Question

a) (13 pts) A stock price is currently $49. It is known that at the end of 6 months it will be either $57 or $42. The risk-free rate of interest with continuous compounding is 11% per year. Calculate the value of a 6-month European call option on the stock with an exercise price of $48 using both the no-arbitrage arguments and risk-neutral valuation arguments. Show that they provide the same answers. b) (12 pts) Consider an option on a non-dividend-paying stock when the stock price is $42, the exercise price is $41, the annual risk-free interest rate is 2.5%, the volatility is 20% per annum, and the time to maturity is a quarter. (a) What is the price of the option according to the Black-Scholes-Merton formula if it is a European call? (b) What is the price of the option if it is an American call? (c) What is the price of the option if it is a European put?

Answer 1

Solution:-(a)
Calaculation of Call option with risk Neutral Valuation
Step 1:- Given factors
Current market price	SP0	$49.00
Exercise price	EP	$48.00
Lower Future spot price	FP1	lower of SP0	$42.00
Higher Future spot price	FP2	higher of SP0	$57.00
Risk free rate of return(e^.11*6/12)=e^0.055	e^rt	11%	1.0565
Time	t	6 Months	0.50
Step 2:-
Calculation of Risk Nuetral Probability
us	FP2/SP0	1.16
ds	FP1/SP0	0.86
P(Risk Nuetral Probability)	(R-ds)/(u-ds)	0.65
Step 3:-
Binomial Tree
	Probability=0.65	$57.00	Exercise(Cu)	$9.00
		NODE B
$49.00
NODE A
	Probability=0.35	$42.00	Lapse(Cd)	$0.00
		NODE C
Step 4:-
Value of Call option	Cu*p+Cd*(1-p)/R	$5.85
Calaculation of Call option with no-arbitrage arguments
Current market price	$49.00
Present value of Exercise Price
(48*e^-0.11*6/12)=(48*e^-0.055)=(48*0.9465)	$45.43
Value of call option	Current market price-Present value of Exercise price
Value of call option	$3.57
So, value of call option throuth both approach is different.

Only part (a) answar given. I will provide part (b) answar if i found this question again or inquired.