Binomial tree

For large message sizes, where the latency term can be ignored, Rabenseifner s algorithm should provide improved performance relative to the binomial tree algorithm, especially when the number of processes is large. 

An all-reduce operation can be performed as an all-to-one reduction followed by a one-to-all broadcast, and, using the binomial tree algorithm for... 

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. 

The first step determines the binomial inflation rate tree according to the inflation expectations. The binomial tree is used for pricing a hypothetical annual... 

FIGURE 6.9 Binomial tree of coupon payments and principal repayment. 

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. 

The valuation of a conventional bond can be performed also using a binomial tree. On maturity, the bond must be priced at par value plus the semiannual coupon payment equal to 2.75. Therefore, the value of a conventional IxHid at maturity tio must be equal to 102.75. The value of the bond in other nodes prior to maturity is calculated using the semi-aimual discount rate of 4.02%. For instance, at node the pricing is given by Equation (9.6) ... 

FIGURE 9.5 The valuation of a conventional bond using the binomial tree. 

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). 

For instance, at maturity, the stock price can have a maximum value of 11.51 and a minimum value of 0.35, according to the assumed volatility. Therefore, at higher node, the value of option will be equal to 8.91 as the difference between the stock price of 11.51 and conversion price or strike price of 2.6. In contrast, at lower node, because the stock price of 0.35 is lower than conversion price of 2.6, the option value will be equal to 0. Particular situation is in the middle of the binomial tree in which in the upstate the stock price is 2.84 and downstate is 1.41, meaning that in the first case the option is in the money, while in the second case is out of the money. 

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. 

Take into account that in qiecial situations, for instance, with the implementation of stock barrier, the binomial tree requires a higher number of steps. Conversely, B S and binomial results can be very different. 

The first one, is to consider the dividend payment as a percentage above the stock price in each node i of the binomial tree as follows (Equation 9.17) ... 

In the example shown in Section 9.3.1, we consider an underlying asset that does not pay dividends. We suppose now a stock that pays 1% of dividends in years 1, 2, 3, 4 and 5 in terms of dividend yield. As explained before, the introduction of dividends decreases the stock price tree. Therefore, at the end of the binomial tree, the stock price will be lower. For instance, as shown in Figure 9.26, the highest stock price tree without the inclusion of dividends is 11.51. 

Consider the following example. We assume to have two hypothetical bonds, a treasury bond and a callable bond. Both bonds have the same maturity of 5 years and pay semiannual coupons, respectively, of 2.4% and 5.5%. We perform a valuation in which we assume a credit spread of 300 basis points and an OAS spread of 400 basis points above the yield curve. Table 11.1 illustrates the prices of a treasury bond, conventional bond and callable bond. In particular, considering only the credit spread we find the price of a conventional bond or option-free bond. Its price is 106.81. To pricing a callable bond, we add the OAS spread over the risk-free yield curve. The price of this last bond is 99.02. We can now see that the OAS spread underlines the embedded call option of the callable bond. It is equal to 106.81-99.02, or 7.79. In Section 11.2.3, we will explain the pricing of a callable bond with the OAS methodology adopting a binomial tree. 

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. 

In the next section, we will show that the bond s value is estimated by assessing the value of the option-free bond and the value of the embedded option using the binomial tree. The same factors that are implemented into B S formula are used for the binomial tree. They are ... 

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 ... 

The binomial tree determines the theoretical bond s price ... 

The proposed methodology determines the value of a callable bond using the binomial tree. The example assumes a 5-year callable bond with the following conditions (Table 11.2) ... 

In the first step, we determine the interest rate path in which we create a risk-neutral recombining lattice with the evolution of the 6-month interest rate. Therefore, the nodes of the binomial tree are for each 6-month interval, and the probability of an upward and downward movement is equal. The analysis of the interest rate evolution has a great relevance in callable bond pricing. We assume that the interest rate follows the path shown in Figure 11.4. In this example, we assume for simplicity a 2-year interest rate. We suppose that the interest rate starts at time tg and can go up and down following the geometric random walk for each period. The interest rate rg at time tg changes due to two main variables ... 

The first one, is the volatihty that determines the width of the binomial tree. Higher the volatility, greater the up and down movements from the prior nodes ... 

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%. 

Determine the Value of an Option-Free Bond After determining the evolution of the interest rates, we calculate the value of the option-free bond. In this case, we develop a binomial tree by ignoring the call feature in which at maturity the value of bond is 100. Although the final value could be equal to 104 (principal plus coupons), we consider at maturity the bond s ex coupon value. In fact, at year 5 the bond s price is 100. 

The process is repeated until time to for which the bond s price is 106.13. The binomial tree is shown in Figure 11.9. 

Conversely, if the option is held the value is discounted at each node at the risk-free rate of the binomial tree. It is given by (11.8) ... 

Determine the Value of a Callable Bond Since the option is held by the issuer, the option element decreases the value of the bond. Therefore, the value of a callable bond is found as an option-free bond less the option element according to Formula (11.3). For the hypothetical bond, the price is 106.13-2.31 or 103.82. This is shown in Figure 11.11. The binomial tree shows that at maturity the option free and callable bond have the same price, or 100. Before the maturity, if the interest rates go down, the callable bond s values are less than an option-free bond, and in particular when the embedded option is deeply in the money, the callable values equal the strike price according to the caU schedule. Conversely, when the interest rates go up, the option free and callable bonds have the same price. 

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