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Jump Diffusion Convertible Bond Valuation There is a rich literature on the subject of convertible bonds. Arguably, the first widely adopted model among practitioners is the one presented by Goldman Sachs (1994) and then formalized by Tsiveriotis and Fernandes (1998). The Goldman Sachs’ solution is a simple one factor model with an equity binomial tree to value convertible bonds. The model considers the probability of conversion at every node. If the convertible is certain to remain a bond, it is then discounted by a risky discount rate that reflects the credit risk of the issuer. If the convertible is certain to be converted, it is then discounted by the risk-free interest rate that is equivalent to default free. Tsiveriotis and Fernandes (1998) argue that in practice one is usually uncertain as to whether the bond will be converted, and thus propose dividing convertible bonds into two components: a bond part that is subject to credit risk and an equity part that is free of credit risk. A simple description of this model and an easy numerical example in the context of a binomial tree can be found in Hull (2003). Grimwood and Hodges (2002) indicate that the Goldman Sachs model is incoherent because it assumes that bonds are susceptible to credit risk but equities are not. Ayache, et al. (2003) conclude that the Tsiveriotis-Fernandes model is inherently unsatisfactory due to its unrealistic assumption of stock prices being unaffected by bankruptcy. To correct this weakness, Davis and Lischka (1999), Andersen and Buffum (2004), Bloomberg (2009), and Carr and Linetsky (2006) etc., propose a jump-diffusion model to explore defaultable stock price dynamics. They all believe that under a risk-neutral measure the expected rate of return on a defaultable stock must be equal to the risk-free interest rate. The jump-diffusion model characterizes the default time/jump directly. The jump-diffusion model was first introduced by Merton (1976) in the market risk context for modeling asset price behavior that incorporates small day-to-day diffusive movements together with larger randomly occurring jumps. Over the last decade, people attempt to propagate the model from the market risk domain to the credit risk arena. Although both the structural jump-diffusion model and the reduced-form model contain jumps, these jumps have different meanings: A jump in the structural jump-diffusion model corresponds to a sudden change in the asset value that may or may not cause the firm to default, whereas a jump in the reduced-form model represents the default event itself. In this paper, we mainly discuss the reduced-form jump-diffusion models. At the heart of the jump-diffusion models lies the assumption that the total expected rate of return to the stockholders is equal to the risk-free interest rate under a risk-neutral measure. Because of their hybrid nature, convertible bonds attract different type of investors. Especially, convertible arbitrage hedge funds play a dominant role in primary issues of convertible debt. In fact, it is believed that hedge funds purchase 70% to 80% of the convertible debt offered in primary markets. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing (i.e., the model prices are on average higher than the observed trading prices) (see Ammann, et al (2003), Choi, et al. (2009), Loncarski, et al. (2009), etc.). However, Agarwal, et al. (2007) and Batta, et al. (2007) argue that the excess returns from convertible arbitrage strategies are not mainly due to underpricing, but rather partly due to illiquid. Calamos (2011) believes that arbitrageurs in general take advantage of volatility. A higher volatility in the underlying equity translates into a higher value of the equity option and a lower conversion premium. Multiple views reveal the complexity of convertible arbitrage, involving taking positions in the convertible bond and the underlying asset that hedges certain risks but leaves managers exposed to other risks for which they reap a reward. This article makes a theoretical and empirical contribution to the study of convertible bonds. In contrast to the above mentioned literature, we present a model that is based on the probability distribution (or intensity) of a default jump (or a default time) rather than the default jump itself, as the default jump is usually inaccessible (see Duffie and Huang (1996), Jarrow and Protter (2004), etc). Valuation under our risky model can be solved by common numerical methods, such as, Monte Carlo simulation, tree/lattice approaches, or partial differential equation (PDE) solutions. The PDE algorithm is elaborated in this paper, but of course the methodology can be easily extended to tree/lattice or Monte Carlo. Using the model proposed, we conduct an empirical study of convertible bonds. We obtain a data set from FinPricing (2015). The data set contains 164 convertible bonds and 2 years of daily market prices as well as associated interest rate curves, credit curves, stock prices, implied Black-Scholes volatilities and recovery rates. The empirical results show that the model prices fluctuate randomly around the market prices, indicating the model is quite accurate. Our empirical evidence does not support a systematic underpricing hypothesis. A similar conclusion is reached by Ammann and Wilde (2008) who use a Monte-Carlo simulation approach. Moreover, market participants almost always calibrate their models to the observed market prices using implied convertible volatilities. Therefore, underpricing may not be the main driver of profitability in convertible arbitrage. It is useful to examine the basics of the convertible arbitrage strategy. A typical convertible bond arbitrage employs delta-neutral hedging, in which an arbitrageur buys a convertible bond and sells the underlying equity at the current delta (see Choi, et al. (2009), Loncarski, et al. (2009), etc.). With delta neutral positions, the sign of Gamma is important. If Gamma is negative, the portfolio profits so long as the underlying equity remains stable. If Gamma is positive, the portfolio will profit from large movements in the stock price in either direction (see Somanath (2011)). Jump diffusion for pricing convertibles |