Pricing of vanilla and first generation exotic options in the local stochastic volatility framework: survey and new results

Alexander Lipton; Andrey Gal; Andris Lasis

Home/Archive/arXiv

arXiv · 1312.5693

Pricing of vanilla and first generation exotic options in the local stochastic volatility framework: survey and new results

Alexander Lipton·Andrey Gal·Andris Lasis

View original source ↗Read source document ↗Export metadata ↓

Abstract

Stochastic volatility (SV) and local stochastic volatility (LSV) processes can be used to model the evolution of various financial variables such as FX rates, stock prices, and so on. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes. Many issues remain, though, including the efficacy of the standard alternating direction implicit (ADI) numerical methods for solving SV and LSV pricing problems. In general, the amount of required computations for these methods is very substantial. In this paper we address some of these issues and propose a viable alternative to the standard ADI methods based on Galerkin-Ritz ideas. We also discuss various approaches to solving the corresponding pricing problems in a semi-analytical fashion. We use the fact that in the zero correlation case some of the pricing problems can be solved analytically, and develop a closed-form series expansion in powers of correlation. We perform a thorough benchmarking of various numerical solutions by using analytical and semi-analytical solutions derived in the paper.

Explore related subjects

q-fin.CP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Alexander Lipton, Andrey Gal, Andris Lasis. 2013-12-19. Pricing of vanilla and first generation exotic options in the local stochastic volatility framework: survey and new results. https://arxiv.org/abs/1312.5693

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Risk-Neutral Crash Frontier: Sharp Joint Bounds on Crash Probability and Conditional Depth from Option Bid-Ask Quotes

Index put prices are the market's quotes for crash insurance, and a put's value equals the probability of a crash times the expected shortfall given one. The market therefore prices the product of likelihood and depth, not the factors, and finitely many bid and ask quotes leave a range of ways to split it. Fitting one density hides that range, and bounds computed one factor at a time can combine into scenarios that no single risk-neutral distribution could produce. We characterize the set of probability and loss pairs that one distribution can generate while pricing every quote inside its spread, with depth as their ratio, and call its boundary the risk-neutral crash frontier. Partitioning the state space at the quoted strikes and the threshold, with one coordinate for mass at the threshold, makes the set the exact projection of a finite linear system, with no price grid. Linear programs trace the frontier and price any portfolio of digital and put payoffs sharply, each bound certified by a static super-replicating portfolio of cash, forward, and quoted options. In weekly SPX cross sections from 2013 to 2023, a median 37 percent of the scenarios that separate bounds admit are jointly infeasible. Extending the quote set from the eight strikes nearest the threshold to the complete put wing shrinks it by a further 5.4 to 18.2 percent.

q-fin.CP↗

arXiv2026

Simulation of stochastic volatility models via operator splitting schemes

The standard Euler discretization schemes for numerical option pricing under stochastic volatility models are known to exhibit high biases and potential unreliability. The alternative use of the exact (unbiased) simulation approach invariably involves numerical evaluation of integrated variance (and / or volatility) conditional on terminal variance (volatility) value. To resolve the technical challenge, most simulation schemes either employ the tedious Fourier inversion of conditional characteristic function or numerical approximation by moment matched distribution. We propose a general framework of constructing efficient and reliable simulation schemes for stochastic volatility models via the Strang operator splitting approximation. The simulation procedure completely circumvents the necessity of evaluation of conditional integrated variance (and / or) volatility. Our simulation schemes compete favorably well with most existing exact simulation schemes and the biased Euler schemes in terms of accuracy, efficiency, reliability and ease of implementation. Extensive numerical tests were conducted to illustrate the versatility and success of our operator splitting approach for most common stochastic volatility models, such as the Heston-type models, lifted Heston model, Hull-White model, and Barndorff-Nielsen and Shephard model. We also establish the proof of second-order convergence of the operator splitting schemes.

q-fin.CP↗

arXiv2026

Efficient simulation schemes for pricing options under the Ornstein--Uhlenbeck driven stochastic volatility model

We develop an efficient Monte Carlo simulation scheme for pricing options under the Ornstein-Uhlenbeck driven stochastic volatility model via the operator splitting approach. With an ingenious splitting of the governing stochastic differential equations, our operator splitting scheme admits analytic solutions in all sub-steps, so its implementation is simplified to require simulation of a few normal variates. This resolves the two typical numerical challenges in other simulation schemes, namely, sampling of conditional integrated variance and pathwise inverse integral transform of characteristic functions. There are three pioneering simulation schemes that attempt to overcome the above two numerical challenges. These include the Hilbert interpolation scheme of Zeng et al. (2023), Karhunen-Lo`eve expansion scheme of Choi (2025) and moment matching scheme based on the Inverse Gaussian distribution of Brignone and Sgarra (2026). We performed numerical tests to compare accuracy-speed performance of pricing options using our operator splitting scheme with these three pioneering schemes. We found that our scheme competes favorably well in terms of accuracy-speed tradeoff among all these schemes, in particular for pricing path dependent options with a large number of monitoring instants. The performance of our scheme can be well enhanced by martingale-preserving control variates and variance reduction via conditioning.

q-fin.CP↗