Implementing an Agent-Based General Equilibrium Model
|
|
- Caitlin Eaton
- 6 years ago
- Views:
Transcription
1 Implementing an Agent-Based General Equilibrium Model
2 1 2 3
3 Pure Exchange General Equilibrium We shall take N dividend processes δ n (t) as exogenous with a distribution which is known to all agents There are a large number of agents with differing utilities who trade claims to these cash flows Relevant prices are the interest rate, r(t), and a price of risk vector, θ(t) Markets clear: in aggregate all cash flows are consumed and all claims are held
4 Bayesian Equilibrium Agents know the possible types of other investors in the market but do not know the wealth of each type Each agent knows there may be many others of his own type and so knowing his own wealth does not help him infer the distribution of wealth across types All agents start out with homogeneous beliefs about this distribution Observing equilibrium r(t) and θ(t) provides the information for updating this distribution
5 Stochastic Discount Factors Recall that the stochastic factor, H(t) gives the time t price of any asset which pays cash flow ξ at T as It s dynamics are 1 H(t) E t[ξh(t )] dh(t) H(t) = r(t)dt θ(t) dw (t) What we are looking for is the dynamics of H(t) in terms of the observables
6 A Simple Example Suppose that all agents have log utility over consumption but different rates of time preference There is only one risky asset and agents own proportions w k of it. [ ] T max E e ρ k t log(c k (t))dt c k (t) 0 subject to [ T ] [ T ] E H(t)c k (t)dt = E H(t)w k δ(t)dt 0 0
7 Solution to Agent s Problems The solutions to the agents problem are given by where c k (t) = 1 H(t) e ρ k t γ k x k γ k = ρ k 1 e ρ k T and x k is the total starting wealth of agents of type k. By starting wealth we mean the value of the endowment stream of this type
8 Aggregation Since in equilibrium all cash flows are consumed we must have that δ(t) = 1 e ρ k t γ k x k H(t) Investors know that this must hold but they don t know the x k But their beliefs must be consistent with this market clearing condition δ(t) = 1 H(t) k e ρ k t γ k E t [x k ] k
9 Updating Let Y k (t) = E t [x k ] be the common beliefs of agents at time t about the starting wealth of type k This must a non-negative martingale so it follows that it is an exponential martingale dy k (t) Y k (t) = v k(t) dw (t) This extra uncertainty must be reflected in the stochastic discount factor dynamics
10 Equilibrium Applying the Ito formula to the market clearing condition and matching coefficients we obtain r(t) = K k=1 e ρ k t γ k Y k (t)ρ k K k=1 e ρ k t γ k Y k (t) + µ δ (t) θ(t) θ(t) K k=1 θ(t) = σ δ (t) e ρ k t γ k Y k (t)v k (t) K k=1 e ρ k t γ k Y k (t) If all the ρ k were the same and the v k were zero then we would have the classical result
11 Why is this Interesting? Notice that even with log-normal endowment shocks we have stochastic interest rates and risk prices in this model Empirical research suggests that a large portion of the variation in prices that we observe is due to time varying risk-prices But the source of the change in risk prices has been hard to identify Here we find that at least part of it is due to aggregate uncertainty about market structure
12 The Goal We want to be able to show how shocks propegate from one asset class to another, i.e. subprime CMOs to general stock market This means we need more than one risky asset We expect that the mechanism is that losses in one asset class changes the wealth of one type of investor disproportionately which causes large changes in risk pricing So investor types must be quite different from eachother We may even need to constrain some investors
13 Other Utility Functions The model just presented is the only one that can be solved analytically Non-log investors are necessary for realism consider the HARA class We can solve for optimal consumption and investment behavior for HARA investors but only with certain class of distributions for r(t) and θ(t) So we must assume that even with updating we always stay within this class
14 Agents with Metacognition In computer science intelligent agents are able to choose from a set of actions based on observed data In this work the agents are smarter than that in that they know their beliefs might be wrong and can adjust Agents also observe the results of their actions (equilibrium r(t) and θ(t)) and determine if this is consistent with their observations and beliefs introspection If not then they update their beliefs before taking new actions reflection
15 Gaussian State Vector Agents know that endowment growth rate is linear in a state vector Y which satisfies They believe that dy (t) = K (Θ Y (t)dt + ΣdW (t) r(t) = d 0 + d 1 Y (t) + Y (t) d 2 Y (t) and θ(t) = θ 0 + θ 1 Y (t)
16 Updating Investors are assumed to see Y (t) each period Equilibrium r(t) and θ(t) are also observed The parameters of the functions that relate r(t) and θ(t) to Y (t) are updated each period to reconcile these observations Ideally we would like this to take place during the market-clearing process, but the computational burden is high
17 Consumption Choice HARA investors optimal consumption c k (t) = X k(t) G k (t, T ) where X k (t) is current wealth and G k (t, T ) T t e ρ k β k (s t) F k (t, s)ds where β k is a risk tolerance parameter and F is a function of Y and t which solves a parabolic PDE
18 Exponential Quadratic Forms The Guassian state vector and the assumed forms of µ δ, r(t) and θ(t) guarantee that this PDE has a solution of the form ( F k (t, T ) = exp C(τ) + D(τ) Y (t) + 1 ) 2 Y (t) Q(τ)Y (t) where τ = T t The function C, D, and Q satisfy a particular set of ODEs
19 Investment Decision A HARA investor chooses investments according to ( π k (t) = β k X(t) σ(t) ) 1 ( θ(t) + Xk (t) σ(t) ) 1 gk (t, T ) where g k (t, T ) is the volatility of G k (t, T ) But to know σ(t) we need to be able to compute prices of risky assets This is another set of PDEs
20 Market Clearing The market clearing r(t) and θ(t) are determined by numerical search The observed Y (t) and the updated parameters determine agent demands for consumption and investment So at each iteration in finding market clearing we need to solve a set of ODEs and do several numerical integrations
21 Open Questions We do not yet know how to incorporate the extra variation caused by updating into the H(t) We can make guesses about it s magnitude and incorporate those guesses But then we may have to run the model long enough to calibrate these guesses to reality Lots of CPU crunching ahead of us
QI SHANG: General Equilibrium Analysis of Portfolio Benchmarking
General Equilibrium Analysis of Portfolio Benchmarking QI SHANG 23/10/2008 Introduction The Model Equilibrium Discussion of Results Conclusion Introduction This paper studies the equilibrium effect of
More informationSPDE and portfolio choice (joint work with M. Musiela) Princeton University. Thaleia Zariphopoulou The University of Texas at Austin
SPDE and portfolio choice (joint work with M. Musiela) Princeton University November 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 Performance measurement of investment strategies 2 Market
More informationHeterogeneous Firm, Financial Market Integration and International Risk Sharing
Heterogeneous Firm, Financial Market Integration and International Risk Sharing Ming-Jen Chang, Shikuan Chen and Yen-Chen Wu National DongHwa University Thursday 22 nd November 2018 Department of Economics,
More informationComprehensive Exam. August 19, 2013
Comprehensive Exam August 19, 2013 You have a total of 180 minutes to complete the exam. If a question seems ambiguous, state why, sharpen it up and answer the sharpened-up question. Good luck! 1 1 Menu
More informationθ(t ) = T f(0, T ) + σ2 T
1 Derivatives Pricing and Financial Modelling Andrew Cairns: room M3.08 E-mail: A.Cairns@ma.hw.ac.uk Tutorial 10 1. (Ho-Lee) Let X(T ) = T 0 W t dt. (a) What is the distribution of X(T )? (b) Find E[exp(
More informationProblem Set 3. Thomas Philippon. April 19, Human Wealth, Financial Wealth and Consumption
Problem Set 3 Thomas Philippon April 19, 2002 1 Human Wealth, Financial Wealth and Consumption The goal of the question is to derive the formulas on p13 of Topic 2. This is a partial equilibrium analysis
More informationArbitrageurs, bubbles and credit conditions
Arbitrageurs, bubbles and credit conditions Julien Hugonnier (SFI @ EPFL) and Rodolfo Prieto (BU) 8th Cowles Conference on General Equilibrium and its Applications April 28, 212 Motivation Loewenstein
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationMulti-dimensional Term Structure Models
Multi-dimensional Term Structure Models We will focus on the affine class. But first some motivation. A generic one-dimensional model for zero-coupon yields, y(t; τ), looks like this dy(t; τ) =... dt +
More informationHousehold income risk, nominal frictions, and incomplete markets 1
Household income risk, nominal frictions, and incomplete markets 1 2013 North American Summer Meeting Ralph Lütticke 13.06.2013 1 Joint-work with Christian Bayer, Lien Pham, and Volker Tjaden 1 / 30 Research
More informationThe Life Cycle Model with Recursive Utility: Defined benefit vs defined contribution.
The Life Cycle Model with Recursive Utility: Defined benefit vs defined contribution. Knut K. Aase Norwegian School of Economics 5045 Bergen, Norway IACA/PBSS Colloquium Cancun 2017 June 6-7, 2017 1. Papers
More informationFINANCIAL OPTIMIZATION. Lecture 5: Dynamic Programming and a Visit to the Soft Side
FINANCIAL OPTIMIZATION Lecture 5: Dynamic Programming and a Visit to the Soft Side Copyright c Philip H. Dybvig 2008 Dynamic Programming All situations in practice are more complex than the simple examples
More informationAsset Pricing and Equity Premium Puzzle. E. Young Lecture Notes Chapter 13
Asset Pricing and Equity Premium Puzzle 1 E. Young Lecture Notes Chapter 13 1 A Lucas Tree Model Consider a pure exchange, representative household economy. Suppose there exists an asset called a tree.
More informationConvergence of Life Expectancy and Living Standards in the World
Convergence of Life Expectancy and Living Standards in the World Kenichi Ueda* *The University of Tokyo PRI-ADBI Joint Workshop January 13, 2017 The views are those of the author and should not be attributed
More informationLecture 4. Finite difference and finite element methods
Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation
More informationThe Equity Premium Puzzle, the consumption puzzle and the investment puzzle with Recursive Utility: Implications for optimal pensions.
The Equity Premium Puzzle, the consumption puzzle and the investment puzzle with Recursive Utility: Implications for optimal pensions. Knut K. Aase Norwegian School of Economics 5045 Bergen, Norway IAALS
More informationConsumption and Portfolio Decisions When Expected Returns A
Consumption and Portfolio Decisions When Expected Returns Are Time Varying September 10, 2007 Introduction In the recent literature of empirical asset pricing there has been considerable evidence of time-varying
More informationChapter 5 Macroeconomics and Finance
Macro II Chapter 5 Macro and Finance 1 Chapter 5 Macroeconomics and Finance Main references : - L. Ljundqvist and T. Sargent, Chapter 7 - Mehra and Prescott 1985 JME paper - Jerman 1998 JME paper - J.
More informationApplication of Stochastic Calculus to Price a Quanto Spread
Application of Stochastic Calculus to Price a Quanto Spread Christopher Ting http://www.mysmu.edu/faculty/christophert/ Algorithmic Quantitative Finance July 15, 2017 Christopher Ting July 15, 2017 1/33
More informationMACROECONOMICS. Prelim Exam
MACROECONOMICS Prelim Exam Austin, June 1, 2012 Instructions This is a closed book exam. If you get stuck in one section move to the next one. Do not waste time on sections that you find hard to solve.
More informationBACHELIER FINANCE SOCIETY. 4 th World Congress Tokyo, Investments and forward utilities. Thaleia Zariphopoulou The University of Texas at Austin
BACHELIER FINANCE SOCIETY 4 th World Congress Tokyo, 26 Investments and forward utilities Thaleia Zariphopoulou The University of Texas at Austin 1 Topics Utility-based measurement of performance Utilities
More informationLimits to Arbitrage. George Pennacchi. Finance 591 Asset Pricing Theory
Limits to Arbitrage George Pennacchi Finance 591 Asset Pricing Theory I.Example: CARA Utility and Normal Asset Returns I Several single-period portfolio choice models assume constant absolute risk-aversion
More informationRECURSIVE VALUATION AND SENTIMENTS
1 / 32 RECURSIVE VALUATION AND SENTIMENTS Lars Peter Hansen Bendheim Lectures, Princeton University 2 / 32 RECURSIVE VALUATION AND SENTIMENTS ABSTRACT Expectations and uncertainty about growth rates that
More informationManaging Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives
Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives Simon Man Chung Fung, Katja Ignatieva and Michael Sherris School of Risk & Actuarial Studies University of
More informationThe Measurement Procedure of AB2017 in a Simplified Version of McGrattan 2017
The Measurement Procedure of AB2017 in a Simplified Version of McGrattan 2017 Andrew Atkeson and Ariel Burstein 1 Introduction In this document we derive the main results Atkeson Burstein (Aggregate Implications
More informationLocal Volatility Dynamic Models
René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Columbia November 9, 27 Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding
More information1 Dynamic programming
1 Dynamic programming A country has just discovered a natural resource which yields an income per period R measured in terms of traded goods. The cost of exploitation is negligible. The government wants
More informationOn modelling of electricity spot price
, Rüdiger Kiesel and Fred Espen Benth Institute of Energy Trading and Financial Services University of Duisburg-Essen Centre of Mathematics for Applications, University of Oslo 25. August 2010 Introduction
More informationExplaining the Boom-Bust Cycle in the U.S. Housing Market: A Reverse-Engineering Approach
Explaining the Boom-Bust Cycle in the U.S. Housing Market: A Reverse-Engineering Approach Paolo Gelain Norges Bank Kevin J. Lansing FRBSF Gisle J. Navik Norges Bank October 22, 2014 RBNZ Workshop The Interaction
More informationThe Macroeconomics of Universal Health Insurance Vouchers
The Macroeconomics of Universal Health Insurance Vouchers Juergen Jung Towson University Chung Tran University of New South Wales Jul-Aug 2009 Jung and Tran (TU and UNSW) Health Vouchers 2009 1 / 29 Dysfunctional
More informationForward Dynamic Utility
Forward Dynamic Utility El Karoui Nicole & M RAD Mohamed UnivParis VI / École Polytechnique,CMAP elkaroui@cmapx.polytechnique.fr with the financial support of the "Fondation du Risque" and the Fédération
More information1 Asset Pricing: Replicating portfolios
Alberto Bisin Corporate Finance: Lecture Notes Class 1: Valuation updated November 17th, 2002 1 Asset Pricing: Replicating portfolios Consider an economy with two states of nature {s 1, s 2 } and with
More informationLabor Economics Field Exam Spring 2011
Labor Economics Field Exam Spring 2011 Instructions You have 4 hours to complete this exam. This is a closed book examination. No written materials are allowed. You can use a calculator. THE EXAM IS COMPOSED
More informationCarnegie Mellon University Graduate School of Industrial Administration
Carnegie Mellon University Graduate School of Industrial Administration Chris Telmer Winter 2005 Final Examination Seminar in Finance 1 (47 720) Due: Thursday 3/3 at 5pm if you don t go to the skating
More informationSTOCHASTIC INTEGRALS
Stat 391/FinMath 346 Lecture 8 STOCHASTIC INTEGRALS X t = CONTINUOUS PROCESS θ t = PORTFOLIO: #X t HELD AT t { St : STOCK PRICE M t : MG W t : BROWNIAN MOTION DISCRETE TIME: = t < t 1
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationIdentifying Long-Run Risks: A Bayesian Mixed-Frequency Approach
Identifying : A Bayesian Mixed-Frequency Approach Frank Schorfheide University of Pennsylvania CEPR and NBER Dongho Song University of Pennsylvania Amir Yaron University of Pennsylvania NBER February 12,
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationThe Risky Steady State and the Interest Rate Lower Bound
The Risky Steady State and the Interest Rate Lower Bound Timothy Hills Taisuke Nakata Sebastian Schmidt New York University Federal Reserve Board European Central Bank 1 September 2016 1 The views expressed
More informationGreek parameters of nonlinear Black-Scholes equation
International Journal of Mathematics and Soft Computing Vol.5, No.2 (2015), 69-74. ISSN Print : 2249-3328 ISSN Online: 2319-5215 Greek parameters of nonlinear Black-Scholes equation Purity J. Kiptum 1,
More informationEconomics 742 Brief Answers, Homework #2
Economics 742 Brief Answers, Homework #2 March 20, 2006 Professor Scholz ) Consider a person, Molly, living two periods. Her labor income is $ in period and $00 in period 2. She can save at a 5 percent
More informationLecture 18. More on option pricing. Lecture 18 1 / 21
Lecture 18 More on option pricing Lecture 18 1 / 21 Introduction In this lecture we will see more applications of option pricing theory. Lecture 18 2 / 21 Greeks (1) The price f of a derivative depends
More informationGeneralized Multi-Factor Commodity Spot Price Modeling through Dynamic Cournot Resource Extraction Models
Generalized Multi-Factor Commodity Spot Price Modeling through Dynamic Cournot Resource Extraction Models Bilkan Erkmen (joint work with Michael Coulon) Workshop on Stochastic Games, Equilibrium, and Applications
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationOptimal investments under dynamic performance critria. Lecture IV
Optimal investments under dynamic performance critria Lecture IV 1 Utility-based measurement of performance 2 Deterministic environment Utility traits u(x, t) : x wealth and t time Monotonicity u x (x,
More informationImplementing the HJM model by Monte Carlo Simulation
Implementing the HJM model by Monte Carlo Simulation A CQF Project - 2010 June Cohort Bob Flagg Email: bob@calcworks.net January 14, 2011 Abstract We discuss an implementation of the Heath-Jarrow-Morton
More informationLecture 2: Stochastic Discount Factor
Lecture 2: Stochastic Discount Factor Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013 Stochastic Discount Factor (SDF) A stochastic discount factor is a stochastic process {M t,t+s } such that
More informationEXAMINING MACROECONOMIC MODELS
1 / 24 EXAMINING MACROECONOMIC MODELS WITH FINANCE CONSTRAINTS THROUGH THE LENS OF ASSET PRICING Lars Peter Hansen Benheim Lectures, Princeton University EXAMINING MACROECONOMIC MODELS WITH FINANCING CONSTRAINTS
More informationLecture 5: Review of interest rate models
Lecture 5: Review of interest rate models Xiaoguang Wang STAT 598W January 30th, 2014 (STAT 598W) Lecture 5 1 / 46 Outline 1 Bonds and Interest Rates 2 Short Rate Models 3 Forward Rate Models 4 LIBOR and
More informationCredit Risk : Firm Value Model
Credit Risk : Firm Value Model Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe and Karlsruhe Institute of Technology (KIT) Prof. Dr. Svetlozar Rachev
More informationPricing in markets modeled by general processes with independent increments
Pricing in markets modeled by general processes with independent increments Tom Hurd Financial Mathematics at McMaster www.phimac.org Thanks to Tahir Choulli and Shui Feng Financial Mathematics Seminar
More informationThe Self-financing Condition: Remembering the Limit Order Book
The Self-financing Condition: Remembering the Limit Order Book R. Carmona, K. Webster Bendheim Center for Finance ORFE, Princeton University November 6, 2013 Structural relationships? From LOB Models to
More informationLecture 3: Review of mathematical finance and derivative pricing models
Lecture 3: Review of mathematical finance and derivative pricing models Xiaoguang Wang STAT 598W January 21th, 2014 (STAT 598W) Lecture 3 1 / 51 Outline 1 Some model independent definitions and principals
More informationContinuous Time Finance. Tomas Björk
Continuous Time Finance Tomas Björk 1 II Stochastic Calculus Tomas Björk 2 Typical Setup Take as given the market price process, S(t), of some underlying asset. S(t) = price, at t, per unit of underlying
More informationOptimal asset allocation under forward performance criteria Oberwolfach, February 2007
Optimal asset allocation under forward performance criteria Oberwolfach, February 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 References Indifference valuation in binomial models (with
More informationAdvanced topics in continuous time finance
Based on readings of Prof. Kerry E. Back on the IAS in Vienna, October 21. Advanced topics in continuous time finance Mag. Martin Vonwald (martin@voni.at) November 21 Contents 1 Introduction 4 1.1 Martingale.....................................
More informationA Model of Financial Intermediation
A Model of Financial Intermediation Jesús Fernández-Villaverde University of Pennsylvania December 25, 2012 Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, 2012 1 / 43
More information4. Black-Scholes Models and PDEs. Math6911 S08, HM Zhu
4. Black-Scholes Models and PDEs Math6911 S08, HM Zhu References 1. Chapter 13, J. Hull. Section.6, P. Brandimarte Outline Derivation of Black-Scholes equation Black-Scholes models for options Implied
More informationModern Dynamic Asset Pricing Models
Modern Dynamic Asset Pricing Models Lecture Notes 7. Term Structure Models Pietro Veronesi University of Chicago Booth School of Business CEPR, NBER Pietro Veronesi Term Structure Models page: 2 Outline
More informationOPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF FINITE
Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 005 Seville, Spain, December 1-15, 005 WeA11.6 OPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF
More information1 Consumption and saving under uncertainty
1 Consumption and saving under uncertainty 1.1 Modelling uncertainty As in the deterministic case, we keep assuming that agents live for two periods. The novelty here is that their earnings in the second
More informationThe Black-Scholes Equation using Heat Equation
The Black-Scholes Equation using Heat Equation Peter Cassar May 0, 05 Assumptions of the Black-Scholes Model We have a risk free asset given by the price process, dbt = rbt The asset price follows a geometric
More informationOptimal Investment with Deferred Capital Gains Taxes
Optimal Investment with Deferred Capital Gains Taxes A Simple Martingale Method Approach Frank Thomas Seifried University of Kaiserslautern March 20, 2009 F. Seifried (Kaiserslautern) Deferred Capital
More informationA potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples
1.3 Regime switching models A potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples (or regimes). If the dates, the
More informationA Model with Costly Enforcement
A Model with Costly Enforcement Jesús Fernández-Villaverde University of Pennsylvania December 25, 2012 Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 1 / 43 A Model with Costly
More informationdt+ ρσ 2 1 ρ2 σ 2 κ i and that A is a rather lengthy expression that we may or may not need. (Brigo & Mercurio Lemma Thm , p. 135.
A 2D Gaussian model (akin to Brigo & Mercurio Section 4.2) Suppose where ( κ1 0 dx(t) = 0 κ 2 r(t) = δ 0 +X 1 (t)+x 2 (t) )( X1 (t) X 2 (t) ) ( σ1 0 dt+ ρσ 2 1 ρ2 σ 2 )( dw Q 1 (t) dw Q 2 (t) ) In this
More informationExercises on the New-Keynesian Model
Advanced Macroeconomics II Professor Lorenza Rossi/Jordi Gali T.A. Daniël van Schoot, daniel.vanschoot@upf.edu Exercises on the New-Keynesian Model Schedule: 28th of May (seminar 4): Exercises 1, 2 and
More informationECON 6022B Problem Set 2 Suggested Solutions Fall 2011
ECON 60B Problem Set Suggested Solutions Fall 0 September 7, 0 Optimal Consumption with A Linear Utility Function (Optional) Similar to the example in Lecture 3, the household lives for two periods and
More informationValuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model
Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 1(23) Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility
More informationMonetary Economics Final Exam
316-466 Monetary Economics Final Exam 1. Flexible-price monetary economics (90 marks). Consider a stochastic flexibleprice money in the utility function model. Time is discrete and denoted t =0, 1,...
More informationStochastic modelling of electricity markets Pricing Forwards and Swaps
Stochastic modelling of electricity markets Pricing Forwards and Swaps Jhonny Gonzalez School of Mathematics The University of Manchester Magical books project August 23, 2012 Clip for this slide Pricing
More informationLinearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing
Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing Liuren Wu, Baruch College Joint work with Peter Carr and Xavier Gabaix at New York University Board of
More informationWhat Can Rational Investors Do About Excessive Volatility and Sentiment Fluctuations?
What Can Rational Investors Do About Excessive Volatility and Sentiment Fluctuations? Bernard Dumas INSEAD, Wharton, CEPR, NBER Alexander Kurshev London Business School Raman Uppal London Business School,
More informationOption pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard
Option pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard Indifference pricing and the minimal entropy martingale measure Fred Espen Benth Centre of Mathematics for Applications
More informationSTATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics. Ph. D. Preliminary Examination: Macroeconomics Fall, 2009
STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics Ph. D. Preliminary Examination: Macroeconomics Fall, 2009 Instructions: Read the questions carefully and make sure to show your work. You
More informationdt + ρσ 2 1 ρ2 σ 2 B i (τ) = 1 e κ iτ κ i
A 2D Gaussian model (akin to Brigo & Mercurio Section 4.2) Suppose where dx(t) = ( κ1 0 0 κ 2 ) ( X1 (t) X 2 (t) In this case we find (BLACKBOARD) that r(t) = δ 0 + X 1 (t) + X 2 (t) ) ( σ1 0 dt + ρσ 2
More informationOil Price Uncertainty in a Small Open Economy
Yusuf Soner Başkaya Timur Hülagü Hande Küçük 6 April 212 Oil price volatility is high and it varies over time... 15 1 5 1985 199 1995 2 25 21 (a) Mean.4.35.3.25.2.15.1.5 1985 199 1995 2 25 21 (b) Coefficient
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postponed exam: ECON4310 Macroeconomic Theory Date of exam: Monday, December 14, 2015 Time for exam: 09:00 a.m. 12:00 noon The problem set covers 13 pages (incl.
More informationProblem set 5. Asset pricing. Markus Roth. Chair for Macroeconomics Johannes Gutenberg Universität Mainz. Juli 5, 2010
Problem set 5 Asset pricing Markus Roth Chair for Macroeconomics Johannes Gutenberg Universität Mainz Juli 5, 200 Markus Roth (Macroeconomics 2) Problem set 5 Juli 5, 200 / 40 Contents Problem 5 of problem
More informationExchange Rates and Fundamentals: A General Equilibrium Exploration
Exchange Rates and Fundamentals: A General Equilibrium Exploration Takashi Kano Hitotsubashi University @HIAS, IER, AJRC Joint Workshop Frontiers in Macroeconomics and Macroeconometrics November 3-4, 2017
More informationA note on the term structure of risk aversion in utility-based pricing systems
A note on the term structure of risk aversion in utility-based pricing systems Marek Musiela and Thaleia ariphopoulou BNP Paribas and The University of Texas in Austin November 5, 00 Abstract We study
More informationInvestigation of Dependency between Short Rate and Transition Rate on Pension Buy-outs. Arık, A. 1 Yolcu-Okur, Y. 2 Uğur Ö. 2
Investigation of Dependency between Short Rate and Transition Rate on Pension Buy-outs Arık, A. 1 Yolcu-Okur, Y. 2 Uğur Ö. 2 1 Hacettepe University Department of Actuarial Sciences 06800, TURKEY 2 Middle
More informationHomework 3: Asset Pricing
Homework 3: Asset Pricing Mohammad Hossein Rahmati November 1, 2018 1. Consider an economy with a single representative consumer who maximize E β t u(c t ) 0 < β < 1, u(c t ) = ln(c t + α) t= The sole
More informationFinancial Risk Management
Financial Risk Management Professor: Thierry Roncalli Evry University Assistant: Enareta Kurtbegu Evry University Tutorial exercices #4 1 Correlation and copulas 1. The bivariate Gaussian copula is given
More informationExtend the ideas of Kan and Zhou paper on Optimal Portfolio Construction under parameter uncertainty
Extend the ideas of Kan and Zhou paper on Optimal Portfolio Construction under parameter uncertainty George Photiou Lincoln College University of Oxford A dissertation submitted in partial fulfilment for
More informationTangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More informationPractical example of an Economic Scenario Generator
Practical example of an Economic Scenario Generator Martin Schenk Actuarial & Insurance Solutions SAV 7 March 2014 Agenda Introduction Deterministic vs. stochastic approach Mathematical model Application
More informationInterest rate models and Solvency II
www.nr.no Outline Desired properties of interest rate models in a Solvency II setting. A review of three well-known interest rate models A real example from a Norwegian insurance company 2 Interest rate
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More informationProblem set 1 Answers: 0 ( )= [ 0 ( +1 )] = [ ( +1 )]
Problem set 1 Answers: 1. (a) The first order conditions are with 1+ 1so 0 ( ) [ 0 ( +1 )] [( +1 )] ( +1 ) Consumption follows a random walk. This is approximately true in many nonlinear models. Now we
More informationDiverse Beliefs and Time Variability of Asset Risk Premia
Diverse and Risk The Diverse and Time Variability of M. Kurz, Stanford University M. Motolese, Catholic University of Milan August 10, 2009 Individual State of SITE Summer 2009 Workshop, Stanford University
More informationINTERTEMPORAL ASSET ALLOCATION: THEORY
INTERTEMPORAL ASSET ALLOCATION: THEORY Multi-Period Model The agent acts as a price-taker in asset markets and then chooses today s consumption and asset shares to maximise lifetime utility. This multi-period
More informationASSET PRICING WITH LIMITED RISK SHARING AND HETEROGENOUS AGENTS
ASSET PRICING WITH LIMITED RISK SHARING AND HETEROGENOUS AGENTS Francisco Gomes and Alexander Michaelides Roine Vestman, New York University November 27, 2007 OVERVIEW OF THE PAPER The aim of the paper
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationA simple wealth model
Quantitative Macroeconomics Raül Santaeulàlia-Llopis, MOVE-UAB and Barcelona GSE Homework 5, due Thu Nov 1 I A simple wealth model Consider the sequential problem of a household that maximizes over streams
More informationBasics of Asset Pricing. Ali Nejadmalayeri
Basics of Asset Pricing Ali Nejadmalayeri January 2009 No-Arbitrage and Equilibrium Pricing in Complete Markets: Imagine a finite state space with s {1,..., S} where there exist n traded assets with a
More informationOptimal robust bounds for variance options and asymptotically extreme models
Optimal robust bounds for variance options and asymptotically extreme models Alexander Cox 1 Jiajie Wang 2 1 University of Bath 2 Università di Roma La Sapienza Advances in Financial Mathematics, 9th January,
More informationFinal exam solutions
EE365 Stochastic Control / MS&E251 Stochastic Decision Models Profs. S. Lall, S. Boyd June 5 6 or June 6 7, 2013 Final exam solutions This is a 24 hour take-home final. Please turn it in to one of the
More informationStochastic Partial Differential Equations and Portfolio Choice. Crete, May Thaleia Zariphopoulou
Stochastic Partial Differential Equations and Portfolio Choice Crete, May 2011 Thaleia Zariphopoulou Oxford-Man Institute and Mathematical Institute University of Oxford and Mathematics and IROM, The University
More information