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92
Option pricing when underlying stock returns are discontinuous
 Journal of Financial Economics
, 1976
"... The validity of the classic BlackScholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying ..."
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Cited by 688 (1 self)
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The validity of the classic BlackScholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the moregeneral cast when the underlying stock returns are gcncrated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black&holes formula in that it does not dcpcnd on investor prcfcrenccs or knowledge of the expcctsd return on the underlying stock. Morcovcr, the same analysis applied to the options can bc extcndcd to the pricingofcorporatc liabilities. 1. Intruduction In their classic paper on the theory of option pricing, Black and Scholcs (1973) prcscnt a mode of an:llysis that has rcvolutionizcd the theory of corporate liability pricing. In part, their approach was a breakthrough because it leads to pricing formulas using. for the most part, only obscrvablc variables. In particular,
Numerical Valuation of High Dimensional Multivariate American Securities
, 1994
"... We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted ..."
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Cited by 104 (0 self)
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We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted expected cash flows under the modified riskneutral information process. Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either latticebased techniques or finite difference approximations of the BlackScholes diffusion equation. However, these methods cannot be used for highdimensional problems, since their memory requirement is exponential in the
Randomization and the American Put
 The Review of Financial Studies
, 1998
"... Conference. In particular, I am grateful to an unknown RFS referee, Kerry Back, Michael Brennan, Darrell Du e, ..."
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Cited by 73 (1 self)
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Conference. In particular, I am grateful to an unknown RFS referee, Kerry Back, Michael Brennan, Darrell Du e,
Alternative characterizations of American put options
 Mathematical Finance
, 1992
"... Viswanathan, and the participants of workshops at Vanderbilt University and Cornell University. The first two authors are grateful for financial support from Banker’s Trust. We are particularly grateful to Henry McKean for many valuable discussions. Alternative Characterizations of American Put Opti ..."
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Cited by 56 (2 self)
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Viswanathan, and the participants of workshops at Vanderbilt University and Cornell University. The first two authors are grateful for financial support from Banker’s Trust. We are particularly grateful to Henry McKean for many valuable discussions. Alternative Characterizations of American Put Options We derive alternative representations of the McKean equation for the value of the American put option. Our main result decomposes the value of an American put option into the corresponding European put price and the early exercise premium. We then represent the European put price in a new manner. This representation allows us to alternatively decompose the price of an American put option into its intrinsic value and time value, and to demonstrate the equivalence of our results to the McKean equation. Alternative Characterizations of American Put Options The problem of valuing American options continues to intrigue finance theorists. For example, in
Option pricing under a double exponential jump diffusion model
 Management Science
, 2004
"... Analytical tractability is one of the challenges faced by many alternative models that try to generalize the BlackScholes option pricing model to incorporate more empirical features. The aim of this paper is to extend the analytical tractability of the BlackScholes model to alternative models with ..."
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Cited by 55 (3 self)
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Analytical tractability is one of the challenges faced by many alternative models that try to generalize the BlackScholes option pricing model to incorporate more empirical features. The aim of this paper is to extend the analytical tractability of the BlackScholes model to alternative models with jumps. We demonstrate a double exponential jump diffusion model can lead to an analytic approximation for Þnite horizon American options (by extending the BaroneAdesi and Whaley method) and analytical solutions for popular pathdependent options (such as lookback, barrier, and perpetual American options). Numerical examples indicate that the formulae are easy to be implemented and accurate.
Pricing of American PathDependent Contingent Claims
, 1994
"... We consider the problem of pricing pathdependent contingent claims. Classically, this problem can be cast into the BlackScholes valuation framework through inclusion of the pathdependent variables into the state space. This leads to solving a degenerate advectiondiffusion Partial Differential Eq ..."
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Cited by 46 (1 self)
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We consider the problem of pricing pathdependent contingent claims. Classically, this problem can be cast into the BlackScholes valuation framework through inclusion of the pathdependent variables into the state space. This leads to solving a degenerate advectiondiffusion Partial Differential Equation (PDE). Standard Finite Difference (FD) methods are known to be inadequate for solving such degenerate PDE. Hence, pathdependent European claims are typically priced through MonteCarlo simulation. To date, there is no numerical method for pricing pathdependent American claims. We first establish necessary and sufficient conditions amenable to a Lie algebraic characterization, under which degenerate diffusions can be reduced to lowerdimensional nondegenerate diffusions on a submanifold of the underlying asset space. We apply these results to pathdependent options. Then, we describe a new numerical technique, called Forward Shooting Grid (FSG) method, that efficiently copes with de...
Pricing and Hedging American Options: A Recursive Integration Method
 Review of Financial Studies
, 1996
"... In this paper, we present a new method for pricing and hedging American options along with an efficient implementation procedure. The proposed method is efficient and accurate in computing both option values and various option hedge parameters. We demonstrate the computational accuracy and efficienc ..."
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Cited by 39 (3 self)
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In this paper, we present a new method for pricing and hedging American options along with an efficient implementation procedure. The proposed method is efficient and accurate in computing both option values and various option hedge parameters. We demonstrate the computational accuracy and efficiency of this numerical procedure in relation to other competing approaches. We also suggest how the method can be applied to the case of any American option for which a closedform solution exists for the corresponding European option. A variety of financial products such as fixedincome derivatives, mortgagebacked securities and corporate securities have earlyexercise or Americanstyle features that significantly influence their valuation and hedging. Considerable interest exists, therefore, in both academic and practitioner circles, in methods of valuation and hedging Americanstyle options that are conceptually sound, as well as efficient in their implementation. It has been recognized early in the ...
On the American option problem
 Math. Finance
, 2005
"... We show how the changeofvariable formula with local time on curves derived recently in [17] can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium repre ..."
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Cited by 24 (7 self)
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We show how the changeofvariable formula with local time on curves derived recently in [17] can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation. This settles the question raised in [15] (dating back to [13]). 1.
Investment timing under incomplete information
 Math. Oper. Res
, 2005
"... We study the decision of when to invest in a project whose value is perfectly observable but driven by a parameter that is unknown to the decisionmaker exante. This problem is equivalent to an optimal stopping problem for a bivariate Markov process. Using filtering and martingale techniques, we sh ..."
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Cited by 20 (1 self)
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We study the decision of when to invest in a project whose value is perfectly observable but driven by a parameter that is unknown to the decisionmaker exante. This problem is equivalent to an optimal stopping problem for a bivariate Markov process. Using filtering and martingale techniques, we show that the optimal investment region is characterized by a continuous and nondecreasing boundary in the valuebelief state space. This generates pathdependency in the optimal investment strategy. We further show that the decisionmaker always benefit from an uncertain drift relative to an average drift situation, and that the value of the option to invest is not globally increasing with respect to the volatility of the value process.
Optimal exercise boundary for an American put option
 Appl. Math. Fin
, 1998
"... The optimal exercise boundary near the expiration time is determined for an American put option. It is obtained by using Green's theorem to convert the boundary value problem for the price of the option into an integral equation for the optimal exercise boundary. This integral equation is solve ..."
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Cited by 20 (0 self)
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The optimal exercise boundary near the expiration time is determined for an American put option. It is obtained by using Green's theorem to convert the boundary value problem for the price of the option into an integral equation for the optimal exercise boundary. This integral equation is solved asymptotically for small values of the time to expiration. The leading term in the asymptotic solution is the result of Barles et al. [1]. An asymptotic solution for the option price is obtained also.