Article

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

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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.

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... Except an example, how to generate the daily indicator is not explicitly described in that paper. Liu (2008) proposes another approximation method (ARB). The idea is that at any state of the stock, if the stock goes backward m steps on a tree, all historical m-step stock paths can be generated, based on counting the number of paths which satisfy the soft-call trigger condition, the probability of trigger can then be calculated. ...
... The idea is using an auxiliary state variable to keep track of the historical path, but instead of tracking the exactly path, here we will only track the number of days stock is above the barrier (It will be called level in this article). The idea is similar to the ARB tree proposed by Liu (2008), instead of counting the trigger probably of stock at every stock grid point, here we will generate the conditional trigger probability based on the new auxiliary state variable. If walk backward on the tree, this probability can be calculated out. ...
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Liu, Q., 2007. China's convertible bond market, Chapter 6 in China's Financial Markets: An Insider's Guide to How the Markets Work (Neftci, Salih. N. and Michelle Y. Menager-Xu eds).
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Calamos, J. P., 1988. Convertible Securities: The Latest Instruments, Portfolio Strategies, and Valuation Analysis. McGraw-Hill, NY.