Jr-Yan Wang

Pricing Convertible Bonds Subject to Default Risk

Convertible bonds are commonplace securities, but valuing them properly is tricky. In addition to being exposed to interest rate risk like any bond in a stochastic interest rate environment, they contain both an option to convert them into shares of the issuing firm, and also exposure to the risk of default. In this article, Hung and Wang present a lattice technique that allows relatively straightforward valuation, even in the presence of these three sources of risk. After describing their technique in general, they put it to use to evaluate a convertible bond issued by Lucent.

Loss aversion and the term structure of interest rates

This article studies how the loss averse behaviour affects the term structure of real interest rates. Since the pro-cyclical conditional expected marginal rate of substitution, implied from the US consumption data, is consistent with the proposition of loss aversion, we incorporate the loss averse behaviour of prospect theory into the consumption-based asset pricing model. Motivated by the similarity between habit formation and the prospect theory utility, habit formation is exploited to determine endogenously the reference point of this behavioural finance utility. The highly curved characteristic of the term structure of real interest rates can thus be captured by the additional consideration of loss aversion. This model also fits the downward sloping volatility of the real yield curve in the data of US Treasury Inflation-Protection Securities (TIPS). Moreover, depending on the effective risk attitude of the representative agent with the loss averse behaviour of prospect theory, our model is capable of generating a normal or an inverted yield curve.

Asset Prices Under Prospect Theory and Habit Formation

In this paper, we develop a consumption-based asset pricing model motivated by prospect theory, where habit formation determines the endogenous reference point. This exploits the similarity between habit formation and prospect theory. Both emphasize that the investor does not care about the absolute amount of gain or loss, but rather compares the gain or the loss experienced to a benchmark. The results show that when taking peoples loss averse attitude over consumption into consideration, our model is capable of resolving the equity premium puzzle.

Variance Reduction for Multivariate Monte Carlo Simulation

As derivative securities become more complex, solution by Monte Carlo simulation is an increasingly necessary tool. But Monte Carlo methods are computationally demanding, and the size of the simulation sample required to achieve reasonable accuracy rapidly escalates beyond what is feasible given current technology when multiple stochastic factors are involved. Variance reduction techniques help considerably. One of the simplest is use of antithetic variables, which imposes on the simulated data the true constraint that the distribution from which the sample is to be drawn is symmetric. Moment matching goes further, by constraining the mean and variance of the sample to match the desired values. In this article, Wang goes further still, to show how to force simulated multivariate vectors to have the right correlations. In the first step a multivariate sample of independent variables is simulated and its sample correlation matrix is calculated. This will be close to the identity matrix but not exactly equal, due to sampling noise. Cholesky factorization of the sample correlation matrix is then used to transform the initial sample of slightly correlated factors into one whose elements are perfectly uncorrelated in the sample. A second Cholesky factorization of the target correlation matrix that is desired for the variables transforms them into a sample with exactly the correct correlations. As Wang demonstrates, this produces a sharp increase in performance for multivariate Monte Carlo problems, even when the simulation sample is constructed from nonrandom low-discrepancy sequences rather than by stochastic simulation.

An Ingenious, Piecewise Linear Interpolation Algorithm for Pricing Arithmetic Average Options

Pricing arithmetic average options continues to intrigue researchers in the field of financial engineering. Since there is no analytical solution for this problem until present, developing an efficient numerical algorithm becomes a promising alternative. One of the most famous numerical algorithms for pricing arithmetic average options is introduced by Hull and White [10]. In this paper, motivated by the common idea of reducing the nonlinearity error in the adaptive mesh model [7] and the adaptive quadrature numerical integration method [6], the logarithmically equally-spaced placement rule in the Hull and Whites model is replaced by an adaptive placement method, in which the number of representative average prices is proportional to the degree of curvature of the option value as a function of the arithmetic average price. Numerical experiments verify the superior performance of our method in terms of reducing the interpolation error. In fact, it is straightforward to apply this method to any pricing algorithm with the techniques of augmented state variables and the piece-wise linear interpolation approximation.

Using forward Monte-Carlo simulation for the valuation of American barrier options

This paper extends the forward Monte-Carlo methods, which have been developed for the basic types of American options, to the valuation of American barrier options. The main advantage of these methods is that they do not require backward induction, the most time-consuming and memory-intensive step in the simulation approach to American options pricing. For these methods to work, we need to define the so-called pseudo critical prices which are used to determine whether early exercise should happen. In this study, we define a new and more flexible version of the pseudo critical prices which can be conveniently extended to all fourteen types of American barrier options. These pseudo critical prices are shown to satisfy the criteria of a sufficient indicator which guarantees the effectiveness of the proposed methods. A series of numerical experiments are provided to compare the performance between the forward and backward Monte-Carlo methods and demonstrate the computational advantages of the forward methods.

A Modified Reduced-Form Model with Time-Varying Default and Recovery Rates and Its Applications in Pricing Convertible Bonds

Reduced-form models of default risk require estimates of the recovery rate, or equivalently, the loss given default. In many cases, this is simply set at a fixed recovery rate of 40%. But the 40% rate is often far from the realized recovery amount in practice, and empirical research shows that recoveries are negatively related to ex ante default probabilities and the rate varies over time. In this article, the authors incorporate a regression-based estimate of the current recovery rate from historical data, modeling it as a function of the default probability and, potentially, macroeconomic data such as growth of the S&P Index. They then build this relationship into a lattice structure and show how it can be applied to price convertible bonds.