A convertible bond (or preferred share) is a hybrid security, part debt and part equity. Its valuation is derived from both the level of interest rates and the price of the underlying equity. Several modeling approaches are available to value these complex hybrid securities such as Binomial Tree, Partial Differential Equation and Monte Carlo simulation. One of the earliest pricing convertible bond approaches was the Binomial Tree model originally developed by Goldman Sachs [1,2] and this model allows for an efficient implementation with high accuracy. The Binomial Tree model is flexible enough to support the implementation of bespoke exotic features such as redemption and conversion by the issuer, lockout periods, conversion and retraction by the share owner etc.
In this post, we will summarize the key steps in pricing convertible bond method using the Binomial Tree approach. Detailed description of the method and examples are provided in references [1,2]. Generally, the value of a convertible bond with embedded features depends on:
where S0 is the stock price at the valuation date; u and d are the up and down jump magnitudes. The superscript j refers to the time step and i to the jump. The up and down moves are calculated as and where is the stock volatility, and is the time step. The risk neutral probability of the up move, u, is and the probability the down move is 1p After building a binomial tree for the common stock price, the convertible bond price is then determined by starting at the end of the stock price tree where the payoff is known with certainty and going backward until the time zero (valuation date). At each node, Pj,i the value of the convertible is where m denotes the conversion ratio. If the bond is callable, the payoff at each node is The payoff of a putable bond is Here C and P are the call and put values respectively; r denotes the riskfree rate. The above equations are the key algorithms in the binomial tree approach. However, there are several considerations that should be addressed due to the complexities of the derivative features
References [1] Valuing Convertible Bonds as Derivatives, Quantitative Strategies Research Notes, Goldman Sachs, November 1994. [2] Pricing Convertible Bonds, Kevin B. Connolly, Wiley, 1998.
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