Question

You own 250 call options on Exxon that expire in 6 months. The current stock price of EXXON is 240. The interest rate is 2 percent, the standard deviation of EXXON stock is 28 percent, and the strike price is 250. Using the Black-Sholes formula, graph the value of your options as the standard deviation goes from 20 to 50 percent.

Answer 1

Computation of value of call using Black Scholes model:
C0	= S0*N(d1) - (X/e^rt)*N(d2)
where,
S0	= Stock Price = 240
X	= Strick Price = 250
r	= Interest Rate = 2% p.a
t	= 0.5 year
SD	= Standard Deviation
d1	= (log(S0/X)+(r+SD^2/2)t)/(SD*√t))
d2	= (log(S0/X)+(r - (SD^2/2))t)/(SD*√t))
Basic Calculations:
log(S0/X)	= Log(240/250)
	= log (0.96)
	= - 0.01773
e^rt	= e^(0.02*0.5)
	= e^0.01
	= 1.0101
X/e^rt	= 250/1.0101
	= 247.50
At SD	d1	d2	N(d1)	N(d2)	C0
0.20	0.02	-0.13	0.0080	0.4483	-109.034
0.25	0.04	-0.13	0.0160	0.4483	-107.114
0.28	0.06	-0.14	0.0239	0.4443	-104.228
0.30	0.07	-0.14	0.0279	0.4443	-103.268
0.35	0.09	-0.15	0.0359	0.4404	-100.383
0.40	0.11	-0.17	0.0439	0.4325	-96.5078
0.45	0.13	-0.18	0.0517	0.4286	-93.6705
0.50	0.15	-0.20	0.0596	0.4207	-89.8193