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32
Maximum process problems in optimal control theory
, 2001
"... Given a standard Brownian motion (Bt)t 0 and the equation of motion: dXt = vt dt + p 2 dBt ..."
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Given a standard Brownian motion (Bt)t 0 and the equation of motion: dXt = vt dt + p 2 dBt
American options with lookback payoff,” Working
, 2005
"... Abstract. We examine the early exercise policies and pricing behaviors of oneasset American options with lookback payoff structures. The classes of option models considered include floating strike lookback options, Russian options, fixed strike lookback options and pricing model of dynamic protecti ..."
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Abstract. We examine the early exercise policies and pricing behaviors of oneasset American options with lookback payoff structures. The classes of option models considered include floating strike lookback options, Russian options, fixed strike lookback options and pricing model of dynamic protection fund. For each class of the American lookback options, we analyze the optimal stopping region, in particular the asymptotic behavior at times close to expiration and at infinite time to expiration. The interrelations between the price functions of these American lookback options are explored. The mathematical technique of analyzing the exercise boundary curves of lookback options at infinitesimally small asset value is also applied to the American twoasset minimum put option model. Key words. Lookback options, American feature, free boundary problems, twoasset minimum put option
Russian Options for a Diffusion with Negative Jumps
, 2001
"... Closed solutions to the problem of pricing a Russian option when the underlying process is a diffusion with negative jumps are obtained. More precisely, the underlying process is assumed to have the form of a Wiener process with drift and negative mixedexponentially distributed jumps driven by a Po ..."
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Closed solutions to the problem of pricing a Russian option when the underlying process is a diffusion with negative jumps are obtained. More precisely, the underlying process is assumed to have the form of a Wiener process with drift and negative mixedexponentially distributed jumps driven by a Poisson process. This result generalizes those of Shepp and Shiryaev (1993) for the Wiener process and Gerber, Michaud and Shiu (1995) for purejumps process.
Valuing American continuousinstallment options
 Hokkaido University
, 2007
"... Installment options are weakly pathdependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing American continuousinstallment options written on dividendpaying assets. The s ..."
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Installment options are weakly pathdependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing American continuousinstallment options written on dividendpaying assets. The setup is a standard BlackScholesMerton framework where the price of the underlying asset evolves according to a geometric Brownian motion. The valuation of installment options can be formulated as an optimal stopping problem, due to the flexibility of continuing or stopping to pay installments as well as the chance of early exercise. Analyzing cash flow generated by the optimal stop, we can characterize asymptotic behaviors of the stopping and early exercise boundaries close to expiry. Combining the PDE and Laplace transform approaches, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping and early exercise boundaries. Abelian theorems of Laplace transforms enable us to obtain a concise result for the perpetual case. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the boundaries.
Numerical Methods for Optimal Stopping Using Linear and NonLinear Programming
 In Series: Lecture Notes in Control and Information Sciences 280. Proceedings of a Workshop ”‘Stochastic Theory and Control”’, held in
, 2002
"... Computational methods for optimal stopping problems are presented. The rst method to be described is based on a linear programming approach to exit time problems of Markov processes and is applicable whenever the objective function is a unimodal function of a threshhold parameter which speci es a ..."
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Computational methods for optimal stopping problems are presented. The rst method to be described is based on a linear programming approach to exit time problems of Markov processes and is applicable whenever the objective function is a unimodal function of a threshhold parameter which speci es a stopping time. The second method, using linear and nonlinear programming techniques, is a modi cation of a general linear programming approach to optimal stopping problems recently proposed by S. Rohl. Both methods are illustrated by solving Shiryaev's quickest detection problem for Brownian motion.
The British Russian Option
"... Following the economic rationale of [10] and [11] we present a new class of lookback options (by first studying the canonical ‘Russian ’ variant) where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the ‘best prediction ’ of the Eur ..."
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Following the economic rationale of [10] and [11] we present a new class of lookback options (by first studying the canonical ‘Russian ’ variant) where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the ‘best prediction ’ of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British Russian option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimise his losses. The practical implications of this protection feature are most remarkable as not only is the option holder afforded a unique protection against unfavourable stock price movements (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. We derive a closed form expression for the arbitragefree price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British Russian option that leads to the conclusions above and shows that with the contract drift properly selected the British Russian option becomes a very attractive alternative to the classic European/American Russian option. 1.
time uncertainty
, 2007
"... This paper studies optimal stopping problems for general diffusion processes with an uncertain time horizon and with different levels of information based on which the decision to stop or to continue is made. First, corresponding value functions are compared and related explicitly to their counterp ..."
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This paper studies optimal stopping problems for general diffusion processes with an uncertain time horizon and with different levels of information based on which the decision to stop or to continue is made. First, corresponding value functions are compared and related explicitly to their counterparts without the time uncertainty. Second, to analyze optimal stopping strategies, characterization results regarding connectivity of the “stopping ” and “continuation ” regions are derived. In particular, sufficient conditions for the wellknown “threshold type ” optimal stopping rule are given, based on which explicit solutions are derived for problems with standard American call/put type payoff functions.
A Method For Computing Double Band Policies For Switching Between Two Diffusions
, 1996
"... : We develop a method for computing the optimal double band [b; B] policy for switching between two diffusions with continuous rewards and switching costs. The two switch levels [b; B] are obtained as perturbations of the single optimal switching point a of the control problem with no switching cos ..."
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: We develop a method for computing the optimal double band [b; B] policy for switching between two diffusions with continuous rewards and switching costs. The two switch levels [b; B] are obtained as perturbations of the single optimal switching point a of the control problem with no switching costs. More precisely, we find that in the case of average reward problems the optimal switch levels can be obtained by intersecting two curves: a) the function, fl(a), which represents the long run average reward if we were to switch between the two diffusions at a and switches were free and b) an horizontal line whose height depends on the size of the transaction costs. Our semianalytical approach reduces, for example, the solution of a problem recently posed by Perry and BarLev [20] to the solution of one nonlinear equation. A Method for Computing Double Band Policies for Switching Between Two Diffusions Florin Avram 1 Fikri Karaesmen 2 1 Department of Mathematics, Northeastern Uni...
Applied Stochastic Processes, 2003, 554, Rm 552, Mon 6:209PM
"... d the book and do the problems you will have mastered all that you will need to get a good grade. An important class of stochastic processes, or "random functions", X(t), satisfy E[X(t + s)jX(t)] = X(t) (1) which is called the martingale hypothesis; we say that the stochastic process is a martinga ..."
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d the book and do the problems you will have mastered all that you will need to get a good grade. An important class of stochastic processes, or "random functions", X(t), satisfy E[X(t + s)jX(t)] = X(t) (1) which is called the martingale hypothesis; we say that the stochastic process is a martingale. The reason people believe that stock prices follow the martingale law (1) is rather deep: people believe that all knowledge of the future has already been incorporated into the price of the stock so that the price is a "martingale" and no trends remain. That is, the expected value of the price at any time in the future is simply the present value. They believe that arbitrageurs have already correctly found the right value and they have removed all profits; this assumes the stock market is allknowing and wise and there is no way to obtain any additional information. To make the notion of martingale more precise we need to know some more probability theory. A random variable is a funct