Learning outcomes

After this module you can:

1. 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.
2. 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.
3. 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 $S_0e^{(\mu-\sigma^2/2)T}$.
4. Derive the Black–Scholes PDE by the delta-hedging argument and solve it by computing the risk-neutral expectation as an explicit integral.
5. 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.
6. 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 $\Theta+\tfrac12\sigma^2S^2\Gamma+(r-q)S\Delta-rV=0$.
7. 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.
8. 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.