option pricing theory

This study addresses the monotonicity of normalized implied volatility coordinates within a finite quoted option chain under no-arbitrage conditions and derives model-independent variance identities. Relying solely on static no-arbitrage assumptions—including discrete strike comparisons, convexity, monotonicity, and call-put parity—the authors provide the first purely discrete proof of monotonicity for normalized coordinates in both the Black–Scholes and Bachelier frameworks, without requiring continuous quotes or differentiability. The main contributions are twofold: (1) a rigorous discrete verification of monotonicity in these two canonical implied volatility models, and (2) the introduction of a normal variance identity that serves as the natural counterpart to Fukasawa’s log-normal result, thereby establishing a model-independent theoretical foundation for volatility derivatives.

This study addresses the challenge of enforcing no-arbitrage constraints in modeling option-implied information—specifically, implied densities and implied volatility—by proposing a shallow neural network approach. The method reinterprets implied volatility as a differentiable, pointwise correction mapping from the Black–Scholes pseudo-density to the risk-neutral density and jointly models both quantities by embedding differentiable no-arbitrage constraints directly into the architecture. The work introduces the first shallow neural representation framework that intrinsically incorporates no-arbitrage conditions, demonstrating that deep networks are unnecessary for this task. Empirical results show that a single-hidden-layer network efficiently and accurately approximates both implied densities and implied volatility, outperforming deeper or wider architectures in terms of both accuracy and computational efficiency.

This study addresses the challenge of constructing option price surfaces that are simultaneously smooth and strictly arbitrage-free across both time-to-maturity and strike dimensions. To this end, the authors propose an efficient and flexible nonparametric method that directly calibrates to market quotes via linear programming, ensuring smoothness and absence of arbitrage under only simple positivity constraints, while naturally accommodating bid–ask spread bounds. The key innovation lies in the introduction of an equivalent parametrization in terms of positive “discrete local volatility,” which substantially simplifies the constraint structure. Compared to existing approaches that fit implied volatility surfaces, the proposed method significantly reduces computational cost and demonstrates strong empirical performance and practicality when applied to S&P 500 index option data.

This study addresses the challenge of robustly extracting risk-neutral densities from near-expiry options, which is hindered by low premiums, wide bid–ask spreads, and asynchronous quotes that undermine conventional approaches. To overcome these issues, the authors propose a model-free two-stage framework: first, the ARIES strategy eliminates static arbitrage opportunities under market depth constraints; second, the SEDEx method recovers the density by incorporating smoothness and maximum entropy principles while respecting bid–ask bounds. Notably, this approach explicitly treats the bid–ask spread as a fundamental market constraint and uniquely integrates executable arbitrage filtering with entropy regularization, substantially enhancing stability in processing short-dated option data. Empirical validation on both synthetic Heston-generated data and real-world SPX options demonstrates the method’s ability to reconstruct stable and accurate implied volatility smiles.

This paper addresses the challenge of rigorously modeling non-smooth price paths under the no-arbitrage paradigm in financial derivative pricing. Methodologically, it extends the universal approximation theorem to the tensor algebra space of non-geometric rough paths for the first time, constructing a polynomial-based functional approximation framework for rough paths and introducing a novel analyticity assumption on signature payoff functions—thereby unifying signature methods with Itô integration theory. The contributions are threefold: (1) establishing the first universal approximation result for non-geometric rough paths; (2) providing a rigorous mathematical foundation for signature-driven derivative pricing; and (3) substantially improving modeling accuracy and theoretical consistency for path-dependent options and other complex instruments, thereby bridging a critical gap between mathematical finance and stochastic analysis.

This study addresses the challenge of simultaneously achieving precise local shape control and strict static no-arbitrage compliance in implied volatility curve modeling. To this end, it proposes a parsimonious and interpretable parametric approach grounded in the risk-neutral distribution. By introducing parameters that exhibit stable cross-maturity patterns, the method directly governs local curvature characteristics—such as convexity and concavity—while inherently satisfying no-arbitrage constraints. The resulting model flexibly accommodates diverse curvature patterns and supports both term structure interpolation and dynamic modeling. Empirical validation on a two-year dataset of S&P 500 options, encompassing over 250,000 calibrated volatility curves, demonstrates the stability, generalizability, and high fidelity of the proposed parameterization in capturing complex market dynamics.

This study investigates whether existing consistency conditions for multi-maturity option prices under bounded bid–ask spreads are sufficient to preclude model-free arbitrage. By refining the definition of admissible quotes and integrating the completeness condition for single-maturity options, the authors construct the first minimal two-maturity counterexample that satisfies all known consistency conditions yet admits a pathwise arbitrage, thereby disproving the sufficiency conjecture of Gerhold and Gülüm. Furthermore, employing techniques such as convex envelopes, concave maximization, and extremal envelopes within the geometry of bounded spreads, they explicitly derive closed-form two-maturity basket operators, precisely characterize the model-free arbitrage cone, and uncover a high-dimensional open region of admissible quotes generated by the counterexample, fully describing the structure of two-maturity executable, arbitrage-free markets.

Existing methods struggle to construct risk-neutral marginal distributions from arbitrage-free option prices that simultaneously satisfy no butterfly arbitrage, no calendar spread arbitrage, exact market price recovery, efficient sampling, and full support. This work proposes an explicit construction method that exactly fits observed option prices within the range of quoted strikes via piecewise probability mass allocation, while extrapolating beyond this range using closed-form power-law tails that satisfy necessary boundary conditions. The approach uniquely achieves, within a unified framework, strict absence of static arbitrage, exact calibration to market prices, analytical expressions for both density and quantile functions, and efficient Monte Carlo sampling. Experiments on synthetic SSVI surfaces and S&P 500 market data demonstrate its computational efficiency, robustness, and practical utility, effectively bridging the gap between option pricing models and downstream applications.

This work addresses the high computational complexity associated with pricing multi-strike options and computing their Greeks under stochastic volatility Bachelier-type models. We propose an efficient numerical method grounded in elementary linear algebra that overcomes the limitations of traditional pointwise evaluation. By performing only a finite number of expectation calculations, the method simultaneously yields option prices and Greeks for infinitely many strikes within a well-defined convergence interval. Specifically, we explicitly characterize this convergence interval for the SABR model and demonstrate the method’s efficacy through numerical experiments on both the SABR and rough Bergomi models. Results show that the approach achieves significant gains in computational efficiency while preserving high accuracy, thereby enabling scalable batch computation across a continuum of strikes.

This study addresses the computational efficiency and numerical stability challenges in solving for Black-Scholes implied volatility by proposing a highly efficient algorithm that guarantees monotonic convergence and production-grade robustness. Building upon Jäckel’s out-of-the-money normalization and tail-stabilized log-price formulation, the method employs the Choi–Huh–Su L3 lower bound as an initial seed, enhanced with a cubic Euler–Chebyshev iteration scheme. It further integrates floating-point engineering techniques, including Bachelier limit handling, saturated price correction, and Jäckel–Newton refinement. For the first time, monotonic convergence theory is rigorously combined with practical numerical strategies, achieving accelerated convergence without overshooting. The proposed ThiopheneIV+ variant further reduces reference price bias in high-precision regimes. Experiments demonstrate that the algorithm outperforms Jäckel’s Let’s Be Rational Java implementation in speed while maintaining comparable accuracy on regular grids.