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 $x,y,\alpha ,\beta ,\mu ,{\sigma }^2,\dots$
  Random variables (scalars or vectors/matrices): upper case $X,Y,A,B,M,{\Sigma }^2,\dots$
  Examples: 
  $X\sim N\left(\mu ,{\sigma }^2\right)$;
  ${\Sigma }^2\sim \mathrm{Wishart}\left(\nu ,{\sigma }^2\right)$;
  $x$ (realized random variable);
  $\beta$ (coefficient).
  Exception: ${ϵ}_t$ is realized invariant; ${\epsilon }_t$ (not $E_t$) is invariant as random variable.

• Dummy counters: $n=1,\dots ,\bar{n}$ (similar for $i,j,k,l,t,\dots$).
  We do not use upper case $n=1,\dots ,N$, to avoid confusion with random variables (such as $N_t$ Poisson counter).
  We do not use other letters $n=1,\dots ,j$ (except in special cases), to be more succinct and clear.
  Examples: 
  ${\sum }_{n=1}^{\bar{n}}{x}_n$ (sum);
  for $i=1,\dots ,\bar{ı}$, $x_i\leftarrow y_i$ (pseudo-code).

• Scalars (random or non-random): regular math font $x,X,\alpha ,\beta ,\dots$
  Vectors and matrices (random or non-random): bold math font $x,X,\alpha ,\beta ,\dots$
  Examples: 
  $X\equiv {\left(X_1,\dots ,X_{\bar{n}}\right)}^{\text{'}}$ (random vector);
  $\beta \equiv {\left\{{\beta }_{n,k}\right\}}_{n=1,\dots ,\bar{n}}^{k=1,\dots ,\bar{k}}$ (non-random matrix).

• Symmetric, positive (semi)definite matrix: square bold math $s^2,{\sigma }^2,{\Sigma }^2,\dots$, where $s,\sigma ,\Sigma ,\dots$are the respective symmetric Riccati roots (50.179).
  Examples:
  $s^2$ is a generic symmetric and positive (semi)definite $\bar{n}\times \bar{n}$ matrix;
  $s_{\mathrm{vol}}\equiv s^{\mathrm{vol}}\equiv \mathrm{vol}\left(s^2\right)$ denotes the “volatility” vector extracted from $s^2$, i.e. $s_{\mathrm{vol}}\equiv \sqrt{\mathrm{diag}\left(s^2\right)}$;
  $c^2\equiv \mathrm{corr}\left(s^2\right)$ denotes the “correlation” matrix extracted from $s^2$, i.e. $c^2\equiv \mathrm{Diag}\left(1.∕s_{\mathrm{vol}}\right)s^2\mathrm{Diag}\left(1.∕s_{\mathrm{vol}}\right)$;
  ${\sigma }^2$ in $X\sim N\left(\mu ,{\sigma }^2\right)$ (multivariate counterpart of univariate $X\sim N\left(\mu ,{\sigma }^2\right)$);
  ${\Sigma }^2\sim \mathrm{Wishart}\left(\nu ,{\sigma }^2\right)$ (matrix-variate counterpart of univariate ${\Sigma }^2\sim \mathrm{Gamma}\left(k,\theta \right)$).

• Date/point-in-time: Latin letter $t,s,u,\dots$
  Period/time span: Greek letter $\tau ,\upsilon ,\dots$
  Examples: 
  $t_{\mathrm{now}}=$ 12/01/2014, 13h:07m:04s:472ms (current time);
  $t_{\mathrm{hor}}=$ 12/02/2014, 00h:00m:00s:000ms (investment horizon);
  $t_{\mathrm{start}}=$ 01/07/2010, 00h:00m:00s:000ms (inception date/vintage);
  $t_{\mathrm{end}}=$ 01/07/2018, 00h:00m:00s:000ms (time of expiry/maturity date);
  $\tau =3$ years (time to expiry/maturity);
  $\alpha =$ 2.4 years (age of a contract).

• Value at time $t$ of one unit of an instrument: $V_t$ (in general value is not the same as price).
  Examples:
  $V_t^{\mathrm{stock}}$ is value ($=$ price) of one share of stock;
  $V_t^{\mathrm{bond}}$ is value ($=$ dirty price, $\ne$ price) of a one-dollar notional coupon bond;
  $V_t^{\mathrm{zcb}}$ is value ($=$ price) of a one-dollar notional zero-coupon bond;
  $V_t^{\mathrm{fwd}}$ is value ($\ne$ price) of one forward contract;
  $V_t^{\mathrm{futures}}$ is value ($\ne$ price) of one futures contract;
  $V_t^{\mathrm{index}}$ is value ($=$ price) of one index (e.g. the S&P 500 index);
  $V_t^{\mathrm{call}}$ is value ($=$ price) of one call option.
  Exception: in market microstructure we use prices $P_t$, because they are more common and equivalent (23.65).

• Short form for a process sampled at discrete times (time series) $x.=\left\{...,x_{t_i},x_{t_{i+1}},x_{t_{i+2}},...\right\}$.
  Example:
  $x.=\left\{\dots ,x_{t-1},x_t,x_{t+1,}\dots \right\}$ is a time series of values sampled at unit steps.

• Time interval for flow variables [W]: $t\to u$
  Examples: 
  $R_{t\to u}$ (return from $t$ to $u$);
  ${\Pi }_{t\to u}\equiv {\left({\Pi }_{1,t\to u},\dots ,{\Pi }_{\bar{n},t\to u}\right)}^{\text{'}}$ (P&L of $\bar{n}$ instruments from $t$ to $u$).

• Time interval for processes with multiple monitoring times: $t⇝u$
  Examples: 
  $X_{t⇝u}\equiv \left(X_t\left|X_{t+1}\right|\dots ,X_{u-1}|X_u\right)$ (univariate stochastic path from $t$ to $u$, with equal-lag sampling);
  $x_{t⇝u}\equiv \left(x_t\left|x_{t+1}\right|\dots \left|x_{u-1}\right|x_u\right)$ (multivariate path from $t$ to $u$, with equal-lag sampling).

• Time: throughout the present work, time $t$ is a continuum, or

  $$t\in R\text{.}$$

  (2)

  - On a microscopic scale, such as in market microstructure, events (such as trades) occur at discrete and randomly spaced points $T_k$ in the time continuum, or $T_k\in R$. Then a value ${\tilde{X}}_k$ (such as a trading price) occurring at time $T_k$ is the mark associated with the point $T_k$ 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 $t\in R$. On this scale we have continuous-time stochastic processes ${\left\{X_t\right\}}_{t\in R}$, which are assumed right continuous with left limits (cadlag [W]). We slice time using the half close-half open interval convention $\left[t,u\right)$. Under this setup the last value of a process in $\left[t,u\right)$ (such as the closing price over a day) is defined as the left limit.
  - For estimation purposes, we sample continuous time processes ${\left\{X_t\right\}}_{t\in R}$ over a discrete time grid ${\left\{X_{t_k}\right\}}_{k\in N}$. The grid is typically equally spaced, or $t_0,t_1=t_0+\Delta ,t_2=t_0+2\Delta ,\dots$, 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 $t_0=0,t_1=1,t_2=2,\dots$. Regardless, we consider these ${\left\{t_k\right\}}_{k\in N}$ as instances of a continuum $t\in R$.
  - For valuation (or “pricing”) purposes, we focus on two discrete points in time: the current valuation time $t_{\mathrm{now}}$ and the payoff horizon $t_{\mathrm{hor}}$. 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 $t\in \left[t_{\mathrm{now}},t_{\mathrm{hor}}\right]$, although we only focus on the two snapshots $t_{\mathrm{now}}$ and $t_{\mathrm{hor}}$.

Operators and special functions

Calculus

See Chapter 50.7 for details.

Probability and general distribution theory