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.