Christian Bayer

Pricing Under Rough Volatility

From an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum 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.

The proof of Tchakaloff's Theorem

We provide a simple proof of Tchakalos Theorem on the exis- tence of cubature formulas of degree m for Borel measures with moments up to order m. The result improves known results for non-compact support, since we do not need conditions on (m + 1)st moments.

From rough path estimates to multilevel Monte Carlo

New classes of stochastic differential equations can now be studied using rough path theory (see, e.g., [T. J. Lyons, M. Caruana, and T. Levy, Differential Equations Driven by Rough Paths, Springer, Berlin, 2007] or [P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014]). In this paper we investigate, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense. Our focus lies on numerical implementations, and more specifically on the savings possible via multilevel methods. Our analysis relies on a subtle combination of pathwise estimates, Gaussian concentration, and multilevel ideas. Numerical examples are given which both illustrate and confirm our findings.

On Nonasymptotic Optimal Stopping Criteria In Monte Carlo Simulations

We 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.

Cubature on Wiener space in infinite dimension

We prove a stochastic Taylor expansion for stochastic partial differential equations (SPDEs) and apply this result to obtain cubature methods, i.e. high-order weak approximation schemes for SPDEs, in the spirit of Lyons and Victoir (Lyons & Victoir 2004 Proc. R. Soc. A 460, 169–198). We can prove a high-order weak convergence for well-defined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of Lévy processes of finite type; here the results seem to be new, even in finite dimension. Several numerical examples are added.

A Functional Limit Theorem for Limit Order Books with State Dependent Price Dynamics

We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. 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 in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics.

Semi-closed form cubature and applications to financial diffusion models

Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka [Adv. Math. Econ., 2004, 6, 69–83] and Lyons–Victoir [Proc. R. Soc. Lond. Ser. A, 2004, 460, 169–198], involve the solution to numerous auxiliary ordinary differential equations (ODEs). With focus on the Ninomiya–Victoir algorithm [Appl. Math. Finance, 2008, 15, 107–121], which corresponds to a concrete level 5 cubature method, we study some parametric diffusion models motivated from financial applications, and show the structural conditions under which all involved ODEs can be solved explicitly and efficiently. We then enlarge the class of models for which this technique applies by introducing a (model-dependent) variation of the Ninomiya–Victoir method. Our method remains easy to implement; numerical examples illustrate the savings in computation time.

Adaptive weak approximation of reflected and stopped diffusions

Abstract We study the weak approximation problem of diffusions, which are reflected at a subset of the boundary of a domain and stopped at the remaining boundary. First, we derive an error representation for the projected Euler method of Costantini, Pacchiarotti and Sartoretto [Costantini et al., SIAM J. Appl. Math., 58(1):73–102, 1998], based on which we introduce two new algorithms. The first one uses a correction term from the representation in order to obtain a higher order of convergence, but the computation of the correction term is, in general, not feasible in dimensions d > 1. The second algorithm is adaptive in the sense of Moon, Szepessy, Tempone and Zouraris [Moon et al., Stoch. Anal. Appl., 23:511–558, 2005], using stochastic refinement of the time grid based on a computable error expansion derived from the representation. Regarding the stopped diffusion, it is based in the adaptive algorithm for purely stopped diffusions presented in Dzougoutov, Moon, von Schwerin, Szepessy and Tempone [Dzougoutov et al., Lect. Notes Comput. Sci. Eng., 44, 59–88, 2005]. We give numerical examples underlining the theoretical results.

Simulation of forward-reverse stochastic representations for conditional diffusions.

In this paper we derive stochastic representations for the finite dimensional distributions of a multidimensional diffusion on a fixed time interval, conditioned on the terminal state. The conditioning can be with respect to a fixed point or more generally with respect to some subset. The representations rely on a reverse process connected with the given (forward) diffusion as introduced in Milstein, Schoenmakers and Spokoiny [Bernoulli 10 (2004) 281-312] in the context of a forward-reverse transition density estimator. The corresponding Monte Carlo estimators have essentially root-

Smoothing the payoff for efficient computation of Basket option prices

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