Binomial Tree Problems
Perhaps the most difficult problems I’ve encountered in Fin 622 are the binomial tree model questions. Initially these can be intimidating, but with a little practice and repetition they become quite simple. The binomial tree model is introduced in Chapter 12 and is a useful tool to evaluate expected premiums on PUT/CALL options on assets. Let’s go through an example.
Problem:
An asset worth $200 can increase or decrease by 25% in one period. The Call option associated with the underlying asset can be exercised at the end of 2 periods. What initial premium does the binomial tree model predict for the call option if the strike price is $250. Assume a risk free interest rate of 10%.
Solution:
First we must calculate the asset worth for the 2 future periods. It is given in the problem that the value of the asset will either increase or decrease by 25% in each period. This gives us a model tree shown below:
Next we must calculate the risk neutral probabilities, pi, associated with each of the possible asset values. This can be done by using equation below:
Where rf is the risk free rate, d is the ratio of next period’s stock price to the current period’s price in the down state, and u is the ratio of next period’s stock price to the current period’s price in the up state. This is given in the text book on page 272.
Thus, u would be equal to 250 / 200 = 1.25, and d = 150 / 200= 0.75. Plug those values into the equation, along with the risk free interest rate of 10%, or 0.10, to calculation a risk neutral probability of 0.7, which can be applied in all of the up states. Since we know that pi in the up state plus pi in the down state equals 1, all of the risk neutral probabilities for down states must be equal to 1 – 0.7 = 0.3.
Next we must use the binomial tree model once again to calculate the value of the Call option using the risk neutral probabilities we just calculated. Here is where our knowledge of Call options must be applied. A Call option allows us to buy an asset at a predetermined price. Hopefully, the predetermined price, or strike price, is less than the current market value of the asset. For example, if an asset is currently selling on the market for $100, and we have a Call option to buy the asset at $75, then we stand to make $25 by purchasing the asset at the strike price of $75 and then selling to the market at $100. However, should the unfortunate occur and the strike price on the Call option is higher than the current market value, then we would simply let the option expire.
Applying this strategy to our binomial model tree gives the following values:
Example: $312.5 - $250 = 87.5
Example: $187.5- $250 = -$62.5 (since we would never exercise a Call if we were going to lose money the -$62.5 essentially goes to $0).
The last step is to take the calculated values, apply the risk neutral probabilities, and discount them to find the NPV of the option at each node. This is accomplished by using the following equations:
Finally we can calculate the value of the Call option by using the same equation and substituting the values we just calculated, shown below:
