Chapter 8 - Brownian Motion And Martingales Flashcards
Brownian motion (intro)
Brownian motion is a continious-time stochastic process with a continous state space.
It can be seen as the continuous version of a simple symmetric random walk.
Standard Brownian motion is also called the Wiener process.
Martingale
In simple terms, a martingale is a stochastic process for which its current value is the optimal estimator of its future value.
- the expected future value is the current value
- the expected change in the process is zero
- the process has “no drift”
Importance of martingales for modern financial theory
Martingales’ importance for modern financial theory cannot be overstated. In fact, the whole theory of pricing and hedging of financial derivatives is formulated in terms of martingales.
Risk-neutral probability distribution
If the discounted price of a financial asset is a martingale when calculated using a particular probability distribution, the probability distribution can be described as risk-neutral.
Comment on the mutual compatibility of the key properties of Brownian motion
It is not an easy task to prove that the properties are mutually compatible. A surprising fact (which shows how tight the constraints are) is that either condition 3 or 4 can be dropped from the definition as it can be shown to be a consequence of the other three properties.
Is Brownian motion differentiable?
A result frequently quoted in books on Brownian motion is that (with probability 1) the sample path of a Brownian motion is not differentiable anywhere.
The proof of this is beyond the scope of the syllabus, but a weaker result may be proved to illustrate the usefulness of the time inversion property.
Two main limitations in using Brownian motion to model market indices in the long run
However successful the Brownian motion may be for describing the movement of market indices in the short run, it is useless in the long run.
- A standard Brownian motion is certain to become negative eventually.
- The Brownian model predicts that daily movements of size 100 or more occur just as frequently when the process is at level 100 as when it is at level 1000.
Two steps for the analysis of path properties of geometric Brownian motion St
- Taking the logarithm of observations
2. Performing analysis using techniques appropriate to Brownian motion
Optimal estimator property of conditional expectations
The importance of conditional expectations comes from the following property.
E[X|Y] is the optimal estimator of X based on Y1, Y2,…,Yn in the sense that for every function h: E{(X - E[X|Y])^2} <= E{(X - h(Y))^2}.
Y denotes the vector Y1, Y2,…,Yn
What is the most useful property of a martingale?
E[Xn] = E[X0] for all n
