This is the centrepiece of Note 1. We derive the Black–Scholes PDE by the delta-hedging argument — the continuous-time version of the replication story of market model and risk-neutral valuation — then connect the PDE back to the expectation formula via the Feynman–Kac theorem, and state the closed-form Black–Scholes price. The note ends where the model meets reality: the only unobservable input is , and inverting the formula for it gives implied volatility, the market's common language for option prices. The smile/skew in implied volatility is simultaneously the model's most useful output and the proof of its deficiency.
Learning objectives. By the end of this note you should be able to:
- derive the Black–Scholes PDE step by step (Itô → kill the noise → no-arbitrage) and explain why disappears;
- state Feynman–Kac in both directions and use it to solve simple terminal-value PDEs;
- state the Black–Scholes formula (with and without dividends) and interpret each term probabilistically;
- define implied volatility, compute it by bisection and Newton–Raphson, and compare the two methods;
- describe the construction and uses of the implied volatility surface.
In this lesson
- 4.1 Derivation of the Black–Scholes PDE by delta-hedging
- 4.2 Feynman–Kac: the bridge between PDE and expectation
- 4.3 The Black–Scholes formula
- 4.4 Deficiencies of Black–Scholes and implied volatility
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