# What is an acceptable error on implied volatility?

Given an implied volatility surface (on equity indexes) and a calibrated model, what is the range of error on implied volatility a trader would accept ?

This obviously depends on the model used to calibrate. I am interested in classical stochastic volatility models (Heston, SABR, e.g.) to begin with.

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Dealing with model error under stochastic volatility (in a more formal way) you could use the UVM (Uncertain Volatility Framework). Here are what i think are the most seminal references:

Avellenada et al (1995) Pricing And Hedging Derivative Securities In Markets With Uncertain Volatilities http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.3736
Lyons (1995) Uncertain Volatility and the risk-free synthesis of derivatives http://it.scribd.com/doc/40177754/Lyons-Uncertain-Volatility-and-Risk-Free-Synthesis-of-Derivatives-Appl-math-Finance

But you can also find some review of the topic (just google UVM).

This answers somehow lead you into the field of stochastic control problems which can be quite technical. One thing i remembered is: If you assume your volatility to be bounded $\sigma_t \in [\sigma_{\min}, \sigma_{\max}]$ you can price a contingent claim under two different viewpoints. One that is beneficial for the long side and one that is beneficial for the short side (by simply assuming in each case that the "true" volatility always has the most beneficial value for the respective side). All else being equal this should give you the arbitrage-free bid/ask prices in this framework.

So if you turn the problem around here you could price an existing contingent claim by means of your stochastic volatility model and end up somewhere between the bid and the ask i would say you should be okay.

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Euh... why don't you ask him? You better make sure that the error is smaller than the bid-offer probably smaller than half of that. Now bid offer can vary on maturity and underlyer and can go from 0.1% to 5%...

Anyhow - the trader will look at the vega of any trade he needs to price and multiply that by the vol error and perhaps add the result to the price so he won't care really whether it's 0.01% of 0.5%.

And btw what product are you pricing, that requires stoch vol?

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I am pricing variance swaps using SV models. The model has to reprice vanillas though, so I am looking at a metric to tell if the model is close enough to the market. The error between my SV implied volatility surface and the market one in basis points seems like a good start to me. –  vanna Jul 27 '12 at 8:34

I presume you are trying to do that so you can price exotic variance? The general thought process is that no matter how hard you try, you would not be able to perfectly fit the wings using one of these models (not within the bid/offer). However, most probably you are just trying to replicate the dynamics, so best thing is to come up with some sort of adjuster framework. Anyway, my 0.02 Vega.

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