## Documentation |

**active return**Amount of return achieved in excess of the return produced by an appropriate benchmark (for example, an index portfolio).

**active risk**Standard deviation of the active return. Also known as the

**tracking error**.**American option**An option that can be exercised any time until its expiration date. Contrast with European option.

**amortization**Reduction in value of an asset over some period for accounting purposes. Generally used with intangible assets. Depreciation is the term used with fixed or tangible assets.

**annuity**A series of payments over a period of time. The payments are usually in equal amounts and usually at regular intervals such as quarterly, semiannually, or annually.

**antithetic sampling**A variance reduction technique that pairs a sequence of independent normal random numbers with a second sequence obtained by negating the random numbers of the first. The first sequence simulates increments of one path of Brownian motion, and the second sequence simulates increments of its reflected, or antithetic, path. These two paths form an antithetic pair independent of any other pair.

**arbitrage**The purchase of securities on one market for immediate resale on another market to profit from a price or currency discrepancy.

**basis**Day count basis determines how interest accrues over time for various instruments and the amount transferred on interest payment dates. The calculation of accrued interest for dates between payments also uses day count basis. Day count basis is a fraction of

`Number of interest accrual days`/`Days in the relevant coupon period`. Supported day count conventions and basis values are:Basis Value

Day Count Convention

`0`actual/actual (default) — Number of days in both a period and a year is the actual number of days.

`1`30/360 SIA — Year fraction is calculated based on a 360 day year with 30-day months, after applying the following rules: If the first date and the second date are the last day of February, the second date is changed to the 30th. If the first date falls on the 31st or is the last day of February, it is changed to the 30th. If after the preceding test, the first day is the 30th and the second day is the 31st, then the second day is changed to the 30th.

`2`actual/360 — Number of days in a period is equal to the actual number of days, however the number of days in a year is 360.

`3`actual/365 — Number of days in a period is equal to the actual number of days, however the number of days in a year is 365 (even in a leap year).

`4`30/360 PSA — Number of days in every month is set to 30 (including February). If the start date of the period is either the 31st of a month or the last day of February, the start date is set to the 30th, while if the start date is the 30th of a month and the end date is the 31st, the end date is set to the 30th. The number of days in a year is 360.

`5`30/360 ISDA — Number of days in every month is set to 30, except for February where it is the actual number of days. If the start date of the period is the 31st of a month, the start date is set to the 30th while if the start date is the 30th of a month and the end date is the 31st, the end date is set to the 30th. The number of days in a year is 360.

`6`30E /360 — Number of days in every month is set to 30 except for February where it is equal to the actual number of days. If the start date or the end date of the period is the 31st of a month, that date is set to the 30th. The number of days in a year is 360.

`7`actual/365 Japanese — Number of days in a period is equal to the actual number of days, except for leap days (29th February) which are ignored. The number of days in a year is 365 (even in a leap year).

`8`actual/actual ICMA — Number of days in both a period and a year is the actual number of days and the compounding frequency is annual.

`9`actual/360 ICMA — Number of days in a period is equal to the actual number of days, however the number of days in a year is 360 and the compounding frequency is annual.

`10`actual/365 ICMA — Number of days in a period is equal to the actual number of days, however the number of days in a year is 365 (even in a leap year) and the compounding frequency is annual.

`11`30/360 ICMA — Number of days in every month is set to 30, except for February where it is equal to the actual number of days. If the start date or the end date of the period is the 31st of a month, that date is set to the 30th. The number of days in a year is 360 and the compounding frequency is annual.

`12`actual/365 ISDA — The day count fraction is calculated using the following formula:

`(Actual number of days in period that fall in a leap year / 366)`+`(Actual number of days in period that fall in a normal year / 365 )`.`13`bus/252 — The number of days in a period is equal to the actual number of business days. The number of business days in a year is 252.

**beta**The price volatility of a financial instrument relative to the price volatility of a market or index as a whole. Beta is commonly used with respect to equities. A high-beta instrument is riskier than a low-beta instrument.

**binomial model**A method of pricing options or other equity derivatives in which the probability over time of each possible price follows a binomial distribution. The basic assumption is that prices can move to only two values (one higher and one lower) over any short time period.

**Black-Scholes model**The first complete mathematical model for pricing options, developed by Fischer Black and Myron Scholes. It examines market price, strike price, volatility, time to expiration, and interest rates. It is limited to only certain kinds of options.

**Bollinger band chart**A financial chart that plots actual asset data along with three other bands of data: the upper band is two standard deviations above a user-specified moving average; the lower band is two standard deviations below that moving average; and the middle band is the moving average itself.

**bootstrapping, bootstrap method**An arithmetic method for backing an implied zero curve out of the par yield curve.

**Brownian motion**A zero-mean continuous-time stochastic process with independent increments (also known as a

*Wiener process*).**building a binomial tree**For a binomial option model: plotting the two possible short-term price-changes values, and then the subsequent two values each, and then the subsequent two values each, and so on over time, is known as "building a binomial tree." See also

**binomial model**.**call****a.**An option to buy a certain quantity of a stock or commodity for a specified price within a specified time. See also**put**.**b.**A demand to submit bonds to the issuer for redemption before the maturity date.**c.**A demand for payment of a debt.**d.**A demand for payment due on stock bought on margin.**callable bond**A bond that allows the issuer to buy back the bond at a predetermined price at specified future dates. The bond contains an embedded call option; that is, the holder has sold a call option to the issuer. See also

**puttable bond**.**candlestick chart**A financial chart usually used to plot the high, low, open, and close price of a security over time. The body of the "candle" is the region between the open and close price of the security. Thin vertical lines extend up to the high and down to the low, respectively. If the open price is greater than the close price, the body is empty. If the close price is greater than the open price, the body is filled. See also

**high-low-close chart**.**cap**Interest-rate option that guarantees that the rate on a floating-rate loan will not exceed a certain level.

**clean price**The price of a bond excluding any interest that has accrued since issue or the most recent coupon payment.

**collar**Interest-rate option that guarantees that the rate on a floating-rate loan will not exceed a certain upper level nor fall below a lower level. It is designed to protect an investor against wide fluctuations in interest rates.

**convexity**A measure of the rate of change in duration; measured in time. The greater the rate of change, the more the duration changes as yield changes.

**correlation coefficient**A statistic in which the covariance is scaled to a value between minus one (perfect negative correlation) and plus one (perfect positive correlation).

**coupon**Detachable certificate attached to a bond that shows the amount of interest payable at regular intervals, usually semiannually. Originally coupons were actually attached to the bonds and had to be cut off or "clipped" to redeem them and receive the interest payment.

**coupon dates**The dates when the coupons are paid. Typically a bond pays coupons annually or semiannually.

**covariance**A measure of the degree to which returns on two assets move in tandem. A positive covariance means that asset returns move together; a negative covariance means they vary inversely.

**day count convention**A convention used to determine the number of days between two coupon dates, which is important in calculating accrued interest and present value when the next coupon payment is less than a full coupon period away. See also

**basis****delta**The rate of change of the price of a derivative security relative to the price of the underlying asset; that is, the first derivative of the curve that relates the price of the derivative to the price of the underlying security.

**depreciation**Reduction in value of fixed or tangible assets over some period for accounting purposes. See also

**amortization**.**derivative**A financial instrument that is based on some underlying asset. For example, an option is a derivative instrument based on the right to buy or sell an underlying instrument.

**diffusion**The function that characterizes the random (stochastic) portion of a stochastic differential equation.

*See also***stochastic differential equation**.**discretization error**Errors that may arise due to discrete-time sampling of continuous stochastic processes.

**drift**The function that characterizes the deterministic portion of a stochastic differential equation.

*See also***stochastic differential equation**.**duration**The expected life of a fixed-income security considering its coupon yield, interest payments, maturity, and call features. As market interest rates rise, the duration of a financial instrument decreases. See also

**Macaulay duration**.**efficient frontier**A graph representing a set of portfolios that maximizes expected return at each level of portfolio risk. See also

**Markowitz model**.**efficient portfolio**Portfolios satisfying the criteria of minimum risk for a given level of return and maximum return for a given level of risk. See also

**Markowitz model**.**elasticity**See

**Lambda**.**Euler approximation**A simulation technique that provides a discrete-time approximation of a continuous-time stochastic process.

**European option**An option that can be exercised only on its expiration date. Contrast with American option.

**face value**The maturity value of a security. Also known as par value, principal value, or redemption value.

**fixed-income security**A security that pays a specified cash flow over a specific period. Bonds are typical fixed-income securities.

**floor**Interest-rate option that guarantees that the rate on a floating-rate loan will not fall below a certain level.

**forward rate**The future interest rate of a bond inferred from the term structure, especially from the yield curve of zero-coupon bonds, calculated from the growth factor of an investment in a zero held until maturity.

**future value**The value that a sum of money (the present value) earning compound interest will have in the future.

**gamma**The rate of change of delta for a derivative security relative to the price of the underlying asset; that is, the second derivative of the option price relative to the security price.

**greeks**Collectively, "greeks" refer to the financial measures beta, delta, gamma, lambda, rho, theta, and vega, which are sensitivity measures used in evaluating derivatives.

**high-low-close chart**A financial chart usually used to plot the high, low, open, and close price of a security over time. Plots are vertical lines whose top is the high, bottom is the low, open is a short horizontal tick to the left, and close is a short horizontal tick to the right.

**implied volatility**For an option, the variance that makes a call option price equal to the market price. Given the option price, strike price, and other factors, the Black-Scholes model computes implied volatility.

**internal rate of return****a.**The average annual yield earned by an investment during the period held.**b.**The effective rate of interest on a loan.**c.**The discount rate in discounted cash flow analysis.**d.**The rate that adjusts the value of future cash receipts earned by an investment so that interest earned equals the original cost. See also**yield**.**issue date**The date a security is first offered for sale. That date usually determines when interest payments, known as coupons, are made.

**Ito process**Statistical assumptions about the behavior of security prices. For details, see the book by Hull in Derivatives Pricing and Yields.

**key rate duration**Key rate duration measures the sensitivity of a portfolio's (or security's) value in relation to changes in specific maturities of the zero or spot curve.

**Lambda**The percentage change in the price of an option relative to a 1% change in the price of the underlying security. Also known as elasticity.

**long position**Outright ownership of a security or financial instrument. The owner expects the price to rise in order to make a profit on some future sale.

**lower partial moment**A model for the moments of asset returns that fall below a minimum acceptable level of return.

**Macaulay duration**A widely used measure of price sensitivity to yield changes developed by Frederick Macaulay in 1938. It is measured in years and is a weighted average-time-to-maturity of an instrument. The Macaulay duration of an income stream, such as a coupon bond, measures how long, on average, the owner waits before receiving a payment. It is the weighted average of the times payments are made, with the weights at time T equal to the present value of the money received at time T.

**Markowitz model**A model for selecting an optimum investment portfolio, devised by H. M. Markowitz. It uses a discrete-time, continuous-outcome approach for modeling investment problems, often called the mean-variance paradigm. See also

**efficient portfolio**and**efficient frontier**.**mean****a.**A number that typifies a set of numbers, such as a geometric mean or an arithmetic mean.**b.**The average value of a set of numbers.**modified duration**The Macaulay duration discounted by the per-period interest rate; that is, divided by (1+rate/frequency).

**Monte-Carlo simulation**A mathematical modeling process. For a model that has several parameters with statistical properties, pick a set of random values for the parameters and run a simulation. Then pick another set of values, and run it again. Run it many times (often 10,000 times) and build up a statistical distribution of outcomes of the simulation. This distribution of outcomes is then used to answer whatever question you are asking.

**moving average**A price average that is adjusted by adding other parametrically determined prices over some time period.

**moving-averages chart**A financial chart that plots leading and lagging moving averages for prices or values of an asset.

**normal (bell-shaped) distribution**In statistics, a theoretical frequency distribution for a set of variable data, usually represented by a bell-shaped curve symmetrical about the mean.

**odd first or last period**Fixed-income securities may be purchased on dates that do not coincide with coupon or payment dates. The length of the first and last periods may differ from the regular period between coupons, and thus the bond owner is not entitled to the full value of the coupon for that period. Instead, the coupon is prorated according to how long the bond is held during that period.

**on-the-run treasury bonds**The most recently auctioned issue of a U.S. Treasury bond or note of a particular maturity.

**option**A right to buy or sell specific securities or commodities at a stated price (exercise or strike price) within a specified time. An option is a type of derivative.

**point and figure chart**A financial chart usually used to plot asset price data. Upward price movements are plotted as X's and downward price movements are plotted as O's.

**present value**Today's value of an investment that yields some future value when invested to earn compounded interest at a known interest rate; that is, the future value at a known period in time discounted by the interest rate over that time period.

**principal value**See

**par value**.**proportional sampling**A stratified sampling technique that ensures that the proportion of random draws matches its theoretical probability. One of the most common examples of proportional sampling involves stratifying the terminal value of a price process in which each sample path is associated with a single stratified terminal value such that the number of paths equals the number of strata.

*See also***stratified sampling**.**purchase price**Price actually paid for a security. Typically the purchase price of a bond is not the same as the redemption value.

**put**An option to sell a stipulated amount of stock or securities within a specified time and at a fixed exercise price. See also

**call**.**puttable bond**A bond that allows the holder to redeem the bond at a predetermined price at specified future dates. The bond contains an embedded put option; that is, the holder has bought a put option. See also

**callable bond**.**Quant**A quantitative analyst; someone who does numerical analysis of financial information in order to detect relationships, disparities, or patterns that can lead to making money.

**redemption value**See

**par value**.**regression analysis**Statistical analysis techniques that quantify the relationship between two or more variables. The intent is quantitative prediction or forecasting, particularly using a small population to forecast the behavior of a large population.

**rho**The rate of change in a derivative's price relative to the underlying security's risk-free interest rate.

**return proxy**The proxy for return is a function that characterizes either the gross benefits or net benefits associated with portfolio choices.

**risk proxy**The proxy for risk is a function that characterizes either the variability or losses associated with portfolio choices.

**sensitivity**The "what if" relationship between variables; the degree to which changes in one variable cause changes in another variable. A specific synonym is volatility.

**settlement date**The date when money first changes hands; that is, when a buyer actually pays for a security. It need not coincide with the issue date.

**Sharpe ratio**The ratio of the excess return of an asset divided by the asset's standard deviation of returns.

**short sale, short position**The sale of a security or financial instrument not owned, in anticipation of a price decline and making a profit by purchasing the instrument later at a lower price, and then delivering the instrument to complete the sale. See also

**long position**.**spread**For options, a combination of call or put options on the same stock with differing exercise prices or maturity dates.

**standard deviation**A measure of the variation in a distribution, equal to the square root of the arithmetic mean of the squares of the deviations from the arithmetic mean; the square root of the variance.

**stochastic differential equation**A generalization of an ordinary differential equation, with the addition of a noise process, that yields random variables as solutions.

**straddle**A strategy used in trading options or futures. It involves simultaneously purchasing put and call options with the same exercise price and expiration date, and it is most profitable when the price of the underlying security is very volatile.

**strata***See***stratified sampling**.**stratified sampling**A variance reduction technique that constrains a proportion of sample paths to specific subsets (or

*strata*) of the sample space.**strike price**See

**exercise price**.**swaption**A swap option; an option on an interest-rate swap. The option gives the holder the right to enter into a contracted interest-rate swap at a specified future date. See also

**swap**.**term structure**The relationship between the yields on fixed-interest securities and their maturity dates. Expectation of changes in interest rates affects term structure, as do liquidity preferences and hedging pressure. A yield curve is one representation in the term structure.

**theta**The rate of change in the price of a derivative security relative to time. Theta is usually very small or negative since the value of an option tends to drop as it approaches maturity.

**tracking error**See

**active risk**.**Treasury bill**Short-term U.S. government security issued at a discount from the face value and paying the face value at maturity.

**Treasury bond**Long-term debt obligation of the U.S. government that makes coupon payments semiannually and is sold at or near par value in $1000 denominations or higher. Face value is paid at maturity.

**vega**The rate of change in the price of a derivative security relative to the volatility of the underlying security. When vega is large, the security is sensitive to small changes in volatility.

**volatility****a.**Another general term for sensitivity.**b.**The standard deviation of the annualized continuously compounded rate of return of an asset.**c.**A measure of uncertainty or risk.**Wiener process***See***Brownian motion**.**yield****a.**Measure of return on an investment, stated as a percentage of price. Yield can be computed by dividing return by purchase price, current market value, or other measure of value.**b.**Income from a bond expressed as an annualized percentage rate.**c.**The nominal annual interest rate that gives a future value of the purchase price equal to the redemption value of the security. Any coupon payments determine part of that yield.**yield curve**Graph of yields (vertical axis) of a particular type of security versus the time to maturity (horizontal axis). This curve usually slopes upward, indicating that investors usually expect to receive a premium for securities that have a longer time to maturity. The benchmark yield curve is for U.S. Treasury securities with maturities ranging from three months to 30 years. See also

**term structure**.**yield to maturity**A measure of the average rate of return that will be earned on a bond if held to maturity.

**zero curve, zero-coupon yield curve**A yield curve for zero-coupon bonds; zero rates versus maturity dates. Since the maturity and duration (Macaulay duration) are identical for zeros, the zero curve is a pure depiction of supply/demand conditions for loanable funds across a continuum of durations and maturities. Also known as spot curve or spot yield curve.

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