Optimal utility function based interpretation of delta hedging.

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: MATH where the is the time horizon, $W_{T}$ is the final wealth and $U\,\$ is a concave function. $U\left( x\right)$ is slightly increasing for large because we would like to make money but not too much because of risk aversion. Also, $U\,\$ 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:

$F_{t}$ is the equilibrium (optimal trading strategy, expected utility maximizing) price of an option,

Definition

$S_{t}$ is the stock price,

$M_{t}$ 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: MATH for any $W_{t}$ .

Claim

MATH

Proof

We calculate MATH We substitute upper case values for the lower case values: MATH and MATH We equate and : MATH The -terms cancel and we arrive to the claim.

Notation

We use the notation MATH The state variable is . MATH MATH

Claim

MATH

Proof

We calculate MATH MATH MATH Hence, MATH MATH The function does not depend on directly (because the maximization removes the dependency). However, the variable depends on : MATH hence MATH

Therefore, MATH MATH Taking the linear combination MATH Note that MATH hence MATH and the claim MATH follows.

We obtained the Black-Scholes equation.

Notation.

Index.

Contents.