Question

A call option on a stock expires in 4 months with a strike price of 75. The price of the stock is 80, and the interest rate is 4 percent. Graph the value of this option as the standard deviation of the underlying stock goes from 10 to 50 percent.

Answer 1

We use Black-Scholes Model to calculate the value of the call option.

The value of a call option is:

C = (S0 * N(d1)) - (Ke-rt * N(d2))

where :

S0 = current spot price

K = strike price

N(x) is the cumulative normal distribution function

r = risk-free interest rate

t is the time to expiry in years. This is (4/12), or 0.3333.

d1 = (ln(S0 / K) + (r + σ2/2)*T) / σ√T

d2 = d1 - σ√T

σ = standard deviation of underlying stock returns. At 10% standard deviation, this is 0.10.

First, we calculate d1 and d2 as below :

· ln(S0 / K) = ln(80 / 75). We input the same formula into Excel, i.e. =LN(80 /75)

· (r + σ2/2)*T = (0.04 + (0.102/2)*0.3333

· σ√T = 0.10 * √0.3333

d1 = 1.3776

d2 = 1.3199

N(d1) and N(d2) are calculated in Excel using the NORMSDIST function and inputting the value of d1 and d2 into the function.

N(d1) = 0.9158

N(d2) = 0.9066

Now, we calculate the values of the call and put options as below:

C = (S0 * N(d1))   - (Ke-rt * N(d2)), which is (80 * 0.9158) - (75 * e(-0.04 * 0.3333))*(0.9066)    ==> $6.1755

At 10% standard deviation,

Value of call option is $6.1755

In this way, we calculate the value of the call option at each standard deviation from 10% to 50%

The graph is as above