Learning outcomes

After this module you should be able to:

1. 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.
2. 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 $V(\sigma_r)-V(\sigma_i)$.
3. 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.
4. Define the variance risk premium, derive its sign from a stochastic discount factor, and summarise the evidence (Carr–Wu 2009 and the Indian evidence).
5. 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.
6. 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.
7. 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.
8. 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.