e consider choosing optimal
trading strategy under the real world probability measure. The optimality is
defined in terms of maximization of utility function of final
wealth:
where the
is the time horizon,
is the final wealth and
is
a concave function.
is slightly increasing for large
because we would like to make money but not too much because of risk aversion.
Also,
is
sharply decreasing for negative
because we do not like to loose money. We are going to show that such setup
leads to delta hedging if the market is complete.
The reference for this section is
[Yang]
.
Notation
We introduce the following notations:
is the equilibrium (optimal trading strategy, expected utility maximizing)
price of an option,
Definition
is the stock price,
is the amount on the margin account.
The sum
is the "wealth".
The pair
is trading strategy.
We introduce the lower case notation for all processes as follows:
for any
.
Claim
Claim
We obtained the BlackScholes equation.
