**Multiperiod Binomial Valuation of a Derivative**

To calculate a multiperiod binomial valuation of a derivative, you must first understand how to use the binomial method of valuing a derivative. This method assumes that at any point in time, a derivative has only two possible values for the next period of time. One may refer to these states as the “up” state and the “down” state. Typically and intuitively, the “up” state represents a better value position, but the mathematics works the same no matter which value you select for the “up” state.

The tracking portfolio must first be determined by solving two simultaneous equations in order to find the beta (Β) and delta (Δ) values. The first of these equations deals with the “up” values. The equation is ΔS_{u} + Β(1+r_{f}) = V_{u}, where:

Δ is the number of units of the underlying asset

S_{u} is the value of the underlying asset in the “up” state

Β is the number of dollars in the risk free security

r_{f} is the risk free rate

V_{u} is the value of the derivative in the “up” state.

Similarly there is a down node governed by the equation ΔS_{d} + Β(1+r_{f}) = V_{d}.

The data for V_{u}, V_{d} , S_{u}, S_{d} and r_{f} must be provided.

Subtracting the formulas for the up and down nodes and solving for Δ gives the formula:

Δ = (V_{u} – V_{d})/(S_{u} - S_{d}).

You substitute this value into either the up or down node equation to solve for Β:

Β = (V_{u} – ΔS_{u} ) /(1+r_{f}).

Knowing B and Δ, the value of the derivative can be calculated by the equation V=ΔS+Β, where S is the cost per share of the underlying asset.

Multiperiod Binomial Valuation merely extends the concept to multiple instances of binomials. You must start at the last set of nodes and work your way back to the current value. For example, assume you have the following data for an asset:

The initial price is $100. In the next period of time, it can be $110 (up node) or $90 (down node). In the second period of time, it can be $115 (up-up node), $105 (up-down node), $102 (down-up node) or $85 (down-down node). A derivative based on the asset is valued at $15 in the up-up node, $5 in the up-down node, $2 in the down-up node and $0 in the down-down node. The risk free rate is 5%.

You must look at the problem as being three instances of binomial valuations. The first time period contains an up and a down node which gives the first binomial. Each of these nodes has an up and a down node giving you two additional binomials. You would begin by calculating the Δ for the up-up/up-down binomial. Using (V_{u} – V_{d})/(S_{u} - S_{d}) would yield a Δ of:

(15-5)/(115-105) = 1

Then using (V_{u} – ΔS_{u} ) /(1+r_{f}) would give a B of:

(115 – (1)*(15))/(1+0.05)=-95.2

Substituting these values into V=ΔS+Β would give a derivative value at the up node of (1)*(110)-95.2 = 14.76

Similarly, the down-up/down-down binomial’s Δ and B are 0.02 and 0 to give a derivative value at the down node of 1.76.

Using these, calculate up and down derivative values. You can then solve the original derivative value using the first binomial (Δ and B are 0.65 and -54 to give a derivative value at the origin node of $11).

You may use this method for any arbitrary number of time periods.