Girsanov's Theorem

The Radon-Nikodym derivative dQ/dP is the mathematical object that defines the density of the new measure Q with respect to the original measure P. It acts as a likelihood ratio, re-weighting the probability of each path. For a Brownian motion with drift μ, the derivative takes the form of an exponential martingale:

• Form: Z_t = exp(-∫θ_s dW_s - ½∫θ_s² ds)
• Requirement: The process Z_t must be a true martingale, not just a local martingale
• Novikov's Condition: A sufficient condition ensuring Z_t is a martingale: E[exp(½∫θ_s² ds)] < ∞
• Interpretation: Z_t re-weights paths where the drift was positive downward and paths where the drift was negative upward

Girsanov's theorem states that under the new measure Q, a Brownian motion with drift dX_t = μ dt + dW_t becomes a standard Brownian motion. The drift μ is absorbed into the Brownian motion itself:

• Original process (P): dS_t = μ S_t dt + σ S_t dW_t
• Transformed process (Q): dS_t = r S_t dt + σ S_t dW̃_t
• Relationship: dW̃_t = dW_t + θ_t dt where θ_t = (μ - r)/σ is the market price of risk
• Key insight: Only the drift changes; the volatility σ remains invariant under equivalent measure changes
• Diffusion invariance: The quadratic variation <X>_t is preserved across equivalent measures

Two probability measures P and Q are equivalent if they agree on which events have zero probability. This equivalence is critical for financial applications:

• Null set agreement: P(A) = 0 ⟺ Q(A) = 0 — impossible events remain impossible
• Path continuity: Both measures assign probability one to continuous paths for Brownian motion
• Finite time horizon: Equivalence holds on [0, T] for finite T; may break down on infinite horizons
• No arbitrage connection: The existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage (First Fundamental Theorem of Asset Pricing)
• Completeness: If the equivalent martingale measure is unique, the market is complete and all derivatives can be perfectly hedged

Girsanov's theorem extends naturally to multi-dimensional Brownian motion, essential for pricing derivatives on multiple correlated assets:

• Vector process: dX_t = μ_t dt + σ_t dW_t where W_t is an n-dimensional Brownian motion
• Multi-dimensional shift: dW̃_t = dW_t + θ_t dt where θ_t is an n-dimensional vector of risk premia
• Correlation preservation: The instantaneous correlation structure between assets is determined by the quadratic covariation, which is invariant under equivalent measure changes
• Incomplete markets: When there are more Brownian motions than traded assets, multiple equivalent martingale measures exist, requiring additional criteria (e.g., minimal entropy) to select a pricing measure
• Stochastic correlation: If volatility is itself stochastic, the change of measure can also adjust the drift of the volatility process

Girsanov's theorem is the mathematical backbone of risk-neutral pricing, enabling the valuation of options, futures, and exotic derivatives:

• Black-Scholes derivation: The theorem justifies replacing μ with r in the geometric Brownian motion, leading to the classic pricing PDE
• Interest rate models: In the Heath-Jarrow-Morton (HJM) framework, Girsanov transformations connect the forward rate dynamics under different measures (risk-neutral, forward, spot)
• Change of numéraire: Moving from the risk-neutral measure to a forward measure involves a Girsanov transformation where the numéraire changes from the money market account to a zero-coupon bond
• Foreign exchange: In FX markets, the theorem handles the transition between domestic and foreign risk-neutral measures, introducing the quanto adjustment
• Credit derivatives: Used to switch between the physical default probability measure and the risk-neutral survival measure for pricing credit default swaps

While theoretically powerful, applying Girsanov's theorem in practice requires careful attention to its mathematical boundaries:

• Martingale failure: If Novikov's condition is violated, the Radon-Nikodym derivative is only a strict local martingale, not a true martingale, leading to pricing anomalies and arbitrage opportunities
• Jump processes: The theorem in its standard form applies to continuous processes; for Lévy processes with jumps, a generalized version (Jacod's theorem) is required, and the change of measure can alter both drift and jump intensity
• Estimation risk: The market price of risk θ_t is unobservable and must be estimated from market prices, introducing model risk
• Infinite horizon: On infinite time horizons, equivalent measures may become singular, requiring the use of local equivalence concepts
• Numerical implementation: In Monte Carlo simulations, the change of measure is implemented via likelihood ratio weighting rather than actual path transformation, which can increase variance