Rough Heston Model

Fast reliable pricing and calibration of the rough Heston model ArXiv ID: 2508.15080 “View on arXiv” Authors: Svetlana Boyarchenko, Marco de Innocentis, Sergei Levendorskiĭ Abstract The paper is an extended and modified version of the preprint S.Boyarchenko and S.Levendorskiĭ Correct implied volatility shapes and reliable pricing in the rough Heston model". We combine a modification of the Adams method with the SINH-acceleration method S.Boyarchenko and S.Levendorskii (IJTAF 2019, v.22) of Fourier inversion (iFT) to price vanilla options under the rough Heston model. For moderate or long maturities and strikes near spot, thousands of prices are computed in several milliseconds (ms) in Matlab on a Mac with moderate specs, with relative errors $\lesssim 10^{"-4"}$. Even for options close to expiry and far-OTM, the pricing takes a few tens or hundreds of ms. We show that, for the calibrated parameters in El Euch and Rosenbaum (Math.Finance 2019, v.29), the model implied vol surface is much flatter and fits the market data poorly; thus the calibration in op.cit. is a case of ghost calibration’’ (M.Boyarchenko and S.Levendorskiĭ, Quant. Finance 2015, v.15): numerical error and model specification error offset each other, creating an apparently good fit that vanishes when a more accurate pricer is used. We explain how such errors arise in popular iFT implementations that use fixed numerical parameters, yielding spurious smiles/skews, and provide numerical evidence that SINH acceleration is faster and more accurate than competing methods. Robust error control is ensured by a general Conformal Bootstrap principle that we formulate; the principle is applicable to many Fourier-pricing methods. We outline how this principle and our method enable accurate calibration procedures that are hundreds of times faster than approaches commonly used in the industry. Disclaimer: The views expressed herein are those of the authors only. No other representation should be attributed. ...

Stochastic Optimal Control of Iron Condor Portfolios for Profitability and Risk Management ArXiv ID: 2501.12397 “View on arXiv” Authors: Unknown Abstract Previous research on option strategies has primarily focused on their behavior near expiration, with limited attention to the transient value process of the portfolio. In this paper, we formulate Iron Condor portfolio optimization as a stochastic optimal control problem, examining the impact of the control process ( u(k_i, τ) ) on the portfolio’s potential profitability and risk. By assuming the underlying price process as a bounded martingale within $[“K_1, K_2”]$, we prove that the portfolio with a strike structure of $k_1 < k_2 = K_2 < S_t < k_3 = K_3 < k_4$ has a submartingale value process, which results in the optimal stopping time aligning with the expiration date $τ= T$. Moreover, we construct a data generator based on the Rough Heston model to investigate general scenarios through simulation. The results show that asymmetric, left-biased Iron Condor portfolios with $τ= T$ are optimal in SPX markets, balancing profitability and risk management. Deep out-of-the-money strategies improve profitability and success rates at the cost of introducing extreme losses, which can be alleviated by using an optimal stopping strategy. Except for the left-biased portfolios $τ$ generally falls within the range of [“50%,75%”] of total duration. In addition, we validate these findings through case studies on the actual SPX market, covering bullish, sideways, and bearish market conditions. ...

High order approximations and simulation schemes for the log-Heston process ArXiv ID: 2407.17151 “View on arXiv” Authors: Unknown Abstract We present weak approximations schemes of any order for the Heston model that are obtained by using the method developed by Alfonsi and Bally (2021). This method consists in combining approximation schemes calculated on different random grids to increase the order of convergence. We apply this method with either the Ninomiya-Victoir scheme (2008) or a second-order scheme that samples exactly the volatility component, and we show rigorously that we can achieve then any order of convergence. We give numerical illustrations on financial examples that validate the theoretical order of convergence. We also present promising numerical results for the multifactor/rough Heston model and hint at applications to other models, including the Bates model and the double Heston model. ...

A generalization of the rational rough Heston approximation ArXiv ID: 2310.09181 “View on arXiv” Authors: Unknown Abstract Previously, in [“GR19”], we derived a rational approximation of the solution of the rough Heston fractional ODE in the special case λ= 0, which corresponds to a pure power-law kernel. In this paper we extend this solution to the general case of the Mittag-Leffler kernel with λ\geq 0. We provide numerical evidence of the convergence of the solution. ...

Efficient option pricing in the rough Heston model using weak simulation schemes ArXiv ID: 2310.04146 “View on arXiv” Authors: Unknown Abstract We provide an efficient and accurate simulation scheme for the rough Heston model in the standard ($H>0$) as well as the hyper-rough regime ($H > -1/2$). The scheme is based on low-dimensional Markovian approximations of the rough Heston process derived in [“Bayer and Breneis, arXiv:2309.07023”], and provides weak approximation to the rough Heston process. Numerical experiments show that the new scheme exhibits second order weak convergence, while the computational cost increases linear with respect to the number of time steps. In comparison, existing schemes based on discretization of the underlying stochastic Volterra integrals such as Gatheral’s HQE scheme show a quadratic dependence of the computational cost. Extensive numerical tests for standard and path-dependent European options and Bermudan options show the method’s accuracy and efficiency. ...

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 $H > -1/2$. ...