# Black-Scholes volatility implied by stock prices only

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?