SeminarInaugural day: Monday October 16th, 2006 200607 Regents' Lecture by Dr. Bruno Dupire(Bloomberg and NYU)
Past
20122013 seminars 20132014 SeminarsUpcoming
Monday, November 18, South Hall 5607F, 3:305PM, Refreshments
served at 3:15 PM Title: BismutElworthyLi Formulas for Diffusions with Irregular Drift Coefficients Abstract: One of the most pertinent applications of Malliavin calculus in Mathematical Finance is the representation of the socalled Greeks, which are sensitivities of option prices w.r.t. to involved parameters, as expectation functionals of the payoff function times a socalled Malliavin weight. In particular for nonsmooth payoff functions this representation yields a numerically efficient way to compute Greeks. In the case of the Delta (sensitivity w.r.t. the initial value of the underlying diffusion), which is of special interest for hedging purposes, the above mentioned representation is also referred to as BismutElworthyLi type formula. However, in the existing literature the underlying Itˆo diffusion is assumed to have smooth coefficients. For example, an extended OrnsteinUhlenbeck process with regime switching mean reversion rate, an important model in electricity price modelling, is not included in this class of Itˆo diffusions. In this presentation we demonstrate how to generalize the BismutElworthyLi type formula to Itˆo diffusions with irregular drift coefficients. To this end, we study the theoretical questions of existence and Malliavin differentiability of strong solutions of stochastic differential equations with irregular drift coefficients. Using techniques from white noise analysis and a compactness criteria based on Malliavin calculus we develop a new method for the construction of strong solutions of SDE’s with irregular drift coefficients. Further, this approach yields the additional important result that the constructed strong solutions are Malliavin differentiable and Sobolev differentiable in their initial conditions. This insight together with some ”local time variational calculus” finally enables us to extend the corresponding BismutElworthyLi representation. past Monday, November 18, South Hall 5607F, 3:305PM, Refreshments
served at 3:15 PM Title: BismutElworthyLi Formulas for Diffusions with Irregular Drift Coefficients Abstract: One of the most pertinent applications of Malliavin calculus in Mathematical Finance is the representation of the socalled Greeks, which are sensitivities of option prices w.r.t. to involved parameters, as expectation functionals of the payoff function times a socalled Malliavin weight. In particular for nonsmooth payoff functions this representation yields a numerically efficient way to compute Greeks. In the case of the Delta (sensitivity w.r.t. the initial value of the underlying diffusion), which is of special interest for hedging purposes, the above mentioned representation is also referred to as BismutElworthyLi type formula. However, in the existing literature the underlying Itˆo diffusion is assumed to have smooth coefficients. For example, an extended OrnsteinUhlenbeck process with regime switching mean reversion rate, an important model in electricity price modelling, is not included in this class of Itˆo diffusions. In this presentation we demonstrate how to generalize the BismutElworthyLi type formula to Itˆo diffusions with irregular drift coefficients. To this end, we study the theoretical questions of existence and Malliavin differentiability of strong solutions of stochastic differential equations with irregular drift coefficients. Using techniques from white noise analysis and a compactness criteria based on Malliavin calculus we develop a new method for the construction of strong solutions of SDE’s with irregular drift coefficients. Further, this approach yields the additional important result that the constructed strong solutions are Malliavin differentiable and Sobolev differentiable in their initial conditions. This insight together with some ”local time variational calculus” finally enables us to extend the corresponding BismutElworthyLi representation. Monday, November 25, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Francesca Biagini (University of Munich) Title: Mathematical models for the formation of financial bubbles Abstract: The notion of an asset price bubble has two ingredients. One is the observed market price of a given financial asset, the other is the asset's intrinsic value, and the bubble is defined as the difference between the two. The intrinsic value, also called the fundamental value of the asset, is usually defined as the expected sum of future discounted dividends. In the first part of the talk we study a flow in the space of equivalent martingale measures and focus on the corresponding shifting perception of the fundamental value of a given asset in an incomplete financial market model. This allows us to capture the birth of a perceived bubble and to describe it as an initial submartingale which then turns into a supermartingale before it falls back to its initial value zero. In the second part of the talk we examine the impact of overconfidence on bubbles formation in the framework of reducedform models for credit risk. We assume that the wealth associated to a defaultable asset may be strongly affected by the trading activity of overconfident investors, who believe the asset to be safe and provoke an alteration of its estimated value. Since the value process changes under this influence, the underlying pricing measure has also to readapt determining a switch in the space of the equivalent martingale measures. In this way we provide a constructive approach to explain bubbles formation as well as motivate a dynamics in the space of equivalent martingale measures at microeconomic level. Tuesday, November 26, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Marcel Nutz (Columbia University) Title: On Model Uncertainty in Discrete Time Abstract: We study the problems of arbitrage, superhedging and utility maximization in a nondominated model of a discretetime ﬁnancial market. We show that absence of arbitrage in a quasisure sense is equivalent to the existence of a suitable family of martingale measures, that a superhedging duality holds, and that optimal strategies for robust utility maximization exist. If time permits, some consequences for martingale theory will also be discussed. Based on joint works with Mathias Beiglböck and Bruno Bouchard. Wednesday, February 5, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Tim Leung (Columbia University) Title: Pricing Derivatives with Counterparty Risk and Collateralization: A Fixed Point Approach Abstract: We study a valuation framework for financial contracts subject to reference and counterparty default risks with collateralization requirement. A fixed point approach is proposed to analyze the marktomarket contract value with counterparty risk provision, and show that it is a unique bounded and continuous fixed point via contraction mapping. This leads us to develop an accurate iterative numerical scheme for valuation. Specifically, we solve a sequence of linear inhomogeneous PDEs, whose solutions converge to the fixed point price function. We apply our methodology to compute the bid and ask prices for both defaultable equity and fixedincome derivatives, and illustrate the nontrivial effects of counterparty risk, collateralization ratio and liquidation convention on the bidask spreads. (joint work with Jinbeom Kim (Barclays)) Monday, February 10, South Hall 5607F, 56:30PM, Refreshments served at 3:15 PM Dr. Sebastian Jaimungal (University of Toronto) Title: Robust Market Making Abstract: Because market makers (MMs) acknowledge that their models are incorrectly specified, in this paper, we allow for ambiguity in their choices to make their models robust to misspecification in (i) the arrival rate of market orders (MOs), (ii) the fill probability of limit orders, and (iii) the dynamics of the fundamental value of the asset they deal. We demonstrate that MMs adjust their quotes to reduce inventory risk and adverse selection costs. Moreover, robust market making increases the Sharpe ratio of market making strategies and allows for the MM to fine tune the tradeoff between the mean and the standard deviation of expected profits. Our framework adopts the robust optimal control approach of Hansen and Sargent (2007) and we provide analytical solutions for the robust optimal strategies as well as a verification theorem. We also find that in many circumstances ambiguity averse MMs act differently from MMs who are risk averse. Monday, Febrary 24, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Stéphane Loisel, (Lyon 1, visiting Cornell) Title: Ruin probability for some particular correlated claims, for worsening risks, and risks with infinite mean. Abstract: We first give explicit formulas for the infinite time ruin probability for some particular correlated claim amounts or interarrival times. We then investigate asymptotics of ruin probabilities when claim distribution is worsening over time, due to phenomena like sectorial inflation or global warming. We end up with some results in the case where claim amounts have infinite mean. If time permits, we’ll make the link between some ruin problem and some longevity online changepoint detection problem. Monday, March 10, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Andrey Sarantsev (Ph.D student at dept of Mathematics University of Washington, Seattle) Title: Market Models with Splits and Mergers Abstract: We study models of regulatory breakup but with a fluctuating number of companies.
If at some moment the share of the total market capitalization of a company
reaches a certain threshold, then the company is split into two random parts.
This can be viewed as a consequence of antitrust legislation.
Companies are also allowed to merge when an exponential clock rings.
Under certain condition, the quantity of stocks does not go to infinity in finite time,
and the model does not admit arbitrage. Monday, April 7, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Pavel Shevchenko Title: A generalized grouped tcopula with multiple parameters of degrees of freedom and analysis of tail dependence in currency carry trades. Joint work with Gareth Peters, Matthew Ames, Guillaume Bagnarosa and Xiaolin Luo Abstract: The tcopula is a popular dependence structure often used in risk management as it allows for modelling the tail dependence between risks and it is simple to simulate and calibrate. The use of a standard tcopula is often criticized due to its restriction of having a single parameter for the degrees of freedom (dof) that may limit its capability to model the tail dependence structure in a multivariate case. To overcome this problem, the grouped tcopula was proposed in the literature, where risks are grouped a priori in such a way that each group has a standard tcopula with its specific dof parameter. To avoid a priori grouping, which is often difficult in practice, recently we proposed a generalized grouped tcopula, where each group consists of one risk factor. We present characteristics, simulation and calibration procedures for the generalized tcopula, including Markov chain Monte Carlo method for estimation and Bayesian model selection. This generalized grouped tcopula is significantly different from the standard tcopula in terms of risk measures such as tail dependence, valueatrisk and expected shortfall. Using historical data of foreign exchange (FX) rates as a case study, we found that Bayesian model choice criteria overwhelmingly favor the generalized tcopula when compared to the grouped and standard tcopulas. We demonstrate the impact of model choice on the conditional ValueatRisk for portfolios of major FX rates. In addition, using this and other copula models, we analyse tail dependence in baskets of high and low interest rate currencies used for carry trade investment strategies. Monday, May 19, South Hall 5607F, 3:305PM, Refreshments
served at 3:15 PM Title:On Control Invariance methods and their use in Finance Abstract:Asset transactions in the financial market can be viewed as controls influenc ing the dynamic evolution of an uncertain system. Uncertainty is traditionally described in terms of a stochastic control process driven by random noise. Re cent developments in control theory recognized the fact that in physics, biology, engineering, and other fields including finance, probability distributions are es sentially unknown  or at best largely contingent on ad hoc assumptions  and moved in the direction of obtaining essentially distributionfree results. These developments led to the notion of robustness and robust controlinvariance. The unknown random variables affecting the system’s trajectory are assumed to lie in a given set. All prior knowledge of randomness is subsumed under the shape of this set. The ensuing exercise is to determine what properties of the dynamic system remain invariant with respect to that set.
20122013 SeminarsFriday, October 19 Title: Optimal Stopping for Spectrally Negative Levy Processes and Applications in Finance Abstract: We consider a class of infinitetime horizon optimal stopping problems for spectrally negative Levy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoffs via the scale function. We obtain and show the equivalence of the continuous/smooth fit condition and the firstorder condition for maximization over threshold levels. Extensions to multiplestopping and applications in Leland’s endogenous default model are also discussed. Monday October 22, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Hoi Ying Wong (Department of Statistics, The Chinese University of Hong Kong) Title: Application and Implication of Cointegration in Asset Pricing Abstract: Cointegration is a useful econometric tool for identifying assets which share a common
equilibrium. The importance of cointegration has become recognized and resulted in a
Nobel Prize in Economics for Granger in 2003. In this talk, I will report several
recent advances of asset pricing theories based on continuoustime cointegration
dynamics. It covers cointegrated pairstrading using classical meanvariance portfolio
theory, cointegration option pricing with stochastic correlations using Fourier
analysis, and (if time allows) the hedging with mortality risk in insurance products.
Our theories predict that 1. if cointegrated assets are liquidly traded, then there
exists a statistical arbitrage opportunity; 2. If the assets are not traded or not
liquidly traded, their corresponding derivatives securities, in particular futures
contracts, exhibit stochastic convenient yields which are partially driven by
cointegrating factors; and 3. As human mortality is not traded by its nature and the
national mortality rate is cointegrated with the mortality rate of an individual
insurance companys client pool, cointegration techniques enhance the hedging of
mortality risk with national mortality bonds. Empirical studies are performed to
validate the use of the developed theories and numerical methods. Monday October 29, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Mykhaylo Shkolnikov (University of California, Berkeley) Title: Asymmetrically colliding Brownian particles in stochastic portfolio theory and beyond Abstract:We will discuss systems of Brownian particles on the real line, which interact by splitting the local times of collisions among themselves in an asymmetric manner. These can be identified with the collections of ordered processes in a Brownian particle system, in which the drift coefficients, the diffusion coefficients, and the collision local times for the individual particles are assigned according to their ranks. Such processes can be viewed as generalizations of those arising in firstorder models for equity markets in the context of stochastic portfolio theory, and are able to correct for several shortcomings of such models while being equally amenable to computations. We also show that, in addition to being of interest in their own right, such systems of Brownian particles arise as universal scaling limits of systems of jump processes on the integer lattice with local interactions. In particular, this result extends the convergence of TASEP to its continuous analogue. This is joint work with Ioannis Karatzas and Soumik Pal.
Monday November 12, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Jianfeng Zhang (USC) Title: Viscosity Solutions of Fully Nonlinear Parabolic PathDependent PDEs Abstract: In this talk we introduce a notion of viscosity solutions for Path Dependent PDEs (PPDEs for short). Such new type of PDEs include, among others: Backward SDEs (semilinear PPDE), Gmartingales and Second Order BSDEs (Path dependent HJB equations), Path dependent BellmanIssacs equation (corresponding to zerosum stochastic differential game), Backward Stochastic PDEs, and (Forward) Stochastic PDEs. Compared to the standard approach in the literature on those topics, a viscosity theory enables us to: (i) mimic the PDE language to characterize the solutions to those SDEs; (ii) provide a unified treatment to all those subjects and extend to even more general framework; (iii) establish the wellposedness result (existence, uniqueness, comparison, stability) under weaker conditions. Monday, January 28, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Ronnie Sircar (Princeton University) Title: Analysis of VIX Options & Leveraged ETF Options Abstract: Financial Crisis notwithstanding, new products continue to be developed in financial markets to try and capture risks such as those from changing volatility. In the spirit of analyzing these and quantifying their risks, we study them through stochastic, asymptotic and empirical analysis of the data. A main focus is to test the relations between the new products and existing more liquid ones such as S&P 500 options themselves, and unleveraged ETFs. Models used for the analysis include multiscale stochastic volatility, and Heston stochastic volatility with regime shifts. Monday Feburary 25, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Richard Sowers (University of of Illinois Urbana Champaign) Title: Scales in High Frequency Trading Abstract:Highfrequency trading has led to a number of new problems in mathematical finance. In particular, it highlights Wednesday, March 13, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Dmitry Kramkov (Carnegie Mellon) Title:Integral representation of martingales motivated by the problem of endogenous completeness in financial economics Abstract:Let Q and P be equivalent probability measures and
let psi be a Jdimensional vector of random variables such that
d Q/ d P and psi are defined in terms of a
weak solution X to a ddimensional stochastic differential equation.
Motivated by the problem of endogenous completeness in financial
economics we present conditions which guarantee that every local
martingale under Q is a stochastic integral with respect to
the Jdimensional martingale S_t \set
E^{Q}[psiF_t]. While the drift b=b(t,x)
and the volatility sigma = sigma(t,x) coefficients for X need to
have only minimal regularity properties with respect to x, they are
assumed to be analytic functions with respect to t. We provide a
counterexample showing that this $t$analyticity assumption for
sigma cannot be removed. Wednesday, May 8, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Shige Peng (Shandong University, Global Scholar of Princeton) Title: BSDE, PDE, Nonlinear Expectation and Model Uncertainty Abstract:Nonlinear FeynmanKac formula tells us that, when the coefficients depend only on the state of the path of a Brownian, then a Backward Stochastic Differential Equation (BSDE) becomes a quasilinear PDE of parabolic type. This reveals that, in general, a BSDE is in fact a new type of PDE called path dependent PDE, in which the continuous path plays the role of state variable x. The nonlinear semigroup associated to this PDE is a nonlinear expectation. Monday, May 13, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Harish Bhat (UC Merced) Title: Markov Tree Option Pricing Abstract:The Markov Tree (MT) model generalizes the binomial tree by allowing firstorder dependence on past outcomes. We first review an asymptotic argument showing that the distribution of log returns generated by the MT model can be closely approximated by a mixture of normals with three scalar parameters. Using this distribution, we develop closedform expressions for the price of a European call option and the corresponding delta hedge. We then use five years worth of daily option data to compare the MT model's empirical, outofsample performance against that of the BlackScholes model and Heston's stochastic volatility model. This comparison includes versions of the MT and BlackScholes models in which volatilities are either dependent or independent of option strike and maturity. Our findings indicate that, of the models and fitting procedures considered, the MT model yields the most accurate and least risky singleinstrument hedges.
Wednesday, September 28, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Alexandra Chronopoulou (PSTATUCSB) Title: Parameter Estimation for Fractional SDEs Abstract: We consider the parameter estimation problem for a multidimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H > 1/2, with nonlinear random drift and diffusion coefficients. Due to the intractability of the likelihood function, we propose the maximizer of a partial likelihood as the estimator of the parameters of the model. We show how to compute this estimator using Malliavin calculus techniques and approximation results. We study the computational efficiency of our method and we provide rates of convergence for the approximation task. For a particular class of fractional SDEs, we establish consistency of the proposed estimator. We apply our methodology to the estimation of the parameters of the fractional BlackScholes model using S&P 500 data. Monday October 10, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Gerard Brunick (PSTAT, UCSB) Title: Weak uniqueness for a class of degenerate diffusions with continuous covariance Abstract: Motivated by the problem of calibrating linear pricing rules to the market prices of options, we provide a new weak uniqueness result for degenerate diffusions. In particular, we consider pathdependent stochastic differential equations where the diffusion coefficient is a function of both the current location of the process and the running integral of the process, and we show that uniqueness holds for continuous, strictly positivedefinite diffusion coefficients. These results combine tools from the theory of singular integrals on Lie groups with the localization machinery of Stroock and Varadhan. Monday October 17th, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Stephan Sturm (ORFE, Princeton University) Title: Portfolio optimization under convex incentive schemes: Duality theory and stochastic volatility Abstract: We consider the utility maximization problem from the point of view of a portfolio manager paid by a convex incentive scheme. This problem departs from classical portfolio optimization theory since the implied utility function is no more concave which produces some interesting phenomena. Using duality theory, we are able top prove existence and uniqueness of the optimal wealth in general (incomplete) semimartingale markets as long as the unique optimizer of the dual problem has no atom with respect to the Lebesgue measure. In many cases, this fact is independent of the incentive scheme and depends only on the structure of the set of equivalent local martingale measures. As example we discuss stochastic volatility models and show that existence and uniqueness of an optimizer are guaranteed as long as the market price of risk satisfies a certain (Malliavin)smoothness condition. This is joint work with Maxim Bichuch. Monday, November 21, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Srikanth Iyer (PSTATUCSB) Title: Pricing Credit Derivatives in a Markov Modulated Reduced Form Model Abstract: Numerous incidents in the financial world have exposed the need for the design and analysis of models for correlated default timings. Some models have been studied in this regard which can capture the feedback in case of a major credit event. We extend the research in the same direction by proposing a new family of models having the feedback phenomena and capturing the effects of regime switching economy on the market. The regime switching economy is modeled by a continuous time Markov chain. The Markov chain may also be interpreted to represent the credit rating of the firm whose bond we seek to price. We model the default intensity in a pool of firms using the Markov chain and a risk factor process. We price some singlename and multiname credit derivatives in terms of certain transforms of the default and loss processes. These transforms can be calculated explicitly in case the default intensity is modeled as a linear function of a conditionally affine jump diffusion process. In such a case, under suitable technical conditions, the price of credit derivatives are obtained as solutions to a system of ODEs with weak coupling, subject to appropriate terminal conditions. Solving the system of ODEs numerically, we analyze the credit derivative spreads and compare their behavior with the nonswitching counterparts. We show that our model can easily incorporate the effects of business cycle. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low floating interest rate, high default intensity rate and high volatility. We also model the effects of firm restructuring on the credit spread, in case of a default. Monday, January 9, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Konstantinos Spiliopoulos (Brown University) Title: Recent results on systemic risk in large financial networks & more Abstract: The past several years have made clear the need to better understand the behavior of risk in large interconnected financial networks. Interconnections often make a system robust, but they can act as conduits for risk. In this talk, I will present recent results on modeling the dynamics of correlated default events in the financial market. An empirically motivated system of interacting point processes is introduced and we study how different types of risk, like contagion and exposure to systematic risk, compete and interact in largescale systems. A law of large numbers for the loss from default is proven and used for approximating the distribution of the loss from default in large, potentially heterogenous portfolios. Large deviation arguments are then used to identify the way that atypically large (i.e. ``rare'') default clusters are most likely to occur. The results give insights into how different sources of default correlation interact to generate typical and atypical portfolio losses. Time permitting, I will discuss briefly recent general results on large deviations and MonteCarlo methods for multiple scale systems. Questions of interest include qualitative and quantitative descrpition of transitions probabilities between different equilibrium states of a given system. The results can potentially have applications in scientific disciplines such as chemistry and are also related to certain stochastic volatility models. Wednesday, January 11, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Tomoyuki Ichiba (PSTAT) Title: Planar Diffusions with rankbased characteristics Abstract: We construct a diffusion process with values in the plane and with rankbased drift/dispersive characteristics. We compute the transition probabilities of this process and the order statistics, discuss pathwise uniqueness and strength of related stochastic differential equations, and study its dynamics under a timereversal. We also show that the planar diffusion can be represented in terms of onedimensional diffusion with bangbang drift driven by a standard Brownian motion, its local time accumulated at the origin, and an independent standard Brownian motion, in a form which can be construed as a twodimensional analogue of the stochastic equation satisfied by the socalled skew Brownian motion. This is a joint work with E. Robert Fernholz, Ioannis Karatzas, and Vilmos Prokaj. Thursday, January 12, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Sergey Nadtochiy (Oxford Man Institute) Title: MARKETBASED APPROACH TO MODELING DERIVATIVES PRICES ABSTRACT. Most classical models for derivatives prices focus on prescribing the time evolution of the underlying stochastic factors. The prices of derivatives are then computed, for example, via the riskneutral expectations. As markets developed and many derivative contracts became liquidly traded, it appeared necessary, in order to avoid creating arbitrage opportunities and to fully exploit the information given by the market, to calibrate such models so that they reproduce the observed derivatives prices. However, the calibration results may vary significantly from day to day, implying that none of the calibrated models can be used to describe the future time evolution of the derivatives prices and, in particular, study the risks associated with them. The idea of the marketbased approach is to model the derivatives prices directly, as the prices of generic financial assets. This approach allows to start a model from an arbitrary combination of derivatives prices currently observed in the market, without having to change (recalibrate) the model. In this presentation, I will outline the main problems associated with the construction of a marketbased model and will present the general methodology which provides solutions to these problems. I will also give an overview of the existing constructions of the marketbased models, starting with the famous HeathJarrowMorton theory, and show how these results agree with the general method. Finally, I will illustrate the theory by constructing (both mathematically and numerically) a family of marketbased models for the European call options of multiple strikes and maturities. Wednesday, January 18, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Yan Dolinsky (ETH Zurich) Title: Numerical Schemes for Stochastic Volatility Models Abstract: We present a tree based approximations for stochastic volatility models, such as the Stein and Stein model, Heston model etc. The importance of such numerical schemes follows from the wide use of stochastic volatility models among practitioners and the fact that for these models analytical solutions usually are not available. We show how to calculate efficiently options prices (European and American) with general type of payoffs such as Vanilla Options, Barrier Options, Lookback Options, Asian options etc. Our main tool is the weak convergence approach which allows to approximate diffusion processes by correlated random walks. Friday, January 20, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Gerard Brunick (PSTAT, UCSB) Title:Optimal Investment in the Presence of Highwater Mark Fees Abstract: In this talk, we will consider the problem of optimal asset allocation for an agent who may invest in a money market fund, a stock, and a hedge fund. We model the risky assets as correlated geometric Brownian motions and we assume that our investor maximizes discounted CRRA utility from consumption on an infinite horizon. We further suppose that the investment in the hedge fund is subject to a proportional performance fee that is assessed each time the cumulative profittodate derived from the investment in the hedge fund eaches a new running maximum. We will see that this problem reduces to the optimal control of a reflected diffusion. We will examine the regularity of the associated HamiltonJacobiBellman equation and show the existence of optimal controls. Finally, we will examine some qualitative properties of the optimal investment strategy. Monday Jan 23, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Alexandra Chronopoulou (PSTAT, UCSB) Title:Stochastic volatility models with longmemory in discrete and continuous time Abstract: We consider a continuous time stochastic volatility model with long memory in which the stock price is described by a Geometric Brownian motion with volatility that follows a fractional OrnsteinUhlenbeck process. In addition, we study two discrete time models: a discretization of the continuous model via an Euler scheme and a discrete model in which the returns are a zero mean iid sequence where the volatility is a fractional ARIMA process. Using a particle filtering algorithm we estimate the empirical distribution of the unobserved volatility process for all three models. Based on the volatility filter, we construct a multinomial recombining tree for option pricing. We also discuss appropriate parameter estimation techniques for each model. For the long memory parameter, we compute an implied value by calibrating the model with real data. Finally, we compare the different models using simulated data and we price options on the S&P 500 index. Monday Feb. 13, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Henry Schellhorn (Claremont Graduate University) Title: A Noarbitrage Model of Liquidity in Financial Markets involving Brownian Sheets Abstract: We consider a dynamic market model where buyers and sellers submit limit orders. If at a given moment in time, the buyer is unable to complete his entire order due to the shortage of sell orders at the required limit price, the unmatched part of the order is recorded in the order book. Subsequently these buy unmatched orders may be matched with new incoming sell orders. The total number of buy orders D(p,t) entered in the system at time t is a collection of stochastic processes indexed by a continuous parameter p, the limit price, and its dynamics are modelled by a stochastic partial differential equation (SPDE) driven by a Brownian sheet. The same type of dynamics hold for the total number of sell orders S(p,t). The curves D and S are related to but not identical to the demand and supply curves, and thus the exogenous data is not identical to the exogenous data specified in other models (such as Cetin, Jarrow, and Protter 2004, or CJP). The equilibrium price process is the (limit) price at which these two curves are equal at all times. We derive the SPDE of the equilibrium process and then provide necessary and sufficient conditions for the existence of a riskneutral measure where this process is a local martingale. This result is important in finance because the first fundamental theorem of asset pricing in the CJP model holds if and only if there is a measure where the equilibrium price process is a local martingale. Interestingly, the riskneutral measure is not unique in our model, which shows that more economic conditions are needed to close it. The noarbitrage conditions we obtain are applicable to a wide class of models, in the same way that the HeathJarrowMorton conditions apply to a wide class of interest rate models. We implement and calibrate several parametric models and analyze empirically when arbitrage occurred using real order book data. We also investigate the effectiveness of some arbitrage strategies based on these parametric models. Wed, February 29, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Joscha Diehl (TU Berlin) Title: Rough path theory: a quick overview and some applications Abstract: Differential equations of the form dY = f(Y) dX are covered by classical theory in the case where X is a smooth path. Rough path theory treats such equations when the driving signal X is very irregular in time, e.g. only Hölder continuous for some exponent larger then zero. It turns out that in general the information given by the path itself is not sufficient to build a satisfying theory. The missing piece is some kind of 'higher order' process, which can be thought of as encoding iterated integrals of the path against itself. I will sketch the main ideas of the theory, using an approach that only requires knowledge of undergraduate mathematics. Wednesday, March 14, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Ronnie Sircar (Princeton University) Title: A Feedback Model for the Financialization of Commodities Prices Abstract: Tang and Xiong (2009) discuss the financial markets as a result of increased index investing activity in the past decade. They find empirical evidence of increased exposure of commodities prices to shocks to other asset classes. We build a feedback model to try and capture some of these effects in which traditional economic demand for a commodity, oil say, is perturbed by the influence of portfolio optimizers. The analysis reveals correlation effects proportional to the long or short positions of the investors, along with a lowering of volatility. Monday, April 16, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Marco Frittelli (University of Milano, visiting UCSB) Title: Two applications of quasiconvex analysis Abstract: We introduce a class of law invariant quasiconvex risk measures based on an appropriate family of acceptance sets. As a particular case, we propose a generalization of the classical notion of the V@R, that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. Our second application concerns the evaluation of the quality of the scientific research. We introduc a family of performance measures, called Scientific Research Measures (SRM) that are based on the whole distribution of citations. These SRM are: flexible to fit peculiarities of different areas and seniorities; inclusive, as they comprehend several popular indices;coherent, as they share the same structural properties; calibrated to the particular scientific community. Monday, May 14, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Sachin Adlakha (Caltech) Title: Mean Field Equilibria in Large Scale Stochastic Games Abstract:We study stochastic games with many interacting players. In contrast to studying Markov perfect equilibrium (MPE), we consider mean field equilibrium (MFE), where players’ strategies depend only on the long runaverage state of their competitors. We study the conditions under which MFE exists and show that the same conditions ensure that MFE is a good Monday, May 21, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Yuri Saporito (PSTATUCSB) Title: Functional It Weights for Greeks Abstract: Functional Ito calculus is an extension of the classical Ito calculus to functions of path and it was first presented by Dr. Bruno Dupire in his paper "Functional Ito Calculus". In this talk I will present an application of this theory to the computation of Greeks, which are sensitivities of derivatives prices with respect to model parameters. This gives us an alternative to the Malliavin calculus approach presented by Fournie et al in the classical paper "Applications of Malliavin calculus to Monte Carlo methods in finance".
20102011 SeminarsMONDAY September 27, South Hall 5607F, 3:305PM, Refreshments served at 3:15 PM Dr. Jeremy Staum (Northwestern University) Title: Systemic Risk Components and Deposit Insurance Premia Abstract: In light of recent events, there have been several proposals to establish
a theory of financial system risk management analogous to portfolio risk management.
One important aspect of portfolio risk management is risk attribution, the process
of decomposing a portfolio risk measure into components that are attributed to
individual assets or activities. We consider the total premium required to insure
all deposits in the banking system as the systemic risk measure. The component of
this risk measure attributable to a bank could serve as the bank's deposit insurance
premium. The richer structure of a banking system, compared to a portfolio, makes
the theory of systemic risk components more complicated than the theory of portfolio
risk components. We propose a scheme for systemic risk attribution that could be MONDAY October 11, South Hall 5607F, 3:305 PM, Refreshments served at 3:15 PM Prof. Tomoyuki Ichiba (PSTAT/CRFMSUCSB) Title: Modeling a systemic risk of frozen interbank lending Abstract: In this talk we propose a simple model of banking system and analyze a systemic risk of frozen interbank lending. The monetary reserves of banks are modeled as a system of interacting Feller diffusion. The model is simple enough for mathematical analysis, yet captures how lending preferences of banks cause possible multiple bank failures. We quantify the lending preference from one bank to another as a function of all the reserves. We find an extreme example that only k out of n banks can survive. This banking system induces a class of random graph processes in continuous time. We may extend this toy model for understanding the systemic risk and for preventing crisis in the banking system. (joint work with JeanPierre Fouque and Chunkai Gao). MONDAY October 25, South Hall 5607F, 3:305 PM, Refreshments served at 3:15 PM Prof. Marco Frittelli (Math. Dept., Milano Univ., visiting PSTATCRFMS/UCSB) Title: On quasiconvex dynamic risk measures Abstract: We introduce conditional quasiconvex risk measures and provide their dual representation. This generalizes the representation of quasiconvex real valued risk measures and of conditional convex risk measures. These results are applied in the study of the theory of dynamic risk measures and of performance indices. MONDAY November 15, South Hall 5607F, 3:305 PM, Refreshments served at 3:15 PM Dr. Ramon van Handel (ORFE, Princeton) Title: The ergodic theory of nonlinear filters Abstract: The goal of nonlinear filtering is to estimate a process of interest given noisy and incomplete observations. Nonlinear filtering methods play a role in a wide variety of applications, ranging from robotics to finance. In many applications, one is interested in understanding the performance of the nonlinear filter (as well as related Monte Carlo algorithms or sequential decision procedures) over a long time horizon. Mathematically, this requires an understanding of the ergodic theory of nonlinear filters. Such a theory was first developed in a classic paper of H. Kunita (1971). Unfortunately, the key part of the proof in this paper contains a fundamental measuretheoretic error, which lies at the heart of the ergodicity problem for nonlinear filters. In this talk, I will discuss recent progress in understanding the general ergodic theory of nonlinear filters. I will introduce the central measuretheoretic identity and outline its proof under very general assumptions, by means of the theory of Markov chains in random environments. I will also discuss two surprising counterexamples where the filter fails to be ergodic. The upshot is that the ergodicity of classical nonlinear filtering problems is now largely resolved, but nonlinear filtering problems with infinite dimensional signals (such as appear in applications to weather prediction or data assimilation) remain a mystery. MONDAY November 22, South Hall 5607F, 3:305 PM, Refreshments served at 3:15 PM Dr. Olaf Menkens, Dublin City University (DCU), Ireland Title: Optimising Proportional Reinsurance Using a Worst Case Scenario Approach Abstract: This presentation considers the problem of an insurance company to optimise its reserve process by proportional reinsurance. Usually, the reinsurance level will be determined by a ruin probability constraint or by minimising the ruin probability (see e.g. Hipp and Vogt (2003), Schmidli (2001, 2002, and 2004), or Eisenberg and Schmidli (2008)). Instead of conditioning on the ruin probability, this presentation will maximise the controlled reserve process by a worstcase scenario approach. The worstcase scenario approach has been introduced in the context of portfolio optimisation by Korn and Wilmott (2002). This approach has been extended so far in various ways (e.g. considering different utility function (Korn and Menkens (2005)), optimising investment portfolio of an insurance company (Korn (2005)), in a stochastic differential game context (Korn and Steffensen (2007)). We start by making the socalled small claims assumption, that is the claims will be modelled as a Brownian motion with drift. Second, the claims will be modelled as the sum of a Brownian motion with drift and a Poisson process and third, claims will be modelled as a Poisson process. Results will be computed, analysed, and compared with the results of minimising the ruin probability. This is work in progress and joint research with Ralf Korn (TU Kaiserslautern) and Mogens Steffensen (U of Copenhagen). MONDAY January 31, 2011, South Hall 5607F, 3:305 PM, Refreshments served at 3:15 PM Jorge Zubelli (IMPA, Rio, Brazil) Title: A ConvexRegularization Framework for LocalVolatility Calibration in Derivative Markets Abstract: We discuss a unified framework for the calibration of local volatility models that makes use of recent tools of convex regularization of illposed Inverse Problems. The key aspect of the present approach is that it addresses in a general and rigorous way the issue of convergence and sensitivity of the regularized solution when the noise level of the observed prices goes to zero. In particular, we present convergence results that include convergence rates with respect to noise level in fairly general contexts and go well beyond the classical quadratic regularization. This is joint work with Otmar Scherzer (Vienna) and Adriano De Cezaro (UFRGS). MONDAY February 14, 2011, South Hall 5607F, 3:305 PM, Refreshments served at 3:15 PM Matt Lorig (UCSB) Title: Time changed Fast MeanReverting Stochastic Volatility Models Abstract: We introduce a class of randomly timechanged fast meanreverting stochastic
volatility
models and, using spectral theory and singular perturbation techniques, we
derive an
approximation for the prices of European options in this setting. Three
examples of random MONDAY April 11, 2011, South Hall 5607F, 3:305 PM, Refreshments served at 3:15 PM Sylvain Rubenthaler (U of Nice) Title: Particle systems, Kalman interacting filter, definitions and proof of convergence. Abstract: I will talk about particle systems used to approximate conditional laws. I will present the classic system and something called the interacting Kalman filter. I will give some elements of proof of why this last system has a good performance uniformly in time. Finally, I will present some examples. MONDAY May 9, 2011, South Hall 5607F, 3:305 PM, Refreshments served at 3:15 PM Yuri Saporito (PSTATUCSB) Title: Functional Ito's Calculus a la Dupire Abstract: This talk will present some results of an extension of the Itô Calculus to functionals of the current path of a stochastic process. These results were first presented by Dr. Bruno Dupire in his paper "Functional Itô Calculus", which is the main reference for the talk. We will also discuss the possible applications to Finance. MONDAY May 23, 2011, South Hall 5607F, 3:305 PM, Refreshments served at 3:15 PM Prof. Rodney Garratt (Economics, UCSB) Title: Mapping systemic risk in the international banking network Abstract: Systemic risk among the network of international banking groups arises when financial stress threatens to crisscross many national boundaries and expose imperfect international coordination. To assess this risk, we apply an information theoretic map equation due to Martin Rosvall and Carl Bergstrom to partition banking groups from 21 countries into modules. The resulting modular structure reflects the flow of financial stress through the network, combining nodes that are most closely related in terms of the transmission of stress. The modular structure of the international banking network has changed dramatically over the past three decades. In the late 1980s four important financial centres formed one large supercluster that was highly contagious in terms of transmission of stress within its ranks, but less contagious on a global scale. Since then the most influential modules have become significantly smaller and more broadly contagious. The analysis contributes to our understanding as to why defaults in US subprime mortgages had such large global implications. WEDNESDAY June 1, South Hall 5607F, 3:30 PM, Refreshments served at 3:15 PM Prof. Bernt Oksendal (University of Oslo) THURSDAY June 2, South Hall 5607F, 9:3010:45AM, NOTE unusual day and time Prof Matheus Grasselli (McMaster University, Canada) Past 20092010 SeminarsMONDAY September 21, South Hall 5607F, 3:154:45 PM, Refreshments served at 3:00 PM Dr. Tomoyuki Ichiba (PSTAT/CRFMS Postdoctoral Fellow, UCSB) Title: Hybrid Atlas Models (Part I) We study Atlas type models of equity markets with local characteristics that depend on both name and rank, and in ways that explain a stability of empirical capital distribution. This study involves rankings of continuous semimartingales and the local times of the differences between adjacent processes. By a comparison argument with Bessel processes we derive a representation of the market in terms of the reflected Brownian motion that yields invariant distribution of market capital shares under some assumptions. The class of resulting expected capital distribution curves seem rich enough to explain the empirical curves. We also discuss various portfolios including universal portfolios. Topics: (sub)martingale problems, rankings of continuous semimartingales, attainability of diffusions, collisions of Brownian particles, reflected Brownian motions, stochastic stability, average occupation time, capital distributions, stochastic portfolio theory. MONDAY September 28, South Hall 5607F, 3:154:45PM, Refreshments served at 3:00 PM Dr. Tomoyuki Ichiba (PSTAT/CRFMS Postdoctoral Fellow, UCSB) Title: Hybrid Atlas Models (Part II) We study Atlas type models of equity markets with local characteristics that depend on both name and rank, and in ways that explain a stability of empirical capital distribution. This study involves rankings of continuous semimartingales and the local times of the differences between adjacent processes. By a comparison argument with Bessel processes we derive a representation of the market in terms of the reflected Brownian motion that yields invariant distribution of market capital shares under some assumptions. The class of resulting expected capital distribution curves seem rich enough to explain the empirical curves. We also discuss various portfolios including universal portfolios. Topics: (sub)martingale problems, rankings of continuous semimartingales, attainability of diffusions, collisions of Brownian particles, reflected Brownian motions, stochastic stability, average occupation time, capital distributions, stochastic portfolio theory. MONDAY October 12, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Adam Tashman (PSTAT, UCSB) Title:
Option Pricing Under a StressedBeta Model Joint work with J.P. Fouque. MONDAY October 19, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Winslow Strong (PSTAT, UCSB) Title:
Regulation and the Removal of Arbitrage in Strongly Markovian Equity Market Models MONDAY October 26, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Peter Van De Zilver (PIMCO,
Newport Beach, CA) MONDAY November 16, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Hao Xing (Boston University) MONDAY January 25, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM David German (Claremont McKenna College) MONDAY February 22, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Erhan Bayraktar (Univ. of Michigan) WEDNESDAY Mar. 3, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Dr. Nizar Touzi (Ecole Polytechnique, France) Title: Wellposedness of Second Order Backward SDEs Abstract: We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs. In particular, we provide a fully nonlinear extension of the FeynmanKac formula. We discuss various applications to probabilistic numerical methods, and hedging under market illiquidity. MONDAY April 26, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Kostas Kardaras (Boston University) MONDAY May 17, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Kasper Larsen (Carnegie Mellon University) WEDNESDAY June 2, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Eric Hillebrand (Department of Economics, Louisiana State University) Joint work with Ambar N. Sengupta and Pierre Xu.
PAST 20082009 SeminarsMONDAY September 29, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Robert Gingrich (PIMCO, Newport Beach, CA) Title:
SYSTEMIC CREDIT RISK:
WHAT IS THE MARKET TELLING US? [paper] MONDAY October 13, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM David Nualart (Kansas University) Title:
Fractional Brownian motion: Stochastic calculus and applications. MONDAY October 20, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Ioannis Karatzas (Columbia University) Title: VOLATILITY STABILIZATION, DIVERSITY AND ARBITRAGE IN STOCHASTIC FINANCE [slide] We'll start this talk with an overview of the modern theory of portfolios, based on Stochastic Analysis. We shall introduce then the notion of relative arbitrage. and provide simple, descriptive and easytotest criteria for the existence of such arbitrage in equity markets. These criteria postulate essentially that the excess growth rate of the market portfolio, a positive quantity that can be estimated or even computed from a given market structure, be "sufficiently large". We show that conditions satisfying these criteria are manifestly present in the US equity market, and construct explicit portfolios under these conditions. One such condition, market diversity, emerges when the volatility structure is bounded. MONDAY October 27, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Raj Sau (UCSB) Title: HEDGING IN MODELS WITH JUMPS AND TRANSACTION COSTS  a survey Abstract: Transaction costs invalidate BlackScholes argument, since continuous revision implies infinite trading. Hence it is necessary to rebalance in discrete time steps, which results in hedging error. The hedging strategies can be classified into two main categories, timebased and movebased (utility) strategies. The goal is to minimize the variance of hedging error. If the asset price follows a jump diffusion, the market is in general incomplete. Kennedy, Forsyth Vetzal devise a strategy that simultaneously eliminates diffusion risk, and minimizes an objective that is a linear combination of jump risk and transaction cost. In this survey I review the above hedging methods. WEDNESDAY, November 5, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Michael Ludkovski (UCSB) Title: Optimal Risk Sharing under Distorted Probabilities Abstract: We study optimal risk sharing among n agents endowed with distortion risk measures. Risk sharing under thirdparty constraints is also considered. We obtain an explicit formula for Pareto optimal allocations. In particular, we find that a stoploss or deductible risk sharing is optimal in the case of two agents and several common distortion functions. This extends recent result of Jouini et al. (2006) to the problem with unbounded risks and market frictions. In the first part of my talk I will give a brief survey of distortion risk measures and its relation to other risk preferences. I will then MONDAY January 12, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Josh Davis (PIMCO, Newport Beach, CA) Title: Robust Bayesian Portfolio Construction FRIDAY January 23, South Hall 4607, 2PM3PM Konstantinos Spiliopoulos (University of Maryland ) Title: ReactionDiffusion Equations with Nonlinear Boundary Conditions in Narrow Domains Abstract: We will consider the second initial boundary problem in narrow domains of width $\epsilon\ll 1$ for linear second order differential equations with nonlinear boundary conditions. Using probabilistic methods we show that the solution of such a problem converges as $\epsilon \downarrow 0$ to the solution of a standard reactiondiffusion equation in a domain of reduced dimension. This reduction allows to obtain some results concerning wave front propagation in narrow domains. In particular, we describe conditions leading to jumps of the wave front. In addition, an important and interesting problem, which is related to the previous one and will be presented here, is the Wiener process with instantaneous reflection in a narrow tube which, in contrast to before, is assumed to be nonsmooth asymptotically. MONDAY January 26, PUBLIC LECTURE at 4PM, Corwin Pavilion, followed by a reception. [Event Flyer] Arnold Miyamoto (CitiGroup, Managing Director CitiFX) Title: Overview on Foreign Exchange and Capital Markets Abstract: Arnold will present an overview of Foreign Exchange Sales & Trading from its historical beginnings to today's environment. As uantitative strategies are at the center of how Wall Street works, he will also discuss where quants are used and the opportunities. Lastly, he will reflect on the past 11/2 years in the financial markets with a view toward what is on the near horizon. Monday February 9, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Prof. Masaaki Kijima, Graduate School of Economics, Tokyo Metropolitan University Title:
Risk Management and the Pricing of CDOs Based on the Multivariate Wang Transform WEDNESDAY March 4, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Prof. Henry Schellhorn (Claremont Graduate University) Title: An Algorithm for the Pricing of PathDependent American Options Using Malliavin Calculus Abstract: We propose a recursive scheme to calculate backward the values of conditional expectations of functions of path values of Brownian motion. This scheme is based on the ClarkOcone formula in discrete time. We construct an algorithm based on our scheme to efficiently calculate the price of American options on securities with pathdependent payoffs. Our algorithm can be combined with regressionbased Monte Carlo methods, like the LongstaffSchwartz algorithm. In this case, our algorithm remedies the decrease of performance experienced by regressionbased methods when the number of basis functions, or regressands, needs to be quite large, because of pathdependence. Monday April 6, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Mihai Sirbu (UT Austin) Title: Asymptotic analysis of utilitybased prices and hedging strategies for utilities defined on the whole real line Abstract: We perform an asymptotic expansion of utilitybased prices and hedging
strategies for small number of contingent claims, in the framework of
optimal investment with general utilities defined on the whole real line.
Conceptually, this follows previous joint work with Dmitry Kramkov and
relies on using the risktolerance wealth processes as the natural
numeraire for pricing and hedging. The quantitative analysis generalizes
the work of Henderson for the case of exponential utility and basisrisk
model and the qualitative part is related to the work of Mania &
Schweizer, Becherer and Kallsen & Rheinlander. The presentation is based
on work in progress. Monday April 13, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Soumik Pal (University of Washington) Title: Two models of equity markets and the distribution of market
weights Monday April 20, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Carole Bernard (University of Waterloo) Title: Pathdependent inefficient strategies and how to make them
efficient Monday April 27, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Xin Guo (UC Berkeley) Title: Connecting singular controls and switching controls, with
applications Monday May 4, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Paolo Guasoni (Boston University) Title: Two and Twenty: what Incentives? Monday May 18, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Matheus Grasselli (McMaster University) Title: Chaotic Interest Rate Model Calibration Abstract: The Wiener chaos approach to interest rates was introduced a few years ago by Hughston and Rafailidis as an axiomatic framework to model positive interest rates, continuing a line of research started by the seminal Flesaker and Hughston model and including the elegant potential approach of Rogers and others. Apart from ensuring positivity, one appealing feature of the chaotic approach is its hierarchical way to introduce randomness into a model through different orders of chaos expansions. We propose a systematic way to calibrate Wiener chaos models to market data, and compare the performance of expansions of different orders in the presence of interest rate derivatives of increased complexity. This is joint work with Tsunehiro Tsujimoto. WEDNESDAY May 20, South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Prof. Marco Frittelli (University of Milano, Italy) Title: Conditional certainty equivalent and representation of risk measures Monday June 1, South Hall 5607F, 4PM [note the unusual time] Yingying Fan (Marshall School of Business, USC) Title : Testing and Detecting Jumps Based on a Discretely Observed Process Abstract: We propose a new nonparametric test for detecting the presence of jumps in asset prices using discretely observed data. Compared with the test statistic in A\"{i}tSahalia and Jacod (2007), our new test statistic enjoys the same asymptotic properties but has smaller variance. These results are justified both theoretically and numerically. Thanks to the reduction of the variance, we also propose a new test procedure to identify the locations of jumps. The problem of jump identification thus reduces to a multiple comparison problem. We employ the False Discovery Rate (FDR) approach to control the type I error. Simulation studies and real data analysis further demonstrate the power of the newly proposed test method. This is a joint work with Professor Jianqing Fan.
PAST 20072008 SeminarsWEDNESDAY 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 quasiexplicitly 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 mathematicalfinance. Real data are assumed to be observations of a continuoustime process at discrete timepoints. 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 Taylorexpansion of the Kolmogorovforward equation in the spirit of AitSahalia(1999, 2002). Properties of maximumlikelihood estimators are illustrated by simulation. Some aspects of applying the approximation to Bayesian inference and statisticalsurveillance (changepointdetection) 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 noarbitrage 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. 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 quasivariational 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 exante 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 expost 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 volatilityofvariance function for an uncorrelated stochastic volatility model, so as to be consistent with an observed symmetric smallmaturity smile, by solving an Abel Volterra integral equation. We show how to adapt the methodolgy for implied volatility skews. We also discuss Lewis's smalltime asymptotics for the derivatives of the implied volatility At theMoney, and the largex 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 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. Friday,
January 18th, South Hall 5607F, 3:15PM, refreshments served at 3:00PM 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 Tuesday,
January 22nd, South Hall 5607F, 10:00AM, refreshments served at 9:45am. 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 piecewisedeterministic 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 Utilitybased 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 markettraded 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 utilitybased 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 freeboundary problems of reactiondiffusion type. In addition, we examine the holder's hedging strategies that involve a combination of dynamic trading of correlated assets and static positions in markettraded 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 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. Tuesday February 19, 10AM, Sobel room, refreshments at 9:45 Tom Hurd (McMaster University, Canada) 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 Monday
February 25, South Hall 5607F, 3:15PM, refreshments served at 3:00PM Distribution densities in stochastic volatility models
Stochastic volatility models were introduced in 1980s1990s. In these
models, the volatility of a stock is described by a stochastic process.
For instance, in the HullWhite model, the volatility is a geometric
Brownian motion, in the SteinStein model, the absolute value of an
OrnsteinUhlenbeck 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 meansquare
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. Monday March 3, South Hall 5607F, 3:15PM, refreshments served at 3:00PM 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 regimeswitching 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 19902005, and its performance is compared against the prevalent betaneutral hedging strategy. The merger arbitrage index exhibited strong evidence of a regimeswitching process, and the proposed model offered an improved fit relative to standard regression techniques. Mixture hedging was more effective at reducing downside risk than betaneutral hedging. Maximum drawdown was 5.50% for the mixture strategy, versus 5.98% for betaneutral 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 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 benchmarkcalibrated 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 OptionAdjusted 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 interestrate 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 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. Monday
April 21, South Hall 5607F, 3:15PM, refreshments served at 3:00PM 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. Monday, May 5 at McCune Conference Room HSSB 6020 at 3:15pm Monday
June 9, South Hall 5607F, 3:15PM, refreshments served at 3:00PM An Introduction to Malliavin Calculus for Lévy Processes and Applications to Finance The purpose of this lecture is to give a nontechnical, yet rigorous introduction to Malliavin calculus for Lévy processes and its applications to finance. The lecture consists of two parts:
PAST 20062007 SeminarsMonday September 25: South Hall 5607F, 3:15 PM, Refreshments served at 3:00 PM Jose FigueroaLopez (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 FigueroaLopez (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 "Smalltime and tail asymptotics for stochastic volatility models" We discuss tail/large strike asymptotics for a Dupiretype local volatility model, and for a diffusion process subordinated to an independent stochastic clock. We also discuss smalltime asymptotics for these classes of models, with applications to volatility derivatives, and the general pstochastic 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 "ContinuousTime PrincipalAgent Problems with Moral Hazard and/or Adverse Selection" Abstract: We study the continuous time principalagent problems with moral hazard and/or adverse selection. We will discuss the firstbest, secondbest, and thirdbest 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 forwardbackward 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 semiexplicit. 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 "Optimal stopping in regimeswitching models" In the talk, a general framework for pricing of perpetual American and real options in regimeswitching 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 finitestate 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 pricetaking 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 regimeswitching models.
The talk is based on 3 joint papers with Svetlana Boyarchenko: Monday,
November 20, South Hall 5607F, 3:15 PM, Refreshments served at 3:00
PM We consider multiname default modeling using a reduced form modeling approach. The model is based on a multiscale Vasicek or OrnsteinUhlenbeck 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 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 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 Paolo
Guasoni( Boston University)
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. Jan. 29: Rene Carmona (Princeton, ORFE) HJM: a Unified Approach to Fixed Income, Credit and Equity Markets Monday,
Feb. 12, Mack
Galloway (PSTAT, UCSB) 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 March
12, 3:15
pm, Refreshments served at 3 pm, South Hall 5607F
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 CremerLundberg model, and with three major problems in mind: ruin probability, optimization with reinsurance, and equitylinked insurance pricing, we show how martingale theory, large deviation, stochastic control theory, backward stochastic differential equations (with jumps), nonlinear FeynmanKac 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 Huiju Zhang, The Robert H. Smith School of Business, University of Maryland, College Park Title: Applying Model Reference Adaptive Search to Americanstyle Option Pricing This paper considers the application of stochastic optimization methods to Americanstyle option pricing. We apply a randomized optimization algorithm called Model Reference Adaptive Search (MRAS) to pricing Americanstyle options through parameterizing the early exercise boundary. Numerical results are provided for pricing Americanstyle call and put options written on underlying assets following geometric Brownian motion and Merton jumpdiffusion processes. We also price Americanstyle Asian options written on underlying assets following geometric Brownian motion. The results from the MRAS algorithm are compared with the crossentropy (CE) method, and MRAS is found to be an efficient method. April
9, Gordan Zitkovic (Mathematics, UT Austin) Title: Stability and equilibria in incomplete markets A study of the continuity properties of the demand function of riskaverse 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 fixedpoint 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) 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 selfaffecting affine point process. In our model, risk is driven by economywide 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 KaoChih Syao. Wednesday,
April 25: Dr.
Bruno Dupire, Bloomberg and NYU. 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: Thursday,
April 26: Dr.
Bruno Dupire, Bloomberg and NYU. An Idiot's Guide to Option Pricing Monday
May 21, 3:15 pm, Refreshments served at 3 pm, South Hall 5607F Receding Horizon Control Methods in Financial Engineering Abstract: Receding horizon control (also known as Model Predictive Control) is a methodology in which online 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 semidefinite 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 Risk Management and Solvency  Mathematical Methods in Theory and Practice Abstract Slides
PAST 20052006 SeminarsMay
17 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).
May
22 (Monday) Ronnie Sircar (Princeton University) Impact of Risk Aversion on Credit Derivatives The market in creditlinked 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 everincreasing 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 1530% 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 utilityindifference valuation for credit derivatives to quantify this.
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. 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) 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 meanreversion 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 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 QuasiMonte Carlo methods are presented. FRIDAY
Jan 11, at 2:00PM: Qing Zhang (U. of Georgia) Abstract:
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 setup can be informally stated as follows: under the assumption that a related doubly re°ected BSDE admits a solution under some riskneutral measure, the stateprocess multiplied by the default indicator process is the minimal superhedging 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 followup 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 setup 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 superhedging 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


