Mathematical finance

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

δ

, one stock unit produces cash

δ S_{t} d t

, so its total gains are

d S_{t} + δ S_{t} d t

. A self-financing portfolio must include this cash in its budget equation.

Financial volatility ( $σ$ )

Financial volatility quantifies variability of asset returns. In the Black-Scholes model,

σ

is the coefficient of the Brownian increment in the stock return

d S_{t} / S_{t} = μ d t + σ d W_{t}

, so its instantaneous log-return variance rate is

σ^{2}

. For deterministic time-varying volatility, the variance over an interval is the integral of that squared coefficient.

Drawdown (economics, $D_{t} = max_{s \leq t} y_{s} - y_{t}$ )

The absolute decline from a previous running peak of a cumulative-value process. For positive peak values a relative decline can also be defined, but that is a different unit convention. Flat intervals do not end a drawdown, and a brief recovery that stays below the previous peak does not necessarily end it either. A maximum consecutive strictly negative run is therefore not generally a maximum drawdown.

Finite-horizon drawdown of a two-sign-memory process ( $D_{m a x} (N) = 2 ⌊ N /2 ⌋ + 2 (N \geq 1)$ )

For increments

X_{t} = a_{t - 2} a_{t - 1} + a_{t}

with freely admissible starting sign history, the worst absolute financial drawdown over

N

steps is the displayed value. A four-state recursion

d_{n + 1} (b, c) = min_{a = \pm 1} [d_{n} (a, b) + ab + c]

computes the minimum cumulative increment, with zero initial costs. The minimum is attained by nonpositive increments:

(- 2, 0)^{k}, - 2

for odd

N = 2 k + 1

, and

(- 2, 0)^{k - 1}, - 2, - 2

for even

N = 2 k

. The horizon-free bound is infinite. At most two strictly negative increments can be consecutive, but confusing that losing-run bound with drawdown would be incorrect.

Discounted integral ( $\int_{0}^{\infty} f (t + s) e^{- rs} d s$ )

An integral of future flows weighted by a specified discount factor, such as

\int_{0}^{\infty} f (t + s) e^{- rs} d s

. For an exponentially growing flow, convergence requires the discount rate to exceed its growth rate.

Option pricing

Determining the value of a European option or American option from a specified financial market model and its permitted exercise rules. Risk-neutral pricing, replicating strategies and optimal stopping address different parts of this problem.

Financial return ( $R = (V_{1} - V_{0} + D) / V_{0}$ )

For initial asset value

V_{0} > 0

, final value

V_{1}

, and intervening cash distribution

D

, the net financial return is

(V_{1} - V_{0} + D) / V_{0}

. Its gross return is

1 + R = (V_{1} + D) / V_{0}

. In a one-period portfolio with initial-value weights

w_{i}

summing to one, the net return is

\sum_{i} w_{i} R_{i}

. The expected return is its expected value; a realized return need not equal it.

Financial market

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.

Stylized facts of financial returns

Recurring empirical regularities, such as heavy tails, weak return autocorrelation and volatility clustering, observed over specified datasets and sampling scales. They are modelling targets rather than exact universal laws. In particular, absence of linear autocorrelation does not imply independence or absence of nonlinear prediction.

Frictionless market

A financial market model that omits trading frictions such as transaction costs and a difference between buying and selling prices. The exact idealization must still specify permitted holdings, short selling, and borrowing and lending terms; the worked binomial market allows divisible holdings and unrestricted borrowing at the bank growth factor.

Financial asset

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.

Strike price ( $K$ )

The fixed price parameter at which an option permits an underlying trade. A European call option pays

(S_{T} - K)^{+}

at maturity; a European put option pays

(K - S_{T})^{+}

. Specifying

K

therefore determines how terminal prices translate into the financial payoff.

Credit derivative

A credit derivative transfers exposure to deterioration of credit quality or a specified credit event of a reference borrower without necessarily transferring the underlying loan or bond. Credit default swaps transfer a defined loss exposure; total return swaps transfer a wider asset return. Funded structures such as credit-linked notes combine the exposure with an investor's initial principal. Documentation specifies the reference entity, obligations, credit events and settlement; a hedge can leave counterparty credit risk and mismatch with the original exposure.

Synthetic CDO

A synthetic collateralized debt obligation creates portfolio credit exposure using credit default swaps rather than necessarily owning a pool of cash loans or bonds. Losses are allocated to tranches with different attachment and exhaustion levels. A tranche can be funded, unfunded or partly funded depending on structure. Individual default probabilities alone do not price it: dependence among defaults and recovery outcomes changes the portfolio loss distribution.

Credit tranche loss function ( $ℓ_{A, D} (L) = min ((L - A)^{+}, D - A)$ )

For aggregate monetary loss

L

and monetary attachment/exhaustion levels

A < D

, this is the loss allocated to the tranche. It is zero until attachment, rises with losses between

A

and

D

, and is capped at

D - A

. Expected tranche losses and surviving tranche notional determine protection and premium legs. For basket products whose payoff occurs at a first default, the joint default-time distribution matters as well as the final loss distribution.

Credit-linked note

A funded note whose coupons or principal depend on specified credit events of a reference entity or portfolio. In a simple structure, the investor supplies principal and receives enhanced coupon for economically selling credit protection; a reference default reduces principal repayment. The investor may also face the note issuer's own credit risk. This is not automatically the same exposure as directly holding the reference borrower's bond.

Total return swap ( $asset income and price change \leftrightarrow funding leg$ )

The total-return receiver obtains a reference asset's income and price gains, pays its price losses, and pays an agreed funding leg to the total-return payer. This supplies economic exposure without buying the reference asset outright. The transfer includes ordinary market-price risk as well as credit risk, distinguishing it from a credit default swap. Both parties can have counterparty credit risk as the swap's replacement value changes.

Credit default swap ( $premium payments \leftrightarrow N (1 - R) at default$ )

The protection buyer pays a contractual premium in exchange for compensation upon a defined credit event of the reference entity. In a simple cash-settled contract of notional

N

, a default loss with fractional recovery

R

gives payment

N (1 - R)

. Physical settlement instead exchanges an eligible defaulted obligation for par. Payments, accrued premium, maturity and eligible obligations depend on the contract. The protection buyer need not own the reference debt; the seller earns premium while taking the specified credit risk.

First-to-default swap ( $τ_{first} = min_{i} τ_{i}$ )

A basket credit contract whose protection payment is triggered by the first specified default among its reference entities. The loss and settlement terms refer to the defaulting name according to the contract. Premium generally stops at that first default or maturity. Pricing requires the joint default-time distribution; separate marginal default probabilities alone do not determine the first-default law.

Continuous-premium credit-default swap spread ( $s = (1 - R) \frac{\int _{0}^{T} B ( t ) λ ( t ) G ( t ) d t}{\int _{0}^{T} B ( t ) G ( t ) d t}$ )

Assume deterministic discount factors, deterministic risk-neutral default intensity

λ

, fixed fractional recovery

R

, and continuous premium until default or maturity. Survival is

G (t) = exp (- \int_{0}^{t} λ (u) d u)

. Expected discounted protection is

N (1 - R) \int B λ G

, while premium is

N s \int BG

. Equating them gives the par spread. Constant intensity gives

s = (1 - R) λ

. Discrete premiums, default accrual, stochastic recovery and counterparty losses require changing the corresponding cash-flow expectations.

Risk-free asset ( $R = 1/ B$ )

For a fixed horizon, a risk-free asset has a deterministic terminal payoff. If the price of a unit terminal payoff is

B > 0

, its gross return is

R = 1/ B

. In a finite state market, the sum of the Arrow state prices is

B

. Risk-free does not mean that a longer-maturity bond has a deterministic resale price before its maturity.

Value at risk ( $VaR_{p} (L) = in f {ℓ : P (L \leq ℓ) \geq p}$ )

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.

Compound-Poisson annual loss quantile ( $VaR_{p} (L) \approx λ μ + Φ^{- 1} (p) λ (σ^{2} + μ^{2})$ )

Let annual losses be

L = \sum_{i = 1}^{N} X_{i}

with

N \sim Poisson (λ)

independent of identically distributed nonnegative loss given default amounts of mean

μ

and variance

σ^{2}

. Conditional expectation and variance give

E L = λ μ

and

Var L = λ (σ^{2} + μ^{2})

. A many-event normal approximation yields the displayed quantile; the actual loss distribution has a zero atom and depends on the entire severity law. For rare events or extreme tails, the moment-based normal approximation need not be accurate.

Credit risk

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.

Merton model ( $D_{T} = min (V_{T}, F), E_{T} = (V_{T} - F)^{+}$ )

In the basic structural model, the firm's asset value

V

follows geometric Brownian motion with constant volatility and a single debt obligation of face value

F

matures at

T

. Default at maturity occurs when

V_{T} < F

. Equity is a call on firm assets; debt is the risk-free face claim minus a put on those assets, since

min (V_{T}, F) = F - (F - V_{T})^{+}

. This links debt valuation to asset value, volatility and capital structure. The basic model does not permit default before maturity; richer structural models add a default barrier or more detailed obligations.

Counterparty credit risk

The risk that a trading counterparty defaults while owing a positive replacement value or settlement amount. It differs from the reference borrower's default exposure in a credit default swap. A credit hedge may fail precisely when its protection seller is distressed. Collateral, netting and margining change exposures and loss amounts; they do not turn every mismatched hedge into a risk-free claim.

Credit spread ( $s = y_{credit} - y_{benchmark}$ )

A yield difference between a credit-risky instrument and a chosen benchmark, with maturity and other terms matched as appropriate. The benchmark and quotation convention must be specified. The spread may compensate for default losses, default-risk premiums, liquidity and other effects; it is not generally identical to a physical default probability or loss rate.

Credit spread option ( $N A_{T} (s_{T} - K)^{+}$ )

An option with payoff linked to widening or narrowing of a specified credit spread. A simple spread-widening payoff has the displayed form, where

A_{T}

is an annuity or settlement scale. This transfers spread-mark-to-market risk and need not require an actual default event. Pricing depends on spread volatility, discounting, default treatment and the settlement convention.

Credit rating

A credit rating is an ordinal assessment of an issuer's or obligation's creditworthiness. Agency scales group credits into broad investment-grade and speculative-grade classes, with default categories and rating transitions. A rating is not itself a calibrated default probability, and identical ratings do not establish independence or constant intensities of defaults.

Trinomial tree ( $p_{d} = ka / h^{2} - kb / (2 h), p_{m} = 1 - 2 ka / h^{2}, p_{u} = ka / h^{2} + kb / (2 h)$ )

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

2 ak

, 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.

Competitive equilibrium with one productive asset

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} θ_{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.

Dominated martingale measure ( $Q ≪ P, E_{Q} Y = 0$ )

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 \geq 0

E_{P} Z = 1

and

E_{P} [Z Y] = 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.

Incomplete market

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.

Budget constraint ( $S_{0}^{T} θ + x = w_{0}$ )

A restriction that the initial cost of all allocated resources equals the available budget. In a portfolio with risky holdings

θ

and bank holdings

x

, the displayed equation is the budget identity. Barring borrowing makes

x \geq 0

, equivalently

S_{0}^{T} θ \leq w_{0}

; banning the bank account instead imposes equality.

Discrete dividend budget equation ( $C_{t} + S_{t} θ_{t + 1} = (S_{t} + d_{t}) θ_{t}$ )

When a productive stock pays

d_{t}

before trading at ex-dividend price

S_{t}

, an investor entering with

θ_{t}

shares receives

d_{t} θ_{t}

and sells or buys shares to leave with

θ_{t + 1}

. The displayed budget constraint accounts for the available consumption and the cost of the new holding. Here

θ_{t}

is known before date

t

, while

C_{t}

and

θ_{t + 1}

may depend on the information revealed at that date.

Arithmetic stock model with constant volatility ( $d S_{t} = μ d t + σ d W_{t}$ )

This additive price model has constant absolute volatility and can take negative stock values. With constant cash interest rate

r

, its market price of risk is

(μ - r S_{t}) / σ

. Under an equivalent risk-neutral measure, its drift becomes

r S_{t}

, so its conditional law is Gaussian with mean

e^{r (T - t)} S_{t}

and variance

σ^{2} (e^{2 r (T - t)} - 1) / (2 r)

. It is distinct from a multiplicative Black–Scholes diffusion.

Call price in an arithmetic stock model with interest ( $C = (s - K e^{- r τ}) Φ (d) + ν ϕ (d)$ )

Let

ν = σ (1 - e^{- 2 r τ}) / (2 r)

and

d = (s - K e^{- r τ}) / ν

, where

τ = T - t

. The displayed discounted Gaussian positive-part expectation prices a call in the arithmetic stock model with constant volatility. Its delta is

Φ (d)

. The stock holding

C_{s}

and bank-account holding

(C - s C_{s}) / B

replicate the payoff with nonnegative wealth. A nonnegative discounted-wealth supermartingale bound proves this is the least replication capital.

Delta bound in an arithmetic stock model ( $0 \leq Δ = Φ (d) \leq 1$ )

The delta of the call price in an arithmetic stock model with interest is the normal distribution function at its standardized discounted moneyness. It lies strictly between zero and one before maturity. Its maturity limit is the call payoff derivative away from the strike, an exceptional event of probability zero under the Gaussian pricing law.

Local volatility ( $σ (t, s)$ )

A local volatility model uses a deterministic function of current time and spot as its diffusion volatility,

σ (t, S_{t})

. It can reproduce a surface of European call option prices through the Dupire equation.

Local volatility model ( $d S_{t} = S_{t} (r d t + σ (t, S_{t}) d W_{t})$ )

The stock diffusion

d S_{t} = S_{t} (r d t + σ (t, S_{t}) d W_{t})

has spot-dependent local volatility. With positive volatility and the Brownian filtration, Brownian martingale representation theorem converts discounted payoff martingales into stock gains.

Exponential claim in a driftless square-root model ( $V (t, S) = exp (\frac{v S}{1 - v ( T - t ) /2})$ )

In the zero-interest local volatility model

d S_{t} = S_{t} d W_{t}^{Q}

, the risk-neutral pricing value of the European contingent claim

e^{v S_{T}}

, for

v > 0

and

v (T - t) < 2

, is

V (t, S_{t}) = exp (\frac{v S _{t}}{1 - v ( T - t ) /2}) .

The compound Poisson transition law of a driftless square-root diffusion proves finiteness and this conditional-expectation formula. Alternatively, substituting

V (t, S) = e^{a (t) S + b (t)}

into the pricing equation for a local volatility model gives

a^{'} + a^{2} /2 = 0

and

b^{'} = 0

, with terminal values

a (T) = v, b (T) = 0

. Hence

a (t) = v / [1 - v (T - t) /2]

and

b (t) = 0

. The conditional-expectation calculation verifies that this formal solution is the price of the unbounded claim. The delta hedge holds

a (t) V

stocks and

(1 - a (t) S_{t}) V / B_{t}

bank units.

Pricing equation for a local volatility model ( $V_{t} + r S V_{S} + \frac{1}{2} σ (S)^{2} S^{2} V_{SS} = r V$ )

For a local volatility model

d S_{t} = S_{t} (r d t + σ (S_{t}) d W_{t}^{Q})

with bank account

B_{t} = B_{0} e^{r t}

, a nonnegative classical solution

V

V_{t} + r S V_{S} + \frac{1}{2} σ (S)^{2} S^{2} V_{SS} = r V, V (T, S) = g (S),

defines a possible price of a European contingent claim. Indeed, Itô formula gives

d (V (t, S_{t}) / B_{t}) = B_{t}^{- 1} σ (S_{t}) S_{t} V_{S} (t, S_{t}) d W_{t}^{Q}

, a local martingale. Thus the same equivalent local martingale measure works for the market augmented by this price. For an admissible trading strategy whose discounted wealth is bounded below, the discounted gains are a local martingale bounded below, hence a supermartingale by localization and the Fatou lemma. A zero-cost terminal gain that is nonnegative and strictly positive with positive probability would then contradict the supermartingale expectation inequality. This proves the absence of arbitrage for such strategies. If

V

is bounded on the finite time interval, its discounted value is a bounded martingale and consequently

V (t, S_{t}) = B_{t} E_{Q} [g (S_{T}) / B_{T} ∣ F_{t}]

. Nonnegativity alone guarantees the local martingale calculation, not this latter equality for every possible solution.

Explicit log-price scheme for local-volatility pricing ( $u_{t} + a u_{xx} + b u_{x} = 0, a = σ^{2} /2, b = r - σ^{2} /2$ )

With

x = lo g S

and

u = e^{r (T - t)} V (e^{x}, t)

, a local volatility pricing equation becomes the displayed backward equation. Central finite differences and the explicit Euler method give

u_{i}^{j - 1} = u_{i}^{j} + k [a_{i}^{j} D_{xx} u_{i}^{j} + b_{i}^{j} D_{x} u_{i}^{j}]

. Nonnegative weights require

k σ_{i}^{2} / h^{2} \leq 1

and

∣ b_{i} ∣ h \leq σ_{i}^{2}

. Under these conditions the update is a convex combination, yielding maximum-norm stability. With smooth coefficients, positive diffusion, fixed positive remaining maturity and suitable consistent boundary treatment, its ordinary error is

O (k + h^{2})

. Volatility bounds alone do not imply all this regularity or a Greek error expansion.

Finite-difference option Greeks ( $Δ = e^{- r (T - t) - x} u_{x}, Γ = e^{- r (T - t) - 2 x} (u_{xx} - u_{x})$ )

The logarithmic coordinate requires the displayed chain-rule factors for option delta and option gamma. If only a value error bound

∥ e_{h} ∥_{\infty} = O (h^{2})

is known, central differentiation gives

D_{x} e_{h} = O (h)

and

D_{xx} e_{h} = O (1)

. Thus first and second Greek convergence do not follow with the same order from a value bound alone. With a smooth spatial error expansion

e_{h} = h^{2} E + o (h^{2})

and corresponding derivative control, both Greeks can instead retain second-order convergence away from a nonsmooth payoff or boundary. Refinement and Greek-specific regularity are essential to distinguish these situations.

Delta replication from a local-volatility pricing equation ( $h^{S} = V_{S}, h^{B} = (V - S V_{S}) / B$ )

V

satisfies the pricing equation for a local volatility model, hold

h_{t}^{S} = V_{S} (t, S_{t})

stocks and

h_{t}^{B} = [V (t, S_{t}) - S_{t} V_{S} (t, S_{t})] / B_{t}

units of the bank account. Their value is

V (t, S_{t})

. The pricing equation for a local volatility model and Itô formula give

d V (t, S_{t}) = r V (t, S_{t}) d t + σ (S_{t}) S_{t} V_{S} (t, S_{t}) d W_{t}^{Q} = h_{t}^{B} d B_{t} + h_{t}^{S} d S_{t} .

Thus this is a self-financing strategy, with terminal value

g (S_{T})

. It is a delta hedge, and nonnegative

V

makes its wealth an admissible trading strategy. The stock and bank wealth amounts are

S_{t} V_{S}

and

V - S_{t} V_{S}

, rather than the asset-unit holdings.

Dupire equation ( $C_{T} = \frac{1}{2} K^{2} σ (T, K)^{2} C_{KK} - rK C_{K}$ )

For a non-dividend-paying stock and constant interest rate

r

, discounted call prices satisfy

C_{T} = \frac{1}{2} K^{2} σ (T, K)^{2} C_{KK} - rK C_{K}

. Strike differentiation recovers the discounted terminal density as

C_{KK}

; differentiating the discounted payoff identity supplies the maturity derivative. Put-call parity implies the same equation for put prices.

Local volatility recovery from call prices ( $σ^{2} = 2 (C_{T} + rK C_{K}) / (K^{2} C_{KK})$ )

Where

C_{KK} > 0

, rearranging the Dupire equation gives

σ (T, K)^{2} = 2 (C_{T} + rK C_{K}) / (K^{2} C_{KK})

. The strike curvature represents discounted density, while the adjusted maturity derivative gives the local diffusion contribution.

Brownian representation replication in a local volatility market ( $π_{t} = h_{t} / (S_{t} σ (t, S_{t}))$ )

For a square-integrable discounted claim martingale

d M = h d W

and discounted stock

d S = S σ d W

, choose stock holdings

h / (S σ)

. The remaining wealth lies in the bond. A nonnegative claim makes this replication admissible; discounted admissible wealth is a supermartingale, establishing the minimal initial cost.

Credit default ( $τ$ )

Failure to meet a financial obligation, modeled by a random default time. A zero-recovery default model sets the affected asset value to zero after this time.

Loss given default ( $LGD$ )

Loss given default is the loss conditional on a default, measured as an amount or as a fraction of exposure under the chosen convention. In an independent marked default model the loss amounts are independent marks attached to the events of a Poisson process. Their raw second moment, not just their variance, controls the aggregate compound Poisson distribution variance.

Default time ( $τ$ )

The random time at which credit default occurs. In an observed default model it is a stopping time for the market filtration.

Default intensity ( $G (t) = exp (- \int_{0}^{t} λ (u) d u)$ )

In a deterministic-intensity model,

λ (t)

is the instantaneous conditional default rate given survival and

G (t)

is the survival probability. Risk-neutral intensity is calibrated for pricing and can differ from the physical intensity used for forecasting losses. More general stochastic-intensity models condition on the underlying filtration and require joint expectations with discounting and recovery.

Investment portfolio ( $H = (H^{1}, \dots, H^{n})$ )

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.

Portfolio diversification

Spreading a portfolio across assets can reduce risk that is not shared by them. For centered asset risks with covariance matrix

Γ

, the variance of the weighted risk is

w^{T} Γ w

. Equal weights across

n

independent risks with common variance

s^{2}

give variance

s^{2} / n

, whereas equal weights across risks that are all the same random variable leave variance

s^{2}

. Thus small individual weights do not by themselves remove common risk; the covariance criterion for diversification of factor residuals identifies sufficient dependence and weight conditions.

Short (finance)

Selling a borrowed asset and later buying it back to return it. A negative portfolio weight represents a short position. The position benefits from falling prices and loses from rising prices; its sale proceeds may be invested in the bank account. Unrestricted short positions are a modelling assumption, not a guarantee of unlimited practical borrowing.

Expected return ( $E R$ )

The expected value of a random financial return

R

. For asset return vector

R

with mean

μ

, a portfolio of weights

w

has expected return

w^{T} μ

by linearity of expectation. This measures average payoff growth rather than a guaranteed realized gain.

Portfolio wealth ( $w_{t}$ )

The value of an investment portfolio, obtained by summing each holding times its current asset price. Consumption removes money from portfolio wealth; a self-financing portfolio changes its value only through asset gains. Nonnegative portfolio wealth is a common admissibility restriction in an investment-consumption problem.

Brownian portfolio exposures ( $d w = \dots d t + \sum_{j} y_{j} d W^{j}$ )

Brownian portfolio exposures are the coefficients

y_{j}

of independent Brownian motions in the wealth equation. An invertible asset volatility matrix lets one optimize directly over these exposures. The drift risk premium is then the dot product of exposures with the market price of risk vector.

Sharpe ratio ( $S = (E R - r) / σ_{R}$ )

The expected excess return of a portfolio divided by its return standard deviation. Under the capital asset pricing model, the ratio for one asset equals its correlation with the market times the market's ratio.

Market portfolio

In a one-period normal-return model with common beliefs and unrestricted exponential utility investors, their risky holdings are proportional to

V^{- 1} b

, where

b

is the expected excess-gain vector and

V

its positive definite covariance matrix. The aggregate risky holding defines the common market portfolio.

Sign of the normalized tangency portfolio ( $w = z / (1^{T} z), z = Σ^{- 1} (μ - r 1)$ )

Assume a positive definite covariance matrix and nonzero excess-mean vector

d = μ - r 1

. The Cauchy-Schwarz inequality gives

∣ d^{T} x ∣ \leq d^{T} Σ^{- 1} d x^{T} Σ x

, with the positive-premium efficient direction

x = cz

c > 0

. If

s = 1^{T} z \neq = 0

, its normalized risky portfolio has Sharpe ratio

sgn (s) d^{T} Σ^{- 1} d

. Thus negative normalization reverses the efficient orientation. If

s = 0

, the efficient risky direction is self-funded and cannot be normalized to a unit risky budget.

Capital market line ( $m = r + θ σ$ )

In mean-variance optimization with unrestricted borrowing/lending, the upper efficient ray through the risk-free point has slope

θ = (μ - r 1)^{T} Σ^{- 1} (μ - r 1)

. If

1^{T} Σ^{- 1} (μ - r 1) > 0

, it touches the risky efficient frontier at the positive-premium normalized market portfolio. The Sharpe ratio there equals the slope. If this normalization is negative, the normalized tangency portfolio instead has negative excess return; the upper ray uses a short position in that portfolio and is not tangent to the upper risky branch.

Capital asset pricing model

The expected excess return obeys

E (R_{i} - r) = β_{i} E (R_{M} - r)

in the mean-variance market model. The beta of an asset is a covariance-to-market-variance ratio.

Beta of an asset ( $β_{i}$ )

For a nondegenerate market return

R_{M}

β_{i} = Cov (R_{i}, R_{M}) / Var (R_{M})

. It measures the asset's market-related exposure and enters the capital asset pricing model.

Stock

A stock is a traded ownership interest in a company. In a mathematical finance model, its price is a stochastic process and a self-financing portfolio may hold positive or negative numbers of its shares.

Log-mean-reverting price ( $d S_{t} = α (μ - lo g S_{t}) S_{t} d t + σ S_{t} d W_{t}$ )

For

α > 0

σ > 0

, this positive stochastic process is the exponential of an Ornstein-Uhlenbeck process. The Itô formula gives

d lo g S_{t} = α (μ - lo g S_{t}) d t + σ d W_{t}, μ = μ - \frac{σ ^{2}}{2 α} .

The mean level of the logarithm differs from

μ

by the quadratic variation correction. In a stock model, specifying this under a physical probability measure does not by itself specify risk-neutral pricing. A non-dividend-paying traded stock has drift

r S

under a risk-neutral measure; a different drift under that measure would require a corresponding dividend or carry convention.

Conditional mean of a log-mean-reverting price ( $F (t, T) = exp [a lo g S_{t} + (1 - a) μ + σ^{2} (1 - a^{2}) / (4 α)]$ )

For a log-mean-reverting price, set

a = e^{- α (T - t)}

. The Ornstein-Uhlenbeck transition distribution gives conditional log-mean

a lo g S_{t} + (1 - a) μ

and log-variance

σ^{2} (1 - a^{2}) / (2 α)

. The moment-generating function of a normal distribution therefore gives the displayed formula for

F (t, T) = E [S_{T} ∣ F_{t}]

. It is a martingale in

t

for fixed

T

, by the tower property of conditional expectation, and the Itô formula yields

d F (t, T) = σ e^{- α (T - t)} F (t, T) d W_{t} .

This uses the same probability measure as the specified conditional expectation. It describes futures pricing only when that measure is an appropriate pricing measure, with the required cash-flow conventions. For a non-dividend-paying traded stock and constant interest rate, the forward contract delivery price follows the usual carry relation instead.

Continuous dividend yield ( $d D_{t} = δ S_{t} d t$ )

A proportional continuous dividend payment at rate

δ

per unit time relative to the ex-dividend stock price. The total gain of one share is

d S_{t} + δ S_{t} d t

. Under a risk-neutral measure with short rate

r

, a geometric stock diffusion therefore has ex-dividend drift

(r - δ) S_{t}

. A fixed cash dividend at a specified date is a different convention and generally requires an ex-dividend price jump.

Banking stock dividends versus reinvesting them ( $A_{t} = \int_{0}^{t} δ S_{u} e^{r (t - u)} d u, X_{t} = S_{t} + A_{t}$ )

For a stock with constant continuous dividend yield

δ

, 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} d t + δ S_{t} d t

, 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^{- δ (T - t)}

in the Black-Scholes model with dividends. It is replicated by

n e^{- δ (T - t)}

shares with dividends reinvested in new shares: the share count grows at rate

δ

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.

Geometric stock index ( $J_{t} = \prod_{i = 1}^{n} (S_{t}^{i})^{1/ n}$ )

For positive stocks satisfying

d S_{t}^{i} / S_{t}^{i} = \sum_{j} σ_{ij} d W_{t}^{j} + μ_{i} d t

, let

q = n^{- 1} 1

and

V = σ σ^{T}

. The Itô formula gives

lo g (J_{t} / J_{0}) = q^{T} σ W_{t} + (q^{T} μ - \frac{1}{2 n} tr V) t .

Thus its volatility is

σ^{T} q

. This geometric average is generally not the value of the constant equal-weight self-financing portfolio: its instantaneous drift is

q^{T} μ - tr (V) / (2 n) + q^{T} V q /2

, whereas the fully invested equal-weight portfolio has drift

q^{T} μ

Defaultable stock ( $S_{t}$ )

A stock model whose price can be killed or reduced at a default time. The zero-recovery default model combines continuous price dynamics before default with a jump to zero.

Zero-recovery default model ( $S_{t} = 1_{{t < τ}} e^{λ t} S_{t}$ )

For an independent exponential distribution default time of rate

λ

and a continuous nonnegative martingale

S

, this process is a martingale. The compensating pre-default factor

e^{λ t}

balances the survival probability

e^{- λ t}

. The strict survival inequality gives a càdlàg price process.

Numéraire

A numéraire is a strictly positive traded asset used as the unit in which all other asset prices are expressed. Changing the numéraire changes the associated martingale measure while preserving no-arbitrage prices.

Change of numeraire ( $\frac{d Q ^{M}}{d Q ^{N}}_{F_{t}} = \frac{M _{t} / N _{t}}{M _{0} / N _{0}}$ )

M, N

are strictly positive traded numeraires and

M / N

is a mean-one martingale after normalizing its initial value under

Q^{N}

, this ratio defines

Q^{M}

. Conditional change of measure then makes every

Q^{N}

-martingale asset price

S / N

into the corresponding

Q^{M}

-martingale

S / M

, when the required integrability holds. In particular a martingale exchange rate under one currency's bank-account measure has a reciprocal martingale under the other currency's measure when their interest rates coincide.

Stock-numeraire measure in the Black-Scholes model ( $d Q^{S} / d Q = e^{- ρT} S_{T} / S_{0}$ )

For a non-dividend-paying stock in the Black-Scholes model, use density

d Q^{S} / d Q = e^{- ρT} S_{T} / S_{0}

. Under

Q^{S}

W_{t}^{S} = W_{t}^{Q} - σ t

is Brownian motion, and

lo g S_{t} = lo g S_{0} + (ρ + σ^{2} /2) t + σ W_{t}^{S}

. Thus

e^{- ρT} E_{Q} [S_{T} H] = S_{0} E_{Q^{S}} H

. This change of numeraire is particularly useful for payments delivered in stock units.

Numéraire portfolio

In a one-period market, a numéraire portfolio has strictly positive value at both dates and can therefore be used to convert initial consumption into terminal wealth.

Numéraire strategy

A numéraire strategy is a self-financing portfolio with zero consumption whose wealth is strictly positive at every date. Its wealth process can carry consumption received at an earlier date forward to a later date.

Arbitrage ( $V_{0} = 0, V_{T} \geq 0 P -a.s., P (V_{T} > 0) > 0$ )

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.

No-arbitrage ( $V_{0} = 0, V_{T} \geq 0 a.s. ⟹ V_{T} = 0 a.s.$ )

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 ( $E r_{i} - R = \sum_{j} b_{ij} λ_{j}$ )

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.

Covariance criterion for diversification of factor residuals ( $Var (w^{T} ϵ) = w^{T} Γ w$ )

For centered residual random variables

ϵ_{i}

with covariance matrix

Γ

, the residual of a portfolio has variance

w^{T} Γ w

. Pairwise uncorrelated residuals with uniformly bounded variances and weights

∣ w_{i} ∣ \leq W / n

give

Var (w^{T} ϵ) \leq 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} ∣ \leq W / n

and

\sum_{i, j} ∣ Γ_{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.

Exact factor pricing without a traded risk-free asset ( $E r = λ_{0} 1 + B λ$ )

Suppose asset returns have the exact representation

r = a + B f

, 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 span (1, col B)

. Taking expectations proves the displayed pricing relation, with

λ

equal to the coefficients of

B

a

plus

E f

. When

[1 B]

has full column rank,

λ_{0}

is the return of a unit-cost zero-exposure portfolio and

λ_{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.

Exact factor pricing without idiosyncratic risk ( $a - R 1 \in rowspan B$ )

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 - R 1)^{T} v + \sum_{j} (B v)_{j} f_{j}

. If

B v = 0

, a nonzero constant term would create an arbitrage after choosing the sign of

v

. Hence

a - R 1

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.

Singular-volatility drift arbitrage ( $σ^{T} v = 0, v^{T} μ > 0$ )

If a constant-coefficient investment model has a vector

v

with

σ^{T} v = 0

and

v^{T} μ > 0

, investment in direction

v

produces deterministic positive gains without initial capital. For

A = σ σ^{T}

, absence of such a direction is equivalent to

μ \in Range A

, because

ker A = ker σ^{T}

and

Range A = (ker A)^{⊥}

. 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.

Law of one price

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.

Strict local martingale failure of the law of one price

Fix

T > 0

, let

X

be Brownian motion, and let

τ = in f {u : X_{u} = - 1}

. Set

q (t) = t / (T - t)

for

t < T

and

B_{t} = 1, S_{t} = 2 + X_{q (t) \land τ} (t < T), S_{T} = 1.

The 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 \neq = 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.

Investment-consumption arbitrage

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.

Pure-investment arbitrage

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

Consumption

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.

Terminal-consumption arbitrage

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.

Arbitrage in a one-period Gaussian market

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.

Self-financing portfolio

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.

Self-financing condition ( $d V_{t} = ϕ_{t} d S_{t} + ψ_{t} d B_{t}$ )

For holdings with value

V_{t} = ϕ_{t} S_{t} + ψ_{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.

Portfolio with constant stock value in bond units ( $V_{t} / B_{t} = w_{0} / B_{0} + θ lo g (S_{t} / S_{0}) + θ (σ^{2} /2 - ρ) t$ )

In the Black-Scholes model, take

B_{t} = e^{- ρ (T - t)}

as a zero-coupon bond numéraire. Holding the fixed stock value

θ B_{t}

requires

g_{t} = θ B_{t} / S_{t}

shares. The self-financing conditions for smooth stock and bond holdings give

h_{t} = \frac{w _{0}}{B _{0}} - θ + θ lo g (S_{t} / S_{0}) + θ (σ^{2} /2 - ρ) t

units 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} + θ σ W_{t}^{Q}

: the discounted gain is a constant multiple of Brownian motion.

Constant-proportion portfolio ( $π_{t} = γ V_{t}$ )

A constant-proportion portfolio holds a fixed fraction

γ

of its portfolio wealth in the stock. In the Black-Scholes model with bank rate

ρ

, initial wealth

V_{0}

, and no consumption, its value is

V_{t} = V_{0} (S_{t} / S_{0})^{γ} exp ((1 - γ) (ρ + \frac{1}{2} γ σ^{2}) t) .

The stock holding is

γ V_{t} / S_{t}

and the remaining value

(1 - γ) V_{t}

is invested in the bank account. The Itô formula verifies

d V_{t} / V_{t} = [ρ + γ (μ - ρ)] d t + γσ d W_{t}

when the physical stock drift is

μ

Self-financing conditions for smooth stock and bond holdings

For

p (x, t) = xg (x, t) + B_{t} h (x, t)

and deterministic

d B_{t} = ρ B_{t} d t

, 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} + σ^{2} x^{2} g_{x} /2 = 0

. Conversely these two identities give

d p = g d S + h d B

. A value PDE alone does not verify arbitrary prescribed holdings.

Discounted dividend gains ( $Y_{t} = S_{t} / B_{t} + \int_{0}^{t} D_{s} / B_{s} d s$ )

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

π

satisfies

d (X / B) = π \cdot d Y

. This follows by differentiating

X / B

, substituting

d X = ϕ d B + π \cdot (d S + D d t)

and using

X = ϕB + π \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.

Multi-period dividend pricing identity ( $B_{r} V_{r} = E_{Q} [B_{n} V_{n} + \sum_{j = r + 1}^{n} B_{j} D_{j} ∣ F_{r}]$ )

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.

Dividend-inclusive portfolio return ( $μ_{total} = μ_{price} + δ / S$ )