CHARACTERIZATION OF OPTIMAL STOPPING REGIONS OF AMERICAN ASIAN AND LOOKBACK OPTIONS

A general framework is developed to analyze the optimal stopping (exercise) regions of American path-dependent options with either the Asian feature or lookback feature. We examine the monotonicity properties of the option values and stopping regions with respect to the interest rate, dividend yield, and time. From the ordering properties of the values of American lookback options and American Asian options, we deduce the corresponding nesting relations between the exercise regions of these American options. We illustrate how some properties of the exercise regions of the American Asian options can be inferred from those of the American lookback options.     

A NEW METHOD OF PRICING LOOKBACK OPTIONS

A new method for pricing lookback options (a.k.a. hindsight options) is presented, which simplifies the derivation of analytical formulas for this class of exotics in the Black-Scholes framework. Underlying the method is the observation that a lookback option can be considered as an integrated form of a related barrier option. The integrations with respect to the barrier price are evaluated at the expiry date to derive the payoff of an equivalent portfolio of European-type binary options. The arbitrage-free price of the lookback option can then be evaluated by static replication as the present value of this portfolio. We illustrate the method by deriving expressions for generic, standard floating-, fixed-, and reverse-strike lookbacks, and then show how the method can be used to price the more complex partial-price and partial-time lookback options. The method is in principle applicable to frameworks with alternative asset-price dynamics to the Black-Scholes world.     

BOUNDARY EVOLUTION EQUATIONS FOR AMERICAN OPTIONS

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.     

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.     

NONREPLICATION OF OPTIONS

In this paper, we study the replication of options in security markets X with a finite number of states. Specifically, we prove that in security markets without binary vectors, for any portfolio, at most m ? 3 options can be replicated where m is the number of states. This is an essential improvement of the result of Baptista where it is proved that the set of replicated options is of measure zero. Additionally, we extend the results of Aliprantis and Tourky on the nonreplication of options by generalizing their condition that markets are strongly resolving. Our results are based on the theory of lattice‐subspaces and positive bases.     

PERFECT AND PARTIAL HEDGING FOR SWING GAME OPTIONS IN DISCRETE TIME

The paper introduces and studies hedging for game (Israeli) style extension of swing options considered as multiple exercise derivatives. Assuming that the underlying security can be traded without restrictions, we derive a formula for valuation of multiple exercise options via classical hedging arguments. Introducing the notion of the shortfall risk for such options we study also partial hedging which leads to minimization of this risk.     

DYNAMIC SPANNING: ARE OPTIONS AN APPROPRIATE INSTRUMENT?1

Ross (1976) has shown, in a static framework, how options can complete financial markets. This paper examines the possible extensions of Ross's idea in a dynamic setup. Surprisingly enough, we find that the answer is very sensitive to the choice of the stochastic model for the underlying security returns. More specifically we obtain the following results: In a discrete-time model, classical European options typically become redundant with some probability (Proposition 2.1). Obnly path dependent (“exotic”) options may generate dynamic spanning (Proposition 4.1). In a continuous-time model with stochastic volatility of the underlying security, and under reasonable assumptions, a European option is always a good instrument for completing markets (Proposition 5.2).     

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.     

MONTE CARLO METHODS FOR THE VALUATION OF MULTIPLE-EXERCISE OPTIONS

We discuss Monte Carlo methods for valuing options with multiple-exercise features in discrete time. By extending the recently developed duality ideas for American option pricing, we show how to obtain estimates on the prices of such options using Monte Carlo techniques. We prove convergence of our approach and estimate the error. The methods are applied to options in the energy and interest rate derivative markets.     

PORTFOLIOS OF AMERICAN OPTIONS UNDER GENERAL PREFERENCES: RESULTS AND COUNTEREXAMPLES

We consider the optimal exercise of a portfolio of American call options in an incomplete market. Options are written on a single underlying asset but may have different characteristics of strikes, maturities, and vesting dates. Our motivation is to model the decision faced by an employee who is granted options periodically on the stock of her company, and who is not permitted to trade this stock. The first part of our study considers the optimal exercise of single options. We prove results under minimal assumptions and give several counterexamples where these assumptions fail—describing the shape and nesting properties of the exercise regions. The second part of the study considers portfolios of options with differing characteristics. The main result is that options with comonotonic strike, maturity, and vesting date should be exercised in order of increasing strike. It is true under weak assumptions on preferences and requires no assumptions on prices. Potentially the exercise ordering result can significantly reduce the complexity of computations in a particular example. This is illustrated by solving the resulting dynamic programming problem in a constant absolute risk aversion utility indifference model.     

股票期权计划及其在我国的应用

本文介绍了股票期权计划作为一种新的长期激励制度的特征和实施的内容,并从人力资本产权理论、团队生产理论、风险理论和契约理论四方面阐述了股票期权计划的理论基础,进而结合我国的具体实际分析了在中国实施股票期权计划面临的一些限制性因素,从而就我国实行股票期权计划提出了一些建议。     

GAME CALL OPTIONS REVISITED

In this paper, having been inspired by the work of Kunita and Seko, we study the pricing of δ‐penalty game call options on a stock with a dividend payment. For the perpetual case, our result reveals that the optimal stopping region for the option seller depends crucially on the dividend rate d. More precisely, we show that when the penalty δ is small, there are two critical dividends 0 < d₁ < d₂ < ∞ such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if d < d₁; (2) a singleton if d∈ [d₁, d₂]; or (3) an empty set if d > d₂. When d∈ [d₁, d₂], the value function is not continuously differentiable at the optimal stopping boundary for the option seller, therefore our result in the perpetual case cannot be established by the free boundary approach with smooth‐fit conditions imposed on both free boundaries. For the finite time horizon case, the dependence of the optimal stopping region for the option seller on the time to maturity is exhibited; more precisely, when both δ and d are small, we show that there are two critical times 0 < T₁ < T₂ < T, such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if t < T₁; (2) a singleton if t∈ [T₁, T₂]; or (3) an empty set if t > T₂. In summary, for both the perpetual and the finite horizon cases, we characterize in terms of model parameters how the optimal stopping region for the option seller shrinks when the dividend rate d increases and the time to maturity decreases; these results complete the original work of Emmerling for the perpetual case and Kunita and Seko for the finite maturity case. In addition, for the finite time horizon case, we also extend the probabilistic method for the establishment of existence and regularity results of the classical American option pricing problem to the game option setting. Finally, we characterize the pair of optimal stopping boundaries for both the seller and the buyer as the unique pair of solutions to a couple of integral equations and provide numerical illustrations.     

DEFAULTABLE OPTIONS IN A MARKOVIAN INTENSITY MODEL OF CREDIT RISK

This paper is a follow‐up to “Valuation and Hedging of Defaultable Game Options in a Hazard Process Model” by the same authors. In the present paper we give user friendly assumptions ensuring that the general conditions in the previous paper are satisfied. We also give a systematic procedure to construct suitable intensity models of credit risk, and, in the Markovian case, we provide a variational inequality approach to the pre‐default pricing problem. We finally illustrate our results on a study of defaultable convertible bonds.     

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.     

PRICING A CLASS OF EXOTIC OPTIONS VIA MOMENTS AND SDP RELAXATIONS

We present a new methodology for the numerical pricing of a class of exotic derivatives such as Asian or barrier options when the underlying asset price dynamics are modeled by a geometric Brownian motion or a number of mean-reverting processes of interest. This methodology identifies derivative prices with infinite-dimensional linear programming problems involving the moments of appropriate measures, and then develops suitable finite-dimensional relaxations that take the form of semidefinite programs (SDP) indexed by the number of moments involved. By maximizing or minimizing appropriate criteria, monotone sequences of both upper and lower bounds are obtained. Numerical investigation shows that very good results are obtained with only a small number of moments. Theoretical convergence results are also established.     

CALLABLE PUTS AS COMPOSITE EXOTIC OPTIONS

Introduced by Kifer (2000) , game options function in the same way as American options with the added feature that the writer may also choose to exercise, at which time they must pay out the intrinsic option value of that moment plus a penalty. In Kyprianou (2004) an explicit formula was obtained for the value function of the perpetual put option of this type. Crucial to the calculations which lead to the aforementioned formula was the perpetual nature of the option. In this paper we address how to characterize the value function of the finite expiry version of this option via mixtures of other exotic options by using mainly martingale arguments.     

TRUE UPPER BOUNDS FOR BERMUDAN PRODUCTS VIA NON-NESTED MONTE CARLO

We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic "true" stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale can be employed for computing tight dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator. Numerical experiments indicate the efficiency of the proposed algorithms.     

OPTIONS AND EFFICIENCY IN MULTIDATE SECURITY MARKETS

This paper extends the work of Ross (1976; Q. J. Econ. (90)1, 75–89) to multidate security markets. First, we show that if a primitive security separates states at the terminal date, then there exist multiperiod European options on that security generating dynamically complete markets. Second, we show that if a primitive security conditionally separates states at the terminal date, then there exist multiperiod European options on that security generating generically dynamically complete markets provided that certain conditions hold. Third, we show that there are economies for which the minimum number of multiperiod European options on a primitive security generating generically dynamically complete markets is relatively large. Finally, we show that in these economies, a relatively small number of multiperiod European options on possibly different portfolio strategies of primitive securities generates generically dynamically complete markets.     

Monte Carlo Evaluation of Greeks for Multidimensional Barrier and Lookback Options

In this paper, we consider the problem of the numerical computation of Greeks for a multidimensional barrier and lookback style options: the payoff function depends in a rather general way on the minima and maxima of the coordinates of the d -dimensional underlying asset process. Using Malliavin calculus techniques, we derive additional weights that enable computation of the Greeks using Monte Carlo simulations. Numerical experiments confirm the efficiency of the method. This work is a multidimensional extension of previous results (see Gobet and Kohatsu-Higa 2001 ).     

STOCHASTIC VOLATILITY MODELS AND THE PRICING OF VIX OPTIONS

In this paper, we examine and compare the performance of a variety of continuous‐time volatility models in their ability to capture the behavior of the VIX. The “3/2‐ model” with a diffusion structure which allows the volatility of volatility changes to be highly sensitive to the actual level of volatility is found to outperform all other popular models tested. Analytic solutions for option prices on the VIX under the 3/2‐model are developed and then used to calibrate at‐the‐money market option prices.