Self-financing portfolios
Consider a portfolio of just two assets: . The portfolios value is
Note that equation (2) is not naturally true, it is called the self-financing condition only satisfied under certain circumstances.
Using the ItΗ product rule, we have
So the equation (2) is equivalent to
For the discrete time:
Example 1: Blackβs formula
We can choose
then is self-financing.
Example 2: Black-Scholes
where is constant.
The portfolio value satisfies
Define
then
which is equivalent to equation (2). In particular,
Funding costs
Three different interest rates
- : interest on cash collateral
- : interest on risky collateral
- : interest on unsecured loans
Typically, we have
Goal: price a call in the interest rates setup here.
Consider the following portfolio:
- sell a call on stock at time 0 for
- post collateral
- hold shares of
- borrow to fund stock position, secure against stock position
The portfolio value is
and
Goal: price , i.e. price the
LHS is equal to
RHS is equal to
Then
Extreme cases:
- β no collateral
- β full collateral
No Collateral
and . Note that is the growth rate of under .
Transforming from to we have
thus is a martingale under , that is,
where can be calculated using the B-S formula.
As for the second case
Full Collateral
thus
and .
Transforming from to we have
thus is a -martingale, and
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