The valuation of multivariate contingent claims under transformed trinomial approaches

This study develops a transformed-trinomial approach for the valuation of contingent claims written on multiple underlying assets. Our model is characterized by an extension of the Camara and Chung (J Futur Mark 26: 759–787, 2006) transformed-binomial model for pricing options with one underlying asset, and a discrete-time version of the Schroder (J Finance 59(5): 2375–2401, 2004) model. However, unlike the Schroder model, our model can facilitate straightforward valuation of American-style multivariate contingent claims. The major advantage of our transformed-trinomial approach is that it can easily tackle the volatility skew observed within the markets. We go on to use numerical examples to demonstrate the way in which our transformed-trinomial approach can be utilized for the valuation of multivariate contingent claims, such as binary options.     

The term structure of interest rates: Alternative approaches and their implications for the valuation of contingent claims

One of the most active areas of research in financial economics has been the modeling of the term structure of interest rates and its relationship to the pricing of contingent claims. There is a vast array of issues in the area, as well as a variety of perspectives, ranging from theoretical to practical. This article provides a general framework for the analysis of issues in the modeling of the term structure. Specifically, this article provides an overview of the conceptual issues and the empirical evidence in the area, based on an examination of five seminal models by Black, Scholes, and Merton; Vasicek; Cox, Ingersoll, and Ross; Ho and Lee; and Heath, Jarrow, and Morton. The article provides a synthesis of the area and suggests directions for future research.     

International money and stock market contingent claims

We develop a unified approach with closed-form solutions for pricing bonds, stocks, currencies and their derivatives. The specification assumes a fundamental risk factor represented by a stochastic positive definite matrix following a Wishart autoregressive (WAR) process. By assuming a volatility-in-mean specification for the domestic stock returns and the relative changes of exchange rates, and a domestic stochastic discount factor exponential affine with respect to the fundamental risk, it is possible to derive closed form solutions for the term structures of interest rates and for the risk-neutral probabilities while keeping the flexibility of the model. In particular:

i) The domestic and foreign term structures are jointly affine and correspond to Wishart quadratic term structures, which can ensure the positivity of interest rates;

ii) In this framework where the stock price follows a model with stochastic volatility, we obtain explicit or quasi-explicit formulas for futures and forward contracts, swaps and options. This extends results by

Heston (1993)

and

Ball and Roma (1994)

.

Optimal replication of contingent claims under portfolio constraints

We determine the minimum cost of super-replicating a nonnegativecontingent claim when there are convex constraints on portfolioweights. We show that the optimal cost with constraints is equalto the price of a related claim without constraints. The relatedclaim is a dominating claim, that is, a claim whose payoffsare increased in an appropriate way relative to the originalclaim. The results hold for a variety of options, includingsome path-dependent options. Constraints on the gamma of thereplicating portfolio, constraints on the portfolio amounts,and constraints on the number of shares are also considered.     

Private acquisition gains: A contingent claims explanation

This paper studies announcement returns of Western European acquisitions of private and public targets. It uses a contingent claims perspective to offer a new explanation for the difference in abnormal returns between acquirers of private and public targets. In this context, an acquisition is analogous to buying a call option and the value of the acquirer increases with uncertainty about its growth prospects (options). We test this idea by studying the relation between announcement returns and acquirer's characteristics that proxy for the existence of growth options. Consistent with the contingent claims hypothesis, the private acquisition gains are associated with the combined effects of growth options (having higher runup before the acquisition announcement) with low level of leverage (near-all equity capital) and with uncertainty (measured by age and analyst coverage of acquirers).     

Convergence from discrete- to continuous-time contingent claims prices

This article generalizes the Cox, Ross, and Rubinstein (1979)binomial option-pricing model, and establishes a convergencefrom discrete-time multivariate multinomial models to continuous-timemultidimensional diffusion models for contingent claims prices.The key to the approach is to approximate the N-dimensionaldiffusion price process by a sequence of N-variate, (N+1)-nomialprocess. It is shown that contingent claims prices and dynamicreplicating portfolio strategies derived from the discrete timemodels converge to their corresponding continuous-time limits.     

Explicit tests of contingent claims models of mortgage default

This paper provides explicit and powerful tests of contingent claims approaches to modeling mortgage default. We investigate a model of frictionless default (i.e., one in which transactions costs, reputation costs, and moving costs play no role) and analyze its implications-the relationship between equity and default, the timing of default, its dependence upon initial conditions, and the severity of losses. Absent transactions costs and other market imperfections, economic theory makes well-defined predictions about these various outcomes.The empirical analysis is based upon two particularly rich bodies of micro data: one indicating the default and loss experience of all mortgages purchased by the Federal Home Mortgage Corporation (Freddie Mac), and a large sample of all repeat sales of single family houses whose mortgages were purchased by Freddie Mac since 1976.     

Interest-only and principal-only mortgage strips as interest-rate contingent claims

Interest-only (IO) and principal-only (PO) mortgage strips are valued in a stochastic interest-rate environment. The prepayment rate of the underlying mortgages is affected by two considerations not present in the pure financially rational model: (1) The property owner's holding period is assumed to follow a Gamma distribution, resulting in the possibility of prepayment due to the sale of the property (i.e., prepayment that is too early based on market interest rates); and (2) borrowers are assumed to face heterogeneous transaction costs related to refinancing the existing mortgage, and delay refinancing when market conditions make it optimal to do so (refinancing too late). Properties of IO/PO strips are identified by the finite difference method.     

Hazard rate for credit risk and hedging defaultable contingent claims

We provide a concise exposition of theoretical results that appear in modeling default time as a random time, we study in details the invariance martingale property and we establish a representation theorem which leads, in a complete market setting, to the hedging portfolio of a vulnerable claim. Our main result is that, to hedge a defaultable claim one has to invest the value of this contingent claim in the defaultable zero-coupon.Received: April 2003Mathematics Subject Classification: 91B24, 91B29, 60G46JEL Classification: G10The authors would like to thank D. Becherer and J.N. Hugonnier for interesting discussions and the anonymous referee whose pertinent questions on the first version of this paper help them to clarify the proofs. All remaining errors are ours.     

Pension regulation and the market value of pension liabilities: A contingent claims analysis using Parisian options

We analyze the market-consistent valuation of pension liabilities in a contingent claim framework whereby a knock-out barrier feature is applied to capture early regulatory closure of a pension plan. We investigate two cases which we call “immediate closure procedure” and “delayed closure procedure”. In an immediate closure procedure, when the assets value hits the regulatory boundary, the pension plan is terminated immediately. Whereas in a delayed closure procedure, a grace period is given to the pension fund for reorganization and recovery before premature closure is executed. The framework is then used to construct fair pension deals. Furthermore, we provide rules for deriving the optimal recovery period in pension regulation using utility analysis and interconnect the recovery period to the regulatory liquidation probability.     

Convex duality in optimal investment and contingent claim valuation in illiquid markets

This paper develops duality theory for optimal investment and contingent claim valuation in markets where traded assets may be subject to nonlinear trading costs and portfolio constraints. Under fairly general conditions, the dual expressions decompose into three terms, corresponding to the agent’s risk preferences, trading costs and portfolio constraints, respectively. The dual representations are shown to be valid when the market model satisfies an appropriate generalization of the no-arbitrage condition and the agent’s utility function satisfies an appropriate generalization of asymptotic elasticity conditions. When applied to classical liquid market models or models with bid–ask spreads, we recover well-known pricing formulas in terms of martingale measures and consistent price systems. Building on the general theory of convex stochastic optimization, we also obtain optimality conditions in terms of an extended notion of a “shadow price”. The results are illustrated by establishing the existence of solutions and optimality conditions for the nonlinear market models recently proposed in the literature. Our results allow significant extensions including nondifferentiable trading costs which arise, e.g., in modern limit order markets where the marginal price curve is necessarily discontinuous.     

Convergence of numerical methods for valuing path-dependent options using interpolation

One method for valuing path-dependent options is the augmented state space approach described in Hull and White (1993) and Barraquand and Pudet (1996), among others. In certain cases, interpolation is required because the number of possible values of the additional state variable grows exponentially. We provide a detailed analysis of the convergence of these algorithms. We show that it is possible for the algorithm to be non-convergent, or to converge to an incorrect answer, if the interpolation scheme is selected in appropriately. We concentrate on Asian options, due to their popularity and because of some errors in the previous literature. This revised version was published online in June 2006 with corrections to the Cover Date.     

Valuation of general contingent claims: Existence,uniqueness, and comparisons of solutions

The Black-Scholes equation for the price u(x,t) of a call option for a single share of common stock with dividend policy d(x,t) is $\text{1}\text{2}\text{σ}{}^2\text{x}{}^2\text{u}{}_{\mathrm{xx}}\text{+(rx?d(x,t))u}{}_{\mathrm{x}}\text{?ru?u}{}_{\mathrm{t}}\text{=0, 0<x, 0<t<T}$, with boundary conditions u(x,0)=max(0,x?E), 0≤x, u(0,t)=0, 0≤t≤T.The coefficients are unbounded and the equation is not uniformly parabolic. We prove an existence (and recall a uniqueness) theorem for a class of equations with boundary conditions that includes the Black-Scholes equation. These may be used to show that an American option must, or will not, sell for the same price as a European option.     

Projections of pension fund solvency under alternative valuation regimes

This paper examines the impact of three alternative valuation regimes on perceived pension fund solvency. Deterministic valuation assumes smoothed valuation of assets and liabilities. National valuation is based on market valuation of assets and on smoothed valuation of liabilities. International valuation marks assets and liabilities to market values. Using closed-form methods based on the funding ratio return, we exemplify the dramatic effect that the choice of valuation approach has on long-horizon solvency projections.     

The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours

In this paper, we present a dynamic programming algorithm for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) under a general Lévy processes framework. The GMWB gives the policyholder the right to make periodical withdrawals from her policy account even when the value of this account is exhausted. Typically, the total amount guaranteed for withdrawals coincides with her initial investment, providing then a protection against downside market risk. At each withdrawal date, the policyholder has to decide whether, and how much, to withdraw, or to surrender the contract. We show how different policyholder’s withdrawal behaviours can be modelled. We perform a sensitivity analysis comparing the numerical results obtained for different contractual and market parameters, policyholder behaviours and different types of Lévy processes.