Please could someone explain how the greeks (especially the delta) of a multi-underlying autocallable product (i.e. an autocall on a basket) change when the correlation of the underlyings fluctuates?
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The impact will be uncertain as stated here due to 2 opposite effects:
Increasing correlation would increase the overall basket volatility, thus tends to push the option price higher
Increasing correlation would decrease the Forward price, thus tends to push the option price lower
Now, point 1 is in itself not 100% clear if you price with stochastic vol models that exhibit a so decorrelation effect. Since SV has vols fluctuate as opposed to deterministic, you get more variation in the price of the underlyings. Hence, your effective realized correlation is smaller in SV. Even that is not the end of the story because Autocallables are short volatility, so when Vol is reduced the price goes up (all else equal). This effect is called bi-locality. The paper I linked demonstrates that for large vol-vol correlation, the bi-locality effect is the one that dominates. In the absence of correlation between the variances, decorrelation dominates.
In terms of what the greeks look like in general, that is in itself quite difficult to answer for various reasons. Greeks will be bump and reprice, with all the complications and implementation choices. Ignoring these, it still remains difficult to answer as there is a number of factors affecting this. You can find simple and intuitive charts here.