Derivatives

Accreting Swap A swap with a notional principal that increases over time, matching the growing size of an expanding debt or investment.

American Option An American option can be exercised at any time up to and including its expiration date, giving the holder more flexibility and typically higher value than a European option.

American vs European Option How the right to exercise at any time (American) or only at maturity (European) affects pricing, hedging, and strategy.

Amortizing Swap A swap with a notional principal that decreases over time, matching the repayment schedule of a declining debt.

Asian Option An Asian option's payoff depends on the average price of the underlying asset over a set period, rather than just the final price, reducing the impact of price spikes.

Asian Option Pricing: How Averaging Reduces Volatility Exposure Asian average rate option pricing: arithmetic and geometric averaging reduce realized volatility, enabling lower option premiums and reduced hedging costs.

Asset Swap A package combining a fixed-coupon bond with an interest-rate swap to convert it into a synthetic floating-rate instrument.

Asset-or-Nothing Option A digital option that delivers the underlying asset itself rather than a fixed cash amount when in-the-money.

Assignment (Options) The process of obligating a short option seller to fulfill their contract when a long option holder exercises.

At-the-Money At-the-money describes an option whose strike price equals the underlying asset's current price, making it the most sensitive to volatility and direction.

Average Strike Option An average strike option sets its exercise price as the arithmetic mean of the underlying asset's price over a sampling window, reducing pricing uncertainty.

Average-Price vs Average-Strike Asian Options Average-price and average-strike Asian options reduce volatility impact differently. Average-price options average the asset's price over time; average-strike options fix the strike as the mean price. Learn when each lowers hedging costs.

Bachelier Model The normal-distribution option pricing formula, ideal for near-zero or negative interest rates and forwards.

Backwardation Backwardation is a situation where futures prices are lower for distant delivery dates than for near-term dates, signaling immediate scarcity or convenience value.

Barone-Adesi Whaley Model A quadratic approximation for American option pricing that decomposes value into European and early-exercise components, widely used by practitioners.

Barrier Option A barrier option becomes active or worthless when the underlying asset's price crosses a predetermined level, making it cheaper and more exotic than vanilla options.

Barrier Option Pricing: Knock-In and Knock-Out Mechanics Barrier option pricing accounts for automatic activation or cancellation when spot price hits a threshold; explains knock-in/knock-out mechanics and Greek sensitivities.

Basis (Futures) Basis is the difference between a futures contract price and the spot price of the underlying asset, measuring the market's valuation of carrying the asset forward.

Basis Risk The risk that the spot-futures spread (basis) changes unexpectedly, leaving a hedger worse off—the residual exposure that remains after locking in a futures price.

Basis Swap A swap that exchanges cash flows tied to two different floating-rate indices to profit from or neutralize differences in their rates.

Basket Derivative Derivative instrument with multiple underlying assets, allowing hedging or speculation on asset-class relationships.

Basket Option A basket option is a derivative whose payoff depends on the performance of multiple underlying assets, often enabling correlation-based strategies at lower cost than owning individual options.

Bear Call Spread A net-credit option strategy using short and long calls at different strikes, designed to profit from sideways or declining markets with defined risk.

Bear Put Spread A debit strategy using short and long puts at different strikes, used to bet on a stock declining or staying flat with defined maximum loss.

Bermuda Option A Bermuda option can be exercised on specific dates (not just at expiration), sitting between the flexibility of American options and the simplicity of European options.

Bermudan Option vs American Option A Bermudan option allows exercise on a discrete set of dates, whereas an American option can be exercised any time before expiry—a difference reflected in both value and pricing.

Bid-Ask Spread in Derivatives Markets Explained Understand why bid-ask spreads are wider in derivatives than equities: liquidity depth, counterparty risk, and how spreads affect the true cost of trading options and futures.

Bid-Ask Spread in Options The gap between buy and sell prices in options, often wider than equities due to lower volume and higher hedging costs.

Binary Option A binary option is a derivative with a predetermined fixed payoff that either occurs in full or not at all, depending on whether the underlying asset crosses a strike price.

Binary Option Settlement Risk: How All-or-Nothing Payoffs Create Cliff Risk Binary options have discontinuous payoffs at the strike; delta hedging fails near expiry, creating cliff risk and settlement volatility.

Binomial Model vs Black-Scholes: When Each Is More Accurate Compare binomial and Black-Scholes option pricing: when each is more accurate, computational trade-offs, and suitability for American vs European options.

Binomial Option Pricing The binomial option pricing model values options by building a tree of possible future stock prices and working backward to calculate option value at each node.

Bjerksund-Stensland Model An analytic approximation formula for American option pricing that avoids the computational cost of binomial trees and full partial differential equations.

Black-76 Model Fischer Black's 1976 extension of Black-Scholes for pricing futures options and interest-rate derivatives.

Black-Scholes Delta Calculation Explained With an Example How to calculate delta using the Black-Scholes formula with a worked example, showing the precise inputs and step-by-step computation.

Black-Scholes Model The Black-Scholes model is the foundational formula for pricing European options, using stock price, strike, time, volatility, and interest rates to calculate fair value.

Black-Scholes Model Assumptions and Their Limitations The Black-Scholes formula rests on five key assumptions—constant volatility, no dividends, European exercise, efficient markets, no friction—that all break down in real trading.

Box Spread An advanced option strategy combining a bull call spread and bear call spread with overlapping strikes, designed to lock in a risk-free profit if entry and exit prices are favorable.

Broken Wing Butterfly An asymmetric butterfly spread with unequal wing widths that shifts risk to one side and reduces or eliminates upfront cost.

Bull Call Spread A two-leg option strategy that caps both risk and profit by buying and selling calls at different strikes, typically used when expecting modest upside.

Bull Put Spread A credit strategy using short and long puts at different strikes, designed to profit when the stock stays above the short put strike through expiration.

Bull Put Spread for Income Generation How to use a bull put spread for income: strike selection, risk management, and return-on-capital calculation for conservative traders.

Bull Spread (Call) Multi-leg options strategy combining a long call and short higher-strike call to reduce upfront cost and cap profit potential.

Butterfly Spread A limited-risk option strategy using three different strikes, designed to profit from minimal price movement and high time decay.

Buy-Write A strategy of simultaneously buying stock and selling call options on that same stock, combining stock ownership with upside cap for income generation.

Buying Options vs Selling Options: Which Side Wins Over Time Buyers of options pay premium and risk unlimited losses on the wrong forecast. Sellers collect premium but face unlimited loss. Learn the statistical edge.

Calculating Option Break-Even at Expiration: Worked Examples How to calculate option break-even price at expiration for calls and puts. Step-by-step formulas with dollar examples for long and short positions.

Calendar Spread An option strategy that profits from time decay and volatility expansion by selling a near-term option and buying a longer-dated one at the same strike.

Calendar Spread Futures Explained A calendar spread in futures buys one contract expiry and sells another, profiting from changes in the price curve between maturities.

Calendar Spread in a Low-Volatility Environment Calendar spreads gain when implied volatility rises. Learn why low VIX conditions create opportunities and how to choose expiration gaps.

Calibrating a Stochastic Volatility Model to the Implied Volatility Smile How to fit stochastic volatility model parameters to market option prices and the trade-offs in calibration.

Call Backspread An option strategy that sells fewer near-the-money calls and buys a greater number of out-of-the-money calls to profit from explosive upside moves.

Call Option A call option is a contract giving the holder the right, but not the obligation, to buy an underlying asset at a fixed price on or before a specific date.

Call Ratio Spread An advanced option strategy that buys one call and sells multiple calls at a higher strike, designed to enhance returns in sideways markets but with unlimited loss risk above the short calls.

Call Spread A multi-leg options strategy combining a long call and short call to define risk and reward boundaries.

Call Spread Options strategy of buying a call at a lower strike and selling a call at a higher strike for net reduced premium cost.

Call Spread Strategy Limited-risk options strategy combining a long call with a short call at a higher strike, capping both profit and loss.

Call Swaption Option that grants the right, but not obligation, to enter into an interest rate swap as the fixed-rate payer.

Callable Swap A swap contract where one party has the right to terminate the agreement early on specified dates, shifting exercise risk to the counterparty.

Capped Call Option A capped call option limits maximum payoff at a preset ceiling price; issuers use capping to reduce premium cost or manage risk, creating a defined profit range.