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
is the time horizon,
is the final wealth and
a concave function.
is slightly increasing for large
because we would like to make money but not too much because of risk aversion.
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
We introduce the following notations:
is the equilibrium (optimal trading strategy, expected utility maximizing)
price of an option,
is the stock price,
is the amount on the margin account.
is the "wealth".
is trading strategy.
We introduce the lower case notation for all processes as follows:
We obtained the Black-Scholes equation.