Heston Calibration and Arbitrage-Free Option-Implied Densities

# Using Heston volatility model to derive option-implied densities. Correct or no?


# Using Heston volatility model to derive option-implied densities. Correct or no?












I am working with complementary option data (end-of-day quotes) for Bitcoin, obtained from the Deribit exchange. My objective is to extract smooth risk-neutral densities. Initially, I attempted to numerically second-differentiate the call surface (for a given day) to directly obtain the risk-neutral densities. This approach turned out to be problematic, even after applying various filtering methods, such as removing low-volume options and options beyond certain moneyness or spreads. My research suggests that most options data is too noisy to directly extract smooth and no-arbitrage densities. Consequently, I decided to use the following procedure (that I think is more or less consistent with industry) for each day:

- Remove zero volume options from the call surface.

- Calibrate a Heston stochastic volatility model to the call surface, obtaining the 5 parameters. The exact procedure was followed from https://www.youtube.com/watch?v=Jy4_AVEyO0w .

- Feed the estimated Heston parameters back into the Heston model and generate a set call options that expire in 3 months on a denser strike grid of (0.05 current trading price, 3x current trading price), using equally spaced intervals of L/1000 where L is the difference between the ends of the interval.

Now this is where I may run into trouble with my understanding. My understanding is that these newly generated option prices will be (1) arbitrage free (because its a heston model), and (2) By definition of calibration these heston prices will be as close as possible to the observed market prices as possible.

- I quantify the validity of my heston parameters by computing the average absolute percent error between observed call options, and the predicted heston parameters for a given option surface obtained in step 1. I summarize this in a table. For instance on average the my errors are about 3.24%.

- I numerically differentiate the heston call prices that I simulated to obtain the risk neutral densities at 3month maturity. I clip the density the moment the probability reaches 10^-4, or a value very close to zero. Finally I renormalize the distribution so that the probability sums to 1. A sample of the densities is presented.

- Now, I want to claim that the densities that I generated, are reasonably close to what the market is saying, and that all I did was do the minimum possible adjustment necessary make sure that the densities follow established financial principles of Heston. My friend in academia however is not convinced because I use a parametric method and that densities are possibly mis-specified if the model is inappropriate. Additionally he says non-parametric methods for extracting the densities will be more correct.

6B. From what I read, the "industry" standard (not sure if that is necessarily the best) is to convert option prices to IV, and then find a way to interpolate the IV smile in a way that is consistent with no arbitrage. This is typically done parametrically using the SABR model. Now convert those IVs back to call options and numerically differentiate. To me it seems like I'm conceptually doing the same thing except I'm instead of parametrizing the IV space, I'm doing so in the call space and with the heston model. The output of the heston model is as close to the observed prices as possible, but with minors adjustments so that the entire curve is arbitrage free.

## Answer by Jesper Tidblom (score 2)

https://quant.stackexchange.com/a/80279

This is my two cents:

Everything depends on the applications of the result and what is important to you there. Is a good fit to the market data very essential, or is having an arbitrage free interpolation important? Or something else? Different calibrated models will give different densities which will affect the results. There is no unique, correct, density.

Sure, the Heston Model will give you an arbitrage free density, but the model has only five parameters, so you will in general not get a very good fit to the data. Also, what is your input to calibrate the model? You should avoid calibrating the Heston model to option data for just one maturity. The reason for this is that some parameters in the Heston model concerns the time dependency of the Heston model. If you just use one maturity, the calibration will typically become unstable.

If you calibrate the Heston model to options with several maturities you will typically get a much more stable calibration. This can be accomplished so that the initial guesses is not that much of a problem anymore (but it takes a lot of care with the details). However, if you calibrate an entire surface and then use this to interpolate a smile, you will have a pretty bad fit to market data. Sure, you get no arbitrage, but this might not be good enough for your application.

The SABR model is more suitable if you want to calibrate and interpolate option data for one maturity only. This model also has the problem that it only has a few parameters though, so the fit to the market data is typically not that good. Also there are problems with the standard so called Hagan formulas to compute the Black implied volatility. While they are very good approximations in most cases, they are just approximations. This can result in an error and even arbitrage and negative densities in extreme cases.

If you just want to interpolate option prices to get densities I would, as a first step, try to remove any arbitrage in your set of prices. This relatively new article is taking care of that and only uses the elementary simplex method: https://ora.ox.ac.uk/objects/uuid:77110176-4528-42c1-85eb-2f78b94514e8/files/smw22v582j

The reason you might want to remove arbitrage first is that some models are sensitive to arbitrage in the input. This will then typically make the calibration step fail or become unstable.

Also there is an interesting series of articles about more direct arbitrage free interpolation with an excellent fit to option prices by authors like C.W. Oosterlee and F. Le Floc'h. In his latest article Le Floc'h presents an arbitrage free interpolation with excellent fit to market prices where the density is of class $C^3$. This seems to be what you are looking for. This is a very recent article which has not been published in some Journal yet, but can be found on the Arxiv. There are a few misprints, but the math is pretty straightforward so it is not hard to spot the few mistakes.

https://arxiv.org/abs/2305.13791

Otherwise you can look in the list of references in the above mention article to see their previous work.