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).

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)

The 7th Berlin Workshop on Mathematical Finance for Young Researchers provides a forum for PhD students, postdoctoral researchers, and young faculty members from all over the world to discuss their research in an informal atmosphere. Keynote lectures will be given by

• Sara Biagini (Rome)
• Luciano Campi (Milano)
• Giorgio Ferrari (Bielefeld)
• Mete H. Soner (Princeton)
• Luitgard Veraart (London)

We also invite up to 20 contributed talks from young researchers. The deadline for abstract submission is May 20. Notification of acceptance will be sent by May 31. Accommodation for speakers will be arranged. 
Limited support for travel expenses may be available upon request. Here a link to the workshop webpage.