Question

►A stock price is currently $50. It is known that at the end of six months it will be either $46 or $54. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of $48? What is the value of a six-month American put option with a strike price of $48?

Answer 1

Using Risk neutral model for option pricing,

S0*e^(r*t) = P*UP + (1-P)*LP

where S0 is current spot price = $50

r is risk free rate = 5% p.a.

t is no. of years = 0.5

P is probability of price increase

(1-P) is probability of price decrease

UP is upper price = $54

LP is lower price = $46

Solving the above equation,

50*e^(0.05*0.5) = P*54 + (1-P)*46

50*1.0253 = 54P - 46P - 46

or 8P = 51.265 - 46

or P = 5.265/8

So, P = 0.66

and (1-P) = 0.34

Price of option = P*Payoff from upper price + (1-P)*Payoff from lower price

Here, K is strike Price = 48

For call option,

Payoff from upper price (If UP > K) = UP - K

= 54 - 48

= 6

Payoff from lower price (LP < K) = 0

Therefore, Price of call option = 0.66*6 + 0.34*0

= $3.96

For put option,

Payoff from upper price (If UP > K) = 0

Payoff from lower price (LP < K) = K - LP

= 48 - 46

= 2

Therefore, Price of call option = 0.66*0 + 0.34*2

= $0.68