Continuous time bond pricing

Brennan M, Schwartz E (1979) A Continuous Time Approach to the Pricing of Bonds. Journal of Banking and Finance 3 133-155. 

Brennan, M., Schwartz, E., 1979. A continuous time approach to the pricing of bonds. J. Bank. Financ. 3, 134. 

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. 

Analysts and researchers frequendy work with logarithms of yields and prices, or continuously compounded rates. One advantage of the logarithmic approach is that it converts the nonlinear relationship expressed in (3-2) into a linear one. The zero-coupon bond price equation in continuous time is... 

This section is an introduction to bond pricing in continuous time. Chapter 4 presents a background on price processes. 

Expression (3.13) states that the bond price is a function of the continuously compounded interest rate. The r ht-hand side is the discount factor at time t. At t= T—that is, on the redemption date—the discount factor is 1, which is the redemption value of the bond and hence the price of the bond at that time. 

Brennan, M., and E. Schwartz. 1979. A Continuous Time Approach to the Pricing of Bonds, journal of Banking and Finance -------. 1980. Conditional Predictions of Bond Prices and Returns. Journal of... 

The first property that asset prices, which can be taken to include interest rates, are assumed to follow is that they are part of a continuous process. This means that the value of any asset can and does change at any time and from one point in time to another, and can assume any ffactiOTi of a unit of measurement. It is also assumed to pass through every value as it changes so, for example, if the price of a bond moves from 92.00 to 94.00, it must also have passed through every point in between. This feature means that the asset price does not exhibit jumps, which in fact is not the case in many markets, where price processes do exhibit jump behaviour. For now, however, we may assume that the price process is continuous. 

The continuously compounded constant spot rate is r as before. An investor has a choice of purchasing the zero-coupon bond at price P(t, T), which will return the sum of 1 at time T, or of investing this same amount of cash in the money market account, and this sum would have grown to 1 at time T. We know that the value of the money market accoxmt is given by Me If M must have a... 

In 1995 the SNDO launched the second inflation-linked bond, another zero but with a shorter maturity of 10 years (No. 3002, 0% 2004). At this time the SNDO decided to replace the common price anc-tions with multiple price auctions. Moreover, the SNDO opened a noncompetitive facility for small volumes in the auctions, so that small investors could enter the market. In February 1996 the SNDO launched two new bonds a 5-year zero-coupon bond (3003, 0% 2001) and a 12-year coupon bond (3101, 4% 2008). In June 1996 the 24-year coupon bond (3102, 4% 2020) was launched. The market continued to grow rapidly in 1997 and 1998. 

In the academic literature, the risk-neutral price of a zero-coupon bond is expressed in terms of the evolution of the short-term interest rate, r t)—the rate earned on a money market account or on a short-dated risk-free security such as the T-bill—which is assumed to be continuously compounded. These assumptions make the mathematical treatment simpler. Consider a zero-coupon bond that makes one payment, of 1, on its maturity date T. Its value at time ris given by equation (3.14), which is the redemption value of 1 divided by the value of the money market account, given by (3.12). 

This process is continued all the way back to time to to give a start price of 98.23 for the bond part of the convertible. 

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