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?
