An option that can be exercised at any time up to expiry. Worth at least as much as the European option with the same terms.

A Monte Carlo variance-reduction technique that pairs every random draw with its negative.

An option whose strike equals the current spot price. Gamma, Vega and Theta all peak here.

An option on the average price over the option's life rather than the final price. Cheaper than the vanilla because the average is less volatile.

An option that is knocked out or knocked in when the spot touches a barrier level. Knock-in plus knock-out equals the vanilla.

Pays a fixed amount, or one share, if the option finishes in the money. Also called a binary option.

A trade with no cost and no risk that still makes money. Pricing theory assumes none exists, which pins down prices such as put-call parity.

A 1987 approximation for American options that adds an early-exercise premium to the European price.

A lattice in which the price moves up or down each step. Prices are found by rolling back from expiry.

A closed-form American approximation assuming a flat early-exercise boundary.

The 1973 closed-form price of a European option under constant volatility and lognormal prices.

The right, but not the obligation, to buy the underlying at the strike price.

The rate at which Delta decays with time. Matters over weekends and near expiry.

The rate at which Gamma changes with time.

A Monte Carlo variance-reduction technique that corrects the estimate using a correlated quantity with a known expectation. For arithmetic Asians the geometric Asian is the classic choice.

The value of holding an American option rather than exercising it now. Exercise is optimal where the payoff exceeds it.

The classic binomial parameterisation with u = exp(σ√dt) and d = 1/u.

A finite-difference scheme averaging explicit and implicit steps. Stable and second-order accurate in time.

The sensitivity of the option price to the underlying price. Also the hedge ratio.

The continuous rate at which the underlying pays dividends. Lowers the forward price, hurting calls and helping puts.

The extra value of an American option over its European twin.

An option that can be exercised only at expiry.

A finite-difference scheme where each new value comes directly from old ones. Simple but only stable with small time steps.

Solving the pricing partial differential equation numerically on a grid of spot and time.

The rate at which Delta changes with the underlying price. The curvature of the option value.

The price process assumed by Black-Scholes: continuous, with lognormal returns and constant volatility.

The volatility that makes the Black-Scholes price match an observed market price. How options are quoted.

A finite-difference scheme that solves a tridiagonal system each step. Stable for any step size.

An option that would pay off if exercised now: a call with spot above strike or a put with spot below.

The payoff from exercising immediately. The option's value above this is time value.

A binomial parameterisation that places a node on the strike and converges much faster than others.

Least-squares Monte Carlo: estimates continuation values by regression to price American options by simulation.

Pricing by simulating many possible futures and averaging the discounted payoffs.

An option that would pay nothing if exercised now.

A Monte Carlo sensitivity computed by differentiating the payoff along each simulated path rather than by bumping.

The right, but not the obligation, to sell the underlying at the strike price.

C − P = S e^(−qT) − K e^(−rT). Holds without any model assumption.

The sensitivity of the option price to the risk-free interest rate.

The probability measure under which every asset grows at the risk-free rate. Prices are discounted expectations under it.

The continuously compounded return on a riskless deposit over the option's life.

The pattern of implied volatility varying with strike. Evidence that markets do not believe in constant volatility.

The rate at which Gamma changes with the underlying price.

The statistical uncertainty of a Monte Carlo estimate. Shrinks with the square root of the number of paths.

The price at which the option holder may buy or sell the underlying.

The sensitivity of the option price to the passage of time. Usually negative: time decay.

The option value above intrinsic value, reflecting the chance of a better outcome before expiry.

A lattice with up, flat and down moves each step. Equivalent to an explicit finite-difference grid.

The sensitivity of Delta to volatility, equally of Vega to spot.

The sensitivity of the option price to volatility.

The annualised standard deviation of the underlying's log returns.

The sensitivity of Vega to volatility: convexity in volatility.

The sensitivity of Gamma to volatility.

A call and a put bought at the same strike. Profits from a large move in either direction and loses the two premiums if the spot stays put.

A straddle built with an out-of-the-money call and put. Cheaper than a straddle but needs a bigger move to pay off.

A long option and a short option of the same type and maturity at different strikes. Caps both the profit and the loss.

One low-strike option, two short middle-strike options and one high-strike option. Pays most if the spot expires at the middle strike.

A short put spread below the spot and a short call spread above it. Collects premium if the spot stays inside the range.

The spot at expiry where a position's profit is exactly zero, after the premium paid or received.

Owning the underlying and selling a call against it. The premium is income; the upside above the strike is given away.

The pattern of implied volatility rising away from the at-the-money strike. Evidence that the market prices fatter tails than the lognormal model.

A smile that is lopsided: higher implied volatility for low strikes than high ones, typical of equity indices.

How implied volatility changes with maturity at a fixed strike.

Implied volatility as a function of both strike and maturity. Traders read option prices off it.

The price agreed today for delivery at a future date: the spot grown at the rate less the yield. Strikes above the forward are out-of-the-money calls; below it, out-of-the-money puts.

A model in which volatility is itself random rather than a fixed number. Explains the smile without abandoning no-arbitrage pricing.

The standard stochastic-volatility model: variance follows a mean-reverting square-root process correlated with the spot. Priced by integrating its characteristic function.

The σ parameter of Heston: how much the variance itself fluctuates. Zero recovers Black-Scholes; larger values lift the wings of the smile.

2κθ > σ² in Heston. When it holds the variance process never reaches zero.

The tendency of a process to be pulled back towards a long-run level. In Heston the variance reverts to θ at speed κ.

The empirical tendency for volatility to rise when prices fall. Captured in Heston by a negative correlation ρ, which produces the equity skew.

A price process that diffuses like Black-Scholes but also gaps at random times. Merton's version has lognormal jump sizes and a closed-form series price.

A counting process in which events arrive independently at a constant rate λ. The number of events in time T has mean λT.

The λk subtracted from the drift in a jump model so that the expected return still equals the risk-free rate after allowing for the average jump.

Heston plus Merton jumps: stochastic volatility for the long-dated smile and jumps for the short-dated one. Priced through the product of the two characteristic functions.

The expected value of exp(iu ln S_T) as a function of u. Known in closed form for Heston, Merton and Bates, and enough to price any European option by one integral.

An option whose payoff uses the highest or lowest spot seen over its life. Floating-strike lookbacks are never out of the money.

Two ways to average a path. The geometric average of lognormal prices is lognormal, which gives a closed form; the arithmetic average, used in real contracts, has none.

Checking a barrier or recording an average only at fixed times rather than continuously. It makes barriers harder to hit and extremes less extreme.

A knock-in plus a knock-out option with the same barrier, strike and maturity equals the vanilla.

The date on which a stock starts trading without the right to the next dividend; the price drops by roughly the dividend.

Pricing with known cash dividends by removing their present value from the spot and treating the rest as lognormal.

Only ever optimal just before an ex-dividend date, when the dividend to be captured exceeds the time value given up.

A volatility that is a deterministic function of the spot level and time. Dupire's formula derives the unique one that reproduces a given implied-volatility surface.

Local variance equals a calendar spread divided by a butterfly spread of call prices: (∂C/∂T + (r−q)K ∂C/∂K + qC) / (½K² ∂²C/∂K²).

The second derivative of the call price with respect to strike is the discounted risk-neutral density of the spot at expiry.

For short maturities the local-volatility skew is about twice the implied-volatility skew.

The annualised standard deviation of past log returns. What actually happened, as opposed to implied volatility, which is what the market expects.

The tendency of calm and turbulent periods to persist, visible in a rolling-window volatility chart.

A measure of how fat the tails of a distribution are. Excess kurtosis above zero means more extreme moves than a normal distribution.

The fall from a running peak to a subsequent trough, as a percentage. The maximum drawdown is the worst such fall in the sample.

Choosing a model's parameters so that its prices match the market's, usually by minimising the squared difference in implied volatility across strikes and maturities.

A derivative-free optimiser that walks a simplex of trial points downhill. Robust and simple; the workhorse for small calibration problems.

The square root of the average squared miss. In calibration it is quoted in volatility points, where one point is one percentage point of implied volatility.

Holding delta units of the underlying against an option so that small moves in the spot do not change the value of the combined position.

What a delta-hedged option earns from the spot moving: ½ S² Γ times the squared move, which averages to ½ σ² S² Γ per unit of time.

The profit or loss a discretely hedged position picks up between rebalances. Zero on average, with a spread that shrinks with the square root of the rebalancing interval.

Being short options and delta-hedged, so that the position profits if realised volatility turns out below the implied volatility sold.

Stochastic alpha, beta, rho: a per-maturity smile model with a closed-form implied volatility, standard on rates and FX desks.

How the at-the-money volatility moves when the forward moves, set in SABR by β: lognormal at 1, normal at 0.

The asymptotic formula for SABR implied volatility that makes the model fit instantly.

Longstaff and Schwartz's method for American options by simulation: regress the value of waiting on the state, exercise where the payoff beats it, and roll back.

A contract paying realised variance minus a fixed strike. Pure volatility exposure with no premium and no delta.

A payoff of minus twice the log of the price ratio. Delta-hedged it accumulates realised variance, and it is replicated by options weighted by 1/K².

The size of a variance swap quoted so that a one-point move in volatility near the strike is worth about that amount.