Definitions & Notation

The idea behind this page is to be able to put in one place mathematical or financial type definitions, which can then be referenced from other places on the site.

Notation
Since it is difficult and not very pretty to use much inline math in a LaTeX Rendered blog, we will often omit clarifying statements and instead agree to use the following notation.

\displaystyle \boldsymbol{Y}_t Market variable
\displaystyle \boldsymbol{S}_t Traded security
\displaystyle \boldsymbol{U}_t Influence factor
\displaystyle \big( \Omega , \mathcal{F} , P \big) Probability space
\displaystyle \mathbb{F} = \big( \mathcal{F}_t \big) Filtration
\displaystyle B_t Numeraire
\displaystyle \mathrm{DF}_T^t Discount factor
\displaystyle \boldsymbol{\phi}_t Portfolio
\displaystyle V_t(\phi) Portfolio value process
\displaystyle \sigma = \big( \sigma^{(kl)} \big) Volatility matrix
\displaystyle \mathfrak{C} = \sigma^\dagger \cdot \sigma Covariance matrix
\displaystyle W_t Wiener process (Risk factor)

So by writing, say \boldsymbol{\phi}_t , it is understood that it refers to a portfolio (trading strategy).

Definition (Market)
By a theoretical-market we mean the following collection of mathematical constructions:

  1. A K-dimensional stochastic vector process, called the variables of the market. Among the market variables, there are traded-assets, usually occupying the first (N+1) components. The remaining components are called (non-traded) influence factors.
  2. Underlying is a probability-space and a filtration generated by the market variables:
    \displaystyle \mathcal{F}_t = \sigma \big( \boldsymbol{Y}_t \big)
  3. Furthermore, the market has a numeraire, chosen among the traded securities to be the unit of wealth. This means that the observable price processes are the ratios of the S’s to the numeraire and one refers to the market consisting of the price processes:
    \displaystyle \boldsymbol{Z}_t = \frac{\boldsymbol{S}}{B_t}

    as the normalized market.

Definition (Generalized Black-Scholes)
Consider a market containing an N risk-factor independently Wiener-driven vector process and equally many traded securities with dynamics

\displaystyle \textrm{d} S_t = S_t \cdot \alpha (t,S_t) \, \textrm{d} t + S_t \cdot \big( \sigma (t,S_t) \, \textrm{d} W_t \big)

(the dot refers to componentwise multiplication). The coefficients are real functions

\displaystyle \alpha : \mathbb{R}_+ \times \mathbb{R}^N \rightarrow \mathbb{R}^N, \quad \sigma : \mathbb{R}_+ \times \mathbb{R}^N \rightarrow \mathbb{R}^{N \times N}

such that the volatility matrix is nonsingular for all arguments. Furthermore assume no (non-traded) influence factors and an additional traded security with (non-risky / bank-account) dynamics

\displaystyle \textrm{d} B_t = B_t r \, \textrm{d} t

With the bank-account as Numeraire, the above market is called the Generalized Black-Scholes market. It turns out to be arbitrage-free and complete.

More definitions inserted here.