Question

The current stock price is 50. One period later, its value becomes three possible outcomes: {35,55, 60}. The value of $10 invested in the risk-free asset becomes $10.5 in one period. Find the upper and lower bound of a call with a strike price of 50 and maturing at the end of one period.

Answer 1

Sol:

Stock current Price = $50

Expected to increase over next period = 55, 60

Expected to decrease over next period = 35

Risk free rate = 10.5/10 x 100 = 105 = 5%

CMP as on expiry can be:-

35 or 55 or 60

Therefore probability of the call option will be:

p1= (CMP(1+r)-S2) / (S1-S2)  

S1 = High CMP as on expiry

S2 = Low CMP as on expiry

Stock price expected to increase over next period = 55

p1 = (50 x (1+0.05) - 35) / (55 - 35)

p1 = (50 x 1.05) - 35) / (55 - 35)

p1 = 17.5 / 20 = 0.875

p2 = 1 - 0.875 = 0.125

Stock price expected to increase over next period = 60

p1 = (50 x (1+0.05) - 35) / (60 - 35)

p1 = (50 x 1.05) - 35) / (60 - 35)

p1 = 17.5 / 25 = 0.70

p2 = 1 - 0.70 = 0.30

Value of one period call option:-

Call Option premium for price 55 = 55 - 50 = 5

Call Option premium for price 60 = 60 - 50 = 10

Call Option premium for <= 50 = 0

Value of one period call option for price 55 = (5 x 0.875) /(1+0.05)

Value of one period call option for price 55 = 4.375 / 1.05 = $4.17

Value of one period call option for price 60 = (10 x 0.70) /(1+0.05)

Value of one period call option for price 60 = 7 / 1.05 = $6.67

Therefore upper bound of the call with a strike price of 50 will be $4.17 if stock price closes at $55 at the end of the period.

Upper bound of the call with a strike price of 50 will be $6.67 if stock price closes at $60 at the end of the period.

Since a call option cannot sell below its intrinsic value, its value cannot be negative. Therefore the lower bound for the call option will be ZERO if stock price closes at $35 at the end of the period.