In this paper, we present a new strategy for pricing average value options, i.e. options whose payoff depends on the average price of the underlying asset over a fixed period leading up to the maturity date. Such options are of particular interest and importance for thinly-traded assets (e.g. crude oil), since price manipulation is inhibited, and both the investor and issuer enjoy a welcome degree of protection from the vagaries of the market. These options are often implicit in a bond contract, although they also appear in a straightforward form. Our results suggest that the price of an average-value option will always be lower than that of a standard European option. Our pricing strategy involves Monte Carlo simulation with variance reduction elements and offers an enhanced pricing method to both arbitragers and hedgers, as well as to the issuers of such bonds.

A Pricing Method for Options Based on Average Asset Values

Applying the technology of option pricing and contingent claims modeling to credit risk is one of the major growth areas in derivatives research these days. It is an old idea to treat default as a firm9s management rationally exercising the shareholders9 option to go bankrupt rather than to make a required payment to the debtholders. But only in the last few years has this insight been extended and widely applied to practical bond valuation problems. Convertible bonds have also become much more popular in recent years, as the weak and volatile stock market has combined with low interest rates to create an environment in which the conversion option can be very attractive. One critical uncertainty in credit risk analysis that may not be fully appreciated by those who have not worked in this area, is what the payoff to bondholders will actually be in the event of a default. In this article, Ayache, Forsyth, and Vetzal explore the interactions among the multiple options embedded in convertibles that are subject to default risk, call provisions, and put provisions, in addition to the conversion option. Along with results that illustrate the interactions among these multiple options, the authors provide a straightforward valuation technique for convertibles, by applying principles of linear complementarity to the discretization of the fundamental PDE of contingent claims pricing.

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Valuation of Convertible Bonds With Credit Risk

Convertible bonds can be difficult to value, given their hybrid nature of containing elements of both debt and equity. Further complications arise due to the frequent presence of additional options such as callability and puttability, and contractual complexities such as trigger prices and “soft call” provisions, in which the ability of the issuing firm to exercise its option to call is dependent upon the history of its stock price. This paper explores the valuation of convertible bonds subject to credit risk using an approach based on the numerical solution of linear complementarity problems. We argue that many of the existing models, such as that of Tsiveriotis and Fernandes (1998), are unsatisfactory in that they do not explicitly specify what happens in the event of a default by the issuing firm. We show that this can lead to internal inconsistencies, such as cases where a call by the issuer just before expiry renders the convertible value independent of the credit risk of the issuer, or situations where the implied hedging strategy may not be self-financing. By contrast, we present a general and consistent framework for valuing convertible bonds assuming a Poisson default process. This framework allows various models for stock price behaviour, recovery, and action by holders of the bonds in the event of a default. We also present a detailed description of our numerical algorithm, which uses a partially implicit method to decouple the system of linear complementarity problems at each timestep. Numerical examples illustrating the convergence properties of the algorithm are provided.

/pdf/the-valuation-of-convertible-bonds-with-credit-risk-vdioek0yg6.pdf

The Valuation of Convertible Bonds With Credit Risk

We propose and empirically study a pricing model for convertible bonds based on Monte Carlo simulation. The method uses parametric representations of the early exercise decisions and consists of two stages. Pricing convertible bonds with the proposed Monte Carlo approach allows us to better capture both the dynamics of the underlying state variables and the rich set of real-world convertible bond specifications. Furthermore, using the simulation model proposed, we present an empirical pricing study of the US market, using 32 convertible bonds and 69 months of daily market prices. Our results do not confirm the evidence of previous studies that market prices of convertible bonds are on average lower than prices generated by a theoretical model. Similarly, our study is not supportive of a strong positive relationship between moneyness and mean pricing error, as argued in the literature.

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Simulation-Based Pricing of Convertible Bonds

Simulation-based pricing of convertible bonds

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.

/pdf/pricing-convertible-bonds-subject-to-default-risk-2fg5wli1aj.pdf

Pricing Convertible Bonds Subject to Default Risk

In contrast to the constant exercise boundary assumed by Broadie and Detemple (1996) [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies 9, 1211–1250], we use an exponential function to approximate the early exercise boundary. Then, we obtain lower bounds for American option prices and the optimal exercise boundary which improve the bounds of Broadie and Detemple (1996). With the tight lower bound for the optimal exercise boundary, we further derive a tight upper bound for the American option price using the early exercise premium integral of Kim (1990) [Kim, I.J., 1990. The analytic valuation of American options. Review of Financial Studies 3, 547–572]. The numerical results show that our lower and upper bounds are very tight and can improve the pricing errors of the lower bound and upper bound of Broadie and Detemple (1996) by 83.0% and 87.5%, respectively. The tightness of our upper bounds is comparable to some best accurate/efficient methods in the literature for pricing American options. Moreover, the results also indicate that the hedge ratios (deltas and gammas) of our bounds are close to the accurate values of American options.

http://homepage.ntu.edu.tw/~jryanwang/papers/Tight%20Bounds%20on%20American%20Option%20Prices.pdf

Tight bounds on American option prices

This paper studies the valuation and hedging problems of forward-start rainbow options (FSROs). By combining the characteristics of both multiple assets and forward-start feature, this new type of derivative has many potential applications, for instance, to incorporate the reset provision in rainbow options for investors or hedgers or design more effective executive compensation plans. The main contribution of this paper is a novel martingale pricing technique for options whose payoffs are associated with multiple assets and time points. Equipped with this technique, the analytic pricing formula and the formulae of the delta and gamma of the FSRO are first derived. We conduct numerical experiments to verify these formulae and examine the characteristics of the FSRO’s price and Greek letters. To demonstrate the importance and general applicability of the proposed technique, we also apply it to deriving the pricing formula for the discrete-sampling lookback rainbow options.

The valuation of forward-start rainbow options

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 people's loss averse attitude over consumption into consideration, our model is capable of resolving the equity premium puzzle.

http://homepage.ntu.edu.tw/~jryanwang/papers/Asset%20Prices%20Under%20Prospect%20Theory%20and%20Habit%20Formation.pdf

Asset Prices Under Prospect Theory and Habit Formation

Since there is no analytic solution for arithmetic average options until present, developing an efficient numerical algorithm becomes a promising alternative. One of the most famous numerical algorithms is introduced by Hull and White (J Deriv 1:21–31, 1993). Motivated by the common idea of reducing the nonlinearity error in the adaptive mesh model in Figlewski and Gao (J Financ Econ 53:313–351, 1999) and the adaptive quadrature method, we propose an adaptive placement method to replace the logarithmically equally-spaced placement rule in the Hull and White’s model by placing more representative average prices in the highly nonlinear area of the option value as the function of the arithmetic average stock price. The basic idea of this method is to design a recursive algorithm to limit the error of the linear interpolation between each pair of adjacent representative average prices. Numerical experiments verify the superior performance of this method for reducing the interpolation error and hence improving the convergence rate. To show that the adaptive placement method can improve any numerical algorithm with the techniques of augmented state variables and the piece-wise linear interpolation approximation, we also demonstrate how to integrate the adaptive placement method into the GARCH option pricing algorithm in Ritchken and Trevor (J Finance 54:377–402, 1999). Similarly great improvement of the convergence rate suggests the potential applications of this novel method to a broad class of numerical pricing algorithms for exotic options and complex underlying processes.

http://homepage.ntu.edu.tw/~jryanwang/papers/Adaptive%20Placement%20Method%20on%20Pricing%20Arithmetic%20Average%20Options.pdf

Jr-Yan Wang

Papers

Pricing Convertible Bonds Subject to Default Risk

Tight bounds on American option prices

The valuation of forward-start rainbow options

Asset Prices Under Prospect Theory and Habit Formation

Adaptive placement method on pricing arithmetic average options