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Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in insurance, pension, finance, investment, psychology, medicine, and other industries and professions.

In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets. Proposed by economist Stephen Ross in 1976, it is widely believed to be an improved alternative to its predecessor, the capital asset pricing model (CAPM). APT is founded upon the law of one price, which

An Asian option (or average value option) is a special type of option contract. For Asian options, the payoff is determined by the average underlying price over some pre-set period of time. This is different from the case of the usual European option and American option, where the payoff of the option contract depends on the price of the underlying instrument at exercise; Asian options a

In financial economics, asset pricing refers to the formal development of the principles used in pricing, together with the resultant models. The treatment develops the interrelated paradigms of general equilibrium asset pricing and rational asset pricing, the latter corresponding to risk neutral pricing.

The term asset swap has a number of different meanings:

The Bachelier model is a model of an asset price under Brownian motion presented by Louis Bachelier in his PhD thesis The Theory of Speculation (French: Théorie de la spéculation), published 1900. It is also called the normal model equivalently (as opposed to log-normal model or Black–Scholes model). One early criticism of the Bachelier model is that the probability distribution which h

In options trading, the backspread is the converse strategy to the ratio spread and is also known as the reverse ratio spread. Using calls, a bullish strategy known as the call backspread can be constructed and with puts, a strategy known as the put backspread can be constructed.

Backwardation is a market condition where the price of a commodity's forward or futures contract price is trading below the spot price.

A barrier option is an option whose payoff is conditional upon the underlying asset's price breaching a barrier level during the option's lifetime.

A basket option is a financial derivative, more specifically an exotic option, whose underlying is a weighted sum or average of different assets that have been grouped together in a basket. A basket option is similar to an index option, where a number of stocks have been grouped together in an index and the option is based on the price of the index, but differs in that the members and we

In options trading, a bear spread is a bearish, vertical spread options strategy that can be used when the options trader is moderately bearish on the underlying security.

In finance, the beta (β or market beta or beta coefficient) is a statistic that measures the expected increase or decrease of an individual stock price in proportion to movements of the stock market as a whole. Beta can be used to indicate the contribution of an individual asset to the market risk of a portfolio when it is added in small quantity. It refers to an asset's non-diversifiabl

A binary option is a financial exotic option in which the payoff is either some fixed monetary amount or nothing at all. The two main types of binary options are the cash-or-nothing binary option and the asset-or-nothing binary option. The former pays some fixed amount of cash if the option expires in-the-money while the latter pays the value of the underlying security. They are also cal

In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting, which in general does not exist for the BOPM.

The Black model (also known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, interest rate cap and floors, and swaptions. It was first presented in a paper written by Fischer Black in 1976.

In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) § Interest rate derivatives. It is a one-factor model; that is, a single stochastic factor—the short rate—determines the future evolution of all interest rates. It was the first model to com

In mathematical finance, the Black–Scholes equation, also called the Black–Scholes–Merton equation, is a partial differential equation (PDE) governing the price evolution of derivatives under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.

In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC.

In options trading, a box spread is a combination of positions that has a certain (i.e., riskless) payoff, considered to be simply "delta neutral interest rate position". For example, a bull spread constructed from calls (e.g., long a 50 call, short a 60 call) combined with a bear spread constructed from puts (e.g., long a 60 put, short a 50 put) has a constant payoff of the difference

In options trading, a bull spread is a vertical spread options strategy that is designed to profit from a moderate rise in the price of the underlying security.

In finance, a butterfly (or simply fly) is a limited risk, non-directional options strategy that is designed to have a high probability of earning a limited profit when the future volatility of the underlying asset is expected to be lower (when long the butterfly) or higher (when short the butterfly) than that asset's current implied volatility.

In finance, a calendar spread (also called a time spread or horizontal spread) is a spread trade involving the simultaneous purchase of futures or options expiring on a particular date and the sale of the same instrument expiring on another date. These individual purchases, known as the legs of the spread, vary only in expiration date; they are based on the same underlying market and str

In finance, a call option (often simply a "call") is an option contract that gives its buyer the right, but not the obligation, to buy an agreed quantity of a particular commodity, security, or other financial instrument from the seller at a specified price. The asset or instrument to which the option relates is known as the underlying, and the specified purchase price is known as the st

A callable bull/bear contract, or CBBC in short form, is a derivative financial instrument that provides investors with a leveraged investment in underlying assets, which can be a single stock, or an index. CBBC is usually issued by third parties, mostly investment banks, but neither by stock exchanges nor by asset owners. It was first introduced in Europe and Australia in 2001, and it i

Capital market line (CML) is the tangent line drawn from the point of the risk-free asset to the feasible region for risky assets. The tangency point M represents the market portfolio, so named since all rational investors (minimum variance criterion) should hold their risky assets in the same proportions as their weights in the market portfolio.

In mathematics, the Chan–Karolyi–Longstaff–Sanders process (abbreviated as CKLS process) is a stochastic process with applications to finance. In particular it has been used to model the term structure of interest rates. The CKLS process can also be viewed as a generalization of the Ornstein–Uhlenbeck process. It is named after K. C. Chan, G. Andrew Karolyi, Francis A. Longstaff, and Ant

In finance, the Chen model is a mathematical model describing the evolution of interest rates. It is a type of "three-factor model" (short-rate model) as it describes interest rate movements as driven by three sources of market risk. It was the first stochastic mean and stochastic volatility model and it was published in 1994 by Lin Chen, PhD from Harvard, former economist of Federal Res

In finance, a chooser option is a special type of option contract. It gives the purchaser a fixed period to decide whether the derivative will be a European call or put option.

In finance, a collar is an option strategy that limits the range of possible positive or negative returns on an underlying to a specific range. A collar strategy is used as one of the ways to hedge against possible losses and it represents long put options financed with short call options. The collar combines the strategies of the protective put and the covered call.

In economics, a complete market (a.k.a. Arrow-Debreu market or complete system of markets) is a market with two conditions:

A compound option or split-fee option is an option on an option. The exercise payoff of a compound option involves the value of another option. A compound option then has two expiration dates and two strike prices. Usually, compounded options are used for currency or fixed income markets where insecurity exists regarding the option's risk protection. Another common business application t

A conditional variance swap is a type of variance swap or swap derivative product that allows investors to take exposure to volatility in the price of an underlying security but only while the underlying security is within a pre-specified price range. This instrument can be useful for hedging complex volatility exposures, making a bet on the volatility levels contained in the skew of th

A condor is a limited-risk, non-directional options trading strategy consisting of four options at four different strike prices. The buyer of a condor earns a profit if the underlying is between or near the inner two strikes at expiry, but has a limited loss if the underlying is near or outside the outer two strikes at expiry. Therefore, long condors are used by traders who expect the un

In mathematical finance, the constant elasticity of variance model (CEV) is a stochastic volatility model, although technically it would be classed more precisely as a local volatility model, that attempts to capture stochastic volatility and the leverage effect. The model is widely used by practitioners in the financial industry, especially for modelling equities and commodities. It was

A constant maturity swap (CMS) is a swap that allows the purchaser to fix the duration of received flows on a swap.

Constant proportion portfolio investment (CPPI) is a trading strategy that allows an investor to maintain an exposure to the upside potential of a risky asset while

Contango is a situation in which the futures price (or forward price) of a commodity is higher than the spot price. In a contango situation, arbitrageurs or speculators are "willing to pay more for a commodity [to be received] at some point in the future than to purchase the commodity immediately. This may be due to people's desire to pay a premium to have the commodity in the future rat

In finance, a contract for difference (CFD) is a financial agreement between two parties, commonly referred to as the "buyer" and the "seller." The contract stipulates that the buyer will pay the seller the difference between the value of an asset at the time the contract was initiated and the current value of the asset. If the asset's price increases from the opening to the closing of t

A correlation swap is an over-the-counter financial derivative that allows one to speculate on or hedge risks associated with the observed average correlation, of a collection of underlying products, where each product has periodically observable prices, as with a commodity, exchange rate, interest rate, or stock index.

A covered option is a financial transaction in which the holder of securities sells (or "writes") an options contract, either a "call option" or a "put option", against stock that they own or are short, respectively.

In finance, a covered warrant (sometimes called naked warrant) is a type of warrant that has been issued without an accompanying bond or equity. Like a normal warrant, it allows the holder to buy or sell a specific amount of equities, currency, or other financial instruments from the issuer at a specified price at a predetermined date.

In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of interest rates. It is a type of "one factor model" (short-rate model) as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was introduced in 1985 by John C. Cox, Jonathan E. Ingersoll and Stephen A.

In finance, a default option, credit default swaption or credit default option is an option to buy protection (payer option) or sell protection (receiver option) as a credit default swap on a specific reference credit with a specific maturity. The option is usually European, exercisable only at one date in the future at a specific strike price defined as a coupon on the credit default s

In finance, a credit spread, or net credit spread is an options strategy that involves a purchase of one option and a sale of another option in the same class and expiration but different strike prices. It is designed to make a profit when the spreads between the two options narrows.

In finance, a currency swap (more typically termed a cross-currency swap, XCS) is an interest rate derivative (IRD). In particular it is a linear IRD, and one of the most liquid benchmark products spanning multiple currencies simultaneously. It has pricing associations with interest rate swaps (IRSs), foreign exchange (FX) rates, and FX swaps (FXSs).

A currency future, also known as an FX future or a foreign exchange future, is a futures contract to exchange one currency for another at a specified date in the future at a price (exchange rate) that is fixed on the purchase date; see Foreign exchange derivative.

In finance, delta neutral describes a portfolio of related financial securities, in which the portfolio value remains unchanged when small changes occur in the value of the underlying security (having zero delta). Such a portfolio typically contains options and their corresponding underlying securities such that positive and negative delta components offset, resulting in the portfolio's

In derivatives trading, the term diagonal spread is applied to an options spread position that shares features of both a calendar spread and a vertical spread. It is established by simultaneously buying and selling equal amount of option contracts of the same type (call options or put options) but with different strike prices and expiration dates.

In macroeconomics, discretionary policy is an economic policy based on the ad hoc judgment of policymakers as opposed to policy set by predetermined rules. For instance, a central banker could make decisions on interest rates on a case-by-case basis instead of allowing a set rule, such as Friedman's k-percent rule, an inflation target following the Taylor rule, or a nominal income targe

In finance, a dividend future is an exchange-traded derivative contract that allows investors to take positions on future dividend payments. Dividend futures can be on a single company, a basket of companies, or on an Equity index. They settle on the amount of dividend paid by the company, the basket of companies, or the index during the period of the contract.

Employee stock options (ESO or ESOPs) is a label that refers to compensation contracts between an employer and an employee that carries some characteristics of financial options.

An energy derivative is a derivative contract based on (derived from) an underlying energy asset, such as natural gas, crude oil, or electricity. Energy derivatives are exotic derivatives and include exchange-traded contracts such as futures and options, and over-the-counter (i.e., privately negotiated) derivatives such as forwards, swaps and options. Major players in the energy derivati

In finance, an equity derivative is a class of derivatives whose value is at least partly derived from one or more underlying equity securities. Options and futures are by far the most common equity derivatives, however there are many other types of equity derivatives that are actively traded.

An equity swap is a financial derivative contract (a swap) where a set of future cash flows are agreed to be exchanged between two counterparties at set dates in the future. The two cash flows are usually referred to as "legs" of the swap; one of these "legs" is usually pegged to a floating rate such as LIBOR. This leg is also commonly referred to as the "floating leg". The other leg of

An equity-linked note (ELN) is a debt instrument, usually a bond issued by a financial institution such as an investment bank or a subsidiary of a commercial bank. ELNs are liabilities of the issuer, but the final payout to the investor is based on an unrelated company's stock price, a stock index or a group of stocks or stock indices. The underlying stocks typically have large market ca

The owner of an option contract has the right to exercise it, and thus require that the financial transaction specified by the contract is to be carried out immediately between the two parties, whereupon the option contract is terminated. When exercising a call option, the owner of the option purchases the underlying shares (or commodities, fixed interest securities, etc.) at the strike

In finance, an exotic option is an option which has features making it more complex than commonly traded vanilla options. Like the more general exotic derivatives they may have several triggers relating to determination of payoff. An exotic option may also include a non-standard underlying instrument, developed for a particular client or for a particular market. Exotic options are mor

In finance, the expiration date of an option contract (represented by Greek letter tau, τ) is the last date on which the holder of the option may exercise it according to its terms. In the case of options with "automatic exercise", the net value of the option is credited to the long and debited to the short position holders.

Fairmat is a free-of-charge multi-platform software that allows to model financial contracts (e.g. a derivative contract) or projects with many contingencies (e.g. a Real Options model) by decomposing it into basic parts. Complex structures and dependencies are modelled using a graphical interface. Virtually any pay-off function and asset class( from interest rate derivatives to equity-l

Financial correlations measure the relationship between the changes of two or more financial variables over time. For example, the prices of equity stocks and fixed interest bonds often move in opposite directions: when investors sell stocks, they often use the proceeds to buy bonds and vice versa. In this case, stock and bond prices are negatively correlated.

Financial econometrics is the application of statistical methods to financial data. Financial econometrics is a branch of financial economics, in the field of economics. Areas of study include capital markets, financial institutions, corporate finance and corporate governance. Topics often revolve around asset valuation of individual stocks, bonds, derivatives, currencies and other fina

Finite difference methods for option pricing are numerical methods used in mathematical finance for the valuation of options. Finite difference methods were first applied to option pricing by Eduardo Schwartz in 1977.

Fixed-income attribution is the process of measuring returns generated by various sources of risk in a fixed income portfolio, particularly when multiple sources of return are active at the same time.

A foreign exchange derivative is a financial contract whose value changes depending on currency exchange rates.

In finance, a foreign exchange option (commonly shortened to just FX option or currency option) is a derivative financial instrument that gives the right but not the obligation to exchange money denominated in one currency into another currency at a pre-agreed exchange rate on a specified date. See Foreign exchange derivative.

In finance, a foreign exchange swap, forex swap, or FX swap is a simultaneous purchase and sale of identical amounts of one currency for another with two different value dates (normally spot to forward) and may use foreign exchange derivatives. An FX swap allows sums of a certain currency to be used to fund charges designated in another currency without acquiring foreign exchange risk. I

In finance, a forward contract, or simply a forward, is a non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed on in the contract, making it a type of derivative instrument. The party agreeing to buy the underlying asset in the future assumes a long position, and the party agreeing to sell the asset in the future assumes a sh

The forward price (or sometimes forward rate) is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, the forward price can be expressed in terms of the spot price and any dividends. For forwards on non-tradeables, pricing the forward may be a complex task.

In finance, a forward rate agreement (FRA) is an interest rate derivative (IRD). In particular, it is a linear IRD with strong associations with interest rate swaps (IRSs).

The FTSE MTIRS Indices are designed to accurately move in direct correlation to OTC Interest Rate Swaps market with a total of 45 indices covering the USD curve from 2 years to 30 years including spreads and butterflies. FTSE MTIRS Indices account for changes to both fixed and floating rates and are rebalanced daily.

A fund derivative (see also derivative) is a financial structured product related to a fund, normally using the underlying fund to determine the payoff. This may be a private equity fund, mutual fund or hedge fund. Purchasers obtain exposure to the underlying fund (or funds) whilst improving their risk profile over a direct investment.

The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical finance, provide necessary and sufficient conditions for a market to be arbitrage-free, and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss. Though arbitrage opportunities do exis

Fundamentally based indexes or fundamental indexes, also called fundamentally weighted indexes, are indexes in which stocks are weighted according to factors related to their fundamentals such as earnings, dividends and assets, commonly used when performing corporate valuations. This fundamental weight may be calculated statically, or it may be adjusted by the security's fundamental to m

In probability theory, Girsanov's theorem or the Cameron-Martin-Girsanov theorem explains how stochastic processes change under changes in measure.

In mathematical finance, the Greeks are the quantities (known in calculus as partial derivatives; first-order or higher) representing the sensitivity of the price of a derivative instrument, such as an option or future, to changes in one or more underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most

In probability theory and statistics, a Hawkes process is an age-dependent branching process driven by immigration from an inhomogeneous Poisson process. The process, named after Alan G. Hawkes, is also called a self-exciting point process., i.e., the occurrence of an event increases the likelihood of another event occurring. It has arrivals at times

The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curves – instantaneous forward rate curves in particular (as opposed to simple forward rates). When the volatility and drift of the instantaneous forward rate are assumed to be deterministic, this is known as the Gaussian Heath–Jarrow–Morton (HJM) model of forward rates. For direct mode

In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.

In financial mathematics, the Hull–White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates. It is relatively straightforward to translate the mathematical description of the evolution of future interest rates onto a tree or lattice and so interest rate de

In financial mathematics, the implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (usually Black–Scholes), will return a theoretical value equal to the price of the option. A non-option financial instrument that has embedded optionality, such as an interest rate cap, can also have an impl

In economics, incomplete markets are markets in which there does not exist an Arrow–Debreu security for every possible state of nature. In contrast with complete markets, this shortage of securities will likely restrict individuals from transferring the desired level of wealth among states.

In finance, inflation derivative (or inflation-indexed derivatives) refers to an over-the-counter and exchange-traded derivative that is used to transfer inflation risk from one counterparty to another. See Exotic derivative.

An inflation swap is an over-the-counter derivative in which one party pays a fixed inflation rate and the other pays the realised change in a price index such as CPI, RPI or HICP. Markets trade two main structures: the zero-coupon inflation swap, which settles once at maturity, and the year-on-year inflation swap, which pays periodic coupons. Users include pension funds, insurers and ot

In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of different interest rate indices that can be used in this definition.

An interest rate future is a futures contract (a financial derivative) with an interest-bearing instrument as the underlying asset. It is a particular type of interest rate derivative. Examples include Treasury-bill futures, Treasury-bond futures and Eurodollar futures.

In finance, an interest rate swap (IRS) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations with forward rate agreements (FRAs), and with zero coupon swaps (ZCSs).

In finance, the intrinsic value of an asset or security is its value as calculated with regard to an inherent, objective measure. As a distinction, the asset's price is determined relative to other similar assets.

In finance an iron butterfly, also known as the ironfly, is the name of an advanced, neutral-outlook, options trading strategy that involves buying and holding four different options at three different strike prices. It is a limited-risk, limited-profit trading strategy that is structured for a larger probability of earning smaller limited profit when the underlying stock is perceived to

The iron condor is an options trading strategy utilizing two vertical spreads – a put spread and a call spread with the same expiration and four different strikes. A long iron condor is essentially selling both sides of the underlying instrument by simultaneously shorting the same number of calls and puts, then covering each position with the purchase of further out of the money call(s)

Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance, in stochastic differential equations, and more recently even in machine learning.

A jelly roll, or simply a roll, is an options trading strategy that captures the cost of carry of the underlying asset while remaining otherwise neutral. It is often used to take a position on dividends or interest rates, or to profit from mispriced calendar spreads.

Jump diffusion is a stochastic process that involves jumps and diffusion. It is a type of Lévy process. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, and pattern theory and computational vision.

The Korn–Kreer–Lenssen model (KKL model) is a discrete trinomial model proposed in 1998 by Ralf Korn, Markus Kreer and Mark Lenssen to model illiquid securities and to value financial derivatives on these.

In finance, a ladder, also known as a Christmas tree, is a combination of three options of the same type (all calls or all puts) at three different strike prices. A long ladder is used by traders who expect low volatility, while a short ladder is used by traders who expect high volatility. Ladders are in some ways similar to strangles, vertical spreads, condors, or ratio spreads.

In quantitative finance, a lattice model is a numerical approach to the valuation of derivatives in situations requiring a discrete time model. For dividend paying equity options, a typical application would correspond to the pricing of an American-style option, where a decision to exercise is allowed at the closing of any calendar day up to the maturity. A continuous model, on the othe

The LIBOR market model, also known as the BGM Model (Brace Gatarek Musiela Model, in reference to the names of some of the inventors) is a financial model of interest rates. It is used for pricing interest rate derivatives, especially exotic derivatives like Bermudan swaptions, ratchet caps and floors, target redemption notes, autocaps, zero coupon swaptions, constant maturity swaps and

A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level

Lookback options, in the terminology of finance, are a type of exotic option with path dependency, among many other kind of options. The payoff depends on the optimal (maximum or minimum) underlying asset's price occurring over the life of the option. The option allows the holder to "look back" over time to determine the payoff. There exist two kinds of lookback options: with floating st

In investing and finance, the low-volatility anomaly is the observation that low-volatility securities have higher returns than high-volatility securities in most markets studied. In other words, assets whose prices or returns change wildly pay worse than assets whose prices or returns are steady. This is an example of a stock market anomaly since it contradicts the central prediction of

Low-volatility investing is an investment style that buys stocks or securities with low volatility and avoids those with high volatility. This investment style exploits the low-volatility anomaly. According to financial theory risk and return should be positively related, however in practice this is not true. Low-volatility investors aim to achieve market-like returns, but with lower ris