Bond valuation would be relatively simple if interest rates exhibit little day-to-day volatility. One could value a bond by discounting each of its cash flows at its own zero-coupon ("spot") rate. This procedure is equivalent to discounting the cash flows at a sequence of one-period forward rates. However, investors having bonds with one or more embedded options may result in volatile interest rates, a historically steep yield curve, and complex bond structures. These make valuation of bonds with embedded options, a complicated process. Therefore, the framework used for valuing bonds in a relatively stable interest rate environment is inappropriate for valuing bonds with embedded options.
In building a valuation model for bonds with embedded option, we need to consider the future cash flows which in turn depend on the changing future interest rates. The future interest rate is incorporated into a valuation model by assuming a few interest rates changes considering volatility. With the assumed interest rates volatility, an interest rate "tree" representing possible future interest rates is constructed. From interest rate tree we can obtain interest rates that are used to generate the cash flows and also to compute the present value of the same.
An interest rate model is a probabilistic description of how interest rates can change during the life of the bond. An assumption about the relationship between the level of short-term interest rates and the interest rate volatility, (measured by the standard deviation), is made to build the interest rate model. Interest rate models can be classified as 'one-factor' model and 'two-factor' model. When only one interest rate is involved, it is known as one factor model. When more than one interest rate changes are considered, i.e., if a model considers both short-term and long-term interest rates, it is called two-factor model.
With interest model and interest rate volatility in place, an interest rate tree can be developed. Binomial model is an option valuation method, which is developed based on the assumption that probability of each possible price follows a binomial distribution and that prices can either move to higher level or a lower level with time until the option expires (over any short time period). This model reduces possibilities of price changes, removes the possibility for arbitrage, assumes a perfectly efficient market, and shortens the duration of the option. Under these simplifications, it is able to provide a mathematical valuation of the option at each point in time specified. A valuation model built on the assumption of three possible rates is known as trinomial models. A more complex model is to be considered if there are more than three possible rates in the next period. Whatever may be our assumption about the interest rates, an interest rate tree must be capable of producing an arbitrage-free value i.e., it must be able to produce a value for the on-the-run Treasury issue, that is equal to its observed, market price. Once an interest rate tree is constructed, the next thing to do is to use this to value a bond with embedded option.