Question

You are a research analyst at JPMorgan Investment Management. You are collecting and analyzing data to find the value for a six month European call option and put option on BAC stock. The stock price of BAC is currently $80. The strike price is $78. It is known that at the end of six months it will be either $84 or $76. The risk-free interest rate is 5% per year.
Please choose all correct answers.

	1.	The value of the put option on the stock is $0.48
	2.	The put option delta is -0.5
	3.	The value of the call option on the stock is $2.81
	4.	The value of the put option on the stock 0.90
	5.	The put option delta is 0.5
	6.	The risk neutral probability of an up movement is 0.7532
	7.	The value of the put option on the stock 1.90
	8.	The call option delta is 0.05
	9.	The value of the call option on the stock $38.1
	10.	The risk neutral probability of an up movement is 0.1908
	11.	The call option delta is 0.5
	12.	The call option delta is 0.75
	13.	The put option delta is 0.05
	14.	The value of the put option on the stock 2.90
	15.	The risk neutral probability of an up movement is 0.08
	16.	The risk neutral probability of an up movement is 0.25
	17.	The value of the call option on the stock is $4.41
	18.	The put option delta is -0.25.

Answer 1

I can help with the calculation of the value of the call and put option, and the risk neutral probability of an up movement.

Value of the call option using the risk neutral binomial model :

Current stock price: S = 80

StrikePrice: X = $ 78

Upside Price: uS = $84

Downside Price: dS = $76

interest rate: r = 5% per year = 2.5% for 6 months

Value of call on expiry at higher price (uS): C1 = uS - X = 84 - 78 = 6

Value of call on expiry at lower price (dS): C2 = dS - X = 76 - 78 = 0 (since the call option will not be exercised if stock price falls below strike price)

u = uS / S = 84 / 80 = 1.05

d = dS / S = 76 / 80 = 0.95

Probability of upside price: p = (1 + r) - d / u - d = (1 + 0.025) - 0.95 / 1.05 - 0.95 = 0.075 / 0.1 = 0.75

Probability of downside price: 1-p = 1-0.75 = 0.25

Value of the call = C1 x p + C2 x (1-p) / 1 + r

= 6 x 0.75 + 0 x 0.25 / 1.025

= 4.5/ 1.025 = 4.390

Value of the put option :

C1 = X - uS = 50 - 55 = 0 (since the put option will lapse at the higher price)

C2 = X - dS = 50 - 45 = 5

Value of the put = C1 x p + C2 x (1-p) / 1 + r

= 0 x 0.75 + 6 x 0.25 / 1.025

= 1.4631

As regards the value of the call and the risk neutral probability of up movement (p), our values are closest to the respective values in option 17 - $4.41 and option 16 - 0.25 . Hence options 17 and 16 are correct, as regards these two values.