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.     

Nonparametric,conditional pricing of higher order multivariate contingent claims

This paper describes and applies a nonparametric model for pricing multivariate contingent claims. Multivariate contingent claims are contracts whose payoffs depend on the future prices of more than one underlying variable. The pricing however of these kinds of contracts represents a challenge. All known models are adaptations of earlier ones that have been introduced to price plain vanilla calls and puts. They are imposing strong assumptions on the distributional properties of the underlying variables. In contrast, this study adopts a methodology that relaxes such restrictions. Following [Barone-Adesi, G., Bourgoin, F., Giannopoulos, K., 1998. Don’t Look Back, Risk 11 (August), 100–104; Barone-Adesi, G., Engle, R., Mancini, L., 2004. GARCH Options in Incomplete Markets, mimeo, University of Applied Sciences of Southern Switzerland; Long, X., 2004. Semiparametric Multivariate GARCH Model, mimeo, University of California, Riverside], multivariate pathways for a set of underlying variables are constructed before the option payoffs are computed. This enables the covariances, in addition to the means and variances, to be modelled in a dynamic and nonparametric manner. The model is particular suitable for options whose payoffs depend on variables that are characterised by high nonlinearities and extremes and on higher order multivariate options whose underlying variables are more unlikely to conform to a common theoretical distribution.     

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.     

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.     

Empirical evaluation of asset pricing models: Arbitrage and pricing errors in contingent claims

Hansen and Jagannathan (1997) have developed two measures of pricing errors for asset-pricing models: the maximum pricing error in all static portfolios of the test assets and the maximum pricing error in all contingent claims of the assets. In this paper, we develop simulation-based Bayesian inference for these measures. While the literature reports that the time-varying extensions substantially reduce pricing errors of classic models on the standard test assets, our analysis shows that the reduction is much smaller based on the second measure. Those time-varying models have large pricing errors on the contingent claims of the test assets because their stochastic discount factors are often negative and admit arbitrage opportunities.     

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.     

The valuation of contingent claims using alternative numerical methods

This study compares the computational accuracy and efficiency of three numerical methods for the valuation of contingent claims written on multiple underlying assets; these are the trinomial tree, original Markov chain and Sobol–Markov chain approaches. The major findings of this study are: (i) the original Duan and Simonato (2001) Markov chain model provides more rapid convergence than the trinomial tree method, particularly in cases where the time to maturity period is less than nine months; (ii) when pricing options with longer maturity periods or with multiple underlying assets, the Sobol–Markov chain model can solve the problem of slow convergence encountered under the original Duan and Simonato (2001) Markov chain method; and (iii) since conditional density is used, as opposed to conditional probability, we can easily extend the Sobol–Markov chain model to the pricing of derivatives which are dependent on more than two underlying assets without dealing with high-dimensional integrals. We also use ‘executive stock options’ (ESOs) as an example to demonstrate that the Sobol–Markov chain method can easily be applied to the valuation of such ESOs.     

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.     

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.     

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.     

An evaluation of contingent immunization

This paper tests the effectiveness of contingent immunization, a stop loss strategy that allows portfolio managers to take advantage of their ability to forecast interest rate movements as long as their forecasts are successful, but switches to a pure immunization strategy should the stop loss limit be encountered. This study uses actual daily transactions in the Spanish Treasury market covering the period 1993–2003 and uses performance measures that accounts for skewness and kurtosis as well as mean variance. The main result of this paper is that contingent immunization provides excellent performance despite its simplicity.