Notations and Ito Formula
Multiplication table
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Ito Formula
Denote , , and is a real function of real variable, then
where .
For example,
Martingale Representation Theorem
Informally, the sources of uncertainty is from Brownian motions (). Then any martingale w.r.t. filtration of is sum of deterministic and stochastic integrals w.r.t. .
Suppose are independent BMs on the space . is generated by on . Denote as a martingale w.r.t. , that is
then
Lévy Theorem
Same settings as above. Denote are two martingales. If , , , , Then are independent standard BMs.
Proof
If , then for any . And thus
if , then
if , then
Besides, since
and are independent (because their Generation Function are independent).
Girsanov Theorem
Girsanov in discrete time
The theorem explains how stochastic processes change under changes in measure.
For example, consider 3 tosses of a coin. Denote as the probability of getting a H on toss i, which are independent for 3 tosses and .
The the probability for each output is as below
HHH | ||
HHT | … | |
HTH | … | |
HTT | … | |
THT | … | |
TTH | … | |
TTT |
Denote as the number of H’s in the first 2 tosses, then
Radon–Nikodym Random Variable () is defined as, for example,
Radon-Nikodym Martingale:
transformation
- Expectation:
- Conditional Expectation and Bayes Rule: suppose is -measurable, then
where is -measurable.
Girsanov in continuous time
- ,
- almost surely, and
Define
- as a martingale
then
- Bayes
if is -measurable
Equivalence:
as for , since , thus is valid. Besides
This implies
Girsanov for Brownian motion (1D)
- suppose ,
- define as a 1D Brownian motion
Define
Claim:
- almost surely, and for any adapted process ( is -measurable)
Explanation for the Claim
Denote
then
As for , it has derivatives . Using the Ito formula, we have
therefore is a martingale.
Claim: is a -Brownian motion.
Proof:
Using the Lévy Theorem:
- is a martingale under (to be proved)
As for the third condition, using the Bayes rule
if the is a martingale. To prove that
Remark: Girsanov for BM of multi-dimensions
- Suppose are independent BMs
- define
then almost surely and .
Remark:
- The Girsanov theorem and the Radon-Nikodym derivative cannot maintain correlations ( but might not equal to using the Radon-Nikodym derivative based on and ),
- The Girsanov theorem can maintain the independence.
Girsanov in Black-Scholes model
Asset prices assumption in the BS model
Suppose there are n time intervals during the period from 0 to . Interval timestamps are denoted by . Denote
- : the portfolio value at
- : the number of shares in risky asset at
Then
For the continuous trading:
Example: We have an European call option, payoff function of which is . We want to hedge a short position of the European call, that is, we need to decide and so that .
Solution: Define a s.t. and .
Since , we have
where
while
Because of , the term and term should be the same, that is
Therefore, we should use the “Delta Hedge”.
For the term:
We want to solve this Black-Scholes PDE and find (using the change of measure).
We have that . We aim to define a and a corresponding so that
Choose
and
Then
which is a martingale.
Because of the no arbitrage price of the call option, i.e.
thus
Suppose we have the
and , then and . Applying the Itô formula we have
and thus
therefore
Note that is equal to
then
for the -term, the integral is equal to
the other term is equal to
and thus .
General Models: Brownian Framework
Suppose and independent BMs .
Assume the underlying asset prices follow
where .
Denote the interest rate process as , and all processes are adapted. Then 1$ risk-free asset will worth $ at time . The discount process is thus
Suppose there is a risk-neutral measure s.t. (discounted primary asset prices) are martingales. Then is also a martingale where is the portfolio value.
According to our martingale assumption, the term should be eliminated, i.e.
thus (market price of risk equations, MPR)
1st FTAP (Fundamental Theorem of Asset Pricing)
A model is arbitrage free there exists a risk-neutral probability measure (MPR) have a solution .
For example, suppose we have two assets
Claim:
- arbitrage Sharpe ratio 1 = Sharpe ratio 2
2nd FTAP (completeness)
Every contract in terms of the primary assets can be replicated by a portfolio in the primary assets. Suppose there are replication portfolios , then the portfolio value has
Arbitrage means: with positive probability.
It can be proved that “Model compete”
unique risk-neutral measure
MPR have a unique solution (unique )
Example of an incomplete model
where .
Suppose there is portfolio having stocks, then the portfolio’s value follows
From the risk-neutral measure perspective, the discount process is
then
Note that
Denote , and
Define
then
and thus
and is a martingale under because
To verify the 1st FTAP (i.e. the mode has no arbitrage), we claim that there are infinite number of risk-neutral measures because we can define
and define
Note that
which are two independent brownian motions. And is also a risk-neutral measure because
but ; in fact, there are risk-neutral measures for different .
And thus the market is not complete.
From the replication perspective:
We cannot hedge a European call under this setting.
under the measure , we have
The call price at time is
Consider the replication:
Therefore, this equation does not have a solution.
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