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Dynamic model of Loan Portfolio with Lévy Asset Prices
Šmíd, Martin
We generalize the well known Merton-Vasicek (KMV) model of a loan portfolio value in two ways: we assume a L' evy process of the debtors' assets' value (instead of the Gaussian one) and we model a dynamics of the portfolio value so that the debts may last several periods (instead of a single one). Our model is computable by simulation.
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Pravděpodobnostní vlastnosti spojité dvojité aukce - rovnoměrný případ
Šmíd, Martin
We study probabilistic properties of a zero intelligence model of a limit order market, very similar to those of /citet{Maslov00} and /citet{Smith03}. We (recursively) describe the distributions of the order books and the best quotes. Based on these theoretical results, a procedure for statistical inference of the model may be designed and the evolution of the process may be simulated more efficiently then by the crude simulation of all the events.
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Pravděpodobnostní vlastnosti spojité dvojité aukce - rovnoměrný případ
Šmíd, Martin
We study probabilistic properties of a zero intelligence model of a limit order market, very similar to those of Maslov [2000] and Smith at. al. [2003]. We (recursively) describe the distributions of the order books and the best quotes. Based on these theoretical results, we propose a procedure for statistical inference of the model and we show the way how to simulate the evolution of the process more efficiently then by the simulation of all the events. We prove that the price increments may be fat tailed if there is no initial order book but they are thin-tailed given a non-empty (Poisson distributed) order book. Finally, we show that, by a discretization of our model, the model of Smith et. al. [2003] is obtained and that a majority of our results is applicable to this model too.
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Zjednodušení modelu Smitha a Farmera
Šmíd, Martin
We formulate a model of a limit order market, very similar to this of Smith at al. [2003]. We describe analytically the conditional distribution of the state of the market (i.e. the shapes of the limit order books) given the history of the bid and ask quotes. Thanks to this result, we may construct estimates of (usually unknown) shape of the order book based on a history of the quotes (which is usually available). Further, we are able to test statistically the validity of the model. In addition, our results allow us to specify the (conditional) distribution of the inter-event times and the jumps of the quotes which may help us to speed up eventual simulations of the system significantly.
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Optimální strategie na trhu s limitními objednávkami
Šmíd, Martin
We define a decision problem of an investor, trading continuously at a limit order market, maximizing a utility from his wealth at a random time horizon. We show that, in special cases (e.g. risk neutrality, quadratic or exponential utility function), the problem may be factorized and, given additional restrictions, it may even be solved.
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Markovská vlastnost trhu s limitními objednávkami
Šmíd, Martin
We formulate a rigorousmathematical description of a limit order market (we describe the state of the market by means of atomic measures). Further, we state sufficient conditions for the evolution of the market to be Markov; in particular, all the agents should either trade randomly or use strategies dependent only on the current state of the market and on external random elements.
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Predpovídání ve dvojité aukci se spojitým časem
Šmíd, Martin
Recently, the continuous double auction, i.e. the trading mechanism used in the majority of the financial markets, is the subject of an extensive study. In the present paper, a model of the continuous double auction with the completely random flow of the limit orders is studied. The main result of the paper is an approximate formula for the distribution of the market price and the traded volume at the time s given the information available at t < s.
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