Gerard Brunick (UT Austin)

Constructing Markov models for barrier options

In this talk we will present an approach for modeling barrier options which is analogous to the "local volatility" approach for modeling European options.  In place of the forward equation, we employ a recent weak existence result which can handle the path-dependence in the option payoff.  We then use the martingale problem of Stroock and Varadhan to conclude that the resulting model is Markov.

Coskun Cetin (Cal. State Sacramento).

Optimal portfolio and stopping time problem for the acquision of a partially hedgeable house

Abstract: We consider the problem of the optimal time to purchase a house by a risk-averse investor whose objective is to maximize the expected utility from wealth at some fixed horizon. The house purchase is financially attractive (due to the tax advantages, for example), which provides an incentive to buy as soon as possible; however its value is only partially correlated with financial markets and, therefore, house price risk cannot be perfectly hedged, which provides an incentive to delay purchase. We characterize the solution to the problem in the case of CARA utility and provide some examples to study the trade-off between the two conflicting incentives.

Mark Davis (Imperial College)

Counterparty risk via Bessel bridge

Counterparty risk concerns evaluating potential losses on a financial trade conditioned on one's counterparty defaulting at a specific time in the future. Often, default can be modelled as the hitting time of a barrier by Brownian motion. Generally this barrier is not flat, but we show how to reformulate the problem so that it is flat. Then conditioning on default at a specific time is equivalent to replacing the BM by a BES(3) bridge. We suppose that the factors determining the value of the trade are further BMs, correlated with the first. Then we can calculate the conditional value of the trade in terms of SDEs whose 'inputs' are the Bessel bridge and additional independent BMs. An example relating to interest rate swaps will be presented

Noureddine El Karoui (UC Berkeley)

High-dimensionality effects in the Markowitz problem and other quadratic programs: risk underestimation

It is often the case in statistics and various branches of applied mathematics that one wishes to solve optimization problems involving parameters that are estimated from data. It is therefore natural to try to characterize the relationship between the solution of the optimization problem involving estimated parameters (the sample version) and the solution we would get if we knew the actual value of the parameters (the population version). An example of particular interest is the classical Markowitz portfolio optimization problem - an instance of a quadratic program with linear equality constraints. I will discuss some of these questions in the large dimensional setting (i.e portfolios with many assets), when the optimization is performed over vectors of size p (i.e the number of assets), and p is comparable to n, the number of observations we use to get our estimates. From a practical standpoint, this asymptotic setting (p and n go to infinity while p/n does not go to zero) tries to capture the difficulties arising from the fact that we have limited amount of data to estimate the parameters appearing in the problem. I will present results showing that the high-dimensionality of the data (i.e p/n not small) implies significant and quantifiable risk underestimation. I will also consider the question of robustness of the conclusions to various distributional assumptions, focusing particularly on understanding the sensitivity of the results to heavy-tails. Finally, I will discuss the impact of working with dependent observations. Though random matrix theory plays a key role in the analysis, no prior knowledge of it will be assumed.

Alex Eydeland (Morgan Stanley)

An overview of the Quantitative Finance program and the role of a Commodity Strategist at Morgan Stanley

In this presentation we introduce a new structure of analytical and technological environment that is currently being implemented at Morgan Stanley. In particular, the presentation will be focused on the activities of the strategists and modelers within the Quantitative Finance program at Morgan Stanley. As an illustration, the main discussion will be concentrated on the work of commodity strategists, the challenges they face, typical problems they solve, and the modeling tools they use.

David German (Claremont McKenna College)

Illiquid Markets and Demand-driven Prices

We study a financial model with a non-trivial price impact effect. In this model we consider the interaction of a large investor trading in an illiquid security, and a market maker who is quoting prices for this security. We assume that the market maker quotes the prices such that by taking the other side of the investor's demand, the market maker will arrive at maturity with maximal expected wealth. Within this model we concentrate on two major issues: evaluation of contingent claims, and hedging.

Kay Giesecke (Stanford)

Exact Simulation of Point Processes With Stochastic Intensities

Point processes with stochastic intensities are ubiquitous in finance, insurance and other areas. They can be simulated from standard Poisson arrivals by time-scaling with the cumulative intensity, whose path is typically generated with a discretization method. However, discretization introduces bias into the simulation results. This paper proposes a method for the exact simulation of point processes with stochastic intensities. The method is based on a projection argument and leads to unbiased estimators. It is illustrated for a point process whose intensity follows an affine jump-diffusion process.

Tomoyuki Ichiba (UCSB)

Hybrid Atlas Models

We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stability of the capital distribution. Ergodic properties and rankings of processes are examined with reference to the theory of reflected Brownian motions in polyhedral domains. In the context of such models, we discuss properties of various investment strategies, including the so-called growth-optimal and universal portfolios. This is a joint work with R. Fernholz, A. Banner, V. Papathanakos and I. Karatzas (INTECH and Columbia University).

Burhan Izgi (Mimar Sinan U., Visiting Cal. State Sacramento)

Monte Carlo Simulations for Option Pricing In Incomplete Markets

We consider an incomplete financial market setting with one risky stock and one locally predictable bond based on CIR interest rate model. We compute the fair price of the European options using the Monte Carlo simulations and present some examples. We also provide a sensitivity analysis of some of the model parameters.

Matthew Lorig (UCSB)

A Fast Mean-Reverting Correction to Heston's Stochastic Volatility Model

We propose a multi-scale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular pertubative expansion is then used to obtain an approximation for European option prices. Our results are illustrated numerically and with option data.

Joint work with J.-P. Fouque

Rafael Mendoza-Arriaga (UT Austin)

Modeling Correlated Defaults

We present a novel framework for modeling correlated defaults where it is possible to capture the so-called ?contagion effect??. Time changing multi-parameter Markov processes with multivariate subordinators leads to jump-diffusion processes that are correlated through their jump measures. When unpredictable shocks arrive, the default intensity of multiple firms will shift simultaneously, which can trigger multiple defaults. The modeling framework presented can be used for modeling both Systemic and Contagion default risk in an analytically tractable way.

Soumik Pal (University of Washington, Seattle)

The Importance of being strict

Local martingales which are not martingales (hence strict) have always had a marginal presence in Probability theory and Mathematical Finance. They do not exist in discrete Probability, and thus they are treated as technical hiccups, only to be smoothed by localization. However, several new theories in recent times have stressed on the importance of being strict: whether it is digging for relative arbitrages in Stochastic Portfolio Theory, or a foundation of the theory of financial bubbles. We intend to show that, far from being an anomaly, strict local martingales capture a fundamental probabilistic phenomenon. They appear whenever we apply a change of measure (on a filtration) which is dominated by, but not equivalent to, the existing measure. Thus they appear in such diverse areas as diffusions conditioned to have a certain boundary behavior, size-bias sampling, and non-colliding particles used in Random Matrix Theory. Is there a way to avoid being strict under a change of measure ? The answer is yes, if you do not hit zero before the martingale, representing the Radon-Nikodym derivative, hits zero.

Based on joint work with Philip Protter.

Sergio Pulido (Cornell)

Bubbles and Futures contracts in markets with short-selling constraints

The current financial crisis, product of the burst of the alleged real estate bubble, has brought back the attention of the financial and academic community to the study of the causes and implications of asset price bubbles.  In recent works Jarrow, Protter and Shimbo (2006, 2008) and Cox and Hobson (2005) developed an arbitrage-free pricing theory for bubbles in complete and incomplete markets. These papers approach the subject by using the insights and tools of mathematical finance, rather than equilibrium arguments where substantial structure, such as investor optimality and market clearing mechanisms, has to be imposed. In their framework, bubbles occur because the market's valuation measure is a local martingale measure which is not a martingale measure and hence the discounted asset's price is above the expectation of its future cash-flows. The existence of bubbles does not contradict the condition of no free lunch with vanishing risk (NFLVR), because *short-selling onstraints*, given by an admissibility condition on the set of trading strategies, do not allow investors to make a riskless profit from the overpriced securities. In an attempt to combat sharp and extreme declines in certain stock prices related to the bursting of these bubbles, both the British and the American government imposed temporary bans on the short selling of certain categories of stocks. This affects the liquidity of stocks, and has other less obvious effects.The aim of this talk is to explain how the previous work extends to models where some assets cannot be sold short whatsoever and explore how financial instruments such as *futures contracts* behave in such models.

Johannes Ruf  (Columbia)

Optimal trading strategies under arbitrage

Explicit formulas for optimal trading strategies in terms of minimal required initial capital are derived to replicate a given terminal wealth in a continuous-time Markovian context. To achieve this goal this talk does not assume the existence of an equivalent local martingale measure. Instead a new measure is constructed under which the dynamics of the stock price processes simplify. It is shown that delta hedging does not depend on the no free lunch with vanishing risk'' assumption. However, in the case of arbitrage the problem of finding an optimal strategy is directly linked to the non-uniqueness of the partial differential equation corresponding to the Black-Scholes equation. The recently often discussed phenomenon of bubbles'' is a special case of the setting in this talk.

Rituparna Sen (UC Davis)

Hedging options in the incomplete market with stochastic volatility

We show that it is possible to avoid the discrepancies of continuous path models for stock prices and still be able to hedge options if one models the stock price process as a birth and death process. One needs the stock and another market traded derivative to hedge an option in this setting. However, unlike in continuous models, number of extra traded derivatives required for hedging does not increase when the intensity process is stochastic. We obtain parameter estimates using Generalized Method of Moments and describe the Monte Carlo algorithm to obtain option prices. We show that one needs to use ltering equations for inference in the stochastic intensity setting. We present real data applications to study the performance of our modeling and estimation techniques.

Winslow Strong (UCSB)

Regulation, Diversity, and Arbitrage

Motivated by R. Fernholz's result on the existence of relative arbitrage with respect to the market portfolio in diverse markets, we show that when diversity is maintained by a certain kind of antitrust regulation that such markets may be free of relative arbitrage. The type of regulation considered here redistributes the wealth of the companies in the economy when their relative capitalizations enter a forbidden region. Sufficient conditions under which such a regulated model may be free of relative arbitrage are presented. As an example, using the volatility-stabilized market as a premodel and applying this regulation procedure we create a regulated model which is both diverse and free of relative arbitrage.

Option Pricing  Under a Stressed-Beta Model

Empirical studies have concluded that stochastic volatility is an important component of option prices. We introduce a regime-switching mechanism into a continuous-time Capital Asset Pricing Model (CAPM) which naturally induces stochastic volatility in the asset price. Under this {\it Stressed-Beta model}, the mechanism is relatively simple: the slope coefficient - which measures asset excess returns relative to market excess returns - switches between two values, depending on the market being above or below a given level.  After specifying the model, we use it to price  European options on the asset.  Interestingly, these option prices are given explicitly as integrals with respect to known densities. We find that the model is able to produce a volatility skew, which is a prominent feature in  option markets. This opens the possibility of forward-looking calibration  of the slope coefficients, using options data, as illustrated in the paper.

Joint work with J.-P. Fouque

Antoine Toussaint (Stanford)

Simulation by Approximate Thinning of Self-Exciting Point Process driven by Diffusion Processes

Guoliang Wu (UT Austin)

Smooth fit principle for impulse control problems

Impulse control problems have wide applications in economics and engineering. Regularity study helps finding closed- form solutions and provides useful insight for numerical approxi- mation when the former is impossible. In this talk we will derive the W^{2,p} -regularity (in particular, C^1 smooth-fit principle) of the value functions for impulse controls of multidimensional diffusions and jump diffusions, using a viscosity solution approach and other
PDE tools.

Lihong Xia (UNC Charlotte)

Mingxin Xu (UNC Charlotte)

Infinite Horizon Optimal Search Problem with Hiring and Firing Options

Abstract: As in the classic `Secretary Problem': the candidates arrive sequentially.  In this paper, they are represented with i.i.d. Ito diffusion processes.  Two interwoven sequences of optimal stopping times are decided
which signify the hiring and firing times of each candidate.  The goal is to choose the stopping times to maximize expected sum of benefit and cost when the time horizon is infinite.  The optimality conditions in terms of Verification Theorem, Least Superharmonic Majorant, and Variational Inequalities are given.  The solution for the simple Brownian case with linear cost/benefit functions is calculated which results in a new two-one threshold strategy giving rise to the corresponding decisions of hiring/letting go-firing.

Thaleia Zariphopoulou (UT Austin)

Stochastic pdes and portfolio choice

In this talk, I will present an SPDE that models the performance of investment strategies in markets with Ito price dynamics. I will also discuss the new concept of performance volatility as well as examples for specific volatility classes.

Meeting schedule

Statistics & Applied Probability
University of California
Santa Barbara, California 93106-3110
(805) 893-2129