Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

18.04.2024

16. Uhr c.t.

Alexandre Pannier (Université Paris Cité)

A path-dependent PDE solver based on signature kernels

Abstract: We develop a provably convergent kernel-based solver for path-dependent PDEs (PPDEs). Our numerical scheme leverages signature kernels, a recently introduced class of kernels on path-space. Specifically, we solve an optimal recovery problem by approximating the solution of a PPDE with an element of minimal norm in the signature reproducing kernel Hilbert space (RKHS) constrained to satisfy the PPDE at a finite collection of collocation paths. In the linear case, we show that the optimisation has a unique closed-form solution expressed in terms of signature kernel evaluations at the collocation paths. We prove consistency of the proposed scheme, guaranteeing convergence to the PPDE solution as the number of collocation points increases. Finally, several numerical examples are presented, in particular in the context of option pricing under rough volatility. Our numerical scheme constitutes a valid alternative to the ubiquitous Monte Carlo methods. Joint work with Cristopher Salvi (Imperial College London).

18.04.2024

17 Uhr c.t.

Nils Detering (Universität Düsseldorf)

Local Volatility Models for Commodity Forwards

Abstract: 

We present a dynamic model for forward curves in commodity markets, which is defined as the solution to a stochastic partial differential equation (SPDE) with state-dependent coefficients, taking values in a Hilbert space H of real valued functions. The model can be seen as an infinite dimensional counterpart of the classical local volatility model frequently used in equity markets. We first investigate a class of point-wise operators on H, which we then use to define the coefficients of the SPDE. Next, we derive growth and Lipchitz conditions for coefficients resulting from this class of operators to establish existence and uniqueness of solutions.  We also derive conditions that ensure positivity of the entire forward curve. Finally, we study the existence of an equivalent measure under which related traded, 1-dimensional projections of the forward curve are martingales.

Our approach encompasses a wide range of specifications, including a Hilbert-space valued counterpart of a constant elasticity of variance (CEV) model, an exponential model, and a spline specification which can resemble the S shaped local volatility function that well reproduces the volatility smile in equity markets. A particularly pleasant property of our model class is that the one-dimensional projections of the curve can be expressed as one dimensional stochastic differential equation. This provides a link to models for forwards with a fixed delivery time for which formulas and numerical techniques exist. In a first numerical case study we observe that a spline based, S shaped local volatility function can calibrate the volatility surface in electricity markets. 

Joint work with Silvia Lavagnini (BI Norwegian Business School)

02.05.2024

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02.05.2024

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16.05.2024

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16.05.2024

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30.05.2024

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30.05.2024

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13.06.2024

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27.06.2024

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27.06.2024

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11.07.2024

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11.07.2024

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