A Priestley space is a compact topological partially ordered set that is totally order-disconnected: if $x\nleqq y$, a clopen upward-closed set contains $x$ and omits $y$. A Priestley dual space is a space of this kind constructed from the prime filters of a bounded distributive lattice.

Infinitary logic permits infinitary syntactic operations, such as infinite conjunctions, together with an interpretation of its formulas. The infinitary language $L_{\kappa ,\lambda }$ specifies the permitted conjunction and quantifier lengths.

A mathematical definition fixes the meaning of a term or symbol using previously specified concepts. A definition by a universal property can still require a mathematical proof that an object with that property exists. For example, the definition of a product in a category specifies its mapping behavior, while existence is a further question about that category.

An axiom is a sentence admitted as a starting assumption of a formal theory. An axiom schema specifies a collection of such sentences, usually one for each formula of an allowed form. A model of a first-order theory satisfies all its axioms; a relative consistency construction can show that the other axioms do not imply a particular omitted one.

Proof theory studies formal proofs and the deductive systems that produce them. It examines which conclusions follow from axioms, how proofs can be transformed, and what those transformations reveal about the chosen logic. Natural deduction organizes a proof through introduction and elimination rules for logical connectives and quantifiers.

Mean-variance optimization balances expected linear return against a positive-definite quadratic variance penalty.

Modern portfolio theory constructs portfolios by comparing expected return with variance risk and the effects of diversification.

Implied volatility is the volatility input of an option-pricing model that reproduces an observed option price. Black-Scholes implied volatility specializes this inversion to the Black-Scholes model.

The Greeks are sensitivities of a financial claim value to model inputs, expressed through partial derivatives. The option delta measures sensitivity to the underlying price; a delta hedge uses that sensitivity to cancel first-order price exposure.

The binomial options pricing model values a contingent claim on a two-outcome asset-price tree using risk-neutral probability and backward option pricing. Its underlying discrete-time binomial market distinguishes the market assumptions from the pricing procedure.

A risk-neutral measure $Q$ is a probability measure equivalent to the physical probability measure $P$, meaning they agree on which events have zero probability. For assets without intermediate payments, prices expressed in the chosen numéraire must be martingales under $Q$. For a positive bank account $B$ and a stock without dividends, this requires $S/B$ to be a martingale; with dividends, use discounted dividend gains. In an unrestricted finite-state one-period financial market, existence of such a measure is equivalent to absence of arbitrage, by the fundamental theorem of asset pricing.

Physical probability $p=P(\text{up})$ describes the modeled frequency or belief about an up move. The risk-neutral probability $q=Q(\text{up})$ is constrained by traded prices. In the worked binomial market, $q=5/8$ even if $p=1/2$ or $p=4/5$. Calling $Q$ risk-neutral does not assume that investors have no risk aversion; it changes the weights used for risk-neutral pricing.

A restriction that the initial cost of all allocated resources equals the available budget. In a portfolio with risky holdings $\theta$ and bank holdings $x$, the displayed equation is the budget identity. Barring borrowing makes $x\ge 0$, equivalently $S_0^T\theta \le w_0$; banning the bank account instead imposes equality.

A market is incomplete if some contingent claim cannot be replicated by a self-financing strategy. In a one-period finite-state market, attainable payoff vectors form the span of the traded assets' terminal payoff vectors. If this span is a proper subspace of the full payoff space, the market is incomplete. For example, two linearly independent asset-payoff vectors on three states leave at least one payoff direction unattainable. Absence of arbitrage only requires an appropriate positive state-price density; it does not imply that every claim is attainable.

Suppose a productive asset pays total consumption ${\delta }_t$, while price-taking agents value their consumption shares $p_t^i=c_t^i/{\delta }_t$ through discounted utility functions. Marginal pricing by a common state-price density ${\xi }_t$ gives $e^{-{\rho }_{i}t}{U}_i^{\prime }(p_t^i)/{\delta }_t=y_i{\xi }_t$. Writing $b(t)={\xi }_t{\delta }_t$, market clearing reduces to the scalar equation ${\sum }_i{I}_i(y_i{e}^{{\rho }_{i}t}b(t))=1$, where $I_i$ is the inverse marginal utility. Its unique positive solution is deterministic. Therefore the consumption shares are deterministic even when dividend levels are random.

With finite fundamental prices, the asset's price is $S_t={\delta }_{t}b(t{)}^{-1}{\int }_t^{\infty }b(s)ds$. Initial budget equations determine the multipliers. Identical preferences and discount rates make the shares constant and equal to initial ownership fractions. The aggregate benchmark is taken as given in the price-taking first-order conditions.

For discounted one-period asset gains $Y$, a dominated martingale measure is a probability measure $Q$ absolutely continuous with respect to the physical probability measure $P$, under which the gains have zero expectation. Its Radon-Nikodym derivative $Z$ satisfies $Z\ge 0$, ${\mathbb{E}}_{P}Z=1$ and ${\mathbb{E}}_P[ZY]=0$. An equivalent martingale measure additionally has $Z>0$ almost surely. Absolute continuity permits some positive-probability physical states to receive zero pricing probability; equivalence does not.

A competitive dividend economy specifies the joint law of total output, an information filtration, utility functions, discount factors, initial ownership shares and trading restrictions. Its derived objects are a common ex-dividend price, individually optimal consumption and share holdings satisfying ${\sum }_j{C}_t^j=d_t$ and ${\sum }_j{\theta }_t^j=1$. Each investor obeys the discrete dividend budget equation and, at an unconstrained interior optimum, marginal utility pricing in a dividend economy. A common price is required; a unique common state-price density or arbitrary contingent transfers require extra market completeness or implementability hypotheses.

A trinomial tree allows three next-step states. For the explicit log-price scheme for local-volatility pricing these are $x-h,x,x+h$, with the displayed weights. They sum to one; if nonnegative they are transition probabilities and make backward pricing a conditional average. Their first two increment moments are $bk$ and $2ak$, matching the diffusion to first order in the time step. Coefficients can depend on the node, so the tree need not have constant transition probabilities.

Credit risk is the risk of loss through a counterparty's inability or failure to meet a financial obligation, or deterioration in its credit quality. A loss model distinguishes credit default frequency from loss given default; shared exposures or systematic factors may produce dependence beyond an independent Poisson model.

Value at risk is the displayed upper loss quantile for a specified horizon and confidence level. It is distinct from a two-sided confidence interval for a model parameter and does not by itself describe the severity of losses exceeding that quantile.

A traded claim or instrument carrying a current price and future payments or resale value. In a one-period financial market, its terminal payoff is a random variable and a portfolio combines payoffs linearly through its asset holdings. A stock, a zero-coupon bond and a bank account are basic examples. Dividends require treating the full gain process rather than only resale prices.

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.