This question represents the third (& final) installment of my two earlier ones regarding least squares optimization, which centered on constructing an optimized vega hedge portfolio. In this last & final step, I am trying to introduce another risk sensitivity, so called rega, in addition to vega, which effectively means that I am adding quantities to the optimization/
One of the challenges is that the rega is in different order of magnitude really (vega risk is typically reported 5 – 10 x as large as rega risk). I have tried to address this by scaling the weights for the rega risk up so that it matches - in order of magnitude - approximately the vega risk (specifically, I scale it by the square root of the maximum vega/maximum rega).
The attached spreadsheet illustrates the set-up; I have tried a variety of different set-ups in order to tackle the problem, but none seems to deliver a satisfactory fit really:
Case 1: just optimizes for vega only; as the example shows the resulting hedge portfolio would imply a substantial net rega exposure
Case 2: optimizes for vega & rega, with both exposures equally weighted. There is still a significant net rega exposure (215k – 324k) across all spot steps, and the fit for vega is relatively poor locally (e.g. 323k at +2%)
Case 3: Weighted <> scales up the rega weights. The net rega exposure seems to be better now, yet the vega fit is extremely poor (difference of 800k at +1.0% step).
Case 4: same set-up as Case 3, but optimized for max error; fit is equally poor.
Case 5: I added a 6th hedging instrument & optimized for vega & rega in a weighted fashion. Rega fit looks better now but vega still relatively poor (particularly at +1.0% & +2.0% nodes)
Case 6: as above but using the max error as objective function. Vega fit at +1.0% & +2.0% nodes has been improved but still rather poor.
Case 7: here, I used a different approach; instead of minimizing the exposure at the respective nodes, I tried to minimize the vega & rega exposure at the 0% nodes only, as well as the changes in the exposure between any two nodes. Vega fit is very poor here (especially at +1.% node, which shows a deviation of 599k)
Overall, case 5 & 6 (i.e. which include a 6th instrument) seem to yield the most promising result, but they still seem far from satisfactory. I am therefore wondering whether there may be any suggestion as to further improve the fit; either by
(1) Calibrating the weightings
(2) Changing the objective function
(3) Changing the optimization routine (fmarshal hinted in another post that extending the optimization to the weights may improve the fit)/
The main additional challenge seems to risk units for rega & sega would differ from the vega units (e.g. rega & sega numbers would typically be in the order of 1000-50000, so some sort of normalization would have to be applied which I figured may be done by using multiplicators to scale the numbers up to the same order of magnitude as the vega numbers. Alternatively, I would weigh the errors for rega/sega much more heavily).