Volatility – Sisi Tang

Implied Volatility

Option price in the Black-Scholes-Merton formula can be seen as a function of volatility $\sigma$ if interest rate $r$ and strike price $K$ are known. Market quote of option price gives the volatility which is called implied volatility. Market convention is annualized volatility.

Realized Volatility

Observing the prices of a stock every day in a year, we can compute the realized volatility.

Suppose the stock price follows geometric Brownian Motion, i.e.,

$S(t)=S_0e^{\alpha-\frac{1}{2}\sigma^2t + \sigma W(t)},$

where $W(t)$ is the standard Brownian Motion, and time $t$ takes year as unit.

Then, for $n=1,2, \ldots, 251$ , the number of stock trading days of NYSE in a year.

$\frac{S_{n}}{S_{n-1}} = e^{\frac{\sigma^2}{251}+\sigma\left(W\left(\frac{n}{251}\right)-W\left(\frac{n-1}{251}\right)\right)} \overset{\sD}{=} e^{\frac{\sigma^2}{251}}e^{\sigma W\left(\frac{1}{251}\right)}.$