# 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 $\tilde{H}(0)=1$.
• What you are asking is unclear to me: in what respect would that be a $Delta$ hedge if $Delta$ is not zero? – Quantuple Oct 5 '16 at 17:58
• By a hedge I mean an admissible strategy; a progressively measurable map wrt the SDE process $X_t$ modeling the market – AIM_BLB Oct 5 '16 at 21:26