Quantitative Finance

We present a neural network based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models -including the rough volatility family- and a range of derivative contracts. The aim of neural networks in this work is an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. We highlight how this perspective opens new horizons for quantitative modelling: The calibration bottleneck posed by a slow pricing of derivative contracts is lifted. This brings several numerical pricers and model families (such as rough volatility models) within the scope of applicability in industry practice. The form in which information from available data is extracted and stored influences network performance: This approach is inspired by representing the implied volatility and option prices as a collection of pixels. In a number of applications we demonstrate the prowess of this modelling approach regarding accuracy, speed, robustness and generality and also its potentials towards model recognition.

With the recent advancement in Deep Reinforcement Learning in the gaming industry, we are curious if the same technology would work as well for common quantitative financial problems. In this paper, we will investigate if an off-the-shelf library developed by OpenAI can be easily adapted to mean reversion strategy. Moreover, we will design and test to see if we can get better performance by narrowing the function space that the agent needs to search for. We achieve this through augmenting the reward function by a carefully picked penalty term.

In this paper, we present a novel computational framework for portfolio-wide risk management problems, where the presence of a potentially large number of risk factors makes traditional numerical techniques ineffective. The new method utilises a coupled system of BSDEs for the valuation adjustments (xVA) and solves these by a recursive application of a neural network based BSDE solver. This not only makes the computation of xVA for high-dimensional problems feasible, but also produces hedge ratios and dynamic risk measures for xVA, and allows simulations of the collateral account.

As is known, an option price is a solution to a certain partial differential equation (PDE) with terminal conditions (payoff functions). There is a close association between the solution of PDE and the solution of a backward stochastic differential equation (BSDE). We can either solve the PDE to obtain option prices or solve its associated BSDE. Recently a deep learning technique has been applied to solve option prices using the BSDE approach. In this approach, deep learning is used to learn some deterministic functions, which are used in solving the BSDE with terminal conditions. In this paper, we extend the deep-learning technique to solve a PDE with both terminal and boundary conditions. In particular, we will employ the technique to solve barrier options using Brownian motion bridges.

We consider the problem of determining the Lévy exponent in a Lévy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure P , consists of a pricing kernel { π t } t≥0 together with one or more non-dividend-paying risky assets driven by the same Lévy process. If { S t } t≥0 denotes the price process of such an asset then { π t S t } t≥0 is a P -martingale. The Lévy process { ξ t } t≥0 is assumed to have exponential moments, implying the existence of a Lévy exponent ψ(α)= t −1 logE( e α ξ t ) for α in an interval A⊂R containing the origin as a proper subset. We show that if the initial prices of power-payoff derivatives, for which the payoff is H T =( ζ T ) q for some time T>0 , are given for a range of values of q , where { ζ t } t≥0 is the so-called benchmark portfolio defined by ζ t =1/ π t , then the Lévy exponent is determined up to an irrelevant linear term. In such a setting, derivative prices embody complete information about price jumps: in particular, the spectrum of the price jumps can be worked out from current market prices of derivatives. More generally, if H T =( S T ) q for a general non-dividend-paying risky asset driven by a Lévy process, and if we know that the pricing kernel is driven by the same Lévy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the Lévy exponent up to a transformation ψ(α)→ψ(α+μ)−ψ(μ)+cα , where c and μ are constants.

We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the process both in moderate interest rate environments and in near-zero interest rate regimes such as the currently prevalent ones. The parabolic Kolmogorov pricing equations for the SABR model are degenerate at the origin, yielding non-standard partial differential equations, for which conventional pricing methods ---designed for non-degenerate parabolic equations--- potentially break down. We derive here the appropriate analytic setup to handle the degeneracy of the model at the origin. That is, we construct an evolution triple of suitably chosen Sobolev spaces with singular weights, consisting of the domain of the SABR-Dirichlet form, its dual space, and the pivotal Hilbert space. In particular, we show well-posedness of the variational formulation of the SABR-pricing equations for vanilla and barrier options on this triple. Furthermore, we present a finite element discretization scheme based on a (weighted) multiresolution wavelet approximation in space and a θ -scheme in time and provide an error analysis for this discretization.

In this short note we develop a model for discounting. A focus of the model is the discounting, when discount factors cannot be derived from market products. That is, a risk-neutralizing trading strategy cannot be performed. This is the case, when one is in need of a risk-free (default-free) discounting, but default protection on funding providers is not traded. For this case, we introduce a default compensation factor ( exp(+ λ ~ T) ) that describes the present value of a strategy to compensate for default (like buying default protection would do). In a second part, we introduce a model, where the survival probability depends on the required notional. This model is different from the classical modelling of a time-dependent survival probability ( exp(−λT) ). The model especially allows that large liquidity requirements are instantly more likely do default than small ones. Combined the two approaches build a framework in which discounting (valuation) is non-linear. The non-linear discounting presented here has several effects, which are relevant in various applications: * If we consider the question of default-free valuation, i.e., factoring in the cost of default protection, the framework can will lead to over-proportional higher values (or cost) for large projects (or damages). The framework can lead to the effect that discount-factors for very large liquidity requirements or projects are an increasing function of time. It may even lead to discount factors larger than one. This may have relevance in the assessment of event like climate change. * For the valuation of defaultable products, e.g., like a defaultable swap, the framework leads to the generation of a continuum of (defaultable) par rate curves (interest rate curve) and the valuation of a payer and a receiver swap differs by more than just a sign.

In this paper, we attempt to introduce the Bellman principle for a discrete time multi-period mean-variance model. Based on this new take on the Bellman principle, we obtain a dynamic time-consistent optimal strategy and related efficient frontier. Furthermore, we develop a varying investment period discrete time multi-period mean-variance model and obtain a related dynamic optimal strategy and an optimal investment period. This paper compares the highlighted dynamic optimal strategies of this study with the 1/n equality strategy, and shows that we can secure a higher return with a smaller risk based on the dynamic optimal strategies.

This paper expands the work on distributionally robust newsvendor to incorporate moment constraints. The use of Wasserstein distance as the ambiguity measure is preserved. The infinite dimensional primal problem is formulated; problem of moments duality is invoked to derive the simpler finite dimensional dual problem. An important research question is: How does distributional ambiguity affect the optimal order quantity and the corresponding profits/costs? To investigate this, some theory is developed and a case study in auto sales is performed. We conclude with some comments on directions for further research.

This paper investigates calculations of robust funding valuation adjustment (FVA) for over the counter (OTC) derivatives under distributional uncertainty using Wasserstein distance as the ambiguity measure. Wrong way funding risk can be characterized via the robust FVA formulation. The simpler dual formulation of the robust FVA optimization is derived. Next, some computational experiments are conducted to measure the additional FVA charge due to distributional uncertainty under a variety of portfolio and market configurations. Finally some suggestions for future work, such as robust capital valuation adjustment (KVA) and margin valuation adjustment (MVA), are discussed.