North Carolina State University Financial Mathematics Program

(1) Rigorously measure baseline liabilities and performance towards meeting this goal;
(2) Evaluate key risk and cost drivers;
Identify strategic opportunities for reducing costs;
(3) Minimize the burden on client staff in providing information for complying with the initiative;
(4) Conduct the analysis consistent with the client’s reserve reporting requirements.

Abstract: Option pricing and hedging in a complete market are well-studied with nice results using martingale theories. However, they remain as open questions in incomplete markets. In particular, when the underlying processes involve jumps, there could be infinitely many martingale measures which give an interval of no-arbitrage prices instead of a unique one. Consequently, there is no martingale representation theorem to produce a perfect hedge. The question of picking a particular price and executing a hedging strategy according to some reasonable criteria becomes a non-trivial issue and an interesting question.

Abstract: The utility indifference pricing mechanism is an alternative to traditional no arbitrage valuation methods for derivative contracts in incomplete markets. It yields the price by comparing the maximal expected utilities with and without trading the derivative. In markets with stochastic volatility, the price turns out to be the solution of a certain quasilinear PDE. We study this price using bounds, asymptotic approximations and numerical solutions, and use these to describe its relation to the usual mechanism. The problem is connected to the Merton problem of portfolio optimization with derivative securities, and we describe how the assumption of exponential utility can make it computationally and analytically tractable.

Abstract: The purpose of the talk is to present new theoretical results on the asymptotic properties of the optimal filter of a Hidden Markov Model, and to report on numerical experiments with the particle filters. We shall consider engineering applications such as in-car intelligent navigation systems, and financial applications such as volatility tracking and fixed income affine model fitting.

Abstract: The first part of the talk will review some of the traditional financial derivatives traded on the energy markets. In particular, we shall discuss the pricing and the hedging of temperature options and spark spreads. The second part of the talk will concentrate on the mathematical theory of instruments with multiple American exercises. This research effort is motivated by the widespread use of swing options, and the fact that despite the existence of several numerical investigations, no mathematical theory seems to be existing for these instruments.

Abstract: We highlight several special features of Energy Portfolio VaR implementation and application that are intimately related with the volatile behavior of energy commodity prices and volatility term structures.

Abstract: Weather has a direct impact on energy consumption as well price levels. These relations will be illustrated in light of their importance in understanding and projecting energy consumption levels when pricing and structuring energy contracts.

Monday, October 28, 2002
Tao Pang, Dept. of Mathematics, NCSU
A Stochastic Portfolio Optimization Model

Abstract: A stochastic portfolio optimization model on an infinite time horizon is considered. The goal is to choose the optimal strategy for investment and consumption rate to maximize the total expected, discounted utility. A dynamic programming principle is used to derive the dynamics programming equation for the value function. The sub/super solution method is used to get the existence of the solution. The solution is then used to obtain the optimal control policies for investment and consumption rate. This is a generalization of the classical Merton's problem. The background of Merton's problem will be given in the beginning of the talk. I will then talk about some technical details about the problem I dealt with.

Friday, January 11, 2002
Steven P. Clark, Department of Economics, University of Virginia
Managerial Utility Versus Stockholder Wealth Maximization in Dividend Optimization Models

Abstract: One of the central features of the modern corporation is the separation of ownership and control. It has become apparent over the last 30 years that this principal-agent problem has profound implications for the behavior of the firm. The dividend decision is one such example where there is an embedded principal-agent problem. Managers may resist distributing dividends at the level which would maximize the wealth of the stockholders as prescribed by classical theory. A plausible explanation for this behavior is that managers tend to be risk averse since much of their wealth is tied up in the company in the form stock options, performance incentives and firm-specific human capital. In addition, with higher levels of retained earnings, managers may pursue projects that fulfil personal goals rather than operating only value maximizing projects. There has recently been much interest in the rigorous mathematical modeling of the dividend decision. However, for the most part, attention has focused on neoclassical models. Although they frequently admit very elegant solutions, they do not address important principal-agent issues. Thus researchers in mathematical finance and financial economists view the dividend problem very differently. In this talk, we model the dividend problem from the perspective of a utility-maximizing manager. Managerial preferences are influenced by aspects of the principle-agent problem. The level of the firm's reserves are modeled as a diffusion and the dividends accrue as a local time. It is particularly instructive to formulate the optimal policy in terms of the Green function of the diffusion. This approach makes transparent the trade-offs involved in determining the optimal dividend decision.

Monday, January 14, 2002
Michael Taksar, Department of Applied Mathematics, SUNY Stony Brook
Author's website: http://www.ams.sunysb.edu/~taksar/
Singular Stochastic Control and Related PDE with Gradient Constraints in Portfolio Optimization Models in Mathematical Finance

Abstract: In the modern mathematical finance the stock prices are modeled by stochastic differential equations, whose solutions produce logarithmic Brownian motions. This is the backbone of what is nowadays became the classical Black-Scholes option pricing theory and Merton's investment/ consumption theory. We consider a dynamical portfolio optimization model in the spirit of the latter. The portfolio consists of several risky assets (Stocks) and one risk-free asset (Bond). The rate of return on Bond is constant while the rate of return of Stocks is governed by SDE of the logarithmic Brownian motion type. Funds can be transferred from one asset to another, however such transaction involves penalty (brokerage fees) proportional to the size of the transaction. The objective is to find the policy which maximizes the expected rate of growth of funds. The main mathematical tool in the solution of this problem is singular stochastic control theory. In this theory the control functionals are represented by processes of bounded variation, and the optimal control consists of functionals which reflect the process from an a priori unknown boundary. They are continuous but singular (not absolutely continuous) with respect to time. The analytical part of the solution to singular control is related to a free boundary problem for an elliptic PDE with gradient constraints, similar to the ones encountered in elastic-plastic torsion problems. The existence of the classical C^2 solution cannot be proved in general but one can show an existence of viscosity solution to this equation. The optimal policy is to keep the vector of fractions of funds invested in different assets in an optimal (a priori unknown) boundary. We show how to find these boundaries explicitly in the case of one risky and one risk-free asset when the problem becomes one dimensional. In this case the free boundary problem can be reduced to a Stephan problem for an ODE.

Wednesday, January 16, 2002
Tao Pang, Brown University
An Optimal Investment-Consumption Policy Model in an Infinite Time Horizon

Abstract: An optimal investment-consumption policy model is considered. The goal is to maximize the expected utility of consumption in an infinite time horizon. The utility function is HARA with exponent less than 1. The problem can be reformulated as an infinite time horizon stochastic control problem. The dynamic programming equations for different HARA exponents are studied. I obtain some estimates for the solutions of each equation. This can be used to derive an optimal control policy.

Friday, January 18, 2002
Petter Wiberg, University of Toronto
An Introduction to Value-at-risk Simulation: Background, Modeling, and Algorithms

Abstract: The value-at-risk is the maximum loss that a portfolio might suffer over a given holding period with a certain confidence level. In recent years, value-at-risk has become a benchmark for measuring financial risk used by both practitioners and regulators. In this seminar, we discuss value-at-risk from a modeling and simulation perspective. We present a new efficient algorithm for computing value-at-risk and the value-at-risk gradient for portfolios of derivative securities. In particular, we discuss dimensional reduction of the model, perturbation theory, and applications to hedging of derivatives portfolios.

Abstract: The empirical analysis of asset pricing models abstracts from a variety of frictions and institutional factors. In this paper we present a generic statistical fully nonparametric procedure designed to benchmark the sampling frequency where microstructure noise contaminates the distributional characteristics of asset prices. The tests are based on rank transformations of normalized returns. The basic insights underlying the tests exploit (1) stochastic process theory of semimartingale processes, (2) results from continuous record asymptotic theory and (3) results from rank-based statistics. Stocks with different liquidity profiles and foreign exchange are used to benchmark the sampling frequency where microstructure noise ceases to affect asset price dynamics.

Monday, April 22, 2002
Mark Fisher, Atlanta Federal Reserve Bank (Author's web site: http://www.markfisher.net/~mefisher/)
An Analysis of the Doubling Strategy: The Countable Case (http://www.markfisher.net/~mefisher/papers/countable_doubling_strategy.pdf)

Abstract: We analyze the doubling strategy in static and dynamic settings with a countable state space. We apply the no-arbitrage and no-free-lunch definitions of Kreps (1981), which (in the dynamic setting) put the focus on the gain produced by a self-financing trading strategy, rather than on the strategy itself. By applying the Krepsian notions of no arbitrage andno free lunches to dynamic models, instead of the notions common in standard practice, we avoid the situation where there areno free lunches at the same time there are arbitrage opportunities. Depending on the topological space one adopts, the doubling strategy is either (i) not in the space of payouts (and hence not a free lunch), (ii) in the space and a free lunch, or (iii) in the space but not a free lunch. In the latter case, which requires `near risk-neutrality', the doubling strategy has a bubble component in the sense of Gilles and LeRoy (1997). We propose a new approach to specifying the space of marketedpayouts in dynamic security market models. We use the deflated gains process to reduce a dynamic model to equivalent static model. To this static model, we apply the analysis of Kreps (1981) and establish viability for a given topology. We use this topology to form the (dynamic) space of marketed payouts in such a way as to ensure that it is a subset of the closure of the (static) marketed subspace. Consequently, there can be no arbitrages or free lunches in the space of marketed payouts. Depending on the space and topology one adopts, the doubling strategy is either (i) not in the space of marketed securities or (ii) in the space but not a free lunch. In the latter case, the doubling strategy is a bubble in the sense of Gilles and LeRoy (1997). The entire analysis applies equally well to Ponzi schemes.

Friday, April 26, 2002
Craig Pirrong, Assoc. Prof. of Finance & Watson Family Chair of Commodity and Financial Risk Mgmt; Director, Risk Mgmt. Center, College of Business, Oklahoma State Univ.
Author's web site: http://www2.bus.okstate.edu/fin/pirrong
The Price of Power (http://www2.bus.okstate.edu/fin/pirrong/acpow4old.pdf)

Abstract: Pricing contingent claims on power presents numerous challenges due to (1) the nonlinearity of power price processes, and (2) the seasonal and intraday variations in prices. We propose and implement an equilibrium model in which the spot price of power is a function of two state variables: demand (load or temperature) and fuel price. In this model, any power derivative price must satisfy a PDE with boundary conditions that reflect capacity limits and the non-linear relation between load and the spot price of power. Moreover, since power is non-storable and load (or temperature) is not a traded asset, the power derivative price involves a market price of risk. Using inverse problem techniques and power forward prices from the PJM market, we solve for this market price of risk function. The market price of risk represents a substantial fraction (as much as one-third) of power forward prices for delivery during peak demand summer months. This is plausibly due to the extreme right skewness of power prices; this induces left skewness in the payoff to short forward positions, and a large risk premium is required to induce traders to sell power forwards. The existence of this huge risk premium suggests that the power derivatives market is not fully integrated with the broader financial markets.

Monday, April 29, 2002
Jianqing Fan, Professor of Statistics, UNC-Chapel Hill
Author's web site: http://www.stat.unc.edu/faculty/fan.html
Semiparametric Methods for Prediction of VaR

Abstract: Value at Risk is a fundamental tool for managing market risks. It measures the worst loss to be expected of a portfolio over a given time horizon under normal market conditions at a given confidence level. Calculation of VaR frequently involves estimating the volatility of return processes and quantiles of standardized returns. In this paper, several semiparametric techniques are introduced to estimate the volatilities of the market prices of a portfolio. In addition, both parametric and nonparametric techniques are proposed to estimate the quantiles of standardized return processes. The newly proposed techniques also have the flexibility to adapt automatically to the changes in the dynamics of market prices over time. Their statistical efficiencies are studied both theoretically and empirically. The combination of newly proposed techniques for estimating volatility and standardized quantiles yields several new techniques for evaluating multiple period VaR. The performance of the newly proposed VaR estimators is evaluated and compared with some of existing methods. Our simulation results and empirical studies endorse the newly proposed time-dependent semiparametric approach for estimating VaR.

Monday Sept. 17, 2001
Quentin Kerr, NCSU
Managing Energy Derivatives

Abstract: With the new and innovative development in energy markets, pricing energy derivatives becomes a hot topic in recent years. In the traditional Black-Scholes option pricing formulas, the underlying asset price is assumed to follow a geometric Brownian motion model with the constant incremental rate and volatility. However, the constant incremental rate and volatility will not be proper assumptions in energy markets, in particular, electricity markets. In my talk, I will propose a mean-reverting SDE with jumps to model the underlying spot price in energy markets using electricity market data as an example and also show how to price Futures/forward, options and other derivatives such as Caps, Floors, Collars and Swaptions based on the jump-diffusion mean-reverting model. Finally, I will discuss the possible application in the weather derivative market using the mean-reverting SDE model.

Monday, October 29, 2001
Claudio Albanese, Mathematics, University of Toronto
Author's website: http://www.math.utoronto.ca/albanese/
Integrability by Quadratures of Diffusion Equations

Abstract: The problem of classifying integrable heat equations was first addressed by Lie in 1887, who identifes the largest family solveable by group theoretical methods. Later analysis by Bluman and Ibragimov showed that all models in Lie's family can be reduced to the Wiener process and have quadratic diffusions. In 1977, a 2-parameter family of integrable diffusions that doesn't fit into Lie's classification scheme appeared in a finance paper by Cox and Ross. In this case, integrability is achieved by reduction to a Bessel process. In this talk, we describe two 7-parameter families of diffusion models which integrate in terms of hypergeometric functions, reduce to a Feller process and admit as particular cases all previously known solveable models in the literature. This class of solutions has a variety of applications to financial engineering and a rich mathematical structure. Key to the extension of the classical integrable cases is the construction of a symmetry grupoid preserving the quantum Poisson structure of a non-commutative manifold, in place of an ordinary Lie group symmetry relating the various solveable models.

Friday Feb. 2, 2001
Jianfeng Zhang, Purdue University
A Numerical Scheme for Backward Stochastic Differential Equations

Abstract: The theory of Backward Stochastic Differential Equations (BSDEs) has received strong attention in the past decade, and applications have been found in various fields, especially in Mathematical Finance. However, to date only few numerical methods have been developed for BSDEs, and all the existing results either require strong smoothness assumptions on coefficients or lack a good rate of convergence. In this talk we propose a new numerical scheme for a fairly large class of BSDEs whose terminal value (contingent claim) are allowed to depend on the history of a forward diffusion (price of the underlying assets). Our scheme approximates both components of the adapted solution (hedging price and hedging strategy) by step processes. We show that, under "minimum" regularity conditions on the coefficients, this scheme converges strongly in $L^2$, with rates $\sqrt{\log n\over n}$ for functional-type terminals, and $1\over\sqrt{n}$ for function-type terminals. Some other features of the scheme will also be discussed.

Friday Feb.16, 2001
Vladimir Pozdnyakov, The Wharton School, University of Pennsylvania
A Bound on LIBOR Futures Prices For HJM Yield Curve Models

Abstract: We prove that for a large class of widely used term structure models there is a simple theoretical upper bound for value of LIBOR futures prices. When this bound is compared to observed futures prices, one nevertheless finds that the theoretical bound is sometimes violated by market prices. The main consequence of this observation is that virtually all of the important fixed income models have theoretical implications that are sometimes at odds with market realities, at least when they are applied to futures markets.

Monday Feb.19, 2001
Jan Vecer, University of Michigan
Trading with Safety Net

Abstract: In this talk, we study the case when the trader wants to be insured for the case of the loss resulting from his trading strategy. This is a typical situation which financial institutions (banks, investment funds, pension funds or hedging funds) have to face: they are implementing some trading strategy which include active trading in the stock and the money market. Since these institutions are in some extent responsible for the outcome of their trading, it is desirable that they would have only limited liability when the loss occurs. They can buy an option contract which would let them to keep the profits, but they would be forgiven the loss. Using the probability techniques combined with the techniques of the stochastic optimal control (Hamilton-Jacobi-Bellman equations), we can compute price of this type of product. Moreover, we can determine what is the optimal trading strategy for the holder of the contract and the hedging strategy for the seller of the contract. In many cases, the price of this type of insurance is surprisingly cheap.

Friday Feb.23, 2001
Reade Ryan, UCLA
Optimal Exit from a Risky Project with Noisy Returns

Abstract: We consider the problem of when to exit an investment project when the project's expected profit rate $\mu$ never changes over time but $\mu$ is not directly observable. Specifically, $\mu$ takes the value $\mu_H > 0$ when in the high state and $\mu_L < 0$ when in the low state, and the initial probability $p_0$ that the project is in the high state is known. The cumulative profit up to time $t$ is $X_t$ where $\{X_t: t\geq 0\}$ is a Brownian motion with drift $\mu$ and variance $\sigma^2$. The decision-maker's problem is to select a stopping time $\tau$ which determines when to exit the project: her goal is to maximize the expected discounted profit up to time $\tau$. Using the theory of stochastic differential equations, we show that it is optimal to exit only when the posterior probability $P_t$ of being in the high state falls below a critical number $p^*$, which is an explicit function of the problem parameters $\mu_H$, $\mu_L$, $\sigma^2$, and the discount rate $\alpha$. The expected discounted return up to time $\tau$ can be strictly positive even if the expected profit rate at time 0 is negative because the ability to exit after observing the profit process for a period of time imparts an option value to the project. The most surprising comparative statics result is that the expected discounted profit increases as $\mu_L$ decreases provided that $\mu_L$ is sufficiently negative.

Monday, March 5, 2001
Knut Solna, Dept. of Mathematics, UC Irvine
Modeling and Estimation of S&P 500 Data

Abstract: We consider high frequency S&P 500 historical price data and analyze these with a view toward identifying important time scales and systematic features. In particular we estimate the rate of mean-reversion of volatility. The data shows a periodic behavior that depends on both maturity dates and also the trading hour. We examine briefly the implications of this for modeling and option pricing.

Monday, January 24, 2000
Jean-Pierre Fouque, Dept. of Mathematics, NCSU (joint with George Papanicolaou (Stanford) and Ronnie Sircar (Michigan))
Correction to the Heat Equation Motivated by Stochastic Volatility Model

Abstract: The heat equation in Physics or the Black-Scholes equation in Finance are partial differential equations of parabolic type. They involve a parameter, the diffusion constant or the volatility, which one might want to perturb in order to describe more general situations. In the context of Finance it is natural to do so at the level of the stochastic evolution of the underlying asset by introducing randomness in the volatility. In the context of Physics this means that the ``virtual'' Brownian particle associated to heat diffusion is moving in a random medium. At this stage the quantities of interest described by these PDE's strongly depend on how the extra randomness is modeled. Another very natural ingredient coming from the financial context is the fact that the stochastic volatility is ``running'' on a fast time-scale. This separation of scales leads to a singular perturbation theory of the initial PDE. The leading order term is the usual solution of the corresponding constant coefficient PDE and the first correction is explicitly computed. It involves third derivatives and is universal in the sense that it is essentially model independent. Applications in finance will be discussed and perturbations of the heat equation are proposed.

Friday, February 25, 2000
Paul Fackler, Dept. of Agricultural and Resource Economics, NCSU
Specification Issues for Multivariate Affine Diffusion Models

Abstract: Affine diffusion models provide a tractable framework for pricing many financial assets, including bonds, futures and European options. Affine diffusions have affine instantaneous mean and variance but not all affine model are admissible. Building on recent work by Duffie and Kan and Dai and Singleton, necessary and sufficient characterizations of admissible affine diffusions are developed and simple operational methods for testing admissibility are presented. In addition, the properties of classes of affine diffusions that are closed under affine transformations are discussed along with implications for model identification.

Friday, March 24, 2000
Marco Avellaneda, Robert Buff, Craig Friedman, Nicolas Grandchamp, Lukasz Kruk, and Joshua Newman
Weighted Monte Carlo: A New Technique for Calibrating AssetPricing Models

Abstract: A general approach for calibrating Monte Carlo models to the market prices of benchmark securities is presented. Starting from a given model for market dynamics (price diffusion, rate diffusion, etc.), the algorithm corrects for price misspecifications and finitesample effects in the simulation by assigning "probability weights'' to the simulated paths. The choice of weights is done by minimizing the KullbackLeibler relative entropy distance of the posterior measure to the empirical measure. The resulting ensemble prices the given set of benchmark instruments exactly or in the sense of least squares. We discuss pricing and hedging in the context of these weighted Monte Carlo models. A significant reduction of variance is demonstrated theoretically as well as numerically. Concrete applications to the calibration of stochastic volatility models and termstructure models with up to forty benchmark instruments are pre sented. The construction of implied volatility surfaces and forwardrate curves and the pricing and hedging of exotic options are investigated through several examples.