Binomial tree model

As explained in the introduction, the value of a convertible bond is the sum of two main components, the option-free bond and a call option on underlying security. The value of the option-free bond, or bond floor, is determined as the sum of future payments (coupon and principal at maturity). Therefore, the bond component is influenced by three main parameters, that is the maturity, the coupon percentage on par value and the yield to maturity (discount rate). Differently, the value of a call option can be found mainly through two option pricing models, Black Scholes model and binomial tree model. 

Cox et al. (1979) developed a binomial tree model in which they determine the option value through a discrete time formula. 

Determining the Value of an Embedded Option The value of an embedded option is found through the binomial tree model. The first step is to forecast the value of the underlying security in which the price S of a security can move, respectively, in the upstate and downstate with a probability of p and 1 p. The change in price occurs in discrete time interval At and will depend on the level of volatility assumed. An option written on the asset, with maturity T will move in discrete steps as the movements of the share prices. The process can be carried on for any number of time intervals (Figure 9.6). 

To calculate the value of these bonds, it is preferable to use the binomial tree model. The value of a straight bond is determined as the present values of expected cash flows in terms of coupon payments and principal repayment. For bonds with embedded options, since the main variable that drives their values is the interest rate, the binomial tree is the most suitable pricing model. 

The binomial tree model evaluates the return of a bond with embedded option by adding a spread to the risk-free yield curve. Generally, the price obtained by the model is compared to the one exchanged in the market. If the theoretical price is different, the model can be calibrated with three key elements. The first ones are the volatility and drift factor. They allow to calibrate the model interest rate path in order to obtain the equality with the market yield curve. The third one is the spread applied over the yield curve. Generally, when volatility and drift are correctly calibrated, the last element to select in order to obtain the market parity is the spread. Conventionally, banks define it in the following way ... 

Any models using implied forward rates to generate future prices for options underlying bonds would be assuming that the future interest rates implied by the current yield curve will actually occur. An analysis built on this assumption would, like yield-to-worst analysis, be inaccurate, because the yield curve does not remain static and neither do the rates implied by it therefore future rates can never be known with certainty. To avoid this inaccuracy, a binomial tree model assumes that interest rates fluctuate over time. These models... 

As with the previous models, the key factor is the short-rate. Using the binomial tree approach, a one-step tree is used to derive the current short-rate to the short-rates one period in the future. These derived rates are then used to derive rates two periods away, and so on. 

Therefore, we propose an example in which we price an inflation-linked bond by using a binomial tree. Conventionally, this type of pricing model is not implemented in the reality, but it allows to understand the impact of the embedded option on bond s value. 

In 1973, Myron Scholes and Fisher Black developed a model known as B S model for valuing options. Like the binomial tree, in the B S model the option value depends mainly on the price of the underlying asset, volatility, interest rate, time to expiration and dividend yield. Because in this chapter, we propose the value of a cOTivertible as the sum of the straight bond and call option, the... 

Therefore, the model is easy to implement and gives similar results as the binomial tree. Because B S works in continuous compounding while the binomial tree in discrete time, the models give the same results only if the binomial tree has a high number of steps. The more periods in binomial tree are implemented, the nearer is the value that we get in both models. Consider the convertible bond pricing shown in Section 9.3.1. In that analysis we estimate the value of a call option using the binomial tree, obtaining a value per call of 0.46. 

Figure 9.22 confirms the sensitivity analysis of the share price implemented by Connolly (1998) in which in some area the binomial tree overvalues the B S model and in other area not. 

FIGURE 9.22 The comparison between the value of Black Scholes model and binomial tree. 

After drift adjustment, the new binomial tree is plotted in Figure 11.7. Increasing the slope of the model yield curve, all interest rates are greater than before. The maximum interest at time ts is now 10.52%. 

Equation (4.21) states that the dynamics of the forward-rate process, beginning with the initial rate/(0, J), are specified by the set of Brownian motion processes and the drift parameter. For practical applications, the evolution of the forward-rate term structure is usually derived in a binomial-type path-dependent process. Path-independent processes, however, have also been used, as has simulation modeling based on Monte Carlo techniques (see Jarrow (1996)). The HJM approach has become popular in the market, both for yield-curve modeling and for pricing derivative instruments, because it matches yield-curve maturities to different volatility levels realistically and is reasonably tractable when applied using the binomial-tree approach. 

A better measure of the relative value of a bond with an embedded option is the constant spread that, when added to all the short rates in the binomial tree, makes the bond s theoretical (model-derived) price equal to its observed market price. The constant spread that satisfies this requirement is the option-adjusted spread. It is option adjusted because it refiects the option feature attached to the bond. 

Application of the binomial model requires a binomial tree detailing the price outcomes from the start period, which is shown at FIGURE 13.3. In the case of a convertible bond this will refer to the prices for the underlying asset, which is the ordinary share of the issuing company. 

Ahu, H., and Chen, J. J. (1997). Tree-structured logistic model for over-dispersed binomial data with application to modeling developmental effects. Biometrics, S3 435-455. 

One popular way of turning theory into practice is to use a tree approach to modelling. The tree can be either binomial or trinomial in its construction. To illustrate the idea consider first the binomial approach. The tree could be set up to reflect observed or estimated market short rates and the data provided in Exhibit 18.2 will help to demonstrate this idea. 

The binomial model evaluates a bond s return by measuring the extent to which it exceeds those determined by the risk-free short rates in the tree. The spread between these returns is the bond s incremental return at a specified price. Determining the spread involves the following steps ... 

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