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Stochastic Calculus / Change of Measure, Girsanov, and Risk-Neutral Pricing

The Martingale Representation Theorem and Market Completeness

In the previous note, Assumption 3 stated that the discounted portfolio value V~t\tilde{V}_t could be written as a stochastic integral V~0+∫0tAs dWs\tilde{V}_0 + \int_0^t A_s \, dW_s. The Martingale Representation Theorem (MRT) says Assumption 3 is always satisfied — in a market driven by a single Brownian motion, every square-integrable martingale with respect to the Brownian filtration is a stochastic integral. The MRT is therefore the theorem that makes the Black-Scholes market complete: every contingent claim can be replicated, there is a unique equivalent martingale measure, and the replicating portfolio can always be constructed from the integral representation.

The finance consequence is concrete: if you can compute the Itô form of the Q\mathbb{Q}-martingale V~t=EQ[V~T∣Ft]\tilde{V}_t = \mathbb{E}^{\mathbb{Q}}[\tilde{V}_T \mid \mathcal{F}_t], you have the delta-hedging ratio at=At/(σS~t)a_t = A_t/(\sigma \tilde{S}_t) in closed form. For a European call, this gives at=Φ(d1)a_t = \Phi(d_1).

Learning objectives. By the end of this note you should be able to:
  • State the Martingale Representation Theorem for the Brownian filtration.
  • Explain the binomial analogy that motivates why the theorem holds.
  • Derive the hedging ratios for a European call from the MRT.
  • State the completeness theorem for the Black-Scholes market and explain why it fails with stochastic volatility.
  • Explain why the MRT is equivalent to the uniqueness of the EMM.

In this lesson

  1. The Martingale Representation Theorem
  2. Proof Idea via the Binomial Model
  3. Finding AtA_t via Itô's Formula
  4. Completeness of the Black-Scholes Market

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