Exercise Regions And Efficient Valuation Of American Lookback Options

This paper presents an efficient method to compute the values and early exercise boundaries of American fixed strike lookback options. The method reduces option valuation to a single optimal stopping problem for standard Brownian motion and an associated path-dependent functional, indexed by one parameter in the absence of dividends and by two parameters in the presence of a dividend rate. Numerical results obtained by this method show that, after a space-time transformation, the stopping boundaries are well approximated by certain piecewise linear functions with a few pieces, leading to fast and accurate approximations for American lookback option values. An explicit decomposition formula for American lookback options is derived and applied not only to the development of these approximations but also to the asymptotic analysis of the early exercise boundary near the expiration date.     

ON THE AMERICAN OPTION PROBLEM

We show how the change-of-variable formula with local time on curves derived recently in Peskir (2002) 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 Myneni (1992) and dating back to McKean (1965) .     

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.     

THE EARLY EXERCISE PREMIUM REPRESENTATION OF FOREIGN MARKET AMERICAN OPTIONS1

The note deals with the pricing of American options related to foreign market equities. the form of the early exercise premium representation of the American option's price in a stochastic interest rate economy is established. Subsequently, the American fixed exchange rate foreign equity option and the American equity-linked foreign exchange option are studied in detail.     

CONVEXITY OF THE EXERCISE BOUNDARY OF THE AMERICAN PUT OPTION ON A ZERO DIVIDEND ASSET

We show that the optimal exercise boundary for the American put option with non-dividend-paying asset is convex. With this convexity result, we then give a simple rigorous argument providing an accurate asymptotic behavior for the exercise boundary near expiry.     

A QUANTIZATION TREE METHOD FOR PRICING AND HEDGING MULTIDIMENSIONAL AMERICAN OPTIONS

We present here the quantization method which is well-adapted for the pricing and hedging of American options on a basket of assets. Its purpose is to compute a large number of conditional expectations by projection of the diffusion on optimal grids designed to minimize the (square mean) projection error ( Graf and Luschgy 2000 ). An algorithm to compute such grids is described. We provide results concerning the orders of the approximation with respect to the regularity of the payoff function and the global size of the grids. Numerical tests are performed in dimensions 2, 4, 5, 6, 10 with American style exchange options. They show that theoretical orders are probably pessimistic.     

American options on assets with dividends near expiry

Explicit expressions valid near expiry are derived for the values and the optimal exercise boundaries of American put and call options on assets with dividends. The results depend sensitively on the ratio of the dividend yield rate D to the interest rate r . For D > r the put boundary near expiry tends parabolically to the value rK / D where K is the strike price, while for D ≤ r the boundary tends to K in the parabolic-logarithmic form found for the case D =0 by Barles et al. (1995) and by Kuske and Keller (1998) . For the call, these two behaviors are interchanged: parabolic and tending to rK / D for D < r , as was shown by Wilmott, Dewynne, and Howison (1993) , and parabolic-logarithmic and tending to K for D ≥ r . The results are derived twice: once by solving an integral equation, and again by constructing matched asymptotic expansions.     

ON THE TIMING OPTION IN A FUTURES CONTRACT

The timing option embedded in a futures contract allows the short position to decide when to deliver the underlying asset during the last month of the contract period. In this paper we derive, within a very general incomplete market framework, an explicit model independent formula for the futures price process in the presence of a timing option. We also provide a characterization of the optimal delivery strategy, and we analyze some concrete examples.     

ANAYTICAL SOLUTIONS FOR THE PRICING OF AMERICAN BOND AND YIELD OPTIONS1

In this paper we use the Cox, Ingersoll, and Ross (1985b) single-factor, term structure model and extend it to the pricing of American default-free bond puts. We provide a quasi-analytical formula for these option prices based on recently established mathematical results for Bessel bridges, coupled with the optimal stopping time method. We extend our results to another interest rate contingent claim and provide a quasi-analytical solution for American yield option prices which illustrates the flexibility of our framework.     

OPTIMAL MULTIPLE STOPPING AND VALUATION OF SWING OPTIONS

The connection between optimal stopping of random systems and the theory of the Snell envelop is well understood, and its application to the pricing of American contingent claims is well known. Motivated by the pricing of swing contracts (whose recall components can be viewed as contingent claims with multiple exercises of American type) we investigate the mathematical generalization of these results to the case of possible multiple stopping. We prove existence of the multiple exercise policies in a fairly general set-up. We then concentrate on the Black–Scholes model for which we give a constructive solution in the perpetual case, and an approximation procedure in the finite horizon case. The last two sections of the paper are devoted to numerical results. We illustrate the theoretical results of the perpetual case, and in the finite horizon case, we introduce numerical approximation algorithms based on ideas of the Malliavin calculus.     

Pricing Options On Risky Assets In A Stochastic Interest Rate Economy1

This paper studies contingent claim valuation of risky assets in a stochastic interest rate economy. the model employed generalizes the approach utilized by Heath, Jarrow, and Morton (1992) by imbedding their stochastic interest rate economy into one containing an arbitrary number of additional risky assets. We derive closed form formulae for certain types of European options in this context, notably call and put options on risky assets, forward contracts, and futures contracts. We also value American contingent claims whose payoffs are permitted to be general functions of both the term structure and asset prices generalizing Bensoussan (1984) and Karatzas (1988) in this regard. Here, we provide an example where an American call's value is well defined, yet there does not exist an optimal trading strategy which attains this value. Furthermore, this example is not pathological as it is a generalization of Roll's (1977) formula for a call option on a stock that pays discrete dividends.     

Monte Carlo valuation of American options

This paper introduces a dual way to price American options, based on simulating the paths of the option payoff, and of a judiciously chosen Lagrangian martingale. Taking the pathwise maximum of the payoff less the martingale provides an upper bound for the price of the option, and this bound is sharp for the optimal choice of Lagrangian martingale. As a first exploration of this method, four examples are investigated numerically; the accuracy achieved with even very simple choices of Lagrangian martingale is surprising. The method also leads naturally to candidate hedging policies for the option, and estimates of the risk involved in using them.     

NONCONVEXITY OF THE OPTIMAL EXERCISE BOUNDARY FOR AN AMERICAN PUT OPTION ON A DIVIDEND‐PAYING ASSET

We prove that when the dividend rate of the underlying asset following a geometric Brownian motion is slightly larger than the risk‐free interest rate, the optimal exercise boundary of the American put option is not convex.     

存款保险的期权定价模型构造及实证研究

存款保险定价是存款保险制度建设中的核心内容,保险定价效率直接影响制度的功效。碍于现金流贴现估价模型的局限性,从期权的角度阐述了存款保险与期权的关系,指出存款保险合同实质上就是一份看跌期权,从理论和实证两方面论述了如何运用Black-Schole期权定价模型确定存款保险价格的问题,对实践中存款保险的合理定价和制度建设具有重要的指导意义。