Question

Question #3: You would like to be holding a protective put position on the stock of XYZ Company to lock in a guaranteed minimum value of $100 at year-end. XYZ currently sells for $100. Over the next year, the stock price will either increase by 10% or decrease by 10%. The T-Bill rate is 5.0%. Unfortunately, no put options are traded on XYZ Co. Suppose the desired put option were traded. How much would it cost to purchase?

Answer 1

Solution

If the price of the stock rose by 10% in one year, the stock price would be $100* 1.10 = $110. Since the strike price of the put is $100, it will be $0, if the stock price increases by 10%.

On the other hand, if the stock price falls by 10% in the next year, the price of the stock will be $90, and the put option with the strike price of $100 will have an intrinsic value of $10.

We can find out the value of the put option by using the binomial option pricing model. First, we have to find out the probability of both possibilities

The probability of the increase in stock value can be found out using the following formula: ((1+r)-d)/(u-d), where,

r is the risk-free rate of return. u is the rate at which the stock may increase, in this case, 1+10% or 1.10 and d is the rate at which it will decrease or 1-10%= 0.9

The probability of an upmove in the stock is ((1+0.05)-0.90)/(1.10-0.90) = 0.15/0.2 = 0.75 or 75% probability. Hence, the probability of a down move is 1-0.75 or 0.25 or 25%.

Now that we know the probability of each move, we can calculate the value of the put option using the binomial option pricing model.

The cost of the put = ((probability of stock increase * intrinsic value of put if stock increases) + (probability of stock decrease * intrinsic value of put if stock decreases)) / (1 + risk-free rate)^n n is the duration

Hence, the cost of the put = (0.75 * 0) + (0.25 * $10) / 1.05 = $2.38