Glossary: Derivative Pricing Terminology

This glossary provides concise definitions of derivative pricing terms used throughout the Derivative Pricing and Models series. Terms are alphabetized for quick reference.

Terms A-D

American option: An option that can be exercised at any time before expiration, requiring specialized pricing methods beyond Black-Scholes. See also: European option.

Arbitrage: Risk-free profit from pricing inconsistencies; no-arbitrage principles underpin all derivative pricing models.

At-the-money (ATM): An option whose strike price equals the current underlying price; ATM options have the highest vega sensitivity.

Binomial tree: A discrete-time pricing model that constructs a lattice of possible price paths; converges to Black-Scholes as steps increase.

Black-Scholes model: The foundational option pricing formula assuming constant volatility, no dividends, and log-normal returns; produces closed-form solutions for European options.

Calibration: The process of fitting model parameters to match observed market prices; typically minimizes squared error between model and market implied volatilities.

Convexity (gamma): The rate of change of delta with respect to underlying price; measured in units of delta per dollar move.

Crank-Nicolson: A finite difference method that averages explicit and implicit schemes; unconditionally stable and second-order accurate in time.

Delta: The sensitivity of option price to underlying price changes; expressed as a ratio between 0 and 1 for calls, -1 and 0 for puts.

Dupire local volatility: A volatility model where instantaneous volatility is a deterministic function of spot price and time; calibrates exactly to the implied volatility surface.

Terms E-I

European option: An option exercisable only at expiration; priced by Black-Scholes or similar closed-form models.

Explicit method: A finite difference scheme that calculates future values directly from current values; simple but conditionally stable.

Forward variance: The market-implied variance for a future time period, derived from the term structure of implied volatility.

Gamma: See Convexity.

Greeks: Collective term for option price sensitivities: delta, gamma, theta, vega, rho, and higher-order derivatives.

Heston model: A stochastic volatility model where variance follows a mean-reverting square-root process; captures smile and term structure dynamics.

Hull-White model: A short-rate model with mean reversion to a time-dependent level; widely used for interest rate derivative pricing.

Implied volatility (IV): The volatility input that equates a pricing model's output to the observed market price; quoted in annualized percentage terms.

Implicit method: A finite difference scheme requiring solution of a system of equations at each timestep; unconditionally stable but more computationally intensive.

In-the-money (ITM): An option with intrinsic value; for calls, strike below spot; for puts, strike above spot.

Terms L-O

LIBOR Market Model (LMM): A term structure model that directly models forward rates rather than short rates; used for pricing exotic interest rate derivatives.

Local volatility: Instantaneous volatility as a function of spot and time; deterministic once calibrated. See also: Dupire local volatility.

Log-normal distribution: The probability distribution assumed for asset prices in Black-Scholes; implies returns are normally distributed.

Mean reversion: The tendency of a variable (such as volatility) to return toward a long-term average; parameterized by speed (kappa) and level (theta).

Model risk: The risk of loss arising from model errors, inappropriate assumptions, or incorrect implementation; governed by SR 11-7 and similar regulations.

Monte Carlo simulation: A numerical method that estimates prices by averaging outcomes across randomly generated paths; essential for path-dependent and multi-asset derivatives.

No-arbitrage: The principle that risk-free profits cannot exist in efficient markets; forms the foundation of derivative pricing theory.

Numeraire: A reference asset used to normalize prices in risk-neutral valuation; changing numeraire can simplify certain pricing problems.

Out-of-the-money (OTM): An option with no intrinsic value; for calls, strike above spot; for puts, strike below spot.

Terms P-R

Path-dependent option: A derivative whose payoff depends on the price history, not just the final price; examples include Asian options and barriers.

Put-call parity: The relationship linking put and call prices to forward price: C - P = PV(F - K); violations indicate arbitrage opportunities.

QuantLib: An open-source C++ library for derivative pricing; industry standard for quantitative finance implementation.

Replication: Constructing a portfolio that exactly matches an option's payoffs; the replication cost equals the no-arbitrage price.

Rho: The sensitivity of option price to interest rate changes; measured in dollars per 1% rate change.

Risk-neutral measure: A probability measure under which expected returns equal the risk-free rate; used for pricing rather than prediction.

RMSE (Root Mean Square Error): A measure of calibration accuracy; typical thresholds are < 0.5 vols for equity volatility models.

Terms S-V

SABR model: A stochastic volatility model with parameters alpha (volatility level), beta (CEV exponent), rho (correlation), and nu (vol of vol); widely used for interest rate smile.

Skew: The asymmetry in implied volatility across strikes; negative skew means OTM puts have higher IV than OTM calls.

Smile: The pattern of implied volatility varying with strike; typically U-shaped for equity indices.

Stochastic volatility: Models where volatility itself is a random process; examples include Heston and SABR.

Term structure (volatility): The pattern of implied volatility across expiration dates; can be upward-sloping (contango) or inverted.

Theta: The sensitivity of option price to time passage; typically negative for long option positions, measured in dollars per day.

Variance swap: A derivative that pays the difference between realized and implied variance; used for volatility exposure.

Vega: The sensitivity of option price to implied volatility changes; measured in dollars per 1% volatility change.

Volatility of volatility (vol of vol): A parameter in stochastic volatility models controlling how much volatility varies; higher values produce fatter smile wings.

Volga: The sensitivity of vega to volatility changes (second derivative of price with respect to volatility); important for smile dynamics.

Abbreviations

Related Articles

For detailed treatment of the Black-Scholes model, see Black-Scholes Model Inputs and Outputs.

For understanding volatility surfaces, review Implied Volatility Surface Basics.

This glossary is updated quarterly to reflect evolving terminology and market conventions.