Estimating Annualized Stock Volatility from Weekly Prices

# Calculating historical Volatility for the Black Scholes Model


# Calculating historical Volatility for the Black Scholes Model












Below is a problem from the book "Options, Futures, and other Derivatives" by John C. Hull. I did the problem but I am fairly sure that my answer is wrong. I am hoping that somebody can tell me where I went wrong? Thanks, Bob

Problem: Suppose that observations on a stock price(in dollars) at the end of each $15$ consecutive weeks are as follows: $30.2$, $32.0$, $31.1$, $30.1$, $30.2$, $30.3$, $30.6$, $33.0$, $32.9$, $33.0$, $33.5$, $33.5$, $33.7$, $33.5$, $33.2$ Estimate the stock price volatility. Answer: Let the closing prices be denoted by $S$. \begin{eqnarray*} u_i &=& \ln{ \bigg( \frac {S_1} {S_{i-1}} \bigg) } \\ \end{eqnarray*} Using R, I find that: \begin{eqnarray*} u &=& 0.057893978 \,\, -0.028528084 \,\, -0.032682647 \\ && 0.003316753 \,\, -0.006644543 \,\, -2.302585093 \\ && 2.322387720 \,\, 0.075507553 \,\, -0.003034904 \\ && 0.003034904 \,\, 0.015037877 \,\, 0.000000000 \\ && 0.005952399 \,\, -0.005952399 \,\, -0.008995563 \\ \end{eqnarray*} Now using $R$, I find that the standard deviation of $u$ is $0.8744864$. Call that value $s$. I will call the volatility of the stock to be $\sigma$. Now let $\tau$ be the length of time we observed the value of the stock for. \begin{eqnarray*} \sigma &=&\frac{s}{\sqrt{\tau}} \\ \tau &=& \frac{14}{52} = 0.2692308 \\ \sigma &=& \frac{0.8744864}{\sqrt{ 0.2692308}} \\ \sigma &=& 1.6853523 \\ \end{eqnarray*} This number seems way off to me. What did I do wrong? It also seems strange to me that in the last step you are dividend by $\sqrt{\tau}$ but that is the procedure given in the book.

## Answer by Magic is in the chain (score 0, accepted)

https://quant.stackexchange.com/a/41848

Does this sound more reasonable?

To annualise the weekly std dev, you need to multiply by the square root of 52.

Let me know if you spot any typo.

Re-comment, 14 is the number of return observations, so it is already incorporated in the calculation of weekly std dev (2.88%). I think the author might have implied that if the observations that you use for stddev calculations have an interval of tau in years (daily=1/252, weekly=1/52, monthly=1/12), then you divide the computed std dev by sqrt(tau), which is sqrt(1/52) in your case. Now dividing by 1/sqrt(52) is same as multiplying by sqrt(52), and that’s exactly the factor you need to multiply 2.88% by to get the answer.

## Answer by Ivan (score 0)

https://quant.stackexchange.com/a/41846

The division is correct but more often you’ll find a multiplication by the inverse of $\sqrt{\tau}$ instead.

Your vector of returns $u$ has gone wrong somehow, there is simply no way you would have numbers over the 0.05 area with the prices provided.

Additionally you would calculate the standard deviation of $\ln{\frac{S_i}{S_{i-1}}}$, do you have a typo there or a logical error ?