Question

1. Estimate the historical volatility of a stock using the following data. The data show the stock price at the end of each of week over ten consecutive weeks.

12.0, 13.3 11.2, 12.0, 12.7, 13.1, 13.3, 14.5, 14.9, 14.1

  

2. What is the Black-Scholes-Merton price of a European call option when the stock price is $75, the strike price is $80, the risk-free rate is 2% (based on continuous compounding), the volatility is 50% and the time to expiration is 2 years?

Answer 1

1]

Return in each week = (current week price - previous week price) / previous week price

Historical volatility = annualized standard deviation of weekly returns

annualized standard deviation = weekly standard deviation * 52 (there are 52 weeks in a year)

Weekly standard deviation of returns is calculated using STDEV.S function in Excel

Historical volatility = 59.31%

Historical volatility = 59.31%

Historical volatility = 59.31%

Historical volatility = 59.31%

Historical volatility = 59.31%

2]

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

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

d2 = d1 - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d1 and d2 as below :

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

· (r + σ2/2)*T = (0.02 + (0.502/2)*2

· σ√T = 0.50 * √2

d1 = 0.3189

d2 = -0.3883

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) = 0.6251

N(d2) = 0.3489

Now, we calculate the values of the call option as below:

C = (S0 * N(d1))   - (Ke-rT * N(d2)), which is (75 * 0.6251) - (80 * e(-0.02 * 2))*(0.3489)    ==> $20.0624

Value of call option is $20.0624