Stochastic Volatility

Error bound for the asymptotic expansion of the Hartman-Watson integral ArXiv ID: 2504.04992 “View on arXiv” Authors: Unknown Abstract This note gives a bound on the error of the leading term of the $t\to 0$ asymptotic expansion of the Hartman-Watson distribution $θ(r,t)$ in the regime $rt=ρ$ constant. The leading order term has the form $θ(ρ/t,t)=\frac{“1”}{“2πt”}e^{"-\frac{1"}{“t”} (F(ρ)-π^2/2)} G(ρ) (1 + \vartheta(t,ρ))$, where the error term is bounded uniformly over $ρ$ as $|\vartheta(t,ρ)|\leq \frac{“1”}{“70”}t$. ...

Mathematical Modeling of Option Pricing with an Extended Black-Scholes Framework ArXiv ID: 2504.03175 “View on arXiv” Authors: Unknown Abstract This study investigates enhancing option pricing by extending the Black-Scholes model to include stochastic volatility and interest rate variability within the Partial Differential Equation (PDE). The PDE is solved using the finite difference method. The extended Black-Scholes model and a machine learning-based LSTM model are developed and evaluated for pricing Google stock options. Both models were backtested using historical market data. While the LSTM model exhibited higher predictive accuracy, the finite difference method demonstrated superior computational efficiency. This work provides insights into model performance under varying market conditions and emphasizes the potential of hybrid approaches for robust financial modeling. ...

A multi-factor model for improved commodity pricing: Calibration and an application to the oil market ArXiv ID: 2501.15596 “View on arXiv” Authors: Unknown Abstract We present a new model for commodity pricing that enhances accuracy by integrating four distinct risk factors: spot price, stochastic volatility, convenience yield, and stochastic interest rates. While the influence of these four variables on commodity futures prices is well recognized, their combined effect has not been addressed in the existing literature. We fill this gap by proposing a model that effectively captures key stylized facts including a dynamic correlation structure and time-varying risk premiums. Using a Kalman filter-based framework, we achieve simultaneous estimation of parameters while filtering state variables through the joint term structure of futures prices and bond yields. We perform an empirical analysis focusing on crude oil futures, where we benchmark our model against established approaches. The results demonstrate that the proposed four-factor model effectively captures the complexities of futures term structures and outperforms existing models. ...

Pricing Quanto and Composite Contracts with Local-Correlation Models ArXiv ID: 2501.07200 “View on arXiv” Authors: Unknown Abstract Pricing composite and quanto contracts requires a joint model of both the underlying asset and the exchange rate. In this contribution, we explore the potential of local-correlation models to address the challenges of calibrating synthetic quanto forward contracts and composite options quoted in the market. Specifically, we design on-line calibration procedures for generic local and stochastic volatility models. The paper concludes with a numerical study assessing the calibration performance of these methodologies and comparing them to simpler approximations of the correlation structure. ...

Heath-Jarrow-Morton meet lifted Heston in energy markets for joint historical and implied calibration ArXiv ID: 2501.05975 “View on arXiv” Authors: Unknown Abstract In energy markets, joint historical and implied calibration is of paramount importance for practitioners yet notoriously challenging due to the need to align historical correlations of futures contracts with implied volatility smiles from the option market. We address this crucial problem with a parsimonious multiplicative multi-factor Heath-Jarrow-Morton (HJM) model for forward curves, combined with a stochastic volatility factor coming from the Lifted Heston model. We develop a sequential fast calibration procedure leveraging the Kemna-Vorst approximation of futures contracts: (i) historical correlations and the Variance Swap (VS) volatility term structure are captured through Level, Slope, and Curvature factors, (ii) the VS volatility term structure can then be corrected for a perfect match via a fixed-point algorithm, (iii) implied volatility smiles are calibrated using Fourier-based techniques. Our model displays remarkable joint historical and implied calibration fits - to both German power and TTF gas markets - and enables realistic interpolation within the implied volatility hypercube. ...

The AI Black-Scholes: Finance-Informed Neural Network ArXiv ID: 2412.12213 “View on arXiv” Authors: Unknown Abstract In the realm of option pricing, existing models are typically classified into principle-driven methods, such as solving partial differential equations (PDEs) that pricing function satisfies, and data-driven approaches, such as machine learning (ML) techniques that parameterize the pricing function directly. While principle-driven models offer a rigorous theoretical framework, they often rely on unrealistic assumptions, such as asset processes adhering to fixed stochastic differential equations (SDEs). Moreover, they can become computationally intensive, particularly in high-dimensional settings when analytical solutions are not available and thus numerical solutions are needed. In contrast, data-driven models excel in capturing market data trends, but they often lack alignment with core financial principles, raising concerns about interpretability and predictive accuracy, especially when dealing with limited or biased datasets. This work proposes a hybrid approach to address these limitations by integrating the strengths of both principled and data-driven methodologies. Our framework combines the theoretical rigor and interpretability of PDE-based models with the adaptability of machine learning techniques, yielding a more versatile methodology for pricing a broad spectrum of options. We validate our approach across different volatility modeling approaches-both with constant volatility (Black-Scholes) and stochastic volatility (Heston), demonstrating that our proposed framework, Finance-Informed Neural Network (FINN), not only enhances predictive accuracy but also maintains adherence to core financial principles. FINN presents a promising tool for practitioners, offering robust performance across a variety of market conditions. ...

Pricing Multi-strike Quanto Call Options on Multiple Assets with Stochastic Volatility, Correlation, and Exchange Rates ArXiv ID: 2411.16617 “View on arXiv” Authors: Unknown Abstract Quanto options allow the buyer to exchange the foreign currency payoff into the domestic currency at a fixed exchange rate. We investigate quanto options with multiple underlying assets valued in different foreign currencies each with a different strike price in the payoff function. We carry out a comparative performance analysis of different stochastic volatility (SV), stochastic correlation (SC), and stochastic exchange rate (SER) models to determine the best combination of these models for Monte Carlo (MC) simulation pricing. In addition, we test the performance of all model variants with constant correlation as a benchmark. We find that a combination of GARCH-Jump SV, Weibull SC, and Ornstein Uhlenbeck (OU) SER performs best. In addition, we analyze different discretization schemes and their results. In our simulations, the Milstein scheme yields the best balance between execution times and lower standard deviations of price estimates. Furthermore, we find that incorporating mean reversion into stochastic correlation and stochastic FX rate modeling is beneficial for MC simulation pricing. We improve the accuracy of our simulations by implementing antithetic variates variance reduction. Finally, we derive the correlation risk parameters Cora and Gora in our framework so that correlation hedging of quanto options can be performed. ...

What events matter for exchange rate volatility ? ArXiv ID: 2411.16244 “View on arXiv” Authors: Unknown Abstract This paper expands on stochastic volatility models by proposing a data-driven method to select the macroeconomic events most likely to impact volatility. The paper identifies and quantifies the effects of macroeconomic events across multiple countries on exchange rate volatility using high-frequency currency returns, while accounting for persistent stochastic volatility effects and seasonal components capturing time-of-day patterns. Given the hundreds of macroeconomic announcements and their lags, we rely on sparsity-based methods to select relevant events for the model. We contribute to the exchange rate literature in four ways: First, we identify the macroeconomic events that drive currency volatility, estimate their effects and connect them to macroeconomic fundamentals. Second, we find a link between intraday seasonality, trading volume, and the opening hours of major markets across the globe. We provide a simple labor-based explanation for this observed pattern. Third, we show that including macroeconomic events and seasonal components is crucial for forecasting exchange rate volatility. Fourth, our proposed model yields the lowest volatility and highest Sharpe ratio in portfolio allocations when compared to standard SV and GARCH models. ...

Fast Deep Hedging with Second-Order Optimization ArXiv ID: 2410.22568 “View on arXiv” Authors: Unknown Abstract Hedging exotic options in presence of market frictions is an important risk management task. Deep hedging can solve such hedging problems by training neural network policies in realistic simulated markets. Training these neural networks may be delicate and suffer from slow convergence, particularly for options with long maturities and complex sensitivities to market parameters. To address this, we propose a second-order optimization scheme for deep hedging. We leverage pathwise differentiability to construct a curvature matrix, which we approximate as block-diagonal and Kronecker-factored to efficiently precondition gradients. We evaluate our method on a challenging and practically important problem: hedging a cliquet option on a stock with stochastic volatility by trading in the spot and vanilla options. We find that our second-order scheme can optimize the policy in 1/4 of the number of steps that standard adaptive moment-based optimization takes. ...

Log Heston Model for Monthly Average VIX ArXiv ID: 2410.22471 “View on arXiv” Authors: Unknown Abstract We model time series of VIX (monthly average) and monthly stock index returns. We use log-Heston model: logarithm of VIX is modeled as an autoregression of order 1. Our main insight is that normalizing monthly stock index returns (dividing them by VIX) makes them much closer to independent identically distributed Gaussian. The resulting model is mean-reverting, and the innovations are non-Gaussian. The combined stochastic volatility model fits well, and captures Pareto-like tails of real-world stock market returns. This works for small and large stock indices, for both price and total returns. ...