Volume 15, Issue 4

In this paper we propose an extension of the Libor market model with a high-dimensional specially structured system of square root volatility processes, and give a road map for its calibration. As such the model is well suited for Monte Carlo simulation of derivative interest rate instruments. As a key issue, we require that the local covariance structure of the market model is preserved in the stochastic volatility extension. In a case study we demonstrate that the extended Libor model allows for stable calibration to the cap-strike matrix. The calibration algorithm is FFT based, so fast and easy to implement.

The Soboĺ sequence [USSR Comput. Math. and Phy. 7: 86–112, 1967, USSR Comput. Math. and Phy. 16: 236–242, 1976] is one of the standard quasirandom sequences, and is widely used in Q uasi- M onte C arlo (QMC) applications. QMC methods are a variant of ordinary M onte C arlo (MC) methods that employ highly uniform quasirandom numbers in place of the pseudorandom numbers used in ordinary M onte C arlo (MC) [Caflisch, Acta Numerica 7: 1–49, 1998]. The error of MC methods is asymptotically , where N is the number of samples, while QMC methods can have an error bound which behaves as well as O ((log N ) s N –1 ), for s -dimensional problems. The common practice in MC of using a predetermined error criterion as a deterministic termination condition, is almost impossible to achieve in QMC without extra technology. In order to provide such dynamic error estimates for QMC methods, several researchers [Owen, Randomly permuted (t, m, s)-nets and (t, s)-sequences, 1995, Tezuka, Uniform Random Numbers, Theory and Practice, Kluwer Academic Publishers, 1995] proposed the use of R andomized QMC (RQMC) methods, where randomness can be brought to bear on quasirandom sequences through scrambling and other related randomization techniques [Chi and Lorenzo, A Parallel Cellular Automata Model to Forecast the Effects of Urban Growth in North Florida, 2007, Chi and Mascagni, Efficient Generation of Parallel Quasirandom Sequences via Scrambling, 2007, Vandewoestyne, Chi, Mascagni and Cools, An Empirical Investigation of Different Scrambling Methods for Faure Sequences, 2007]. Besides providing practical error estimates, another byproduct of scrambled quasirandom numbers is that they furnish a natural way to generate quasirandom numbers in parallel and distributed applications. In this paper, we propose a new algorithm for scrambling the Soboĺ sequence based on the permutation (scrambling) of several groups of binary digits from the individual Soboĺ numbers. Most of the current scrambling methods either randomize a single digit at each iteration, or randomize a group of digits through linear operations. In contrast, our multiple-digit scrambling is efficient and fast because it permutes small groups of digits. We implemented this new Soboĺ scrambling algorithm in the software context of the well-known Scalable P arallel R andom N umber G enerators (SPRNG) library [Mascagni and Srinivasan, ACM Transactions on Mathematical Software 26: 436–461, 2000, Mascagni, Scalable Parallel Random Number Gernerators Library (SPRNG)] library.

In [Stochastic Analysis: 331–346, 1991, Annals of Probability 26: 267–307, 1998] Kurtz and Protter resp. Jacod and Protter specify the asymptotic error distribution of the Euler method for stochastic differential equations (SDEs) with smooth coefficients growing at most linearly. The required differentiability and linear growth of the coefficients rule out some popular SDEs as for instance the Cox–Ingersoll–Ross (CIR) model, the Heston model, or the stochastic Brusselator. In this article, we partially extend one of the fundamental results in [Jacod and Protter, Annals of Probability 26: 267–307, 1998], so that also the mentioned examples are covered. Moreover, we compare by means of simulations the asymptotic error distributions of the CIR model and the geometric Brownian motion with mean reversion.

In this paper we introduce a class of statistics for the functional estimation of the spot volatility in the setting of frequency observed diffusion processes which may be disturbed by microstructure noise. We show that the limit theorems for the estimation of the spot volatility and the cross spot volatility of the statistics are still valid even if we add jump processes of finite or infinite activity to the underlying diffusion process. These statistics extend the quadratic variational approach and are related to the concept of multipower variation, which is used in the problem of estimating the integrated volatility.