Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models

summary

The gist

Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models show that Fourier inversion techniques can be effectively extended to a broad class of flexible models previously

In short

The research extends Fourier inversion to complex polynomial Ornstein-Uhlenbeck volatility models, including quintic and one-factor Bergomi types, which are more flexible than standard affine models. It connects the joint Fourier-Laplace functional of log-price and variance to solving an infinite-dimensional system of Riccati equations. This allows for efficient pricing of options and volatility swaps using numerical schemes based on these transforms.

Key concepts

Polynomial Ornstein-Uhlenbeck (OU) Volatility Models: These are stochastic models where the volatility of a price follows a power series (polynomial) of an Ornstein-Uhlenbeck process. The OU process itself is a common model for mean reversion, meaning it tends to pull the volatility back towards a long-term average value over time. These models are more flexible than simpler affine models.
Fourier-Laplace Transform: These mathematical tools convert complex stochastic processes (like log-prices) into the frequency domain using Fourier transforms, and then further transform them using Laplace transforms. This transformation is crucial because it simplifies the analysis of pricing problems by turning complex differential equations into algebraic ones, making solutions easier to find.
Infinite-Dimensional System of Riccati Equations: This is a set of coupled ordinary differential equations that exist in an infinite number of dimensions. The paper shows that the joint Fourier-Laplace functional describing the log-price and integrated variance can be solved by finding the solution to this system. Solving these equations provides a direct mathematical link between market observables and the model's structure.
Fourier Inversion for Pricing: This technique uses the Fourier transform results to evaluate derivative prices, such as European call options. Instead of complex integration over price paths, this method involves applying an inverse Fourier transform to a specific integral formula. This provides a computationally efficient way to determine option prices within the context of these flexible volatility models.

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This episode discusses

The paper

Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models · Read on arXiv

Eduardo Abi Jaber, Shaun (Xiaoyuan) Li, Xuyang Lin

Ecole Polytechnique · AXA Investment Managers · Université Paris 1 Panthéon-Sorbonne, CES

We consider the Fourier-Laplace transforms of a broad class of polynomial Ornstein-Uhlenbeck (OU) volatility models, including the well-known Stein-Stein, Schöbel-Zhu, one-factor Bergomi, and the recently introduced quintic OU models motivated by the SPX-VIX joint calibration problem. We show the connection between the joint Fourier-Laplace functional of the log-price and the integrated variance, and the solution of an infinite-dimensional system of Riccati equations. Next, under some non-vanishing conditions on the Fourier-Laplace transforms, we establish an existence result for such Riccati equations and show how the Fourier-Laplace functional can be seen as the limit of a sequence of entire functions. On the practical side, we develop a numerical scheme to solve the stiff infinite-dimensional system of Riccati equations and demonstrate the efficiency and accuracy of the scheme for pricing SPX options and volatility swaps using Fourier and Laplace inversions, with specific examples of the quintic OU and the one-factor Bergomi models and their calibration to real market data.

Transcript

Introduction to the show: ident: Quantitative Finance. The day in quantitative finance: models of markets, risk and prices.

Ruth: We're looking at the paper: "Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models".

Warren: Here's the gist of the paper. Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models show that Fourier inversion techniques can be effectively extended to a broad class of flexible models previously considered infeasible as they fall beyond the conventional class of affine diffusions.

Ruth: The paper establishes a connection between the joint Fourier-Laplace functional of the log-price and the integrated variance, and this connection is solved by an infinite dimensional Riccati equation.

Warren: Model Class and Theoretical Framework The research considers a class of polynomial Ornstein-Uhlenbeck (OU) models where the volatility of the log-price is defined as a power series of an OU process.

Ruth: The authors focus on verifying that Fourier techniques can be extended to models like the one-factor Bergomi model and the recently introduced Quintic OU model, which fit SPX-VIX joint calibration problems.

Paper summary: Ruth: Looking at the "Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models" paper, the main point is that they've successfully shown how Fourier inversion techniques can be applied to a broad class of flexible models previously thought too complicated for them.

Warren: They focus on these polynomial OU volatility models, like the one-factor Bergomi and the Quintic OU model, which fit those tough SPX and VIX joint calibration problems.

Iris: The core contribution is showing that the joint Fourier-Laplace functional of the log-price is solved by an infinite dimensional system of Riccati equations.

Ruth: They then prove that a solution to this Riccati system exists under certain conditions, which is important because it gives us a way to find the pricing structure.

Warren: Practically, they developed a numerical scheme—combining variation of constants and implicit Euler—to solve those stiff infinite dimensional Riccati equations efficiently for pricing options and volatility swaps.

Iris: Their work on real market data calibration for the Quintic OU model shows that their numerical method is stable across various realistic parameters values.

Ruth: So what this means in plain terms is that we have a robust framework for applying Fourier pricing to these flexible stochastic volatility models, allowing us to price derivatives and even q-volatility swaps.

Warren: It opens the door for real-time evaluation of thousands of derivatives because you can calibrate the SPX and VIX smiles together using this method.

Conclusion: Ruth: So we've looked at how these polynomial OU models, like that one for SPX volatility, can actually be priced using Fourier and Laplace transforms, and now we're wrapping up what this paper is all about.

Warren: It boils down to taking a bunch of these complex models—the Stein-Stein ones, the quintic OU—and showing that even though they are super flexible, you can handle them with a specific mathematical tool.

Iris: The authors prove that the joint functional of the log-price and its integrated variance isn't just some abstract thing; it actually solves an infinite dimensional system of Riccati equations.

Ruth: Exactly. That connection is the bridge between what we observe in markets—the price movements—and this very specific set of differential equations that govern how things evolve over time.

Warren: It also gives us a way to check if a solution even exists for these equations, which is crucial because if it doesn't exist, then pricing with this method is impossible.

Iris: They establish the existence result under some non-vanishing conditions on those transforms, meaning they found the right mathematical path to follow.

Ruth: And numerically, they built a way to actually solve that infinite dimensional system using variation of constants and implicit Euler schemes for practical application.

Warren: So, for someone just listening to this show, what does this mean? It means we can use these advanced techniques to price real options like SPX calls and volatility swaps much more accurately than before.

Iris: The calibration aspect is also strong; they successfully used these Fourier methods to fit the Quintic OU model directly onto real market data.

Ruth: It shows that this framework isn't just theoretical math; it works when you plug in parameters from the actual market, which makes it much more useful for understanding how markets actually behave.

Warren: It opens up a whole new avenue for pricing complex derivatives where those older, simpler models just couldn't cope with the flexibility of real-world volatility.

Iris: Next time we talk about this, we'll look at what happens when you change the assumptions in those initial Riccati equations.

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