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I'm looking at this paper: https://doi.org/10.1057/jdhf.2011.25, which is on pricing autocallable structured product. The author uses the Black-Scholes equation to describe the product's dynamic value, that is $$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial S^2}+(r-q)S\frac{\partial V}{\partial S}-(r+CDS)V=0, $$ where $CDS$ is the credit default swap spread of the issuer and $V$ is the product value. My question is why it is valid to use BS model to price this kind of structured product?

And, if possible, could anyone tell me what should I do if I want to find out what the Delta and Vega profile of this product looks like? Furthermore, how am I supposed to hedge this product?

I am quite new to quant finance and if there is any mistake in my description, please point it out. Thank you!

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Short answer: Do not use BS for AC

Long answer: There are plenty of question here about this already. Typically it is priced via Monte Carlo but that is not a model, just an implementation of some model (LV, SV, SLV).

Standard would be to use SLV but even there are issues with decorrelation and bilocality if you look at basket AC (which are very common).

There is also an interesting (fun) tweet about AC hedging.

Edit: for Greeks, without a sophisticated tool from a bank or vendor, you will not find anything better than this in my opinion.

With a proper tool, it will be bump and reprice. The tweet I included above explains some of the problems you will face with regards to Greeks (or hedging).

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