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
 Nonrandom 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 , (pseudocode).
 Scalars (random or nonrandom): regular math font
Vectors and matrices (random or nonrandom): bold math font
Examples:
(random vector);
(nonrandom matrix).
 Symmetric, positive (semi)definite matrix: square bold math , where are the respective symmetric Riccati roots (51.8).
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 );
(matrixvariate counterpart of univariate ).
 Date/pointintime: 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 onedollar notional coupon bond;
is value ( price) of a onedollar notional zerocoupon 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 (24.62).
 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 equallag sampling);
(multivariate path from to , with equallag 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 continuoustime stochastic processes , which are assumed right continuous with left limits (cadlag [W]). We slice time using the half closehalf 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 optionreplicating 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 [W] 

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

canonical basis 



Fourier transform (Table 38.1) 

inverse Fourier transform (Table 38.1) 

integration operator applied times (36.20) 

inversecall transformation with parameter (E.1.29) 

positive part function 

error function [W] 

change (difference, increment) 

lag operator (2.45) 

exponentially weighted moving average at time with halflife 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 

trace of the square matrix , i.e. 

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



pseudoinverse (52.70) of the matrix 

Euclidean (vector) norm 

Frobenius (matrix) norm 

elementwise 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 (52.84) 

convolution [W] 

discrete convolution (38.68) 

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 





Calculus

multivariate function 

gradient 

Hessian 

Jacobian 




See Chapter 52.4 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

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 semidefinite (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) 

(Matrixvariate) 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 

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

Chisquared 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 

InverseWishart distribution with degrees of freedom , dispersion parameter 

historical with flexible probabilities distribution 

scenarioprobability distribution 





() The gamma distribution has different equivalent parametrizations [W]. Furthermore, the gamma is a special case of Wishart (32.372).
() 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, equallyspaced points) 

index for financial instruments 

index for risk drivers 

index for invariants 

index for Monte Carlo scenarios 

index for factors 

index for categories (e.g. credit ratings) 




Risk drivers

vector of risk drivers at time 
