Arbitrage ($V_0=0,V_T\ge 0P\text{-a.s.},P(V_T>0)>0$)

An arbitrage is an allowed self-financing portfolio with zero initial wealth, nonnegative terminal portfolio wealth almost surely under the physical probability measure $P$, and strictly positive terminal wealth with positive probability. In a finite model where every state has positive probability, nonnegativity means no loss in any state. Positive expected return alone does not establish arbitrage: losses may still occur.

For the binomial market example, suppose the European call option sells for 13 while its replicating portfolio costs $250/21$. Sell the call, buy its replicating portfolio, and lend the surplus $13-250/21=23/21$ in the bank account. The initial net cost is zero. At maturity the call obligation and portfolio cancel in either state, leaving $(21/20)(23/21)=23/20$. A price below $250/21$ permits the reversed construction, using short selling and borrowing. This is the law of one price made operational; the assumed trading permissions are essential.

A market satisfies no-arbitrage when it admits no admissible self-financing strategy with zero initial cost, nonnegative terminal wealth almost surely, and a strictly positive gain with positive probability. The admissibility restriction excludes doubling strategies. Identical attainable cash flows then have the same price.

Arbitrage pricing theory relates expected returns to systematic factor loadings and their risk premiums. With idiosyncratic noise, a diversification or asymptotic argument needs additional assumptions. With no idiosyncratic noise and spanned factors, the formula is an exact consequence of law of one price: portfolios matching factor exposures can differ only by a constant payoff, which absence of arbitrage forces to be zero.

For centered residual random variables ${ϵ}_i$ with covariance matrix $\Gamma$, the residual of a portfolio has variance $w^T\Gamma w$. Pairwise uncorrelated residuals with uniformly bounded variances and weights $|w_i|\le W/n$ give $\mathrm{Var}(w^Tϵ)\le s^2{W}^2/n$, hence convergence to zero in mean-square convergence. For nonnegative weights summing to one, the sharper bound is $s^{2}W/n$. With correlated residuals, $|w_i|\le W/n$ and ${\sum }_{i,j}|{\Gamma }_{ij}|=o(n^2)$ also suffice. Individual bounded variances alone do not: if every residual equals the same nondegenerate centered $Z$, every fully invested portfolio retains residual $Z$. A one-sided upper bound on signed weights is insufficient even for uncorrelated residuals because it allows a fixed large short selling.

Suppose asset returns have the exact representation $r=a+Bf$, allowing unrestricted long and short portfolios. A position $v$ with $1^{T}v=0$ and $B^{T}v=0$ costs zero and has constant financial payoff $a^{T}v$. Absence of arbitrage therefore forces $a^{T}v=0$, since either sign of $v$ is allowed. The orthogonal complement identity gives $a\in \mathrm{span}(1,\mathrm{col}B)$. Taking expectations proves the displayed pricing relation, with $\lambda$ equal to the coefficients of $B$ in $a$ plus $\mathbb{E}f$. When $[1B]$ has full column rank, ${\lambda }_0$ is the return of a unit-cost zero-exposure portfolio and ${\lambda }_j$ is the expected return of a zero-cost portfolio with exposure one to factor $j$ and zero to the others. Without that rank condition the coefficients need not be unique, and a traded zero-exposure portfolio need not exist.

For returns $r_i=a_i+{\sum }_j{b}_{ji}{f}_j$, fund a risky position $v$ by a risk-free asset. Its zero-cost payoff is $(a-R1{)}^{T}v+{\sum }_j(Bv{)}_j{f}_j$. If $Bv=0$, a nonzero constant term would create an arbitrage after choosing the sign of $v$. Hence $a-R1$ annihilates $\ker B$ and lies in the row span of $B$. Taking expectations gives exact factor-premium pricing. Full row rank makes the coefficients unique; if exposures have smaller rank, only the effective spanned factors are identified.

If a constant-coefficient investment model has a vector $v$ with ${\sigma }^{T}v=0$ and $v^T\mu >0$, investment in direction $v$ produces deterministic positive gains without initial capital. For $A=\sigma {\sigma }^T$, absence of such a direction is equivalent to $\mu \in \mathrm{Range}A$, because $\ker A=\ker {\sigma }^T$ and $\mathrm{Range}A=(\ker A{)}^{\perp }$. Scaling this direction makes any unbounded increasing terminal utility function unbounded on a positive horizon. Thus a finite investment value cannot be inferred from a volatility matrix which is allowed to be singular without this additional condition.

The law of one price requires traded assets or portfolios with the same terminal payoff to have the same earlier price. In a model where every discounted traded price is a true martingale under a common equivalent measure, it follows by taking the conditional expectation of that terminal payoff. With only local martingale prices and a lower-bound restriction on admissible trading strategies, it can fail: the seemingly profitable short position may have unbounded possible losses before maturity.

Fix $T>0$, let $X$ be Brownian motion, and let $\tau =\inf \{u:X_u=-1\}$. Set $q(t)=t/(T-t)$ for $t<T$ andThe hitting time is finite almost surely, so each price path reaches one before $T$ and stays there. Stopping additionally on hitting $n+2$ makes the stock price a bounded martingale; these stopping times increase to $T$. Hence $S$ is a positive local martingale, and the original probability measure is an equivalent local martingale measure, excluding admissible arbitrage. Nevertheless $S_T=B_T$ while $S_0=2\ne 1=B_0$. The zero-cost short-stock/long-two-cash portfolio has terminal gain one but wealth $2-S_t$, which has no deterministic lower bound. It is not admissible, resolving the apparent contradiction.

An investment-consumption arbitrage starts with zero capital, has nonnegative consumption at every date, and has strictly positive consumption at some date with positive probability.

A pure-investment arbitrage has zero initial cost, no intervening consumption, and a nonnegative terminal payoff that is strictly positive with positive probability.

Consumption in a self-financing portfolio model is wealth withdrawn for use outside the portfolio. With no initial capital, a one-period position of nonpositive initial cost and nonnegative terminal payoff permits nonnegative consumption at both dates.

A terminal-consumption arbitrage has zero initial cost and a nonnegative terminal payoff that is strictly positive with positive probability. Allowing negative initial cost gives the broader one-period notion in which some consumption may occur immediately.

If the terminal price vector is normal with covariance matrix $V$, every portfolio outside $\ker V$ has a terminal value with full support on the real line. Arbitrage can therefore arise only from portfolios in $\ker V$, whose terminal values are deterministic.

A portfolio is self-financing when every change in its asset holdings is paid for entirely by selling or buying assets within the portfolio, with no external cash added or removed.

For holdings with value $V_t={\varphi }_t{S}_t+{\psi }_t{B}_t$, the condition says that all gains and losses arise from the held assets, with no external cash transfer. The stochastic-integrability and admissibility requirements on the holdings are part of a valid trading strategy. Differentiating the value identity alone does not establish self-financing, since changes of holdings can contribute additional terms.

In the Black-Scholes model, take $B_t=e^{-\rho (T-t)}$ as a zero-coupon bond numéraire. Holding the fixed stock value $\theta B_t$ requires $g_t=\theta B_t/S_t$ shares. The self-financing conditions for smooth stock and bond holdings giveunits of the bond. Since $V_t=g_t{S}_t+h_t{B}_t$, this yields the displayed wealth. Under the risk-neutral measure, $V_t/B_t=w_0/B_0+\theta \sigma W_t^Q$: the discounted gain is a constant multiple of Brownian motion.

A constant-proportion portfolio holds a fixed fraction $\gamma$ of its portfolio wealth in the stock. In the Black-Scholes model with bank rate $\rho$, initial wealth $V_0$, and no consumption, its value isThe stock holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma )V_t$ is invested in the bank account. The Itô formula verifies $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$ when the physical stock drift is $\mu$.

For $p(x,t)=xg(x,t)+B_{t}h(x,t)$ and deterministic $d{B}_t=\rho B_{t}dt$, Itô formula shows that a self-financing portfolio must have $p_x=g$. This is equivalent to $x{g}_x+B_t{h}_x=0$. Differentiating it gives $p_{xx}=g_x$, so the remaining gain identity is $x{g}_t+B_t{h}_t+{\sigma }^2{x}^2{g}_x/2=0$. Conversely these two identities give $dp=gdS+hdB$. A value PDE alone does not verify arbitrary prescribed holdings.

If $B$ is a positive riskless numéraire of finite variation, $S$ the ex-dividend price vector and $D$ its cash-dividend rate, a self-financing portfolio with risky holdings $\pi$ satisfies $d(X/B)=\pi \cdot dY$. This follows by differentiating $X/B$, substituting $dX=\varphi dB+\pi \cdot (dS+Ddt)$ and using $X=\varphi B+\pi \cdot S$. If $Y$ is a martingale under an equivalent measure, discounted wealth is a local martingale. A deterministic lower bound on discounted wealth makes it a supermartingale, ruling out arbitrage. The cumulative dividends here are discounted at their own payment times.

For an integrable discrete trading strategy under an equivalent martingale measure, discounted portfolio wealth plus accumulated discounted strategy payments is a martingale. Conditioning its terminal value gives the displayed identity. A formula involving future payments alone requires terminal liquidation, $V_n=0$, or inclusion of that terminal value in the final payment. For an attainable claim paid only at maturity, liquidation yields its usual conditional discounted expected value.

A share paying a continuous dividend rate ${\delta }_t$ produces dollar gains $d{S}_t+{\delta }_{t}dt$. If its price follows a geometric Brownian motion with drift ${\mu }_{\mathrm{price}}$ and its dividend yield ${\delta }_t/S_t$ is constant, its drift in an investment-consumption problem is their sum. The self-financing portfolio equation uses this total return, even if the agent consumes the dividends rather than reinvesting them. Omitting the dividend yield changes both the optimal stock fraction and the equilibrium interest rate.

A self-financing portfolio is admissible when its wealth satisfies the prescribed lower-bound restriction, typically a deterministic bound below. Nonnegative wealth is sufficient. This condition permits supermartingale bounds on discounted gains and excludes doubling strategies.

Relative to a positive numéraire, a predictable self-financing strategy with risky holdings ${\theta }_t$ and discounted risky prices $X_t$ has discounted valueThus its gains are the discrete stochastic integral of its holdings against discounted price increments.

An integrable predictable strategy whose discounted gains ${\varphi }_n\Delta {\tilde{S}}_n$ are integrable preserves the martingale property of discounted prices in finite discrete time. Bounded holdings suffice. Without integrability the claim is false: let ${\tilde{S}}_0=3/2$, ${\tilde{S}}_1=1+U$, ${\tilde{S}}_2=1+U+\epsilon /2$, where $U$ is uniform on $(0,1)$ and $\epsilon$ is an independent fair sign. Holding zero stock first and $1/U$ shares in the second period, funded by the bank, gives zero-initial-cost terminal wealth $\epsilon /(2U)$, which is not integrable. Replication pricing requires the corresponding true-martingale condition, not merely formal self-financing.