9.1 Three volatilities · M09 Volatility trading

Motivation

A trader who says "vol is cheap" is comparing two numbers. Unless you know which two, the statement means nothing. Most retail option losses in India come from buying or selling without that comparison.

Setup

Let $S_t$ be the underlying (for Nifty options, think of the Nifty future, which removes the dividend question; see M07 on cost of carry). Notation follows M08: rates continuously compounded on ACT/365, $q$ a continuous dividend yield (futures-implied $q=r-\ln(F/S)/T$ ), $N(\cdot)$ and $\varphi(\cdot)$ the standard normal cdf and density, theta per year, vega per 1.00 of volatility. Write $\sigma$ for annualised volatility of log returns. Three different objects share the name.

Realised (historical) volatility $\sigma_r$ : a statistic of past returns over a stated window, for example 20 trading days. It is backward-looking and measurable.
Implied volatility $\sigma_i$ : the number which, put into Black–Scholes, reproduces an option's market price (M08 · 8.10; for Nifty, Black-76 on the future, M08 · 8.8). It is a price, quoted in volatility units. Each strike and expiry has its own.
Forecast volatility $\hat\sigma$ : your estimate of the realised volatility over the option's remaining life, from a model such as GARCH (M13) or from realised-vol estimators of this section.

An option trade is a bet on the gap between the implied volatility you trade at and the realised volatility that will occur. 9.3 makes that precise.

Time conventions Realised volatility is annualised with 252 trading days. Option pricing in India commonly uses calendar time, $\tau$ in days/365, and India VIX is computed on calendar minutes (9.6). The mismatch matters most for short-dated options: a weekly Nifty option bought on Friday loses three calendar days of time value between Friday's close and Monday's close, a span that contains only one trading day of variance. In this module $\tau$ is calendar years unless stated.