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Back to the Future: An Approximate Solution for N Out of M Soft-Call Option
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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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.
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.
“Soft-call” in convertible bonds (CBs) usually means
that the bond can be recalled by the issuer only if the stock price has
previously closed above a specified trigger price for any 20 out of any
30 consecutive trading days. It is not an easy optionality to value and
no method has been implemented besides Monte Carlo. The problem is not
very well suited to Monte Carlo due to a large number of possible
permutations of stock price closes above or below the trigger over a
year period (i.e., 2260) with the result that a Monte Carlo
valuation requires a trade off between being slow and not smooth. The
soft-call feature is typically modeled in the CB industry by presuming
that the bond is called as soon as stock touches the trigger price.
After discussion of the exact solution of this problem (requiring
valuation of component derivatives on, of order, 2260 grids),
a simple algorithm is presented to approximately value this feature for
the general n out of m case of soft-call. The algorithm requires merely
a subtle change to the call feature of the one-touch model and only one
running of a grid or tree and hence it is very fast. The method boils
down to making the bond “1-touch” callable on some days and
not on others, the precise sequence being a function of the 29 day stock
price close history. It gives smooth functional output (the theoretical
price jumps from day to day) and very compelling qualitative results.
The results are accurate to a dime on the dollar of benefits due to
provisional call, and this is determined by comparison to the exact
solution for easily calculated cases
Convertible bonds can be difficult to value, given that they incorporate elements of both debt and equity. Further complications arise with the presence of additional options such as callability and puttability, and contractual complexities such as trigger prices and soft call provisions, when the ability of the issuing firm to exercise its option to call depends on the history of its stock price. This article explores the valuation of convertible bonds subject to credit risk using an approach based on the numerical solution of linear complementarity problems. Models that do not explicitly specify what happens in the event of a default by the issuer can lead to internal inconsistencies, such as a call by the issuer just before expiration rendering the convertible value independent of the credit risk of the issuer, or implied hedging strategies that are not self-financing. A general and consistent framework for valuing convertible bonds assuming a Poisson default process allows various models for stock price behavior, recovery, and action by holders of the bonds in the event of a default. The numerical algorithm uses a partially implicit method to decouple the system of linear complementarity problems at each time step. Numerical examples illustrating the convergence properties of the algorithm are provided.