Glossario delle opzioni

0DTE

zero days to expiration

An option that expires on the current trading day; little time remains for the thesis to work, while gamma and execution risk can change quickly.

Affine process

A Markov process whose conditional exponential transform has a logarithm affine in the current state on its valid argument domain, often governed by generalized Riccati equations.

American-style option

An option that may generally be exercised on any eligible business day through expiration, unlike a European-style contract that is exercisable only at expiration.

Assignment

The process that requires an option writer to fulfill the contract after an exercise notice is allocated; it can create or remove an underlying position.

At the money

ATM

A call or put whose strike is near the underlying price; it has little intrinsic value and often substantial sensitivity to time and volatility.

Ask price

offer

The lowest displayed price at which a seller is currently willing to sell; it is a quote, not a guarantee that every order size will fill there.

AM versus PM settlement

AM settlement commonly uses component opening prices for an expiration-day value, while PM settlement commonly uses closing prices; last trading times can differ.

Aggregate option premium

Quoted premium per unit multiplied by contract multiplier and contract quantity, plus commissions, fees, and execution slippage.

Assignment funding shortfall

An account lacks sufficient cash, shares, or approved credit to support the stock or cash obligation created by short-option assignment.

Bid price

The highest displayed price at which a buyer is currently willing to buy; selling at the bid prioritizes execution over price improvement.

Bid-ask spread

The gap between the best displayed bid and ask, which is a practical trading cost and a signal of how uncertain an immediate fill may be.

Bayesian inference

The combination of a likelihood and prior into a posterior distribution for parameters and predictions, conditional on the chosen model, prior, and observed data.

Bias–variance tradeoff

The relationship between systematic prediction error from model mismatch and instability caused by sensitivity to the training sample.

Break-even price

The underlying price at which a specified payoff equals zero under stated assumptions; expiration break-even is not the same as the price needed to exit profitably before expiration.

Backwardation

A downward-sloping futures curve in which nearer contracts trade above later ones; in VIX markets it often accompanies near-term stress but does not guarantee a future decline.

Black–Scholes model

BSM model

A benchmark model deriving European option value from no-arbitrage dynamic replication under assumptions such as continuous paths, constant volatility, and ideal trading.

Binomial tree

binomial model

A discrete pricing model that gives the underlying an up or down node at each step and works backward from terminal payoffs to option value.

Block Maxima

An EVT method that divides observations into fixed blocks, retains one maximum per block, and models those maxima with a generalized extreme value distribution.

Bisection method

A robust root-finding method repeatedly halving a volatility interval that brackets the solution, slower but independent of derivatives.

Backspread

reverse ratio spread

An unequal option spread that sells fewer contracts and buys more of the same type, generally trading a limited-loss region for large-move potential in one direction.

Box spread

A bull call spread plus bear put spread using the same two strikes and expiration, producing the strike-width fixed payoff at expiration with European-style contracts.

Bullish risk reversal

Long out-of-the-money call plus short same-expiration out-of-the-money put, using put premium to finance upside participation while retaining stock-like downside risk.

Broken-wing butterfly

skip-strike butterfly · BWB

A same-type, same-expiration 1-by-2-by-1 butterfly whose middle-to-wing strike distances are unequal, producing larger expiration tail risk on one side.

Bull call spread break-even

The expiration stock price equal to the lower long-call strike plus the opening net debit, before fees.

Bear put spread break-even

The expiration stock price equal to the higher long-put strike minus the opening net debit, before fees.

Bull put spread break-even

The expiration stock price equal to the higher short-put strike minus the opening net credit, before fees.

Bear call spread break-even

The expiration stock price equal to the lower short-call strike plus the opening net credit, before fees.

Options buying-power requirement

Account capacity reserved by a broker for an option position; it can change with strategy, markets, concentration, and house rules and is not a loss cap.

Broker risk liquidation

A broker-initiated closing trade when an account breaches margin, buying-power, concentration, expiration, or delivery requirements under its agreement and risk policy.

Bankruptcy option contract

An option after its issuer files for bankruptcy that continues against existing shares or later OCC terms; filing alone does not immediately cancel the contract or fix its value at zero.

Call option

A contract that gives its holder the right, but not the obligation, to buy the underlying at the strike before or at expiration under the contract terms.

Carr–Madan method

FFT option pricing

A method that exponentially damps call prices in log strike, then uses a complex-shifted characteristic function and Fourier inversion to calculate European option prices.

Cameron–Martin theorem

finite-energy path shift

A result identifying deterministic path translations that preserve equivalence of Wiener measure; admissible directions form a finite-energy Hilbert space.

Central Limit Theorem

CLT

An asymptotic result stating that, under suitable conditions, a correctly centered and scaled sum approaches a normal distribution; the raw observations need not be normal.

Convergence in probability

Convergence in which the probability of exceeding any fixed positive error threshold tends to zero; it is weaker than almost-sure convergence.

Confidence interval

CI

An interval produced by a repeated-sampling procedure that, under model assumptions, covers a fixed true parameter at a stated long-run rate.

Credible interval

Bayesian credible interval

An interval containing a stated proportion of posterior probability for a parameter, conditional on a specified model, prior, and observed data.

Causation

causality

A relationship describing how an outcome would differ under a specified intervention or counterfactual change; observed association alone does not identify it.

Cointegration

The existence of a stationary linear combination among nonstationary series, implying a shared long-run equilibrium restriction under a specified model.

Confounding

Bias arising when a common cause affects both the studied variable and outcome, mixing a target effect with association through other paths.

Correlation

A measure of statistical co-movement between variables under a defined population, sample, and transformation; it does not identify causal direction.

Characteristic function

The complex transform φ(u)=E[e^{iuX}], which exists for every real frequency and uniquely determines the probability distribution of X.

Coherent risk measure

A capital requirement satisfying monotonicity, subadditivity, positive homogeneity, and cash translation invariance, with a worst-case expectation representation.

Copula

A function that joins marginal distributions into a joint distribution while isolating their dependence structure; equal correlation need not imply equal joint tails.

COS method

Fourier-cosine pricing

An option-pricing method that expands a terminal density on a finite interval, obtains cosine coefficients from its characteristic function, and combines them with payoff coefficients.

Cornish–Fisher expansion

An asymptotic method that corrects a reference normal quantile with standardized cumulants such as skewness and kurtosis; truncation can make quantiles nonmonotone.

Contract multiplier

The factor that converts a quoted per-unit option price into contract value; a standard U.S. equity option usually uses 100, while adjusted contracts may differ.

Contango

An upward-sloping futures curve in which later contracts trade above nearer ones; it describes relative prices, not guaranteed daily roll return or spot direction.

Censoring

A data condition in which a unit is observed but its exact event time is known only before, after, or within an observed time interval.

Competing risks

A time-to-event setting with mutually exclusive first events where one event prevents the target event from occurring first.

Cumulative incidence function

CIF

The absolute probability that a specified cause occurs first by a horizon when competing events are possible.

Covariance matrix

A symmetric matrix placing each variable's variance on the diagonal and pairwise covariances in the off-diagonal entries.

Covered ratio spread

Long stock combined with two short calls at a lower strike and one long call at a higher strike, capping reward while retaining substantial stock downside.

Covered strangle

covered combination

Long stock plus a higher-strike short call and cash-secured lower-strike short put, allowing shares to be sold on a rise or doubled on a decline.

Conversion arbitrage

conversion

Long stock plus a matching long put and short call, replacing directional stock exposure with a fixed expiration value before financing, dividends, and costs.

Call broken-wing butterfly

One lower-strike long call, two middle-strike short calls, and one higher-strike long call with unequal wing widths.

Call diagonal spread

Common long form buying a farther-dated lower-strike call and selling a nearer-dated higher-strike call, with directional, volatility, time, and assignment exposure.

Call calendar spread

Call time spread

A long time spread selling a nearer call and buying a farther call at the same strike, driven by relative decay and volatility across expirations.

Covered call break-even

For a simultaneous buy-write, the expiration stock price equal to starting stock cost minus call premium received, before fees.

Cash-secured put effective cost

The per-share economic acquisition price after assignment, equal to put strike minus opening premium received, before costs.

Collar net premium

Protective-put premium paid minus covered-call premium received; positive is a net debit and negative is a net credit.

Covered call stock coverage

Matching owned shares satisfy the short-call delivery obligation and offset upside option loss, while leaving substantial stock downside.

Selling call extrinsic value

Closing a long call at market can recover intrinsic and remaining extrinsic value, while early exercise normally forfeits that time value.

Covered call roll cumulative P&L

The old call's realized close result plus all prior credits and the replacement call cash flow, without treating a new roll credit as erasing an old loss.

Cash-secured put roll cumulative P&L

Separate realized P&L on the old put from new-put credit, then recalculate assigned cost, maximum loss, and funding term at the new strike.

Cash versus physical settlement

Cash settlement pays the contract's settlement-value difference, while physical settlement delivers or receives the specified shares after exercise or assignment.

Cash account options trading

Trading within an approved option level in an account that does not borrow from the broker and must fully pay or provide the eligible deliverable for its obligations.

Days to expiration

DTE

The number of calendar days remaining until expiration, used as a compact time-horizon label rather than a complete measure of time-decay risk.

Difference-in-differences

DiD · DD

A quasi-experimental method subtracting the comparison group's contemporaneous change from the treated group's change to construct an untreated counterfactual.

Delta

An estimate of how much an option value may change for a small underlying move, all else equal; it is a sensitivity and not a promised probability or hedge result.

Degrees of freedom

df

The Student-t parameter ν governing tail thickness and moment existence; smaller values produce heavier tails, and the distribution's scale is not necessarily its standard deviation.

Dynkin's formula

generator identity

An identity expressing the expected change of a suitable function of a Markov process as the expected time integral of its infinitesimal generator.

Doob–Meyer decomposition

A unique decomposition, under suitable conditions, of a submartingale into a martingale and a predictable increasing process.

Donor pool

The candidate untreated units eligible to receive weights when constructing a synthetic counterfactual for a treated unit.

Black–Scholes d1

d₁

A standardized Black–Scholes quantity combining forward moneyness with half total variance whose normal CDF is related to delta.

Black–Scholes d2

d₂

The quantity d1 minus volatility times square-root time, whose normal CDF links to risk-neutral expiration ITM probability in the model.

Diagonal spread

Simultaneously buying and selling same-type options with both different strikes and different expirations, combining vertical direction with calendar term exposure.

Delisted stock option

An unexpired option whose underlying left its primary exchange; trading may become closing-only while exercise, assignment, and delivery obligations can continue.

Expiration

The date and process after which an option no longer exists; trading cutoffs, exercise instructions, settlement, and broker procedures can occur at different times.

Equivalent martingale measure

risk-neutral measure

A pricing measure equivalent to the physical measure under which properly discounted tradable assets are martingales; its existence is tied to no arbitrage.

Esscher transform

exponential tilting

A change of measure that reweights outcomes by an exponential factor and normalizes by the moment-generating function; finance may select its parameter with a martingale condition.

Edgeworth–Gram–Charlier expansion

cumulant expansion

A family of reference-density corrections expressed with cumulants and Hermite polynomials; Edgeworth orders terms asymptotically, while Gram–Charlier is a formal series.

Extrinsic value

time value

The portion of premium above intrinsic value, reflecting remaining time, expected uncertainty, rates, dividends, and supply and demand.

Expected Shortfall

ES · CVaR

The average loss from a chosen VaR quantile through the worst tail under a precise convention; it measures severity beyond the threshold rather than only its location.

Error correction model

ECM · VECM

A dynamic model combining short-run changes with adjustment toward a lagged long-run equilibrium error, commonly used for cointegrated systems.

Endogeneity

Correlation between a regressor and the regression disturbance, causing OLS to mix a target effect with omitted pathways, simultaneity, or measurement error.

Event study

A design estimating outcome paths by time relative to an event; causal interpretation depends on its comparison group and identifying assumptions.

Extreme Value Theory

EVT

The asymptotic theory of sample maxima, minima, and high-threshold exceedances used to estimate rare outcomes in a distribution's far tail.

Explained variance

A component's eigenvalue as a share of the sum of all eigenvalues, describing how much sample variance that direction retains.

Eigenvalue

The scale factor applied by a linear transformation along an eigenvector; for a covariance matrix it measures variance in that direction.

Eigenvector

A nonzero direction preserved by a linear transformation and scaled only by its eigenvalue, often representing a common risk direction.

Effective leverage

The amplification of underlying exposure measured through an option or position's percentage sensitivity, changing with delta and option value.

Equity versus index option

Equity options reference shares and are commonly physically settled; index options reference index points and are commonly cash settled, subject to each specification.

Exercise and assignment tax basis

The U.S. federal tax rules that attach option premium to the basis of stock purchased or the amount realized on stock sold after exercise or assignment.

ETF liquidation option

An open option whose ETF deliverable becomes a fixed cash amount after fund liquidation, with later-dated series potentially accelerated under OCC rules.

Forward volatility

The annualized volatility implied for a future interval by subtracting total variance at two maturities; it is a priced quantity, not a promise of realized volatility.

Family-wise error rate

FWER

The probability of falsely rejecting at least one true null hypothesis within a predefined family of tests.

False discovery rate

FDR

The expected proportion of false rejections among all rejections, with the proportion conventionally set to zero when there are no rejections.

Fourier pricing error

The combined domain, truncation, quadrature, aliasing, interpolation, and complex-arithmetic error created when an exact infinite transform is evaluated numerically.

Finite difference method

FDM

A numerical method that approximates a pricing partial differential equation on price and time grids and works backward from the terminal payoff.

Feynman–Kac formula

A representation that links a suitable linear parabolic PDE to a conditional expectation of discounted terminal and running payoffs along diffusion paths.

Feller condition

square-root boundary condition

A parameter inequality making zero unattainable from a positive start for a CIR-type square-root diffusion; failure allows boundary contact, not negative exact states.

Fixed effects

A panel method identifying coefficients from changes within the same unit while absorbing unit characteristics that do not vary over time.

Filtering

Real-time inference recursively updating the current latent-state distribution using observations available through the current date.

Factor model

A model decomposing asset returns into exposure to a small set of common factors and an asset-specific residual.

Factor loading

factor beta

The sensitivity or common-component weight connecting an asset return to a factor, conditional on the factor's scale and model identification.

Factor return

The realized return of a specified common factor over a period, multiplied by an asset's loading to form its common return component.

Forward-split adjusted option

An option changed after a forward stock split through contract count, strike, or shares per deliverable so the position's aggregate economics remain broadly continuous.

Gamma

An estimate of how much delta may change for a small move in the underlying, all else equal; it often becomes more concentrated near expiration and near the money.

Gamma exposure

GEX

A modeled estimate of aggregate option gamma translated into potential hedging sensitivity; its sign and size depend on assumptions about positions and dealer behavior.

Gamma squeeze

A feedback process in which option hedging demand can require additional underlying purchases as price rises; it cannot be confirmed without knowing net gamma and actual hedging behavior.

Girsanov theorem

A result that shifts Brownian drift through an absolutely continuous change of measure under suitable conditions while preserving the diffusion structure.

Gärtner–Ellis theorem

A theorem that, under regularity conditions, turns a limiting scaled log moment-generating function into a large-deviation rate through a Legendre–Fenchel transform.

GARCH

volatility clustering model

A time-series model that updates conditional variance from past squared shocks and prior variance; it models volatility persistence, not return direction.

Granger causality

A temporal predictive relation in which past values of one series improve prediction of another beyond a specified information set and model.

Gamma scalping

Repeatedly rebalancing a delta hedge around an option position with gamma to realize path-dependent trading P&L, subject to theta, volatility, gaps, and costs.

Historical volatility

realized volatility

A backward-looking measure of how much an underlying actually moved over a chosen period, unlike implied volatility inferred from current option prices.

Heteroskedasticity

Variation in the conditional error variance across regressor values or states, invalidating an equal-variance covariance formula.

Hamilton–Jacobi–Bellman equation

HJB equation

The infinitesimal dynamic-programming equation that optimizes running reward plus the controlled generator of future value to determine a value function and policy.

Heston model

stochastic volatility model

An option-pricing model with mean-reverting square-root variance and correlated spot and variance shocks, allowing volatility to evolve randomly.

Hedging error

The difference between a hedge portfolio and its target option value or P&L, caused by discrete rebalancing, jumps, model misspecification, costs, or liquidity.

Hausman test

A comparison of estimators with different efficiency properties under a null; rejection signals model conflict, while nonrejection does not prove validity.

Hazard rate

hazard function

The instantaneous event rate at a time conditional on remaining event-free until then, not a fixed-interval probability.

Hazard ratio

HR

A ratio of instantaneous event rates between groups at the same time among units still at risk, not a cumulative risk ratio.

Hidden Markov model

HMM

A model generating observations from finite discrete latent states with Markov transition probabilities and state-dependent emissions.

Implied correlation

A risk-neutral price of average co-movement inferred from index and component option variances; it is not a guaranteed forecast of future realized correlation.

Implied volatility

IV

The volatility input consistent with an option's market price under a pricing model; it reflects priced uncertainty, not a direct forecast of direction.

IV crush

A sharp decline in implied volatility, often after a known event, that can reduce option premium even when the underlying moves in the expected direction.

In the money

ITM

A call with strike below the underlying price or a put with strike above it; the label describes intrinsic value, not whether the trade is profitable.

Incomplete market

A market model in which available assets cannot replicate every contingent claim; martingale measures are generally nonunique and residual risk needs another valuation criterion.

Infinite variance

The absence or divergence of a probability law's second central moment; a finite dataset still has a finite computed sample variance and cannot rule this out by itself.

Intrinsic value

The immediate exercise value of an option: max(spot minus strike, 0) for a call and max(strike minus spot, 0) for a put, before costs.

Itô's lemma

Itô formula

A stochastic chain rule expanding a function's change into time, first-order, and second-order terms, with the extra term arising from Brownian quadratic variation.

IV percentile

The percentage of observations in a lookback window below the current IV; the result depends on the data series, window, and treatment of missing or extreme values.

IV rank

The current IV's position between a lookback-period low and high; one outlier can distort it, so it should not be read as a standalone signal.

Importance sampling

Monte Carlo under a proposal distribution that visits important outcomes more often, with likelihood-ratio weights restoring the target expectation and potentially reducing variance.

Instrumental variable

IV

A source that moves an endogenous regressor and, under independence and exclusion restrictions, affects the outcome only through that regressor.

Inverse probability weighting

IPW · IPTW

Weighting observations by the inverse probability of their received treatment to create a pseudo-population for a target estimand.

Interrupted time series

ITS

A quasi-experimental design projecting a preintervention path from repeated observations and evaluating postintervention level or trend departures.

Idiosyncratic risk

specific risk

Asset-specific return variation left unexplained by common factors, which may be diversifiable when residual dependence is limited.

Implied volatility solver

A numerical procedure finding the volatility that makes an option-pricing model equal a selected market price while other inputs remain fixed.

Iron condor maximum loss

For equal-width wings, one spread width minus the opening net credit, multiplied by the contract multiplier, at expiration before costs.

Iron condor adjustment

A trade closing, moving, or rebuilding one side or the full condor, realizing current P&L while opening new strike, volatility, and assignment exposure.

Iron condor pin risk

Uncertainty about assignment and final stock quantity when the underlying finishes near either short strike at expiration.

Iron butterfly maximum loss

For a symmetric short iron butterfly, body-to-wing width minus opening net credit, multiplied by the contract multiplier, before costs.

Iron butterfly adjustment

A trade closing, moving, or rebuilding part or all of an iron fly, realizing existing P&L and creating a new body, wing, and assignment exposure.

Iron butterfly pin risk

Assignment and final stock-direction uncertainty when expiration price finishes near the common short call-and-put body strike.

In-the-money option expiration outcome

The stock or cash result after an expiring option has intrinsic value and is processed under exercise instructions, exercise-by-exception, broker policy, and settlement terms.

IRA options trading

Trading approved option strategies inside a U.S. individual retirement account under the custodian's policy, using assets and permissions within that account.

Jump risk

gap risk

The risk that the underlying moves discontinuously between tradable prices, preventing a continuous hedge from adjusting through intermediate levels and invalidating local approximations.

Jelly roll

A four-option trade that sells a synthetic stock combo in one expiration and buys the matching combo in another, isolating financing value between maturities.

Kolmogorov equations

forward equation · backward equation · Fokker–Planck equation

The forward equation evolves state distributions with the adjoint generator; the backward equation evolves conditional future payoffs or event probabilities with the generator.

Kurtosis

tail thickness

A fourth-power statistic describing the shape of standardized returns; high kurtosis often accompanies more extreme observations but does not specify direction or one tail.

Kaplan–Meier estimator

product-limit estimator

A nonparametric survival-curve estimator multiplying conditional survival proportions across event-time risk sets while accommodating right censoring.

Kalman filter

A recursion predicting and updating state means and error covariances in a linear Gaussian state-space model.

Kalman gain

A dynamic weight mapping a new observation innovation into a state revision based on predicted-state and measurement uncertainty.

Liquidity

The ability to trade a reasonable size near a competitive price; spreads, displayed depth, activity, contract specifics, and market conditions matter together.

Local volatility

The instantaneous volatility assigned by a local-volatility model as a function of time and the current underlying level to fit today's European option price surface.

Local martingale

strict local martingale

A process that becomes a martingale after each stop in an increasing localizing sequence; it is strict when the unstopped process is not a true martingale.

Lévy process

jump process

A càdlàg process with stationary independent increments and stochastic continuity, described by drift, Brownian variance, and a Lévy jump measure.

Large deviation principle

LDP

Exponential probability bounds for a family of variables at an extreme scale, governed by a speed and rate function with separate statements for open and closed sets.

Laplace’s method

steepest descent

A large-parameter integral method that uses local behavior near a dominant real extremum or contributing complex saddle, with curvature determining the leading factor.

Law of Large Numbers

LLN

A limit theorem under which a sample average approaches a population mean or corresponding expectation as the sample grows, given suitable moment and dependence conditions.

Level change

An immediate vertical shift in an outcome at intervention relative to its projected prior trend, distinct from a slope change.

Latent state

A time-varying quantity used by a model to summarize system dynamics that is inferred from data rather than observed directly.

Long box spread

A box bought for a net debit to receive the fixed strike-width payoff at expiration, economically resembling fixed-term lending through options.

Long call break-even

The expiration stock price equal to the call strike plus premium paid per share, before fees.

Long put maximum profit

For stock at zero, the put strike minus premium paid per share, before costs.

Long call versus short put asymmetry

Both can be bullish, but the call pays a debit for limited loss and uncapped upside while the short put earns limited credit against substantial downside.

Long put versus short call asymmetry

Both can be bearish, but the put limits loss and gains on a large decline while an uncovered short call earns limited credit against unlimited upside loss.

Last trading day

The final date an expiring contract can trade on its exchange, which may precede the expiration date and must be checked in the product specification.

Long option loss limit

For a standalone standard option held only as a contract before exercise, economic loss is generally limited to total premium actually paid plus transaction costs; post-exercise asset risk is separate.

Long straddle break-even

Either expiration stock price equal to the common strike plus or minus the total net debit paid for the call and put.

Long strangle break-even

The call strike plus total debit or the put strike minus total debit, the two expiration zero-P&L prices.

Mark price

A platform's reference value for an option, often based on the bid-ask midpoint or a model; it is useful for estimation but is not necessarily executable.

Martingale

A stochastic process whose next value has conditional expectation equal to its current value under specified information and measure; it need not be constant or riskless.

Max pain

A calculation that finds the expiration price producing the lowest aggregate intrinsic payout for a chosen set of open interest; it is not an observed dealer target or a reliable price forecast.

Model risk

calibration risk

The risk that model structure, data, parameters, numerics, or misuse causes valuation, sensitivity, or hedging results to differ materially from reality.

Moment explosion

The loss of finiteness of a power expectation of an asset price at a given horizon under a specified model and measure; individual price paths need not explode.

Maximum entropy principle

MaxEnt

The rule of selecting the highest-entropy probability law among those satisfying explicit normalization and expectation constraints, relative to a support and reference measure.

Maximum Likelihood Estimation

MLE

Selection of parameter values that maximize the likelihood of observed data under a specified probability model; likelihood is not the probability that a parameter is true.

Multiple testing

multiple comparisons

Testing several hypotheses or specifications within one research decision, so valid individual p-values can still produce many chance winners.

Mean reversion

A model-dependent tendency for a variable to move back toward a fixed or changing reference; it does not automatically imply stationarity or tradability.

Multicollinearity

Strong linear dependence among included regressors that makes their separate coefficients imprecise and sensitive to the sample.

Martingale representation theorem

A theorem expressing suitable martingales as stochastic integrals against specified driving processes, linking representable risk to dynamic hedging.

Monte Carlo simulation

A numerical method that generates many paths from a specified random model and estimates derivative value and sampling error from discounted payoffs.

Matching

A design method pairing treated units with comparable untreated units using pretreatment characteristics or a propensity score.

Measurement equation

observation equation

An equation mapping the current latent state into observed data while adding measurement noise or temporary disturbance.

Merger-adjusted option

An unexpired option whose share deliverable changes after a completed merger to the cash, acquirer shares, or other property payable to target shareholders under OCC terms.

Numerical Greeks

Price sensitivities approximated by repricing, simulation derivatives, or adjoint methods; results depend on bump size, model, numerical error, and implementation.

Normal distribution

Gaussian distribution

A symmetric bell-shaped law determined by mean and variance, with tails decaying exponentially in squared distance; it is a benchmark, not a guaranteed model for financial extremes.

Novikov and Kazamaki conditions

stochastic-exponential criteria

Sufficient criteria making a continuous local-martingale stochastic exponential a true expectation-one martingale suitable as a probability density.

Numeraire

numéraire

A strictly positive tradable asset used as the unit of value; dividing other prices by it and using the associated measure makes relative prices martingales.

Newey–West standard error

HAC standard error

A standard error using a kernel and maximum lag to remain asymptotically consistent under heteroskedasticity and bounded serial dependence.

Newton–Raphson method

Newton method

A root-finding method updating volatility by price error divided by vega, converging quickly near a solution when vega is adequate.

Negative option-position P&L

An option position can show a loss below zero relative to its cost or opening credit without the premium or intrinsic value of a standard single option becoming negative.

Open interest

OI

The number of option contracts reported open after clearing; it is updated with a lag and does not by itself reveal whether current trading is opening, closing, bullish, or bearish.

Optional stopping theorem

optional sampling theorem

A theorem preserving martingale expectation relations at stopping times under sufficient conditions such as boundedness or uniform integrability.

Overfitting

Learning sample-specific noise or accidental structure as if it were repeatable signal, producing strong in-sample results that fail to generalize.

Option chain

A table of contracts organized by expiration, strike, and call or put, usually including quotes, activity, implied volatility, and Greeks from a chosen data source.

Option premium

The option's quoted price per unit, composed of intrinsic and extrinsic value; multiply it by the contract multiplier to estimate contract value.

Out of the money

OTM

A call with strike above the underlying price or a put with strike below it; it has no intrinsic value but may still have extrinsic value.

Ordinary least squares

OLS

A regression estimator selecting coefficients that minimize the sum of squared residuals between observed outcomes and linear fitted values.

Overlap

common support · positivity

The condition that units across the relevant covariate range have a meaningful probability of either treatment state.

Option elasticity

lambda · omega · gearing

The local percentage option-value change per 1% underlying-price change, calculated as delta times spot divided by option value.

Option return denominator

The stated capital base used for a percentage return, such as premium paid, maximum risk, cash collateral, or stock value; those bases are not interchangeable.

Option liquidation value

The position-side estimate of cash received or paid to close now using executable bids or asks, size, slippage, and costs rather than a displayed midpoint or mark.

Options extended-hours session

An exchange session outside regular trading hours for designated option products, with symbol eligibility, order types, liquidity, and broker access requiring separate confirmation.

Option exercise-instruction cutoff

The latest time a holder may give a broker exercise or do-not-exercise instructions, distinct from the option trading close, OCC processing, and final settlement.

Option exercise settlement value

The official underlying value calculated under a cash-settled option's expiration method; its difference from strike times the multiplier determines cash, not the option's last trade.

Option assignment notification

Account confirmation after OCC allocates an exercise notice to a clearing member and the broker assigns that obligation to a customer short the same option series.

Out-of-the-money option expiration outcome

The usual unexercised termination of an option whose expiration reference creates no intrinsic value, subject to contrary instructions and product or broker rules.

Option trade settlement cycle

The time between an option trade date and final premium payment; U.S. listed equity option trades generally complete settlement on the next business day under T+1.

Option day trade

Opening and closing a position in the same option security on one trading day, counted and margined under the FINRA framework then used by the broker and its order records.

Options account approval

A broker's account-specific decision about whether, and for which strategies, a customer may trade options after reviewing objectives, experience, and financial capacity.

Options position transfer

Moving an open listed option contract in kind from a delivering brokerage account to a receiving brokerage willing and able to carry that contract and its risk.

Option order rejection

An option order fails account, funding, format, product, session, or risk checks and therefore does not become a valid working market order.

Option order fill

An accepted option order matches an opposite-side market order under its price, quantity, time, and other execution conditions.

Option margin loan

The distinction between fully paying long-option premium and later incurring interest only when exercise, assignment, or other positions create an actual broker loan balance.

Option tax treatment

The classification and reporting of option gains or losses according to whether a contract is closed, expires, is exercised, or is assigned, plus its holding period and product type.

Option wash sale

A loss on stock, securities, or a related option may be deferred when substantially identical securities or a contract or option to acquire them is obtained within the 30-day window before or after the loss sale.

Option tax straddle

A set of offsetting positions in actively traded property that substantially reduces another position's loss risk and may trigger loss-deferral and holding-period rules.

Option Form 1099-B reporting

Broker reporting of option closing, expiration, settlement, or Section 1256 activity that a taxpayer reconciles with actual records and Form 8949 or Form 6781.

Option symbol change

The migration of an open option to a new root after a company or stock ticker change; a pure symbol change ordinarily leaves strike, expiration, multiplier, and standard deliverable intact.

Probability of profit

POP

A model-based estimate that a defined position will show profit at a specified time under stated assumptions; it is not the same as expected return or probability of touching a price.

Probability of touch

The modeled probability that the underlying reaches a specified level at least once before a deadline; it differs from finishing beyond it or earning a profit.

P-value

Under a specified null model and analysis rule, the probability of a test statistic at least as incompatible with that model as the observed value.

Pairs trading

A relative-value strategy taking opposing positions in two related assets to seek profit from convergence after a price divergence.

Panel data

longitudinal data

Data repeatedly observing the same firms, people, or other units over dates, combining between-unit differences with within-unit change.

Parallel trends

The DiD assumption that average untreated potential outcomes for treated and comparison groups would have followed the same change path without treatment.

Pontryagin maximum principle

PMP

Necessary optimal-control conditions using a Hamiltonian and backward costate along a candidate path, complementary to HJB's value-function view across states.

Peaks Over Threshold

POT

An EVT method that retains every observation above a high threshold and commonly models exceedance frequency and excess size with a generalized Pareto tail.

Put/call ratio

P/C ratio

Put volume divided by call volume for a defined universe and period; product mix, hedges, spreads, closing trades, and exercise-related activity can distort a directional reading.

Put option

A contract that gives its holder the right, but not the obligation, to sell the underlying at the strike before or at expiration under the contract terms.

Propensity score

The probability of receiving treatment given observed pretreatment covariates, used to design groups balanced on those measured characteristics.

Principal component analysis

PCA

A dimension-reduction method rotating correlated variables into orthogonal linear combinations ordered by the sample variance they explain.

Principal component

A weighted direction through the original variables, ordered to maximize remaining variance subject to being orthogonal to earlier components.

Put broken-wing butterfly

One higher-strike long put, two middle-strike short puts, and one lower-strike long put with unequal wing widths.

Put diagonal spread

Common long form buying a farther-dated higher-strike put and selling a nearer-dated lower-strike put, with potential long-stock assignment from the short leg.

Put calendar spread

Put time spread

A long time spread selling a nearer put and buying a farther put at the same strike, with potential long-stock assignment from the near short put.

Protective put maximum loss

Stock cost minus put strike plus protective-put premium, the expiration loss cap during the hedge term before costs.

Protective put total position

The combined long stock and matched long put, which limits downside during the hedge term while retaining stock upside less premium.

Selling put extrinsic value

Closing a long put at market can recover intrinsic and remaining extrinsic value, while exercise sells the deliverable and can forfeit time value.

Quadratic variation

The limit of sums of squared process increments over increasingly fine partitions, measuring accumulated second-order path variation and generating Itô corrections.

Q-value

Under a chosen method and assumptions, an estimated minimum false discovery rate threshold at which a particular result can be called a discovery.

Risk-neutral probability

A pricing probability that makes discounted asset and derivative values consistent; it embeds risk preferences and insurance premia rather than forecasting real-world frequencies.

Realized volatility

Volatility calculated from price changes that occurred under a stated return, sampling-window, and annualization rule; different conventions can produce different values.

Radon–Nikodym derivative

likelihood ratio · density

A density describing how one absolutely continuous measure reweights states relative to another, converting expectations between the measures.

Regression coefficient

The fitted linear outcome difference associated with one unit of a regressor while holding the other included regressors fixed.

Residual

The observed outcome minus a fitted value estimated on the same sample; it is not a directly observed population disturbance.

Robust standard error

An estimate of coefficient sampling uncertainty under a stated variance or dependence structure that does not change the point estimate or cure endogeneity.

Random effects

A panel model combining within and between variation by treating unit components as random and orthogonal to regressors in every period.

Regression discontinuity design

RDD · RD design

A design using a treatment-probability jump at a known running-variable cutoff plus potential-outcome continuity to identify a local effect.

Running variable

assignment variable · forcing variable

The score, size, or ratio locating a unit relative to the known cutoff that governs treatment assignment in an RDD.

Regime switching

A model allowing parameters to change with an unobserved discrete regime whose evolution is described by transition probabilities.

Risk-neutral ITM probability

A model probability under a pricing measure that the underlying finishes beyond a strike, not a physical forecast or probability of profit.

Ratio spread

A same-type, same-expiration option spread with unequal long and short quantities; a common one-long, two-short structure leaves an extra uncovered short option and tail risk.

Reverse conversion

reversal arbitrage · reversal

Short stock plus a matching long call and short put, a three-sided parity trade whose economics depend heavily on stock borrow, dividends, assignment, and execution.

Risk reversal

A directional same-expiration combination that buys one out-of-the-money option wing and sells the opposite wing; the bullish form buys a call and sells a put with substantial downside obligation.

Realized versus unrealized option P&L

Unrealized P&L estimates an open position from a reference mark; realized P&L uses actual closing, expiration, or settlement cash flows with multiplier, quantity, and costs.

Reverse-split adjusted option

A nonstandard option after a reverse split that usually keeps contract count, strike, and multiplier but reduces deliverable shares and may include cash for a fraction.

Rights-offering adjusted option

A nonstandard option whose deliverable OCC may change after a rights distribution to include transferable subscription rights, which can later be removed or replaced under the event memo.

Settlement

The method used to satisfy an exercised or assigned contract, commonly through shares or a cash amount; style and timing depend on the product specification.

Statistical significance

A decision that a test crossed a prespecified significance level; it does not equal effect size, economic importance, or hypothesis probability.

Stationarity

Time invariance of specified distributional features of a stochastic process; weak stationarity requires constant mean and lag-dependent covariance.

Self-financing strategy

A trading strategy that receives no external cash after inception; all rebalancing is funded by sales, purchases, and cash flows within the portfolio.

Slippage

The difference between an expected reference price and the actual execution price, including the effect of spread, depth, volatility, order size, and delay.

Snell envelope

The smallest supermartingale dominating a stopping payoff process; its value solves the optimal stopping problem and its contact set suggests an optimal stop.

Skew carry

skew roll-down

Conditional P&L from a position moving across the volatility surface as time and spot change; it is not fixed income or a guaranteed return.

Static arbitrage

A same-time fixed portfolio exploiting inconsistent option and underlying prices to create no negative terminal payoff and a positive payoff in some states.

State price

Arrow–Debreu price

Today's price of a claim paying one unit only in a specified future state, combining time discounting with how that state is valued.

Stable distribution

alpha-stable distribution

A probability family whose weighted independent copies remain in the same family after location and scale adjustment; non-Gaussian members have power tails and infinite variance.

Stochastic discount factor

pricing kernel · SDF

A random variable that weights future payoffs by time and state value to produce today's price; asset returns satisfy its pricing moment condition.

Stochastic exponential

Doléans–Dade exponential

The multiplicative process solving dZ = Z₋dX; for a continuous local martingale it is the ordinary exponential corrected by half its quadratic variation.

Superhedging price

The least initial capital needed for an admissible self-financing strategy whose terminal wealth covers a claim in every relevant scenario.

Student-t distribution

t distribution

A symmetric location-scale family with tail thickness governed by degrees of freedom, polynomial tail decay, and a normal limit as the degrees of freedom increase.

SABR model

stochastic alpha beta rho model

A volatility model with correlated random forward and volatility processes, using alpha, beta, rho, and vol-of-vol to shape smile levels and dynamics.

Saddlepoint approximation

A method that solves for the cumulant-generating-function tilt centered at a target, then uses local curvature to approximate a density or tail probability.

Strike price

The fixed contract price at which the holder may buy the underlying with a call or sell it with a put, subject to the option's terms.

Sticky delta

A scenario in which implied volatility at the same delta or relative moneyness stays fixed as spot moves, causing the skew curve to move with the underlying.

Sticky strike

A scenario in which implied volatility at each absolute strike stays fixed as spot moves, so the at-the-money option travels along the existing skew.

Synthetic control

A method weighting untreated donors to reproduce a treated unit's pretreatment path and extend its no-treatment counterfactual.

Segmented regression

A regression using an intervention indicator and postintervention time to separate changes in a series' level and slope.

Survival analysis

time-to-event analysis

Statistical methods for time from a defined origin to an event that retain partial information from censored observations.

Survival function

The probability that an event time exceeds a specified horizon, meaning a unit remains event-free beyond that time.

Subdistribution hazard

A constructed hazard used by Fine–Gray regression to link covariates directly to a cause's cumulative incidence function.

State-space model

A time-series model using a transition equation for latent dynamics and a measurement equation connecting states to noisy observations.

Smoothing

Full-sample inference revising an earlier latent-state estimate using observations that arrived after that date.

Standard normal CDF

N(x)

A function returning the probability that a standard normal variable is no greater than an input, converting d1 and d2 into pricing weights.

Short straddle

Selling a call and put with the same strike and expiration; maximum gain is limited to premium while upside loss is unlimited and downside loss can be substantial.

Short strangle

Selling a higher-strike call and lower-strike put with one expiration, creating a wider profit range but limited reward and large or unlimited tail risk.

Stock repair

Buying a near-current-price call and selling two higher-strike calls against underwater stock to lower a recovery target, while retaining stock downside and expiration risk.

Short box spread

The reversed box that receives cash initially and owes the fixed strike-width payoff at expiration, economically resembling collateralized fixed-term borrowing.

Short calendar spread

Reverse calendar

A net-credit time spread buying a nearer option and selling a farther same-strike option, which can leave an uncovered farther short after near expiry.

Short call unlimited loss

The theoretically unlimited loss from an uncovered short call because the stock price has no fixed upper bound.

Short put maximum loss

For stock at zero, the put strike minus premium received per share, before costs.

Section 1256 option tax

Eligible nonequity options are generally deemed sold at year-end fair market value and their capital gain or loss is split 60% long-term and 40% short-term under U.S. federal rules.

Spinoff-adjusted option

A nonstandard option whose deliverable after a spinoff can combine parent shares, distributed company shares, and cash for fractions under OCC's event terms.

Special-dividend adjusted option

An option changed by OCC for a qualifying non-ordinary cash dividend through a lower strike or added cash deliverable; regular dividends normally are not adjusted.

Stock-dividend adjusted option

An outstanding option changed after the issuer distributes additional shares of itself through contract count, strike, share deliverable, or cash in lieu of a fraction.

Short straddle downside loss

For a stock-like underlying at zero, the common strike minus total opening credit per share, before costs.

Short strangle downside loss

For a stock-like underlying at zero, the short-put strike minus total opening credit per share, before costs.

Theta

An estimate of the option value change associated with one day passing, all else equal; actual decay is nonlinear and other inputs rarely remain unchanged.

Tail dependence

The limiting probability that one variable is extreme given that another is extreme in the selected tail; ordinary correlation does not determine it.

Tail index

extreme value index

A shape quantity governing tail decay and extreme extrapolation; sources may mean the EVT shape ξ or its reciprocal power exponent α, so the convention matters.

Type I error

false positive

The event of falsely rejecting a null hypothesis that is actually true, with its probability controlled under the test assumptions by the test level.

Time decay

The erosion of an option's extrinsic value as expiration approaches, all else equal; it is usually nonlinear and can be offset by price or volatility changes.

Truncation

A sampling mechanism in which a unit appears only if its event time satisfies an entry rule, leaving some source-population units entirely unseen.

Transition equation

An equation describing how a latent state evolves from its prior value, known inputs, and process disturbance.

Tender-offer option

An option while its underlying shares are subject to a tender or exchange offer; the offer normally does not adjust the contract, and exercise and tendering are separate steps.

Underlying

The asset, index, or other reference on which an option's value and settlement terms are based.

Unit root

An autoregressive characteristic root at one, causing shocks to persist in a series' level and invalidating ordinary stationary asymptotics.

Unconfoundedness

conditional exchangeability

The identifying assumption that treatment assignment is independent of potential outcomes after conditioning on observed pretreatment covariates.

Underlying notional exposure

Underlying price or index level times contract multiplier and quantity, a reference exposure that is not premium, delta-equivalent exposure, or maximum loss.

Buying options with unsettled funds

Using proceeds from a security sale that has not completed settlement to fund a new option trade, whose availability and resale timing depend on account type, funding source, and broker rules.

Vega

An estimate of how much an option value may change for a one-point change in implied volatility, all else equal; it does not measure realized volatility directly.

Variance

The average squared deviation of returns from their mean; its square root is standard deviation, so variance aggregates differently from volatility percentages.

Variance swap

A derivative whose settlement is based on realized variance minus a pre-agreed variance strike, multiplied by variance notional and any contract-specific adjustments.

Variational inequality

free-boundary problem

A condition combining value dominance over immediate exercise with the pricing equation in the continuation region, jointly determining an American option and its exercise boundary.

VIX

Cboe Volatility Index

A constant 30-day expected-volatility measure for the S&P 500 calculated from a strip of SPX option prices; it is not the IV of an individual stock or contract.

Volatility cone

A set of historical realized-volatility ranges or percentiles across rolling horizons, used to place current implied volatility in context rather than declare absolute value.

Volatility skew

The pattern of different implied volatilities across strikes for the same expiration, reflecting asymmetric demand and priced tail risk rather than one universal IV.

Volatility risk premium

VRP

Compensation embedded in implied volatility relative to subsequent realized volatility; it is a historical tendency, not a guaranteed profit in every period.

Volatility of volatility

vol-of-vol · VVIX

The variability of volatility itself; VVIX uses VIX option prices to measure 30-day implied volatility of VIX forward prices.

Vomma

volga · vega convexity

A second-order sensitivity measuring how vega changes as implied volatility changes; it describes curvature in option value with respect to IV, not a directional forecast.

Option volume

The number of contracts traded during a session; it resets each day and cannot alone determine whether trades opened or closed positions.

Vega conditioning

How option sensitivity to volatility controls inversion stability, with low vega turning small price errors into large implied-volatility differences.

Option writer

option seller

The party that sells an option and assumes the contract obligation if assigned; a closing sale by a holder is not the same as writing a new option.

Weak instrument

An instrument with insufficient conditional relevance to supply enough exogenous regressor variation, destabilizing IV estimation and ordinary inference.

Within estimator

An estimator subtracting each unit's time averages to remove time-invariant unit effects and use only within-unit variation.

Weekend theta

holiday theta

Model-estimated time-value change across a non-trading calendar interval, which markets may partly price before the close rather than charge at a fixed reopening moment.

Wheel strategy

options wheel

A cycle of selling cash-secured puts and, after share assignment, covered calls, retaining stock downside, capped upside, and assignment risk.

25-delta risk reversal

25-delta RR · 25-delta skew

A market convention comparing the price or implied volatility of same-expiration 25-delta put and call wings; the sign depends on the stated quote direction.