Convertible Bond Pricing with Stochastic Volatility
The aim of this paper is to compare the performance of different pricing models in valuing bonds with callable and convertible features. Additionally, we wish to provide a theoretical foundation and derivations of the models as we move through the paper. Much of the foundations for our approach to convertible bonds pricing, including optimal conditions for call and conversion, can be attributed to Ingersoll (1976) and Brennan and Schwartz (1977). These fundamental pricing conditions can then be built upon to arrive at more elaborate and numerically sophisticated models with the objective of more accurately pricing derivative securities. The Black-Scholes (BS) model is the most commonly used model in valuing short term derivative instruments, such as equity derivatives, for example. As for longer term securities, such as convertible bonds, movements in volatility and interest rates are likely to have a compounding effect. Consequently, we conjecture that that allowing for stochastic volatility and stochastic interest rates within the pricing of these longer term instruments is preferable. Additionally, given the much larger size of the fixed income derivatives markets when compared to other derivatives, it seems that the answer as to which pricing model is preferable carries significance. As to the findings with regard to equity derivatives, Bakshi, Cao, and Chen (1997) conclude that "taking stochastic volatility into account is of the first order importance in improving on the BS formula", but "going from the SV to the SVSI does not necessarily improve the fit much further." Firstly we shall look at pricing equity derivatives and convertible bonds using a more basic BS framework, then comparing this to the more complex SV and SVSI models later on in the paper. As for numerical pricing procedures, we concentrate on the use of the ADI finite difference method in order to estimate derivative values. Given the multiple variables that we wish to model, including firm value, volatility, and interest rates, we want a pricing procedure that is both accurate and computationally efficient. Whilst the ADI method is ideal for this situation, monte-carlo simulation is also an attractive approach to pricing convertible bonds. Indeed, in the case where the value of the option is path dependant, monte-carlo simulation is the ideal choice. To see examples of finite difference techniques used in the context of convertible bonds pricing, see Andersen and Buffum (2002). Alternatively, for a look into monte-carlo simulation, see Lvov, Yigitbasioglu, and Bachir (2004).