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I would like some insight as to how to value modified rainbow options on multiple assets:

For example: A multi asset option, Call GOOG with $S_t$ \$1600 that you may exercise if and only if you also exercise a put on TSLA with $S_t$ \$600 and a call on the SPY with $S_t$ \$400, I have read Jan Stuller's answer to a similar question, but not exactly sure how to generalize this for the option structure above on $n$ securities.

As well, how would a pricing model for a option such as the one above model changes in the correlations of the underlying securities and their volatilities? How would one define the Greeks for an option like this one?

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For more than two underlyings, look here.

It is not a traditional basket option, just external barriers. Usually modelled with Monte Carlo (and a model of choice, or whatever is available). Ideally, SLV.

Greeks will be bump and reprice. In terms of the shape of greeks, this is difficult to answer as there is a number of factors affecting this. You can find simple and intuitive charts here.

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