Monte Carlo

A High-Level Framework for Practically Model-Independent Pricing ArXiv ID: 2512.15718 “View on arXiv” Authors: Marco Airoldi Abstract We present a high-level framework that explains why, in practice, different pricing models calibrated to the same vanilla surface tend to produce similar valuations for exotic derivatives. Our approach acts as an overlay on the Monte Carlo infrastructure already used in banks, combining path reweighting with a conic optimisation layer without requiring any changes to existing code. This construction delivers narrow, practically model-independent price bands for exotics, reconciling front-office practice with the robust, model-independent ideas developed in the academic literature. ...

Efficient Importance Sampling under Heston Model: Short Maturity and Deep Out-of-the-Money Options ArXiv ID: 2511.19826 “View on arXiv” Authors: Yun-Feng Tu, Chuan-Hsiang Han Abstract This paper investigates asymptotically optimal importance sampling (IS) schemes for pricing European call options under the Heston stochastic volatility model. We focus on two distinct rare-event regimes where standard Monte Carlo methods suffer from significant variance deterioration: the limit as maturity approaches zero and the limit as the strike price tends to infinity. Leveraging the large deviation principle (LDP), we design a state-dependent change of measure derived from the asymptotic behavior of the log-price cumulant generating functions. In the short-maturity regime, we rigorously prove that our proposed IS drift, inspired by the variational characterization of the rate function, achieves logarithmic efficiency (asymptotic optimality) by minimizing the decay rate of the second moment of the estimator. In the deep OTM regime, we introduce a novel slow mean-reversion scaling for the variance process, where the mean-reversion speed scales as the inverse square of the small-noise parameter (defined as the reciprocal of the log-moneyness). We establish that under this specific scaling, the variance process contributes non-trivially to the large deviation rate function, requiring a specialized Riccati analysis to verify optimality. Numerical experiments demonstrate that the proposed method yields substantial variance reduction–characterized by factors exceeding several orders of magnitude–compared to standard estimators in both asymptotic regimes. ...

Function approximations for counterparty credit exposure calculations ArXiv ID: 2507.09004 “View on arXiv” Authors: Domagoj Demeterfi, Kathrin Glau, Linus Wunderlich Abstract The challenge to measure exposures regularly forces financial institutions into a choice between an overwhelming computational burden or oversimplification of risk. To resolve this unsettling dilemma, we systematically investigate replacing frequently called derivative pricers by function approximations covering all practically relevant exposure measures, including quantiles. We prove error bounds for exposure measures in terms of the $L^p$ norm, $1 \leq p < \infty$, and for the uniform norm. To fully exploit these results, we employ the Chebyshev interpolation and show exponential convergence of the resulting exposure calculations. As our main result we derive probabilistic and finite sample error bounds under mild conditions including the natural case of unbounded risk factors. We derive an asymptotic efficiency gain scaling with $n^{“1/2-\varepsilon”}$ for any $\varepsilon>0$ with $n$ the number of simulations. Our numerical experiments cover callable, barrier, stochastic volatility and jump features. Using 10,000 simulations, we consistently observe significant run-time reductions in all cases with speed-up factors up to 230, and an average speed-up of 87. We also present an adaptive choice of the interpolation degree. Finally, numerical examples relying on the approximation of Greeks highlight the merit of the method beyond the presented theory. ...

Tensor train representations of Greeks for Fourier-based pricing of multi-asset options ArXiv ID: 2507.08482 “View on arXiv” Authors: Rihito Sakurai, Koichi Miyamoto, Tsuyoshi Okubo Abstract Efficient computation of Greeks for multi-asset options remains a key challenge in quantitative finance. While Monte Carlo (MC) simulation is widely used, it suffers from the large sample complexity for high accuracy. We propose a framework to compute Greeks in a single evaluation of a tensor train (TT), which is obtained by compressing the Fourier transform (FT)-based pricing function via TT learning using tensor cross interpolation. Based on this TT representation, we introduce two approaches to compute Greeks: a numerical differentiation (ND) approach that applies a numerical differential operator to one tensor core and an analytical (AN) approach that constructs the TT of closed-form differentiation expressions of FT-based pricing. Numerical experiments on a five-asset min-call option in the Black-Sholes model show significant speed-ups of up to about $10^{“5”} \times$ over MC while maintaining comparable accuracy. The ND approach matches or exceeds the accuracy of the AN approach and requires lower computational complexity for constructing the TT representation, making it the preferred choice. ...

Adaptive Multilevel Stochastic Approximation of the Value-at-Risk ArXiv ID: 2408.06531 “View on arXiv” Authors: Unknown Abstract Crépey, Frikha, and Louzi (2023) introduced a multilevel stochastic approximation scheme to compute the value-at-risk of a financial loss that is only simulatable by Monte Carlo. The optimal complexity of the scheme is in $O({"\varepsilon"}^{"-5/2"})$, ${"\varepsilon"} > 0$ being a prescribed accuracy, which is suboptimal when compared to the canonical multilevel Monte Carlo performance. This suboptimality stems from the discontinuity of the Heaviside function involved in the biased stochastic gradient that is recursively evaluated to derive the value-at-risk. To mitigate this issue, this paper proposes and analyzes a multilevel stochastic approximation algorithm that adaptively selects the number of inner samples at each level, and proves that its optimal complexity is in $O({"\varepsilon"}^{"-2"}|\ln {"\varepsilon"}|^{“5/2”})$. Our theoretical analysis is exemplified through numerical experiments. ...