Finite Financial Markets
In this post I am going to explain how finite financial markets are modelled in math.
Throughout the remainder of this post, let be a finite probability space with measure . We also have a filtration , where is fixed. That means that the are -algebras on , where . Alternatively the partition induced by the -algebras on gets finer as increases. Intuitively this means that more information is available as the time increases. We furthermore let , i.e., at the beginning we have no information, and the powerset of .
In this setting a finite financial market is a -valued -adapted stochastic process such that and for .
What do all those complicated words mean?
- stochastic process means, that each vector is a random variable .
- -adapted means that each random variable is -measurable.
That special stuff about is because we want to have a point of reference if we have portfolios where we hold a linear combination of the and want to compare them.
We now have modelled a market with different assets . Now we want to model some trading strategies.
We say that a stochastic process is predictable, if is -measurable for all . That means intuitively, that is known at time , but not before. This is how we want to allocate our cash.
A trading strategy is now a -valued predictable stochastic process . Now represents how much of the asset we want to hold between time and . The filtration of is shifted, because we can decide how much of we want to hold between time and only based on the knowledge at time up to . We cannot see into the future.
We now want that no additional money comes out of nowhere. That means that . If this is the case, then we call the strategy self-financing. Note that the strange brackets denote the dot-product, i.e., . Also the stochastic process defined by denotes the wealth process, where is the seed capital. This directly models our wealth at each point in time.
We can also define the wealth by the sum of the gains. . 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.