Part IV · Derivatives · M09 · week 5 · 18 h

Volatility and option trading

Realised-volatility estimators, the P&L of a delta-hedged option, the variance risk premium, the smile, variance swaps and VIX, option strategies with exact profit conditions, expiry-day gamma and why short volatility has negative skew.

Start reading · Learning outcomes

Builds on M02 Statistics of returnsM03 Kelly and ruinM07 Forwards and futuresM08 Option pricing

Prerequisites M02 (returns, heavy tails, volatility clustering), M03 (Kelly, ruin), M07 (futures, cost of carry), M08 (Black–Scholes, Greeks, implied volatility). Maths: Itô's lemma, the reflection principle for Brownian motion, change of measure via a stochastic discount factor. Time: ~18 hours (notes 9 h, problems 6 h, code 3 h). Primary texts: Sinclair, Volatility Trading (2nd ed., 2013); Gatheral, The Volatility Surface (2006); Hull, Options, Futures, and Other Derivatives (11th ed.), chapters on Greeks and volatility smiles; Natenberg, Option Volatility & Pricing (2nd ed., 2015).

M08 priced an option given a volatility. This module is about the volatility itself: how to measure it, what you earn or lose when the market's number differs from what happens, why the market's number is usually too high, and why collecting that difference produces a return distribution that ruins people who size it badly. Everything here is about Nifty options unless stated otherwise, because that is where Indian option volume is.

Learning outcomes

Compute close-to-close, Parkinson, Garman–Klass, Rogers–Satchell and Yang–Zhang volatility from OHLC data, derive $E[R^2]=4\ln 2\,\sigma^2 T$ for the Brownian range, and state each estimator's relative efficiency and failure mode.
Derive the instantaneous P&L of a delta-hedged option, $\tfrac12\Gamma S^2(\sigma_r^2-\sigma_i^2)\,dt$ , and its discrete daily form; derive the exact expected P&L under GBM, and prove that with $\mu=r$ it equals the Black–Scholes price difference $V(\sigma_r)-V(\sigma_i)$ .
Explain, with the proofs, why hedging at realised volatility locks in a deterministic P&L while hedging at implied volatility gives a path-dependent one.
Define the variance risk premium, derive its sign from a stochastic discount factor, and summarise the evidence (Carr–Wu 2009 and the Indian evidence).
Derive the replication of a variance swap by a static strip of out-of-the-money options weighted by $1/K^2$ , and connect it to the VIX and India VIX formula.
For verticals, straddles, strangles, iron condors, calendars, ratio spreads and covered calls, write the payoff, sign each Greek, and state the exact condition under which the position makes money, both unhedged at expiry and delta-hedged.
Derive the $1/(\sigma S\sqrt{2\pi\tau})$ blow-up of at-the-money gamma and quantify its consequences on a Nifty Tuesday expiry.
Show by simulation that systematic option selling has strongly negative skew, prove that negative skew pushes the Kelly fraction below the mean–variance value, and size a short-volatility position accordingly.

Sections ≈ 2 h 55 min of study

Foundation 5 sections · ≈ 35 min
Core 9 sections · ≈ 1 h 50 min
Advanced 2 sections · ≈ 30 min

Learning outcomes Foundation 5′
9.1 Three volatilities Foundation 5′
9.2 Measuring realised volatility Advanced 20′
9.3 The P&L of a delta-hedged option Core 20′
Figure Selling an option and hedging its delta
9.4 The variance risk premium Core 10′
Lab India VIX against realised Nifty volatility
9.5 The smile and the skew Core 10′
Figure The skew and the risk-neutral density
9.6 Variance swaps and the VIX Advanced 10′
9.7 Option strategies: payoffs, Greeks and exact conditions Core 15′
9.8 Expiry-day gamma Core 10′
Figure Expiry-day gamma
9.9 Why systematic option selling has negative skew, and how to size it Core 15′
Figure Short strangles: win rate, skew and Kelly
9.10 India specifics (as of 6 Nov 2026) Foundation 10′
9.11 A checklist before any option trade Foundation 5′
9.12 Case studies Core 20′
Summary of key results Foundation 10′
Readings and resources Core 5′
Facts that change Core 5′