I was solving Problem 2.47 from T.F. Crack's "Heard on the Street". I think that the answer given in the book is not correct and I would be thankful if you tell me, where I am mistaken.

Question 2.47. You have 30 days of "representative" stock price data. How do you calculate historical volatility $\widehat σ^2$ to use in Black-Scholes?

In the Black-Scholes model stock prices follow a geometric Brownian motion $$ \frac{dS_t}{S_t} = \mu dt + σ dW_t. $$ Solving this SDE I get the following expression for one-day continuously compounded returns: $$ \log\frac{S_{t+∆ t}}{S_t} = (\mu-\frac{σ^2}{2})∆\!t + σ \sqrt{∆ t} Z_t, $$ where $Z_t$ are independent random variables from $N(0,1)$, $∆ t = 1/250 \approx 0.004$ and $\sqrt{∆ t} \approx 0.063$. Using the observations I estimate $(\mu - σ^2/2)∆\!t$ and $σ^2 ∆\!t$ by the sample mean and variance: $$ M := \frac{1}{29}\sum_{i=1}^{29}X_i, \\ V := \frac{1}{28} \sum_{i=1}^{29}(X_i - M)^2, $$ where $X_i$ are the observed continuously compounded returns $\log(S_{t+∆ t}/S_t)$ for the period of $30$ days. Then I use $V/{∆t}$ as an estimator of $σ^2$. However, solution given in the book does not use the quantity $∆ t$ at all, and the quantity $V$ is used as an estimator of historical volatility. Are they mistaken?


Correct you will need to annualise the daily volatility, which is done by dividing the volatility calculated using daily returns by the square root of one over number of days in a year, dt in your example. Which is the same thing as multiplication by the square root of the number of days in a year. Usually assumed to be 252 but vary.

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