Article

Back to the Future: An Approximate Solution for N Out of M Soft-Call Option

Authors:
  • bloomberg
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Abstract

In convertible bond market, it is very common to protect the conversion privilege from being called away too soon by using soft-call constraint, or to protect the bond being converted too soon by using provision convert constraint. The first option will protect the bond holder; the second will be benefit to bond issuer. Both constrains have the common feature that the option can be exercise only when the underlying stock closes above a pre-set barrier for any n or more days over m consecutive trading days up to the exercise day. This feature brings challenge for pricing. This paper will propose an approximation solution by Looking Backward (LB) method. In order to illustrate the idea more clearly, I will focus on the Black model stock dynamic using binomial tree based on Cox-Ross-Rubinstein scheme. The results are compared with the exactly solution given by the author in [1]. The extension to other numerical method such as PDE with more general stock dynamic will also be discussed, and the numerical scheme will be laid out. The idea of the method can be applied to the pricing of other path dependent instruments in general.

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Article
In convertible bond market, it is very common to protect the conversion privilege from being called away too early by using a soft-call constraint. It stipulates is that the option can be exercise only when the underlying stock closes above a pre-set barrier for any n or more days over m consecutive trading days. Such soft-call option feature makes pricing a very challenging task. Up to now, no exact solution to this problem has been reported in literatures, which not only brings complications to convertible bond pricing but also accounting/taxing/regulation. This paper proposes an exact numerical solution to this soft-call pricing problem based on Black stock dynamic, which can provide a benchmark for other approximation approaches.
Article
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.
Conference Paper
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Article
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