Arbitrage-Free Interpolation and Modeling of Option Surfaces

# Interpolating option surface while respecting no-arbitrage conditions


# Interpolating option surface while respecting no-arbitrage conditions












I have data of market prices of puts and calls, in addition to the corresponding strikes and times to maturity.

I can also find the risk-free rate and the price of the underlying.

How can I interpolate the option market price surface while respecting the no-arbitrage conditions?

I’m struggling in particular to impose the constraints related to the partial derivatives.

Thanks

## Answer by Andrea (score 0)

https://quant.stackexchange.com/a/80544

Once you normalise properly to remove drift effects and dividends, you will need a convex-decreasing in space and increasing in time interpolation.

(basically do in moneyness (to fwd) and not absolute strike)

The only example I've ever found is this: https://www.researchgate.net/publication/228872089_An_Arbitrage-free_Interpolation_of_Volatilities

## Answer by Jesper Tidblom (score 0)

https://quant.stackexchange.com/a/80561

I am interested in the same topic. Often you have some kind of stochastic model which you calibrate to fit the option data as closely as possible and then calculate the implied volatility surface implied by that model. In this way you (typically, depending on the model) get an arbitrage free surface. However, then you typically only match the market prices approximately, so this is not really interpolating the prices.

If you just consider one time to expiry for your option, that is one smile, then there is quite a bit of research on that topic. I recently implemented this article/model by F.Le Floc'h : https://arxiv.org/abs/2305.13791. It interpolates option prices for one maturity in an arbitrage free way and the result is three times differentiable, meaning we get construct a smooth density from the resulting curve.

But if you want a full surface the mentioned article is not suitable. In that case this article of Huge and Andreasen is a classic: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1694972. They don't provide much details in the article, but there are later works expanding on the article. Also using linear interpolation instead of constant interpolation in their calibration procedure stabilized the calibration procedure a bit. But anyway, their procedure gives arbitrage free prices and thereby volatilities for a full surface on a given grid. The implied density implied by their procedure is not as well behaved (it gives some slight oscillations due to the interpolation used) as in the previous article though. It also seems sensitive to arbitrage, which is probably no surprise since it is an impossible task to fit an arbitrage free model exactly to data which is not arbitrage free. But it is a well known and used model that might be a good choice depending on your needs.

I also found a very new article trying to accomplish exactly what you want. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4831218. The author, F Deschâtres, interpolates volatilities, or more precisely variances, directly without using some underlying stochastic model. He calibrates the entire volatility surface in one go, ensuring no arbitrage of any kind. Then he use some quadratic programming solver for his problem, which could be a bit of an obstacle if you don't have one available. I have not implemented the article myself yet, but it looks promising to me. It is a bit technical, but I believe you could find some answers there regarding to your questions on arbitrage conditions.

## Answer by Marwin Steiner (score 0)

https://quant.stackexchange.com/a/85441

Surface Stochastic Volatility Inspired (SSVI) by Gatheral and Jacquier (2014) may be of interest to you. Not interpolation but parametrization of the vol surface into a series of shape parameters, which respect no arbitrage.

There are various extensions, such as extended SSVI (incidentally).