Intrinsic value, time value and parity
A European call's intrinsic value is max(S−K, 0); the market price never falls below it, and the excess is time value — the premium for the chance of further improvement before expiry. Near expiry time value decays to zero (positive theta for the seller). Put-call parity is a model-free identity: C − P = S·e^(−qT) − K·e^(−rT). A violation is arbitrage by construction, which is why the platform locks parity as an acceptance test on every engine release.
The Greeks: the language of options risk
Delta (∂V/∂S) is first-order price sensitivity — and roughly the risk-neutral exercise probability for European options. Delta hedging buys or sells Δ units of the underlying to neutralize it. Gamma (∂²V/∂S²) measures how fast the hedge decays: large gamma means frequent rehedging, paying transaction costs and slippage each time. Vega (∂V/∂σ) is volatility exposure — option buyers are long vega, earning in vol expansion and bleeding in stillness. Theta is time decay: the seller's income stream, paid for with tail risk. Together these four numbers describe any options strategy's risk profile.
Worked example: delta-hedging a call
- Long 10 calls on 100 shares each, delta 0.58 → equivalent position 580 shares.
- Short 580 shares (or 5.8 futures contracts) to neutralize.
- Spot rises $1: option gains ≈ $580; hedge loses ≈ $580 — the book is locally price-immune.
- A large move shifts delta (gamma effect): the hedge must be reset — rehedging frequency is the real cost of delta neutrality.
نتیجه: Delta neutrality is a moment, not a state: gamma sets the rehedging cadence, and managing that cadence is the actual skill of running an options book.
American options have no closed form: the CRR binomial tree compares continuation vs early exercise at every node and converges to BSM as steps increase — a convergence the platform locks in tests at N = 250 and 1000.