Marked doubly stochastic Poisson processes are a particular type of marked point processes that are characterized by the number of events in any time interval as being conditionally Poisson distributed, given another positive stochastic process called intensity. Here we consider a subclass of these processes in which the intensity is assumed to be a deterministic function of another nonexplosive marked point process. In particular, we will investigate an intensity jump process with an exponential decay having an analytic form for the distribution of the times and sizes of the jumps, which can be seen as a generalization of the classical shot noise process. Assuming that the intensity is unobservable, interest here is in its filtering, that is, in the computation of its conditional distribution, over a whole time interval, given an observed trajectory of realized events. Because, in general, this computation cannot be performed analytically, we propose a simulation method that provides an approximate solution, which relies on the reversible-jump Markov chain Monte Carlo algorithm. Interestingly, the proposed filtering algorithm also allows the setup of a likelihood-based procedure for the estimation of the parameters of the model based on stochastic versions of the expectation-maximization (EM) algorithm. The potential of the filtering and estimation methods proposed are illustrated through some simulation experiments as well as on a financial ultra-high-frequency dataset of intraday S&P500 futures prices.

We propose a modeling framework for ultra-high-frequency data on financial asset price movements. The models proposed belong to the class of the doubly stochastic Poisson processes with marks and allow an interpretation of the changes in price volatility and trading activity in terms of news or information arrival. Assuming that the intensity process underlying event arrivals is unobserved by market agents, we propose a signal extraction (filtering) method based on the reversible jump Markov chain Monte Carlo algorithm. Moreover, given a realization of the price process, inference on the parameters can be performed by appealing to stochastic versions of the expectation-maximization algorithm. The simulation methods proposed will be applied to the computation of hedging strategies and derivative prices.

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To model intraday stock price movements we propose a class of marked doubly stochastic Poisson processes, whose intensity process can be interpreted in terms of the effect of information release on market activity. Assuming a partial information setting in which market agents are restricted to observe only the price process, a filtering algorithm is applied to compute, by Monte Carlo approximation, contingent claim prices, when the dynamics of the price process is given under a martingale measure. In particular, conditions for the existence of the minimal martingale measure Q are derived, and properties of the model under Q are studied.Keywords: Minimal martingale measure; news arrival; marked point process; nonlinear filtering; reversible jump Markov chain Monte Carlo; ultra-high frequency data. † Corresponding author. 1250018-1 Int. J. Theor. Appl. Finan. 2012.15. Downloaded from www.worldscientific.com by THE UNIVERSITY OF WESTERN ONTARIO on 04/12/15. For personal use only. 1250018-2 Int. J. Theor. Appl. Finan. 2012.15. Downloaded from www.worldscientific.com by THE UNIVERSITY OF WESTERN ONTARIO on 04/12/15. For personal use only. Monte Carlo Derivative Pricing in a Class of Marked DSPPtrajectory of the price process in any bounded time interval is characterized by a finite (although random) number of changes.An interesting feature of the framework proposed is that it can be interpreted to account for the link between the information release and the changes in price volatility and trading activity, whose existence has been many times suggested in the economic literature (see, among others, Engle and Ng [9] and Kalev et al. [17]). In our model, this link is embodied by the intensity process δ governing the speed of price changes. In particular, if δ is a shot noise process, its sudden increases can be interpreted as perturbations in market activity caused by pieces of news reaching the market, being the size of each increase due to the importance and unexpectedness of the news, and its consequent exponential decays can be interpreted as progressive normalizations due to the absorption of the effect of the news by the market.As far as the problem of pricing a contingent claim is concerned, a basic result of mathematical finance states that for a stochastic process S, representing the discounted stock price, the existence of an equivalent martingale measure, that is, of a measure equivalent to the "natural" probability P, such that S is a local martingale, is essentially equivalent to the absence of arbitrage opportunities (see, for example, Harrison and Kreps [15], Delbaen and Schachermayer [7]). If the price of the risky asset follows a marked point process, the market model is in general incomplete and it can be shown that there exist more then one of such equivalent measures. Thus, the problem of pricing a contingent claim, under the no arbitrage assumption, is reduced to taking expected values under the "right" measure among all existing equivalent martingale measures. One possibility is to choose the so cal...