Principal-Agent Problems in Continuous Time
|
|
- Myron Bradford
- 6 years ago
- Views:
Transcription
1 Principal-Agent Problems in Continuous Time Jin Huang March 11, / 33
2 Outline Contract theory in continuous-time models Sannikov s model with infinite time horizon The optimal contract depends on the agent s continuation value and has a lower and upper boundary A procurement model in finite time horizon The optimal contract depends on the continuation value and the state process 2 / 33
3 Related literature Books: - Discrete-time contract theory: Salanie (1997), Laffont and Martinmort (21), Bolton and Dewatripont (24) - Continuous-time: Cvitanic and Zhang (213) Holmstrong and Milgrom (1987), Schatterler and Sung (1993), Williams (23), DeMarzo and Sannikov (27), Sannikov (28), He (28), Cvitanic and Wang and Zhang (29), He (28), Zhu (212), Cvitanic and Wan and Yang (212) 3 / 33
4 Modeling variations Time horizons: finite or infinite State processes: Brownian motion plus drift, geometric Brownian motion, or mean-reverting Payments: instantaneous payments C t, or lump-sum payment C T at the terminal time Information: moral hazard, adverse selection, and/or learning Modeling approaches: weak or strong formulation 4 / 33
5 The contracting environment of Sannikov s model t [, ) The output (state) process: dx t = A t dt + σdb t Effort: A t Hidden action (moral hazard) Per-period payment: C t Random fluctuations: B t 5 / 33
6 The contracting problem r is the common discounting factor The principal offers C to [ max E r C = max C max A E [ r ] e rt (dx t C t dt) ] e rt (A t C t )dt The agent best-responds with A to [ E r e rt( ) u(c t ) h(a t ) dt ] 6 / 33
7 Principal s constrained problem subject to: (P): E max A,C [ E r ] e rt (A t C t )dt [ r ) ] e (u(c rt t ) h(a t ) dt R (IC): A is agent s optimal response to C. 7 / 33
8 Representations of agent s continuation value Define W t E A [ t r t e rs( u(c s ) h(a s ) ) ds ] By the Martingale Representation Theorem, there is a unique Y such that W t = W +r t ( Ws u(c s )+h(a s ) ) t ds+r Y s dzs A The agent is indifferent between getting a promise of 1. {C s : t s }, or 2. W t 8 / 33
9 Incentive compatibility Proposition (Incentives) The following two conditions are equivalent. 1. A is an optimal response to the payment C. 2. A t (ω) arg max a {Y t (ω)a h(a)}, for almost every (t, ω). It is a continuous-time version of the one-shot deviation principle in that A is optimal if and only if A t is optimal at each moment t. 9 / 33
10 The main idea behind the IC proposition Let A be optimal, and assume that the agent follows A up to time t and switch to an alternative A after time t. V t = r ) t (u(c e rs s ) h(a s) ds + e rt W t Drift of V t : re rt( ) [Y t A t h(a t )] [Y t A t h(a t )] dt 1 / 33
11 Designing the optimal pair (A, C) Proposition (Transversality condition) If E A t [e rs Wt+s ] as s, then W t = W t. W t = W + r t ( Ws u(c s ) + h(a s ) ) ds + r t β sdzs A. Principal must pay the agent eventually. Let γ(a) = {y : a arg max a ya h(a )}. To enforce A, the principal promises W t = W +r t ( Ws u(c s )+h(a s ) ) t ds+r γ(a s )dzs A. 11 / 33
12 Converting the principal s problem into a stochastic control problem Let F (w) = u 1 (w). The principal s problem where [ τ ] max E r e rs (A s C s )ds + e rτ F (W τ ), A,C,τ W t = W + r t ( Ws u(c s ) + h(a s ) ) t ds + r γ(a s )db s. 12 / 33
13 The dynamic programming principal and the HJB equation Let F (w) = max A,C,τ E t [ τ ] r e rs (A s C s )ds + e rτ F (Wτ t,w ). t Following the standard arguments of DPP, F ( ) is a solution of the following ODE: ( ) rf = max a,c r(a c) + r w u(c) + h(a) F r2 γ(a) 2 σ 2 F F () = F (W gp ) = F (W gp ), and F (W gp ) = F (W gp) Let a (w) and c (w) be the maximizers. 13 / 33
14 Description of the optimal contract Theorem (Optimal contract) Let W t = W + r + r t t ( ) W s u(c (W s )) + h(a (W s )) ds γ(a (W s ))db s. The stopping rule τ is the first time W t hits lower boundary W t = or the upper boundary W t = W gp. The payment and requested efforts before τ is A t = a (W t ) and C t = c (W t ) and after τ, A t = and C t = F (W τ ). 14 / 33
15 Features of the contract W t summarizes the past history Lower boundary serves as a punishment scheme for incentives Upper boundary is due to income effect; too costly to compensate for the agent for his effort when W t is too high Probational period for low W t. 15 / 33
16 16 / 33
17 Implementation 1. Find W that is the best starting point for the principal. It may be higher than the agent s reservation value R 2. At time t, calculate W t 3. Once W t is known, so is A t = a (W t ) and C t = c (W t ) 4. Retire the agent once W t hits the boundary 17 / 33
18 The contracting environment of a procurement problem t [, T ] The price (state) process is mean reverting: P t = P + t λ(a s P s )ds + t σdm s. Effort: A t Hidden action (moral hazard); the principal cannot contract on actions Lump-sum payment at terminal time: C T Random fluctuations: M t 18 / 33
19 An example: a bilateral contract to supply ancillary services A supplier can manipulate the market price without detection, for example 1. manipulate supplies 2. through virtual bids 3. by proxy The supplier manipulate the price to maximize payment The utility company seeks to pay as little as possible We want to know how to mollify the agent s incentives to manipulate prices through an adjustment C T 19 / 33
20 The contracting problem The agent s problem: max A [ T ( ) ] E U(C T ) + u(p s D s ) h(a s ) ds The principal s problem: min C T [ T ] E C T + P s D s ds 2 / 33
21 The same problem with the ABM state proces subject to: min A,C T [ T ] E C T + P s D s ds 1. Incentive compatibility constraint in that A is an optimal response to C T 2. The participation contraint [ T ( E U(C T ) + u(ps D s ) h(a s ) ) ] ds R 3. The state process is Arithmetic Brownian (not mean-reverting) P t = P + t A s ds + t σdm s 21 / 33
22 First-best (1) 1. Let F ( ) be the solution of the PDE t F (t, p) = min a pd t λu(pd t ) + λh(a) + a p F (t, p) σ2 pp F (t, p) F (T, p) =, for all p, and a,λ (t, p) be the solution of λh (a) + p F (t, p) =. 2. Let C,λ be the solution of 1 = λu (C T ). 22 / 33
23 First-best (2) 3. And λ is found by the participation constraint [ T ( R U(C,λ ) = E where P,λ t u(p,λ s ) ] D s ) h(a,λ (t, Ps,λ )) ds, = P + t a (t, Ps,λ )ds + t σdm s. Proposition (First-best contract) The first best contract is (C,λ, A ), where A = {a,λ (t, P t ) : t T }. 23 / 33
24 First-best (3): Steps to obtain the contract (numerically) 1. Solve the PDE and obtain a,λ ( ) 2. Obtain C,λ by inverting U( ) 3. Find λ by setting participation constraint at equality 4. Calculate (C,λ, A ). 24 / 33
25 Second-best: agent s problem The agent s continuation value is W t = E A t [ U(C T ) + T which has a representation: W A t = U(C T )+ T Proposition (Incentives) t t ] (u(p s D s ) h(a s ))ds, T (u(p s D s ) h(a s )+Z s A s )ds Z s σdb s. t The following two conditions are equivalent. 1. A is an optimal response to the payment C T. 2. A t (ω) arg max a {Z t (ω)a h(a)}, for almost every (t, ω). 25 / 33
26 The resolution of the problem of agency Let γ(a) (resp. η(z)) be the solution of z h (a) = in terms of z (resp. a). 1. Given C T, the response A t = η(z t ) is optimal, where (W, Z) is the unique solution of the BSDE W t = U(C T )+ T t T (u(p s D s ) h(η(z s ))+Z s η(z s ))ds Z s σdb s. t 2. Given (A, W ), the enforcing C T = J(WT A ) where J is the inverse of U( ), and W A t = W t ( ) u(p s D s ) h(a s ) ds + t γ(a s )σdb A s. 26 / 33
27 Second-best: principal s problem Let F be the solution to the PDE t F = min a pd t + a p F σ2 pp F + (h(a) u(pd t )) w F γ(a)2 σ 2 ww F + γ(a)σ 2 pw F F (T, p, w) = J(w), for all p and w, (1) and a (t, p, w) be the optimizer of min a a p F + h(a) w F γ(a)2 σ 2 ww F + γ(a)σ 2 pw F. 27 / 33
28 Second-best: the contract Theorem (Optimal Contract) Let a (t, p, w) be the minimizer in (1), and the agent is paid C T = J( W T ), where P t = P + t a (s, P s, W s )ds + t σdm s W t = R + [ t h(a (s, P s, W ] s )) u(p s D s ) ds + t γ(a (s, P s, W s ))σdm s. Then, the the contract (A, C T ) = ( {a (t, P t, W t ) : t T }, C T ) is incentive compatible for the agent, and optimal for the principal among all incentive compatible contracts that deliver an initial expected value of at least W to the agent. 28 / 33
29 Features of the contract The pair (P t, W t ) summarizes the past history; P t is needed because the restriction on payments There are no boundaries on which the agent is retired. This is due to finite time horizon The calculation of the contract relies on solving a PDE, as opposed to an ODE in infinite time horizon It is never optimal for the principal to gives the agent an initial value more than R 29 / 33
30 An implementation 1. The agent is asked to perform A 2. By time t, the agent is paid t P sd s ds 3. By time t, the agent is also paid R t ( ) u(p s D s ) h(a s) ds + t γ(a s)σdm s Note that P is observable and M is the standard Brownian motion, hence the payments can be calculated by time t. 3 / 33
31 Proof of the optimal contract theorem Verify four lemmas: 1. (C T, A) is optimal for the agent. 2. Any IC contract delivering R to the agent costs the principal at least F (, P, R). 3. (C T, A) costs the principal F (, P, R). 4. Any IC contract delivering more than R is not optimal for the principal. 31 / 33
32 First-best vs. second-best If U( ) and u( ) is linear, then first-best and second-best are identical Using a different approach, the condition for the second-best payment can be reduced to 1 = Γ T U (C T ), comparing to the first best 1 = λu (C T ), where Γ t λ + t σ 1 A sdm s 32 / 33
33 Thoughts Theories on continuous-time models are relatively developed Not too many completely solved models General frameworks have been developed but there are still limitations (Cvitanic and Zhang, 213) Limited work on applying the theories Computational complexities because general theories relie on solving PDEs and BSDEs (also FBSDEs) 33 / 33
Dynamic Contracts: A Continuous-Time Approach
Dynamic Contracts: A Continuous-Time Approach Yuliy Sannikov Stanford University Plan Background: principal-agent models in economics Examples: what questions are economists interested in? Continuous-time
More informationContract Theory in Continuous- Time Models
Jaksa Cvitanic Jianfeng Zhang Contract Theory in Continuous- Time Models fyj Springer Table of Contents Part I Introduction 1 Principal-Agent Problem 3 1.1 Problem Formulation 3 1.2 Further Reading 6 References
More informationLimited liability, or how to prevent slavery in contract theory
Limited liability, or how to prevent slavery in contract theory Université Paris Dauphine, France Joint work with A. Révaillac (INSA Toulouse) and S. Villeneuve (TSE) Advances in Financial Mathematics,
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationDynamic Principal Agent Models: A Continuous Time Approach Lecture II
Dynamic Principal Agent Models: A Continuous Time Approach Lecture II Dynamic Financial Contracting I - The "Workhorse Model" for Finance Applications (DeMarzo and Sannikov 2006) Florian Ho mann Sebastian
More informationGrowth Options, Incentives, and Pay-for-Performance: Theory and Evidence
Growth Options, Incentives, and Pay-for-Performance: Theory and Evidence Sebastian Gryglewicz (Erasmus) Barney Hartman-Glaser (UCLA Anderson) Geoffery Zheng (UCLA Anderson) June 17, 2016 How do growth
More informationLecture 7: Bayesian approach to MAB - Gittins index
Advanced Topics in Machine Learning and Algorithmic Game Theory Lecture 7: Bayesian approach to MAB - Gittins index Lecturer: Yishay Mansour Scribe: Mariano Schain 7.1 Introduction In the Bayesian approach
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 informationMulti-period mean variance asset allocation: Is it bad to win the lottery?
Multi-period mean variance asset allocation: Is it bad to win the lottery? Peter Forsyth 1 D.M. Dang 1 1 Cheriton School of Computer Science University of Waterloo Guangzhou, July 28, 2014 1 / 29 The Basic
More informationOptimal stopping problems for a Brownian motion with a disorder on a finite interval
Optimal stopping problems for a Brownian motion with a disorder on a finite interval A. N. Shiryaev M. V. Zhitlukhin arxiv:1212.379v1 [math.st] 15 Dec 212 December 18, 212 Abstract We consider optimal
More informationInformation, Risk and Economic Policy: A Dynamic Contracting Approach
Information, Risk and Economic Policy: A Dynamic Contracting Approach Noah University of Wisconsin-Madison Or: What I ve Learned from LPH As a student, RA, and co-author Much of my current work builds
More informationDynamic Agency with Persistent Exogenous Shocks
Dynamic Agency with Persistent Exogenous Shocks Rui Li University of Wisconsin-Madison (Job Market Paper) Abstract Several empirical studies have documented the phenomenon of pay for luck a CEO s compensation
More informationStock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models
Stock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models David Prager 1 1 Associate Professor of Mathematics Anderson University (SC) Based on joint work with Professor Qing Zhang,
More informationAn overview of some financial models using BSDE with enlarged filtrations
An overview of some financial models using BSDE with enlarged filtrations Anne EYRAUD-LOISEL Workshop : Enlargement of Filtrations and Applications to Finance and Insurance May 31st - June 4th, 2010, Jena
More informationHedging with Life and General Insurance Products
Hedging with Life and General Insurance Products June 2016 2 Hedging with Life and General Insurance Products Jungmin Choi Department of Mathematics East Carolina University Abstract In this study, a hybrid
More informationOptimal Contract, Ownership Structure and Asset Pricing
Optimal Contract, Ownership Structure and Asset Pricing Hae Won (Henny) Jung University of Melbourne hae.jung@unimelb.edu.au Qi Zeng University of Melbourne qzeng@unimelb.edu.au This version: January 2014
More informationReputation Games in Continuous Time
Reputation Games in Continuous Time Eduardo Faingold Yuliy Sannikov March 15, 25 PRELIMINARY AND INCOMPLETE Abstract In this paper we study a continuous-time reputation game between a large player and
More informationMASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS.
MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED
More informationContinuous-time Stochastic Control and Optimization with Financial Applications
Huyen Pham Continuous-time Stochastic Control and Optimization with Financial Applications 4y Springer Some elements of stochastic analysis 1 1.1 Stochastic processes 1 1.1.1 Filtration and processes 1
More informationAsset pricing under optimal contracts
Asset pricing under optimal contracts Jakša Cvitanić (Caltech) joint work with Hao Xing (LSE) 1/44 Motivation and overview I Existing literature: either - Prices are fixed, optimal contract is found or
More informationFINANCE RESEARCH SEMINAR SUPPORTED BY UNIGESTION
FINANCE RESEARCH SEMINAR SUPPORTED BY UNIGESTION ʺDynamic Agency and Real Optionsʺ Prof. Sebastian Gryglewicz Erasmus School of Economics, Erasmus University Rotterdam Abstract We model a firm facing a
More informationOn the Optimality of Financial Repression
On the Optimality of Financial Repression V.V. Chari, Alessandro Dovis and Patrick Kehoe Conference in honor of Robert E. Lucas Jr, October 2016 Financial Repression Regulation forcing financial institutions
More informationSingular Stochastic Control Models for Optimal Dynamic Withdrawal Policies in Variable Annuities
1/ 46 Singular Stochastic Control Models for Optimal Dynamic Withdrawal Policies in Variable Annuities Yue Kuen KWOK Department of Mathematics Hong Kong University of Science and Technology * Joint work
More informationAsset Pricing Models with Underlying Time-varying Lévy Processes
Asset Pricing Models with Underlying Time-varying Lévy Processes Stochastics & Computational Finance 2015 Xuecan CUI Jang SCHILTZ University of Luxembourg July 9, 2015 Xuecan CUI, Jang SCHILTZ University
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 informationHomework 2: Dynamic Moral Hazard
Homework 2: Dynamic Moral Hazard Question 0 (Normal learning model) Suppose that z t = θ + ɛ t, where θ N(m 0, 1/h 0 ) and ɛ t N(0, 1/h ɛ ) are IID. Show that θ z 1 N ( hɛ z 1 h 0 + h ɛ + h 0m 0 h 0 +
More informationSupply Contracts with Financial Hedging
Supply Contracts with Financial Hedging René Caldentey Martin Haugh Stern School of Business NYU Integrated Risk Management in Operations and Global Supply Chain Management: Risk, Contracts and Insurance
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 information13.3 A Stochastic Production Planning Model
13.3. A Stochastic Production Planning Model 347 From (13.9), we can formally write (dx t ) = f (dt) + G (dz t ) + fgdz t dt, (13.3) dx t dt = f(dt) + Gdz t dt. (13.33) The exact meaning of these expressions
More informationOptimal Investment for Worst-Case Crash Scenarios
Optimal Investment for Worst-Case Crash Scenarios A Martingale Approach Frank Thomas Seifried Department of Mathematics, University of Kaiserslautern June 23, 2010 (Bachelier 2010) Worst-Case Portfolio
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 informationTHE OPTIMAL ASSET ALLOCATION PROBLEMFOR AN INVESTOR THROUGH UTILITY MAXIMIZATION
THE OPTIMAL ASSET ALLOCATION PROBLEMFOR AN INVESTOR THROUGH UTILITY MAXIMIZATION SILAS A. IHEDIOHA 1, BRIGHT O. OSU 2 1 Department of Mathematics, Plateau State University, Bokkos, P. M. B. 2012, Jos,
More informationOn Using Shadow Prices in Portfolio optimization with Transaction Costs
On Using Shadow Prices in Portfolio optimization with Transaction Costs Johannes Muhle-Karbe Universität Wien Joint work with Jan Kallsen Universidad de Murcia 12.03.2010 Outline The Merton problem The
More informationBasic Arbitrage Theory KTH Tomas Björk
Basic Arbitrage Theory KTH 2010 Tomas Björk Tomas Björk, 2010 Contents 1. Mathematics recap. (Ch 10-12) 2. Recap of the martingale approach. (Ch 10-12) 3. Change of numeraire. (Ch 26) Björk,T. Arbitrage
More informationVolatility Smiles and Yield Frowns
Volatility Smiles and Yield Frowns Peter Carr NYU CBOE Conference on Derivatives and Volatility, Chicago, Nov. 10, 2017 Peter Carr (NYU) Volatility Smiles and Yield Frowns 11/10/2017 1 / 33 Interest Rates
More informationOptimal contracts with reflection
Optimal Contracts with Reflection WP 16-14R Borys Grochulski Federal Reserve Bank of Richmond Yuzhe Zhang Texas A&M University Optimal contracts with reflection Borys Grochulski Yuzhe Zhang December 15,
More informationPath Dependent British Options
Path Dependent British Options Kristoffer J Glover (Joint work with G. Peskir and F. Samee) School of Finance and Economics University of Technology, Sydney 18th August 2009 (PDE & Mathematical Finance
More informationON MAXIMIZING DIVIDENDS WITH INVESTMENT AND REINSURANCE
ON MAXIMIZING DIVIDENDS WITH INVESTMENT AND REINSURANCE George S. Ongkeko, Jr. a, Ricardo C.H. Del Rosario b, Maritina T. Castillo c a Insular Life of the Philippines, Makati City 0725, Philippines b Department
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 informationA No-Arbitrage Theorem for Uncertain Stock Model
Fuzzy Optim Decis Making manuscript No (will be inserted by the editor) A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao Received: date / Accepted: date Abstract Stock model is used to describe
More informationResolution of a Financial Puzzle
Resolution of a Financial Puzzle M.J. Brennan and Y. Xia September, 1998 revised November, 1998 Abstract The apparent inconsistency between the Tobin Separation Theorem and the advice of popular investment
More informationOptimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing
Optimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing Prof. Chuan-Ju Wang Department of Computer Science University of Taipei Joint work with Prof. Ming-Yang Kao March 28, 2014
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 informationExam Quantitative Finance (35V5A1)
Exam Quantitative Finance (35V5A1) Part I: Discrete-time finance Exercise 1 (20 points) a. Provide the definition of the pricing kernel k q. Relate this pricing kernel to the set of discount factors D
More informationOptimal Incentive Contract with Costly and Flexible Monitoring
Optimal Incentive Contract with Costly and Flexible Monitoring Anqi Li 1 Ming Yang 2 1 Department of Economics, Washington University in St. Louis 2 Fuqua School of Business, Duke University January 2016
More informationOptimal Trade Execution: Mean Variance or Mean Quadratic Variation?
Optimal Trade Execution: Mean Variance or Mean Quadratic Variation? Peter Forsyth 1 S. Tse 2 H. Windcliff 2 S. Kennedy 2 1 Cheriton School of Computer Science University of Waterloo 2 Morgan Stanley New
More informationAspects of Financial Mathematics:
Aspects of Financial Mathematics: Options, Derivatives, Arbitrage, and the Black-Scholes Pricing Formula J. Robert Buchanan Millersville University of Pennsylvania email: Bob.Buchanan@millersville.edu
More informationDynamic pricing with diffusion models
Dynamic pricing with diffusion models INFORMS revenue management & pricing conference 2017, Amsterdam Asbjørn Nilsen Riseth Supervisors: Jeff Dewynne, Chris Farmer June 29, 2017 OCIAM, University of Oxford
More informationFINANCE RESEARCH SEMINAR SUPPORTED BY UNIGESTION
FINANCE RESEARCH SEMINAR SUPPORTED BY UNIGESTION Cash and Dynamic Agency Prof. Barney HARTMAN-GLASER UCLA, Anderson School of Management Abstract We present an agency model of cash dynamics within a firm.
More informationWORKING PAPER SERIES
WORKING PAPER SERIES No. 2/22 ON ASYMMETRIC INFORMATION ACROSS COUNTRIES AND THE HOME-BIAS PUZZLE Egil Matsen Department of Economics N-749 Trondheim, Norway www.svt.ntnu.no/iso/wp/wp.htm On Asymmetric
More informationRisk Neutral Measures
CHPTER 4 Risk Neutral Measures Our aim in this section is to show how risk neutral measures can be used to price derivative securities. The key advantage is that under a risk neutral measure the discounted
More informationContinuous-Time Consumption and Portfolio Choice
Continuous-Time Consumption and Portfolio Choice Continuous-Time Consumption and Portfolio Choice 1/ 57 Introduction Assuming that asset prices follow di usion processes, we derive an individual s continuous
More informationDynamic Agency and Real Options
Dynamic Agency and Real Options Sebastian Gryglewicz and Barney Hartman-Glaser January 27, 2014 Abstract We present a model integrating dynamic moral hazard and real options. A riskaverse manager can exert
More informationPart 1: q Theory and Irreversible Investment
Part 1: q Theory and Irreversible Investment Goal: Endogenize firm characteristics and risk. Value/growth Size Leverage New issues,... This lecture: q theory of investment Irreversible investment and real
More informationReplication under Price Impact and Martingale Representation Property
Replication under Price Impact and Martingale Representation Property Dmitry Kramkov joint work with Sergio Pulido (Évry, Paris) Carnegie Mellon University Workshop on Equilibrium Theory, Carnegie Mellon,
More informationOptimal Order Placement
Optimal Order Placement Peter Bank joint work with Antje Fruth OMI Colloquium Oxford-Man-Institute, October 16, 2012 Optimal order execution Broker is asked to do a transaction of a significant fraction
More informationCalculating Implied Volatility
Statistical Laboratory University of Cambridge University of Cambridge Mathematics and Big Data Showcase 20 April 2016 How much is an option worth? A call option is the right, but not the obligation, to
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 informationDynamic Replication of Non-Maturing Assets and Liabilities
Dynamic Replication of Non-Maturing Assets and Liabilities Michael Schürle Institute for Operations Research and Computational Finance, University of St. Gallen, Bodanstr. 6, CH-9000 St. Gallen, Switzerland
More informationQI 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 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 informationA Robust Option Pricing Problem
IMA 2003 Workshop, March 12-19, 2003 A Robust Option Pricing Problem Laurent El Ghaoui Department of EECS, UC Berkeley 3 Robust optimization standard form: min x sup u U f 0 (x, u) : u U, f i (x, u) 0,
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 19 11/20/2013. Applications of Ito calculus to finance
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.7J Fall 213 Lecture 19 11/2/213 Applications of Ito calculus to finance Content. 1. Trading strategies 2. Black-Scholes option pricing formula 1 Security
More informationOptimization Models in Financial Mathematics
Optimization Models in Financial Mathematics John R. Birge Northwestern University www.iems.northwestern.edu/~jrbirge Illinois Section MAA, April 3, 2004 1 Introduction Trends in financial mathematics
More informationLecture 14 Consumption under Uncertainty Ricardian Equivalence & Social Security Dynamic General Equilibrium. Noah Williams
Lecture 14 Consumption under Uncertainty Ricardian Equivalence & Social Security Dynamic General Equilibrium Noah Williams University of Wisconsin - Madison Economics 702 Extensions of Permanent Income
More informationIlliquidity, Credit risk and Merton s model
Illiquidity, Credit risk and Merton s model (joint work with J. Dong and L. Korobenko) A. Deniz Sezer University of Calgary April 28, 2016 Merton s model of corporate debt A corporate bond is a contingent
More informationContinuous-Time Pension-Fund Modelling
. Continuous-Time Pension-Fund Modelling Andrew J.G. Cairns Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Riccarton, Edinburgh, EH4 4AS, United Kingdom Abstract This paper
More informationNo-arbitrage theorem for multi-factor uncertain stock model with floating interest rate
Fuzzy Optim Decis Making 217 16:221 234 DOI 117/s17-16-9246-8 No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate Xiaoyu Ji 1 Hua Ke 2 Published online: 17 May 216 Springer
More informationHigh Frequency Trading in a Regime-switching Model. Yoontae Jeon
High Frequency Trading in a Regime-switching Model by Yoontae Jeon A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Mathematics University
More informationBinomial model: numerical algorithm
Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4
More informationSTATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics. Ph. D. Comprehensive Examination: Macroeconomics Spring, 2009
STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics Ph. D. Comprehensive Examination: Macroeconomics Spring, 2009 Section 1. (Suggested Time: 45 Minutes) For 3 of the following 6 statements,
More informationVolatility Smiles and Yield Frowns
Volatility Smiles and Yield Frowns Peter Carr NYU IFS, Chengdu, China, July 30, 2018 Peter Carr (NYU) Volatility Smiles and Yield Frowns 7/30/2018 1 / 35 Interest Rates and Volatility Practitioners and
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 informationValuation of Asian Option. Qi An Jingjing Guo
Valuation of Asian Option Qi An Jingjing Guo CONTENT Asian option Pricing Monte Carlo simulation Conclusion ASIAN OPTION Definition of Asian option always emphasizes the gist that the payoff depends on
More informationBROWNIAN MOTION II. D.Majumdar
BROWNIAN MOTION II D.Majumdar DEFINITION Let (Ω, F, P) be a probability space. For each ω Ω, suppose there is a continuous function W(t) of t 0 that satisfies W(0) = 0 and that depends on ω. Then W(t),
More informationEffects of Wealth and Its Distribution on the Moral Hazard Problem
Effects of Wealth and Its Distribution on the Moral Hazard Problem Jin Yong Jung We analyze how the wealth of an agent and its distribution affect the profit of the principal by considering the simple
More informationValuing Early Stage Investments with Market Related Timing Risk
Valuing Early Stage Investments with Market Related Timing Risk Matt Davison and Yuri Lawryshyn February 12, 216 Abstract In this work, we build on a previous real options approach that utilizes managerial
More informationLinear Capital Taxation and Tax Smoothing
Florian Scheuer 5/1/2014 Linear Capital Taxation and Tax Smoothing 1 Finite Horizon 1.1 Setup 2 periods t = 0, 1 preferences U i c 0, c 1, l 0 sequential budget constraints in t = 0, 1 c i 0 + pbi 1 +
More informationUtility Indifference Pricing and Dynamic Programming Algorithm
Chapter 8 Utility Indifference ricing and Dynamic rogramming Algorithm In the Black-Scholes framework, we can perfectly replicate an option s payoff. However, it may not be true beyond the Black-Scholes
More informationOption Approach to Risk-shifting Incentive Problem with Mutually Correlated Projects
Option Approach to Risk-shifting Incentive Problem with Mutually Correlated Projects Hiroshi Inoue 1, Zhanwei Yang 1, Masatoshi Miyake 1 School of Management, T okyo University of Science, Kuki-shi Saitama
More informationChapter 7: Portfolio Theory
Chapter 7: Portfolio Theory 1. Introduction 2. Portfolio Basics 3. The Feasible Set 4. Portfolio Selection Rules 5. The Efficient Frontier 6. Indifference Curves 7. The Two-Asset Portfolio 8. Unrestriceted
More informationSensitivity of American Option Prices with Different Strikes, Maturities and Volatilities
Applied Mathematical Sciences, Vol. 6, 2012, no. 112, 5597-5602 Sensitivity of American Option Prices with Different Strikes, Maturities and Volatilities Nasir Rehman Department of Mathematics and Statistics
More informationContinuous Time Mean Variance Asset Allocation: A Time-consistent Strategy
Continuous Time Mean Variance Asset Allocation: A Time-consistent Strategy J. Wang, P.A. Forsyth October 24, 2009 Abstract We develop a numerical scheme for determining the optimal asset allocation strategy
More informationOptimal incentive contracts with job destruction risk
Optimal Incentive Contracts with Job Destruction Risk WP 17-11 Borys Grochulski Federal Reserve Bank of Richmond Russell Wong Federal Reserve Bank of Richmond Yuzhe Zhang Texas A&M University Optimal incentive
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 informationEquivalence between Semimartingales and Itô Processes
International Journal of Mathematical Analysis Vol. 9, 215, no. 16, 787-791 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.215.411358 Equivalence between Semimartingales and Itô Processes
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 informationInsurance against Market Crashes
Insurance against Market Crashes Hongzhong Zhang a Tim Leung a Olympia Hadjiliadis b a Columbia University b The City University of New York June 29, 2012 H. Zhang, T. Leung, O. Hadjiliadis (Columbia Insurance
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 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 informationOptimal trading strategies under arbitrage
Optimal trading strategies under arbitrage Johannes Ruf Columbia University, Department of Statistics The Third Western Conference in Mathematical Finance November 14, 2009 How should an investor trade
More informationRisk Minimization Control for Beating the Market Strategies
Risk Minimization Control for Beating the Market Strategies Jan Večeř, Columbia University, Department of Statistics, Mingxin Xu, Carnegie Mellon University, Department of Mathematical Sciences, Olympia
More informationMartingale Approach to Pricing and Hedging
Introduction and echniques Lecture 9 in Financial Mathematics UiO-SK451 Autumn 15 eacher:s. Ortiz-Latorre Martingale Approach to Pricing and Hedging 1 Risk Neutral Pricing Assume that we are in the basic
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More informationThe British Russian Option
The British Russian Option Kristoffer J Glover (Joint work with G. Peskir and F. Samee) School of Finance and Economics University of Technology, Sydney 25th June 2010 (6th World Congress of the BFS, Toronto)
More informationPricing Dynamic Solvency Insurance and Investment Fund Protection
Pricing Dynamic Solvency Insurance and Investment Fund Protection Hans U. Gerber and Gérard Pafumi Switzerland Abstract In the first part of the paper the surplus of a company is modelled by a Wiener process.
More informationA Structural Model for Carbon Cap-and-Trade Schemes
A Structural Model for Carbon Cap-and-Trade Schemes Sam Howison and Daniel Schwarz University of Oxford, Oxford-Man Institute The New Commodity Markets Oxford-Man Institute, 15 June 2011 Introduction The
More informationArbitrage, Martingales, and Pricing Kernels
Arbitrage, Martingales, and Pricing Kernels Arbitrage, Martingales, and Pricing Kernels 1/ 36 Introduction A contingent claim s price process can be transformed into a martingale process by 1 Adjusting
More informationHedging under Arbitrage
Hedging under Arbitrage Johannes Ruf Columbia University, Department of Statistics Modeling and Managing Financial Risks January 12, 2011 Motivation Given: a frictionless market of stocks with continuous
More informationStructural Models of Credit Risk and Some Applications
Structural Models of Credit Risk and Some Applications Albert Cohen Actuarial Science Program Department of Mathematics Department of Statistics and Probability albert@math.msu.edu August 29, 2018 Outline
More information