Question

In time 0, an investor takes a calendar spread by selling two-year European call option and buying

three-year European call option. These two options have the same strike price of $80 and are for the

same stock that pays no dividends. The two-year option sells for $5 and the three-year option sells for

$7. Two years later, the stock price turns out to be $90. The risk-free rate is 2% per annum. What is

the minimum of the profit from this strategy? (We assume that we sell the longer-term option in year

two)

Answer 1

We will calculate the total inflow and outflow at at time 0 and at time 2 years.

Time 0:

Strike price = $80

Sell two European call option at $5 = inflow of 2 * 5 = +$10

Buy three European call option at $7 = outflow of 3*7 = -$15

so Total cash flow = +$10 - $15 = -$5

Time 2:

Stock price is $90. Since the stock price is greater than the strike price, the European call option will be exercised and the investor will suffer a loss of $90-$80 = $10. (this is because the investor will have to buy from market at $90 and deliver the stock to the call buyer at $80

Since we have two European call options the total loss = 2* 10 = $20

Investor is long 3 European call options that expire one year from now.

The value of these European call options = {Max(0,St-(X/(1+r)^t)), St) (where r is the risk free rate =2 % and t is the time left for expiry = 1 year.

Since in the question we are asked for the minimum profit, we will only consider the lower range value.

= Max(0, 90 - (80/(1+2%)¹) )= 11.5686

If investor sell the 3 European call options at the end of 2 years, then he will receive 3*11.5686 = 34.7059

Total cash flow at the end of 2 years = Gain from selling 3 European call options - loss from exercise of the 2 European call options = 34.7059 - 20 = +14.7059

Hence the total cash flow across Time 0 and Time 2 = -$5 + 14.7059 = $9.7059

Answer:

minimum of the profit from this strategy = $9.7059