Question

Bruzen Co. are trading at $39.50 a share.  BC does not pay dividends. The yearly standard deviation for BC’s returns is 43.05 percent (0.4305). The risk-free interest rate is 2.40 percent compounded continuously. A call option on Bruzen with a 45 strike price and nine months to expiration has a hedge ratio of 0.4678. The risk-adjusted probability that a nine-month 45 call option finishes in the money is 0.3170. Assuming the present value of a risk-free $1 due the end of nine months is $0.97995, determine the price of an American call option on Bruzen having strike price of $45 and nine months to expiration.

Answer 1

Price of an American call option is higher of current intrinsic value (IV)which is =max(0,(S-k)) or its current theoretical price which is = IV +time value(TV) = (Cu×P + Cd×q)×PVF(9)

Where

Cu= payoff of call if it exercises = Su -k

Where

K= strike price =45

Su = upside price potential of stock= S0×e^(sd×(t)^.5)

S0= Current stock price =39.5

Sd=43.05% and t= 9/12 years

Su = 39.5×e^(.4305×(9/12)^.5) = 57.45

Cu= 57.45-45=12.35

Cd= value of call if it gets lapsed = 0 always

P= probability that the call will exercise =.3170

q= probability that the call will lapse= 1-p= 1-.3170=.6830

PVF (9)= present value factor of 1$ standing after 9 months=.97995

Fair value= (12.35×.3170+0×.6830)×.97995=3.836$

Current intrinsic value=max(0,(39.5-45))=0

Price = higher of these two= 3.836  

Why higher because American option is exercisable before maturity also so if we exercise it now we will get 0 and if we sell it we will get 3.836$ so it is better to sell