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Mathematical and Computational Finance / Mathématiques financières
(Org: Tahir Choulli and/et Jie Xiong)

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FELIPE AGUERREVERE, University of Alberta, School of Business
Equilibrium investment strategies and output price behavior: a real-options approach

Most of the real options applications that are concerned with strategic exercise policies focus on the effect of competition on the value of the option to invest and ignore the operating decisions that may arise once the investment is completed. However, operating flexibility is a valuable real option. This paper studies the effect of competitive interactions on investment decisions and on the dynamics of the price of a non-storable commodity. We develop a model of incremental investment with time to build and operating flexibility. We find that an increase in uncertainty may encourage firms to increase their capacity. Furthermore, we show that it may be optimal to invest in additional capacity during periods in which part of the completed capacity is being utilized. Our results on capacity choice and capacity utilization are qualitatively the same under oligopoly and perfect competition. However, the impact of competition on the properties of the endogenous output price is dramatic. In particular, the dynamics of capacity utilization induces heteroskadasticity in the output price process, and the price volatility may be increasing in the number of competitors in the industry. Our results on output price behavior appear to conform to actual behavior in competitive wholesale electricity markets.

CRISTIN BUESCU, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta  T6G 2G1
Optimal portfolio management when there are taxes and transaction costs

(joint work with A. Cadenillas and S. Pliska)

We consider a financial market with a bank account, and a single risky stock modelled as a geometric Brownian motion. The objective is to maximize the long-run growth rate of the wealth resulting after paying transaction costs and taxes. We apply the theory of optimal stopping to determine the optimal investment strategy under the effect of taxes and transaction costs.

ABEL CADENILLAS, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta  T6G 2G1
Optimal manager compensation with choice of effort and volatility

(Joint work with J. Cvitanic and F. Zapatero)

We consider the problem of optimal manager compensation in a continuous-time framework. We assume that the manager can control both the drift and the volatility of the stock. We solve this problem using martingale/duality methods.

JACQUES CARRIERE, Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta
Martingale valuation of cash-flows for insurance and interest models

Using a pricing axiom from financial economics, a martingale valuation method is developed with the properties of numéraire invariance and no-arbitage. The pricing method is then applied to cash-flows in actuarial models like loans, bonds, insurances, annuities, reserves and surplus processes. Special emphasis is given to the modelling of portfolios of defaultable bonds where new results are given. Also, an optimal repayment analysis of a common loan arrangement reveals that the book and market rates have to be equal.

Please send requests to receive a copy of the entire manuscript directly to the author at: jacques@gompertz.math.ualberta.ca

SHUI FENG, McMaster University, Hamilton, Ontario  L8S 4K1
The valuation of some American call options

In this talk, I will review results on the valuation of American call options and American capped options. New results will also be presented on the valuation of American call options on the minimum of two dividend-paying assets. This is joint work with Jerome Detemple and Weidong Tian.

SEBASTIAN FERRANDO, Department of Mathematics, Ryerson University, Toronto, Ontario  M5B 2K3
Haar wavelets systems for pricing and hedging financial derivatives

We use Haar wavelets systems to construct pathwise approximations of simple financial processes. This is in contrast to most techniques which depend on weak approximations to the process. Our constructions provide alternative computational approaches to price and hedge financial derivatives. The approach allows the deployment, in an stochastic process setting, of wavelets techniques such as: compression, search for best Haar packets and denoising.

BOB KIMMEL, Department of Economics, Princeton University, Princeton, New Jersey  08544, USA
Market price of risk specifications for affine models: theory and evidence

We extend the standard specification for the market price of risk for affine yield models of the term structure, and estimate models using the extended specification. For most models, likelihood ratio tests indicate that the extended specification fits the data better than standard specifications, often with huge statistical significance. The square-root process of Feller [1951] was first used in term structure modeling by Cox, Ingersoll, and Ross [1985], who specified the market price of risk as a constant times the square root of the interest rate. With this specification, the interest rate follows an affine diffusion under both the true and risk-neutral probability measures. However, the true and risk-neutral dynamics of the interest rate are still affine under both measures if the market price of risk also includes a term that is inversely proportional to the square root of the interest rate. This specification is never used in financial applications, probably because it does not satisfy either the Novikov or Kazamaki criteria. However, these criteria are sufficient but not necessary for the existence of a Girsanov ratio. We show that the extended specification does not permit arbitrage opportunities, provided that under either measure the probability is zero that the interest rate process achieves the boundary value of zero. We also extend our market price of risk specification to the general multivariate affine yield model of Duffie and Kan [1996], deriving the most general market price of risk specification for which state variable dynamics are linear under both measures. We estimate all affine yield models with three or fewer factors, using the market price of risk specifications of Dai and Singleton [2000], Duffee [2002], and our extended specification. We apply likelihood ratio tests of our specification relative to both the Dai/Singleton and Duffee specifications. There are six models for which our extension is more general than Duffee's specification; we find our extension is statistically significant for five of these models at the conventional 95% confidence level, and at far higher levels for three of the models. The results are particularly strong for affine diffusions with multiple square-root type variables, i.e., using the notation of Dai and Singleton, Am(N) models with m > 1. Although we focus on affine yield models, our extended market price of risk specification also applies to any model in which Feller's process or a multivariate extension is used to model asset prices.

ALI LAZRAK, University of British Columbia
Information neutrality in the stochastic differential utility

This paper develops in a Brownian information setting an approach for analyzing the nonindifference for the timing of resolution of uncertainty, a question that motivates the stochastic differential utility (SDU) due to Duffie and Epstein (1992). For a class of Backward Stochastic Differential Equations (BSDEs) including SDU, we formulate the information neutrality property as an invariance principle when the filtration is coarser (or finer) and characterize this property. Furthermore, we provide a concrete example of heterogeneity in information that illustrates explicitly the neutrality property for some particular BSDEs.

ALEXANDER MELNIKOV, University of Alberta
On the pricing of life-insurance contracts based on risky assets

The talk is devoted to the pricing of so-called equity-linked life-insurance contracts. These investment instruments reflect an innovative trend in traditional insurance toward financial economics. In the talk we consider discrete and continuous time variants of Black-Scholes models and present mean-variance and other methodologies.

PAT MULDOWNEY, University of Ulster
A generalized Black-Scholes equation without Ito calculus

Using modern integration theory based on the generalized Riemann integration of Henstock and Kurzweil, stochastic processes such as those of financial theory can be modelled by elementary methods not involving the Itô calculus. To demonstrate this, the following form of the Black-Scholes partial differential equation is established:

¶Em (f) ¶t +mx  ¶Em (f) ¶x +  1 2 s2 x2  ¶2 Em (f) ¶x2 =r Em(f).

With the underlying asset price process x, with t as the initial time and x the initial value (x(t) = x), and with r an arbitrary discounting parameter, the expectation value E^m(f) is the expected initial value of a discounted derivative asset price process f relative to the probabilities which are induced in the sample space by assuming an arbitrary growth rate m in the underlying process x. This new approach also separates those mathematical features which depend on the martingale property from those which depend on the assumption of geometric Brownian motion.

BRUNO REMILLARD, HEC, Montreal
Discrete time problems in hedging strategies and portfolio optimization

In this talk I will discuss how martingales enters naturally in dynamical selection problems like hedging strategies and investment portfolio. I will also examine the impact of discrete time versus continuous time in hedging strategies, option pricing and portfolio optimization.

LOUIS SECO, University of Toronto
Entropy methods for time series calibration

Determining the structure of distributions is at the heart of financial risk management, specially when dealing with non-gaussian markets that exhibit heavy tail behavior. The problem lies at the ability to extract information on extreme events from a limited sample data set.

This talk will present a methodology that uses entropy maximization methods to provide with a framework that allows for the determination of one-dimensional probabilistic parametric structures.

MICHAEL TAKSAR, Missouri
Ruin probability minimization and dividend distribution optimization in diffusion models

We will consider a model of an insurance companies with different modes of risk and financial control. Different types of reinsurance correspond to the risk reduction techniques of the insurance, while financial control corresponds to a more familiar portfolio rebalancing. There are different objective which the company may pursue. One is the classical minimization of the ruin probabilities. Another one is the dividend pay-out maximization. The later merges with the classical finance issue of utility optimization by a small investor, pioneered by Merton. Diffusion approximation enables one to get a closed form solution to many problems and see the structure of the optimal policy. Mathematically, the problem becomes a mixed singular/regular control of a diffusion process, whose analytical portion corresponds to a solution of nonlinear ordinary or partial differential equations.

RUPPA THULASIRAM, Department of Computer Science, University of Manitoba, Winnipeg, Manitoba
Fast Fourier transform in option pricing-a parallel algorithm

Computational Finance (CF) is relatively young and rapidly growing field. CF knowledge and skills are in increasing demand in the finance industry. This area is a coalescence of many disciplines, such as Mathematics, Computer Science & Engineering, Numerical Techniques and the theory of Financial Economics. Pricing of derivatives is one of the central problems in CF. Since the theory of derivative pricing is highly mathematical, numerical techniques such as lattice approach, finite-difference and finite-element techniques among others have been employed. Recently, Fast Fourier Transform (FFT) has been used for derivative pricing. In this talk we describe the FFT relation to option pricing problem and describe the design and development of a parallel algorithm for option pricing with FFT. For a data size of N and P processors, a blocked data distribution for the FFT in general produces log(N)-log(P) iterations of local communications and log(P) iterations of remote communications. We discuss the performance of our algorithm first and identify some of the shortcomings. We then improve our algorithm by exploiting data locality to reduce communication overheads inherent in the parallel FFT algorithm. Compared to the original FFT algorithm, the modified algorithm with data swap network improves the performance by about 15% for large data sizes.

HAO WANG, University of Oregon, Eugene, Oregon  97403-1222, USA
Calibrated models for heavy-tailed risk factors

We have seen the deficiencies of normal and other thin-tailed distribution models in risk management with extreme observations. In this talk, we will introduce some models to handle heavy-tailed phenomena. First, a calibrated scenario generation model for multivariate risk factors with heavy tailed distribution will be discussed. This model is a generalization of the standard model of scenario generation developed by J. P. Morgan and later on updated by RiskMetrics to simulate risk factors with heavy-tailed distribution. In this model, a rotation method is introduced to calibrate the covariance matrix and a mixture of normal distributions is used to calibrate and fit to each marginal distribution. Then, based on the scenario generation, a non-parametric method is applied to estimate portfolio extreme Value-at-Risk and Value-at-Risk confidence interval. The generalization of Black-Scholes formula is also considered for the stock or asset prices with heavy-tailed distributions.

LIXIN WU, Claremont Graduate University
LIBOR market model: from deterministic to stochastic volatility

LIBOR market model is the benchmark model for interest-rate derivatives. It has been a challenge to extend the standard LIBOR market model so as to cope with the volatility smiles and/or skews that are pronounced in the swaption markets. In this talk we extend the standard LIBOR market model, which takes forward rate or swap rate as state variables, by adopting stochastic volatility. Specifically, we adopt a multiplicative stochastic factor for the volatility functions of all relevant forward rates. The stochastic factor follows a squared-root diffusion process, and it can be correlated with the forward-rate processes. We derive approximate processes for swap rates after the change to forward swap measures, and develop a closed-form formula for swaption prices in terms of Fourier integrals. We then develop a fast Fourier transform algorithm for the implementation of the formula. The approximations are well supported by pricing accuracy. By adjusting the correlation between the forward rates and the volatility in a way consistent with intuition, we can generate volatility smiles or skews of the swaption prices similar to those observed in the markets. Calibration of the model will also be discussed.

YONG ZENG, Department of Mathematics and Statistics, University of Missouri, Kansas City, Missouri  64110, USA
A cass of partially-observed models for micro-movement of asset orices with Bayesian inference via filtering

A general class of micro-movement models that describes the transactional price behavior is proposed. The model ties the sample characteristics of micro-movement and macro-movement in a consistent manner. An important feature of the model is that it can be transformed to a filtering problem with counting process observations. Consequently, the complete information of price and trading time is captured and then utilized in Bayesian inference via filtering for the parameter estimation and model selection. The filtering equations are derived. Recursive algorithms are constructed via the Markov chain approximation method to compute (1)  the approximate posterior for trade-by-trade Bayes estimation and (2)  the Bayes factor for Bayesian model selection. The consistency (or Robustness) of the recursive algorithms is proven. Two models are studied in detail. One is the model built on geometric Brownian motion (GBM) and the other is on the GBM plus jumping stochastic volatility. Simulation results show that the Bayes estimates for time-invariant parameters are consistent, the Bayes estimates for stochastic volatility capture the movement of volatility, and the Bayes factor can effecitvely and efficiently selects the right model. Real-world applications to Microsoft transaction data are also provided.

XINGQIU ZHAO, University of Alberta
Option pricing with modified fractional Brownian motion

The Black-Scholes model introduced by Black and Scholes (1973) and Merton (1973) has become synonymous with modern finance theory. It assumes that the dynamics of stock prices is well described by exponential Brownian motions; that is, stock price returns behave like a sequence of independently idential distributed Gaussian random variables. The hypothesis is not consistent with empirical stock price returns. Many authers (e.g., Mandelbrot (1967), Greene and Fielitz(1977), Lo and Mackinlay (1998), Willinger, Taqqe and Teverovsky (1999), etc.) found empirical evidence of long-range dependence in stock price returns. This paper develops a numerical model for option pricing under the hypothesis that the underlying asset price satisfies a stochastic differential equation driven by a modified fractional Brownian motion. The empirical results indicate that the model is better than the Black-Scholes model.