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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Back on the balance sheet: The tax effects of contingent claims in commercial banking

Contingent claims separate revenue and cost into two different time periods. Revenue comes in the initial origination process, while the cost comes upon completion of the contract in the event of default. With banks increasing contingent claims in recent years, a higher taxable income leads to a shift in a bank's balance sheet toward tax-free income and tax-shielding liabilities. This provides a valuable case-study of corporate finance theories of tax management. This paper builds a model to illustrate the income features of contingent claims. Call Reports from 1990-1996 are examined, and show significant evidence of increases in leverage associated with contingent claims.     

A contingent claims analysis of the interest rate risk characteristics of corporate liabilities

This paper provides a contingent claims analysis of the interest rate risk characteristics of corporate liabilities by identifying Merton's (1973) option pricing model with Vasicek's (1977) mean reverting term structure model. Only a non-zero positive range of duration values for the firms' assets is shown to be consistent with the previous empirical evidence on the interest rate sensitivity of corporate stocks and bonds. Chance's (1990) duration measure is shown to be biased downward under empirically realistic conditions. Theoretical conditions are derived under which the duration of a default-prone zero coupon bond can be either higher or lower than the duration of the corresponding default-free bond. The duration of the default-prone bond of a firm with high (low) interest rate sensitive assets is shown to be an increasing (decreasing) function of the bond's default-risk.     

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.     

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.     

A general class of distortion operators for pricing contingent claims with applications to CAT bonds

ABSTRACT

The current paper provides a general approach to construct distortion operators that can price financial and insurance risks. Our approach generalizes the (Wang 2000) transform and recovers multiple distortions proposed in the literature as particular cases. This approach enables designing distortions that are consistent with various pricing principles used in finance and insurance such as no-arbitrage models, equilibrium models and actuarial premium calculation principles. Such distortions allow for the incorporation of risk-aversion, distribution features (e.g. skewness and kurtosis) and other considerations that are relevant to price contingent claims. The pricing performance of multiple distortions obtained through our approach is assessed on CAT bonds data. The current paper is the first to provide evidence that jump-diffusion models are appropriate for CAT bonds pricing, and that natural disaster aversion impacts empirical prices. A simpler distortion based on a distribution mixture is finally proposed for CAT bonds pricing to facilitate the implementation.     

The valuation of projects under the dividend imputation tax system

This paper proves that a modified weighted average cost of capital (“WACC”) valuation methodology is a rigorous and practicable method of valuing projects and companies under the Australian dividend imputation tax system. This methodology uses an effective tax rate in calculating both the discount rate and the ungeared after tax cash flow. A cash flow after effective corporate tax is shown to be equivalent to a cash plus value of imputation credit stream. Importantly, this valuation methodology is applicable to returns that are non-uniform and of finite duration. Also examined is the discounting of equity returns at the company's cost of equity capital. A worked example is presented to clarify and quantify the effects discussed.     

Valuation of default-sensitive claims under imperfect information

We propose a valuation method for financial assets subject to default risk, where investors cannot observe the state variable triggering the default but observe a correlated price process. The model is sufficiently general to encompass a large class of structural models and can be seen as a generalization of the model of Duffie and Lando (Econometrica 69:633–664, [2001]). In this setting we prove that the default time is totally inaccessible in the market’s filtration and derive the conditional default probabilities and the intensity process. Finally, we provide pricing formulas for default-sensitive claims and illustrate in particular examples the shapes of the credit spreads.      

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.