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Arbitrage-Free Pricing via Equivalent Martingale Measures

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The Formal Theorem

Let (Ω,F,P) (\Omega, \mathcal{F}, \mathbb{P}) be a probability space with a filtration F={Ft}t∈[0,T] \mathbb{F} = \{\mathcal{F}_t\}_{t \in [0,T]} . Let St S_t be the price process of a risky asset. There exists a measure Q∼P \mathbb{Q} \sim \mathbb{P} , called an Equivalent Martingale Measure (EMM), such that the discounted price process S~t=e−rtSt \tilde{S}_t = e^{-rt} S_t is a Q \mathbb{Q} -martingale if and only if the market model is arbitrage-free. Under Q \mathbb{Q} , the price of a derivative VT=H(ST) V_T = H(S_T) is given by:
Vt=e−r(T−t)EQ[H(ST)∣Ft] V_t = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}[H(S_T) | \mathcal{F}_t]

Analytical Intuition.

Imagine the financial market as a vast, turbulent ocean where P \mathbb{P} represents our real-world belief of price movements. Arbitrage is the "free lunch"—a perpetual motion machine in finance. The First Fundamental Theorem of Asset Pricing tells us that if such a free lunch exists, the market is broken. To fix it, we perform a radical change of perspective: we switch to the Q \mathbb{Q} measure, an 'Equivalent Martingale Measure'. In this new world, all assets grow at the risk-free rate r r . It is as if we have flattened the slope of the ocean; in this world, there is no 'downhill' to exploit for riskless profit. Because Q \mathbb{Q} is equivalent to P \mathbb{P} , we agree on which events are possible, but we disagree on their probabilities. This measure is the 'fair' world where the current price Vt V_t is simply the average of future payoffs, discounted back to today. By adjusting our outlook to this balanced state, we strip away the bias of risk appetite, leaving behind the pure, arbitrage-free value of the derivative.
CAUTION

Institutional Warning.

Students often confuse the physical measure P \mathbb{P} with the risk-neutral measure Q \mathbb{Q} . P \mathbb{P} describes actual market behavior, while Q \mathbb{Q} is a mathematical construction used solely for pricing; Q \mathbb{Q} -probabilities are not predictions of future market outcomes.

Academic Inquiries.

01

Why is the discount factor e−rt e^{-rt} necessary?

It accounts for the time value of money, transforming nominal currency into 'today's dollars' to ensure the martingale condition holds under the risk-neutral measure.

02

What does 'equivalent' mean in Q∼P \mathbb{Q} \sim \mathbb{P} ?

It means the measures share the same null sets: P(A)=0 \mathbb{P}(A) = 0 if and only if Q(A)=0 \mathbb{Q}(A) = 0 . They agree on what is possible.

Standardized References.

  • Definitive Institutional SourceShreve, S. E., Stochastic Calculus for Finance II: Continuous-Time Models.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Arbitrage-Free Pricing via Equivalent Martingale Measures: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/advanced-stochastic-processes/arbitrage-free-pricing-via-equivalent-martingale-measures

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