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 (14.440).
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 (0a.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
(1) - 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
|
Iverson brackets [W] |
|
indicator function of a set [W], notation consistent with Iverson brackets |
|
Dirac delta function (point mass) concentrated at [W] |
|
canonical basis |
|
|
|
Fourier transform (Table 16.3) |
|
inverse Fourier transform (Table 16.3) |
|
integration operator applied times (23.41) |
|
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 |
|
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 (14.466) of the matrix |
|
conjugate transpose (14.131) of the complex-valued matrix |
|
complex conjugate of the complex number |
|
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 (14.460) |
|
Kronecker product (14.462) |
|
Kronecker sum [W] |
|
commutation matrix (14.474) |
|
convolution (16.108) |
|
discrete convolution (16.111) |
|
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] |
|
|
|
|
|
Sets
|
ellipsoid with center , shape and radius (when the subscript is dropped) |
|
-dimensional unit sphere |
|
filled ellipsoid with center , shape and radius (when the subscript is dropped) |
|
-dimensional filled unit ball |
|
unit simplex in |
|
-dimensional unit hypercube |
|
|
|
|
Calculus
|
multivariate function |
|
gradient |
|
Hessian |
|
Jacobian |
|
|
|
|
See Chapter 15.1.3 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 |