The literature on direct VIX modeling highlights volatility-of-volatility as an important feature of VIX dynamics. Recent periods of financial market turbulence have been characterized by numerous uncertain events, leading to pronounced spikes in volatility-of-volatility. To capture this feature, this study proposes a new direct VIX model incorporating an additional spike process and derives an analytical characteristic function for pricing VIX derivatives. Our empirical results indicate that the additional spike component is statistically significant and helps capture large, transient spikes in volatility-of-volatility in an economically interpretable way.
Assets with low exposure to market risk often trade at larger liquidity discounts than high-beta, high-volatility assets-a pattern that standard models cannot explain. This article reconciles that anomaly by valuing liquidity as an exchange option-the right to swap an illiquid asset for the market portfolio. When the reinvestment benchmark is the market rather than cash, the relevant risk driver is the exchange volatility between the asset and the market, which measures how much their returns diverge. Exchange volatility is U-shaped in the asset's systematic exposure, minimized at beta = 1: For the empirically prevalent range beta < 1, the discount for lack of marketability declines with beta, whereas for beta > 1 it increases. Calibrations using industry-level unlevered betas and equity volatilities confirm that the model reproduces observed secondary-market patterns across sectors. Low-beta industries such as utilities, infrastructure, and real estate exhibit the highest predicted discounts, whereas assets tracking the market one-for-one face the lowest illiquidity costs. An empirical test using US closed-end fund data confirms the predicted U-shaped relationship between discounts and systematic exposure. The framework isolates the pure cost of illiquidity-the value of reallocation flexibility absent such frictions-and provides a tractable, beta-adjusted formula for valuing thinly traded assets.
This article addresses the valuation of issuer-callable exotic structured products featuring Bermudan-style optionality and contingent coupon and principal payoff mechanisms. These products are structurally complex, often involving multi-asset underlyings, layered barrier conditions, and highly nonlinear coupon rules. The standard valuation approach is the Longstaff-Schwartz algorithm, which uses least-squares regression to approximate continuation values but encounters difficulties in high dimensions. We propose a randomized neural network (RNN) framework where hidden layers are randomly initialized and only the output layer is trained. Numerical experiments on representative structured products show that the RNN approach achieves pricing accuracy comparable to that of fully trained neural networks while offering improved computational efficiency. The RNN-based method thus provides a scalable and effective solution forvaluing complex callable structured products.
Option valuation is a fundamental task in the derivatives market. The implied volatility surface (IVS) plays a crucial role in valuing exotic options, managing risk, identifying arbitrage opportunities, hedging, and calibrating models.Traditional methods for IVS estimation rely stochastic volatility models or direct parametric and nonparametric approaches. We propose a novel, deep learning approach using a transformer-based model to predict movements of the implied volatility surface. Our model predicts the direction and magnitude of implied volatility change for each contract using 12 features (e.g., Greeks, interest rates, and open interest) from the previous nine days. Our model captures time-series and cross-sectional dependencies in market data of various contracts, leveraging a shared parameter space extract meaningful information from past IVS observations and related data. Our approach may be generalized to valuation and forecasting of othersecurities driven by multiple factors such as equityfactors (e.g., size, momentum, quality) and fixed-income factors (e.g., longevity, default risk, callability).
In this article, we investigate the pricing of vulnerable basket-spread options under multivariate variance gamma models. We derive a pricing formula in integral form and, by applying Gauss-Hermite and generalized Gauss-Laguerre quadrature rules, reformulate it as a weighted sum, which allows for efficient numerical implementation. Numerical experiments show that the proposed approximate pricing formula achieves high accuracy. In addition, we investigate the effects of default risk and skewness on option prices.
This study proposes the concept of "integrated options" to systematically capture embedded optionality in Treasury bond futures in China. Building on the cost-of-carry model, we model valuation deviations of deliverable bonds by independent generalized error distributions (GEDs). Using Monte Carlo simulations to generate price trajectories, we quantify integrated option values and derive futures prices via the Newton-Raphson method. Empirical analysis across 23 contracts reveals that the GED-integrated option (GED-IO) model significantly outperforms the CoC benchmark in normal markets. It reduces mean absolute error by 22.98% and mean absolute percentage error by 0.09% for 70% of contracts (16/23), demonstrating enhanced accuracy, convergence, and stability. However, structural failures occur in 7 contracts during intensive monetary easing cycles (2021-2022 and post-2024) due to accelerated policy expectation iteration. This highlights adaptation limitations to policy-risk transmission. Future research should integrate adaptive policy expectations and sentiment-price transmission. Our work provides a methodological foundation for pricing derivatives in China's policy-sensitive markets while framing critical refinements.
We propose a short-term interest rate model integrating the self-exciting Hawkes process with exogenous jump intensity modeled via the Feller-type square-root process. This framework captures jump clustering while introducing level-dependent variance into jump dynamics. Using stochastic Taylor expansions, we derive closed-form moment solutions and apply generalized method of moments estimation. Monte Carlo simulations validate estimator precision. Empirical analysis of US Treasury bill rates demonstrates enhanced capture of extreme events and clustered jumps, with statistical tests favoring our framework over constant-intensity alternatives. Derivative pricing implications are explored, highlighting the model's practical relevance in financial markets.
We propose a novel, distribution-free framework for estimating option Greeks using machine learning. Unlike traditional parametric approaches, our method employs accumulated local effects to capture data-driven sensitivities without imposing functional form restrictions. Validation on synthetic Black-Scholes prices confirms the method's ability to accurately recovertheoretical Greeks. When applied to a comprehensive dataset of 14 million European S&P 500 options, we document notable empirical deviations: Vega sensitivity plateaus at high volatility levels, Gamma exhibits sharper peaks near at-the-money strikes, and Theta decay follows nonlinear patterns. These findings reveal important limitations of conventional models in capturing market-implied sensitivities. The model-agnostic framework, illustrated here with gradient-boosted trees, offers practitioners a robust, parametric assumption-free alternative for risk measurement. Its versatility suggests potential for extension to hedging American options, exotic derivatives, and complex portfolios.
This article proposes a novel framework for forecasting the implied volatility surface of Volatility Index (VIX) options by extending the traditional heterogeneous autoregressive (HAR) model into a convolution-based surface forecasting approach (ConvHAR). Unlike conventional HAR models that generate forecasts at isolated points, the extended approach captures spatial-temporal dependencies across moneyness and time-to-maturity dimensions through convolutional architectures. To further enhance predictive power, we incorporate the implied volatility surface of SPX options as an auxiliary input, motivated by the theoretical and empirical linkages between SPX and VIX markets. Empirical results show that the proposed method outperforms benchmark univariate and pointwise models in forecasting accuracy, particularly when forecasting 5-15 days ahead. Moreover, incorporating cross-market volatility information from SPX leads to further gains in performance. These findings underscore the importance of modeling surface-level dynamics and market interdependencies in volatility forecasting, with practical implications for derivative pricing and risk management.
We propose a tractable SOFR-swaption pricing framework that integrates Jamshidian's decomposition with the one-factor Hull-White model in the extended forward-measure/ extended-bond setting and derives closed-form formulae consistent with SOFR's compounded-in-arrears convention. Model calibration generates a trader-ready 6-surface that supports interpolation across the expiry-tenor grid and facilitates pricing and mark-to-market valuation of less-liquid or bespoke contracts. Empirical results show good fit for short-and intermediate-maturity swaptions but significant errors for long-dated options, highlighting both the model's practical value and its structural limitations.
This article introduces a semi-analytical method for pricing American options on assets (stocks, ETFs) that pay discrete or continuous dividends. The problem is notoriously complex because discrete dividends create abrupt price drops and affect the optimal exercise timing, making traditional continuous dividend models unsuitable. Our approach uses the generalized integral transform (GIT) method introduced by the author and his co-authors in several previous publications, which transforms the pricing problem from a complex partial differential equation with a free boundary into an integral Volterra equation of the second or first kind. In this article, we illustrate this approach by considering a popular geometric Brownian motion model that accounts for discrete cash and proportional dividends using Dirac delta functions. By reframing the problem as an integral equation, we can sequentially solve for the option price and the early exercise boundary, effectively handling the discontinuities caused by the dividends. Our methodology provides a powerful alternative to standard numerical techniques such as binomial trees or finite difference methods, which can struggle with the jump conditions of discrete dividends by losing accuracy or performance. Several examples demonstrate that the GIT method is highly accurate and computationally efficient, avoiding the need for extensive computational grids or complex backward induction steps.
This paper provides evidence that stock returns, after truncation, might be modeled by a special type of continuous mixtures or normals, so-called $q$-Gaussians. Negative binomial distributions might model the counts for extreme returns. A generalized jump-diffusion model is proposed, and an explicit option pricing formula is obtained.
In this article, we develop an option-pricing model for exotic options using generative machine learning techniques. We focus on pricing exotic options with dual underlying assets, utilizing a non-Gaussian Levy process called the generalized normal tempered stable (gNTS) process. This multivariate process, a variation of time-changed multivariate Brownian motion, lacks a straightforward analytic formula for its probability density function. To overcome this, we employ the conditional real-valued non-volume preserving (CRealNVP) model, a type of flow-based generative network, to approximate the probability density function. We then explore no-arbitrage pricing within the gNTS framework for a quanto option. The CRealNVP model is trained to learn the probability density function of the gNTS distribution through a training set derived from Monte Carlo simulations. Subsequently, we use this trained model to estimate the parameters of the gNTS model based on empirical market data. Finally, we propose a methodology to establish an equivalent martingale measure for the gNTS model, enabling us to price the quanto option using the risk-neutral parameters obtained via the CRealNVP model. This comprehensive approach integrates advanced machine learning techniques with financial modeling to enhance the pricing accuracy of complex financial derivatives.
This study proposes an alternative method for estimating the risk-neutral density (RND) from option prices. Building on Birru and Figlewski (2012), by equalizing the density value and the slope between the empirical RND and fitted generalized Pareto distribution (GPD) at one connection point in each tail, the proposed method eliminates kinks at the junction and enhances computational efficiency by over 3.5% on average. Using S&P 500 Index options from January 2016 to August 2023, the results demonstrate that the proposed method slightly enhances predictive performance for future returns, as evidenced by higher out-of-sample R2 using the rolling window approach. Moreover, RND skewness shows positive significance in both 7-day and 30-day returns, implying stronger return expectations under more right-skewed distributions. RND kurtosis also exhibits a positive, albeit comparatively weaker, relationship with 30-day returns only, reflecting the premium demanded for bearing extreme uncertainty in longer horizons. The findings emphasize the importance of smooth RND estimation for effective risk assessment and return prediction in equity index markets.
Understochastic volatility (SV), despite the abundant literature on American option pricing, there is little work on American option hedging. This paper develops a feasible and excellently performing static-hedging-portfolio (SHP) method for hedging and pricing an American option under SV by constructing a portfolio of European options to match the payoff, delta, and vega of the target American option along its early exercise boundary. The novelty of the proposed SHP method is in incorporating the expected variance conditional on the stock price into Chung and Shih's (2009) method and further improvingtheir method by imposing the vega-matching condition. Our numerical analyses show the superiority of the proposed SHP method in effectively hedging and accurately pricing American options in the presence of SV, especially when the vega-matching condition is taken into account. For a large, randomly generated set of American option contracts, the average pricing error (hedging risk, measured by 5% Value at Risk) of the proposed SHP method ranges from 0.11% to 0.14% (0.78% to 0.88%) of the average option value, and the 5% Value at Risk of the proposed SHP method is around 3% that of the widely used dynamic delta-neutral hedging method.
