Binomial model, Financial Management

The option features embedded in many bonds and fixed-income securities have made the binomial interest rate tree approach a valuable model for pricing debt. Binomial model is an option valuation method developed by Cox, Ross, Rubinstein and Sharpe in 1979. This method of pricing options or other equity derivatives is based on the assumption that probability of each possible price follows a binomial distribution and that prices can either move to a higher level or a lower level with time until the option expires.

To value bonds using the binomial model, a binomial interest rate tree is to be constructed first. A binomial interest rate tree is nothing but a graphical representation of the short-term interest rates over a period of time based on some assumption about interest rate volatility. In this tree, each node represents a defined time period, say one year. Each node is represented by the letter T. The current year spot rate for the specified time period, in our example one-year spot rate, is represented by r0. As the model is based on the assumption that each possible price can either move higher or lower, each node gives rise to two options, TH and TL, where H represents higher and L represents lower. (Multiple paths to same node have been avoided to keep the figure simple. For example, HL can be reached in two ways, HL and LH, but only HL is shown in the Table.)

In Table 1, we see that T is the starting point of the interest rate tree, and r0 represents the current 1-year spot rate. It is assumed that the 1-year rate can take two possible values, either higher or lower, in the defined time period, i.e., 1-year in our example and they both have the same probability of occurrence.

         σ       = Assumed volatility of the 1-year rate.

         r1, L      = The lower 1-year rate one year from now.

         r1, H      = The higher 1-year rate one year from now.

Table 1: Binomial Interest Rate Tree

1634_binomial model.png

Now, we can define the relationship between the lower and higher value as follows:

r1, H = r1, L ( e )

e is the base of the natural logarithm, 2.71828.

Let us calculate the values using a hypothetical example; let us assume that the value of r1, L to be 4.7801, σ is 10% per year, then,

r1, H = 4.7801% (e 2*0.10) = 5.8384

In the second year, we find three possible values for the one year rate; they are:

r2, HH  =      1-year rate in second year assuming the higher rate in the first year and the

 higher rate in the second year.

r2, HL   =       1-year rate in second year assuming the higher rate in the first year and the

 lower rate in the second year.

r2, LL    =       1-year rate in second year assuming lower rate in first year and lower rate in

 second year.

r2, HH  is defined as r2, LL (e 4 σ) and r2, HL = r2, LL( e ).

Assuming r2, LL to be 4.8051% and σ as 10%, we can determine r2, HH, r2, HL as follows:

r2, HH = 4.8051% (e 4*0.10) = 7.1683%

r2, HL = 4.8051% (e 2*0.10) = 5.8689%

There are four possible values for the 1-year rate in the third year, they are denoted as r3, HHH, r3, HHL , r3, HLL and r3, LLL.

The relationship between them can be expressed as follows:

r3, HHH = r3, LLL( e )

r3, HHL = r3, LLL(e )

r3, HLL = r3, LLL( e ).

Let us make the Table easier to understand by replacing the notations with the simplified notations.

Table 2: Binomial Interest Rate Tree with One-Year Rates

1897_binomial model1.png

*rt equals forward 1-year lower rate. t year from now.

In valuing option free bonds, we have seen the use of single forward rate, but in valuing bonds with embedded option we use a set of forward rates, as at every level we come with more then one option.

Posted Date: 9/10/2012 6:42:32 AM | Location : United States

Related Discussions:- Binomial model, Assignment Help, Ask Question on Binomial model, Get Answer, Expert's Help, Binomial model Discussions

Write discussion on Binomial model
Your posts are moderated
Related Questions
Differences between Hedge Funds and Mutual Funds Hedge Funds are extremely flexible in their investment options because they use financial instruments generally beyond the reach

LEAMINGER PLC (a) Purchase outright (2) Balancing allowance Tax effect = 93,906 × 30% = 28,172 Finance lease Annuity Factor (AF) at 10% for 4 year

Twelve cases of leukemia are reported in people living in a certain census tract over a 5 year period. Is this number abnormal is only 6.7 cases would be expected based on national

Export/Import Bank (Eximbank) Federal Import-Export Bank, whose mainly function originally was to compensate U.S. exporters for subsidies approved competitors by foreign govern

Six years ago . the singleton company sold a 20 year bond with a 14% annual coupon rate and a 9% call premium. today, singleton called the bonds. the bonds originally were sold at

Functional Classification of Mutual Funds Functional classification of Mutual Funds is based on the basic characteristics of the mutual fund schemes for subscription. Mutual Fu

Question 1 Globalization is a process of international integration that arises due to increasing human connectivity as well as the interchange of products, ideas and other aspe

Determine about the Zero Interest Bonds (ZIBs) Very much alike DDBs, only crucial difference is that these are issued at face values (DDBs are issued at a discount to face valu

Q. What do you mean by Financial Leverage? Financial Leverage: - The financial leverage perhaps defined as the tendency of the residual net profit to vary disproportionately wi

The following information pertains to Fairways Driving Range, Inc.: The company is considering operating a new driving range facility in Sanford, FL. In order to do so, they wi