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The discrete-time fundamental theorem of asset pricing supplies a strictly positive martingale deflator $Y$. Since the maturity-$T$ bond pays one unit at $T$, its deflated price is a martingale:Division by $Y_t>0$ gives the formula.

The proposed variable satisfiesNo arbitrage and the fundamental theorem of asset pricing provide a strictly positive one-period deflator $Y$ with $\mathbb{E}(Y{P}_1)=P_0$. Part c makes such a deflator unique in a complete market, so $Z=Y>0$ almost surely.

The one-period fundamental theorem of asset pricing, applied as a separating-hyperplane theorem to the cone of attainable payoffs, gives the superhedging dualitywhere the supremum is over martingale deflators. The assumed strict inequalities make the right side strictly below ${\xi }_0$. Hence some $H$ satisfies $H\cdot P_0\le {\xi }_0$ and $H\cdot P_1\ge {\xi }_1$ almost surely.

The European call option with strike $5$ pays $2,0,0$ at terminal stock prices $7,5,3$. Its lower-node value is zero, while its upper-node value isA second step of backward option pricing givesBy the fundamental theorem of asset pricing, this discounted expectation under the unique risk-neutral probabilities is the unique no-arbitrage price.

The fundamental theorem of asset pricing says, in this discrete-time formulation, that the market has no arbitrage if and only if it admits a strictly positive martingale deflator. Under a chosen positive numeraire this is equivalent to the existence of an equivalent martingale measure for numeraire-discounted gains.

Let $Y$ be a martingale deflator for the original arbitrage-free market. Part b shows thatis a local martingale. Thus $Y$ also deflates the gains of the added asset whose price and dividend are $({\pi }^H,{\xi }^H)$. It already deflates the original $n$ assets, so it is a martingale deflator for the enlarged market. The fundamental theorem of asset pricing implies that the enlarged market has no arbitrage. This expresses the fact that adding a dynamically replicated asset cannot create an arbitrage.

The cash-discounted stock is a positive continuous local martingale, because its dynamics contain no drift. Applying Itô formula to ${\pi }_t=U(t,v_t,S_t)$, the displayed partial differential equation cancels its drift exactly, leaving another local martingale. Since $U$ is bounded, $\pi$ is in fact a true martingale.

Thus the physical measure itself is an equivalent local martingale measure relative to cash for all three traded assets. The continuous-time fundamental theorem of asset pricing rules out arbitrage, more precisely no free lunch with vanishing risk, in the usual admissible class.

In discounted terms, an arbitrage is a portfolio with zero initial value whose terminal payoff is nonnegative in every state and strictly positive with positive probability. A risk-neutral measure is a probability measure equivalent to the physical measure under which every discounted asset price is a martingale. The one-period fundamental theorem of asset pricing says that a finite one-period market is arbitrage-free exactly when it has a risk-neutral measure.

A European call option with maturity $N$ and strike $K$ paysat time $N$. Let $q$ be the risk-neutral conditional probability of the factor $1+b$. The discounted stock must be a martingale, soand thereforeBoth probabilities are strictly positive. At every node these are the unique probabilities satisfying the martingale condition, so the fundamental theorem of asset pricing gives a unique no-arbitrage price.

If exactly $n$ of the $N$ moves use the factor $1+a$, thenand there are $\left(\genfrac{}{}{0ex}{}{N}{n}\right)$ such paths. Discounted risk-neutral expectation now giveswhere

With zero interest rate, the bank account is constant. The risky asset $S$ is a continuous local martingale by assumption, while the European contingent claim priceis a true martingale by the defining property of conditional expectation. Thus the original probability measure is an equivalent local martingale measure for all traded discounted prices. The fundamental theorem of asset pricing then excludes arbitrage for admissible self-financing strategies.

Let $V_t=\mathbb{E}[P_t{P}_t^{\top }\mid {\mathcal{F}}_{t-1}]$ be positive definite. Market completeness restricts each parent atom to at most $n$ successors, while positive definiteness forces exactly $n$ independent successor price vectors. The positive martingale deflator supplied by the fundamental theorem of asset pricing then has the unique one-step ratio $Z_t=P_t^{\top }{V}_t^{-1}{P}_{t-1}>0$. Its product deflates the prices into martingales.