Research

Importance sampling is one several variance reduction techniques meant to accelerate Monte Carlo simulation. It has been applied successfully and less so to many fields, including reliability, stochastic operations research, and finance. In dynamic models, it is particularly difficult to determine a good importance sampling measure. A specific case of interest is the calculation of the expected value of the integral of a function of the states variables. For diffusions, an original approach based on Malliavin calculus (although not limited to the first Wiener chaos) was given by Newton (1994). For Markov chains, Kuruganti and Strickland (1996) give a formula where the optimal measure is path-dependent. Schellhorn and Kidani (2000) suggest a Markovian scheme, that is, the transition kernel of the importance sampling measure is a function of the current state only. The Markovian nature of the scheme is a clear advantage when the scheme is embedded in an adaptive algorithm, that is, the system learns about itself. It allows the learning algorithm to extract information in a much lower dimensional space. We want to test this algorithm on sizeable problems in various fields of applied mathematics.
>> for more information contact Dr. Henry Schellhorn.

The LIBOR market Model (also called BGM/J, from Brace, Gatarek, Musiela (1997)), is one of the most popular models for pricing interest rate derivatives. In this model interest rates follow a stochastic partial differential equation, where one variable is time, and the other maturity of the loan. One way to implement the BGM/J model is Monte Carlo simulation. However, the slowness of Monte Carlo simulation is one of the main obstacles faced by practitioners when implementing the model. As a consequence, approximation formulas are also widely used for pricing European derivatives. Also, the LIBOR model is often used only to price interest-rate dependent securities, although it could be used in more generality to price securities that pay credit-dependent and currency-specific cash flows. Schellhorn and Chen (2005) suggest an approximation formula for the drift that leads to superior performance. We intend to conduct some further tests of the method, and generalize it to credit risk and currency risk.
>> for more information contact Dr. Henry Schellhorn.

Dynamic models of credit risk and financial risk contagion have been for a long-time focused on one single debtor. The celebrated Black-Scholes (1973) formula was indeed developed to price the debt of one single company. Although the Black-Scholes formula has been applied mostly to price equity options and then other exchange-traded options, the field of structural models of credit risk has been growing steadily. In these models, debt is a put option on the value of the firm. The underlying variable of these models is then the value of the firm. Several authors have added features to the most basic structural models. Among others, Leland (1994) considers the possibility of strategic bankruptcy of the shareholders. Goldstein, Ju and Leland (2001) take as underlying the revenue process of the firm. These structural models have been limited to one firm due to the complexity of the mathematics underlying these models: each firm solves an optimal stopping time problem, and the number of underlying stochastic processes is the number of firms. Recently Cossin and Schellhorn (2006) suggested that, by incorporating the buffering effect of cash management, the optimal stopping problem can be approximately decoupled into a collection of single-firm problems, and an approximate formula results, which extends the results of Goldstein et al. This approximation is an application of the theory of Jackson networks of queues to the financial problem. Our next plan is to apply this theory to price the debt and evaluate the risk on a network of interrelated firms, where each firm is a potential lender/borrower to every other firm in the network. In cases where the approximate formula does not hold, we plan to resort to Monte Carlo simulation. Other research in the field of counterparty risk and contagion includes Frey (2003) and Giesecke (2004). Although they try to assess similar questions to ours, and are more general, their models are less structural, and rely more on statistical inference.
>> For more information contact Dr. Henry Schellhorn.

Combinatorial exchanges have existed for a long time in securities markets. At various points in time traders on the Chicago Mercantile Exchange and Eurex options and futures exchanges could match various types of swaps against each leg of the contract. The years 1999 and 2000 saw an explosion of business-to-business exchanges (B2B). On these exchanges (unlike securities or commodities exchanges) buyers can bid simultaneously on several products that are complementary, and select a particular product based on the difference between the subjective value of its attributes and market price. These are examples of double-sided combinatorial auctions with attributes. An example of a single-sided combinatorial auction (without attributes) is the FCC auction, in which a single seller (the US government) auctioned to several buyers (cellular telephone companies) bandwidth across the US. Buyers were wanted to bid only on bundles of contiguous states rather than placing several independent bids on individual states. Rothkopf et al (1998) showed that the FCC problem, or more generally, the single-sided combinatorial auction was a NP-complete problem, and significant care should be taken in devising algorithms to determine the optimal allocation. In Schellhorn (1997) I exposed a price-based algorithm that turned out to be optimal and particularly efficient when the number of traders is large, for double-sided combinatorial auctions (without attributes). I have recently expanded the algorithm to take care of double-sided combinatorial auctions with attributes. It is based on a Kakutani fixed-point algorithm related to Scarf (1973) algorithm. My intention is to test the efficiency of this algorithm on large real-life examples.
>> For more information contact Dr. Henry Schellhorn.

The knowledge of the structural fold of a protein, while of theoretical interest in its own right, is integral to the process of drug discovery in pharmaceutical research. In order for a candidate drug molecule to be efficacious in targeting a given protein, it must "dock"  in a stable manner on the surface of the active site of the protein. This type of energetic stability can be assessed only if the detailed structures of the drug molecule and its protein target are known to reasonable accuracy. Experimental methods to determine protein structure, while becoming increasingly accurate, are notoriously laborious and time-consuming. Ab initio computational methods, on the other hand, are fast but often intractable for realistic proteins. They can also be highly inaccurate in the absence of information other than the primary sequence of the protein alone. Other homology-based computational methods that rely on the sequence similarity between the protein of unknown structure and other proteins of known structure are fast, and, if the protein of unknown structure is sufficiently similar in sequence with other proteins of known structure, also highly accurate. Over the last decade, advances in these methods that use hidden Markov models (Baum 1972) have enabled reasonably accurate structure prediction even when the sequence similarity is relatively low (Haussler et al. 1993, Krogh et al. 1994, Karplus et al. 1998). Recent extensions of hidden Markov models for predicting protein folds are mainly directed toward increasing the complexity of the alphabet used, i.e., using secondary structure, solvent accessibility and other input data (Raval et al. 2002, Karchin et al. 2003, 2004). In all studies of this problem to date, fold prediction is carried out by estimating the most likely parameters of a hidden Markov model based on sufficiently similar proteins of known structure (the Maximum Likelihood method). However, this method is potentially unreliable (Scott 2002), either because of the on-uniqueness of the optimal parameters (multiple local maxima on the likelihood surface) or because of the existence of parameter values that are only marginally sub-optimal (broad local maxima). We are therefore extending previous work in this area to a more computationally intensive approach by (a) building more complex, higher-order hidden Markov models that capture more accurately the long-range correlations within the protein sequence, and (b) using Bayesian methods for predictions that do not rely on the existence and uniqueness of optimal model parameters. This work is being carried out in collaboration with Steven Lewis, a graduate student at CGU, and John Angus at CGU.

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