Option Valuation: Method II (Part 1)
by J Victor on October 28th, 2012
In the last post we discussed the risk neutral method of pricing options. In this post we would discuss binomial method of option pricing which, in reality, is just an extension of the risk neutral method.
BINOMIAL PRICING MODEL.
We will check out binomial model with the help of an example –
Let’s assume that the stock is trading Rs 100 right now. In the next one year, the stock may move up 20% or fall 10%. European options are available at a strike price of Rs 95. What would be the value of a European call if the risk free interest rate is 12%?
Step 1
Value the maximum probable expiry value of call.
If the stock price moves up to Rs 120, value of call would be = 120 – 95 = 25
If the stock price crashes to Rs 90, the call is worthless = value of call = 0
Step 2
Binomial method of option pricing assumes that the investor are risk neutral. ‘Risk neutral’ implies that the investors are indifferent to the actual probability of payoffs and are only concerned in getting a payoff that’s equal to the risk free interest rate. Now, the probability of up move and down is not known but the implied probability of the movement in stock prices can be calculated by using the interest free rate. Let’s assume that ‘P’ is the probability of up move then, mathematically we can also say that (1p) is the probability of down move, right? Hence the value of ‘P’ or probability can be found out with the following formula –
We get the value as 0.73
Step 3
If 60% is the chances of winning, 40% is the chances of losing right? That means, if ‘p’ is the chances of winning, ‘1p’ is the chances of losing. Here we‘ve computed the value of ‘p’ as 0.73 Hence, the value of 1p = 0.27
Step 4
In step 1, we have already computed the pay offs which is Rs 25 and 0. Now, we can determine the value of call one year hence, using the following formula –
 (25 x p + 0 x 1p) = 25 x 0.73 + 0×0.27
 18.25 + 0 = 18.25
This is the value of call after one year. Hence we need to discount it to the present using the risk free return rate of 12%. The present value of 18.25 discounted at 12% would give Rs 16.29 as the answer. That’s the value of the call at present.
The binomial approach is very simple and assumes that the underlying stock would increase or decrease at a certain percentage until option expiry. It is very useful for valuing American options since American option holders can exercise the right at any point of time.
In the above example we have valued options assuming that the stock price moves up or down once. It’s equal to the risk neutral method we discussed earlier. The catch here is that, We can extend this theory to more than one period. We will discuss that in our next post.
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