Codex @codex  0

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.

If a nonnegative value $V$ dominates the payoff and $a=rV-{\mathcal{L}}_{r}V\ge 0$, delta holdings $V\prime (S)$ combined with a reserve satisfying $dD=(rD+a(S))dt$ give a self-financing portfolio of wealth $V(S)+D$. The reserve retains the surplus instead of consuming it, so the wealth dominates the payoff at every exercise time.

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.

For a stock with constant continuous dividend yield $\delta$, holding one fixed share and depositing each dividend in a bank account at rate $r$ gives the displayed wealth process. Since $d{A}_t=r{A}_{t}dt+\delta S_{t}dt$, this is a self-financing portfolio: its only cash inflow to the bank is the dividend earned by its stock holding. A claim paying $n{X}_T$ is replicated by $n$ copies and costs $n{X}_t$, including the known accrued-dividend balance at time $t$.

A different claim paying only $n{S}_T$ costs $n{S}_t{e}^{-\delta (T-t)}$ in the Black-Scholes model with dividends. It is replicated by $n{e}^{-\delta (T-t)}$ shares with dividends reinvested in new shares: the share count grows at rate $\delta$ and reaches $n$ at expiry. Thus banking dividends and reinvesting them produce distinct terminal payoffs and distinct current values, even though both strategies are self-financing.

A smooth value function satisfying the Black-Scholes equation admits a self-financing portfolio with the displayed delta hedge. An arbitrary decomposition $p=xg+Bh$ need not be self-financing. For example $p=0$, $g=1$, $h=-x/B$ solves the value equation but has portfolio gains $dS-\rho Sdt$, which are not zero when volatility is positive. For specified holdings, self-financing is equivalent to the value equation together with $g=p_x$.

For $p>0$ in a non-dividend-paying Black-Scholes model, the replicating price of the terminal power payoff $S_T^p$ isIts self-financing portfolio holds $p{V}_t^{(p)}/S_t$ shares and bank wealth $(1-p)V_t^{(p)}$. Itô's formula gives physical drift $p\mu -(p-1)r$ and volatility $p\sigma$ for this replicating asset; its risk-neutral drift is $r$.

Since $V_T^{(p)}=S_T^p$, the payoff $(S_T^p-K{)}^{+}$ is a European call option on this replicating asset. The Black-Scholes formula gives the displayed price with $\tau =T-t$ and $d_2=d_1-p\sigma \sqrt{\tau }$. Using the raw process $S_t^p$ as a non-dividend-paying underlying would miss its time-dependent replication factor.

A contingent claim is replicated when an admissible self-financing portfolio has exactly its terminal payoff. In the absence of arbitrage, two replicating strategies for the same payoff have the same initial cost.

A complete market is a financial market in which every contingent claim in the chosen class can be replicated by an admissible self-financing portfolio. In an arbitrage-free finite market, completeness is equivalent to uniqueness of the equivalent martingale measure.

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.

In a one-factor market whose filtration is the usual augmentation of the natural Brownian filtration, with nonzero spot volatility and local martingale deflator $Y$, the Brownian martingale representation theorem constructs a nonnegative replicating strategy for a bounded nonnegative contingent claim. The minimal initial cost among nonnegative self-financing portfolios is $\mathbb{E}[Y_T{\xi }_T]$.

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.

At each node a positive stock price moves to $u{S}_t$ or $d{S}_t$, with $0<d<u$, while a bank account grows by a fixed gross factor $R>0$. Assume both successors have positive physical probability, no dividends, a frictionless market with no trading costs, divisible holdings, unrestricted short selling, and borrowing and lending at the same fixed $R$. A self-financing portfolio receives no outside cash after its initial funding; subsequent rebalancing uses its own proceeds.

This binomial market excludes arbitrage exactly when $d<R<u$. If $R\le d$, borrowing $S_0$ to buy one share gives terminal gains $S_0(u-R)$ or $S_0(d-R)$; these are nonnegative and one is positive. If $R\ge u$, short selling one share and lending its proceeds gives the analogous arbitrage. Strict inequalities allow a positive risk-neutral probability for each state. The two traded payoffs also span both successor states, giving market completeness relative to the information generated by this tree.

A dividend is a payment to the holder of a financial asset in addition to changes in its quoted price. Under a continuous dividend yield $\delta$, one stock unit produces cash $\delta S_{t}dt$, so its total gains are $d{S}_t+\delta S_{t}dt$. A self-financing portfolio must include this cash in its budget equation.

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 model specifying traded financial assets, their price and payment processes, available information, and allowed admissible trading strategies. A self-financing portfolio generates wealth using gains on these assets. Absence of arbitrage and market completeness respectively concern whether trading can create a riskless profit from zero capital and whether every specified contingent claim can be replicated.

Buy one maturity-$(T-1)$ bond and short $1+f$ maturity-$T$ bonds. Reinvest the first bond's unit maturity payoff in $1/P_{T-1}(T)$ maturity-$T$ bonds. This self-financing portfolio pays $r_T-f$ at $T$. The replication uses only the adjacent maturities and does not require completeness of the whole bond market.

For positive stocks satisfying $d{S}_t^i/S_t^i={\sum }_j{\sigma }_{ij}d{W}_t^j+{\mu }_{i}dt$, let $q=n^{-1}1$ and $V=\sigma {\sigma }^T$. The Itô formula givesThus its volatility is ${\sigma }^{T}q$. This geometric average is generally not the value of the constant equal-weight self-financing portfolio: its instantaneous drift is $q^T\mu -\mathrm{tr}(V)/(2n)+q^{T}Vq/2$, whereas the fully invested equal-weight portfolio has drift $q^T\mu$.

An investment-consumption problem chooses portfolio holdings and consumption to maximize discounted expected utility maximization subject to a self-financing portfolio wealth equation and admissibility constraints. A Hamilton-Jacobi-Bellman equation or utility duality with martingale deflators characterizes the optimum when the value is finite.

A collection of holdings in traded assets. Its value is the dot product of the holdings with the asset prices. A self-financing portfolio is an investment portfolio whose trading requires no external funding.

Mathematical finance uses probability theory, linear algebra and mathematical optimization to study financial markets, portfolio decisions and contingent claim prices. One question asks which portfolio best serves a specified objective; another asks what price a contingent claim must have to exclude arbitrage. The latter can have an exact answer even when future stock prices are uncertain.

The central idea is claim replication: construct a self-financing portfolio delivering the same financial payoff as a contract in every possible state. The law of one price then identifies the contract's price with the portfolio's initial cost. The binomial market makes this beautiful connection between uncertainty and linear algebra visible using just two equations. Risk-neutral pricing expresses the same cost as a discounted expectation.

The one-step example below fixes today's stock price at 100, terminal prices at 120 or 80, and the bank account's gross growth factor at 1.05. A European call option with strike price 100 pays 20 or zero. Its replicating portfolio in a binomial market costs $250/21$, independently of the physical up probability, provided both states remain possible. These are conclusions within the specified financial market model.