Binomial model, Financial Management

Assignment Help:

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.


Related Discussions:- Binomial model

Show the compound value of the single flow, Q. Show the Compound Value of t...

Q. Show the Compound Value of the Single Flow ? Compound Value of the Single Flow (Lump Sum):- The process of computing future value becomes very cumbersome if they have to be

Cash flow matching, Cash flow matching strategy is used to build a ...

Cash flow matching strategy is used to build a bond portfolio wherein the cash flows of the bond portfolio exactly match a stream of liabilities. The most s

PROFIT MAXIMIZATION, what are the arguments in favour of profit maximizat...

what are the arguments in favour of profit maximization?

Reinforced concrete design, Q. Reinforced concrete design? In BS8110 fo...

Q. Reinforced concrete design? In BS8110 for reinforced concrete design, it is stated that longer tension lap lengths have to be provided at the top of concrete members. The mo

Explain the mechanism that restores the balance of payments, Explain the me...

Explain the mechanism which restores the balance of payments equilibrium when it is disturbed under the gold standard. Answer:  The adjustment mechanism within the gold standar

Show the working capital in a business, Q. Show the Working capital in a bu...

Q. Show the Working capital in a business? Working capital in a business is essential since of operating cycle. However the need for working capital doesn't come to an end afte

Identify an analytic theme for fictitious business, 1. identify an analytic...

1. identify an analytic theme or goal for a fictitious business or something that you are working on (e.g. Maximize revenue in a car dealership) 2. Build an Enterprise Bus Matri

Re-order point - technique of inventory management, Q. Re-order point - tec...

Q. Re-order point - technique of inventory management? Re-order point: - The re-order point is that stock level at which an order should be placed. Mutually the excessive and i

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd