Malliavin Calculus and the Existence of Delta-Positive Hedges

Summary

The document poses a mathematical question about using Malliavin calculus to construct hedges in a market model driven by a stochastic differential equation. It refers to a proof approach for obtaining an explicit delta-neutral hedge and asks whether a related argument can establish a hedge whose delta stays positive over the full time interval.

The proposed hedge is required to start with unit value and to have terminal delta at least as large as that of every other hedge with the same initial value. No construction, proof, model assumptions, or evidence are provided, so the question remains open in the document. In particular, it does not specify admissibility, the meaning of terminal delta, or conditions under which such a maximizing hedge could exist.

Key ideas

• The document asks whether a Malliavin calculus proof for delta-neutral hedging can be adapted to positive-delta hedging.
• It requires the hedge to maintain positive delta throughout the interval.
• The hedge begins with unit value and must maximize terminal delta among hedges with that initial value.
• The question supplies no solution or assumptions sufficient to establish existence.

Tags

# Using malliavin derivative to find the worst Delta-positive hedge?


# Using malliavin derivative to find the worst Delta-positive hedge?












### Background:

I've heard that Malliavin Calculus can be used to show the explicit form of a delta-neutral hedge (given an SDE driven market model). For example, here is a sketch here on page 21 on how to achieve a $\Delta$-neutral hedge but how would one achieve a $\Delta$-positive hedge.

### Question:

Fix $T>0$. My question is how can this proof strategy be used to show the existence of a hedge $H$ which has:

- positive Delta throughout $[0,T]$

- H(0)=1

- $\Delta H(T)\geq \Delta \tilde{H}(T)$ for every hedge $\tilde{H}$ with $\tilde{H}(0)=1$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.