Question

Consider the following information: Time to expiration = 9 months Standard deviation = 25% per year Exercise price = $35 Stock price = $37 Interest rate = 6% per year Table for N(x) with selected values 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.1 0.2 0.3 0.4 0.5000 0.5040 0.50800.5120 0.5160 0.51990.52390.5279 0.5319 0.5398 0.5438 0.5478 | 0.5517 | 0.5557 0.5596 0.5636 0.5675 0.5714 0.5793 0.5832 0.5871 | 0.5910 0.5948 0.5987 | 0.6026 0.6064 0.6103 0.6179 0.6217 | 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6554 | 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.5359 0.5753 0.6141 0.6517 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.70880.71230.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.76420.76730.77040.77340.7764 0.77940.78230.7852 0.8 0.7881 0.7910 | 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.89620.89800.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.91150.9131 0.9147 0.9162 0.9177 1.4 0.91920.9207 0.9222 0.92360.9251 0.9265 0.9279 0.92920.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.94520.9463 0.9474 0.94840.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1. 70 .9554 0.95640 .9573 0.95820 .9591 0.9599 0.9608 0.96160.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 | 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.97560.9761 0.9767 a. Use the Black-Scholes-Merton formula to find the value of a European call option on the stock. (Hint: Use the Cumulative Normal Distribution Table with interpolation. (10 marks) b. Find the value of a European put option with the same exercise price and expiration as the call option above. (5 marks)

a. Use the Black-Scholes-Merton formula to find the value of a European call option on the stock. [Hint: Use the Cumulative Normal Distribution Table with interpolation.] (10 marks)

b.  Find the value of a European put option with the same exercise price and expiration as the call option above. (5 marks)

Answer 1

1. The Black-Scholes Model; European call option formula

C = SN (d1) - N (d2) Ke ^ (-rt)

Where

C = call value =?

S = current stock price =$37

N = cumulative standard normal probability distribution

t = days until expiration = 9 month or 9/12 = 0.75 years

Standard deviation, SD = 25% = 0.25

K = option strike price = $35

r = risk free interest rate = 6% = 0.06

Formula to calculate d1 and d2 are -

d1 = {ln (S/K) +(r+ σ^2 /2)* t}/σ *√t

= {ln (37/35) + (0.06 + (0.25^2)/2) * 0.75} / 0.25 * √0.75

= 0.5728

d2 = d1 – σ *√t = 0.5728 – 0.25 * √0.75 = 0.3563

Now putting the value in the above formula

C = 37 * N (0.5728) – N (0.3563)* 35 * e^ (-0.06*0.75)

= $5.13

Price of call option is $5.13

2. The Black-Scholes Model; European put option formula

P = Ke^–rt * N(–d2) – SN(-d1)

Where, P = Put value, other values are calculated above

P = 35* e^–(0.06 *0.75) * N(–0.3563) – 37 *N(-0.5728)

= $1.59                                                    [here N(–d1) = 1- N(d1) & N(–d2) = 1- N(d2)]

Price of put option is $1.59