## 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.

25 January 2015

Lesson