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The performance of various vectorizable discrete random-sampling methods, along with the commonly used inverse sampling method, is assessed on a vector machine. Monte Carlo applications involving, one-dimensional, two-dimensional and multi-dimensional probability tables are used in the investigation. Various forms of the weighted sampling method and methods that transform the original probability table are examined. It is found that some form of weighted sampling is efficient, when the original probability distribution is not far from uniform or can be approximated analytically. Table transformation methods, though requiring additional memory storage, are best suited in applications where multidimensional tables are involved.

We propose robust numerical algorithms for pricing variance options and volatility swaps on discrete realized variance under general time-changed Lévy processes. Since analytic pricing formulas of these derivatives are not available, some of the earlier pricing methods use the quadratic variation approximation for the discrete realized variance. While this approximation works quite well for long-maturity options on discrete realized variance, numerical accuracy deteriorates for options with low frequency of monitoring or short maturity. To circumvent these shortcomings, we construct numerical algorithms that rely on the computation of the Laplace transform of the discrete realized variance under time-changed Lévy processes. We adopt the randomization of the Laplace transform of the discrete log return with a standard normal random variable and develop a recursive quadrature algorithm to compute the Laplace transform of the discrete realized variance. Our pricing approach is rather computationally efficient when compared with the Monte Carlo simulation and works particularly well for discrete realized variance and volatility derivatives with low frequency of monitoring or short maturity. The pricing behaviors of variance options and volatility swaps under various time-changed Lévy processes are also investigated.

This paper focuses on the pricing of variance swaps in incomplete markets where the short rate of interest is determined by a Cox–Ingersoll–Ross model and the stock price is determined by a Heston model with simultaneous Lévy jumps. We obtain the pricing kernel and the equivalent martingale measure in an equilibrium framework. We also give new closed-form solutions for the delivery prices of discretely sampled variance swaps under the forward measure, as opposed to the risk neural measure, by employing the joint moment generating function of underlying processes. Theoretical results and numerical examples are provided to illustrate how the values of variance swaps depend on the jump risks and stochastic interest rate.

We consider a parameter estimation problem for one-dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed estimators have the same form and asymptotic properties regardless of the nature of the domain –bounded domain or whole space. The derivation of the estimators and the proofs of their asymptotic properties are based on computations of power variations of some relevant stochastic processes. We use elements of Malliavin calculus to establish the asymptotic normality properties in the case of bounded domain. We also discuss the joint estimation problem of the drift and volatility coefficient. We conclude with some numerical experiments that illustrate the obtained theoretical results.

Stochastic partial differential equations (SPDE) are used for stochastic modelling, for instance, in the study of neuronal behaviour in neurophysiology, in modelling sea surface temparature and sea surface height in physical oceanography, in building stochastic models for turbulence and in modelling environmental pollution. Probabilistic theory underlying the subject of SPDE is discussed in Ito [2] and more recently in Kallianpur and Xiong [11] among others. The study of statistical inference for the parameters involved in SPDE is more recent. Asymptotic theory of maximum likelihood estimators for a class of SPDE is discussed in Huebner, Khasminskii and Rozovskii [7] and Huebner and Rozovskii [8] following the methods in Ibragimov and Khasminskii [9]. Bayes estimation problems for such a class of SPDE are investigated in Prakasa Rao [21,25] following the techniques developed in Borwanker et al. [2]. An analogue of the Bernstein-von Mises theorem for parabolic stochastic partial differential equations is proved in Prakasa Rao [21]. As a consequence, the asymptotic properties of the Bayes estimators of the parameters are investigated. Asymptotic properties of estimators obtained by the method of minimum distance estimation are discussed in Prakasa Rao [30]. Nonparametric estimation of a linear multiplier for some classes of SPDE is studied in Prakasa Rao [26,27] by the kernel method of density estimation following the techniques in Kutoyants [12]. In all the papers cited above, it was assumed that a continuous observation of the random field satisfying the SPDE is available. It is obvious that this assumption is not tenable in practice for various reasons. The question is how to study problem of estimation when there is only a discrete sampling on the random field. A simplified version of this problem is investigated in Prakasa Rao [28,29,30,31]. A review of these and related results is given.