USLV: Unspanned Stochastic Local Volatility Model

We propose a new framework for modeling stochastic local volatility, with potential applications to modeling derivatives on interest rates, commodities, credit, equity, FX etc., as well as hybrid derivatives. Our model extends the linearity-generating unspanned volatility term structure model by Carr et al. (2011) by adding a local volatility layer to it. We outline efficient numerical schemes for pricing derivatives in this framework for a particular four-factor specification (two "curve" factors plus two "volatility" factors). We show that the dynamics of such a system can be approximated by a Markov chain on a two-dimensional space (Z_t,Y_t), where coordinates Z_t and Y_t are given by direct (Kroneker) products of values of pairs of curve and volatility factors, respectively. The resulting Markov chain dynamics on such partly "folded" state space enables fast pricing by the standard backward induction. Using a nonparametric specification of the Markov chain generator, one can accurately match arbitrary sets of vanilla option quotes with different strikes and maturities. Furthermore, we consider an alternative formulation of the model in terms of an implied time change process. The latter is specified nonparametrically, again enabling accurate calibration to arbitrary sets of vanilla option quotes.Comment: Sections 3.2 and 3.3 are re-written, 3 figures adde

The ATM implied skew in the ADO-Heston model

In this paper similar to [P. Carr, A. Itkin, 2019] we construct another Markovian approximation of the rough Heston-like volatility model - the ADO-Heston model. The characteristic function (CF) of the model is derived under both risk-neutral and real measures which is an unsteady three-dimensional PDE with some coefficients being functions of the time $t$ and the Hurst exponent $H$. To replicate known behavior of the market implied skew we proceed with a wise choice of the market price of risk, and then find a closed form expression for the CF of the log-price and the ATM implied skew. Based on the provided example, we claim that the ADO-Heston model (which is a pure diffusion model but with a stochastic mean-reversion speed of the variance process, or a Markovian approximation of the rough Heston model) is able (approximately) to reproduce the known behavior of the vanilla implied skew at small $T$. We conclude that the behavior of our implied volatility skew curve ${\cal S}(T) \propto a(H) T^{b\cdot (H-1/2)}, \, b = const$, is not exactly same as in rough volatility models since $b \ne 1$, but seems to be close enough for all practical values of $T$. Thus, the proposed Markovian model is able to replicate some properties of the corresponding rough volatility model. Similar analysis is provided for the forward starting options where we found that the ATM implied skew for the forward starting options can blow-up for any $s > t$ when $T \to s$. This result, however, contradicts to the observation of [E. Alos, D.G. Lorite, 2021] that Markovian approximation is not able to catch this behavior, so remains the question on which one is closer to reality.Comment: 23 pages, 3 figures, 3 table