Delta Hedging A Note on a Basket of Three Assets
Aug 01, 2005
Here is a solution to the problem of delta hedging a note on a basket of three assets (could be in any payoff structure). If you are long such a note then you need to calculate the correlated or the total delta of the note to hedge it. But once you calculate the correlated delta (using the VCV matrix) and partial deltas then how do you use that to hedge, given there are three stocks. We approach the problem from a different angle. We would treat this as an optimization problem rather than going the correlated delta route.
Let N represent the basket note. Say, the basket note has three stocks that make up its payoff structure: Sa, Sb and S and the trader wants to use these the deltas of these three stocks to hedge the note, i.e. create an overall delta-neutral portfolio. Therefore, P represent the new portfolio which includes Sa, Sb and Sc with weights wa, wb and wc along with the note N. The portfolio P comprises the basket note and the three (component) stocks used to hedge the note.
Form here on it is a simple optimization problem.
We could write,
If we wish to have a delta neutral portfolio, then,
The above equation when expanded is,
This reduces to the simple equation,
where is given by the equation,
Substituting the above, will be,
Now we can substitute the above in equation (3) to get a solution for w,
Once we know , we can calculate the inverse of the matrix and if we know the partial deltas (i.e. ) of the note (from Monte Carlo simulation), then we could calculate the weights on the underlying stocks required to delta hedge the note.
The asset movement is given by the equations,
The expected value of is,
Now plugging in the expected value of in (2),
Now that we have expressed as a function of , we can take the partial derivative,
Taking the limit of the above equation,
Hence we get,
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