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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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. ...
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.
... In other words, the m-out-of-n soft call provision is replaced with the 1-out-of-1 soft call provision. Liu (2008) proposes an auxiliary reverse binomial tree method to handle the m-out-of-n soft call provision embedded in a CB with a constant interest rate. Due to the path dependency in the provision, the number of examined paths on the binomial tree grows exponentially, which makes it computationally infeasible. ...
The Chinese convertible bond market has been developing rapidly in the last 10 years. However, some special characteristics of the Chinese convertible bond, such as the soft call/put provision, cause huge difficulty in the valuation. In this paper, we establish a new valuation model for the Chinese convertible bond, based on the available Chinese market data, through a hybrid willow tree approach with consideration of the underlying stock price, stochastic interest rate, and credit risk of the issuer. We employ the Brownian bridge to handle the special characteristics. Finally, we examine our model prices for the daily market closing prices for 20 Chinese convertible bonds traded from 2007 to 2017. The empirical results show the effectiveness of our valuation model under the historical and implied volatilities of the underlying stock price.
... In other words, it replaces the m out of n with 1 out of 1, but it is different from the Chinese CB contract, which the current trading price is excluded from determining the activation of the provision. Liu (Liu, [2008]) proposed an auxiliary reverse binomial tree method to handle the m out of n soft-call provision embedded in a CB with a constant interest rate. Due to the path-dependency, the number of examined paths on the binomial tree grows exponentially, which makes its computation infeasible for long-term CBs. ...
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Chinese convertible bond market is developing very fast in recent 10 years. However, studies on the pricing model of this emerging market are limited due to some special features embedded in the CB contract different from the major markets. Among these features, the most difficult one is the Parisian-like m out of n soft-call/put provision because of the moving observation window and high path dependency. In this paper, we consider the stock price, stochastic interest rate and credit risk of the issuer on a two-factor willow tree structure. Then, we employ the Brownian bridge at each day to handle the Parisian-like m out of n soft-call/put provision. Finally, we compare the prices computed by our hybrid method with the daily closing prices of 20 Chinese CBs from 2007 to 2017. It illustrates that our pricing model can be a good benchmark for Chinese CBs pricing. Moreover, as a by-product, our method can also calibrate the implied volatility of stock from the market price of its CB, even though no stock option is traded on Chinese market.
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Conference Paper
“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., 2<sup>260</sup>) 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, 2<sup>260</sup> 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.
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  • Q Liu
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).
Convertible Securities: The Latest Instruments, Portfolio Strategies, and Valuation Analysis
  • J P Calamos
Calamos, J. P., 1988. Convertible Securities: The Latest Instruments, Portfolio Strategies, and Valuation Analysis. McGraw-Hill, NY.