Perturbative Approaches for Robust Intertemporal Optimal Portfolio Selection
|
|
- Duane Price
- 5 years ago
- Views:
Transcription
1 Perturbative Approaches for Robust Intertemporal Optimal Portfolio Selection F. Trojani and P. Vanini ECAS Course, Lugano, October 7-13, 2001
2 1 Contents Introduction Merton s Model and Perturbative Solution Approach Preferences for Robustness Perturbative Solutions Some Explicit Computations Conclusions, Further Research
3 2 Introduction, Motivation Only a few intertemporal optimal portfolio problems can be solved explicitly (cf. Kim and Omberg (1996), Chacko and Viceira (1999)) Existence of closed-form solutions depends on assumptions regarding agent s utility functions, the opportunity set dynamics, intermediate consumption, aversion to model misspecification... Perturbation Theory (PT) is based on approximation methods by which approximate analytical expressions can be achieved Kogan and Uppal (2000): PT is a powerful approximation method for financial optimal decision making also
4 2.1 Preferences for Robustness (I) Agents have a reference model in mind which describes the approximateprobabilisticfeaturesofsomeunderlyingstatevariablesprocess; AHS (2000), Maenhout (1999), Lei (2001), Trojani and Vanini (2001), Uppal and Wang (2001) Agents believe in the possibility that the benchmark model could be slightly misspecified Model deviations that are particularly different from the reference model are penalized in their impact on the final decision The entity of the penalization is parameterized by a parameter that is interpreted as the strength of a preference for robustness
5 2.2 Preferences for Robustness (II) Differences through the way by which model deviations are penalized AHS (2000): penalizes deviations proportionally to their relative entropy w.r.t. the reference model Meanhout (1999): penalizes relative entropy, however in a way that is scaled by the current level of indirect utility AHS (1998), Lei (2001), Trojani and Vanini (2001): put a maximal bound on the relative entropy distance of a relevant candidate misspecification The first two approaches produce second order, the third first order risk aversions
6 3 Merton s Model ³ Z P t, ³ Z X t standard BM in R with covariance ρ Price, state and wealth dynamics db t = r t B t dt, dp t = α t P t dt + σ t P t dzt P dx t = ζ t dt + ξ t dzt X dw t = [w t W t (α t r t )+(r t W t C t )] dt + w t W t σ t dzt P Utility and implied objective function u (C t )= Cγ t 1 γ, V γ (W, X) =E Z 0 e δt u (C t ) dt
7 3.1 Vector Notation Covariance matrix Σ t of (dx t,dw t ) 0 canbefactorizedasσ t = Λ t Λ 0 t, where Λ t = ξ t 0 ³ 2 12 ρw t W t σ t 1 ρ w t W t σ t µ 0, Orthogonalization Z t = Zt X,ZX t Z P t = ρzt X + ³ ρ Zt X, gives for the vector valued state variable Y t =(X t,w t ) 0 dy t = µ t dt + Λ t dz t, where µ t =(ζ t,w t W t (α t r t )+(r t W t C t )) 0
8 3.2 Optimization Problem 1. HJB equation ³ c := W C 0 = sup {u (cw ) δj + A W J + A X J + wwρσξj WX } c,w A W = (r + w(α r) C) W W w2 σ 2 W 2 2 W 2 A X = ζ X ξ2 2 X 2 2. Homogeneous functional form for candidate solution: ³ e g(γ,x) W γ 1 J(W, X) = 1 δ γ
9 3.3 Perturbative Approach First order expansion J(W, X) = ³ 1 e g(γ,x) W γ 1 1 δ γ γ 0 δ (ln(w )+g 0(X)) g(γ,x) = g 0 (X)+γg 1 (X)+O(γ 2 ) Approximate optimal Policies c(x) = w(x) = = µ 1 δ eγg(γ,x) 1 γ 1 = δ (1 γ(g0 (X) ln (δ))) + O ³ γ 2 Ã 1 α r 1 γ σ 2 + γ g(γ,x)! ρξ X σ Ã 1 α r 1 γ σ 2 + γ g! 0(X) ρξ + O ³ γ 2 X σ
10 3.4 Remarks g 1 can be neglected in first order analysis g 0 is sufficient to determine the optimal policies up to first order in γ g 0 is obtained from the solution of a log utility agent g 0 (X) =ln(δ) 1+E Z e δt 0 r t à αt r t σ t and is typically easier to compute than the function g.! 2 dt
11 3.5 Summary on Kogan and Uppal (2000) Approach 1. Parameterize the problems under scrutiny and identify a specific parameter value for which the solution is known explicitly 2. Determine a functional form for the solution, such that to first order only the solvable benchmark model enters in the optimality conditions 3. Compute the optimal policies using the functional form of step 2 4. Expand the optimal policies to first order and determine the value function for the explicitly solvable model
12 4 Introducing Preferences for Robustness Step 1: Define the reference model for asset prices and state dynamics Step 2: Define the candidate model misspecifications Step 3: Define the relevant model misspecifications Step 4: Solve a max-min expected utility problem
13 4.1 Reference Model ³ Z P t, ³ Z X t standard BM in R with covariance ρ Price, state and wealth dynamics db t = r t B t dt, dp t = α t P t dt + σ t P t dzt P dx t = ζ t dt + ξ t dzt X dw t = [w t W t (α t r t )+(r t W t C t )] dt + w t W t σ t dzt P Orthogonalization Z t = gives µ Z X t,zx t dy t = µ t dt + Λ t dz t, 0, Z P t = ρz X t + ³ 1 ρ 2 12 Z X t
14 4.2 Model Misspecifications Model contaminations ν =(ν t ) t 0 are modelled as absolutely continuous changes of measure: where Z t = ν t =exp µ Z X t,zx t µ Z t Z t h s dz s h s 2 ds, 0 0 and for a suitable process (hs )= ³ h X s,h P s 0 By Girsanov Theorem, agents are thus concerned only with misspecifications in the drift of risky assets and state dynamics
15 4.3 Relative Entropy as a Measure of Model Discrepancy Relative entropy at time t I t (ν) =E (ν t ln (ν t )), Continuous-time relative entropy d dt I t (ν) = 1 2 h0 th t.
16 4.4 Max Min Expected Utility Problem Optimization problem of a robust agent J (W, X) = R sup C,w inf h E h 0 e δtcγ t 1 γ dt 1 2 h0 h η. A preference for robustness is modelled by a bound η on the rate at which continuous time relative entropy can increase over time Larger η s represent larger preferences for robustness
17 4.4.1 Worst Case Scenario Infimization w.r.t. h yields the implied worst case model (cf. also AHS (1998) Ã!1 2η 2 h = ξj X + ρwwσj W ³ Γ (w) 2 12, 1 ρ wwσj W where Γ (w) =w 2 W 2 σ 2 J 2 W + ξ2 J 2 X +2wW ρξσj W J X
18 4.4.2 Single-Agent Optimization Problem HJB equation (c := C W ) 0 = sup c,w ½ u (cw ) δj + A W J + A X J + wwρσξj WX (2ηΓ (w)) 1 ¾ 2 Homogeneous functional form ³ e g(γ,η,x) W γ 1 J(W, X) = 1 δ γ
19 4.4.3 Robust Optimal Policies Optimal consumption and risky asset allocation c = w = Ã e γg δ! 1 γ 1, 1 1 ³ 1 2η 2 JW 2 Γ J WW 1 (1 γ) α r σ 2 + γ g X µ 2η Γ 1 2 JX ρξ σ. The functional form of c is the same as in the non robust case w is characterized by the solution of an implicit equation through the function Γ (w)
20 5 Perturbative Optimal Policies First order expansion in γ and η 1 2: w (X) = α r σ 2 + γw 1 (X)+(2η) 1 2 w 2 (X)+O 2 γ, η 1 2 g (X) = g 0 (X)+γg 1 (X)+(2η) 1 2 g 2 (X)+O µγ, 2 η 1 2 µ., Optimal policies, G 0 (X) = ³ α r σ 2 + ³ ξ g 0 X c (X) = δ (1 γ (g 0 (X) ln (δ))), w (X) = 1 Ã 1 γ ³ 2η G 0 1 2! α r σ ³ +2 α r σ ρξ g 0 X : γ Ã 2η G 0!1 2 g 0 X ρξ σ
21 5.1 Remark 1 To γ, η 1 2 first order, robustness influences both the myopic and the hedging demand for risky assets µ 1 α r w (X) = µ 1 1 γ 2η 2 σ 2 G 0 (X) {z } MD µ 1 γ 2η 2 G 0 (X) + γ µ 2η 1 2 g 0 X ρξ σ 1 G 0 (X) {z } HD
22 5.2 Remark 2 The robust risky allocation is the portfolio strategy of an investor with a state dependent effective risk aversion 1 γ µ 2η G 0 (X) 1 2 The state dependent effective risk aversion correction depends on the state X only through the risk factors φ = α r σ and ψ = ξ g 0 X The largest relative portofolio corrections are realized when φ, ψ 0, in a neighborhood of the origin in (φ, ψ)-space Robustness affects optimal portfolios precisely when the standard myopic and intertemporal demands for risky assets are small, that is when risk exposure is low.
23 6 Some Explicit Computations Version of Kim and Omberg s (1996) model allowing for intermediate consumption db t = rb t dt, dp t = α t P t dt + σp t dzt P, dx t = λ ³ X X t dt + ξdz X t, where r, σ, ξ, λ, X > 0, and α t = r + σx t. In this model the market price of risk α t r σ process is an Ornstein Uhlembeck
24 6.1 Computation of g 0 Solution g 0 = a 0 + a 1 X a 2X 2, a 0 = ln(δ) 1+ r δ + ξ 2 2δ (δ +2λ) + λx a 1 = (δ + λ)(δ +2λ) > 0, 1 a 2 = δ +2λ > 0. ³ λx 2 δ (δ + λ)(δ +2λ), In this model the two relevant risk factors φ = α r σ, ψ = ξ g 0 X are perfectly correlated
25 6.1.1 Sketch of the Proof. g 0 (X) = ln(δ) 1+E Since E ³ X 2 t = ln(δ) 1+ X 0 = X " Z Ã e δt 0 Z 0 r + 1 µ αt r dt 2 σ X0 = X e µr δt E ³ Xt 2 X 0 = X dt. 2! = Var(X t X 0 = X)+(E (X t X 0 = X)) 2 # = ξ 2 1 e 2λt 2λ a final integration gives the result. + h e λt ³ X + X ³ e λt 1 i 2,
26 6.2 Optimal Policies Proposition 6.2 In the given model the following first order optimal policies hold true for a robust agent µ µ c (X) = δ 1 γ a 0 + a 1 X a 2X 2 ln (δ) w (X) = where 1 Ã 1 γ ³ 2η G 0 1 2! X σ + γ Ã 2η G 0!1 2 ρξ (a 1 + a 2 X) σ G 0 = ³ ξ 2 a ρξa 2 +1 X 2 +2a 1 ξ (ρ + ξa 2 ) X + ξ 2 a 2 1
27 6.3 Conclusions, Further Research PT provides analytical solutions for consumption/portfolio problems where otherwise only numerical results are available Robustness affects basically only the investment side, by reducing both the myopic and the intertemporal exposure to risky assets CR induces significant portfolio modifications already for low risk exposures Work in progress: Relation between MR and ER in the intertemporal setting, impact of the investment horizon (see also Barberis (1999))
Thomas Knispel Leibniz Universität Hannover
Optimal long term investment under model ambiguity Optimal long term investment under model ambiguity homas Knispel Leibniz Universität Hannover knispel@stochastik.uni-hannover.de AnStAp0 Vienna, July
More informationAmbiguity and Information Processing in a Model of Intermediary Asset Pricing
Ambiguity and Information Processing in a Model of Intermediary Asset Pricing Leyla Jianyu Han 1 Kenneth Kasa 2 Yulei Luo 1 1 The University of Hong Kong 2 Simon Fraser University December 15, 218 1 /
More informationUtility Theory CHAPTER Single period utility theory
CHAPTER 7 Utility Theory 7.. Single period utility theory We wish to use a concept of utility that is able to deal with uncertainty. So we introduce the von Neumann Morgenstern utility function. The investor
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationUtility Maximization in Hidden Regime-Switching Markets with Default Risk
Utility Maximization in Hidden Regime-Switching Markets with Default Risk José E. Figueroa-López Department of Mathematics and Statistics Washington University in St. Louis figueroa-lopez@wustl.edu pages.wustl.edu/figueroa
More informationBirgit Rudloff Operations Research and Financial Engineering, Princeton University
TIME CONSISTENT RISK AVERSE DYNAMIC DECISION MODELS: AN ECONOMIC INTERPRETATION Birgit Rudloff Operations Research and Financial Engineering, Princeton University brudloff@princeton.edu Alexandre Street
More informationOn Consistent Decision Making. and the Theory of Continuous-Time. Recursive Utility
On Consistent Decision Making and the Theory of Continuous-Time Recursive Utility Mogens Ste ensen Köln, April 23, 202 /5 Outline: Quadratic and collective objectives Further comments on consistency Recursive
More informationOptimal portfolio strategies under partial information with expert opinions
1 / 35 Optimal portfolio strategies under partial information with expert opinions Ralf Wunderlich Brandenburg University of Technology Cottbus, Germany Joint work with Rüdiger Frey Research Seminar WU
More informationShadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs
Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs Mihai Sîrbu, The University of Texas at Austin based on joint work with Jin Hyuk Choi and Gordan
More informationSpeculation and the Bond Market: An Empirical No-arbitrage Framework
Online Appendix to the paper Speculation and the Bond Market: An Empirical No-arbitrage Framework October 5, 2015 Part I: Maturity specific shocks in affine and equilibrium models This Appendix present
More informationEquilibrium with Transaction Costs
National Meeting of Women in Financial Mathematics IPAM April 2017 Kim Weston University of Texas at Austin Based on Existence of a Radner equilibrium in a model with transaction costs, https://arxiv.org/abs/1702.01706
More informationA new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009
A new approach for investment performance measurement 3rd WCMF, Santa Barbara November 2009 Thaleia Zariphopoulou University of Oxford, Oxford-Man Institute and The University of Texas at Austin 1 Performance
More informationAsset Pricing. Chapter V. Risk Aversion and Investment Decisions, Part I. June 20, 2006
Chapter V. Risk Aversion and Investment Decisions, Part I June 20, 2006 The Canonical Portfolio Problem The various problems considered in this chapter (and the next) max a EU(Ỹ1) = max EU (Y 0 (1 + r
More informationDYNAMIC ASSET ALLOCATION AND CONSUMPTION CHOICE IN INCOMPLETE MARKETS
DYNAMIC ASSET ALLOCATION AND CONSUMPTION CHOICE IN INCOMPLETE MARKETS SASHA F. STOIKOV AND THALEIA ZARIPHOPOULOU The University of Texas at Austin Abstract. We study the optimal investment consumption
More informationInformation and Credit Risk
Information and Credit Risk M. L. Bedini Université de Bretagne Occidentale, Brest - Friedrich Schiller Universität, Jena Jena, March 2011 M. L. Bedini (Université de Bretagne Occidentale, Brest Information
More informationRobust Markowitz portfolio selection. ambiguous covariance matrix
under ambiguous covariance matrix University Paris Diderot, LPMA Sorbonne Paris Cité Based on joint work with A. Ismail, Natixis MFO March 2, 2017 Outline Introduction 1 Introduction 2 3 and Sharpe ratio
More informationA Correction. Joel Peress INSEAD. Abstract
Wealth, Information Acquisition and ortfolio Choice A Correction Joel eress INSEAD Abstract There is an error in my 2004 paper Wealth, Information Acquisition and ortfolio Choice. This note shows how to
More informationIntroduction Optimality and Asset Pricing
Introduction Optimality and Asset Pricing Andrea Buraschi Imperial College Business School October 2010 The Euler Equation Take an economy where price is given with respect to the numéraire, which is our
More informationExact Simulation of Diffusions and Jump Diffusions
Exact Simulation of Diffusions and Jump Diffusions A work by: Prof. Gareth O. Roberts Dr. Alexandros Beskos Dr. Omiros Papaspiliopoulos Dr. Bruno Casella 28 th May, 2008 Content 1 Exact Algorithm Construction
More informationWorst Case Portfolio Optimization and HJB-Systems
Worst Case Portfolio Optimization and HJB-Systems Ralf Korn and Mogens Steffensen Abstract We formulate a portfolio optimization problem as a game where the investor chooses a portfolio and his opponent,
More informationMulti-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct
More informationRobust control and applications in economic theory
Robust control and applications in economic theory In honour of Professor Emeritus Grigoris Kalogeropoulos on the occasion of his retirement A. N. Yannacopoulos Department of Statistics AUEB 24 May 2013
More informationSome Aspects of Universal Portfolio
1 Some Aspects of Universal Portfolio Tomoyuki Ichiba (UC Santa Barbara) joint work with Marcel Brod (ETH Zurich) Conference on Stochastic Asymptotics & Applications Sixth Western Conference on Mathematical
More informationKey words. Ambiguous correlation, G-Brownian motion, Hamilton Jacobi Bellman Isaacs equation, Stochastic volatility
PORTFOLIO OPTIMIZATION WITH AMBIGUOUS CORRELATION AND STOCHASTIC VOLATILITIES JEAN-PIERRE FOUQUE, CHI SENG PUN, AND HOI YING WONG Abstract. In a continuous-time economy, we investigate the asset allocation
More informationA problem of portfolio/consumption choice in a. liquidity risk model with random trading times
A problem of portfolio/consumption choice in a liquidity risk model with random trading times Huyên PHAM Special Semester on Stochastics with Emphasis on Finance, Kick-off workshop, Linz, September 8-12,
More informationGeneralized Hypothesis Testing and Maximizing the Success Probability in Financial Markets
Generalized Hypothesis Testing and Maximizing the Success Probability in Financial Markets Tim Leung 1, Qingshuo Song 2, and Jie Yang 3 1 Columbia University, New York, USA; leung@ieor.columbia.edu 2 City
More informationOptimal Consumption, Investment and Insurance Problem in Infinite Time Horizon
Optimal Consumption, Investment and Insurance Problem in Infinite Time Horizon Bin Zou and Abel Cadenillas Department of Mathematical and Statistical Sciences University of Alberta August 213 Abstract
More informationHamilton-Jacobi-Bellman Equation of an Optimal Consumption Problem
Hamilton-Jacobi-Bellman Equation of an Optimal Consumption Problem Shuenn-Jyi Sheu Institute of Mathematics, Academia Sinica WSAF, CityU HK June 29-July 3, 2009 1. Introduction X c,π t is the wealth with
More informationFundamentals in Optimal Investments. Lecture I
Fundamentals in Optimal Investments Lecture I + 1 Portfolio choice Portfolio allocations and their ordering Performance indices Fundamentals in optimal portfolio choice Expected utility theory and its
More informationOptimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience
Optimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience Ulrich Horst 1 Humboldt-Universität zu Berlin Department of Mathematics and School of Business and Economics Vienna, Nov.
More informationA Malliavin-based Monte-Carlo Approach for Numerical Solution of Stochastic Control Problems: Experiences from Merton s Problem
A Malliavin-based Monte-Carlo Approach for Numerical Solution of Stochastic Control Problems: Experiences from Merton s Problem Simon Lysbjerg Hansen 3st May 25 Department of Accounting and Finance, University
More informationEXPLOSIONS AND ARBITRAGE IOANNIS KARATZAS. Joint work with Daniel FERNHOLZ and Johannes RUF. Talk at the University of Michigan
EXPLOSIONS AND ARBITRAGE IOANNIS KARATZAS Department of Mathematics, Columbia University, New York INTECH Investment Management LLC, Princeton Joint work with Daniel FERNHOLZ and Johannes RUF Talk at the
More informationDynamic Discrete Choice Structural Models in Empirical IO
Dynamic Discrete Choice Structural Models in Empirical IO Lecture 4: Euler Equations and Finite Dependence in Dynamic Discrete Choice Models Victor Aguirregabiria (University of Toronto) Carlos III, Madrid
More informationLinear Programming and the Control of Diffusion Processes
Linear Programming and the Control of Diffusion Processes Andrew Ahn Department of IE and OR, Columbia University, New York, NY 10027, aja2133@columbia.edu. Martin Haugh Department of IE and OR, Columbia
More informationLecture Notes 10: Dynamic Programming
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 81 Lecture Notes 10: Dynamic Programming Peter J. Hammond 2018 September 28th University of Warwick, EC9A0 Maths for Economists Peter
More informationJ. Marín-Solano (UB), M. Bosch-Príncep (UB), J. Dhaene (KUL), C. Ribas (UB), O. Roch (UB), S. Vanduffel (KUL)
BUY AND HOLD STRATEGIES IN OPTIMAL PORTFOLIO SELECTION PROBLEMS: COMONOTONIC APPROXIMATIONS J. Marín-Solano (UB), M. Bosch-Príncep (UB), J. Dhaene (KUL), C. Ribas (UB), O. Roch (UB), S. Vanduffel (KUL)
More informationCorrections to Theory of Asset Pricing (2008), Pearson, Boston, MA
Theory of Asset Pricing George Pennacchi Corrections to Theory of Asset Pricing (8), Pearson, Boston, MA. Page 7. Revise the Independence Axiom to read: For any two lotteries P and P, P P if and only if
More informationPortfolio Selection under Model Uncertainty:
manuscript No. (will be inserted by the editor) Portfolio Selection under Model Uncertainty: A Penalized Moment-Based Optimization Approach Jonathan Y. Li Roy H. Kwon Received: date / Accepted: date Abstract
More informationWealth, Information Acquisition and Portfolio Choice: A Correction
Wealth, Information Acquisition and Portfolio Choice: A Correction Joel Peress INSEAD There is an error in our 2004 paper Wealth, Information Acquisition and Portfolio Choice. This note shows how to correct
More informationTowards Multi-field Inflation with a Random Potential
Towards Multi-field Inflation with a Random Potential Jiajun Xu LEPP, Cornell Univeristy Based on H. Tye, JX, Y. Zhang, arxiv:0812.1944 and work in progress 1 Outline Motivation from string theory A scenario
More informationProving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control
Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control Erhan Bayraktar University of Michigan joint work with Virginia R. Young, University of Michigan K αρλoβασi,
More informationEquilibria in Incomplete Stochastic Continuous-time Markets:
Equilibria in Incomplete Stochastic Continuous-time Markets: Existence and Uniqueness under Smallness Constantinos Kardaras Department of Statistics London School of Economics with Hao Xing (LSE) and Gordan
More informationDiscrete-Time Finite-Horizon Optimal ALM Problem with Regime-Switching for DB Pension Plan
Applied Mathematical Sciences, Vol. 10, 2016, no. 33, 1643-1652 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.6383 Discrete-Time Finite-Horizon Optimal ALM Problem with Regime-Switching
More informationVariance Reduction Techniques for Monte Carlo Simulations with Stochastic Volatility Models
Variance Reduction Techniques for Monte Carlo Simulations with Stochastic Volatility Models Jean-Pierre Fouque North Carolina State University SAMSI November 3, 5 1 References: Variance Reduction for Monte
More informationof space-time diffusions
Optimal investment for all time horizons and Martin boundary of space-time diffusions Sergey Nadtochiy and Michael Tehranchi October 5, 2012 Abstract This paper is concerned with the axiomatic foundation
More informationMarch 16, Abstract. We study the problem of portfolio optimization under the \drawdown constraint" that the
ON PORTFOLIO OPTIMIZATION UNDER \DRAWDOWN" CONSTRAINTS JAKSA CVITANIC IOANNIS KARATZAS y March 6, 994 Abstract We study the problem of portfolio optimization under the \drawdown constraint" that the wealth
More informationShort-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility
Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility José Enrique Figueroa-López 1 1 Department of Statistics Purdue University Statistics, Jump Processes,
More informationPortfolio Optimization with unobservable Markov-modulated drift process
Portfolio Optimization with unobservable Markov-modulated drift process Ulrich Rieder Department of Optimization and Operations Research University of Ulm, Germany D-89069 Ulm, Germany e-mail: rieder@mathematik.uni-ulm.de
More informationOperations Research Letters. On a time consistency concept in risk averse multistage stochastic programming
Operations Research Letters 37 2009 143 147 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl On a time consistency concept in risk averse
More informationStochastic Volatility and Correction to the Heat Equation
Stochastic Volatility and Correction to the Heat Equation Jean-Pierre Fouque, George Papanicolaou and Ronnie Sircar Abstract. From a probabilist s point of view the Twentieth Century has been a century
More informationRobustness and bootstrap techniques in portfolio efficiency tests
Robustness and bootstrap techniques in portfolio efficiency tests Dept. of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic July 8, 2013 Motivation Portfolio selection
More informationConsumption. Consider a consumer with utility. v(c τ )e ρ(τ t) dτ.
Consumption Consider a consumer with utility v(c τ )e ρ(τ t) dτ. t He acts to maximize expected utility. Utility is increasing in consumption, v > 0, and concave, v < 0. 1 The utility from consumption
More informationAn Introduction to Malliavin calculus and its applications
An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart
More informationA comparison of numerical methods for the. Solution of continuous-time DSGE models. Juan Carlos Parra Alvarez
A comparison of numerical methods for the solution of continuous-time DSGE models Juan Carlos Parra Alvarez Department of Economics and Business, and CREATES Aarhus University, Denmark November 14, 2012
More informationUniversity Of Calgary Department of Mathematics and Statistics
University Of Calgary Department of Mathematics and Statistics Hawkes Seminar May 23, 2018 1 / 46 Some Problems in Insurance and Reinsurance Mohamed Badaoui Department of Electrical Engineering National
More informationarxiv: v1 [math.pr] 24 Sep 2018
A short note on Anticipative portfolio optimization B. D Auria a,b,1,, J.-A. Salmerón a,1 a Dpto. Estadística, Universidad Carlos III de Madrid. Avda. de la Universidad 3, 8911, Leganés (Madrid Spain b
More informationLecture 12: Diffusion Processes and Stochastic Differential Equations
Lecture 12: Diffusion Processes and Stochastic Differential Equations 1. Diffusion Processes 1.1 Definition of a diffusion process 1.2 Examples 2. Stochastic Differential Equations SDE) 2.1 Stochastic
More informationUniversity of Warwick, EC9A0 Maths for Economists Lecture Notes 10: Dynamic Programming
University of Warwick, EC9A0 Maths for Economists 1 of 63 University of Warwick, EC9A0 Maths for Economists Lecture Notes 10: Dynamic Programming Peter J. Hammond Autumn 2013, revised 2014 University of
More informationOptimal investment with high-watermark fee in a multi-dimensional jump diffusion model
Optimal investment with high-watermark fee in a multi-dimensional jump diffusion model Karel Janeček Zheng Li Mihai Sîrbu August 2, 218 Abstract This paper studies the problem of optimal investment and
More informationAggregate Risk. MFM Practitioner Module: Quantitative Risk Management. John Dodson. February 6, Aggregate Risk. John Dodson.
MFM Practitioner Module: Quantitative Risk Management February 6, 2019 As we discussed last semester, the general goal of risk measurement is to come up with a single metric that can be used to make financial
More informationRisk Measures in non-dominated Models
Purpose: Study Risk Measures taking into account the model uncertainty in mathematical finance. Plan 1 Non-dominated Models Model Uncertainty Fundamental topological Properties 2 Risk Measures on L p (c)
More informationMortality Surface by Means of Continuous Time Cohort Models
Mortality Surface by Means of Continuous Time Cohort Models Petar Jevtić, Elisa Luciano and Elena Vigna Longevity Eight 2012, Waterloo, Canada, 7-8 September 2012 Outline 1 Introduction Model construction
More informationLAN property for ergodic jump-diffusion processes with discrete observations
LAN property for ergodic jump-diffusion processes with discrete observations Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Arturo Kohatsu-Higa (Ritsumeikan University, Japan) &
More informationDynamic Asset Allocation - Identifying Regime Shifts in Financial Time Series to Build Robust Portfolios
Downloaded from orbit.dtu.dk on: Jan 22, 2019 Dynamic Asset Allocation - Identifying Regime Shifts in Financial Time Series to Build Robust Portfolios Nystrup, Peter Publication date: 2018 Document Version
More informationMFM Practitioner Module: Risk & Asset Allocation. John Dodson. February 4, 2015
& & MFM Practitioner Module: Risk & Asset Allocation February 4, 2015 & Meucci s Program for Asset Allocation detect market invariance select the invariants estimate the market specify the distribution
More informationFunctions of Several Variables: Chain Rules
Functions of Several Variables: Chain Rules Calculus III Josh Engwer TTU 29 September 2014 Josh Engwer (TTU) Functions of Several Variables: Chain Rules 29 September 2014 1 / 30 PART I PART I: MULTIVARIABLE
More informationOrder book resilience, price manipulation, and the positive portfolio problem
Order book resilience, price manipulation, and the positive portfolio problem Alexander Schied Mannheim University Workshop on New Directions in Financial Mathematics Institute for Pure and Applied Mathematics,
More informationMinimization of ruin probabilities by investment under transaction costs
Minimization of ruin probabilities by investment under transaction costs Stefan Thonhauser DSA, HEC, Université de Lausanne 13 th Scientific Day, Bonn, 3.4.214 Outline Introduction Risk models and controls
More informationThe Cake-Eating problem: Non-linear sharing rules
The Cake-Eating problem: Non-linear sharing rules Eugenio Peluso 1 and Alain Trannoy 2 Conference In Honor of Louis Eeckhoudt June 2012 1 Department of Economics, University of Verona (Italy) 2 Aix-Marseille
More informationA Model of Optimal Portfolio Selection under. Liquidity Risk and Price Impact
A Model of Optimal Portfolio Selection under Liquidity Risk and Price Impact Huyên PHAM Workshop on PDE and Mathematical Finance KTH, Stockholm, August 15, 2005 Laboratoire de Probabilités et Modèles Aléatoires
More information1 Bewley Economies with Aggregate Uncertainty
1 Bewley Economies with Aggregate Uncertainty Sofarwehaveassumedawayaggregatefluctuations (i.e., business cycles) in our description of the incomplete-markets economies with uninsurable idiosyncratic risk
More informationLiquidity risk and optimal dividend/investment strategies
Liquidity risk and optimal dividend/investment strategies Vathana LY VATH Laboratoire de Mathématiques et Modélisation d Evry ENSIIE and Université d Evry Joint work with E. Chevalier and M. Gaigi ICASQF,
More informationRobust Mean-Variance Hedging via G-Expectation
Robust Mean-Variance Hedging via G-Expectation Francesca Biagini, Jacopo Mancin Thilo Meyer Brandis January 3, 18 Abstract In this paper we study mean-variance hedging under the G-expectation framework.
More informationCompetitive Equilibria in a Comonotone Market
Competitive Equilibria in a Comonotone Market 1/51 Competitive Equilibria in a Comonotone Market Ruodu Wang http://sas.uwaterloo.ca/ wang Department of Statistics and Actuarial Science University of Waterloo
More informationOptimal Stopping Problems and American Options
Optimal Stopping Problems and American Options Nadia Uys A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master
More informationThe priority option: the value of being a leader in complete and incomplete markets.
s. a leader in s. Mathematics and Statistics - McMaster University Joint work with Vincent Leclère (École de Ponts) Eidgenössische Technische Hochschule Zürich, December 09, 2010 s. 1 2 3 4 s. Combining
More informationApproximation around the risky steady state
Approximation around the risky steady state Centre for International Macroeconomic Studies Conference University of Surrey Michel Juillard, Bank of France September 14, 2012 The views expressed herein
More informationRisk-Sensitive and Robust Mean Field Games
Risk-Sensitive and Robust Mean Field Games Tamer Başar Coordinated Science Laboratory Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, IL - 6181 IPAM
More informationNotes on Recursive Utility. Consider the setting of consumption in infinite time under uncertainty as in
Notes on Recursive Utility Consider the setting of consumption in infinite time under uncertainty as in Section 1 (or Chapter 29, LeRoy & Werner, 2nd Ed.) Let u st be the continuation utility at s t. That
More informationOptimal transportation and optimal control in a finite horizon framework
Optimal transportation and optimal control in a finite horizon framework Guillaume Carlier and Aimé Lachapelle Université Paris-Dauphine, CEREMADE July 2008 1 MOTIVATIONS - A commitment problem (1) Micro
More informationRobust Optimization: Applications in Portfolio Selection Problems
Robust Optimization: Applications in Portfolio Selection Problems Vris Cheung and Henry Wolkowicz WatRISQ University of Waterloo Vris Cheung (University of Waterloo) Robust optimization 2009 1 / 19 Outline
More informationLong-tem policy-making, Lecture 5
Long-tem policy-making, Lecture 5 July 2008 Ivar Ekeland and Ali Lazrak PIMS Summer School on Perceiving, Measuring and Managing Risk July 7, 2008 var Ekeland and Ali Lazrak (PIMS Summer School Long-tem
More informationSemidefinite and Second Order Cone Programming Seminar Fall 2012 Project: Robust Optimization and its Application of Robust Portfolio Optimization
Semidefinite and Second Order Cone Programming Seminar Fall 2012 Project: Robust Optimization and its Application of Robust Portfolio Optimization Instructor: Farid Alizadeh Author: Ai Kagawa 12/12/2012
More informationDecision principles derived from risk measures
Decision principles derived from risk measures Marc Goovaerts Marc.Goovaerts@econ.kuleuven.ac.be Katholieke Universiteit Leuven Decision principles derived from risk measures - Marc Goovaerts p. 1/17 Back
More informationSensitivity analysis of the expected utility maximization problem with respect to model perturbations
Sensitivity analysis of the expected utility maximization problem with respect to model perturbations Mihai Sîrbu, The University of Texas at Austin based on joint work with Oleksii Mostovyi University
More informationOn Minimal Entropy Martingale Measures
On Minimal Entropy Martingale Measures Andrii Andrusiv Friedrich Schiller University of Jena, Germany Marie Curie ITN Outline 1. One Period Model Definitions and Notations Minimal Entropy Martingale Measure
More information4. Conditional risk measures and their robust representation
4. Conditional risk measures and their robust representation We consider a discrete-time information structure given by a filtration (F t ) t=0,...,t on our probability space (Ω, F, P ). The time horizon
More informationA Dynamic Model for Investment Strategy
A Dynamic Model for Investment Strategy Richard Grinold Stanford Conference on Quantitative Finance August 18-19 2006 Preview Strategic view of risk, return and cost Not intended as a portfolio management
More informationGeometry and Optimization of Relative Arbitrage
Geometry and Optimization of Relative Arbitrage Ting-Kam Leonard Wong joint work with Soumik Pal Department of Mathematics, University of Washington Financial/Actuarial Mathematic Seminar, University of
More informationSolution of Stochastic Optimal Control Problems and Financial Applications
Journal of Mathematical Extension Vol. 11, No. 4, (2017), 27-44 ISSN: 1735-8299 URL: http://www.ijmex.com Solution of Stochastic Optimal Control Problems and Financial Applications 2 Mat B. Kafash 1 Faculty
More informationIn the Ramsey model we maximized the utility U = u[c(t)]e nt e t dt. Now
PERMANENT INCOME AND OPTIMAL CONSUMPTION On the previous notes we saw how permanent income hypothesis can solve the Consumption Puzzle. Now we use this hypothesis, together with assumption of rational
More informationWeak solutions of mean-field stochastic differential equations
Weak solutions of mean-field stochastic differential equations Juan Li School of Mathematics and Statistics, Shandong University (Weihai), Weihai 26429, China. Email: juanli@sdu.edu.cn Based on joint works
More informationModern Portfolio Theory with Homogeneous Risk Measures
Modern Portfolio Theory with Homogeneous Risk Measures Dirk Tasche Zentrum Mathematik Technische Universität München http://www.ma.tum.de/stat/ Rotterdam February 8, 2001 Abstract The Modern Portfolio
More informationHigh-dimensional Problems in Finance and Economics. Thomas M. Mertens
High-dimensional Problems in Finance and Economics Thomas M. Mertens NYU Stern Risk Economics Lab April 17, 2012 1 / 78 Motivation Many problems in finance and economics are high dimensional. Dynamic Optimization:
More informationOptimal Risk Sharing in the Presence of Moral Hazard under Market Risk and Jump Risk
Optimal Risk Sharing in the Presence of Moral Hazard under Market Risk and Jump Risk Takashi Misumi Hitotsubashi University takashi.misumi@r.hit-u.ac.jp Koichiro Takaoka Hitotsubashi University k.takaoka@r.hit-u.ac.jp
More informationBOSTON UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES. Dissertation STOCHASTIC CONTROL PROBLEMS WITH PERFORMANCE FEES AND INCOMPLETE MARKETS GU WANG
BOSON UNIVERSIY GRADUAE SCHOOL OF ARS AND SCIENCES Dissertation SOCHASIC CONROL PROBLEMS WIH PERFORMANCE FEES AND INCOMPLEE MARKES by GU WANG B.S., Peking University, 7 M.S., Boston University, Submitted
More informationIntroduction to Algorithmic Trading Strategies Lecture 4
Introduction to Algorithmic Trading Strategies Lecture 4 Optimal Pairs Trading by Stochastic Control Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Problem formulation Ito s lemma
More informationThe Robust Merton Problem of an Ambiguity Averse Investor
The Robust Merton Problem of an Ambiguity Averse Investor Sara Biagini LUISS G.Carli, Roma (joint with Mustafa Ç. Pınar, Bilkent University) Vienna University of Economics and Business, January 27, 2017
More informationMean-field SDE driven by a fractional BM. A related stochastic control problem
Mean-field SDE driven by a fractional BM. A related stochastic control problem Rainer Buckdahn, Université de Bretagne Occidentale, Brest Durham Symposium on Stochastic Analysis, July 1th to July 2th,
More informationAn Uncertain Control Model with Application to. Production-Inventory System
An Uncertain Control Model with Application to Production-Inventory System Kai Yao 1, Zhongfeng Qin 2 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Economics
More information