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mrravioli
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Joined: August 4th, 2017, 5:50 am

### Question about Heston fit in Volatility Surface A Practitioner's Guide (Jatheral)

I'm reading this classic book but the Heston fit in Ch.3 confuses me.

The basic idea of the chapter is:
1. derive implied vol in terms of local vol
2. derive local vol expression for a certain SVM (Heston in this case)
3. with 1 & 2, derive implied vol expression from the SVM
4. with 3 and market data, calibrate parameters in the orginal SVM

When applying the process to Heston, with some approximations and ansatz, Jatheral got (3.17) in the attached pic. However with this form, for a given time to expiration, implied variance is linear in x (log strike), which is obviously not true. In later part of the chapter, Jatheral showed the whole fitted surface and the skew and curvature are obviously there.

Did I miss or misunderstand something here? Helps appreciated.

Thanks!
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Alan
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### Re: Question about Heston fit in Volatility Surface A Practitioner's Guide (Jatheral)

As Gatheral says repeatedly in the chapter (bottom of pg 32, for example) he is trying to derive an approximation valid close-to-money. So, it's meant to be just the first two terms of an approximate Taylor series in the moneyness parameter about the at-the-money point. Perhaps you should compute both of those terms numerically and see how good the approximation is (at various T). Also, exact analytic results are known (for the atm smile level and slope) at both large and small T, so you could compare with those.

Of course, it's possible you have a different book by a different author.

mrravioli
Topic Author
Posts: 6
Joined: August 4th, 2017, 5:50 am

### Re: Question about Heston fit in Volatility Surface A Practitioner's Guide (Jatheral)

As Gatheral says repeatedly in the chapter (bottom of pg 32, for example) he is trying to derive an approximation valid close-to-money. So, it's meant to be just the first two terms of an approximate Taylor series in the moneyness parameter about the at-the-money point. Perhaps you should compute both of those terms numerically and see how good the approximation is (at various T). Also, exact analytic results are known (for the atm smile level and slope) at both large and small T, so you could compare with those.

Of course, it's possible you have a different book by a different author.
Thanks Alan. Got the idea now. I was confused because in the later part of the chapter the author (well, Jim Gatheral. I misspelled his name in op but yes we are talking about the same book...) demonstrated a whole surface and I thought it was generated from the methodology introduced before. I learned several other ways to calibrate Heston and tried to reproduced the one in this book as a practice. Seems the wrong direction...

bearish
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Joined: February 3rd, 2011, 2:19 pm

### Re: Question about Heston fit in Volatility Surface A Practitioner's Guide (Jatheral)

For a more current perspective, you may want to check out Gatheral's work with Rosenbaum and others under the general headline of "rough volatility".