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.     

Dynamic, nonparametric hedging of European style contingent claims using canonical valuation

The canonical valuation, proposed by Stutzer [1996. Journal of Finance 51, 1633–1652], is a nonparametric option pricing approach for valuing European-style contingent claims. This paper derives risk-neutral dynamic hedge formulae for European call and put options under canonical valuation that obey put–call parity. Further, the paper documents the error-metrics of the canonical hedge ratio and analyzes the effectiveness of discrete dynamic hedging in a stochastic volatility environment. The results suggest that the nonparametric hedge formula generates hedges that are substantially unbiased and is capable of producing hedging outcomes that are superior to those produced by Black and Scholes [1973. Journal of Political Economy 81, 637–654] delta hedging.     

A path-dependent approach to security valuation with application to interest rate contingent claims

The last two decades have witnessed a tremendous growth in the volume of assets and liabilities whose cash flows depend, in a variety of ways, on the path of interest rates. Some of these, including floating-rate notes and swap agreements, contractually base cash flows on current and past interest rates and contain caps, floors, and other, more complex features. Others, including mortgages, many corporate bonds, and time deposits, are fixed-rate instruments that contain embedded options, such as those to prepay, call, or withdrawal. The irregular exercise of these options causes cash flows to vary as time proceeds and interest rates rise or fall. This paper develops a state-contingent claims technique for valuing such securities. It is derived from the option-based model of Breeden and Litzenberger (1978) using the transition matrix approach of Banz and Miller (1978). Particular attention is paid to valuing so-called path-dependent securities whose contemporaneous cash flows depend on the historical path of interest rates as well as their current level. A detailed example is provided in which an adjustable-rate mortgage is valued under a variety of economic and security specific assumptions.     

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.     

Model risk of contingent claims

Paralleling regulatory developments, we devise value-at-risk and expected shortfall type risk measures for the potential losses arising from using misspecified models when pricing and hedging contingent claims. Essentially, P&L from model risk corresponds to P&L realized on a perfectly hedged position. Model uncertainty is expressed by a set of pricing models, each of which represents alternative asset price dynamics to the model used for pricing. P&L from model risk is determined relative to each of these models. Using market data, a unified loss distribution is attained by weighing models according to a likelihood criterion involving both calibration quality and model parsimony. Examples demonstrate the magnitude of model risk and corresponding capital buffers necessary to sufficiently protect trading book positions against unexpected losses from model risk. A further application of the model risk framework demonstrates the calculation of gap risk of a barrier option when employing a semi-static hedging strategy.     

Numerical evaluation of multivariate contingent claims

We develop a numerical approximation method for valuing multivariatecontingent claims. The approach is based on an n-dimensionalextension of the lattice binomial method. Closed-form solutionsfor the jump probabilities and the jump amplitudes are obtained.The accuracy of the method is illustrated in the case of Europeanoptions when there are three underlying assets.     

Universal option valuation using quadrature methods

This paper proposes and develops a novel, simple, widely applicable numerical approach for option pricing based on quadrature methods. Though in some ways similar to lattice or finite-difference schemes, it possesses exceptional accuracy and speed. Discretely monitored options are valued with only one timestep between observations, and nodes can be perfectly placed in relation to discontinuities. Convergence is improved greatly; in the extrapolated scheme, a doubling of points can reduce error by a factor of 256. Complex problems (e.g., fixed-strike lookback discrete barrier options) can be evaluated accurately and orders of magnitude faster than by existing methods.     

Pricing and hedging contingent claims using variance and higher order moment swaps

This paper suggests perfect hedging strategies of contingent claims under stochastic volatility and random jumps of the underlying asset price. This is done by enlarging the market with appropriate swaps whose pay-offs depend on higher order sample moments of the asset price process. Using European options and variance swaps, as well as barrier options written on the S&P 500 index, the paper provides clear cut evidence that hedging strategies employing variance and higher order moment swaps considerably improves upon the performance of traditional delta hedging strategies. Inclusion of the third-order moment swap improves upon the performance of variance swap-based strategies to hedge against random jumps. This result is more profound for short-term out-of-the money put options.     

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)

.

Municipal bonds: a contingent claims perspective

The purpose of this paper is to provide an overview of the municipal bond market with an emphasis on the numerous embedded contingent claims. Embedded contingent claims include the standard call features, sinking funds, the advance refunding option, the synthetic advance refunding option, the credit risk option (default risk), marketability, and the numerous tax-related events. Municipal bond investors must carefully assess the relative value of these contingent claims before investing in municipal bonds. Also, due to unique risk premiums within the municipal bond market, it is important to carefully structure the municipal bond holdings, paying particular attention to duration, within the context of an overall financial plan. There appears to be a benefit to lengthening the duration of the municipal bond portion of the portfolio.     

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.     

Optimal investment and contingent claim valuation in illiquid markets

This paper extends basic results on arbitrage bounds and attainable claims to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. Explicit consideration of swap contracts is essential in illiquid markets where the valuation of swaps cannot be reduced to the valuation of cumulative claims at maturity. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of optimal investment under conditions that extend the no-arbitrage condition in the classical linear market model. All results are derived with the “direct method” without resorting to duality arguments.     

On the pricing of contingent claims and the Modigliani-Miller theorem

A general formula is derived for the price of a security whose value under specified conditions is a known function of the value of another security. Although the formula can be derived using the arbitrage technique of Black and Scholes, the alternative approach of continuous-time portfolio strategies is used instead. This alternative derivation allows the resolution of some controversies surrounding the Black and Scholes methodology. Specifically, it is demonstrated that the derived pricing formula must be continuous with continuous first derivatives, and that there is not a ‘pre-selection bias’ in the choice of independent variables used in the formula. Finally, the alternative derivation provides a direct proof of the Modigliani-Miller theorem even when there is a positive probability of bankruptcy.