Notation

We spend a tremendous amount of time on notation and conventions, to make them both intuitive for readers from different disciplines and consistent across a wide set of topics. In the short Section 1 here below we point out a few key tenets that appear throughout all the present work. For the full list of notations, keep reading further below.

Key notation tenets

  • Non-random quantities (scalars or vectors/matrices): lower case
    Random variables (scalars or vectors/matrices): upper case
    Examples: 
    ;
    ;
    (realized random variable);
    (coefficient).
    Exception:  is realized invariant; (not ) is invariant as random variable.
  • Dummy counters: (similar for ).
    We do not use upper case , to avoid confusion with random variables (such as Poisson counter).
    We do not use other letters (except in special cases), to be more succinct and clear.
    Examples: 
    (sum);
    for , (pseudo-code).
  • Scalars (random or non-random): regular math font
    Vectors and matrices (random or non-random): bold math font
    Examples: 
    (random vector);
    (non-random matrix).
  • Symmetric, positive (semi)definite matrix: square bold math , where are the respective symmetric Riccati roots (49.9).
    Examples:
    is a generic symmetric and positive (semi)definite matrix;
    denotes the “volatility” vector extracted from , i.e. ;
    denotes the “correlation” matrix extracted from , i.e. ;
    in (multivariate counterpart of univariate );
    (matrix-variate counterpart of univariate ).
  • Date/point-in-time: Latin letter
    Period/time span: Greek letter
    Examples: 
    12/01/2014, 13h:07m:04s:472ms (current time);
    12/02/2014, 00h:00m:00s:000ms (investment horizon);
    01/07/2010, 00h:00m:00s:000ms (inception date/vintage);
    01/07/2018, 00h:00m:00s:000ms (time of expiry/maturity date);
    years (time to expiry/maturity);
    2.4 years (age of a contract).
  • Value at time of one unit of an instrument: (in general value is not the same as price).
    Examples:
    is value ( price) of one share of stock;
    is value ( dirty price, price) of a one-dollar notional coupon bond;
    is value ( price) of a one-dollar notional zero-coupon bond;
    is value ( price) of one forward contract;
    is value ( price) of one futures contract;
    is value ( price) of one index (e.g. the S&P 500 index);
    is value ( price) of one call option.
    Exception: in market microstructure we use prices , because they are more common and equivalent (23.65).
  • Short form for a process sampled at discrete times (time series) .
    Example:
    is a time series of values sampled at unit steps.
  • Time interval for flow variables [W]:
    Examples: 
    (return from to );
    (P&L of instruments from to ).
  • Time interval for processes with multiple monitoring times:
    Examples: 
    (univariate stochastic path from to , with equal-lag sampling);
    (multivariate path from to , with equal-lag sampling).
  • Time: throughout the present work, time is a continuum, or
    (2)

    - On a microscopic scale, such as in market microstructure, events (such as trades) occur at discrete and randomly spaced points in the time continuum, or . Then a value (such as a trading price) occurring at time is the mark associated with the point in a marked point process.
    - On a macroscopic scale, such as in dynamic strategies, trading within or across instruments can occur at any point in time . On this scale we have continuous-time stochastic processes , which are assumed right continuous with left limits (cadlag [W]). We slice time using the half close-half open interval convention . Under this setup the last value of a process in (such as the closing price over a day) is defined as the left limit.
    - For estimation purposes, we sample continuous time processes over a discrete time grid . The grid is typically equally spaced, or , although it need not be. If the grid is equally spaced, we may shift and rescale, for ease of notation and without loss of generality, the grid to . Regardless, we consider these as instances of a continuum .
    - For valuation (or “pricing”) purposes, we focus on two discrete points in time: the current valuation time and the payoff horizon . If we are pricing a rebalancing strategy, such as an option-replicating strategy, trading and rebalancing is assumed to occur at any point in time , although we only focus on the two snapshots and .

Operators and special functions

,

indicator function of event (31.140)

Dirac delta function (point mass) concentrated at [W]

canonical basis

Fourier transform (Table 49.6)

inverse Fourier transform (Table 49.6)

integration operator applied times (35.20)

inverse-call transformation with parameter (E.1.29)

positive part function

error function [W]

inverse error function [W]

change (difference, increment)

lag operator of order (50.104)

exponentially weighted moving average at time with half-life and trailing window of size

operator that extracts the main diagonal of an matrix , i.e.

operator that maps an vector into an matrix which has on the main diagonal and null entries elsewhere, i.e.

absolute value of the scalar

determinant of the square matrix

cardinality of the set

floor function [W]

ceiling function [W]

trace of the square matrix , i.e.

operator that maps an arbitrary matrix into a vector, stacking the columns of , i.e.

pseudo-inverse (50.82) of the matrix

Euclidean (vector) norm

Frobenius (matrix) norm

element-wise division (if one operand is a scalar and the other is not, the scalar expands into an array of the same size as the other operand)

Hadamard product [W]

Kronecker product [W]

Kronecker sum [W]

commutation matrix (50.100)

convolution [W]

discrete convolution (45.196)

sorting function which allows to obtain from a generic set the set of sorted elements , where and is a permutation of the indexes such that

directly proportional to [W]

Calculus

multivariate function

gradient

Hessian

Jacobian

See Chapter 50.5 for details.

Probability and general distribution theory

real world probability

risk neutral probability

probability density function (pdf) of

cumulative distribution function (cdf) of

characteristic function of

equality in distribution (i.e. )

(or )

conditional pdf of

univariate standard normal pdf

univariate standard normal cdf

multivariate standard normal pdf (correlation matrix: )

multivariate standard normal cdf (correlation matrix: )

univariate standard Student t cdf (degrees of freedom: )

pdf of an elliptical random variable , including normal (), Student () and Cauchy ()

characteristic function of an elliptical random variable , including normal (), Student () and Cauchy ()

information (generator)

prior distribution (pdf)

posterior distribution (pdf)

copula of represented by the pdf of the grades

vector of parameters

Summary statistical features

generic statistical feature of

location feature of

dispersion feature of

expectation vector, each entry is the expectation of the respective entry in

or

expectation vector conditioned on the information

variance vector, each entry is the variance of the respective entry in

standard deviation vector, each entry is the standard deviation of the respective entry in

covariance matrix ()

correlation matrix ()

skewness

kurtosis

mode

median

interquantile range

modal dispersion

quantile at confidence level of

symmetric, positive semi-definite (dispersion)

“volatility” vector

“correlation” matrix

Distributions

Bernoulli distribution with parameter

Beta distribution with shape parameters ,

Cauchy distribution with location parameter , dispersion parameter

Dirichlet distribution with parameter

Elliptical distribution with location parameter , dispersion parameter and generator function

Exponential distribution with rate parameter

Gamma distribution with shape parameter , scale parameter ()

(Multivariate) lognormal distribution with location parameter , dispersion parameter ()

(Multivariate) normal distribution with expectation (vector), (co)variance (matrix)

(Matrix-variate) normal distribution with expectation (vector), dispersion parameter, dispersion parameter

Poisson distribution of intensity

Poisson distribution of intensity , grid size

(Multivariate) Student -distribution with location parameter , dispersion parameter , degrees of freedom

(Matrix-variate) Student -distribution with location parameter , dispersion parameter, dispersion parameter, degrees of freedom

Chi-squared distribution with degrees of freedom

F distribution with degrees of freedom and

(Multivariate) uniform distribution on the set

Wishart distribution with degrees of freedom , dispersion parameter

Inverse-Wishart distribution with degrees of freedom , dispersion parameter

historical with flexible probabilities distribution

scenario-probability distribution

() The gamma distribution has different equivalent parametrizations [W]. Furthermore, the gamma is a special case of Wishart (31.192).

() The (multivariate) shifted lognormal distribution with location parameter , dispersion parameter and shift parameter is denoted by .

Stochastic processes

Compound Poisson process

Poisson process with intensity parameter

Wiener process (standard Brownian motion)

Time conventions and counting indexes

generic point in time (e.g. "12/01/2014, 13h:07m:04s:472ms")

generic point in time

inception time of a contract (such as a swap), which is a point in time

time of maturity/expiry of a contract, which is a point in time (as opposed to the time to maturity)

current point in time

investment horizon , which is a point in time

settlement date corresponding to the trade time

time period between two points in time (e.g. "three months and two milliseconds"), often used for time to maturity/expiry (), or for time to maturity/expiry at the horizon ()

time period between generic time and the inception time of a forward start contract

time index (when time is sampled at discrete, equally-spaced points)

subset of time indexes

index for financial instruments

index for risk drivers