Martingale pricing and the FTAP

Martingale pricing is the modern unifying framework for derivatives pricing. It says: under the risk-neutral measure, every traded asset's discounted price is a martingale, and any payoff can be priced by taking the discounted risk-neutral expectation. This generalises Black-Scholes, gives a clean account of arbitrage, and forms the algebraic basis of fixed-income, FX, equity, and credit derivatives.

The fundamental theorems of asset pricing (FTAP)

1. First FTAP: a market admits no arbitrage if and only if there exists at least one equivalent martingale measure (EMM).
2. Second FTAP: the market is complete (every contingent claim is hedgeable) if and only if the EMM is unique.

These are due to Harrison-Pliska (1981), Harrison-Kreps (1979), and Delbaen-Schachermayer (1994 — the rigorous continuous-time version). Together they justify expectation-based pricing as the unique no-arbitrage price in complete markets.

The pricing formula

Given a contingent claim with payoff X at time T, in a complete market with EMM Q and numeraire N, the time-t price is:

V(t) = N(t) E^Q[X / N(T) | F_t]

Choosing N(t) = e^{rt} (the bank account) gives V(t) = e^{-r(T-t)} E^Q[X | F_t] — the standard risk-neutral pricing formula. Other numeraires (stock, bond, forward measure) simplify different problems.

Change of numeraire

If two assets are both tradable, you can use either as the numeraire. Switching gives Bayes-rule transformations of the EMM. Powerful technique: for options on the maximum of two assets, use one of the assets as numeraire to simplify the integration. Geman-El Karoui-Rochet (1995) formalised this.

Replication and self-financing portfolios

A self-financing portfolio is one where changes in value come only from changes in held asset prices — no money added or removed. Replication: find a self-financing portfolio whose value at T equals the payoff. If found, the option's price equals the replication cost. Complete markets are precisely those where every payoff is replicable.

Equivalent of BS under FTAP

Black-Scholes formula = e^{-r(T-t)} E^Q[(S(T) - K)⁺]. Under Q, S(T) is lognormal; the expectation is computed exactly. The same machinery handles:

• Put: e^{-r(T-t)} E^Q[(K - S(T))⁺].
• Forward: e^{-r(T-t)} E^Q[S(T) - K] = S(t) - K e^{-r(T-t)}. (Linear payoff.)
• Binary call: e^{-r(T-t)} P^Q(S(T) > K) = e^{-r(T-t)} Φ(d_2). (Pays 1 if in the money.)
• Asian, lookback, barrier — all the same recipe with different payoffs.

Why μ disappears

Real-world expected return μ enters portfolio choice and risk management — but never option pricing. The risk-neutral measure cancels μ via Girsanov. The economic intuition: the option is priced by replication, which depends only on how S moves (σ) and the cost of financing (r), not on how much S is expected to drift.

Incomplete markets

Without market completeness (e.g., stochastic vol that can't be perfectly hedged with bonds and stock alone), multiple EMMs exist and option prices are not unique. Practitioners use one of: (a) a 'reasonable' EMM (e.g., minimum-entropy martingale measure), (b) super-replication (find the cheapest portfolio that dominates the payoff in every state), or (c) utility-indifference pricing (price at the level that makes the investor indifferent between selling the option and not).

Beyond European options

• American options: optimal stopping. Price = sup_τ E^Q[discounted payoff at stopping time τ]. Numerical methods include LSM (Longstaff-Schwartz) and PDE PSOR.
• Path-dependent (Asian, barrier): need full path simulation or augmented-state PDE.
• Bermudan: discrete exercise dates; combine PDE with optimal exercise at each date.
• Volatility derivatives (variance swap, VIX option): priced under stochastic-vol EMM or via the static-replication argument of Carr-Madan (1998).

Calibration vs hedging

Calibration: choose model parameters under Q so that the model reproduces market prices of liquid instruments. The resulting Q is implied, not estimated from history. Hedging: under the calibrated model, compute Greeks and rebalance. Real-world P-measure is used for VaR, stress, and risk attribution — not for pricing. The two-measure dichotomy is fundamental.