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 .
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 Europan 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 Balck-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 )
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