Option pricing and the Greeks
Where does an option price come from, and what is it a price of?
12 of 12 chapters published
Chapters
Beginner
- Bounds — what an option cannot costBefore any model, arithmetic alone rules out most prices. The limits need no view on volatility, no probability and no formula — only the fact that a portfolio worth more at expiry must cost more now.
- Put-call parityOne equation ties the call, the put, the underlying and a loan together. It holds with no model and no assumptions about volatility — which makes it both the most reliable result in option pricing and a working test of any chain.
- Replication, and why a price is not a forecastAn option can be rebuilt out of the underlying and a loan. Once it can, its price is whatever that recipe costs — which means the price does not contain anybody's view about where the underlying is going.
Intermediate
- Calibrating the treeThe two-state world becomes a model the moment the two states are set by a volatility rather than invented. That one substitution turns an arithmetic exercise into something you can price a real contract with — badly, at first.
- More steps, and what the tree converges toAdd states and the error falls in a predictable way. Along the way the hedge stops being a fixed holding and becomes something you have to keep adjusting — and the tree gains the one thing a formula cannot easily do.
- From the tree to Black-ScholesShrink the steps to nothing and the binomial average becomes a normal one. The formula that results is not a new idea — it is chapter 5's table written down, and reading its two terms tells you what it is doing.
- What Black-Scholes assumesEight assumptions, each false, and they do not fail equally. Sorting them by how much damage they do is more useful than listing them — and the most important failure is one the formula never claimed to cover.
Advanced
- Delta, and what hedging involvesDelta is the first derivative of the price, the replicating holding and the equivalent position in the underlying — three descriptions of the same number. It is also only a local approximation, which is the whole of the next chapter.
- Gamma and thetaThe two Greeks that are really one. Curvature pays a delta-hedged owner when the underlying moves and decay charges them when it does not — and the model sets the fee so that the breakeven move is exactly the volatility it assumed.
- Vega, and volatility as a traded quantityThe sensitivity to the one input nobody can observe — and the one the model insists is constant. That contradiction is why vega exists, and why an index needs a whole strip of strikes to measure what a single option implies.
- Implied volatility and the smileRun the formula backwards and every option reports its own volatility. They disagree, systematically and since 1987 — which means the market is pricing a distribution the model does not have, and quoting it in the model's language.
- Pricing options in Indian marketsThe last chapter, where the model meets the contract. Lot sizes, strike schemes, corporate-action formulas, margin and a securities transaction tax that rose sharply on 1 April 2026 — each one a term the formula does not have.