# Binomial Model for Option Valuation

The standard DCF discounted cash flow procedure involves two steps, viz. estimation of expected future cash flows and discounting of these cash flows using an appropriate cost of capital. There are problems in applying this procedure to option valuation. While it is difficult (though feasible) to estimate expected cash flows, it is impossible to determine the opportunity cost of capital because the risk of an option is virtually indeterminate as it changes every time the stock price varies.

Since options cannot be valued by the standard DCF method, financial economists struggled to develop a rigorous method for valuing options for many years.

The basic idea underlying a model developed for valuing options is to set up a portfolio which imitates the call option in its payoff. The cost of such a portfolio, which is readily observed, must represent the value of the call option.

The key insight underlying the model may be illustrated through single period binomial (or two state) model. The following assumptions may be employed to develop this model.

1. The stock, currently selling for S, can take two possible values next year, uS or dS (uS>dS)
2. An amount of B can be borrowed or lent at a rate of r the risk free rate. The interest factor (1+r) may be represented for the sake of simplicity, as R.
3. The value of R is greater than d but smaller than u(d<R<u). This condition ensures that there is no risk free arbitrage opportunity.
4. The exercise price is E.

The value of the call option, just before expiration, if the stock price goes up to uS, is
Cu = Max (uS – E,O) ————-eq1

Likewise the value of the call option, just before expiration, if the stock price goes down to dS is
Cd = Max (dS – E,O)—————–eq2

Let us now set up a portfolio consisting of ∆shares of the stock and B rupees of borrowing. Since this portfolio is set up in such a way that it has a payoff identical to that of a call option at time1, the following equations will be satisfied:

Stock price rises: ∆uS – RB =Cu——-eq3
Stock price falls: ∆dS – RB = Cd——-eq4

Solving Eqs 3 and 4 for ∆ and B, we get

∆ = Cu – Cd /S (u—d)

= spread of possible option prices / Spread of possible share prices

B = dCu–uCd / (u—d) R

Since the portfolio (consisting of ∆ shares ad B debt) has the same payoff as that of a call option, the value of the call option is

C = ∆s – B

Note that the value of option is found out by looking at the value of a portfolio of shares and loan that imitates the option in its payoff. So this may refereed to as the option equivalent calculation.

To illustrate the application of the binomial model the following data for Pioneer’s stock:

S= 200, u = 1.4, d= 0.9 E = 220 r= 0.10, R = 1.10

Cu = Max (uS – E,0) = Max (280 –220, 0) =60

Cd = Max (dS – E,0) = Max (180 – 220,0) =0

Given the above data, we can get the values of ∆ and B by using equations

∆ = Cu – Cd / (u–d) S = 60 / 0.5 (200) = 0.6

B = dCu – uCd/(u—d)R = 0.9(60)/0.5(1.10) = 98.18

Thus the portfolio consists of 0.6 of a share plus a borrowing of 98.18 (entailing a repayment of 98.18 (1.10) = 108 after one year. The identity of the payoffs of the portfolio mad call option is shown below:

Portfolio Call Option

When u occurs 1.4x200x 0.6 – 108 =60 Cu =60
When d occurs 0.9x 200x 0.6 – 108 =0 Cd =0

Given the equivalent o the call option and the portfolio the value of the call option

C= ∆S – B = 0.6 x 200 – 98.18 = 21.82

Note that we could establish the value of the call option without any idea about the probability that the stock would go up or come down. An optimistic investor may think that the probability of an upward move in high whereas a pessimistic investor may think that it is low. Yet the two agree that the value of the call option is 21.82. Why? The answer lies in the fact that the current stock price of 200 already incorporates the views of the optimists as well as the pessimists. And the option value in turn, depends on the stock price.