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I need to generate the Volatility Surface of call options on S&P500 index, my dataset contains implied volatilities regarding various expiration dates for various strike prices.

My doubt is, given beta in advance, in order to get a surface should I recalibrate the model parameters (alfa,rho,nu) for every different exipiration date or I have to run the LSQ non linear between the matrix containing the market volatilities and the volatilities by sabr all togheter? (So my surface will be based on a single set of parameters and not one for each maturity)

Thanks for the help

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You are observing the same underlying $S_t$, therefore it has to be one set of parameters for all maturities. You could add a term structure to the parameters , however , since you are using SABR, I assume you use Hagan expansion to generate the implied vols, and for this approximation, the parameters must be constant.

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  • $\begingroup$ Thanks for the answer, the problem is that every application of the model i found online calibrates the model on a single maturity... $\endgroup$ – Giulio Graziani Nov 28 '19 at 15:18
  • $\begingroup$ on which asset class? $\endgroup$ – Canardini Nov 29 '19 at 14:02
  • $\begingroup$ It was interest rate swap $\endgroup$ – Giulio Graziani Dec 2 '19 at 11:05
  • $\begingroup$ For interest swaps, the underlying is a swap rate, and it is a function of its start and end dates. Each of them are martingale under its annuity measure, therefore will have its own SABR parameters $\endgroup$ – Canardini Dec 2 '19 at 13:48

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