Approximating the Embedded M Out of N Day Soft-Call Option of a Convertible Bond: An Auxiliary Reversed Binomial Tree Method

Abstract

A convertible bond (CB) with a right of m out of n day provisional call or soft-call becomes callable given that the underlying stock closes above a pre-set trigger price for any m or more days over the n consecutive trading days up to the current day. It is computationally challenge to value the contribution of this embedded option to the price of CB. This paper proposes an approximation based on the idea of an auxiliary reversed binomial (ARB) tree, and shows that the approach can be efficiently implemented under the Cox-Ross-Rubinstein parameterization. Two important insights emerge from ARB. First, the convertible bond is unconditionally callable at higher stock prices, but uniformly not callable at lower prices. Second, the effect of the soft-call is rather localized around the trigger price. Surprisingly, the simple One-Touch, or one out of one, approximation is found to yield CB prices that are very close to those from ARB, even though ARB is found to a better approximation in almost every aspect. Further numerical results suggest that in order to generate well-behaved CB prices, cautions need to be taken while designing the terms of soft-call. Being independent of the finite difference pricing grid, the proposed ARB tree can also be used in association with the tree method or Monte Carlo simulation, and could in principle be applicable to exotic derivatives with similar embedded options.