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Fundamental theorem of asset pricing

The fundamental theorems of asset pricing constitute foundational results in mathematical finance, linking the economic concept of no arbitrage in financial markets to probabilistic representations that enable derivative pricing and risk management. The first theorem establishes that a market is free of arbitrage opportunities if and only if there exists an equivalent martingale measure under which discounted asset prices are martingales, providing a framework for risk-neutral valuation. The second theorem extends this by stating that such a market is complete—meaning every contingent claim can be perfectly replicated by a self-financing portfolio—if and only if the equivalent martingale measure is unique. The theorems were first formulated in discrete time by J. Michael Harrison and David M. Kreps in 1979, and extended to continuous time by Harrison and Stanley R. Pliska in 1981, formalizing the martingale approach to asset pricing and building on earlier work in stochastic processes and arbitrage theory. In their seminal paper, Harrison and Pliska demonstrated the equivalence between no-arbitrage conditions and the existence of a probability measure equivalent to the physical measure under which asset prices, when discounted by the numeraire, behave as martingales, thus justifying the use of expectation under this measure for pricing. This continuous-time framework was further generalized by Freddy Delbaen and Walter Schachermayer in 1994 to settings with unbounded semimartingale price processes, accommodating more realistic market models without assuming boundedness or continuity of paths. The theorems' implications are profound: the first ensures consistent pricing across assets in arbitrage-free markets, underpinning models like the Black-Scholes formula for option pricing, while the second highlights conditions for perfect hedging, essential for risk-neutral derivatives valuation in complete markets. In incomplete markets, where multiple martingale measures exist, pricing involves selecting from a range of possible values, often using additional criteria like minimal variance or utility maximization. These results have influenced modern quantitative finance, extending to settings with transaction costs, jumps, and stochastic volatility, while serving as benchmarks for detecting arbitrage in empirical market data.

Overview and Historical Context

Core Statement and Implications

The fundamental theorem of asset pricing (FTAP) establishes a precise equivalence between the absence of arbitrage opportunities in a financial market and the existence of an equivalent martingale measure, also known as a risk-neutral probability measure. Specifically, under this measure, the discounted prices of traded assets behave as martingales, meaning their expected future values equal their current values when adjusted for the risk-free rate. This theorem, first rigorously formulated in multiperiod securities markets, holds that a market admits no arbitrage if and only if there exists at least one such measure equivalent to the physical (real-world) probability measure. The FTAP has profound implications for asset pricing theory, directly linking the no-arbitrage condition to risk-neutral valuation, where the price of any derivative or contingent claim is the expected value of its payoff under the risk-neutral measure, discounted at the risk-free rate. In complete markets—where every claim is attainable through dynamic trading—the uniqueness of this measure ensures unique prices for all assets, eliminating ambiguity in valuation. This framework underpins seminal derivative pricing models, such as the Black-Scholes model for European options, which relies on the theorem to derive closed-form solutions by assuming a unique risk-neutral measure for stock prices following geometric Brownian motion. Economically, the FTAP interprets the absence of arbitrage as a guarantee of fair pricing across assets, preventing riskless profits that would otherwise distort market equilibrium. The risk-neutral measure incorporates investors' risk aversion implicitly through probability adjustments, bypassing the need to specify explicit utility functions or individual preferences, thus providing a universal tool for pricing that aligns with observed market behavior. A simple illustration appears in the one-period binomial model, where a stock price can move up or down, and the no-arbitrage condition implies the existence of a unique risk-neutral probability that equates the expected return of the stock to the risk-free rate, enabling consistent pricing of options.

Historical Development

The roots of the fundamental theorem of asset pricing (FTAP) lie in mid-20th-century economic theory, particularly the work on general equilibrium and arbitrage by Kenneth Arrow and Gérard Debreu in the 1950s, who demonstrated the existence of competitive equilibria in markets free of arbitrage opportunities. Building on this, Stephen Ross advanced arbitrage-based pricing in the 1970s through his development of the arbitrage pricing theory, which linked asset returns to systematic risk factors under no-arbitrage assumptions. These foundational ideas shifted focus from static equilibrium models to dynamic pricing in securities markets, setting the stage for probabilistic formulations. The theorem's formalization emerged in the late 1970s amid growing interest in option pricing and stochastic models. J. Michael Harrison and David Kreps provided a pivotal contribution in their 1979 paper, establishing that no-arbitrage conditions in multiperiod securities markets are equivalent to the existence of equivalent martingale measures, thereby bridging arbitrage theory with martingale representations. This was extended by Harrison and Stanley Pliska in 1981, who formalized the theorem for discrete-time models. This work coincided with practical advancements: the 1973 Black-Scholes model implicitly invoked no-arbitrage for deriving option prices via continuous diffusion processes, while the 1979 Cox-Ross-Rubinstein binomial model offered a discrete-time lattice approach that explicitly used risk-neutral probabilities to replicate payoffs without arbitrage. These developments marked a transition from intuitive hedging arguments to rigorous probabilistic tools, influencing derivative pricing practices. In the 1990s, the FTAP achieved greater generality through extensions to continuous-time settings. Robert C. Dalang, Andrew Morton, and Walter Willinger proved a version of the first fundamental theorem in 1990, showing that no-arbitrage in stochastic securities models equates to the existence of equivalent martingale measures for semimartingale price processes. Freddy Delbaen and Walter Schachermayer advanced this further in 1994 with a comprehensive version linking market completeness to the uniqueness of such measures, applicable to broad classes of unbounded asset prices. These milestones solidified the theorem's role in modern mathematical finance, enabling analysis of complex markets beyond simple binomial or diffusion models. Post-1980s syntheses further entrenched the FTAP in the field. Stanley Pliska's 1997 textbook systematized discrete-time formulations, providing a unified framework for no-arbitrage pricing and martingale methods accessible to advanced students and practitioners. Ioannis Karatzas and Steven Shreve's 1998 monograph offered rigorous treatments of continuous-time extensions, emphasizing stochastic calculus applications to incomplete markets and hedging. Together, these contributions propelled the theorem's influence on risk management, portfolio theory, and regulatory frameworks in global finance.

Mathematical Prerequisites

Stochastic Processes and Filtrations

In mathematical finance, the foundational probabilistic framework for modeling asset prices and trading strategies begins with a complete probability space (\Omega, \mathcal{F}, P), where \Omega is the sample space representing all possible outcomes, \mathcal{F} is a \sigma-algebra of events, and P is a probability measure assigning probabilities to those events. This setup ensures that the space is rich enough to handle null sets and their completions, providing a rigorous basis for handling uncertainties in financial markets. A filtration (\mathcal{F}_t)_{t \geq 0} on this probability space is defined as an increasing family of sub-\sigma-algebras \mathcal{F}_t \subseteq \mathcal{F}, where \mathcal{F}_s \subseteq \mathcal{F}_t for s \leq t, representing the flow of information available to market participants over time. In financial contexts, \mathcal{F}_t captures the information revealed up to time t, such as observed prices or news, enabling the modeling of progressively refined knowledge. To address technical issues like jumps in information, filtrations are often assumed to be right-continuous, meaning \mathcal{F}_t = \bigcap_{u > t} \mathcal{F}_u for each t, and augmented by including all P-null sets from \mathcal{F} to ensure completeness. A stochastic process X = (X_t)_{t \geq 0} is said to be adapted to the filtration (\mathcal{F}_t) if, for each t, the random variable X_t is \mathcal{F}_t-measurable, denoted X_t \in \mathcal{F}_t. This adaptation condition ensures that the process at time t depends only on the information available up to t, which is crucial for realistic modeling of observable market quantities like stock prices. In discrete time, stochastic processes often take the form of random walks, where the increments X_{t} - X_{t-1} are independent and identically distributed random variables, providing a simple model for price fluctuations over finite periods. In continuous time, Brownian motion serves as a fundamental example, defined as a continuous adapted process W = (W_t)_{t \geq 0} with independent increments W_t - W_s \sim \mathcal{N}(0, t-s) for t > s, and W_0 = 0, capturing the irregular paths of asset prices under uncertainty. For trading strategies, predictable processes—those measurable with respect to the predictable \sigma-algebra generated by left-continuous adapted processes—are used to represent non-anticipating strategies, while optional processes, measurable with respect to the optional \sigma-algebra from stopping times, model strategies that can react immediately to information. These distinctions allow for precise definitions of admissible trades that avoid foresight.

Equivalent Martingale Measures

In financial mathematics, an equivalent martingale measure, often denoted as Q, is a probability measure on the underlying probability space that is equivalent to the physical (or real-world) measure P, meaning Q and P agree on sets of measure zero. Under Q, the prices of traded assets, when discounted by a numeraire such as the risk-free asset S^0_t (typically a bank account process B_t), form martingales; that is, the discounted processes \tilde{S}_t = S_t / S^0_t satisfy \mathbb{E}^Q[\tilde{S}_T \mid \mathcal{F}_t] = \tilde{S}_t for $0 \leq t \leq T. This property ensures that the expected value of future discounted payoffs equals the current price, providing a foundation for arbitrage-free pricing. Key properties of Q include its role in altering the drift of asset price dynamics to match the risk-free rate while preserving the volatility structure. Equivalence between Q and P is formalized through the Radon-Nikodym derivative dQ/dP, a positive P-integrable random variable that reweights probabilities without introducing arbitrage opportunities. The first fundamental theorem of asset pricing establishes that the existence of such a Q is equivalent to the absence of arbitrage in the market model. In complete markets, where every contingent claim can be perfectly replicated, the equivalent martingale measure Q is unique, which in turn implies unique arbitrage-free prices for all derivatives. This uniqueness stems from the market's ability to hedge all risks, ensuring a single risk-neutral valuation measure. Seminal work by Harrison and Pliska formalized this martingale approach in continuous trading settings, linking measure existence and uniqueness to market completeness.90026-0) For models driven by diffusions, the Girsanov theorem constructs an equivalent martingale measure by transforming the Brownian motion under P into one under Q via a drift adjustment proportional to the market price of risk \theta = (\mu - r)/\sigma, where \mu is the physical drift, r the risk-free rate, and \sigma the volatility. The Radon-Nikodym derivative takes the form of an exponential martingale Z_t = \exp\left\{ -\int_0^t \theta_s \, dW_s - \frac{1}{2} \int_0^t \theta_s^2 \, ds \right\}, shifting the measure without altering the diffusion's quadratic variation. This provides an explicit method to derive risk-neutral dynamics from physical ones. A central pricing implication is captured by the conditional expectation formula under Q: S_t = \mathbb{E}^Q \left[ S_T \frac{B_t}{B_T} \,\Big|\, \mathcal{F}_t \right], where B_t denotes the bank account process accumulating at the risk-free rate, ensuring consistency with no-arbitrage conditions.

Discrete-Time Formulation

Market Model in Discrete Time

The discrete-time market model is formulated on a finite time horizon T \in \mathbb{N}, divided into T discrete periods t = 0, 1, \dots, T. The underlying probability space is (\Omega, \mathcal{F}, \mathbb{P}), equipped with a filtration (\mathcal{F}_t)_{t=0}^T of sub-\sigma-algebras satisfying \mathcal{F}_0 trivial and \mathcal{F}_T = \mathcal{F}, which captures the progressive revelation of market information over time. The market comprises a risk-free asset, modeled as a bond or money market account with price process B_t = (1 + r)^t, where r \geq 0 is the constant risk-free interest rate per period. There is also at least one risky asset with price process S_t = (S_t^1, \dots, S_t^d) for d \geq 1 dimensions, where each component S_t^i is \mathcal{F}_t-adapted and takes values in a finite state space at each time step to facilitate computational tractability and ensure the model remains finite-dimensional. Let \tilde{S}_t = (B_t, S_t^1, \dots, S_t^d) denote the full asset price vector. A trading strategy is specified by a predictable process H = (H^0, H^1, \dots, H^d), where H_t^j denotes the number of units held in the j-th asset at time t, with predictability implying H_t is \mathcal{F}_{t-1}-measurable for t \geq 1. The corresponding portfolio value at time t is given by V_t = H_t \cdot \tilde{S}_t. The self-financing condition ensures that portfolio changes arise solely from asset price movements, without external cash inflows or outflows. This is expressed as \Delta V_t = H_t \cdot \Delta \tilde{S}_t, \quad t = 1, \dots, T, where \Delta \tilde{S}_t = \tilde{S}_t - \tilde{S}_{t-1} and \Delta V_t = V_t - V_{t-1}, or equivalently, V_t = V_0 + \sum_{s=1}^t H_s \cdot (\tilde{S}_s - \tilde{S}_{s-1}). To exclude unrealistic strategies that could generate infinite wealth, such as doubling strategies, admissibility requires V_t \geq 0 almost surely for all t, or more permissively, V_t \geq -c for some constant c > 0 and all t. A contingent claim is an \mathcal{F}_T-measurable random variable X: \Omega \to \mathbb{R} representing a financial payoff at maturity T. Such a claim is attainable if there exists an admissible self-financing strategy H satisfying V_T = X almost surely. The market is said to be complete if every contingent claim is attainable via some admissible strategy.

No-Arbitrage and Risk-Neutral Measures

In discrete-time financial markets, an arbitrage opportunity is defined as a self-financing trading strategy with initial value V_0 = 0, terminal value V_T \geq 0 almost surely, and P(V_T > 0) > 0. The absence of such opportunities is a fundamental assumption ensuring market efficiency. The First Fundamental Theorem of Asset Pricing (FTAP) establishes the equivalence between no-arbitrage and the existence of risk-neutral measures. Specifically, in a discrete-time market model with a numéraire asset B (e.g., a risk-free bond) and traded assets S, the market is arbitrage-free if and only if there exists an equivalent martingale measure Q \sim P (i.e., Q has the same null sets as the physical measure P) under which the discounted asset prices \tilde{S}_t = S_t / B_t are martingales. The proof proceeds in two directions. For sufficiency, assume such a Q exists and suppose, for contradiction, there is an arbitrage strategy with V_0 = 0 and V_T \geq 0 a.s., P(V_T > 0) > 0. The discounted portfolio value \tilde{V}_t = V_t / B_t is a Q-martingale, so E_Q[\tilde{V}_T \mid \mathcal{F}_0] = \tilde{V}_0 = 0. But \tilde{V}_T \geq 0 a.s. and Q(\tilde{V}_T > 0) > 0, implying E_Q[\tilde{V}_T] > 0, a contradiction. For necessity, the absence of arbitrage implies the existence of a strictly positive linear pricing functional on the space of contingent claims, which by the Riesz representation theorem (in finite dimensions) or separating hyperplane arguments corresponds to an equivalent martingale measure making discounted prices martingales; this relies on optional decomposition of attainable claims. A canonical illustration is the one-period binomial model, where the stock price moves from S_0 to S_0 u (up factor u > 1) or S_0 d (down factor $0 < d < 1) with physical probabilities, and the bond grows by factor $1 + r > 0. The market is arbitrage-free if d < 1 + r < u, and the risk-neutral probability is q = \frac{(1 + r) - d}{u - d}, ensuring $0 < q < 1. Under Q, the discounted stock satisfies the martingale property: E_Q \left[ \frac{S_1}{B_1} \;\middle|\; \mathcal{F}_0 \right] = \frac{S_0}{B_0}. This holds as q u + (1 - q) d = 1 + r, replicating the bond return in expectation.

Continuous-Time Formulation

Market Model in Continuous Time

The continuous-time market model is formulated on a complete filtered probability space (\Omega, \mathcal{F}, (\mathcal{F}_t)_{0 \leq t \leq T}, P), where the filtration (\mathcal{F}_t) satisfies the usual conditions of right-continuity and completeness, representing the evolution of observable information over the finite time horizon [0, T]. This setup allows for the modeling of asset prices as adapted stochastic processes with respect to \mathcal{F}_t. The market includes a risk-free asset with price process B_t = B_0 \exp\left(\int_0^t r_s \, ds\right), where B_0 > 0 is the initial value and the short rate r_t is a non-negative \mathcal{F}_t-adapted process that may be deterministic or stochastic. There are also d risky assets with price vector process S = (S^1, \dots, S^d), where each S^i is a semimartingale expressed as S^i_t = S^i_0 + \int_0^t \mu^i_s \, ds + M^i_t, with M^i a local martingale part that may include diffusion and jump components. In diffusion-based models, the risky assets are typically Itô processes driven by a multidimensional Brownian motion W = (W^1, \dots, W^m) adapted to (\mathcal{F}_t), satisfying the stochastic differential equation dS_t = \mu_t \, dt + \sigma_t \, dW_t, where \mu_t is the drift vector and \sigma_t is the volatility matrix, both \mathcal{F}_t-predictable processes. Trading strategies are defined by a vector of predictable processes H = (H^0, H^1, \dots, H^d), where H^i is \mathcal{F}_t-predictable to ensure that holdings are decided based on information available up to but not including time t, and integrable with respect to the asset prices. The associated value process for an initial wealth V_0 is given by V_t = V_0 + \int_0^t H^0_s \, dB_s + \sum_{i=1}^d \int_0^t H^i_s \, dS^i_s, representing the self-financing portfolio value, with the integrals interpreted in the semimartingale sense. Often, analysis is conducted under the discounted value process \tilde{V}_t = V_t / B_t to normalize the risk-free asset to a martingale under suitable measures. Admissibility conditions on strategies exclude those leading to pathological behaviors, such as unbounded below values; typically, a strategy is admissible if \tilde{V}_t \geq -c almost surely for all t \in [0, T] and some constant c > 0 independent of the path. This bounded-below requirement ensures the exclusion of arbitrages like doubling strategies while allowing limited borrowing against the risk-free asset.

Equivalence to No-Arbitrage Conditions

In continuous-time financial markets modeled by semimartingales, the no-arbitrage condition is strengthened to the no free lunch with vanishing risk (NFLVR) condition to ensure the absence of arbitrage opportunities that can be approximated arbitrarily closely in risk. NFLVR posits that there are no self-financing trading strategies with non-negative gains that approach zero risk in the limit, formalized using consistent price systems that bound the value processes from below. The fundamental theorem of asset pricing in this setting states that NFLVR holds if and only if there exists an equivalent local martingale measure Q \sim P under which the discounted asset prices are local martingales. This equivalence links the economic notion of no-arbitrage to the probabilistic structure of measure changes, where the numéraire (typically a risk-free bond) serves as the discounting factor. The proof, due to Delbaen and Schachermayer, employs advanced functional analytic techniques, including separating hyperplane theorems, to link the economic no-arbitrage condition to the existence of a suitable probability measure. For diffusion processes, such as those in the Black-Scholes model, the Girsanov theorem provides an explicit construction of Q by adjusting the drift of the Brownian motion to match the risk-neutral dynamics. This framework accommodates general semimartingale price processes, including those with jumps, by allowing local martingale measures that handle discontinuous paths without requiring bounded variation. In the Black-Scholes model, where asset prices follow geometric Brownian motion, the theorem yields a unique equivalent martingale measure, enabling closed-form pricing of derivatives. Under the equivalent local martingale measure Q, the discounted asset price process \tilde{S}_t = S_t / B_t satisfies the local martingale property: \tilde{S}_t = E_Q \left[ \tilde{S}_T \mid \mathcal{F}_t \right] locally for $0 \leq t \leq T, where B is the numéraire process and \mathcal{F} the filtration.

Extensions and Generalizations

Incomplete Markets and Duality

In incomplete markets, the set of equivalent martingale measures \mathcal{Q} is typically non-singleton, implying that contingent claims cannot be uniquely replicated and thus lack unique arbitrage-free prices. Instead, the absence of arbitrage ensures that acceptable prices for a claim C lie within the interval [\pi_-(C), \pi_+(C)], where \pi_-(C) = \inf_{Q \in \mathcal{Q}} E_Q[C / B_T] represents the subhedging price and \pi_+(C) = \sup_{Q \in \mathcal{Q}} E_Q[C / B_T] the superhedging price, with B_T denoting the numeraire at maturity. The second fundamental theorem of asset pricing characterizes completeness in terms of the uniqueness of equivalent martingale measures: a market is complete if and only if there exists a unique Q \in \mathcal{Q}. This equivalence, originally established in multiperiod settings, highlights that incompleteness arises from insufficient traded assets relative to the uncertainty structure, leading to a continuum of risk-neutral valuations. Duality theory provides a powerful framework for pricing and hedging in such settings, linking superhedging costs directly to expectations under martingale measures. Specifically, the superhedging price satisfies \pi_+(C) = \sup_{Q \in \mathcal{Q}} E_Q[C / B_T], establishing an upper arbitrage bound via convex optimization over the set of measures. For utility maximization, the primal problem of maximizing expected utility E[U(X_T)] over admissible terminal wealth X_T admits a dual formulation as the infimum over a conjugate utility applied to martingale densities, ensuring no-arbitrage constraints are embedded. A seminal result in this duality is the Kramkov-Schachermayer theorem, which establishes a one-to-one correspondence between solutions to the primal utility maximization and the dual minimization problem in incomplete semimartingale markets, provided the utility function's asymptotic elasticity is strictly less than 1. This duality not only yields optimal investment strategies but also connects to infimal convolution representations of the value function, facilitating robust pricing bounds under model uncertainty.

Infinite-Horizon and Infinite-State Settings

In infinite-horizon settings, the fundamental theorem of asset pricing (FTAP) extends the no-arbitrage condition to models where trading occurs over the unbounded time interval [0, \infty). Discounted asset price processes, typically defined as S^\delta_t = S_t / \beta_t where \beta_t is a numéraire such as the risk-free asset accumulator, must satisfy uniform integrability to ensure martingale convergence and prevent long-run arbitrage opportunities. The FTAP in this context asserts that the absence of free lunch with vanishing risk (NFLVR) is equivalent to the existence of an equivalent probability measure Q under which the discounted prices \{S^\delta_t\}_{t \geq 0} are uniformly integrable martingales. This requires a consistent system of martingale measures for finite-horizon submodels, ensuring the limiting measure Q_\infty preserves the martingale property across the entire horizon. For discrete-time models with infinite horizon, Schachermayer (1994) proved that no free lunch with bounded risk (NFLBR) holds if and only if there exists a probability measure Q equivalent to the physical measure P such that the discounted asset prices are martingales under Q. This result relies on constructing a consistent family of equivalent martingale measures for all finite sub-horizons [0, n], n \in \mathbb{N}, and verifying their uniform integrability to obtain the infinite-horizon extension. The key equation involves the limiting measure Q_\infty satisfying \mathbb{E}_Q \left[ S^\delta_\infty \mid \mathcal{F}_t \right] = S^\delta_t, \quad \forall t \geq 0, where the family \{S^\delta_t\}_{t \geq 0} is uniformly integrable under Q, ensuring no asymptotic arbitrage arises from unbounded time. In continuous-time infinite-horizon models based on semimartingales, the FTAP generalizes results from Jacod and Shiryaev (1987), who established conditions for the existence of equivalent martingale measures in general stochastic process settings using characteristics of semimartingales. The theorem requires that NFLVR implies the existence of a measure Q \sim P under which discounted semimartingale prices are uniformly integrable local martingales, with uniform integrability ensuring global martingale properties over [0, \infty). Challenges in these settings include the potential for asymptotic arbitrages, where standard no-arbitrage (NA) fails but no arbitrage of the first kind (NA1)—defined as the absence of sequences of attainable claims converging in probability to a non-negative payoff strictly positive with positive probability—holds, necessitating stronger conditions like NFLVR for the FTAP to apply. For infinite-state spaces, the FTAP accommodates uncountable state spaces, such as Polish spaces, where asset prices evolve as general stochastic processes. Dalang, Morton, and Willinger (1990) showed in finite-horizon discrete-time models that no arbitrage is equivalent to the existence of an equivalent martingale measure, achieved via the Daniell-Kolmogorov extension theorem, which constructs a probability measure on the path space from consistent finite-dimensional distributions. In infinite-horizon extensions, Polish spaces ensure tightness of measure families under Prokhorov's theorem, allowing the extension of martingale measures while maintaining equivalence to P. This framework links briefly to duality in incomplete markets by providing separating hyperplanes for attainable claims, but focuses on dimensionality challenges rather than measure multiplicity. Recent developments include quantitative versions of the FTAP. For instance, Acciaio, Backhoff-Veraguas, and Pammer (2025) introduced a quantitative FTAP that provides bounds on the size of arbitrage opportunities, facilitating analysis in settings with model uncertainty.