In: Accounting
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.
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 |