Calibrating Smile Models When Only One Side of the Smile Is Quoted

# Calibration of a volatility smile model on a partial smile


# Calibration of a volatility smile model on a partial smile












I'm using a well-known SABR model in order to build an implied volatility surface of caps/floors on a very illiquid market which is entirely missing OTM quotes. What happens to SABR implied smile/surface calibrated only on ITM and ATM quotes? I think that it's going to somehow underestimate/overestimate the real volatility in ITM and OTM regions. Will these effects be the same for all smile models? What if one would calibrate it on OTM and ATM without ITM? I'm looking for either quantitative or qualitative explanation.

UPD: I'm talking about a market where there is only one significant market maker providing indicative quotes for caps/floors and there are usually no more than 2-3 trades made per week, so there is no opportunity to calibrate a model to real trades. The interest rate was roughly 5% and the market maker was providing quotes for 4% floors as well as ATM, 6%, 8%, 10% caps. However due to the global inflation current ATM is already set at 12-13% with fixed strikes staying the same, i.e. 6%, 8%, 10% caps are ITM and 4% floor is OTM.

## Answer by BS. (score 1)

https://quant.stackexchange.com/a/70197

You could use an entropy minimization approach where you minimize a combination of a KL divergence between the physical measure and the risk neutral measure under your model, and a calibration error (distance between model prices and market prices). It's basically saying "I want to fit the model to the few option quotes that I observe in the market but at the same time I don't want to deviate too much from the physical probability measure". This will effectively regularize your model calibration by adding information from the physical measure.

Note that for the KL divergence, if you write it in density form, you may find the Breeden-Litzenberger formula useful to compute your risk-neutral density as a function of your model parameters.

By the way, yes, this will mix your usual "risk-neutral tools" with econometrics, but you don't really have a choice as relying solely on ITM & ATM quotes gives rise to an ill-posed calibration problem. One way to see it is, again via the Breeden-Litzenberger formula, when you're fitting a smile it's like you're fitting a marginal risk-neutral density. When you're giving only ITM & ATM quotes, you're effectively under-specifying the density function (as your quotes only specify one side of it, the other side is a pure extrapolation from the parameterized model you use for your smile). So in the absence of OTM quotes, it's better to say okay I'll settle for something not too far away from the physical measure.

An old paper that did precisely that (although not for the same reasons) is this one by Derman & Zou: http://emanuelderman.com/wp-content/uploads/1999/07/strike_adjusted_spread.pdf