Course O · Options and Derivatives

Glossary

A

Arbitrage: a profit locked in with no money of your own and no risk O1 Forwards and futures: the price of waitingalso in O3

B

Backward induction: valuing a tree from its last step back to today, one one-step problem at a time O3 Pricing without a crystal ball: the binomial tree
Bid–ask spread: the gap between the price a dealer pays for an option and the price it charges; a seller receives less than the middle price, a cost on every trade O7 Testing an options idea honestly
Binomial tree: a model of a price that, at each step, goes up by a factor u or down by a factor d, and nothing else O3 Pricing without a crystal ball: the binomial tree
Bisection: a search that keeps a range holding the answer and halves it each try, keeping the half the answer must be in O6 Implied volatility and the smile
Black–Scholes formula: a European option's price when the log price follows a random walk with fixed volatility and no jumps, the copy is adjusted without pause, and trading is free; the binomial tree's limit O4 Black–Scholes, from the tree
Bump and reprice: estimating a Greek by moving one input a little, pricing again, and dividing the change in price by the move O5 The Greeks

C

Cash-and-carry: borrow, buy the underlying and sell it forward; locks in a profit when the forward price is above the no-arbitrage price O1 Forwards and futures: the price of waiting
Continuous compounding: interest added continuously, so money grows by e^rT instead of (1 + r)^T O1 Forwards and futures: the price of waiting
Cost of carry: what it costs to hold the underlying until delivery: interest, plus storage, minus income O1 Forwards and futures: the price of waiting

D

d₁, N(d₁): in the Black–Scholes model, d₁ = d₂ + σ√T; N(d₁) is a call's delta, the shares in the copy (the shares and loan that reproduce its payoff) today O4 Black–Scholes, from the tree
d₂, N(d₂): in the Black–Scholes model, how many standard deviations the middle (median) of the log year-end price sits above the strike; N(d₂) is the q-chance that a call ends above its strike O4 Black–Scholes, from the tree
Daily settlement: a futures contract's gains and losses paid every day between the two sides' deposits; with a fixed interest rate the futures price equals the forward price O1 Forwards and futures: the price of waiting
Delta (Δ): the shares in the replicating portfolio: the option's swing between up and down divided by the share's O3 Pricing without a crystal ball: the binomial treealso in O5
Delta hedging: holding delta shares against an option, so that small moves of the share cancel O8 Capstone: hedge an option and explain the P&L
Dividend: cash a company pays out for each share to whoever holds it O1 Forwards and futures: the price of waiting

E

Error bar: a simulated average plus or minus two standard errors; a bar from fresh paths misses the value endless runs would give about 1 time in 20 O4 Black–Scholes, from the tree
European option: an option that can be used only at expiry O2 What no-arbitrage forbids: option bounds and put–call parityalso in O4
Expectancy: the average result per trade, each outcome weighed by its size: E = p × W − (1 − p) × L O7 Testing an options idea honestly

F

Fat tails: more very large moves than a bell curve with the same spread allows O6 Implied volatility and the smile

G

Gamma (Γ): the change in delta per euro of the share's price; for a European call in the Black–Scholes model, largest with the share near the strike (for this lesson's call, a little below it, at €95.6) O5 The Greeks
Gamma P&L: a delta-hedged option's result from one day: about ½ × gamma × S² × (σ_imp² − σ_real²) × the day in years, for the seller, when moves are small O8 Capstone: hedge an option and explain the P&L

H

Hedge: a holding that cancels a risk, such as selling delta shares against a call you own O5 The Greeks
Hedging error: what a delta hedge leaves because it is rebalanced at intervals, not without pause; in the Black–Scholes model's world its spread grows about √k times with k days between adjustments O8 Capstone: hedge an option and explain the P&L

I

Implied volatility: the volatility at which the Black–Scholes formula gives a European option's price; one exists, and only one, when the price lies strictly inside the no-arbitrage bounds O6 Implied volatility and the smile
Intrinsic value: what an option would pay if its expiry were today O2 What no-arbitrage forbids: option bounds and put–call parity

J

Jump: a move too large and sudden for the hedge to follow, outside what ½ × gamma × move² describes O8 Capstone: hedge an option and explain the P&L

L

Lognormal: a price whose log lands on a bell curve: never below 0, with a long tail of high prices O4 Black–Scholes, from the tree

M

Margin call: a broker's demand for more cash after a loss on a position bought or sold with borrowed money; if it isn't met, the position is closed O7 Testing an options idea honestly
Moneyness: in the money (intrinsic value above zero), at the money (price equal to the strike) or out of the money O2 What no-arbitrage forbids: option bounds and put–call parity

N

N(x): the share of a bell curve's area left of x standard deviations from its middle O4 Black–Scholes, from the tree
Negative skew: results made of many small gains and a few large losses: a long tail on the losing side, as a sold option has O7 Testing an options idea honestly
No-arbitrage band: the range of forward prices at which neither cash-and-carry nor its reverse pays, once rates differ or trades cost money O1 Forwards and futures: the price of waiting
No-arbitrage bounds: the lowest and highest prices a European option can have without leaving a locked-in profit (underlying paying nothing before expiry, one rate, free trades) O2 What no-arbitrage forbids: option bounds and put–call parity
No-arbitrage forward price: the forward price at which no combination of trades locks in an arbitrage; with one rate, free trades and short selling, a single price O1 Forwards and futures: the price of waiting
No-arbitrage price: the only price that allows no arbitrage: in a binomial tree, the replicating portfolio's cost O3 Pricing without a crystal ball: the binomial tree

P

P&L attribution: splitting a result into the parts that made it: here the formula part, a jump, costs and the hedging error O8 Capstone: hedge an option and explain the P&L
Peso problem: averages from a sample in which a rare, large event never happened; with a yearly chance q, the same each year and independent across years, n years miss it with chance (1 − q)ⁿ O7 Testing an options idea honestly
Present value: what a later amount is worth today O2 What no-arbitrage forbids: option bounds and put–call parity
Put–call parity: for European options with the same strike and expiry on an underlying paying nothing before expiry, a call plus the strike's present value costs what a put plus the underlying costs O2 What no-arbitrage forbids: option bounds and put–call parityalso in O4

R

Realised volatility: the volatility a share actually moves with, as against the implied volatility a price was charged at O8 Capstone: hedge an option and explain the P&L
Rebalancing: moving the holding to the new delta, at chosen intervals O8 Capstone: hedge an option and explain the P&L
Replicating portfolio: shares and borrowing (or lending) that pay exactly what an option pays in every outcome of the model O3 Pricing without a crystal ball: the binomial tree
Reverse cash-and-carry: sell the underlying short, lend the money and buy it forward; locks in a profit when the forward price is below the no-arbitrage price O1 Forwards and futures: the price of waiting
Risk-neutral probability (q): in a binomial tree, the weight on "up" that makes the share grow on average at the interest rate; used for pricing, it is not the real chance of up O3 Pricing without a crystal ball: the binomial tree

S

Selling a put: receiving the put's price today and paying, on its last day, the strike minus the share's price if that is above zero: small likely gains, rare large losses O7 Testing an options idea honestly
Short selling: borrowing an asset, selling it now and giving one back later O1 Forwards and futures: the price of waitingalso in O3
Synthetic position: a position copied from other contracts, such as a put from a bought call, the share sold short and a deposit O2 What no-arbitrage forbids: option bounds and put–call parity

T

The Greeks: measures of how much an option's price moves when one input moves a little and the others stay put O5 The Greeks
Theta (Θ): the change in an option's price as time passes, the others fixed; for a European call in the Black–Scholes model, negative O5 The Greeks
Time value: an option's price minus its intrinsic value; gone at expiry O2 What no-arbitrage forbids: option bounds and put–call parity

V

Vega: the change in an option's price per point of volatility; for a European call in the Black–Scholes model, positive O5 The Greeksalso in O6
Volatility (σ): the standard deviation of a share's yearly log return, ln(end price ÷ start price): how widely its yearly returns typically spread O4 Black–Scholes, from the tree
Volatility skew: implied volatility highest at low strikes and falling with the strike; it appears when big falls are more likely, in the prices, than one bell allows O6 Implied volatility and the smile
Volatility smile: implied volatility plotted against strike, lowest in the middle and higher at both ends; in the Black–Scholes setting it appears when the year-end price curve has fatter tails on both sides than one bell O6 Implied volatility and the smile

W

Win rate: the share of trades that made money; on its own it says nothing about the average result O7 Testing an options idea honestly

Φ

φ(x): the bell curve's height at x: e^−x²/2 ÷ √(2π) O5 The Greeks