I. Basic math.
 II. Pricing and Hedging.
 1 Basics of derivative pricing I.
 A. Single step binary tree argument. Risk neutral probability. Delta hedging.
 B. Why Ito process?
 C. Existence of risk neutral measure via Girsanov's theorem.
 D. Self-financing strategy.
 E. Existence of risk neutral measure via backward Kolmogorov's equation. Delta hedging.
 F. Optimal utility function based interpretation of delta hedging.
 2 Change of numeraire.
 3 Basics of derivative pricing II.
 4 Market model.
 5 Currency Exchange.
 6 Credit risk.
 7 Incomplete markets.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## 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: 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".

We introduce the lower case notation for all processes as follows: for any .

Claim

Proof

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

Notation

We use the notation The state variable is .

Claim

Proof

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

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

We obtained the Black-Scholes equation.

 Notation. Index. Contents.