Markovian Approximation

The ATM implied skew in the ADO-Heston model ArXiv ID: 2309.15044 “View on arXiv” Authors: Unknown Abstract 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. ...

Weak Markovian Approximations of Rough Heston ArXiv ID: 2309.07023 “View on arXiv” Authors: Unknown Abstract The rough Heston model is a very popular recent model in mathematical finance; however, the lack of Markov and semimartingale properties poses significant challenges in both theory and practice. A way to resolve this problem is to use Markovian approximations of the model. Several previous works have shown that these approximations can be very accurate even when the number of additional factors is very low. Existing error analysis is largely based on the strong error, corresponding to the $L^2$ distance between the kernels. Extending earlier results by [“Abi Jaber and El Euch, SIAM Journal on Financial Mathematics 10(2):309–349, 2019”], we show that the weak error of the Markovian approximations can be bounded using the $L^1$-error in the kernel approximation for general classes of payoff functions for European style options. Moreover, we give specific Markovian approximations which converge super-polynomially in the number of dimensions, and illustrate their numerical superiority in option pricing compared to previously existing approximations. The new approximations also work for the hyper-rough case $H > -1/2$. ...