44.6:Let S(t) denote the price at time t of a stock that pays dividends continuo

Question

44.6:Let S(t) denote the price at time t of a stock that pays dividends continuously at a rate proportional to its price. Consider a European gap option with expiration date T, T > 0. If the stock price at time T is greater than $100, the payoff is S(T) 90; otherwise, the payoff is zero. You are given: (i) S(0) = $80 (ii) The price of a European call option with expiration date T and strike price $100 is $4. (iii) The delta of the call option in (ii) is 0.2. Calculate the price of the gap option.[answer: $5.20 ]

Explanation / Answer

In a gap option there are 2 strike prices X1 and X2. X1 is the strike price and X2 is known as the trigger price. The trigger price will determine whether or not the option will have a positive payoff, and the strike price then will determine what will be the amount of positive payoff.

In the question given the X1=90 and X2=100. Hence, we will only get a payoff of S-X1 when stock price (S) moves above X2 (100).

The Black Scholes formula to value gap call option is:

C=S0× N(d1)-X1 e-rt×N(d2)…….(1)

Note we are already given information on N(d1) which is also equivalent to delta of call option of 0.20. We know the stock price (S0) to be $ 80. X1 is known to be 90. We need to know the e-rt×N(d2) component.

We are also given the price of European call option with strike price of $ 100 to be $ 4. So we can re write the Black Scholes formula in equation 1 to be:

C=S0× N(d1)-Xe-rt×N(d2)

4=80×0.20-100e-rt×N(d2)

4=16-100 e-rt×N(d2)

e-rt×N(d2)=(16-4)/100

e-rt×N(d2)=12/100=0.12

So now that we know e-rt×N(d2) component to be 0.12 we are going to use formula 1 to find price of gap call option with X1 to be 90.

C=S0× N(d1)-X1 e-rt×N(d2)

=80×0.20-90×0.12

=16-10.80

=5.20

Conclusion: The price of the gap call option is $ 5.20.

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