474 research outputs found

    Dynkin games with Poisson random intervention times

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    This paper introduces a new class of Dynkin games, where the two players are allowed to make their stopping decisions at a sequence of exogenous Poisson arrival times. The value function and the associated optimal stopping strategy are characterized by the solution of a backward stochastic differential equation. The paper further applies the model to study the optimal conversion and calling strategies of convertible bonds, and their asymptotics when the Poisson intensity goes to infinity

    The History of the Quantitative Methods in Finance Conference Series. 1992-2007

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    This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

    Valuation of Convertible Bonds

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    Convertible bonds are hybrid financial instruments with complex features. They have characteristics of both debts and equities, and usually several options are embedded in this kind of contracts. The optimality of the conversion decision depends on equity price, interest rate and default probability of the issuer. The decision making can be further complicated by the fact that most convertible bonds have call provisions allowing the bond issuer to call back the bond at a predetermined call price. In this thesis, we first adopt a structural approach where the Vasicek-model is applied to incorporate interest rate risk into the firm's value process which follows a geometric Brownian motion. Default is triggered when the firm's value hits a lower boundary. The complex nature of the firm's capital structure and information asymmetry may make it hard to model the firm's value and the capital structure. In this case, reduced-form models are applied for the study of convertible bonds. We adopt a parsimonious, intensity-based default model, in which the default intensity is modeled as a function of the pre-default stock price. We first analyze the contract features of the convertible bonds and show that callable and convertible bonds can be decomposed into a straight bond and a game option component. Then the no-arbitrage prices of the European- and American-style callable and convertible bonds are derived. In American-style contracts, the focus is on the analysis of the strategic optimal behavior. The bondholder and issuer choose their stopping times to maximize or minimize the expected payoff respectively. For the bondholder it is optimal to select the stopping time which maximizes the expected payoff given the minimizing strategy of the issuer, while the issuer will choose the stopping time that minimizes the expected payoff given the maximizing strategy of the bondholder. The no-arbitrage price can be approximated numerically by means of backward induction. In the structural model, the recursion is carried out alongside a recombining binomial tree. Whereas in the reduced-form approach, the optimization problem is formulated and solved with the help of the theory of doubly reflected backward stochastic differential equations. In practice, it is often a difficult problem to calibrate a given model to the available data. Determining the volatility of the firm's value process or stock price process is not a trivial problem. We therefore assume that the volatility of the firm's value process/stock price process lies between two extreme values, and combine it with the results on game option in incomplete market to derive certain pricing bounds for callable and convertible bonds. The maximizing strategy of the bondholder and the choice of the most pessimistic pricing measure from his perspective determine the lower bound of the no-arbitrage price. Whereas the minimizing strategy of the issuer and the most pessimistic expectation from his aspect construct the upper bound of the no-arbitrage price. Numerically, to make the computation tractable a constant interest rate is assumed. The pricing bounds can be calculated with recursion alongside a recombining trinomial tree or with the finite-difference method

    Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach

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    We propose an efficient method to evaluate callable and putable bonds under a wide class of interest rate models, including the popular short rate diffusion models, as well as their time changed versions with jumps. The method is based on the eigenfunction expansion of the pricing operator. Given the set of call and put dates, the callable and putable bond pricing function is the value function of a stochastic game with stopping times. Under some technical conditions, it is shown to have an eigenfunction expansion in eigenfunctions of the pricing operator with the expansion coefficients determined through a backward recursion. For popular short rate diffusion models, such as CIR, Vasicek, 3/2, the method is orders of magnitude faster than the alternative approaches in the literature. In contrast to the alternative approaches in the literature that have so far been limited to diffusions, the method is equally applicable to short rate jump-diffusion and pure jump models constructed from diffusion models by Bochner's subordination with a L\'{e}vy subordinator

    Dynkin games with Poisson random intervention times

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    This paper introduces a new class of Dynkin games, where the two players are allowed to make their stopping decisions at a sequence of exogenous Poisson arrival times. The value function and the associated optimal stopping strategy are characterized by the solution of a backward stochastic differential equation. The paper further provides a replication strategy for the game and applies the model to study the optimal conversion and calling strategies of convertible bonds, and their asymptotics when the Poisson intensity goes to infinity

    Optimal stopping problems in mathematical finance

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    This thesis is concerned with the pricing of American-type contingent claims. First, the explicit solutions to the perpetual American compound option pricing problems in the Black-Merton-Scholes model for financial markets are presented. Compound options are financial contracts which give their holders the right (but not the obligation) to buy or sell some other options at certain times in the future by the strike prices given. The method of proof is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. The latter are solved through their associated one-sided free-boundary problems and the subsequent martingale verification for ordinary differential operators. The closed form solution to the perpetual American chooser option pricing problem is also obtained, by means of the analysis of the equivalent two-sided free-boundary problem. Second, an extension of the Black-Merton-Scholes model with piecewise-constant dividend and volatility rates is considered. The optimal stopping problems related to the pricing of the perpetual American standard put and call options are solved in closed form. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. As a result, the explicit algorithms determining the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options, are presented. Third, the optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under different information flows are studied. In this type of contracts, the writers have a right to withdraw the bonds before the holders can exercise them, by converting the bonds into assets. The value functions and the stopping boundaries' expressions are derived in closed-form in the case of observable dividend rate policy, which is modelled by a continuous-time Markov chain. The analysis of the associated parabolic-type free-boundary problem, in the case of unobservable dividend rate policy, is also presented and the optimal exercise times are proved to be the first times at which the asset price process hits boundaries depending on the running state of the filtering dividend rate estimate. Moreover, the explicit estimates for the value function and the optimal exercise boundaries, in the case in which the dividend rate is observable by the writers but unobservable by the holders of the bonds, are presented. Finally, the optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model, in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and its maximum drawdown, are studied. The latter process represents the difference between the running maximum and the current asset value. The optimal stopping times for exercising are shown to be the first times, at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. The closed-form solutions to the equivalent free-boundary problems for the value functions are obtained with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. The optimal exercise boundaries of the perpetual American call, put and strangle options are obtained as solutions of arithmetic equations and first-order nonlinear ordinary differential equations

    Computational Methods for Game Options

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    Game options are American-type options with the additional property that the seller of the option has the right the cancel the option at any time prior to the buyer exercise or the expiration date of the option. The cancelation by the seller can be achieved through a payment of an additional penalty to the exercise payoff or using a payoff process greater than or equal to the exercise value. The main contribution of this thesis is a numerical framework for computing the value of such options with finite maturity time as well as in the perpetual setting. This framework employs the theory of weak solutions of parabolic and elliptic variational inequalities. These solutions will be computed using finite element methods. The computational advantage of this framework is that it allows the user to go from one type of process to another by changing the stiffness matrix in the algorithm. Several types of Levy processes will be used to show the functionality of this method. The processes considered are of pure diffusion type (Black-Scholes model), the CGMY process as a pure jump model and a combination of the two for the case of jump diffusion. Computational results of the option prices as well as exercise, hold and cancelation regions are shown together with numerical estimates of the error convergence rates with respect to the L2 norm and the energy norm
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