This note builds the complete pipeline of computational option pricing, from the definition of an option contract to a working finite-difference pricer with numerically estimated Greeks. It is the foundation on which every later note rests: trees (Note 2), Monte Carlo and stochastic volatility (Note 3) are all alternative ways of computing the same object defined here.
By the end of this note you should be able to:
- define the standard option contracts and derive model-free price bounds and put–call parity from the no-arbitrage principle alone;
- derive the risk-neutral pricing formula from replication in a one-period market, and explain precisely what the risk-neutral measure is (and is not);
- work with Brownian motion, the diffusion model zoo (ABM, GBM, OU) and Itô's lemma, including solving the GBM SDE in closed form;
- derive the Black–Scholes PDE by delta-hedging, connect it to the expectation formula via Feynman–Kac, and state the Black–Scholes formula;
- explain implied volatility, compute it by bisection or Newton–Raphson, and describe how a volatility surface is built;
- write down the explicit, implicit and Crank–Nicolson finite-difference schemes, analyse their stability, and choose boundary conditions;
- define all first- and second-order Greeks and estimate them numerically by bump-and-revalue or pathwise Monte Carlo.
In this lesson
- Contents
- The big picture
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