Derivative (finance)

Financial derivative is a financial contract whose value is derived from the value or performance of an underlying asset, financial instrument, index, rate, or other reference variable.

Derivatives are commonly used for hedging, speculation, arbitrage, portfolio management, and risk transfer. The underlying reference can include equities, bonds, commodities, currencies, interest rates, credit instruments, market indexes, or other financial variables.

Concept

The value of a derivative can be represented in simplified form as:

$V_D=f(X,t,\theta )$

where:

• $V_D$ = value of the derivative;
• $X$ = underlying asset, rate, index, or reference variable;
• $t$ = time;
• $\theta$ = other contract and market parameters.

The exact valuation relationship depends on the type of derivative and its contractual terms.

Unlike a conventional financial asset such as a share, a derivative generally derives its economic value from an underlying reference rather than representing a direct ownership claim on that underlying asset.

Underlying assets and reference variables

Common underlying assets and reference variables include:

• stocks;
• bonds;
• commodities;
• currencies;
• interest rates;
• stock-market indexes;
• credit measures;
• volatility;
• other financial or economic variables.

A derivative can also reference a combination of assets or variables.

Principal types

Major categories of derivatives include:

• forwards;
• futures;
• options;
• swaps.

These instruments differ in their contractual structure, trading venue, settlement method, and risk characteristics.

Forward contracts

A forward contract is an agreement between two parties to buy or sell an underlying asset or reference instrument at a specified price on a future date.

A simplified payoff for a long forward position is:

$\text{Payoff}=S_T-K$

where:

• $S_T$ = price of the underlying asset at expiration;
• $K$ = agreed forward price.

The payoff of a short forward position is:

$\text{Payoff}=K-S_T$

Forward contracts are generally traded over the counter and can be customized according to the requirements of the counterparties.

Futures contracts

A futures contract is a standardized agreement to buy or sell an underlying asset or financial instrument at a specified price according to standardized contractual terms.

Futures are commonly traded on organized exchanges and are generally subject to central clearing and margin requirements.

A simplified long futures payoff at expiration is:

$\text{Payoff}=S_T-K$

Futures are widely used for managing exposure to commodities, equity indexes, interest rates, currencies, and other financial variables.

Options

An option gives its holder the right, but not the obligation, to buy or sell an underlying asset according to specified contractual terms.

A call option gives the holder the right to buy the underlying asset.

A put option gives the holder the right to sell the underlying asset.

The payoff of a long call at expiration is:

$\text{Call Payoff}=max(S_T-K,0)$

The payoff of a long put is:

$\text{Put Payoff}=max(K-S_T,0)$

The buyer generally pays an option premium for these rights.

Swaps

A swap is a derivative contract in which two parties agree to exchange specified cash flows according to predetermined terms.

Common types include:

• interest-rate swaps;
• currency swaps;
• commodity swaps;
• credit default swaps.

For a simplified fixed-for-floating interest-rate swap, the net payment can be represented as:

$\text{Net Payment}=\text{Floating Payment}-\text{Fixed Payment}$

The actual calculation depends on the notional amount, applicable rates, payment frequency, day-count convention, and contractual terms.

Hedging

Derivatives can be used to reduce exposure to adverse movements in an underlying variable.

For example, a company expecting to purchase a commodity in the future may use futures contracts to reduce uncertainty about the future purchase price.

A simplified hedged position is:

$\text{Hedged Position}=\text{Underlying Exposure}+\text{Offsetting Derivative}$

The objective is generally to reduce the sensitivity of the combined position to a specified risk factor.

Speculation

Derivatives can also be used to take positions based on expectations about future market movements.

For example, an investor expecting an increase in the price of an asset may purchase a call option or establish a long futures position.

Derivatives can provide leveraged exposure, meaning that relatively small amounts of initial capital can provide exposure to a larger notional amount.

A simplified leverage measure is:

$\text{Leverage}\approx \frac{\text{Notional Exposure}}{\text{Capital Invested}}$

The actual economic leverage depends on the derivative, its terms, market conditions, and the method used to measure exposure.

Arbitrage

Derivatives can also be used in arbitrage strategies when related financial instruments are priced inconsistently.

A simplified arbitrage relationship is:

$\text{Profit}=\text{Value of Offset}-\text{Cost of Position}$

subject to transaction costs, financing costs, market constraints, and the specific contractual terms.

Arbitrage activity can contribute to price relationships between derivatives and their underlying assets.

Derivative pricing

The value of a derivative depends on the characteristics of the underlying asset and other market and contractual variables.

A simplified pricing relationship is:

$V_D=f(S,K,T,\sigma ,r,q)$

where:

• $S$ = underlying asset price;
• $K$ = strike price or contractual price;
• $T$ = time to expiration or relevant maturity;
• $\sigma$ = volatility;
• $r$ = interest rate;
• $q$ = dividend yield or other carry-related variable.

Not every derivative depends on all of these variables.

No-arbitrage pricing

Many derivative-pricing frameworks are based on no-arbitrage principles.

A simplified no-arbitrage condition is:

$\text{Equivalent Cash Flows}\Rightarrow \text{Equivalent Price}$

If two portfolios generate identical cash flows under the same relevant conditions but have different prices, an arbitrage opportunity may exist, subject to transaction costs and other market frictions.

Margin and collateral

Many derivatives, particularly exchange-traded futures and certain cleared or OTC derivatives, involve margin or collateral requirements.

A simplified margin relationship is:

$\text{Margin}=\text{Collateral Required to Support Position}$

Margin is not necessarily the same as the total economic value or notional value of the derivative.

Changes in the market value of a derivative can generate additional margin requirements.

Notional amount

The notional amount is a reference amount used to calculate payments or economic exposure under many derivative contracts.

For a simplified derivative whose payment depends on a rate difference:

$\text{Payment}=N\times (r_1-r_2)\times t$

where:

• $N$ = notional amount;
• $r_1$ and $r_2$ = applicable rates;
• $t$ = relevant time fraction.

The notional amount does not necessarily represent the amount exchanged between the counterparties.

Exchange-traded and over-the-counter derivatives

Derivatives can be traded on organized exchanges or through OTC markets.

Exchange-traded derivatives generally have standardized:

• contract sizes;
• expiration dates;
• settlement procedures;
• trading rules;
• margin requirements.

OTC derivatives are privately negotiated between counterparties and can generally be customized.

OTC contracts may therefore include customized:

• notional amounts;
• maturity dates;
• settlement terms;
• underlying references;
• payment schedules;
• contractual conditions.

Central clearing

Many standardized derivatives are cleared through a central counterparty (CCP).

The CCP becomes the buyer to every seller and the seller to every buyer, subject to the applicable clearing framework.

A simplified representation is:

$\text{Buyer}\leftrightarrow \text{CCP}\leftrightarrow \text{Seller}$

Central clearing can reduce bilateral counterparty exposure but introduces dependence on the clearing system, margin arrangements, and the financial strength and governance of the CCP.

Settlement

Derivatives can be settled through physical delivery or cash settlement.

Physical settlement involves delivery of the underlying asset or instrument according to the contract.

Cash settlement involves payment of the difference between specified reference values.

A simplified cash-settlement amount is:

$\text{Settlement Amount}=Q\times (S_T-K)$

where $Q$ represents the relevant contract quantity.

The actual settlement calculation depends on the derivative's contractual specifications.

Risk characteristics

Derivatives can involve several forms of financial risk, including:

• Market risk;
• Credit risk;
• Counterparty risk;
• Liquidity risk;
• Basis risk;
• Leverage risk;
• Volatility risk;
• Model risk;
• Settlement risk;
• Operational risk.

The magnitude and nature of these risks depend on the derivative and the position taken.

Basis risk

Basis risk occurs when the derivative used for hedging does not move exactly in line with the underlying exposure.

A simplified basis can be represented as:

$B=S-F$

where:

• $B$ = basis;
• $S$ = spot price;
• $F$ = relevant derivative price.

Changes in the basis can cause a hedge to be imperfect.

Leverage and losses

Derivatives can create substantial economic exposure relative to the capital initially committed.

A simplified relationship is:

$\text{Economic Exposure}>\text{Initial Capital}$

in situations where the derivative provides leveraged exposure.

Leverage can magnify both gains and losses. Certain derivative positions, particularly some forms of uncovered option writing, can expose participants to losses substantially exceeding the initial amount received or deposited.

Derivative markets

Derivative markets exist across several asset classes.

Examples include:

• equity derivatives;
• commodity derivatives;
• foreign-exchange derivatives;
• interest-rate derivatives;
• credit derivatives;
• volatility derivatives.

These markets allow participants to transfer, transform, or assume exposure to different sources of financial risk.

Importance

Derivatives are important financial instruments for transferring and managing risk, discovering prices, obtaining market exposure, and implementing investment and financing strategies. They allow market participants to separate certain financial risks from ownership or direct transactions in the underlying assets.

However, derivatives can also introduce leverage, counterparty exposure, liquidity risk, and other financial risks. Their appropriate use therefore requires an understanding of the underlying exposure, contractual terms, valuation methodology, collateral arrangements, and associated risks.

See also

• Derivative (finance)
• Hedging
• Option (finance)
• Call option
• Put option
• Forward contract
• Futures contract
• Swap (finance)
• Interest rate swap
• Currency swap
• Credit default swap
• Derivative pricing
• Financial risk
• Risk management
• Arbitrage
• Leverage (finance)
• Basis risk
• Counterparty risk
• Liquidity risk
• Margin (finance)
• Central counterparty
• Quantitative finance