+-----------------------------------------------------------------------------------+ | THE FILTERED PROBABILITY SPACE: (Ω, F, {F_t}, P) | | | | [Physical Measure P] ---> Real-world dynamics: dS_t = μ S_t dt + σ S_t dW_t^P | | | | | | Girsanov Theorem / Change of Measure (Radon-Nikodym Derivative) | | | dQ/dP = exp( -∫θ_s dW_s^P - 1/2 ∫θ_s^2 ds ) | | v | | [Risk-Neutral Measure Q] -> Risk-neutral dynamics: dS_t = r S_t dt + σ S_t dW_t^Q| | | | | | Fundamental Theorem of Asset Pricing (FTAP) | | | No Arbitrage <==> Existence of Q | | | Market Completeness <==> Uniqueness of Q | | v | | [Pricing Measure] ------> Derivative Value: V_0 = e^(-rT) E^Q[ Payoff(S_T) ] | +-----------------------------------------------------------------------------------+