Silvia Lavagnini

Mathematics of Operations Research Doi: https://doi.org/10.1287/moor.2023.0213

We propose a novel computational procedure for quadratic hedging in high-dimensional incomplete markets, covering mean-variance hedging and local risk minimization. Starting from the observation that both quadratic approaches can be treated from the point of view of backward stochastic differential equations (BSDEs), we (recursively) apply a deep learning-based BSDE solver to compute the entire optimal hedging strategies paths. This allows us to overcome the curse of dimensionality, extending the scope of applicability of quadratic hedging in high dimension. We test our approach with a classic Heston model and with a multiasset and multifactor generalization thereof, showing that this leads to high levels of accuracy.

International Journal of Theoretical and Applied Finance 24(8) Doi: https://doi.org/10.1142/S0219024921500412

We derive a series expansion by Hermite polynomials for the price of an arithmetic Asian option. This requires the computation of moments and correlators of the underlying asset price which for a polynomial jump–diffusion process are given analytically; hence, no numerical simulation is required to evaluate the series. This allows to derive analytical expressions for the option Greeks. The weight function defining the Hermite polynomials is a Gaussian density with scale [Formula: see text]. We find that the rate of convergence of the series depends on [Formula: see text], for which we prove a lower bound to guarantee convergence. Numerical examples show that the series expansion is accurate but unstable for initial values of the underlying process far from zero, mainly due to rounding errors.

SIAM Journal on Financial Mathematics 12(4) p. 1374-1415 Doi: https://doi.org/10.1137/21M141556X

In the setting of one-dimensional polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for polynomial processes. The developed framework can, for example, be applied in financial pricing, such as for path-dependent options and in a stochastic volatility models context. In applications to options, having closed and compact formulations is attractive for sensitivity analysis and risk management, since Greeks can be derived explicitly.

Digital Finance 3 p. 209-248 Doi: https://doi.org/10.1007/s42521-021-00030-w

We price European-style options written on forward contracts in a commodity market, which we model with an infinite-dimensional Heath–Jarrow–Morton (HJM) approach. For this purpose, we introduce a new class of state-dependent volatility operators that map the square integrable noise into the Filipović space of forward curves. For calibration, we specify a fully parametrized version of our model and train a neural network to approximate the true option price as a function of the model parameters. This neural network can then be used to calibrate the HJM parameters based on observed option prices. We conduct a numerical case study based on artificially generated option prices in a deterministic volatility setting. In this setting, we derive closed pricing formulas, allowing us to benchmark the neural network based calibration approach. We also study calibration in illiquid markets with a large bid-ask spread. The experiments reveal a high degree of accuracy in recovering the prices after calibration, even if the original meaning of the model parameters is partly lost in the approximation step.

Frontiers in Applied Mathematics and Statistics 6 Doi: https://doi.org/10.3389/fams.2020.00037

CARMA(p, q) processes are compactly defined through a stochastic differential equation (SDE) involving q + 1 derivatives of the Lévy process driving the noise, despite this latter having in general no differentiable paths. We replace the Lévy noise with a continuously differentiable process obtained via stochastic convolution. The solution of the new SDE is then a stochastic process that converges to the original CARMA in an L2 sense. CARMA processes are largely applied in finance, and this article mathematically justifies their representation via SDE. Consequently, the use of numerical schemes in simulation and estimation by approximating derivatives with discrete increments is also justified for every time step. This provides a link between discrete-time ARMA models and CARMA models in continuous-time. We then analyze an Euler method and find a rate of convergence proportional to the square of the time step for discretization and to the inverse of the convolution parameter. These must then be adjusted to get accurate approximations.