The validity of the classic Black-Scholes 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 more-general 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,

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 risk-neutral information process. Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the

Conference. In particular, I am grateful to an unknown RFS referee, Kerry Back, Michael Brennan, Darrell Du e,

We consider the problem of pricing path-dependent contingent claims. Classically, this problem can be cast into the Black-Scholes valuation framework through inclusion of the path-dependent variables into the state space. This leads to solving a degenerate advection-diffusion Partial Differential Equation (PDE). Standard Finite Difference (FD) methods are known to be inadequate for solving such degenerate PDE. Hence, path-dependent European claims are typically priced through Monte-Carlo simulation. To date, there is no numerical method for pricing path-dependent American claims. We first establish necessary and sufficient conditions amenable to a Lie algebraic characterization, under which degenerate diffusions can be reduced to lower-dimensional non-degenerate diffusions on a sub-manifold 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...

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. The problem of valuing American options continues to intrigue finance theorists. For example, in the New Palgrave Dictionary of Economics, Ross (1987) writes: This does not mean, however, that there are no important gaps in the (option pricing) theory.

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 closed-form solution exists for the corresponding European option. A variety of financial products such as fixed-income derivatives, mortgage-backed securities and corporate securities have early-exercise or American-style features that significantly influence their valuation and hedging. Considerable interest exists, therefore, in both academic and practitioner circles, in methods of valuation and hedging American-style options that are conceptually sound, as well as efficient in their implementation. It has been recognized early in the ...

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.

A model of an incomplete market with the incorporation of a new notion of "inside information" is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the Black-Scholes exponential Brownian motion model for stock fluctuations. The drift and volatility parameters take different values when the hidden Markov process is in different states. For example, it is 0 when there is no subset of the market which has or which believes it has, extra information. However, the hidden process is in state 1 when information is not equally shared by all, and then the behavior of the members in the subset causes increased fluctuations in the stock price. This model

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We solve the following three optimal stopping problems for dierent kinds of options, based on the Black-Scholes model of stock uctuations: (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more dicult than the closely related one for the Russian option and we show that for a class of utility functions the free boundary is governed by a nonlinear ordinary dierential equation. (ii) A new type of stock option for a company, where the company provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low, thereby making the option more attractive to the employee. We show that the value of this option is given by solving a non-algebraic equation. (iii) A new call option for the option buyer who is risk-averse and gets to choose, a priori, a xed constant l as a \hedge" on a possible downturn of the stock price, where the buyer gets the maximum of l and the price at ...