On Vix Futures in the Rough Bergomi Model
On Vix Futures in the Rough Bergomi Model

On Vix Futures in the Rough Bergomi Model

Volatility, though not directly observed nor traded, is a fundamental object on financial markets, and has been the centre of attention of decades of theoretical and practical research, both to estimate it and to use it for trading purposes. (Page 1)

Most models used for pricing purposes (Heston [19], SABR [17], Bergomi [5]) are constructed under Q and are of Markovian nature (making pricing, and hence calibration, easier). (Page 1)

This approach (Rough Fractional Stochastic Volatility, RFSV for short) opens the door to revisiting classical pricing and calibration conundrums. They, together with the subsequent paper by Bayer, Friz and Gatheral, (see also [1, 12]) in particular showed that these models were able to capture the extra steepness of the implied volatility smile in Equity markets for short maturities, which continuous Markovian stochastic volatility models fail to describe. (Page 1)

One of the key issues in Equity markets is, not only to fit the (SPX) implied volatility smile, but to do so jointly with a calibration of the VIX (Futures and ideally options). (Page 1)

Recently, Bennedsen, Lunde and Pakkanen [4] presented a new simulation scheme for Brownian semistationary (BSS) processes. This method, as opposed to Cholesky, is an approximate method. However, in [4] the authors show that the method yields remarkable results in the case of the rough Bergomi model. (Page 3)

light of this analysis, both methods seem to approximate the required output in a decent manner. Even if the truncated Cholesky approach gives a considerably fast output for each maturity, it is not considered for calibration, since its computational time grows linearly in the number of maturities, making the algorithm too slow for reasonable calibration. Instead, we will use the truncated Cholesky approach as a benchmark for the upcoming approximations. (Page 9)

Dufresne [11] proved that, under certain conditions, an integral of log-normal variables asymptotically converges to a log-normal. This approximation has been widely used for many applications [11], and Dufresne’s result motivates BayerFriz-Gatheral’s assumption. We provide here exact formulae for the mean and variance of this distribution, and compare them numerically to those by Bayer-Friz-Gatheral. (Page 10)

In this final chapter we assess whether the Hurst parameter H obtained through the VIX Futures calibration algorithm is consistent with SPX options. For this purpose, we calibrate the rough Bergomi model to SPX option data by fixing the parameter H and letting the algorithm calibrate ν and ρ. One of the main reasons to fix H is that the hybrid scheme introduced in Section 2.1 remarkably reduces its complexity to O(n), since the O(n log n) complexity of the Volterra is computed only once and reused afterwards. (Page 18)

Following the path set by Bayer, Friz and Gatheral [3], we developed here a relatively fast algorithm to calibrate VIX Futures and the VIX smile, consistently with the SPX smile, in the rough Bergomi model. The clear strength of this model is that only a few parameters are needed, making the (re)calibration robust and stable. From a trader’s point of view, we highlight some potential market discrepancy between the VIX and the SPX, and leave a refined analysis thereof for future research. (Page 20)