I would like to share the youtube video of my online seminar at Minnesota Center for Financial and Actuarial Mathematics and presentation slides.
I discuss the motivation behind introducing Karasinki-Sepp log-normal stochastic volatility (SV) model in our IJATF paper with Parviz Rakhmonov. I briefly highlight the advantages of this model over existing SV models. Then I focus on new features of the model.
For the first time, I formulate the dynamic of log-normal SV model consistent with the forward variance by construction. This formulation enables to automatically fit the model to a given term structure of variance swap strikes implied from market prices. I show that there is a small modification of the closed-form solution presented in our paper so that the existing solution can be applied here as well.
Also for the first time, I introduce the rough formulation of the log-normal SV model. I note that our exponential affine expansion for the classic log-normal SV model can also be applied for the rough version, but it results in a system of multi-variate system of integral equations which is numerically tedious. We need to resort tom Monte-Carlo simulations of this model and Deep Learning for model calibration. This is work in progress so stay tuned.
Finally, I present the model calibration to the time series of implied volatilities of options on Bitcoin traded on Deribit. I touch upon the calibration of mean-reversion parameters using empirical auto-correlation function discussed in our paper. The rest of model parameters: the current level and long-term mean volatility, volatility beta, and volatility-of-volatility are fitted in time series calibration.
Below I show that the model error (the average difference between market and model implied volatility) is less than 1% most of the times. The volatility beta serves as the expected skeweness indicator switching from large negative values during risk-aversion and positive values during risk-seeking periods. This time series construction can serve as a base for relative value analysis and quant trading strategies.
I mention that Python implementation of model is available in stochvolmodels package at Github. See an example of running the log-normal SV model and example of model calibration using the new formulation of term structure consistent with impled variance.
Disclosure
This research is a personal opinion and it does not represent an official view of my current and last employers.
This paper and the post is an investment advice in any possible form.
Importantly, the valuation of options on these assets is not feasible using conventional stochastic volatility models applied in practice such as Heston, SABR, Exponential Ornstein-Uhlenbeck stochastic volatility models, because these models fail to be arbitrage-free (forwards and call prices are not martingals). Curiously enough, the topic of no-arbitrage for SV models with positive return-volatility correlation has not received attention in literature, despite a large number of assets with positive return-volatility correlation.
