Modelling of implied volatility surfaces of crypto assets, such as Bitcoin and Ether, is challenging when applying conventional stochastic volatility (SV) models. On the theoretical side, the dominance of inverse pay-off functions requires stronger conditions for the existence of equivalent martingale measures. On the practical side, the positive correlation between the price-returns and volatility and price may either invalidate some SV models or pose tight restrictions on model parameters.
First, we introduce a market model for the valuation of vanilla and inverse options on crypto assets under the equivalent martingale money market account measure and the inverse spot measure. We derive conditions for the existence of such valuation measures under conventional SV models and we show limitations of popular SV models when the correlation between returns and their volatility dynamics is positive.
Second, we introduce the lognormal SV model with the quadratic drift for arbitrage-free dynamics of implied volatilities. Despite the non-linear drift, we show that the volatility process has a strong solution. We derive the domain model parameters for which the equivalent spot and inverse measures exists.
We then develop an analytic approach for valuation of options on price and quadratic variance based on a new method of affine expansion for the moment generation function of price processes with non-affine and non-linear dynamics. We proof the stability of the expansion method and demonstrate its accuracy in option pricing applications.
We finally illustrate the application of the lognormal SV model for calibrating and modeling of Bitcoin and Ether volatility surfaces using Deribit options exchange data.
This talk is based on joint work with Parviz Rakhmonov.