Question

Problem 12.25. Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is six months a. Calculate u, d, and p for a two step tree b. Value the option using a two step tree. c. Verify that DerivaGem gives the same answer d. Use DerivaGem to value the option with 5, 50, 100, and 500 time steps.

Answer 1

In a European call option with a strike price (X) of 40, the call will be exercised at expiration when the stock price is more than 40.

Formula for back-tracking the call premium from T=0.5 (6 months), to T=0.25 (6 months) to T=0 is:

Call premium in up factor* p + call premium in down factor*(1-p) divided by e to the power Rf*n

European Call Option
Strike price	X	40
Current stock price	S	40
Risk free interest rate per annum	Rf	4%
Length of time step (in years)	n1	0.5	square root =	0.7071
Volatility	σ	30%
Up factor	u	e to the power (σ*square root of n)	e to the power (0.3*.7071) =	1.2363
down factor	d	1/u	1/I9	0.8089
probability (up)	p	e to the power (Rf*n-d)/(u-d)	e to the power (.04*0.5) - .8089/1.2363-.8089	1.0202	0.4944
probability (down)	1-p				0.5056

		3 months			3 months
							61.14	call premium = 61.14 - 40 =	21.1386
				49.45		-
	Stock price	40.00			call premium = 10.3464		40.00	call premium = 0	-
	put premium= ANSWER	call premium = 5.0644		32.35		7.6457
							26.17	call premium	-
Call Price	Stage 1 = T-0.25 months	call premium up move*p+call premium down move*(1-p)/e to the power Rf*n		(21.1386*0.4944+0*0.5056)/1.0101	10.3464
	Stage 1 = T-0 months			(10.3469*.4944 + 0*.5056)/1.0101	5.0644

DerivaGem