Calibration

Heston vol-of-vol and the VVIX ArXiv ID: 2512.19611 “View on arXiv” Authors: Jherek Healy Abstract The Heston stochastic volatility model is arguably, the most popular stochastic volatility model used to price and risk manage exotic derivatives. In spite of this, it is not necessarily easy to calibrate to the market and obtain stable exotic option prices with this model. This paper focuses on the vol-of-vol parameter and its relation with the volatility of volatility index (VVIX) level. Four different approaches to estimate the VVIX in the Heston model are presented: two based on the known transition density of the variance, one analytical approximation, and one based on the Heston PDE which computes the value directly out of the underlying SPX500. Finally we explore their use to improve calibration stability. ...

Volatility Calibration via Automatic Local Regression ArXiv ID: 2509.16334 “View on arXiv” Authors: Ruozhong Yang, Hao Qin, Charlie Che, Liming Feng Abstract Managing exotic derivatives requires accurate mark-to-market pricing and stable Greeks for reliable hedging. The Local Volatility (LV) model distinguishes itself from other pricing models by its ability to match observable market prices across all strikes and maturities with high accuracy. However, LV calibration is fundamentally ill-posed: finite market observables must determine a continuously-defined surface with infinite local volatility parameters. This ill-posed nature often causes spiky LV surfaces that are particularly problematic for finite-difference-based valuation, and induces high-frequency oscillations in solutions, thus leading to unstable Greeks. To address this challenge, we propose a pre-calibration smoothing method that can be integrated seamlessly into any LV calibration workflow. Our method pre-processes market observables using local regression that automatically minimizes asymptotic conditional mean squared error to generate denoised inputs for subsequent LV calibration. Numerical experiments demonstrate that the proposed pre-calibration smoothing yields significantly smoother LV surfaces and greatly improves Greek stability for exotic options with negligible additional computational cost, while preserving the LV model’s ability to fit market observables with high fidelity. ...

Small Volatility Approximation and Multi-Factor HJM Models ArXiv ID: 2506.12584 “View on arXiv” Authors: V. M. Belyaev Abstract Here we demonstrate how we can use Small Volatility Approximation in calibration of Multi-Factor HJM model with deterministic correlations, factor volatilities and mean reversals. It is noticed that quality of this calibration is very good and it does not depend on number of factors. Keywords: Heath-Jarrow-Morton (HJM) Model, Small Volatility Approximation, Calibration, Deterministic Volatility, Term Structure, Fixed Income ...

ADAGE: A generic two-layer framework for adaptive agent based modelling ArXiv ID: 2501.09429 “View on arXiv” Authors: Unknown Abstract Agent-based models (ABMs) are valuable for modelling complex, potentially out-of-equilibria scenarios. However, ABMs have long suffered from the Lucas critique, stating that agent behaviour should adapt to environmental changes. Furthermore, the environment itself often adapts to these behavioural changes, creating a complex bi-level adaptation problem. Recent progress integrating multi-agent reinforcement learning into ABMs introduces adaptive agent behaviour, beginning to address the first part of this critique, however, the approaches are still relatively ad hoc, lacking a general formulation, and furthermore, do not tackle the second aspect of simultaneously adapting environmental level characteristics in addition to the agent behaviours. In this work, we develop a generic two-layer framework for ADaptive AGEnt based modelling (ADAGE) for addressing these problems. This framework formalises the bi-level problem as a Stackelberg game with conditional behavioural policies, providing a consolidated framework for adaptive agent-based modelling based on solving a coupled set of non-linear equations. We demonstrate how this generic approach encapsulates several common (previously viewed as distinct) ABM tasks, such as policy design, calibration, scenario generation, and robust behavioural learning under one unified framework. We provide example simulations on multiple complex economic and financial environments, showing the strength of the novel framework under these canonical settings, addressing long-standing critiques of traditional ABMs. ...

A case study on different one-factor Cheyette models for short maturity caplet calibration ArXiv ID: 2408.11257 “View on arXiv” Authors: Unknown Abstract In [“1”], we calibrated a one-factor Cheyette SLV model with a local volatility that is linear in the benchmark forward rate and an uncorrelated CIR stochastic variance to 3M caplets of various maturities. While caplet smiles for many maturities could be reasonably well calibrated across the range of strikes, for instance the 1Y maturity could not be calibrated well across that entire range of strikes. Here, we study whether models with alternative local volatility terms and/or alternative stochastic volatility or variance models can calibrate the 1Y caplet smile better across the strike range better than the model studied in [“1”]. This is made possible and feasible by the generic simulation, pricing, and calibration frameworks introduced in [“1”] and some new frameworks presented in this paper. We find that some model settings calibrate well to the 1Y smile across the strike range under study. In particular, a model setting with a local volatility that is piece-wise linear in the benchmark forward rate together with an uncorrelated CIR stochastic variance and one with a local volatility that is linear in the benchmark rate together with a correlated lognormal stochastic volatility with quadratic drift (QDLNSV) as in [“2”] calibrate well. We discuss why the later might be a preferable model. [“1”] Arun Kumar Polala and Bernhard Hientzsch. Parametric differential machine learning for pricing and calibration. arXiv preprint arXiv:2302.06682 , 2023. [“2”] Artur Sepp and Parviz Rakhmonov. A Robust Stochastic Volatility Model for Interest Rate Dynamics. Risk Magazine, 2023 ...

Calibrating the Heston model with deep differential networks ArXiv ID: 2407.15536 “View on arXiv” Authors: Unknown Abstract We propose a gradient-based deep learning framework to calibrate the Heston option pricing model (Heston, 1993). Our neural network, henceforth deep differential network (DDN), learns both the Heston pricing formula for plain-vanilla options and the partial derivatives with respect to the model parameters. The price sensitivities estimated by the DDN are not subject to the numerical issues that can be encountered in computing the gradient of the Heston pricing function. Thus, our network is an excellent pricing engine for fast gradient-based calibrations. Extensive tests on selected equity markets show that the DDN significantly outperforms non-differential feedforward neural networks in terms of calibration accuracy. In addition, it dramatically reduces the computational time with respect to global optimizers that do not use gradient information. ...

Evaluating Microscopic and Macroscopic Models for Derivative Contracts on Commodity Indices ArXiv ID: 2408.00784 “View on arXiv” Authors: Unknown Abstract In this article, we analyze two modeling approaches for the pricing of derivative contracts on a commodity index. The first one is a microscopic approach, where the components of the index are modeled individually, and the index price is derived from their combination. The second one is a macroscopic approach, where the index is modeled directly. While the microscopic approach offers greater flexibility, its calibration results to be more challenging, thus leading practitioners to favor the macroscopic approach. However, in the macroscopic model, the lack of explicit futures curve dynamics raises questions about its ability to accurately capture the behavior of the index and its sensitivities. In order to investigate this, we calibrate both models using derivatives of the S&P GSCI Crude Oil excess-return index and compare their pricing and sensitivities on path-dependent options, such as autocallable contracts. This research provides insights into the suitability of macroscopic models for pricing and hedging purposes in real scenarios. ...

Pricing and calibration in the 4-factor path-dependent volatility model ArXiv ID: 2406.02319 “View on arXiv” Authors: Unknown Abstract We consider the path-dependent volatility (PDV) model of Guyon and Lekeufack (2023), where the instantaneous volatility is a linear combination of a weighted sum of past returns and the square root of a weighted sum of past squared returns. We discuss the influence of an additional parameter that unlocks enough volatility on the upside to reproduce the implied volatility smiles of S&P 500 and VIX options. This PDV model, motivated by empirical studies, comes with computational challenges, especially in relation to VIX options pricing and calibration. We propose an accurate \emph{“pathwise”} neural network approximation of the VIX which leverages on the Markovianity of the 4-factor version of the model. The VIX is learned pathwise as a function of the Markovian factors and the model parameters. We use this approximation to tackle the joint calibration of S&P 500 and VIX options, quickly sample VIX paths, and price derivatives that jointly depend on S&P 500 and VIX. As an interesting aside, we also show that this \emph{“time-homogeneous”}, low-parametric, Markovian PDV model is able to fit the whole surface of S&P 500 implied volatilities remarkably well. ...

On the Hull-White model with volatility smile for Valuation Adjustments ArXiv ID: 2403.14841 “View on arXiv” Authors: Unknown Abstract Affine Diffusion dynamics are frequently used for Valuation Adjustments (xVA) calculations due to their analytic tractability. However, these models cannot capture the market-implied skew and smile, which are relevant when computing xVA metrics. Hence, additional degrees of freedom are required to capture these market features. In this paper, we address this through an SDE with state-dependent coefficients. The SDE is consistent with the convex combination of a finite number of different AD dynamics. We combine Hull-White one-factor models where one model parameter is varied. We use the Randomized AD (RAnD) technique to parameterize the combination of dynamics. We refer to our SDE with state-dependent coefficients and the RAnD parametrization of the original models as the rHW model. The rHW model allows for efficient semi-analytic calibration to European swaptions through the analytic tractability of the Hull-White dynamics. We use a regression-based Monte-Carlo simulation to calculate exposures. In this setting, we demonstrate the significant effect of skew and smile on exposures and xVAs of linear and early-exercise interest rate derivatives. ...

Deep Calibration of Market Simulations using Neural Density Estimators and Embedding Networks ArXiv ID: 2311.11913 “View on arXiv” Authors: Unknown Abstract The ability to construct a realistic simulator of financial exchanges, including reproducing the dynamics of the limit order book, can give insight into many counterfactual scenarios, such as a flash crash, a margin call, or changes in macroeconomic outlook. In recent years, agent-based models have been developed that reproduce many features of an exchange, as summarised by a set of stylised facts and statistics. However, the ability to calibrate simulators to a specific period of trading remains an open challenge. In this work, we develop a novel approach to the calibration of market simulators by leveraging recent advances in deep learning, specifically using neural density estimators and embedding networks. We demonstrate that our approach is able to correctly identify high probability parameter sets, both when applied to synthetic and historical data, and without reliance on manually selected or weighted ensembles of stylised facts. ...