Part IV · Derivatives · M08 · week 5 · 32 h

Option pricing: from replication to Black–Scholes

No-arbitrage bounds and parity, binomial replication and the risk-neutral measure, Brownian motion and Itô's lemma, the Black–Scholes PDE and formula, Black-76, the Greeks and implied volatility, with Indian contract specifics.

Start reading · Learning outcomes

Builds on M02 Statistics of returnsM07 Forwards and futures

Prerequisites M02 (log returns, volatility drag, the normal and lognormal distributions), M07 (no-arbitrage forward pricing $F = Se^{(r-q)T}$ , futures mark-to-market, basis). Mathematics: probability to the level of the central limit theorem and conditional expectation; ordinary and partial differential equations; Taylor expansion; Newton's method. Time: ~32 hours (four days of week 5). This is the densest module of the course. Primary texts: Hull, Options, Futures, and Other Derivatives, 11th ed. (Pearson, 2021), Ch. 10–15, 17–19; Shreve, Stochastic Calculus for Finance I (Springer, 2004), Ch. 1–2 and II, Ch. 3–5; Natenberg, Option Volatility and Pricing, 2nd ed. (McGraw-Hill, 2015), for the trader's reading of the Greeks.

Learning outcomes

State and prove put–call parity, the no-arbitrage bounds on European calls and puts, and the theorem that an American call on a non-dividend-paying stock is never exercised early.
Replicate any claim in a one-period binomial model, derive the risk-neutral probability, and price European and American options on an $n$ -step Cox–Ross–Rubinstein tree.
Derive Itô's lemma heuristically, solve the geometric Brownian motion SDE, and prove that $S_T$ is lognormal with $\mathbb E[S_T]=S_0e^{\mu T}$ and median $S_0e^{(\mu-\sigma^2/2)T}$ .
Derive the Black–Scholes PDE by the delta-hedging argument and solve it by computing the risk-neutral expectation as an explicit integral.
Price options on futures with Black-76 and explain why Nifty options are best priced off the Nifty future rather than spot plus a dividend guess.
Derive the five standard Greeks (delta, gamma, vega, theta, rho) analytically, prove $Se^{-qT}\varphi(d_1)=Ke^{-rT}\varphi(d_2)$ , and read the PDE as the identity $\Theta+\tfrac12\sigma^2S^2\Gamma+(r-q)S\Delta-rV=0$ .
Prove existence and uniqueness of implied volatility, and implement a Newton solver that provably converges from the Manaster–Koehler starting point, with a Brent fallback.
Compute rupee Greeks, premium, and STT for a Nifty position with the current lot size, and explain the mechanism alleged in SEBI's 2025 interim order against Jane Street in terms of option delta.

Sections ≈ 2 h 40 min of study

Foundation 4 sections · ≈ 35 min
Core 9 sections · ≈ 1 h 35 min
Advanced 3 sections · ≈ 30 min

Learning outcomes Foundation 5′
8.1 Options: contracts, payoffs and moneyness Foundation 10′
Figure Payoff builder: Nifty options and futures
8.2 No-arbitrage relations that hold in every model Core 15′
8.3 One-period binomial model and the risk-neutral probability Core 10′
8.4 Multi-period trees: Cox–Ross–Rubinstein Advanced 10′
Figure The Cox–Ross–Rubinstein tree
Figure CRR convergence and the odd–even oscillation
8.5 Brownian motion, geometric Brownian motion and Itô's lemma Advanced 10′
8.6 The Black–Scholes PDE by delta hedging Advanced 10′
8.7 Risk-neutral valuation and the Black–Scholes formula Core 10′
8.8 Black-76: options on futures, Nifty and MCX Core 10′
8.9 The Greeks Core 10′
Figure Greeks against spot and time to expiry
8.10 Implied volatility Core 10′
Figure Implied volatility by Newton's method
8.11 India specifics: contracts, settlement and STT Foundation 10′
8.12 Case studies Core 20′
Summary of key results Foundation 10′
Readings and resources Core 5′
Facts that change Core 5′