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Bit of a newbie question; but I see this pop up from time to time.

If we have a volatility surface (e.g. for the S&P500) built from market options what more can we do with it, but price other European options on non-traded strikes and maturities.

More specifically, I see people claim to need the volatility surface to value exotic options and risk management. How do they do this?

Note, as I understand it, if we have a Heston Model (for instance) calibrated to options prices, we can value any exotic option we'd like and compute some gradients to our liking. But, we can't get there from only picking out an implied volatility of the European options.

As an example question: Given the vol surface - how do I price a barrier option on the SPX? How do I compute its sensitivities to risk-factors such as spot, vol, etc..

What am I missing here?

Sinbad The Sailor
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  • That's a lot of questions combined. Most, if not all will be answered here already if you look carefully. Risk (Greeks) are usually bump and reprice (or from a PDE grid), vol surfaces are cleaned and frequently easier to calibrate to, compared to prices (Google for example Local vol calibration). – AKdemy Apr 18 '23 at 17:26
  • I've been looking but haven't found satisfactory answers. But beginning with the pricing then. It shouldn't be the case that one can use the volatility surface explicitly to compute values of exotic derivatives. Instead, I am assuming people refer to using the actual underlying stochastic model (be it with MC simulation or PDEs)? – Sinbad The Sailor Apr 18 '23 at 17:56
  • See here for Heston calibration to a vol surface (not option prices). This answer shows how to compute Greeks with finite difference. There are lots and lots of similar questions in this forum. Also, many option are only quoted in vol (OTC FX, many interest rate options,...) – AKdemy Apr 19 '23 at 15:58

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