In this section we will value a bond using a binomial interest rate tree.

The Binomial Interest Rate Tree

As its name suggests, the binomial interest rate tree models future periods with two distinct scenarios: an "up" scenario and a "down" scenario. The tree "grows" by repeating this assumption at every point in time. This binomial process continues until maturity.

The interest rates in a tree are implied forward rates. They are based on three assumptions:

• An interest rate model that governs the random process of interest rates.
• The assumed level of interest rate volatility.
• The current benchmark yield curve.

Interest rates can take on negative values with positive volatility. We can fix the problem by working with logs. Lognormal distribution of interest rates has two appealing properties:

• Interest rates can never become negative.
• The absolute change in interest rates becomes smaller and smaller once they get too close to zero.

Recombining trees means that an upward-downward sequence leads to the same result as a downward-upward sequence. That is, r_ul = r_lu. At period n there are n + 1 different values.

The relationship between the two adjacent rates:

i_1,H = i_1,Le^{2 σ} where

• σ is the assumed volatility of the one-period rate.
• e is the base of natural logarithms.
• i_1,H is the higher of one-period forward rate one period from now.
• i_1,H is the lower of one-period forward rate one period from now.

The Role of Volatility

For a lognormal distribution the standard deviation of the one-year rate is equal to i₀ σ.

There are two ways to estimate the volatility:

• The historical estimate is based on what happened in the past.
• Use a model (e.g., Black-Scholes-Merton model) and its other inputs (strike price, call option price, time to expiration, etc.) to estimate the standard deviation.