No-Arbitrage Principle and Risk-Neutral Valuation › The Fundamental Theorem of Asset Pricing

The connection between no-arbitrage and risk-neutral pricing is not a coincidence. It is formalized in one of the most important theoretical results in mathematical finance. This theorem establishes that our two key concepts, absence of arbitrage and existence of risk-neutral probabilities, are logically equivalent.

This theorem connects an economic concept (no free lunch) with a mathematical structure (martingale measure). Let's unpack this statement:

Arbitrage-free: No trading strategy can generate risk-free profits from zero initial capital. This is the economic condition we have been assuming throughout.

Equivalent probability measure: A measure $Q$ that assigns positive probability to exactly the same events as $P$. If something is possible under $P$, it's possible under $Q$, and vice versa. This requirement ensures that the risk-neutral world doesn't consider impossible events to be possible, or vice versa.

Martingale: A stochastic process where the expected future value, given current information, equals the current value. Intuitively, a martingale is a "fair game" with no systematic drift. For discounted asset prices:

$$E^Q\left[\frac{S_T}{B_T} \mid \mathcal{F}_t\right] = \frac{S_t}{B_t}$$

where:

• $E^Q$: expectation under the risk-neutral measure
• $S_t, S_T$: stock prices at time $t$ and $T$
• $B_t, B_T$: risk-free bond prices at time $t$ and $T$
• $\mathcal{F}_t$: information available at time $t$

This equation says the discounted stock price is a "fair game" under $Q$: you don't expect it to systematically drift up or down in discounted terms. When we divide by the bond price, we are measuring the stock's value in units of the risk-free asset. Under the risk-neutral measure, this relative value is expected to stay constant.

The Fundamental Theorem has consequences for both theory and practice:

1. Existence of risk-neutral measure: If markets are arbitrage-free (as we assume), then risk-neutral probabilities exist. We can always find a measure $Q$ for pricing. This is comforting: our pricing approach is not an arbitrary trick but rests on solid theoretical foundations.

2. Uniqueness and completeness: If the risk-neutral measure is unique, the market is complete, meaning every derivative can be perfectly replicated. If multiple risk-neutral measures exist, the market is incomplete, and there's a range of arbitrage-free prices. In incomplete markets, different risk-neutral measures correspond to different ways of pricing risk, and the no-arbitrage condition alone doesn't pin down a unique price.

3. Pricing formula: Under $Q$, any derivative paying $f_T$ at time $T$ has price:

$$f_0 = E^Q\left[e^{-rT} f_T\right]$$

where:

• $f_0$: derivative price today
• $f_T$: derivative payoff at maturity
• $r$: risk-free rate
• $T$: time to maturity
• $E^Q$: expectation under the risk-neutral measure

This elegant formula reduces derivative pricing to computing a single expectation, provided we can characterize the risk-neutral dynamics of the underlying. All the complexity of hedging and replication gets absorbed into the change of probability measure.

In our binomial model, the market is complete: there are two states and two assets (stock and bond), so we can perfectly replicate any payoff. The risk-neutral measure is unique, given by the probability $q$ we derived. With two assets and two states, we have exactly enough instruments to span all possible payoffs.