Browsing by Author "Bayer, Christian"
Now showing 1 - 20 of 31
Results Per Page
Sort Options
- ItemAsymptotics for at the money local vol basket options(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Laurence, PeterWe consider a basket or spread option on based on a multi-dimensional local volatility model. Bayer and Laurence [Comm. Pure. Appl. Math., to appear] derived highly accurate analytic formulas for prices and implied volatilities of such options when the options are not at the money. We now extend these results to the ATM case. Moreover, we also derive similar formulas for the local volatility of the basket.
- ItemDeep calibration of rough stochastic volatility models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bayer, Christian; Stemper, BenjaminSparked by Alòs, León und Vives (2007); Fukasawa (2011, 2017); Gatheral, Jaisson und Rosenbaum (2018), so-called rough stochastic volatility models such as the rough Bergomi model by Bayer, Friz und Gatheral (2016) constitute the latest evolution in option price modeling. Unlike standard bivariate diffusion models such as Heston (1993), these non-Markovian models with fractional volatility drivers allow to parsimoniously recover key stylized facts of market implied volatility surfaces such as the exploding power-law behaviour of the at-the-money volatility skew as time to maturity goes to zero. Standard model calibration routines rely on the repetitive evaluation of the map from model parameters to Black-Scholes implied volatility, rendering calibration of many (rough) stochastic volatility models prohibitively expensive since there the map can often only be approximated by costly Monte Carlo (MC) simulations (Bennedsen, Lunde & Pakkanen, 2017; McCrickerd & Pakkanen, 2018; Bayer et al., 2016; Horvath, Jacquier & Muguruza, 2017). As a remedy, we propose to combine a standard Levenberg-Marquardt calibration routine with neural network regression, replacing expensive MC simulations with cheap forward runs of a neural network trained to approximate the implied volatility map. Numerical experiments confirm the high accuracy and speed of our approach.
- ItemDynamic programming for optimal stopping via pseudo-regression(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bayer, Christian; Redmann, Martin; Schoenmakers, John G.M.We introduce new variants of classical regression-based algorithms for optimal stopping problems based on computation of regression coefficients by Monte Carlo approximation of the corresponding L2 inner products instead of the least-squares error functional. Coupled with new proposals for simulation of the underlying samples, we call the approach pseudo regression. We show that the approach leads to asymptotically smaller errors, as well as less computational cost. The analysis is justified by numerical examples.
- ItemForward-reverse EM algorithm for Markov chains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Mai, Hilmar; Schoenmakers, John G.M.We develop an EM algorithm for estimating parameters that determine the dynamics of a discrete time Markov chain evolving through a certain measurable state space. As a key tool for the construction of the EM method we develop forward-reverse representations for Markov chains conditioned on a certain terminal state. These representations may be considered as an extension of the earlier work [1] on conditional diffusions. We present several experiments and consider the convergence of the new EM algorithm.
- ItemFrom rough path estimates to multilevel Monte Carlo(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Friz, Peter K.; Riedel, Sebastian; Schoenmakers, John G.M.Discrete approximations to solutions of stochastic differential equations are well-known to converge with strong rate 1=2. Such rates have played a key-role in Giles multilevel Monte Carlo method [Giles, Oper. Res. 2008] which gives a substantial reduction of the computational effort necessary for the evaluation of diffusion functionals. In the present article similar results are established for large classes of rough differential equations driven by Gaussian processes (including fractional Brownian motion with H > 1=4 as special case).
- ItemA fully adaptive interpolated stochastic sampling method for random PDEs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Anker, Felix; Bayer, Christian; Eigel, Martin; Neumann, Johannes; Schoenmakers, JohnA numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.
- ItemA functional limit theorem for limit order books(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Bayer, Christian; Horst, Ulrich; Qiu, JinniaoWe consider a stochastic model for the dynamics of the two-sided limit order book (LOB). For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges to a continuous-time limit when the order arrival rates tend to infinity, the impact of an individual order arrival on the book as well as the tick size tend to zero. The limits of the standing buy and sell volume densities are described by two linear stochastic partial differential equations, which are coupled with a two-dimensional reflected Brownian motion that is the limit of the best bid and ask price processes.
- 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.
- 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)
- ItemLog-modulated rough stochastic volatility models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Harang, Fabian; Pigato, PaoloWe propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index H. The so-obtained log-modulated fractional Brownian motion (log-fBm) is a continuous Gaussian process even for H = 0. As a consequence, the resulting super-rough stochastic volatility models can be analysed over the whole range of Hurst indices between 0 and 1/2, including H = 0, without the need of further normalization. We obtain the usual power law explosion of the skew as maturity T goes to 0, modulated by a logarithmic term, so no flattening of the skew occurs as H goes to 0.
- 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.
- ItemNumerical smoothing with hierarchical adaptive sparse grids and quasi-Monte Carlo methods for efficient option pricing(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Bayer, Christian; Ben Hammouda, Chiheb; Tempone, Raúl F.When approximating the expectation of a functional of a stochastic process, the efficiency and performance of deterministic quadrature methods, such as sparse grid quadrature and quasi-Monte Carlo (QMC) methods, may critically depend on the regularity of the integrand. To overcome this issue and reveal the available regularity, we consider cases in which analytic smoothing cannot be performed, and introduce a novel numerical smoothing approach by combining a root finding algorithm with one-dimensional integration with respect to a single well-selected variable. We prove that under appropriate conditions, the resulting function of the remaining variables is a highly smooth function, potentially affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e., Brownian bridge and Richardson extrapolation on the weak error). This approach facilitates the effective treatment of high dimensionality. Our study is motivated by option pricing problems, and our focus is on dynamics where the discretization of the asset price is necessary. Based on our analysis and numerical experiments, we show the advantages of combining numerical smoothing with the ASGQ and QMC methods over ASGQ and QMC methods without smoothing and the Monte Carlo approach.
- ItemOn non-asymptotic optimal stopping criteria in Monte Carlo simulations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Hoel, Hakon; von Schwerin, Erik; Tempone, RaúlWe consider the setting of estimating the mean of a random variable by a sequential stopping rule Monte Carlo (MC) method. The performance of a typical second moment based sequential stopping rule MC method is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule.
- ItemOn the probability density function of baskets(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Friz, Klaus K.; Laurence, PeterThe state price density of a basket, even under uncorrelated Black–Scholes dynamics, does not allow for a closed from density. (This may be rephrased as statement on the sum of lognormals and is especially annoying for such are used most frequently in Financial and Actuarial Mathematics.) In this note we discuss short time and small volatility expansions, respectively. The method works for general multi-factor models with correlations and leads to the analysis of a system of ordinary (Hamiltonian) differential equations. Surprisingly perhaps, even in two asset Black–Scholes situation (with its flat geometry), the expansion can degenerate at a critical (basket) strike level; a phenomena which seems to have gone unnoticed in the literature to date. Explicit computations relate this to a phase transition from a unique to more than one “most-likely” paths (along which the diffusion, if suitably conditioned, concentrates in the afore-mentioned regimes). This also provides a (quantifiable) understanding of how precisely a presently out-of-money basket option may still end up in-the-money.
- 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.
- ItemPricing high-dimensional Bermudan options with hierarchical tensor formats(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Bayer, Christian; Eigel, Martin; Sallandt, Leon; Trunschke, PhilippAn efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the ``curse of dimensionality" can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. This discretization allows for a simple and computationally cheap evaluation of conditional expectations. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favourable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent Neural Network based methods.
- 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.
- ItemPricing under rough volatility(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bayer, Christian; Friz, Peter; Gatheral, JimFrom an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum [SSRN 2509457, 2014] previously showed that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated volatility. We analyze in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.