Question

The SP500 index was at 2677 on April. The same day, the options on the SP500 expiring in July with exercise price of 2680 were at 84.6 (Call) and 76.5 (Put). If the SP500 reaches a 2690 in a month (May), calculate the new pricw of the SP500 options maturing in July. Also, calculate the return for holding options alone and compare it with the return on the SP500. Use Black Scholes to calculate the price.

Things to know:

Use 1.69 for STD of SP500

Use 9.5% for the rate

interest rate= risk free rate, usually given in months

Answer 1

Ans: SP500 index was at 2677 on April and same day the options on the SP500 expiring in July with exercise price of 2680 were at 84.6 (Call) and 76.5 put.

Here, Spot price (S) = 2677

Strike Price (K) = 2680

Call = 84.6

Put = 76.5

Risk free rate (r) = 9.5%

Time (T)= July - April = 3 months

By putting all in the calculating Call : C = S*N(d1) - Ke^(-r*T)*N(d2)

84.6 = 2677 * N(d1) - 2680* e^(-0.095*3/12)*N(d2)

84.6 = 2777* N(d1) - 2680*0.976 * N(d2)

=> 84.6 = 2777*N(d1) - 2617.11*N(d2)

=> 2617.11*N(d2) = 2777*N(d1) - 84.6

By putting all values in Put formula; P = Ke^(-r*T)*N(-d2) - S*N(-d1)

76.5 = 2680*e^(-0.095*3/12)*N(-d2) - 2677*N(-d1)

=> 76.5 = 2617.11*N(-d2) - 2677 * N(-d1)

=> 76.5 = 2617.11*(1-N(d2)) - 2677*(1-N(d1))

=> 76.5 = 2617.11 - 2617.11 * N(d2) - 2677 - 2677*N(d1)

=> 76.5 = -59.89 - 2617.11*N(d2) - 2677*N(d1)

=> 76.5+59.89 = - 2617.11*N(d2) - 2677*N(d1)

=> 136.39 = - 2617.11*N(d2) - 2677*N(d1)

Equating both the equation we can find;

136.39 = 2777*N(d1) - 84.6-2677*N(d1)

=> 136.39 + 84.6 = 100*N(d1)

=> 220.99/100 = N(d1)

=> N(d1) = 2.2

By putting value of N(d1), we can find N(d2)

=> 2617.11*N(d2) = 2711* N(d1) - 84.6

=>N(d2) = [(2711*2.2) - 84.6]/2617.11 = 2.24

Now In May, the new spot price is (S) = 2690

And Time (T) = July - May = 2 months

Call (C) = S*N(d1) - Ke^(-r*T)*N(d2)

= 2690 * 2.2 - 2680 * 2.71^(-0.0158)*2.24

= 5918 - 5909.18

= 8.81

Put (P) = Ke^(-r*T)*N(-d2) - S*N(-d1)

= 2680 * 0.984* (1-N(d2)) - 2690 * (1-N(d1))

= 2680 * 0.984* (1-2.24) - 2690 * (1-2.2)

= -3270.03 - (-3228)

= 42.08