Options — Parity, Binomial & BSM + the Greeks

Learning objectives

By the end you can:

1. State put-call parity precisely (for a non-dividend-paying underlying and, extended, for one paying a continuous yield or a known discrete dividend), derive it from a static replication argument, and, given a set of quoted option prices, detect a violation, name the correct arbitrage (conversion or reversal), and compute the riskless profit, per unit and scaled to a real contract (a Nifty lot, an SPX multiplier).
2. Build an n-step binomial tree from scratch: derive the CRR up/down factors and the risk-neutral probability from no-arbitrage, price a European option by full backward induction with every node's stock price and option value shown, and extend the same tree to an American option by checking early exercise at every node.
3. State exactly when early exercise is (and is not) optimal, never for an American call on a non-dividend stock; sometimes for an American call on a dividend payer just before an ex-dividend date; sometimes for an American put even with no dividends, when sufficiently in the money, and prove each claim with a tree, not an assertion.
4. Show numerically that the binomial model converges to Black-Scholes-Merton as the number of steps grows, and explain why (BSM is the continuous-time limit of the same risk-neutral argument).
5. State the BSM formula for a European call and put, compute d1 and d2 from scratch, and list every assumption the formula makes (constant volatility, GBM/lognormal prices, constant risk-free rate, frictionless continuous trading, European exercise, a known dividend treatment), and name, for each one, a concrete way it fails in real markets.
6. Compute and interpret all five Greeks (delta, gamma, vega, theta, rho) from the closed-form BSM formulas, cross-check every one by finite difference (bump-and-reprice), and state the economic meaning of each in plain language and in real currency terms (per lot, per contract).
7. Use the Greeks to explain a realized option P&L via a Taylor-series decomposition, and show (numerically) that the decomposition is a good approximation for a small move and a visibly worse one for a large move, and explain why (the very reason DV1.03 needs gamma-hedging, not just delta-hedging).
8. Carry India (₹, Nifty/Bank Nifty-style index options) and US ($, SPX-style index options) worked examples side by side throughout, each grounded in real, dated, cited contract facts (lot size, multiplier, exercise style, risk-free proxy) with clearly labeled synthetic/illustrative pricing inputs.
9. (R10 duality.) Use an AI coding assistant to build a pricer or Greeks engine fast, and reconcile every one of its outputs against your own by-hand tree, parity check, or closed-form derivative before trusting it with a single rupee or dollar of risk.

Prerequisites & connections

Builds on. DV1.01 (Forwards, Futures & Swaps (Pricing & Valuation) established the whole branch's central method) risk-neutral, no-arbitrage pricing via cost-of-carry, for instruments with linear, unconditional payoffs. This module is where that method meets optionality: a payoff that bends at the strike, so the tidy single-formula replication of a forward gives way to a tree of possibilities that must be walked backward one node at a time. If DV1.01's cost-of-carry argument ("price it so nobody can lock in a free profit") felt mechanical, you already have the right instinct for everything below. M1.09 (Equity, SBC & Consolidation) introduced options only as an accounting object, grant-date fair value from "a Black-Scholes or lattice model," the fair-value hierarchy, and hedge-accounting's fair-value/cash-flow/net-investment buckets; this module is deliberately the other half of that story: it never re-teaches the accounting, and instead builds the pricing engine that produces the number M1.09 told you to trust. M9.01 (Risk & Position Sizing) gave you the practitioner definition of risk (probability × magnitude of permanent impairment) and M3.02 (Risk, Return & the ERP) gave you the academic one (variance, beta, CAPM), this module does not re-derive either; it hands you a new, sharper vocabulary (delta, gamma, vega) for describing a specific kind of risk, the convex risk of an optionality position, that neither of those frameworks was built to quantify. A working comfort with the normal distribution (density and cumulative probability) from ordinary statistics is assumed; QM1.01 (Probability, Distributions, Sampling & Estimation) formalizes the lognormal/normal machinery this module leans on, if you want the fuller treatment.