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- ItemThe invariant distribution of wealth and employment status in a small open economy with precautionary savings(Amsterdam : North-Holland, 2019) Bayer, Christian; Rendall, Alan D.; Wälde, KlausWe study optimal savings in continuous time with exogenous transitions between employment and unemployment as the only source of uncertainty in a small open economy. We prove the existence of an optimal consumption path. We exploit that the dynamics of consumption and wealth between jumps can be expressed as a Fuchsian system. We derive conditions under which an invariant joint distribution for the state variables, i.e., wealth and labour market status, exists and is unique. We also provide conditions under which the distribution of these variables converges to the invariant distribution. Our analysis relies on the notion of T-processes and applies results on the stability of Markovian processes from Meyn and Tweedie (1993a, b,c). © 2019 The Author(s)
- ItemA regularity structure for rough volatility(Oxford [u.a.] : Wiley-Blackwell, 2019) Bayer, Christian; Friz, Peter K.; Gassiat, Paul; Martin, Jorg; Stemper, BenjaminA new paradigm has emerged recently in financial modeling: rough (stochastic) volatility. First observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, rough volatility captures parsimoniously key-stylized facts of the entire implied volatility surface, including extreme skews (as observed earlier by Alòs et al.) that were thought to be outside the scope of stochastic volatility models. On the mathematical side, Markovianity and, partially, semimartingality are lost. In this paper, we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provide a new and powerful tool to analyze rough volatility models.
- ItemRKHS regularization of singular local stochastic volatility McKean--Vlasov models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Bayer, Christian; Belomestny, Denis; Butkovsky, Oleg; Schoenmakers, John G. M.Motivated by the challenges related to the calibration of financial models, we consider the problem of solving numerically a singular McKean-Vlasov equation, which represents a singular local stochastic volatility model. Whilst such models are quite popular among practitioners, unfortunately, its well-posedness has not been fully understood yet and, in general, is possibly not guaranteed at all. We develop a novel regularization approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularized model is well-posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models. Our results are also applicable to more general McKean--Vlasov equations.
- ItemHierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Bayer, Christian; Hammouda, Chiheb Ben; Tempone, Raúl F.The rough Bergomi (rBergomi) model, introduced recently in [4], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet exhibits remarkable fit to empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a timeconsuming task. To overcome this issue, we design a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.
- ItemLow-dimensional approximations of high-dimensional asset price models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Redmann, Martin; Bayer, Christian; Goyal, PawanWe consider high-dimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction (MOR) to obtain a reduced system. MOR has been previously studied for asymptotically stable controlled stochastic systems with zero initial conditions. However, stochastic differential equations modeling price processes are uncontrolled, have non-zero initial states and are often unstable. Therefore, we extend MOR schemes and combine ideas of techniques known for deterministic systems. This leads to a method providing a good pathwise approximation. After explaining the reduction procedure, the error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments. Within the numerics section, the benefit of the algorithm in the context of option pricing is pointed out.
- ItemMarkovian approximations of stochastic Volterra equations with the fractional kernel(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Bayer, Christian; Breneis, SimonWe consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölder-regularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an N-dimensional diffusion process defined as solution to a system of ordinary stochastic differential equation. If the coefficients of the stochastic Volterra equation are Lipschitz continuous, we show that these approximations converge strongly with superpolynomial rate in N. Finally, we apply this approximation to compute the implied volatility smile of a European call option under the rough Bergomi and the rough Heston model.
- ItemStability of deep neural networks via discrete rough paths(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Friz, Peter; Tapia, NikolasUsing rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks. In particular we derive stability bounds in terms of the total p-variation of trained weights for any p ≥ 1.
- ItemPricing American options by exercise rate optimization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Bayer, Christian; Tempone , Raúl F.; Wolfers, SörenWe present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model.
- ItemRandomized optimal stopping algorithms and their convergence analysis(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Belomestny, Denis; Hager, Paul; Pigato, Paolo; Schoenmakers, John G. M.In this paper we study randomized optimal stopping problems and consider corresponding forward and backward Monte Carlo based optimization algorithms. In particular we prove the convergence of the proposed algorithms and derive the corresponding convergence rates.
- ItemOptimal stopping with signatures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Hager, Paul; Riedel, Sebastian; Schoenmakers, John G. M.We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process. We consider classic and randomized stopping times represented by linear functionals of the associated rough path signature, and prove that maximizing over the class of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature. The only assumption on the process is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion which fail to be either semi-martingales or Markov processes.