In the present paper we show that the Binomial-tree approach for pricing, hedging, and risk assessment of Convertible bonds in the framework of the Tsiveriotis-Fernandes model has serious drawbacks. Key words: Convertible bonds, Binomial tree, Tsiveriotis-Fernandes model, Convertible bond pricing, Convertible bond Greeks, Convertible Arbitrage, Delta-hedging of Convertible bonds, Risk Assessment of Convertible bonds.
In the present research, we address a very important and unanswered so far question regarding the Binomial-tree approach to the Tsiveriotis-Fernandes (TF) model for pricing Convertible Bonds (CBs). Namely, does the Binomialtree framework provide accurate pricing, hedging and risk assessment? We show on a set of representative examples that by applying the Binomial-tree methodology one is unable to provide a consistent analysis of the pricing, hedging and risk assessment.
An important feature of the pricing of CBs is that similar to the American options there is no closed form solution, and the numerical computation of the solution is a challenge due to the free boundaries arising. Respectively, in our study we will employ the natural properties of CBs which are usually exploited in practice. Depending on the underlying stock we examine the profile of CB’s price, of CB’s sensitivities, Convertible Arbitrage strategy, and the Monte Carlo VaR estimation.
Convertible bonds are a widely used type of contract, playing a major role in the financing of the companies ( [3], [7], [9]). From a pricing and hedging perspective they are highly complex instruments. They have the early exercise feature of American options but in three forms: the option to be converted, the option to be called and the option to be put. Hence, sometimes they behave like a bond and sometimes like a stock.
Convertible bonds (or simply “convertibles”) are bonds issued by a company where the holder has the option to exchange (to convert) the bonds for the company’s stock at certain times in the future ( [2]). The “conversion ratio” is the number of shares of stock obtained for one bond (this can be a function of time). If the conversion option is executed, then the rights to future coupons are lost. The bonds are almost always callable (i.e., the issuer has the right to buy them back at certain times at predetermined prices).
The holder always has the right to convert the bond once it has been called. The call feature is therefore usually a way of forcing conversion earlier than the holder would otherwise choose. Sometimes the holder’s “call option” is conditional on the price of the company’s stock being above a certain level. Some convertible bonds incorporate a put feature. This right permits the holder of the bond to return it to the issuing company for a predetermined amount.
Throughout the years different convertible bond pricing methodologies were developed. The main development was in the area of modeling the CB’s price dynamics, as well as towards design of numerical methods for evaluating the convertible bond pricing function. The most advanced and popular idea for modeling CB’s price dynamics was introduced in the seminal paper of Tsiveriotis and Fernandes ( [1], [8]). They have proposed to split the convertible bond value into two components: a cash-only part which is subject to credit risk, and an equity part, which is independent of the credit risk. This leads to a pair of coupled partial differential equations under certain constraints (in fact boundary and free boundary conditions) that can be solved to value the price of the convertibles. From numerical point of view Tsiveriotis and Fernandes have proposed explicit finite difference method for solving their system of equations. On the other hand, Hull ( [2]) has proposed to use Binomial-tree approach for solving the same system. More precisely, the Hull approach is based on Cox, Ross and Rubinstein (CRR) tree.
Currently, there are two basic approaches for CB pricing, hedging and risk assessment. The first one that is based on trees (binomial and trinomial) ([2], [7], [9], [10]), and the second one which is based on finite difference techniques ( [1], [5], [4]).
There is a gap in the above studies as they do not provide a complete report on the methodology performance. By the present paper we want to indicate essential drawbacks of the Binomial-tree methodology and mistakes that are made when this methodology is used, in major practice areas as hedging and risk assessment.
The paper is organized as follows: In section 2 we explain the Binomial-tree scheme for approximation of the TF model. Our main results are in section 3 where we provide the performance valuation. Finally, in the Appendix in section 5 we provide a short but closed and informative outline of the model of Tsiveriotis-Fernandes.
We follow the Binomial-tree approximation to the TF model that is widely used in practice (cf. [2]). It involves modeling the issuer’s stock price. It is assumed that the stock price process follows geometric Brownian motion and its dynamics is represented by the usual Binomial-tree of Cox, Ross and Rubinstein.
The life of the tree denoted by T is set equal to the life of the convertible bond denoted also by T. The value of the convertible bond at the final nodes (at time T ) of the tree is calculated based on the conversion option that the holder has at that time T. We then roll
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