Binomial Trees, Part 1

The arbitrage free valuation framework is used extensively in the pricing of securities. The basic principle of the “law of one price” in freely functioning markets drives this analytical framework.

Arbitrage free valuation methods value securities such that no market participant can earn an arbitrage profit in a trade involving that security. An arbitrage transaction involves no initial cash outlay but a positive riskless profit (cash flow) at some point in the future.

There are two types of arbitrage opportunities: value additivity (when the value of whole differs from the sum of the values of parts) and dominance (when one asset trades at a lower price than another asset with identical characteristics).

If the principle of value additivity does not hold, arbitrage profits can be earned by stripping or reconstitution. A five year, 5% Treasury bond should be worth the same as a portfolio of its coupon and principal strips. If the portfolio of strips is trading for less than an intact bond, one can purchase the strips, combine them (reconstituting), and sell them as a bond.

Similarly, if the bond is worth less than its component parts, one could purchase the bond, break it into a portfolio of strips (stripping), and sell those components.

Arbitrage free valuation of a fixed rate, option free bond entails discounting each of the bond’s future cash flows (i.e., each coupon payment and the par value at maturity) using the corresponding spot rate.

The binomial interest rate tree framework assumes that interest rates have an equal probability of taking one of two possible values in the next period (hence the term binomial).

Over multiple periods, the set of possible interest rate paths that are used to value bonds with a binomial model is called a binomial interest rate tree. The diagram in Figure 24.1 depicts a binomial interest rate tree.

Figure 24.1: Binomial Interest Rate Tree

The starting point $i_0$ in this tree is the current 1 period spot rate and is called the root of the tree. The future nodes are indicated with the bold dots (♦). A node is a point in time when interest rates can take one of two possible paths, an upper path, U, or a lower path, L.

Now, consider the node on the right side of the diagram where the interest rate $i_{2,LU}$ appears.

This is the rate that will occur if the initial rate, $i_0$, follows the lower path from node 0 to node 1 to become $i_{1,L}$, then follows the upper of the two possible paths to node 2, where it takes on the value $i_{2,LU}$. At the risk of stating the obvious, the upper path from a given node leads to a higher rate than the lower path. In fact,

Note that an upward move followed by a downward move, or a down then up move, produces the same result: $i_{2,LU}$ = $i_{2,UL}$ (the math term for this is lattice).

The interest rates at each node in this interest rate tree are one period forward rates corresponding to the nodal period. Each forward rate is related to (i.e., is a multiple of) the other forward rates in the same nodal period. Adjacent forward rates (at the same period) are two standard deviations apart.

For the first period, there are two forward rates and hence:

Beyond the first nodal period, non adjacent forward rates are a multiple of the two standard deviations (depending on how separated the forward rates are).

For example,

The relationship among the set of rates associated with each individual nodal period is a function of the interest rate volatility assumed to generate the tree. Volatility estimates can be based on historical data or can be implied volatility derived from interest rate derivatives.

The binomial interest rate tree framework is a lognormal random walk model with two properties: (1) higher volatility at higher rates and (2) non negative interest rates.

Valuing an Option Free Bond with the Binomial Model

Backward induction refers to the process of valuing a bond using a binomial interest rate tree. The term “backward” is used because in order to determine the value of a bond today at Node 0, you need to know the values that the bond can take at the Year 1 node. But to determine the values of the bond at the Year 1 nodes, you need to know the possible values of the bond at the Year 2 nodes.

Thus, for a bond that has N compounding periods, the current value of the bond is determined by computing the bond’s possible values at Period N and working backwards to Node 0.

Because the probabilities of an up move and a down move are both 50%, the value of a bond at a given node in a binomial tree is the average of the present values of the two possible values from the next period. The appropriate discount rate is the forward rate associated with the node.

The following example should make this all clear.