Levy characterization, representation of martingale and change of probability

I am preparing for the final, so I write some notes for the course maths finance.

Levy characterization for Brownian motion: 
If $\phi_s^T \phi_s = Id$, then

$B_t = \int^t_0 \phi_s dW_s$

is a standard Brownian motion

This theorem is very useful and it describes the nature that after a con-formal transform, the BM keeps its properties.

Representation of martingale:

This theorem has different version. The most general version is that for a $\mathcal{F}_t$ adapted local martingale $M_t$, it can be written as 

$M_t = \mathbb{E}[M_t] + \int_0^t H_s dWs$

where $H_t \in \mathbb{H}^2_{loc}$.

This is a mathematical version of perfect duplication theorem. The proof starts from the case $L^2 \text{martingale} \rightarrow L^1 \text{martingale} \rightarrow \text{local martingale}$. It has many application in the stochastic calculus.

Change of probability: 

First we define $Z_T = \exp{(\int_0^T \phi_s dW_s - \frac{1}{2}\phi_s^2 ds)}$. Generally, it's only a local martingale and if it satisfies $\mathbb{E}[Z_T] = 1$, we can define a change of probability

$\frac{d\mathbb{Q}}{d\mathbb{P}} = Z_T$ 

then under the new probability $\mathbb{Q}$, we can define a new BM in the form

$\tilde{B}_t = B_t - \int_0^t \phi_s ds$

We remark that in the case $\phi$ is deterministic, then the there is no problem since in this case, $Z_T$ is well defined of expectation 1. Otherwise, the expectation is not so clear but there is a theorem Novikov, says that if $\exp{(\int_0^T \frac{1}{2}\phi_s^2 ds)} < \infty$, then all the condition is satisfied.

The change of probability  can simplify the formula and has important applications on Monte-Carlo algorithms.

BS formula, PDE and Mote-Carlo

In mathematical finance, a key problem is the price of the derivative, for example the price of the option. This semester, I take a standard master course in X, which gives me so much impression: How a system can train the mathematics financial engineer as quickly as possible. After all, there are beautiful maths in it. Here, I would like to take some point.

Preparation: Martingale, Brownian motion, Ito calculus

Before beginning the study of mathematical finance, some basic knowledge is necessary. Basic probability, large number theory and central limit is far from sufficient, but the the theory Martincale is also important. We can find a previous article in my blog. Then, the trajectory most common in maths-finance is the standard BM. Yes, we can generate some others like Poisson process, Markov process, Levy process and branching process, but the most basic one is the BM, the scaling limit of the simple random walk, who simulate the behavior of large number of investor in the market.

Personally, I believe the research about the random curve itself is interesting enough, but in maths-finance, what makes sense is the Ito calculus, which makes the integration along a random trace has sense. The integration can be defined for H^2 space, local martingale and even semi-martingale, considering adding one term of finite variation. The construction of these space and integration can be seen as a special application of real analysis and functional analysis.

Mathematics weapons: PDE vs Proba, Numeric vs Mote-Carlo, Feyman-Kac

A key formula who plays an important role and make connection of maths-finance with other domain is the Feyman-Kac formula. This formula tells us that, for calculating the expectation of some random variable, we can study a PDE associated using the generator. The inverse procedure is also correct: to treat a type of parabolic equation, we can also study the associated random process. 

In practice, these two branches also lead to different realization: numerical solution or Mote-Carlo method. Personally, the Mote-Carlo method is easier to implement, but is it necessarily better? No idea, because after we will see other examples.

Feyman-Kac can also treat the Dirichlet question, which associate not a heat equation but a Poisson equation.

Application in the context of finance

The maths-finance is not a course pure probability, so we cannot ignore the part practical. For example, in the pricing, what's the principal? If we do not make it clear, those weapons will be abused. 

There are two principle, one is no arbitrage and the other is perfect replication. The perfect replication is simple, there is no option in the nature, so when we vend one option, we have to prepare  some strategy to produce these service so that we can  give the money. However, the production procedure follows also one condition self-finance, from what we see the final PDE. Then, the no arbitrage principal makes sense. That is the price should be exact the expectation. If not, the other has arbitrage principal to make money.

When we draw conclusion, we always say that the price is the expectation after actualization under the risk neutral measure. This simple phrase uses the change of probability, the PDE and other knowledge. We pay attention that the price does not depend on the change of the stock! But just the volatility. The drift is in fact erased in the neutral risk. We know, the final object of pricing is that equilibrium but not profit!

Beyond the Feyman-Kac

From the technique part, but also the maths-financial part, the Feyman-Kac is not enough since the generator is always linear. But for some general case? For example, the optimization problem, what we do?

The answer is BSDE, an advanced method that give a backward method to solve the equation.

In addition, some other option can make the pricing totally into a maths game. The existence and uniqueness of the SDE has also some practical meaning: the count cannot be infinite in debt. So the  finance and maths interact one with the other, just like my tutor Pierre does.