From No-Arbitrage to a Change of Measure
In a finite-state model, the first fundamental theorem relates no arbitrage to the existence of an equivalent martingale measure. In a continuous-time market, asset prices need not be martingales even after discounting: the appropriate condition is usually that they are local martingales under an equivalent measure. The first fundamental theorem for semimartingale markets makes this connection precise, but only after specifying the class of price processes and the allowed trading strategies.
Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{0\le t\le T},\mathbb{P})\) satisfying the usual conditions, with finite horizon \(T\). Let \(S=(S^1,\dots,S^d)\) be the vector of discounted asset prices, assumed to be càdlàg semimartingales. Discounting is with respect to a strictly positive numeraire, so the theorem concerns gains in discounted units. A strategy \(H=(H^1,\dots,H^d)\) is predictable and integrable with respect to \(S\); its cumulative gain is the semimartingale integral \(G_t=(H\cdot S)_t=\sum_{j=1}^d\int_0^t H^j_s\,dS^j_s\). These are semimartingale integrals, not Brownian Ito integrals; the latter require a progressively measurable integrand in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\).
For clarity, the Brownian integrals used in diffusion examples below are Ito integrals. Their quadratic-variation rules include \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\). A Stratonovich integral \(\int H_s\circ dB_s\) is a different integral; when the relevant covariation exists, it equals the Ito integral plus \(\tfrac12[H,B]_t\). The theorem below is about semimartingale prices and does not identify these two integrals.
This definition rules out not only an exact arbitrage but also a sequence of admissible opportunities whose downside vanishes while the limiting payoff remains nonnegative and nonzero. The lower bound in admissibility is imposed on the whole gain process, not just its terminal value. It prevents strategies that finance increasingly large potential losses by doubling down.
The First Fundamental Theorem
The theorem is a standard deep result of mathematical finance; its proof uses functional analysis and semimartingale methods beyond this tutorial. The locally bounded hypothesis is a model assumption, while admissibility is a restriction on trading strategies. Neither should be silently dropped when applying the theorem. In more general semimartingale settings, the precise dual measure condition may need a different formulation; the statement here is specifically the locally bounded version.
Equivalence means that \(\mathbb{Q}\) and \(\mathbb{P}\) have the same null events on \(\mathcal{F}_T\). It does not mean that their probabilities of non-null events agree. Nor does the theorem say that discounted prices must be true martingales: a local martingale is a process that becomes a martingale after suitable localization, and it need not be a martingale without localization. In finance, \(\mathbb{P}\) is the model’s physical probability measure, while \(\mathbb{Q}\) is a risk-neutral measure used to express the no-arbitrage condition.
Why Admissibility Matters Under an ELMM
The following supporting result explains the role of the lower bound. It does not prove the full theorem: the difficult direction of the theorem is the construction of an ELMM from NFLVR, and the equivalence with the closure condition also requires the theorem’s deeper arguments.
Proof. Put \(X_t=M_t+a\), so \(X_t\ge0\) and \(X\) is a nonnegative local martingale. Choose a localizing sequence \(\tau_n\) such that \(X_{t\wedge\tau_n}\) is a martingale and \(\tau_n\) tends to the end of the time interval almost surely. Fatou’s lemma gives \(\mathbb{E}[X_t]\le\liminf_n\mathbb{E}[X_{t\wedge\tau_n}]=\mathbb{E}[X_0]<\infty\). For \(s\le t\), conditional Fatou and the martingale property of the stopped processes give \(\mathbb{E}[X_t\mid\mathcal{F}_s]\le\liminf_n\mathbb{E}[X_{t\wedge\tau_n}\mid\mathcal{F}_s]=\liminf_n X_{s\wedge\tau_n}=X_s\). The last limit holds almost surely. Thus \(X\), and hence \(M=X-a\), is a supermartingale. \(\square\)
Proof. Under \(\mathbb{Q}\), stochastic integration of a predictable integrable strategy against a local martingale gives a local martingale, so \(G=H\cdot S\) is a local martingale starting at zero. Admissibility bounds \(G\) below by a constant, so the proposition makes it a supermartingale. If \(G_T\ge0\), then \(0\le\mathbb{E}^{\mathbb{Q}}[G_T]\le\mathbb{E}^{\mathbb{Q}}[G_0]=0\). A nonnegative random variable with expectation zero is zero almost surely. Equivalence of \(\mathbb{Q}\) and \(\mathbb{P}\) transfers that conclusion to \(\mathbb{P}\), ruling out an arbitrage of this form. \(\square\)
This argument explains why “local martingale” is the right pricing condition for the theorem, and why admissibility cannot be ignored. A local martingale gain that is bounded below is a supermartingale, but a local martingale without such a bound need not have the expectation or conditional-expectation properties needed in this argument.
Worked Example: A Deterministic Drift Creates Arbitrage
Let the single discounted asset be \(S_t=e^{at}\), \(0\le t\le T\), where \(a\ne0\) and \(T>0\). Show directly that NFLVR fails.
If \(a>0\), hold one unit of the asset: \(H_t=1\). The zero-initial-wealth gain is \(G_t=S_t-S_0=e^{at}-1\). Since \(e^{at}\ge1\) on \([0,T]\), this strategy is admissible with lower bound \(0\), and \(G_T=e^{aT}-1>0\). If \(a<0\), short one unit: \(H_t=-1\). Then \(G_t=1-e^{at}\ge0\) on \([0,T]\), and \(G_T=1-e^{aT}>0\). In either case the terminal gain is a strictly positive constant. Its constant sequence lies in \(C\), so the \(L^\infty\)-closure of \(C\) contains a nonzero nonnegative random variable. NFLVR fails.
Check: if \(a=0\), the price is constant and either constant strategy has zero gain, so this particular arbitrage disappears. The strict positivity of the terminal gain in the argument uses both \(a\ne0\) and \(T>0\).
A Diffusion Example: Finding and Checking an ELMM
Worked Example: Removing the Drift of a Discounted Geometric Brownian Price
On a filtered probability space carrying a Brownian motion \(B_t\) under \(\mathbb{P}\), let \(S_0=2\) and \(dS_t=\tfrac14 S_t\,dt+\tfrac12 S_t\,dB_t\), for \(0\le t\le2\). Find an equivalent measure under which \(S\) is a local martingale.
Set \(\theta=\mu/\sigma=(1/4)/(1/2)=1/2\), and define the density process \(Z_t=\exp(-\theta B_t-\tfrac12\theta^2t)\). Novikov’s condition is \(\mathbb{E}^{\mathbb{P}}\!\left[\exp(\tfrac12\int_0^T\theta_s^2\,ds)\right]<\infty\); here its left side is \(\exp(\tfrac12\cdot\tfrac14\cdot2)=e^{1/4}<\infty\). Thus the standard Girsanov theorem gives a probability measure \(\mathbb{Q}\) with density \(Z_2\) relative to \(\mathbb{P}\), equivalent to \(\mathbb{P}\), under which \(B_t^{\mathbb{Q}}=B_t+\tfrac12t\) is Brownian motion. In particular, \(dB_t=dB_t^{\mathbb{Q}}-\tfrac12dt\), and substituting into the SDE yields \(dS_t=\tfrac14S_tdt+\tfrac12S_t(dB_t^{\mathbb{Q}}-\tfrac12dt)=\tfrac12S_t\,dB_t^{\mathbb{Q}}\). The drift has cancelled.
Independent check. Ito’s formula for \(f(x)=\log x\), with \(f'(x)=1/x\) and \(f''(x)=-1/x^2\), gives under \(\mathbb{Q}\) \(d\log S_t=(1/S_t)(\tfrac12S_t\,dB_t^{\mathbb{Q}})+\tfrac12(-1/S_t^2)(\tfrac12S_t)^2dt=\tfrac12dB_t^{\mathbb{Q}}-\tfrac18dt\). Hence \(S_t=2\exp(\tfrac12B_t^{\mathbb{Q}}-\tfrac18t)\). Since \(B_t^{\mathbb{Q}}\sim N(0,t)\), its normal exponential moment gives \(\mathbb{E}^{\mathbb{Q}}[S_t]=2e^{-t/8}e^{(1/2)^2t/2}=2\). So in this example \(S\) is in fact a true martingale, which is stronger than the local-martingale conclusion required by the theorem.
What the Theorem Does Not Say
The theorem is an existence criterion for an equivalent local martingale measure; it is not a completeness theorem. There may be more than one such measure, and the theorem alone does not say that every contingent claim can be replicated or has a unique price. Completeness and uniqueness of martingale measures are separate questions.
It also does not assert that a model’s prices are martingales under \(\mathbb{P}\). In a diffusion model, the drift under \(\mathbb{P}\) describes the physical model, while changing to \(\mathbb{Q}\) changes the drift according to the density process. The stochastic differential equation under each measure must be interpreted with its own Brownian motion. This measure change is not a claim about strong versus weak solutions: a strong solution is constructed from a specified Brownian motion on a given probability space, whereas a weak solution allows the probability space and Brownian motion to be part of the construction. The fundamental theorem concerns the price process and an equivalent measure on the given filtered space.
A common mistake is to replace “local martingale” with “martingale” in the theorem and then infer that every admissible gain has expectation exactly zero. The valid implication used above is that a local-martingale gain bounded below is a supermartingale; the resulting expectation inequality is \(\mathbb{E}^{\mathbb{Q}}[G_T]\le G_0\), not necessarily equality. Another mistake is to check only that a chosen terminal payoff is bounded below. Admissibility requires a deterministic lower bound for the entire gain process at every time.
Worked Example: Testing the Strategy Bound at Every Time
Let \(S_t=e^{at}\) on \([0,T]\), with \(a>0\), and consider the constant strategy \(H_t=-1\). Its gain process is \(G_t=1-e^{at}\). Determine whether it is admissible and compute its lower bound.
Because \(e^{at}\) increases on \([0,T]\), \(G_t\) decreases there. Thus \(G_t\ge G_T=1-e^{aT}\) for every \(t\in[0,T]\). The number \(1-e^{aT}\) is finite and deterministic, so the strategy is admissible with \(a_0=e^{aT}-1\ge0\) in the lower-bound convention \(G_t\ge-a_0\). Its terminal gain is \(1-e^{aT}<0\), so this strategy is not an arbitrage. By contrast, holding \(H_t=1\) gives \(G_t=e^{at}-1\ge0\) and terminal gain \(e^{aT}-1>0\), the arbitrage in the previous example.
Check: the short strategy’s minimum over the full interval is attained at \(t=T\), not at \(t=0\), where the gain is zero. Checking only the initial gain would miss the required lower bound over time.
When to Apply the Theorem
Before invoking the result, check the following points.
- The prices in the theorem are discounted prices, and the horizon is finite.
- The price vector is a locally bounded semimartingale, so stochastic integration and the stated theorem apply.
- The strategies are predictable, integrable with respect to the price process, and admissible through a uniform lower bound on their gains.
- NFLVR uses closure in \(L^\infty\); it is stronger than merely ruling out one exact arbitrage strategy.
- The measure found is equivalent to \(\mathbb{P}\) and makes prices local martingales; it need not make them true martingales.
These checks identify which assumptions belong to the market model and which belong to the trading rule. A failure of local boundedness is a limitation on applying this theorem as stated. A failure of admissibility may allow strategies with uncontrolled downside. Neither issue can be repaired just by finding a convenient drift adjustment in a diffusion equation.
Check Your Understanding
Use the theorem’s precise assumptions and the distinction between local martingales and martingales in your answers.
- State the two directions of the first fundamental theorem for a locally bounded semimartingale market.
- Why does the theorem require equivalence of \(\mathbb{Q}\) and \(\mathbb{P}\), rather than merely requiring the discounted prices to be local martingales under some measure?
- Explain why a local-martingale gain bounded below is a supermartingale, and identify where the lower-bound assumption enters.
- For \(S_t=e^{at}\) with \(a<0\), which constant position gives an arbitrage, and what is its terminal gain?
- Does an ELMM guarantee that every discounted asset price is a true martingale, or that the market is complete? Explain the distinction.