a unique price given by the value of a Dynkin game which is a zero-sum stochastic stopping game. In incomplete markets, the concept of neutral derivative price process [81] of a game contingent claim is given by the dynamic value of a Dynkin game. Thus, pricing a game contingent claim with asymmetric information also reduces to identifying the value of a Dynkin game in this context. A Dynkin game is a zero-sum stochastic stopping game first introduced by Dynkin [29]. The game is set up between two players A and B. Each player can stop the game at any time for an observable payoff. If a player chooses to stop the game, B receives some premium from A. In such a game, B attempts to maximize the amount he receives, while A tries to minimize the payout. As B receives a payoff from A, he has to pay a price to enter the game which can also be considered as the value of the game. A primary problem in such a Dynkin game is to identify suitable conditions on the payoff which ensure the existence of a fair value for the game. A related question is the existence of saddle points i.e. stopping times which are optimal for each player irrespective of the actions of the other. These questions have been well investigated in the literature. In stochastic games, the decisions of the players are based on the information available to them. In a perfect information game, all players have full knowledge. We consider a Dynkin game in which one party has additional information compared to the other one. While the latter makes his decisions based on the public filtration F; the informed player’s filtration is modeled by the initial enlargement of F with G. We study the effect of this information asymmetry on the existence of the value and saddle points of the Dynkin game. A special case of information asymmetry is considered in [99]. There the authors assumed that G is independent of F and worked in a Markovian framework. They show the existence of a value and provide sufficient conditions for the existence of a saddle point. We do not consider the assumption of independency and the game is studied in a general framework. We restrict ourselves to the simple case in which G takes finitely many values. We show that the Dynkin game with asymmetric information has a value. We also provide a necessary and sufficient condition for the existence of a Nash equilibrium. The key tool is a suitable representation of Gstopping times in terms of Fstopping times. This representation allows us to express the conditional value of the game as a sum of the values of Dynkin games in the reference filtration F parametrized by the values of G. We show that a saddle point exists if and only if each parametrized Dynkin game has a saddle point
  9. Keywords:
  10. Asymmetric Information ; Information Cost ; American Contingent Claims ; Game Contingent Claims ; Dynkin Game ; Reflected Backward Stochastic Differential Equations (RBSDE)

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