Volatility interpolation on crypto-assets: motivation, models and results

The volume on crypto options today represents only 2% of the spot, while on traditional markets this volume is much higher than the underlying market. Volatility interpolation models are needed to price these derivatives.

This motivated us to test volatility interpolation models used in traditional finance and to test their robustness on the crypto market. In this blog post, we present the results obtained by two parametric volatility interpolation models (SVI and SABR) on bitcoin options data as well as the precaution to take to cancel arbitrage opportunities.

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Two arbitrage strategies that the model should cancel

When the volatility used is not well calculated, arbitrage opportunities can arise, i.e it is possible for an actor to make a safe profit without taking any risk by buying a well-chosen combination of calls and puts. Here are the two strategies that we want to cancel.

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Butterfly spread arbitrage

Let us introduce what a butterfly spread arbitrage is.
It is a strategy with options which consists of buying and selling simultaneously three similar types of options on the same underlying, with the same expiry T and with strikes K — ε<K<K+ε.

The butterfly strategy consists in buying a call option with strike K — ε,a call option with strike K+ε and selling two call options with strike K.

The payoff is a positive function of underlying spot price:

Therefore an arbitrage exists if the price of this combination of options is negative, i.e if:

where C(K,T) is the Black&Scholes call price for strike K and expiry T.

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Calendar arbitrage

Let C(T,K) be the price of a call with expiry T and strike K. We know if T<T’ there is a calendar spread arbitrage if C(T,K)>C(T’,K). And here is how to exploit it:

• At time t=0 we short the option with maturity T and we long the option with maturity T’, then we have the initial profit x=C(T,K) — C(T’,K).
• At time t=T if the spot price is less than or equal to K the shorted option expires worthless and our total profit is x+max(Spot(t=T’)-K,0).

But else if the spot price is higher than or equal to K we short sell the stock (i.e borrow and sell) and receive K. At time t=T’ if the spot is higher than or equal to K , we buy the stock by paying the amount K and return the stock that we have short sold at T, in this case the final profit is x. But if at time t=T’ the spot price is less than or equal to K, we buy the stock by paying the amount spot price at time t=T’ and we return it to our borrower, but we have received K at t=T, in this case our net profit is x+K-Spot(t=T’).

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Overview of the two models used: SVI and SABR

SVI and SABR are two parametric models to interpolate volatility. Historically, SABR is the preferred model when it comes to interest rate derivatives, but our results lead us to say that it is quite efficient on interpolating volatilities in the crypto assets market.

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SVI

SVI model represents the total variance or the implied volatility observed in the real market. We have to define a loss function to mesure difference between our parametric model and the real data in oder to adjust parameters. Our model should avoid arbitrage opportunity (calendar spread arbitrage and butterfly spread arbitrage). The set of parameters to adjust is:

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The parametric formula is the following:

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SABR

SABR is a stochastic volatility model that allows to obtain the volatility smile on derivative markets. In our study we have taken the formula at order 0. Unlike the SVI model which interpolates the total variance, the SABR model interpolates directly the implied volatility.

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where:

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and:

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Results of these two models to interpolate the volatility on bitcoin

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Input data

First, it is necessary to retrieve the market data for the options on the underlying asset under consideration. In our case it is bitcoin. Here are some rows of data from our csv:

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Results obtained with SVI parameterization on one example

Results obtained with SABR parameterization on one example

Conclusion

In conclusion, the SVI and SABR models, which are respectively used to interpolate the volatility on equity markets and on interest rate products, are effective in reconstructing the volatility surface on crypto assets. However, research is being done to build a formalism adapted to the new market that are the crypto-assets.

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