Quantitative Analysis.
Trading Platform.
Python for Excel.
Author.

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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 the risk neutral measure via Girsanov's theorem.
D.Self-financing strategy.
E.Existence of the 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.
Bibliography.
Forum 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:MATH where the $T$ is the time horizon, $W_{T}$ is the final welth and $U\,\ $is a concave function. $U\left( x\right) $ is slightly increasing for large $x$ because we would like to make more but not too much because of risk aversion. Also, $U\,\ $is sharply decreasing for negative $x$ 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]. I read that book until I realised that I see no reason to believe that the "agent" can actually trade at what the author calls "equilibrium prices". The major part of the difficulty is absence of clear definition of such prices.

Definition

We introduce the following notations:

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

$S_{t}$ is the stock price,

$M_{t}$ is the amount on the margin account.

The sum MATH is the "wealth".

We introduce the lower case notation for all processes as follows: MATH for any $W_{t}$.

The consideration is performed under the real world probability measure.

Claim

MATH

Proof

MATHMATH Express upper case values for the lower case values, thenMATHMATHMATH Hence, the claim follows.

Notation

MATH the state is MATH MATH.

Notation

MATH

Claim

MATH

Proof

We haveMATHMATHMATH Hence,MATHMATH The function $u$ does not depend on the choice of $n$ at $t$. However,MATH henceMATH

Therefore,MATHMATH Taking the linear combinationMATH Note thatMATH henceMATH and the claimMATH follows.

We obtained the Black-Scholes equation.





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