34.5 AN ARBITRAGE SITUATION

The example XYZ convertible bond had a theoretical value of 103.36% according to the binomial model. For the sake of illustration, let us assume that the XYZ convertible bond is quoted by the market at 90% of par. Given the current price and the estimated volatility of the underlying asset (stock), the convertible bond is clearly undervalued. The question is: How can one exploit such a mispricing? Buying the cheap convertible is clearly part of the solution, but it is not sufficient. Simply waiting for market prices to adjust is not an arbitrage because the long convertible position comes with a variety of risks that could easily wipe out the expected gains. To arbitrage, it is necessary to buy the cheap convertible and hedge its risks, a dynamic process that is very similar to what option arbitrageurs do all day long.

The primary risk of holding a long convertible position comes from the potential variations in the underlying stock price. This equity risk can easily be eliminated by selling short an appropriate quantity of the underlying stock. This quantity corresponds to the convertible's delta multiplied by the number of shares into which the bond may be converted. If the stock price gains $1, the convertible bond will gain approximately delta dollars and the short stock position will lose delta dollars, so that the overall variation will be nil. Conversely, if the stock price drops by $1, the convertible bond will lose approximately delta dollars ...

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