A financial instrument changes value when the market changes. A bond can fall in price when interest rates rise, a credit hedge can gain when a borrower’s credit spread widens, and an option can become more valuable when uncertainty increases. Risk sensitivities describe how much of that change comes from a particular market input.

This module introduces DV01 and CS01 for interest-rate and credit-spread changes, followed by delta, gamma and vega for options. It then explains curvature, which measures how a larger move departs from a straight-line approximation. You will compare bonds, swaps, credit default swaps, equities and options rather than assume that the same sensitivity has the same sign for every instrument.

Begin by changing one market input while keeping the others fixed. Compare the repriced value with the sensitivity approximation, and notice when the approximation becomes less reliable. By the end, you should be able to explain what a hedge protects, what it leaves exposed, and why the size and units of a reported sensitivity matter. The examples use simplified pricing models, so their assumptions remain visible beside each calculation.

Continue your learning: Yield curves and bootstrapping · Stochastic calculus · FRTB market risk

Look up a term as you work through the examples. Definitions explain the terminology; the source references and assumptions beside each lesson explain the scope of its calculation.

72 definitions

Accrual

Accrual allocates interest or other income and expense to the period in which it is earned or incurred. Accrued interest can make a settlement amount differ from a quoted clean bond price.

Accrued interest

Accrued interest is interest earned or owed since the last payment date but not yet paid. Its calculation depends on the contract’s day-count and payment conventions.

Actual, hypothetical and risk-theoretical P&L

P&L means profit and loss. Actual P&L follows the applicable realised desk calculation; hypothetical P&L revalues a specified static portfolio; risk-theoretical P&L uses the risk factors represented by the model. Their definitions and exclusions must be matched before comparing them.

Asset

An asset is a resource or contractual right with economic value. For a bank, examples include cash, securities and loans that customers are expected to repay.

At the money

An option is at the money when its strike is close to the relevant underlying price or forward price under the stated convention. Moneyness describes that relationship more generally.

Basis point

One basis point is one hundredth of a percentage point. A rise from 4.00% to 4.01% is a one-basis-point increase.

Basis risk

Basis risk arises when two related prices or rates do not move together as assumed. A hedge tied to one benchmark may therefore leave exposure to a different benchmark, location or contract specification.

Black–Scholes

The Black–Scholes model values suitable European options under assumptions including continuous trading and constant model volatility. Its formula is useful for teaching how time, price and volatility affect option value, but its assumptions are restrictive.

Bond

A bond is a debt security under which an issuer promises specified payments to investors. Its value depends on the timing of those payments, discount rates and the risk of non-payment.

Bootstrapping

Bootstrapping builds a curve sequentially by using market instruments to determine additional discount factors or rates once earlier points are known. The result depends on instrument selection and pricing conventions.

Call and put

A call gives the holder the right to buy the underlying on the contract’s terms. A put gives the right to sell it. The holder’s exercise choice makes the payoff asymmetric.

Carry and opportunity cost

Carry is the ongoing income or expense from holding and financing a position. Opportunity cost compares that choice with a specified alternative, whose credit, capital and liquidity risks may differ.

Cash flow

A cash flow is a payment or receipt at a specified time. Its amount, date, currency and legal owner affect both valuation and liquidity.

Compounding

Compounding describes how interest accumulates on principal and previously earned interest. Under continuous compounding, accumulation is represented by an exponential function rather than a sequence of periodic additions.

Concentration

Concentration means that a large share of exposure or funding depends on a small number of sources or common drivers. The Herfindahl–Hirschman Index, or HHI, sums squared shares to measure size concentration but does not by itself measure behavioural dependence.

Correlation

Correlation summarises a specified relationship between two variables. A linear correlation does not fully describe joint tail behaviour, and a prescribed regulatory correlation is a model parameter rather than a measured relationship for every portfolio.

Coupon

A coupon is an interest payment on a bond or loan. The coupon rate determines the contractual payment; it is not necessarily the yield available to an investor at the current market price.

Credit default swap

A credit default swap exchanges a protection premium for a contractual payment if a specified credit event occurs. The protection buyer generally benefits when the reference credit deteriorates, subject to the contract’s terms.

Credit spread

A credit spread is the additional yield or premium associated with credit exposure relative to a reference rate. It can reflect default risk, liquidity and other market effects, so it is not a direct default probability.

CS01

CS01 measures the value effect associated with a one-basis-point credit-spread move. A bond holder and a buyer of credit protection can have opposite signed CS01 values.

Currency codes

GBP denotes pounds sterling, USD denotes US dollars, EUR denotes euros and JPY denotes Japanese yen. Currency labels matter because values and thresholds must be converted consistently before they are compared or combined.

Curvature

Curvature describes the part of a value change that a first-order sensitivity does not capture. Its regulatory construction differs by framework: FRTB uses specified repricing shocks, whereas SIMM derives a curvature exposure from volatility-weighted vega and expiry scaling.

Delta

Delta is the first-order sensitivity of value to a specified underlying input. A delta hedge offsets that local exposure at the current point but may need adjustment when the market moves.

Discount rate

A discount rate is a rate used to value future cash flows in present terms. The appropriate rate depends on the valuation framework and the characteristics of the cash flow.

Drift and diffusion

Drift describes the predictable local direction of a stochastic process under a chosen probability measure. The diffusion term determines the magnitude of its continuous random fluctuations.

DV01

DV01 measures the value effect associated with a one-basis-point interest-rate move. Some systems report a signed price change and others report a positive loss magnitude, so the lesson’s sign convention must be checked.

Equity

Equity represents an ownership interest. On a bank balance sheet, equity is the residual interest in assets after liabilities; in a market-risk example, an equity position usually means shares in a company.

Expiry

Expiry is the time at which an option’s exercise right ends. A European option can be exercised only at expiry, while other exercise styles allow different choices.

Finite difference

A finite difference estimates a sensitivity by repricing after a small change in an input. The bump size, direction and units determine how the observed price change is converted into a sensitivity.

Floating rate

A floating rate resets according to a benchmark or contractual rule. Repricing occurs when the rate used to calculate a payment is updated, which need not coincide with repayment of principal.

Forward contract

A forward contract fixes today the terms of a transaction to occur at a future date. The value of an existing forward changes as current market prices and financing conditions change.

FRTB

FRTB stands for the Fundamental Review of the Trading Book. It is the market-risk capital framework covered in the selected Basel and jurisdictional documents, including standardised and internal-models approaches.

Funding and rollover risk

Funding supports assets and payment obligations through deposits, borrowing and other sources. Rollover risk is the risk that maturing funding cannot be replaced at an acceptable cost or at all.

Gamma

Gamma measures how delta changes as the underlying changes. Convexity describes curvature more generally; bond convexity usually refers to the second-order relationship between price and yield.

Garman–Kohlhagen

The Garman–Kohlhagen model adapts the Black–Scholes framework to a currency option by including domestic and foreign interest rates. It retains similarly simplified volatility and market assumptions.

Hazard rate

A default hazard rate describes the instantaneous rate of default conditional on survival up to that time. A spread divided by LGD is a simplified hazard approximation, not a full credit-curve calibration.

Hedge

A hedge is a position intended to offset part of another exposure. It can leave basis, timing, legal or non-linear risks and can introduce its own funding and counterparty demands.

Implied volatility

Implied volatility is the volatility parameter that makes a selected pricing model match an observed option price. A volatility smile or surface describes how implied volatility varies across strikes and expiries.

Incremental and standalone risk

Standalone risk measures a position by itself. Incremental risk is the change in portfolio risk when the position is added; it can be negative when the new position hedges an existing exposure.

Interest rate

An interest rate expresses the cost of borrowing or the return on lending over a stated period. Its meaning depends on the currency, term, compounding convention and credit or collateral conditions.

Interest-rate swap

An interest-rate swap exchanges interest payments calculated on a reference notional. A common swap exchanges fixed interest for floating interest, allowing the parties to change their exposure to future rates.

Investment grade and high yield

Investment grade and high yield are credit-quality groupings based on the relevant rating or classification rules. Rating labels such as AA and BBB indicate categories, not guarantees of repayment.

Long and short positions

A long position generally benefits from ownership or a rise in the relevant instrument’s value. A short position has the opposite exposure. For complex derivatives, the precise payment and sign convention should always be checked.

Maturity

Maturity is the date or remaining period until a contract ends or principal becomes due. It can differ from an instrument’s next interest-rate reset or an option’s expiry.

Modified duration

Modified duration measures the approximate proportional price response to a small yield change under its convention. It is distinct from contractual maturity and from supervisory duration used in SA-CCR.

Netting set

A netting set contains transactions that can be combined under the relevant legally recognised agreement and calculation rules. Economic offsets across different legal sets do not automatically qualify as regulatory or collateral offsets.

Notional

Notional is the reference amount used to scale a financial contract’s payments or exposure measures. It is not generally the contract’s market value or the amount that would be lost on default.

Option

An option gives its holder a contractual choice, such as buying or selling an asset on specified terms. Optionality also occurs inside other products, for example when a borrower can repay early.

Payoff

A payoff is the amount a contract produces in a specified outcome, usually at exercise or maturity. Its present value is not generally equal to the payoff because timing and uncertainty also matter.

Percentage point

A percentage point is the arithmetic difference between two percentages. An increase from 2% to 3% is one percentage point, but it is a 50% relative increase.

Poisson process

A Poisson process counts randomly timed events with a specified intensity under its assumptions. Adding jumps allows a model to represent discontinuous changes that a purely continuous diffusion cannot generate.

Portfolio

A portfolio is a collection of positions considered together. Its risk depends on their sizes, relationships and permitted offsets, not just the sum of their standalone risks.

Present value

Present value is the value today of a future payment or stream of payments under the selected discounting assumptions. A payment received later is converted into today’s value using a discount factor.

Principal

Principal is the amount borrowed or invested before interest. A derivative’s notional principal is a reference amount used to calculate payments and may never be exchanged.

Probability of default

Probability of default is the probability that a borrower defaults over a defined horizon and under specified conditions. One-year, remaining-lifetime, market-implied and prudential PDs answer different questions.

Random walk

A random walk is a process formed by adding successive random steps. Appropriate scaling of increasingly frequent independent steps can lead to a Brownian-motion limit.

Receive-fixed and pay-fixed

A receive-fixed swap position receives fixed interest and pays floating interest; a pay-fixed position does the reverse. Their value sensitivities generally have opposite signs under the same contract and market assumptions.

Recovery and LGD

Recovery is the amount obtained after default, taking account of the relevant cash flows and costs. Loss given default, or LGD, is the fraction of exposure lost under the selected timing and discounting convention.

Regulatory capital and RWA

Regulatory capital is eligible loss-absorbing funding measured under the applicable rules. Risk-weighted assets express exposures or requirements on a risk-adjusted basis and are used with capital ratios; an exposure amount is not automatically a capital charge.

Risk class and bucket

A risk class groups related risk types, such as interest rates or equities. A bucket is a further prescribed grouping used in aggregation. The definitions differ across FRTB, SA-CVA and SIMM.

Risk factor

A risk factor is a market or credit quantity whose movement affects a position’s value or loss. Examples include an interest rate, an exchange rate, a credit spread and an implied volatility.

Sensitivities-based method

The sensitivities-based method is the FRTB standardised component for delta, vega and curvature risk. It weights and aggregates relevant sensitivities and repricing effects using the applicable bucket and correlation rules.

Sensitivity

A sensitivity measures how a value changes when a specified input changes, under a stated bump and unit convention. Greeks are familiar option sensitivities such as delta, gamma and vega.

Settlement

Settlement is the completion of a payment or delivery obligation. A settlement lag separates the trade date from completion and can create cash needs and exposure before the exchange is final.

Spot price

The spot price is the price or exchange rate for a transaction with the usual near-term settlement convention. It differs from a forward price agreed for a later transaction.

Strike

The strike is the price specified in an option for buying or selling the underlying. Its position relative to the current underlying price helps determine the option’s value and sensitivities.

Tenor

A tenor is a specified period, such as three months or five years. A curve vertex is a selected maturity point at which a rate or sensitivity is represented.

Underlying

The underlying is the asset, rate, index or other reference quantity on which a derivative’s payments depend. A derivative can depend on more than one underlying risk factor.

Vega

Vega measures how value changes with the volatility input of a pricing model. A vega quoted per one volatility percentage point differs by a factor of 100 from a derivative taken with respect to volatility expressed as a decimal.

Volatility

Volatility measures the variability of a price, return or other risk factor over a stated period. It describes dispersion, not the direction in which the next move will occur.

Volatility smile and skew

A volatility smile or skew describes how quoted implied volatility varies across strikes for a given expiry. Using one constant volatility omits this market pattern and can miss risk when the pattern changes.

Yield curve

A yield curve or term structure describes how rates or discount factors vary with maturity. Projection curves and discount curves can serve different purposes within one valuation.