Calibrating Dupire Local Volatility to Vanilla Option Prices
Summary
The document discusses how to calibrate a Dupire local volatility model to European vanilla options. Market prices are the calibration targets; for strikes and maturities without quoted prices, the answer says there is no generally useful analytic pricing formula, since availability depends on the chosen local volatility parameterization. A numerical pricing method is therefore part of the calibration problem.
One described approach constructs a finite-difference scheme and uses a global optimizer, such as Levenberg–Marquardt, to fit local volatility proxy parameters. The answer points to interpolation research and reports that piecewise linear interpolation was more stable in the author's implementation than piecewise constant interpolation. It also notes that calibration can be difficult over short time steps because price differences across strikes may be on very different scales. The reported calibration quality is based on personal implementation experience, not a general benchmark; methods continue to evolve, and the document does not provide comparative performance data or implementation details.
Key ideas
- Market option prices provide calibration targets for quoted vanilla contracts.
- There is no broadly applicable analytic vanilla pricing formula for arbitrary local volatility parameterizations.
- A finite-difference pricing scheme can be combined with a global optimizer to fit local volatility proxies.
- Piecewise linear interpolation is reported as more stable than piecewise constant interpolation in one implementation.
- Short time steps and differently scaled strike prices can make calibration difficult.
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Full text
# calibration of a local volatility model # calibration of a local volatility model Generally speaking, when calibrating a local volatility model a la Dupire to European vanilla calls, should I use the numerically (PDE or Monte Carlo) solved price for the vanilla call in the cost function or is there an analytical formula for vanillas in local volatility model I could use to speed up the optimization and to reduce the error made in solving numerically for the vanilla call? I would appreciate any references to this problem. Thanks ## Answer by Jesper Tidblom (score 2) https://quant.stackexchange.com/a/61342 Vanillas, as in ordinary call/put options, are given by their market values. If you refer to vanillas on strikes and maturities not found on the market, then there are no general useful analytic formulas as far as I know (I might be wrong, of course). It would depend on the chosen form of the local volatility parameterization. A classical reference on how to interpolate and construct local vol from market prices can be found in the article: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1694972 by Andreasen and Huge. This text is quite brief and contains the main ideas, but don't go too much into finer details. The Master thesis (by a student of Huge) "Calibrating the local volatility model" by Lykke Rasmussen is a nice reference that clears up a lot of the details. This blog post by Le Floch is also very worth looking at https://chasethedevil.github.io/post/dont-stay-flat-with-andreasen-huge-interpolation/ It seems like piecewise linear interpolation of local vol proxys is a lot more stable than the piecewise constant in the original article. I have implemented the methods myself (using the linear interpolation of Le Floch). Basically one is constructing a finite difference scheme and combine it with a global minimizer, like Levenberg-Marquardt, to calibrate certain local vol proxy constants. To get a good stable calibration can be tricky, from my experience, especially calibrating over small time steps where the difference in option prices for different strikes are on completely different scales. But when it works, the calibration is typically excellent. New and refined methods are developed constantly though, so it is worth to keep an eye on the lastest research.
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