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- ItemReinforced optimal control(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Belomestny, Denis; Hager, Paul; Pigato, Paolo; Schoenmakers, John G. M.; Spokoiny, VladimirLeast squares Monte Carlo methods are a popular numerical approximation method for solving stochastic control problems. Based on dynamic programming, their key feature is the approximation of the conditional expectation of future rewards by linear least squares regression. Hence, the choice of basis functions is crucial for the accuracy of the method. Earlier work by some of us [Belomestny, Schoenmakers, Spokoiny, Zharkynbay, Commun. Math. Sci., 18(1):109?121, 2020] proposes to reinforce the basis functions in the case of optimal stopping problems by already computed value functions for later times, thereby considerably improving the accuracy with limited additional computational cost. We extend the reinforced regression method to a general class of stochastic control problems, while considerably improving the method?s efficiency, as demonstrated by substantial numerical examples as well as theoretical analysis.
- ItemPricing options under rough volatility with backward SPDEs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Qiu, Jinniao; Yao, YaoIn this paper, we study the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic partial differential equation (BSPDE). The existence and uniqueness of weak solutions is proved for general nonlinear BSPDEs with unbounded random leading coefficients whose connections with certain forward-backward stochastic differential equations are derived as well. These BSPDEs are then used to approximate American option prices. A deep learning-based method is also investigated for the numerical approximations to such BSPDEs and associated non-Markovian pricing problems. Finally, the examples of rough Bergomi type are numerically computed for both European and American options.
- ItemWeak error rates for option pricing under linear rough volatility(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Bayer, Christian; Hall, Eric; Tempone, Raúl F.In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, H < 1/2, over reasonable timescales. Both time series of asset prices and option-derived price data indicate that H often takes values close to 0.1 or less, i.e., rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only H. We prove rate H + 1/2 for the weak convergence of the Euler method for the rough Stein--Stein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Indeed, the problem of weak convergence for rough volatility models is very subtle; we provide examples demonstrating the rate of convergence for payoff functions that are well approximated by second-order polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof uses Talay--Tubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.