Option valuation: Method II (Part 2)
by J Victor on November 25th, 2012Option pricing – method II (Part 2)
The two stage binomial pricing was a simple method. In our example we assumed that the stock was trading Rs 100 and that in the next one year, the stock may move up 20% or fall 10%. We tried the value the European option that had a strike price of Rs 95 assuming the risk free rate was 12%. If we draw a picture, the price movement and pay off probabilities would have looked like this –
This two stage process can be generalised to incorporate more realistic scenarios. For example, we can assume that the stock would move up or down in a period of 6 months ( instead of 1 year) . From there, for the next 6months, probabilities can be drawn assuming the same rate of change in price. The binomial model assumes that the increase or decrease in price remains as a fixed percentage of the start point. The calculation would become more complex and cumbersome as we break up the year into more short time intervals.
Let’s work out an example. The current stock price is Rs 100 and the risk free rate is 8%. The stock price may move up 10% or fall by 5% every single period (6 months). If a call option is available at an exercise price of 105, what would be the theoritical valuation? Assume that the call has one year validity.
Here , as we see from the graph, the time period of 1 year has been split to two zones. In the First 6 months we worked out the probable movement of share price from Rs 100 to 110 or 95. In the second 6 months, we worked out the possibility of share price moving up/down from 110 or 95.
The pay off at the end is worked out first –which is Rs 16 (Rs 121105). From there the values are worked backwards step by step. Amoung the final prices, only Rs 121 ends up in the money and hence attians a value of Rs 16 . The other two end up out of the money and hence the value is ‘0’.
The second step is the same as we discussed in our previous post. The implied probability under the risk neutral valuation has to be computed. The formula is given below. ‘P’ is the implied probability.
The formula returns a ‘p’ value of 0.60 and (1p) value of 0.40. The pay off of the first year can be caculated as (16 x 0.60) + ( 0 x 0.40) = 9.6. This 9.6 has to be discounted for half a year at the risk free rate of 8%. This gives a figure of 9.23. Hence the position now will be as follows:
For the final column, the p values have to be computed again. Using 0.60 as ‘p’ and 0.40 as ’1p’, the answer would be (9.23 x .60) + (0 x .40) = 5.53. This amount has to be discounted at the risk free rate of 8% per annum. We get 5.32 as the answer (Since 8% is for a year, for 6 months we take half of it i.e. 4% as the discount factor ) So the entire pay off graph would be:
So the value of call at inception is Rs 5.32. This principle of binomial pricing can be extended to any number of periods. Three or four steps can be calculated on papers. But if you extend the periods to more than that, the calculation becomes difficult and you’ll have to use a spreadsheet.
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