Finite Financial Markets

Introduction

In this post I am going to explain how finite financial markets are modelled in math.

Notation

Throughout the remainder of this post, let \(\Omega\) be a finite probability space with measure \(\mu\). We also have a filtration \(\mathcal{F}=\{\mathcal{F}_t\}_{t=0,...,T}\), where \(T\in\mathbb{N}\) is fixed. That means that the \(\mathcal{F}_t\) are \(\sigma\)-algebras on \(\Omega\), where \(\mathcal{F}_t\subset \mathcal{F}_{t+1}\). Alternatively the partition induced by the \(\sigma\)-algebras on \(\Omega\) gets finer as \(t\) increases. Intuitively this means that more information is available as the time \(t\) increases. We furthermore let \(\mathcal{F}_0=\{\emptyset,\Omega\}\), i.e., at the beginning we have no information, and \(\mathcal{F}_T=\mathcal{P}(\Omega)\) the powerset of \(\Omega\).

Markets

In this setting a finite financial market is a \(\mathbb{R}^{d+1}\)-valued \(\mathcal{F}\)-adapted stochastic process \(S=\{S(t)\}_{t=0,...,T}=\{(S_0(t),S_1(t),...,S_d(t))\}_{t=0,...,T}\) such that \(S_0(0)=1\) and \(S_0(t)>1\) for \(t=1,...,T\).
What do all those complicated words mean?

• stochastic process means, that each vector \(S(t)\) is a random variable \(S(t):\Omega\rightarrow\mathbb{R}^{d+1}\).
• \(\mathcal{F}\)-adapted means that each random variable \(S(t)\) is \(\mathcal{F}_t\)-measurable.

That special stuff about \(S_0\) is because we want to have a point of reference if we have portfolios where we hold a linear combination of the \(S_i\) and want to compare them.

Strats

We now have modelled a market with different assets \(S_i\). Now we want to model some trading strategies.

We say that a stochastic process \(\phi=\{\phi(t)\}_{t=1,...,T}\) is predictable, if \(\phi(t)\) is \(\mathcal{F}_{t-1}\)-measurable for all \(t=1,...,T\). That means intuitively, that \(\phi(t)\) is known at time \(t-1\), but not before. This is how we want to allocate our cash.

A trading strategy is now a \(\mathbb{R}^{d+1}\)-valued predictable stochastic process \(\begin{align*} \phi=\{\phi(t)\}_{t=0,...,T}=\{(\phi_0(t),...,\phi_d(t))\}_{t=0,...,T} \end{align*}\). Now \(\phi_i\) represents how much of the asset \(S_i\) we want to hold between time \(t-1\) and \(t\). The filtration of \(\phi\) is shifted, because we can decide how much of \(S_i\) we want to hold between time \(t-1\) and \(t\) only based on the knowledge at time up to \(t-1\). We cannot see into the future.

We now want that no additional money comes out of nowhere. That means that \(\begin{align*} \langle \phi(t), S(t)\rangle=\langle \phi(t+1), S(t)\rangle \end{align*}\). If this is the case, then we call the strategy \(\phi\) self-financing. Note that the strange brackets denote the dot-product, i.e., \(\begin{align*} \langle \phi(t), S(t)\rangle=\sum_{i=0}^d \phi_i(t)\cdot S_i(t) \end{align*}\). Also the stochastic process \(X\) defined by \(X(t)= \langle \phi(t), S(t)\rangle\) denotes the wealth process, where \(X(0)=\langle \phi(1),S(0)\rangle\) is the seed capital. This directly models our wealth at each point \(t\) in time.

We can also define the wealth by the sum of the gains. \(\begin{align*} X(t)=\langle \phi(t), S(t)\rangle=\sum_{\tau=1}^t \langle \phi(\tau), \Delta S(\tau)\rangle=\sum_{\tau=1}^t \langle \phi(\tau), S(\tau)-S(\tau -1)\rangle \end{align*}\). In case you did not guess it already, that last part is a discrete stochastic integral. Meditate on this if you never saw a stochastic integral before.