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.     

VALUATION OF CONTINUOUSLY MONITORED DOUBLE BARRIER OPTIONS AND RELATED SECURITIES

In this paper, we apply Carr's randomization approximation and the operator form of the Wiener‐Hopf method to double barrier options in continuous time. Each step in the resulting backward induction algorithm is solved using a simple iterative procedure that reduces the problem of pricing options with two barriers to pricing a sequence of certain perpetual contingent claims with first‐touch single barrier features. This procedure admits a clear financial interpretation that can be formulated in the language of embedded options. Our approach results in a fast and accurate pricing method that can be used in a rather wide class of Lévy‐driven models including Variance Gamma processes, Normal Inverse Gaussian processes, KoBoL processes, CGMY model, and Kuznetsov's β ‐class. Our method can be applied to double barrier options with arbitrary bounded terminal payoff functions, which, in particular, allows us to price knock‐out double barrier put/call options as well as double‐no‐touch options.     

DUAL REPRESENTATIONS FOR GENERAL MULTIPLE STOPPING PROBLEMS

In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cash flows which are subject to volume constraints modeled by integer‐valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers ( 2012 ), Bender ( 2011a ), Bender ( 2011b ), Aleksandrov and Hambly ( 2010 ), and Meinshausen and Hambly ( 2004 ) on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cash flow structures than the additive structure in the above references. For example, some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for prices of multiple exercise options and illustrate it with a numerical study on the pricing of a swing option in an electricity market.     

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 INVESTMENT WITH AN UNBOUNDED RANDOM ENDOWMENT AND UTILITY-BASED PRICING

This paper studies the problem of maximizing the expected utility of terminal wealth for a financial agent with an unbounded random endowment, and with a utility function which supports both positive and negative wealth. We prove the existence of an optimal trading strategy within a class of permissible strategies—those strategies whose wealth process is a super-martingale under all pricing measures with finite relative entropy. We give necessary and sufficient conditions for the absence of utility-based arbitrage, and for the existence of a solution to the primal problem. We consider two utility-based methods which can be used to price contingent claims. Firstly we investigate marginal utility-based price processes (MUBPP's). We show that such processes can be characterized as local martingales under the normalized optimal dual measure for the utility maximizing investor. Finally, we present some new results on utility indifference prices, including continuity properties and volume asymptotics for the case of a general utility function, unbounded endowment and unbounded contingent claims.     

Double continuation regions for American and Swing options with negative discount rate in Lévy models

In this paper, we study perpetual American call and put options in an exponential Lévy model. We consider a negative effective discount rate that arises in a number of financial applications including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor. We show that in this case a double continuation region arises and we identify the two critical prices. We also generalize this result to multiple stopping problems of Swing type, that is, when successive exercise opportunities are separated by i.i.d. random refraction times. We conduct an extensive numerical analysis for the Black–Scholes model and the jump‐diffusion model with exponentially distributed jumps.     

PRICING OF AMERICAN PATH-DEPENDENT CONTINGENT CLAIMS

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). We first estabilish necessary and sufficient conditions under which degenerate diffusions can be reduced to lower-dimensional nondegenerate diffusions. We apply these results to path-dependent options. Then, we describe a new numerical technique, called forward shooting grid (FSG) method, that efficiently copes with degenerate diffusion PDEs. Finally, we show that the FSG method is unconditionally stable and convergent. the FSG method is the first capable of dealing with the early exercise condition of American options. Several numerical examples are presented and discussed. ²     

A GENERAL FRAMEWORK FOR PRICING CREDIT RISK

A framework is provided for pricing derivatives on defaultable bonds and other credit-risky contingent claims. The framework is in the spirit of reduced-form models, but extends these models to include the case that default can occur only at specific times, such as coupon payment dates. Although the framework does not provide an efficient setting for obtaining results about structural models, it is sufficiently general to include most structural models, and thereby highlights the commonality between reduced-form and structural models. Within the general framework, multiple recovery conventions for contingent claims are considered: recovery of a fraction of par, recovery of a fraction of a no-default version of the same claim, and recovery of a fraction of the pre-default value of the claim. A stochastic-integral representation for credit-risky contingent claims is provided, and the integrand for the credit exposure part of this representation is identified. In the case of intensity-based, reduced-form models, credit spread and credit-risky term structure are studied.     

Viability and Equilibrium in Securities Markets with Frictions

In this paper we study some foundational issues in the theory of asset pricing with market frictions. We model market frictions by letting the set of marketed contingent claims (the opportunity set) be a convex set, and the pricing rule at which these claims are available be convex. This is the reduced form of multiperiod securities price models incorporating a large class of market frictions. It is said to be viable as a model of economic equilibrium if there exist price-taking maximizing agents who are happy with their initial endowment, given the opportunity set, and hence for whom supply equals demand. This is equivalent to the existence of a positive lineaar pricing rule on the entirespace of contingent claims—an underlying frictionless linear pricing rule—that lies below the convex pricing rule on the set of marketed claims. This is also equivalent to the absence of asymptotic free lunches—a generalization of opportunities of arbitrage. When a market for a nonmarketed contingent claim opens, a bid-ask price pair for this claim is said to be consistent if it is a bid-ask price pair in at least a viable economy with this extended opportunity set. If the set of marketed contingent claims is a convex cone and the pricing rule is convex and sublinear, we show that the set of consistent prices of a claim is a closed interval and is equal (up to its boundary) to the set of its prices for all the underlying frictionless pricing rules. We also show that there exists a unique extended consistent sublinear pricing rule—the supremum of the underlying frictionless linear pricing rules—for which the original equilibrium does not collapse when a new market opens, regardless of preferences and endowments. If the opportunity set is the reduced form of a multiperiod securities market model, we study the closedness of the interval of prices of a contingent claim for the underlying frictionless pricing rules.     

Fair bilateral pricing under funding costs and exogenous collateralization

Bielecki and Rutkowski introduced and studied a generic nonlinear market model, which includes several risky assets, multiple funding accounts, and margin accounts. In this paper, we examine the pricing and hedging of contract from the perspective of both the hedger and the counterparty with arbitrary initial endowments. We derive inequalities for unilateral prices and we study the range of fair bilateral prices. We also examine the positive homogeneity and monotonicity of unilateral prices with respect to the initial endowments. Our study hinges on results from Nie and Rutkowski for backward stochastic differential equations (BSDEs) driven by continuous martingales, but we also derive the pricing partial differential equations (PDEs) for path‐independent contingent claims of a European style in a Markovian framework.     

Nonparametric American option pricing

A nonparametric method is introduced to accurately price American-style contingent claims. This method uses only historical stock price data, not option price data, to generate the American option price. The accuracy of this method is tested in a controlled experimental environment under both Black, F and Scholes, M (1973) and Heston, S (1993) assumptions, and an error-metric analysis is performed. These numerical experiments demonstrate that this method is an accurate and precise method of pricing American options under a variety of market conditions. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:717–748, 2008     

Pricing American Stock Options by Linear Programming

We investigate numerical solution of finite difference approximations to American option pricing problems, using a new direct numerical method: simplex solution of a linear programming formulation. This approach is based on an extension to the parabolic case of the equivalence between linear order complementarity problems and abstract linear programs known for certain elliptic operators. We test this method empirically, comparing simplex and interior point algorithms with the projected successive overrelaxation (PSOR) algorithm applied to the American vanilla and lookback puts. We conclude that simplex is roughly comparable with projected SOR on average (faster for fine discretizations, slower for coarse), but is more desirable for robustness of solution time under changes in parameters. Furthermore, significant speedups over the results given here have been achieved and will be published elsewhere.     

STOCHASTIC HYPERBOLIC DYNAMICS FOR INFINITE-DIMENSIONAL FORWARD RATES AND OPTION PRICING

We model the term-structure modeling of interest rates by considering the forward rate as the solution of a stochastic hyperbolic partial differential equation. First, we study the arbitrage-free model of the term structure and explore the completeness of the market. We then derive results for the pricing of general contingent claims. Finally we obtain an explicit formula for a forward rate cap in the Gaussian framework from the general results.     

The Contributions of Professors Fischer Black,Robert Merton and Myron Scholes to the Financial Services Industry

This paper is written as a tribute to Professors Robert Merton and Myron Scholes, winners of the 1997 Nobel Prize in economics, as well as to their collaborator, the late Professor Fischer Black. We first provide a brief and very selective review of their seminal work in contingent claims pricing. We then provide an overview of some of the recent research on stock price dynamics as it relates to contingent claim pricing. The continuing intensity of this research, some 25 years after the publication of the original Black–Scholes paper, must surely be regarded as the ultimate tribute to their work. We discuss jump‐diffusion and stochastic volatility models, subordinated models, fractal models and generalized binomial tree models for stock price dynamics and option pricing. We also address questions as to whether derivatives trading poses a systemic risk in the context of models in which stock price movements are endogenized, and give our views on the ‘LTCM crisis’ and liquidity risk.     

PRICING AND SEMIMARTINGALE REPRESENTATIONS OF VULNERABLE CONTINGENT CLAIMS IN REGIME‐SWITCHING MARKETS

Using a suitable change of probability measure, we obtain a Poisson series representation for the arbitrage‐free price process of vulnerable contingent claims in a regime‐switching market driven by an underlying continuous‐time Markov process. As a result of this representation, along with a short‐time asymptotic expansion of the claim's price process, we develop an efficient novel method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path‐dependent claims that we term self‐decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the predefault price function of European vulnerable claims, which enables us to rigorously deduce Feynman‐Ka? representations for the predefault pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk‐neutral and objective probability measures.     

Equilibrium pricing of contingent claims in tradable permit markets

We advance a model of the tradable permit market and derive a pricing formula for contingent claims traded in the market in a general equilibrium framework. It is shown that prices of such contingent claims exhibit significantly different properties from those in the ordinary financial markets. In particular, if the social cost function kinks at some level of abatement, the forward price, as well as the spot price, can be subject to the so‐called price spike. However, this price‐spike phenomenon can be weakened if a system of banking and borrowing is properly introduced. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:559–589, 2010     

HAZARD PROCESSES AND MARTINGALE HAZARD PROCESSES

In this paper, we build a bridge between different reduced‐form approaches to pricing defaultable claims. In particular, we show how the well‐known formulas by Duffie, Schroder, and Skiadas and by Elliott, Jeanblanc, and Yor are related. Moreover, in the spirit of Collin Dufresne, Hugonnier, and Goldstein, we propose a simple pricing formula under an equivalent change of measure. Two processes will play a central role: the hazard process and the martingale hazard process attached to a default time. The crucial step is to understand the difference between them, which has been an open question in the literature so far. We show that pseudo‐stopping times appear as the most general class of random times for which these two processes are equal. We also show that these two processes always differ when τ is an honest time, providing an explicit expression for the difference. Eventually we provide a solution to another open problem: we show that if τ is an arbitrary random (default) time such that its Azéma's supermartingale is continuous, then τ avoids stopping times.     

Contingent Claims and Market Completeness in a Stochastic Volatility Model

In an incomplete market framework, contingent claims are of particular interest since they improve the market efficiency. This paper addresses the problem of market completeness when trading in contingent claims is allowed. We extend recent results by Bajeux and Rochet (1996) in a stochastic volatility model to the case where the asset price and its volatility variations are correlated. We also relate the ability of a given contingent claim to complete the market to the convexity of its price function in the current asset price. This allows us to state our results for general contingent claims by examining the convexity of their "admissible arbitrage prices."     

First-Order Schemes in the Numerical Quantization Method

The numerical quantization method is a grid method that relies on the approximation of the solution to a nonlinear problem by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first-order scheme based on piecewise linear approximations. Main ingredients are correction terms in the transition probability weights. We emphasize the fact that in the case of optimal quantization, many of these correcting terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and a priori estimates of the errors are proposed.     

Pricing by Arbitrage Under Arbitrary Information

A substantial applications literature on pricing by arbitrage has effectively restricted information to that arising solely from securities markets; return distributions are then governed solely by past prices. We reconsider pricing by arbitrage in markets rendered incomplete by arbitrary information, which, moreover, may influence required returns. We show that contingent claims depending solely on securities' normalized price histories are priced by arbitrage if and only if all risk-adjusted probabilities agree upon the law of primitive securities' normalized prices. We show how existing diffusion-based results can be preserved, and resolve an issue relating to Merton's (1973) stochastic interest rate model.