Stochastic Calculus Applications in Finance

Core Idea

Mathematical finance applies stochastic calculus to price and hedge financial derivatives. The fundamental theorem of asset pricing connects arbitrage-freeness to the existence of an equivalent martingale measure, and market completeness to its uniqueness. Under the Black-Scholes model, Girsanov's theorem constructs the risk-neutral measure, the martingale representation theorem provides the hedging strategy, and the Feynman-Kac formula connects risk-neutral expectations to the Black-Scholes PDE.

Explainer

Mathematical finance is the most prominent application of stochastic calculus. The central problem is pricing and hedging derivatives — financial contracts whose value depends on the evolution of an underlying asset. The Black-Scholes framework, built on geometric Brownian motion and Itô calculus, provides the theoretical foundation. The key insight is that in a complete market, every derivative can be replicated by dynamically trading the underlying asset and a risk-free bond, and the replication cost determines the derivative's price.

The fundamental theorems of asset pricing are the theoretical pillars. The first FTAP states that a market is arbitrage-free if and only if there exists an equivalent martingale measure (EMM) Q under which all discounted asset prices are martingales. The second FTAP states that the market is complete (every contingent claim is attainable) if and only if the EMM is unique. In the Black-Scholes model (one stock, one Brownian motion), Girsanov's theorem constructs the unique EMM by setting θ = (μ-r)/σ and defining Q via the Girsanov density. Under Q, the stock satisfies dS = rS dt + σS dW̃ — the physical drift μ is replaced by the risk-free rate r.

The Black-Scholes formula C = S₀Φ(d₁) - Ke^{-rT}Φ(d₂) for a European call with strike K and maturity T follows from computing E_Q[e^{-rT}max(S_T - K, 0)]. Since S_T is lognormally distributed under Q (from GBM with drift r), this is a direct calculation. The same result can be derived via the Black-Scholes PDE ∂V/∂t + rS(∂V/∂S) + (1/2)σ²S²(∂²V/∂S²) = rV, which is obtained by constructing the delta-hedging portfolio and eliminating risk. The Feynman-Kac formula provides the bridge: the PDE solution equals the risk-neutral expectation.

The replicating portfolio is constructed via the martingale representation theorem. The discounted option price V(t)e^{-rt} is a Q-martingale adapted to the Brownian filtration, so by the MRT, V(t)e^{-rt} = V(0) + ∫₀ᵗ H(s) dW̃(s). Converting to the stock numeraire: hold Δ(t) = ∂V/∂S shares of stock and invest the remainder in bonds. This delta-hedging strategy replicates the option payoff exactly — it is self-financing, and at maturity the portfolio value equals max(S_T - K, 0). The strategy's existence (guaranteed by the MRT) is what justifies using the risk-neutral expectation as the price. Without a replication argument, the expectation under Q would be just one possible price among many.