Seminar

Inaugural day: Monday October 16th, 2006

2006-07 Regents' Lecture by Dr. Bruno Dupire(Bloomberg and NYU)

Past 2006-2007 seminars
Past 2005-2006 seminars

2007-2008 Seminars

Upcoming

Monday April 21, South Hall 5607F, 3:15PM, refreshments served at 3:00PM
Sebastian Jaimungal (University of Toronto)

Hitting Time Problems with Applications to Finance and Insurance

The distribution of the first hitting time of a Brownian motion to a linear boundary is well known. However, if the boundary is no longer linear, this distribution is not in general identifiable. Nonetheless, the boundary and distribution satisfy a variety of beautiful integral equations due to Peskir. In this talk, I will discuss how to generalize those equations and lead to an interesting partial solution to the inverse problem: “Given a distribution of hitting times, what is the corresponding boundary?” By randomizing the starting point of the Brownian motion, I will show how a kernel estimator of the distribution with gamma kernels can be exactly replicated.
Armed with these tools, there are two natural applications: one to finance and one to insurance. In the financial context, the Brownian motion may drive the value of a firm and through a structural modeling approach I will show how CDS spread curves can be matched. In the insurance context, suppose an individual’s health reduces by one unit per annum with fluctuations induced by a Brownian motion and once their health hits zero the individual dies. I will show how life-table data can be nicely explained by this model and illustrate how to perturb the distribution for pricing purposes.
This is joint work with Alex Kreinin and Angelo Valov

Monday June 9, South Hall 5607F, 3:15PM, refreshments served at 3:00PM
Bernt Oksendal (Oslo, Norway)

An Introduction to Malliavin Calculus for Lévy Processes and Applications to Finance

The purpose of this lecture is to give a non-technical, yet rigorous introduction to Malliavin calculus for Lévy processes and its applications to finance. The lecture consists of two parts:
Part 1 deals with the Brownian motion case. We first use the Wiener-Itô chaos expansion theorem to define the Mallavin derivative in this context and then study some of its fundamental properties, including the chain rule and the duality property (integration by parts). Then we apply it to finance. Examples of applications are
(i) the hedging formula in complete markets provided by the Clark-Ocone theorem,
(ii) "parameter sensitivity results", e.g. a numerically tractable computation of the "delta-hedge" and other "greeks" in finance.
Part 2 deals with the general Lévy process case. To some extent a similar presentation of the Malliavin derivative can be given here as in Part 1, but there are also basic differences, for example regarding the chain rule. Examples of applications to finance are
(i) optimal hedging in incomplete markets (based on the Clark-Ocone formula for Lévy processes),
(ii) optimal consumption and portfolio with partial information in a market driven by Lévy processes.
The presentation is mainly based on the forthcoming book
G. Di Nunno, B. Øksendal and F. Proske:
Malliavin Calculus for Lévy Processes and Applications to Finance. Springer 2008 (to appear).

PAST 2007-2008 Seminars

WEDNESDAY October 3, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Stephane Villeneuve (Toulouse, France, visiting PSTAT)

Title: Optimal dividend policy and growth option

We analyse the interaction between dividend policy and investment decision in a growth opportunity of a liquidity constrained firm. This leads us to study a mixed singular control/optimal stopping problem for a diffusion that we solve quasi-explicitly establishing connection with an optimal stopping problem. We characterize situations where it is optimal to postpone dividend distribution in order to invest at a subsequent date in the growth opportunity. We show that uncertainty and liquidity shocks have ambiguous effect on the investment decision.

MONDAY October 8, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Marek Rutkowski (School of Mathematics and Statistics, University of New South Wales)

PDE Approach to Credit Derivatives Paper1 Paper2

WEDNESDAY October 10, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Helgi Tomasson (University of Iceland, visiting PSTAT)

Some Computational Aspects for Inference on Diffusion Processes

The theory of diffusion processes is fundamental for modern mathematical-finance. Real data are assumed to be observations of a continuous-time process at discrete time-points. The statistical toolbox for financial data is briefly reviewed. A computer program, written in R, for approximation of the likelihood function for some simple processes is shown. The approximation is based on a Taylor-expansion of the Kolmogorov-forward equation in the spirit of Ait-Sahalia(1999, 2002). Properties of maximum-likelihood estimators are illustrated by simulation. Some aspects of applying the approximation to Bayesian inference and statistical-surveillance (change-point-detection) are discussed

MONDAY October 15, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Bjorn Flesaker (Bloomberg, NY)

Title: Replication based pricing of default contingent claims slide

The standard market model for single name credit default swap pricing is usually represented as a pure reduced form model where default occurs as the first jump of a Poisson process with deterministic risk neutral intensity. We provide conditions under which a static portfolio of standard credit default swaps along with a money market account balance can be used to replicate a broad class of default contingent claims and demonstrate that the resulting no-arbitrage values are consistent with the standard market model, regardless of the dynamics of the default generating process. The replication based pricing operator, as well as the associated survival contingent money market account balance and the replicating CDS portfolio positions, are fully characterized in terms of second order ordinary differential equations (for the continuous maturity limit) and difference equations (for discrete holdings), and examples of their explicit solutions are given.
This is based on joint work with Peter Carr.

WEDNESDAY October 17, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Hyekyung Min (postdoc, PSTAT)

Title: A Stochastic Control Model of Optimal Dividend and Capital Financing

The stochastic control model, introduced by Peura and Keppo (2006), is considered for valuing a firm whose capital evolves according to Brownian motion with a drift. The firm controls the flow of capitals not only by paying out the dividends but also by raising the capital in the presence of fixed cost (K) and delay (D). A solution to this control problem is obtained by solving a system of quasi-variational inequalities. It is shown that a unique solution exists for all values of K >= 0 and D > 0. The asymptotic behavior of the optimal dividend and capital issue barriers, and the ruin probability and the expected lifetime of the firm following the optimal policy will be discussed.

MONDAY October 29,South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Eric Hillebrand (Economics, LSU)

Pricing an Option on Revenue from an Innovation: An Application to Movie Box Office Revenue

We develop a model for valuing revenue streams from innovations. The stochastic properties of revenue from innovations create a more diffcult environment in which to value options than when the underlying is a security. There is no initial revenue and cumulative revenue cannot decrease. Revenues from innovations are character- ized by different lives and different rates of the resolution of uncertainty. A common deterministic model for predicting revenue from an innovation is due to Bass (1969). We imbed the Bass model in a gamma process, resulting in a stochastic process with moments proportional to the mean of the Bass model. To illustrate this model we choose the valuation of options on movie box o±ce revenue. These options enable film distributors to manage the risk of a movie, and they offer diversification opportunities for investors. We develop the econometric methodology for ex-ante parameter estimation and a Bayesian updating scheme using Markov Chain Monte Carlo simulation as data after release become available. Call prices obtained using MLE parameter estimates from the full data set closely approximate the average discounted value of ex-post call payouts that would have occurred at option maturity.

Monday, November 26, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Martin Forde (PSTAT, UCSB)

Small time and tail asymptotics for stochastic volatility models

We show how to construct a volatility-of-variance function for an uncorrelated stochastic volatility model, so as to be consistent with an observed symmetric small-maturity smile, by solving an Abel Volterra integral equation. We show how to adapt the methodolgy for implied volatility skews. We also discuss Lewis's small-time asymptotics for the derivatives of the implied volatility At the-Money, and the large-x asymptotics for his CEV(p)- volatility model. We also discuss tail asymptotics for such models, using Olver's asymptotics.

Monday, Dec 10, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM
Huyên PHAM, Université Paris 7, Denis Diderot.

Hedging and pricing with execution delay.

We consider impulse control problems in finite horizon for diffusions with decision lag and execution delay. The new feature is that our general framework deals with the important case when several consecutive orders may be decided before the effective execution of the first one.
This is motivated by financial applications in the trading of illiquid assets such as hedge funds.
We show that the value functions for such control problems satisfy a suitable version of dynamic programming principle in finite dimension, which takes into account the past dependence of state process through the pending orders. The corresponding Bellman partial differential equations (PDE) system is derived, and exhibit some peculiarities on the coupled equations, domains and boundary conditions. We prove a unique characterization of the value functions to this nonstandard PDE system by means of viscosity solutions. We then provide an algorithm to find the value functions and the optimal control. This implementable algorithm involves backward and forward iterations on the domains and the value functions, which appear in turn
as original arguments in the proofs for the boundary conditions and uniqueness results. Finally, we give several numerical experiments illustrating the impact of execution delay on trading strategies and on option pricing.

Friday, January 18th, South Hall 5607F, 3:15PM, refreshments served at 3:00PM
Jan Vecer, Department of Statistics, Columbia University

Tradeable Measures of Risk

The main idea of this talk is to introduce Tradeable Measures of Risk as an objective and model independent way of
measuring risk. The present methods of risk measurement, such as the standard Value-at-Risk supported by Basel II, are based on subjective assumptions of future returns. In order to achieve an objective measurement of risk, we introduce a concept of Realized Risk which we define as a directly observable function of realized returns. Predictive assessment of the future risk is given by Tradeable Measure of Risk - the price of a contract which pays its holder the Realized Risk for a certain period. Our definition of the Realized Risk payoff includes a Weighted Average of Ordered Returns, with the following special cases: the worst return, the empirical Value-at-Risk, and the empirical mean shortfall. When Tradeable Measures of Risk of this type are priced and quoted by the market (even over-the-counter, or traded internally within a financial institution), one does not need a model to calculate values of a risk measure since it will be observed directly from the market. We use an option pricing approach to obtain dynamic pricing formulas for these contracts, where we make an assumption about the distribution of the returns. We also discuss the connection between Tradeable Measures of Risk and the axiomatic definition of Coherent Measures of Risk, and provide some convergence results.

Tuesday, January 22nd, South Hall 5607F, 10:00AM, refreshments served at 9:45am.
Mike Ludkovski

Optimal Stopping and Optimal Switching for Hidden Markov Models

We study optimal stopping and optimal switching problems for hidden Markov chains with Poissonian information structures. In our model, the controller maximizes expected rewards that depend on an unobserved Markovian environment with information collected through a (compound) Poisson observation process. Examples of such systems arise in investment timing, reliability theory, sequential tracking, and economic policy making. We solve the problem by performing Bayesian updates of the posterior likelihoods of the unobservable and studying the resulting optimization problem for a piecewise-deterministic process. We then prove the dynamic programming principle and explicitly characterize an optimal strategy. We also provide an efficient numerical scheme and illustrate our results with several computational examples.

This is based on joint work with Semih Sezer and Erhan Bayraktar (U of M).

Monday, January 28th, South Hall 5607F, 3:15PM, refreshments served at 3:00PM
Tim Siu-Tang Leung (ORFE, Princeton University)

Utility-based Valuation of Employee Stock Options

Employee stock options (ESOs) have become an integral component of compensation in the U.S. Financial regulations now require firms to expense these options in their accounting statements. ESOs have a number of complicated characteristics which distinguish them from standard market-traded American call options. Their value is much less due to the suboptimal exercising strategies of the holders, which arise from risk aversion, hedging constraints, and job termination risk. We analyze a utility-based valuation procedure that accounts for the combined effect of all of these factors, along with multiple exercising rights and vesting periods. This leads to the numerical study of a system of nonlinear free-boundary problems of reaction-diffusion type. In addition, we examine the holder's hedging strategies that involve a combination of dynamic trading of correlated assets and static positions in market-traded put options. We find that static hedges induce the ESO holder to delay exercises, and lead to higher ESO costs.

Monday February 11, South Hall 5607F, 3:15PM, refreshments served at 3:00PM
Alok Khare (PSTAT, UCSB)

Present Value Relation and the Volatility Puzzle: A Reexamination

In the seventies and eighties, martingale model for asset prices was tested using variance bounds. The publications by LeRoy and Porter, and Shiller indicated that prices violated the variance bounds implied by the model. The econometric methods of these early contributions were criticized on the grounds that tests produced biased results. Subsequent research using improved econometric methods produced similar findings. The tests are generally built on the premise that value of a firm is discounted value of dividends paid to shareholders. Miller and Modigliani show that this is true only when firms neither issue nor repurchase shares. In general, the value of a firm is the discounted present value of future net cash flows, which consists of dividends and repurchases net of share issues. We employ the variance bound test derived by West on data using this payout measure. Our result is that prices are not excessively volatile when compared to subsequent total net cash flows to shareholders. The data consist of all firms in the DJIA from 1983 to 2005.

The first passage problem for jump diffusions and its implications for credit risk

We begin by reviewing some classic problems in finance that boil down to a problem of first passage of an underlying stochastic process. Although jump diffusions are widely used for modeling financial time series, they have been only slowly adopted in
some areas, like credit risk, perhaps because of the intractable nature of their first passage problem. In this talk I show how this difficulty can be circumvented. Thus freed, we are able to investigate some consequences of adding jumps to structural credit models
for one firm. As an example of what can be done, a class of credit-equity hybrid models is proposed. At the end of the talk, we will move to multivariate credit models and briefly consider the implications of introducing dynamic default correlations through time change.

Monday February 25, South Hall 5607F, 3:15PM, refreshments served at 3:00PM
Archil Gulisashvili (Ohio University)

Distribution densities in stochastic volatility models

Stochastic volatility models were introduced in 1980s-1990s. In these models, the volatility of a stock is described by a stochastic process. For instance, in the Hull-White model, the volatility is a geometric Brownian motion, in the Stein-Stein model, the absolute value of an Ornstein-Uhlenbeck process plays the role of the volatility of a stock, and in the Heston model, the volatility is a square root process. The main object of our interest in this work is the distribution density of the stock price in a stochastic volatility model. We find explicit formulas for leading terms in asymptotic expansions of such densities and give error estimates. Using these results, we compare ``fat tails" of stock price distributions in various stochastic volatility models. We also obtain a sharp asymptotic formula for the law of a mean-square average of the volatility process. As an application of our methods, the asymptotic behavior of the implied volatility is characterized, and a sharp asymptotic formula for the price of an Asian option is obtained.
This is a joint work with E. M. Stein (Princeton University).

Monday March 3, South Hall 5607F, 3:15PM, refreshments served at 3:00PM
Adam Tashman (Ivy Asset Management, New York, NY)

Modeling Risk in Arbitrage Strategies Using Finite Mixtures

Arbitrage strategies produce stable, modest returns punctuated by intervals of dramatically poor performance. The weakness stems from an oversight in modeling: seemingly independent bets infrequently become highly correlated to market variables. Thus, the risk in arbitrage strategies is systematically underestimated, and hedging is not properly implemented. This paper illustrates the fitting and application of a mixture model to a series of hedge fund index returns, for the purpose of more effectively hedging downside risk. The model captures the regime-switching nature of the process in a general setting, free from the assumption of a linear relationship between explanatory and response variables. A logistic regression function is used to predict the acting regime, and linear regression functions relate explanatory variables to the expected hedge fund return in each regime. The covariates considered are stock market returns, volatility of the stock market, the slope of the US swap curve, and credit spreads. The dependent variable under investigation is the HFRI Merger Arbitrage Index. The model is applied in a novel hedging strategy, termed mixture hedging. The strategy is back tested over the period 1990-2005, and its performance is compared against the prevalent beta-neutral hedging strategy. The merger arbitrage index exhibited strong evidence of a regime-switching process, and the proposed model offered an improved fit relative to standard regression techniques. Mixture hedging was more effective at reducing downside risk than beta-neutral hedging. Maximum drawdown was 5.50% for the mixture strategy, versus 5.98% for beta-neutral hedging and 6.46% for an unhedged portfolio. The improvement will be more pronounced if the portfolio is levered.

Monday, March 10, South Hall 5607F, 3:15PM, refreshments served at 3:00PM
Mike Landrigan (Rimrock Capital, San Juan Capistrano, CA).

Portfolio Optimization for Valuing Collateralized Mortgage Obligations

Collateralized Mortgage Obligations (CMO) can have a high degree of variability in cash flows and a complex embedded optionality structure. Because of this, it is generally recognized that a yield to maturity or static spread calculation is not a suitable valuation methodology. To assess the value of a CMO it is common to rely on benchmark-calibrated Monte Carlo simulation of interest rates and projection of cash flows along the various interest rate paths. The often quoted measure of value derived from such simulation is the Option-Adjusted Spread (OAS). We find that OAS has serious flaws and introduce a variance reducing portfolio optimization technique to value CMO. The optimization is applied to liquid, benchmark interest-rate derivatives. In our talk we will introduce CMOs and OAS, present the general framework of our analysis, and discuss results from numerical experiments.

Monday April 7, South Hall 5607F, 3:15PM, refreshments served at 3:00PM
Eckhard Platen (University of Technology Sydney, Australia)

The Law of the Minimal Price

The paper introduces a general market setting under which the Law of One Price does no longer hold. Instead the Law of the Minimal Price will be derived, which for a range of contingent claims provides lower prices than suggested under the currently prevailing approach. This new law only requires the existence of the numeraire portfolio, which turns out to be the portfolio that maximizes expected logarithmic utility. In several ways the numeraire portfolio cannot be outperformed by any nonnegative portfolio. The new Law of the Minimal Price leads directly to the real world pricing formula, which uses the numeraire portfolio as numeraire and the real world probability measure as pricing measure when computing conditional expectations. The pricing and hedging of extreme maturity bonds illustrates that the price of a zero coupon bond, when obtained under the Law of the Minimal Price, can be far less expensive than when calculated under the risk neutral approach.

PAST 2006-2007 Seminars

Monday September 25: South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Jose Figueroa-Lopez (UCSB, PSTAT).

Estimation methods for Levy based models of asset prices (I)

Stock prices driven by Levy processes or other related jump processes have received a great deal of attention in recent years. The scope of these models goes from simple exponential Levy models to stochastic differential equation with Poisson jumps both on the volatility and on the returns. Several calibration methods have been proposed in the literature to deal with these models. In this talk we will review some classical methods, such as approximated maximum likelihood using FFT, and some recent methods using asymptotic limits of ``power'' variations. In particular, I will discuss in some detail nonparametric procedures.

Monday October 9: South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Jose Figueroa-Lopez (UCSB, PSTAT).

Estimation methods for Levy based models of asset prices (II)

Stock prices driven by Levy processes or other related jump processes have received a great deal of attention in recent years. The scope of these models goes from simple exponential Levy models to stochastic differential equation with Poisson jumps both on the volatility and on the returns. Several calibration methods have been proposed in the literature to deal with these models. In this talk we will review some classical methods, such as approximated maximum likelihood using FFT, and some recent methods using asymptotic limits of ``power'' variations. In particular, I will discuss in some detail nonparametric procedures.

Wednesday, October 11, 2006, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM

Stephen LeRoy, UCSB Economics

"Compactifying the Payoff Space: Applications in Financial Economics."

Finance models with a finite number of states and dates have a number of properties that may or may not extend to their infinite counterparts. Whether they do depends how equilibrium is defined. Under sequential equilibrium many properties do not extend to infinite settings. We propose an alternative equilibrium concept that is closer to classical finite Walrasian equilibrium. This involves appending a date called \uffff~H~^ to the finite dates and defining a topology such that the payoff index set so expanded is compact. Payoffs of infinite portfolio strategies are defined as limits of payoffs of finite portfolio strategies. This setup allows a simplified mathematical treatment of a number of topics that are unwieldy when modeled in a setting where the payoff index set is not compact. Topics discussed include Ponzi schemes, payoff bubbles and the doubling strategy.

Monday October 23, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM
Martin Forde (UCSB, PSTAT)

"Small-time and tail asymptotics for stochastic volatility models"

We discuss tail/large strike asymptotics for a Dupire-type local volatility model, and for a diffusion process subordinated to an independent stochastic clock. We also discuss small-time asymptotics for these classes of models, with applications to volatility derivatives, and the general p-stochastic volatilty model which nests the well known Heston and SABR parametrizations.

Monday October 30, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM
Jianfeng Zhang

"Continuous-Time Principal-Agent Problems with Moral Hazard and/or Adverse Selection"

Abstract: We study the continuous time principal-agent problems with moral hazard and/or adverse selection. We will discuss the first-best, second-best, and third-best contracts for problems with general utilities. We take the first order approach, and our main tool is the stochastic maximum principle. The optimal contracts are characterized as solutions to some forward-backward stochastic differential equations. More importantly in economics, in the case with quadratic cost function, we transform the problem to a ``deterministic" optimization problem and then solve the problem semi-explicit. Some examples will be discussed. The talk is based on joint works with Jaksa Cvitanic and Xuhu Wan.

Monday November 13, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM
Sergey Levendorskiy (UT Austin, Economics)

"Optimal stopping in regime-switching models"

In the talk, a general framework for pricing of perpetual American and real options in regime-switching L\'evy models is presented. In each state of the Markov chain, which determines switches from one L\'evy process to another, the payoff stream is a monotone function of the L\'evy process labelled by the state. This allows for additional switching within each state of the Markov chain (payoffs can be different in different regions of the real line). The payoffs and riskless rates may depend on a state, which allows for jumps in prices at moment of switching. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modelled as finite-state Markov chains. We construct iteration procedures and prove that iterations converge to the solution of the optimization problem monotonically. The procedures are numerically efficient even if the number of states is large provided the transition rates are not large w.r.t. the riskless rates. As first applications, we solve exit problems for a price-taking firm, and pricing problems for American options with infinite and finite time horizon. In the latter case, we use a modification of Carr's randomization procedure for regime-switching models.

The talk is based on 3 joint papers with Svetlana Boyarchenko:
1. "American Options in Regime-Switching Models"(September 6, 2006)
2. "Perpetual American Options in Regime-Switching Models" (September 5, 2006).
3. "Exit Problems in Regime-Switching Models"

Monday, November 20, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM
Knut Solna (UC Irvine, Math)
"Multiname and Multiscale Default Modeling"

We consider multiname default modeling using a reduced form modeling approach. The model is based on a multiscale Vasicek or Ornstein-Uhlenbeck model for the hazard rates of the underlying names. Such default modeling is important in the context of pricing for instance Collaterized Debt Obligations (CDOs) and we analyze the impact of volatility time scales on the default distribution and CDO prices.

THURSDAY, December 7, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM
Claudio Albanese (Imperial College London, Math Finance)
" Operator methods and derivative pricing theory"

We outline a mathematical framework for derivative pricing theory based on operator methods and give examples of applications to long dated callable swaps, bespoke synthetic CDOs and credit equity hybrid derivatives.

FRIDAY, Jan. 12
3:15 pm, Refreshments served at 3:00 PM
South Hall 5607F

Alexander Schied, Berlin University of Technology

Aspects of model uncertainty and robustness in finance and economics

Due to the complexity of financial price processes, their mathematical models are often subject to model misspecification. In this talk we present some recent results on the robustness of certain trading strategies with respect to model uncertainty. In the first part, we consider the robustness of the Delta hedging strategy of an exotic derivative with respect to realized volatility when the underlying model is a local volatility model. Our analysis is based on volatility comparison techniques for SDEs. In the second part, we focus on the construction of optimal investment strategies for an investor who is averse against both risk and model uncertainty. Here one can use or combine several techniques including convex duality, nonlinear PDEs, and robust statistical test theory. In some special cases, the problems considered in parts one and two are closely related to each other.

WEDNESDAY, Jan. 24
3:15 pm, Refreshments served at 3 pm, South Hall 5607F

Paolo Guasoni( Boston University)
No Arbitrage and Consistent Prices under Transaction Costs
(Joint work with M. Rasonyi and W. Schachermayer)

Abstract: Consistent Price Systems are the counterparts of martingale measures for models with transaction costs, as they guarantee the absence of arbitrage, and allow to characterize the superreplication prices of contingent claims.
We first establish the existence of CPS for price processes from a
condition on their topological support, which can often be easily
verified. We then exploit this result to characterize superreplication prices of European contingent claims as concave envelopes (face-lifting) of their payoff.
We will also discuss a new version of the Fundamental Theorem of Asset Pricing under Transaction Costs.

Jan. 29: Rene Carmona (Princeton, ORFE) HJM: a Unified Approach to Fixed Income, Credit and Equity Markets

Monday, Feb. 12,
3:15 pm, Refreshments served at 3 pm, South Hall 5607F

Mack Galloway (PSTAT, UCSB)
Option Pricing with Selfsimilar Additive Processes

Abstract (pdf)

February 26, Monday, 3:15 pm, Refreshments served at 3 pm, South Hall 5607F

Alan Lewis (OptionCity.net)

Title: "Geometries and Smile Asymptotics for a Class of Stochastic Volatility Models"

Slides of the talk

Abstract: I discuss techniques for computing the small time asymptotics in stochastic volatility models. These include methods based upon (i) geodesics, (ii) the eikonal equation, and (iii) the characteristic function. Running examples come from the class of models where the diffusion coefficient of the volatility process V(t) is V(t)^p, where p is any real number. Geometries associated to the implied metric are visualized.

March 12, 3:15 pm, Refreshments served at 3 pm, South Hall 5607F
Jin Ma (Purdue University, Mathematics)

Title: "Mathematics, Finance, and Actuarial Sciences"

An emerging trend in actuarial mathematics has been to infuse modern theory of mathematical finance into the study of actuarial problems. Thus a general insurance risk model today often involves at least two types of uncertainties: one from traditional claims, and the other from investment returns. In this talk I will describe how such a trend will give rise to interesting problems in stochastic analysis and stochastic finance.

Starting from the simplest classical Cremer-Lundberg model, and with three major problems in mind: ruin probability, optimization with reinsurance, and equity-linked insurance pricing, we show how martingale theory, large deviation, stochastic control theory, backward stochastic differential equations (with jumps), nonlinear Feynman-Kac formula, and even hot finance topics such as credit risk, indifference pricing, etc., will all find their places in this relatively ``ancient" subject.

Wednesday, April 4, 2007, 3:15 PM
Refreshments served at 3:00 PM
South Hall 5607F (Sobel)

Huiju Zhang, The Robert H. Smith School of Business, University of Maryland, College Park

Title: Applying Model Reference Adaptive Search to American-style Option Pricing

This paper considers the application of stochastic optimization methods to American-style option pricing. We apply a randomized optimization algorithm called Model Reference Adaptive Search (MRAS) to pricing American-style options through parameterizing the early exercise boundary. Numerical results are provided for pricing American-style call and put options written on underlying assets following geometric Brownian motion and Merton jump-diffusion processes. We also price American-style Asian options written on underlying assets following geometric Brownian motion. The results from the MRAS algorithm are compared with the cross-entropy (CE) method, and MRAS is found to be an efficient method.

April 9, Gordan Zitkovic (Mathematics, UT Austin)
3:15 pm, Refreshments served at 3 pm, South Hall 5607F

Title: Stability and equilibria in incomplete markets

A study of the continuity properties of the demand function of risk-averse agents investing in an incomplete market has been initiated in the work of Larsen and Zitkovic (2006), and further developed in Kardaras and Zitkovic (2007). In this talk I will present a summary of those results and present an extension of the fixed-point theorem of Knaster, Kuratowski and Mazurkiewicz beyond the realm of locally convex spaces. Finally, I will present an outline of a research program where the combination of the two is used to estabilsh existence of financial equilibria in incomplete markets.

April 16: Lisa Goldberg (MSCIBARRA)
3:15 pm, Refreshments served at 3 pm, South Hall 5607F

Title: Calibrating Credit From the Top Down

Credit derivatives constitute an important and enormous asset class, which is currently assessed at $24 trillion in outstanding notional value. The risk profile of a typical credit derivative depends on future losses to a portfolio of default sensitive securities. To evaluate and manage credit derivatives, we develop a top model of portfolio loss that is based on a self-affecting affine point process. In our model, risk is driven by economy-wide factors including the portfolio loss process itself, so past defaults influence future loss dynamics. Our specification takes account of the interdependence between the random loss at default and default timing.

In this lecture, we look at the empirical results of calibrating a Hawkes type affine point process to market data. We examine the quality of the model fit as well as the time evolution of model parameters, which are identified with different aspects of financial risk. The material for the talk was developed in collaboration with Eymen Errais, Kay Giesecke and Kao-Chih Syao.

Wednesday, April 25: Dr. Bruno Dupire, Bloomberg and NYU.
3:15 pm, Refreshments served at 3 pm, South Hall 5607F
SLIDES

Applications of the Root Solution of the Skorohod Embedding Problem in Finance

The Skorokhod embedding problem amounts to stopping a Brownian motion to hit a target density; it has interesting implications for finance:
a) any solution leads to a model that is calibrated to the option prices of a given maturity and
b) it provides a rule to sell a (martingale) asset in order to achieve a prescribed wealth distribution. We concentrate on the Root Solution (hitting time of a barrier), which provides a canonical mapping of a density into a stopping region. We examine
1) the implications in terms of options on realized variance
2) new Monte Carlo schemes which confine the increments in both space and time at each time step

Thursday, April 26: Dr. Bruno Dupire, Bloomberg and NYU.
Plublic lecture
Corwin pavilion, UCSB, 4 PM.
SLIDES PHOTOS

An Idiot's Guide to Option Pricing

Monday May 21, 3:15 pm, Refreshments served at 3 pm, South Hall 5607F
Jim Primbs, Stanford University, Management Science and Engineering
SLIDES

Receding Horizon Control Methods in Financial Engineering

Abstract: Receding horizon control (also known as Model Predictive Control) is a methodology in which on-line optimization problems are repeatedly solved in order to construct feedback control laws. Its development was primarily motivated by problems with constraints. Furthermore, advances in optimization methods and computing speed have greatly increased its domain of applicability. In this talk, we present newly developed semi-definite programming based receding horizon approaches for a class of stochastic systems. We then show how these methods can be used to effectively address a number of difficult financial engineering problems such as dynamic hedging of high dimensional options and index tracking under constraints.

FRIDAY, August 10, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM
Doreen Strassburger(Carl von Ossietzky Universitaet of Oldenburg)

Risk Management and Solvency - Mathematical Methods in Theory and Practice Abstract Slides

PAST 2005-2006 Seminars

May 17
3:15 pm, Refreshments served at 3 pm
South Hall 5607F

Valdo Durrleman, Stanford University

"Coupling Smiles"

Abstract: The first part of the talk will be concerned with the link between implied volatilities and the volatility of the underlying asset. Such a link is of practical interest since it relates the fundamental quantity for pricing derivatives (the spot volatility) which is not observable, to directly observable quantities (the implied volatilities). From a mathematical point of view, it relates information about the law of a positive martingale (the implied volatilities), to the representation of its sample paths as an Ito integral (the spot volatility).

In the second part, we look at an application of this result. As a motivating example, consider the world three major currencies, EUR, JPY, and USD, and their three corresponding exchange rates. An elementary arbitrage argument gives any of the three exchange rates as a function of the other two. We are interested in the similar problem for options on these currencies. More precisely, we would like to reconstruct the implied volatility smile of one currency given the other two.

Parts of the talk are based on a joint work with Nicole El Karoui and on discussions with Andres Villaquiran.

May 22 (Monday)
3:15 pm, Refreshments served at 3 pm
South Hall 5607F

Ronnie Sircar (Princeton University)

Impact of Risk Aversion on Credit Derivatives

The market in credit-linked derivative products has grown astonishingly, from $631.5 billion global volume in the first half of 2001 to above $12 trillion through the first half of 2005. They now account for approximately 10% of the total OTC derivatives market. Despite the popularity and ever-increasing complexity of credit risk structured products, the quantitative technology for their valuation (and hedging) has lagged behind. A major limitation of many approaches is the inability to capture and explain high premiums observed in credit derivatives markets for unlikely events, for example the spreads quoted for senior tranches of CDOs written on investment grade firms. When significant yields are offered for protection against the default of 15-30% of US investment grade firms over the next five years, in some sense the ''end of the world as we know it'', we argue that market participants' risk aversion is playing a large role, and we develop the mechanism of utility-indifference valuation for credit derivatives to quantify this.

May 24
3:15 pm, Refreshments served at 3 pm
South Hall 5607F

Levon Goukasian, Pepperdine University

Title: "Optimal Risk Taking with Flexible Income" (joint work with J. Cvitanic of Caltech and F. Zapatero of USC)

We study the portfolio selection problem in the presence of the option to exert costly effort for more income. As expected, investors who have the flexibility to generate future income are willing to take more risk. Additionally, we find that portfolio allocation is not monotonically increasing with time. We also study the dependence of the portfolio allocation on other parameters like interest rate, market price of risk, individual's productivity and employment constraints.

Nan Chen, Columbia University, IEOR

A tale of two simulations

We discuss Monte Carlo methods for two problems central to the pricing and hedging of derivative securities: (i) calculation of "Greeks" (price sensitivities) and (ii) valuation of American options. Both are based on joint work with Paul Glasserman.
Malliavin Greeks without Malliavin calculus: Standard methods for estimating sensitivities differentiate paths or differentiate measures. A recent line of work derives estimators using Malliavin calculus. In implementing a discrete-time approximation of a continuous-time model, one may discretize first and then differentiate or vice-versa. In several important cases the first route produces the same estimators produced by Malliavin calculus, but using only traditional elementary techniques.

Additive and multiplicative duality: Pricing an American option entails solving an optimal stopping problem. Dual formulations replace maximization over stopping times with minimization over martingales. We compare duals based on additive and multiplicative decompositions of positive supermartingales. We establish an equivalence in the quality of the bounds achieved by the two methods, but show that the variance of the multiplicative method is typically much larger.

June 12 (Monday)
3:15 pm, Refreshments served at 3 pm
South Hall 5607F

Suhas Nayak (Stanford University)

Stochastic volatility surface estimation

Abstract: We study the calibration of volatility surfaces in a stochastic volatility environment. Starting with a stochastic volatility model for asset prices, we cast the volatility estimation problem as a variational one and we derive an HJB equation for the volatility surface. We incorporate uncertainty in market prices and we study the asymptotics of the resulting HJB equation in the fast mean-reversion regime. We present numerical solutions from our estimation scheme and find certain parameters of the volatility surface to be both stable in time and stable with respect to our iteration procedure.

June 21, 2006 (Wednesday) 3:15 pm, Refreshments served at 3 pm
South Hall 5607F

Sean Han (Taiwan)

Option pricing, Hedging, and efficient Monte Carlo methods

Abstract:

Under stochastic volatility models, we propose a new and generic control variates method to efficiently evaluate financial derivatives by means of Monte Carlo simulation. These controls are local martingales and are closely related to some imperfect hedging strategies. Therefore, reduced variances obtained from these Monte Carlo methods are not only measurements of computing efficiency but also represent risks associated to some particular trading strategy in the incomplete market. An asymptotic result is obtained to characterize the variance for rescaled stochastic volatility models. The analysis is done through a combination of perturbation techniques and an averaging effect. Several numerical examples, including some lower and upper American option prices, computed by Monte Carlo and/or Quasi-Monte Carlo methods are presented.

FRIDAY Jan 11, at 2:00PM: Qing Zhang (U. of Georgia)
Trading a mean-reverting asset: Buy low and sell high

Abstract:
Trading a financial asset involves a sequence of decisions to buy or sell the asset over time. A traditional trading strategy is to buy low and sell high. However, in practice, identifying these low and high levels is extremely challenging and difficult. In this talk, I will present our ongoing research on characterization of these key levels when the underlying asset price is dictated by a mean-reversion model. Our objective is to buy and sell the asset sequentially in order to maximize the overall profit. Mathematically, this amounts to determining a sequence of stopping times. We establish the associated dynamic programming equations (quasi-variational inequalities) and show that these differential equations can be converted to algebraic-like equations under certain conditions. The two threshold (buy and sell) levels can be found by solving these algebraic-like equations. We provide sufficient conditions that guarantee the optimality of our trading strategy. We also examine the dependence of these threshold levels on various parameters in numerical examples.

MONDAY Jan 14, at 3:15PM: Marek Rutkowski (U. of South Wales)
Marek Rutkowski

Pricing and Hedging of Convertible Bonds with Credit Risk

In our works [3]-[6], we attempt to shed more light on mathematical modeling of convertible bonds, thus continuing the previous research presented, for instance, in Andersen and Bu®um [1], Ayache et al. [2], Davis and Lischka [7], Kallsen and KÄuhn [8], and Kwok and Lau [9].

In [3], we consider the problem of the decomposition of a convertible bond into a bond component and an option component. This decomposition is indeed well established in the case of an `exchange option', when the conversion can only occur at maturity, and there are no put or call clauses. However, it was not previously studied in the general case of a defaultable convertible bond with call and/or put covenants.

In [4], we specify the valuation results for a defaultable game option (in particular, a convertible bond) to the context of default risk model based on the hazard process. The approach is based on the reduction of the information flow from the full ¯ltration to the reference ¯ltration. Our main existence result for hedging strategies in a hazard process set-up can be informally stated as follows: under the assumption that a related doubly re°ected BSDE admits a solution under some risk-neutral measure, the state-process multiplied by the default indicator process is the minimal super-hedging price up to a sigma martingale cost process.

The associated hedging strategies are subsequently analyzed by means of a martingale decomposition of a solution to the related doubly reflected BSDE. It is worth stressing that these decompositions are by no means arti¯cial. On the contrary, they arise naturally in the context of a Markovian framework, which is studied in some detail in the follow-up paper [5]. Under a rather general speci¯cation of the in¯nitesimal generator of a driving Markov factor process, we develop in [5] the variational inequality approach to pricing and hedging of a defaultable game option.

In [6], we consider a Markovian diffusion set-up with default. In this model, we show that a doubly reflected BSDE related to the convertible security has a solution, and we provide the related super-hedging strategy. Moreover, we characterize the price of a convertible security in terms of a viscosity solution of the associated variational inequality and we prove the convergence of a suitable approximation scheme.

References
[1] Andersen, L. and Buffum, L.: Calibration and implementation of convertible bond models. Journal of Computational Finance 7 (2004), 1-34
[2] Ayache, E., Forsyth, P. and Vetzal, K.: Valuation of convertible bonds with credit risk. The Journal of Derivatives, Fall 2003.
[3] Bielecki, T.R., Crepey, S., Jeanblanc, M. and Rutkowski, M.: Arbitrage pricing of defaultable game options with applications to convertible bonds. Forthcoming in Quantitative Finance.
[4] Bielecki, T., Crepey, S., Jeanblanc, M. and Rutkowski, M.: Valuation and hedging of defaultable options in a hazard process model. Submitted.
[5] Bielecki, T.R., Crepey, S., Jeanblanc, M. and Rutkowski, M.: Defaultable options in a Markovian intensity model of credit risk. Forthcoming in Mathematical Finance.
[6] Bielecki, T.R., Crepey, S., Jeanblanc, M. and Rutkowski, M.: Convertible bonds in a defaultable diffusion model. Submitted.
[7] Davis, M. and Lischka, F.: Convertible bonds with market risk and credit risk. In: Applied Probability, R. Chan et al., eds., American Mathematical Society/International Press, 2002, pp. 45-58.
[8] Kallsen, J. and KÄuhn, C.: Convertible bonds: ¯nancial derivatives of game type. In: Exotic Option Pricing and Advanced Levy Models, A. Kyprianou et al., eds., Wiley, 2005, pp. 277-288.
[9] Kwok, Y. and Lau, K.: Anatomy of option features in convertible bonds. Journal of Futures Markets 24 (2004), 513-532.