State-dependent utility maximization in Lévy markets

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "State-dependent utility maximization in Lévy markets"

Transcription

1 State-dependent utility maximization in Lévy markets José Figueroa-López 1 Jin Ma 2 1 University of California at Santa Barbara 2 Purdue University KENT-PURDUE MINI SYMPOSIUM ON FINANCIAL MATHEMATICS, 2007

2 Outline 1 Introduction The portfolio optimization problem Relation to the replication of contingent claims 2 What has been done? Solution using convex duality 3 What is being done here? An explicit" solution for Lévy-driven markets 4 Conclusions

3 The portfolio optimization problem Formulation 1 Market: A risky asset with price process S t. A bond with value process B t. Frictionless with continuous trading. 2 Goal: Allocate an initial wealth w so that to maximize the agent s expected final utility during a finite time horizon [0, T ]. 3 State-dependent utility: U(w, ω) : R + Ω R s.t. Increasing and concave with the wealth w, for each state of nature ω Ω. Constant for wealths w above certain threshold H(ω) > 0. 4 Problem: Maximize E {U(V T, ω)} s.t. the agent s wealth process {V t } 0 t T satisfies V 0 w and V 0. Utility function graph

4 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

5 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

6 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

7 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

8 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

9 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

10 Relation to the replication of contingent claims Optimal partial replication 1 Motivation: w is typically too high". Initial wealth < w = Shortfall risk (sometimes, V T 2 The shortfall minimization problem: < H). Minimize E { L ( (H V T ) +)} s.t. V 0 w and V t 0, where L is increasing, convex, and null at 0. 3 Equivalent formulation: (Föllmer & Leukert, 2000) Maximize E {U (V T, ω)} s.t. V 0 w and V t 0, with U(w; ω) := L(H(ω)) L((H(ω) w) + ). Utility function graph

11 Solution using convex duality The dual problem 1 Primal problem: p (w) := sup E {U(V T, ω)}. V :V 0 w 2 The dual domain Γ: Nonnegative supermartingales {ξ t } t 0 ; 0{ ξ 0 1; } ξ t B 1 t V t is a supermartingale for all {V t } t T. t T 3 The dual problem: d (λ) := inf ξ Γ } E {Ũ(λξT B 1, ω), T where Ũ is the convex dual function of U.

12 Solution using convex duality Relationship between the dual and primal problems Theorem ([KrSch 99], [FllmLkrt, 2000]) 1 Weak duality: p (w) d (λ) + λw, for all λ > 0. 2 Strong duality: p (w) = d (λ ) + λ w, for some λ > 0. 3 The dual problem is attainable at some ξ. 4 The primal problem is attainable at some admissible V. 5 Dual characterization of the optimal final wealth: ( ) V = I λ ξ B 1, T T T where I(z, ω) := (U ) 1 (z) H.

13 Solution using convex duality Refinements for concrete models A natural problem For a specific market model and a given utility function, Can one narrow down the dual domain Γ Γ where to search ξ? 1 Itô incomplete market: (Karatzas et. al. 91) ds t = S t {{b t dt + σ t dw t }. t ξ t = exp 0 G(s)dW s 1 t 2 ds} 0 G(s) 2 = E( 0 G(s)dW s). 2 Lévy market: (Kunita 03) ds t = S t {b t dt + σ t dw t + } h(t, R ( z)ñ(dt, dz) d ξ t = E 0 G(s)dW s + ) F (t, 0 R z)ñ(dt, dz). d

14 Solution using convex duality Refinements for concrete models A natural problem For a specific market model and a given utility function, Can one narrow down the dual domain Γ Γ where to search ξ? 1 Itô incomplete market: (Karatzas et. al. 91) ds t = S t {{b t dt + σ t dw t }. t ξ t = exp 0 G(s)dW s 1 t 2 ds} 0 G(s) 2 = E( 0 G(s)dW s). 2 Lévy market: (Kunita 03) ds t = S t {b t dt + σ t dw t + } h(t, R ( z)ñ(dt, dz) d ξ t = E 0 G(s)dW s + ) F (t, 0 R z)ñ(dt, dz). d

15 Solution using convex duality Refinements for concrete models A natural problem For a specific market model and a given utility function, Can one narrow down the dual domain Γ Γ where to search ξ? 1 Itô incomplete market: (Karatzas et. al. 91) ds t = S t {{b t dt + σ t dw t }. t ξ t = exp 0 G(s)dW s 1 t 2 ds} 0 G(s) 2 = E( 0 G(s)dW s). 2 Lévy market: (Kunita 03) ds t = S t {b t dt + σ t dw t + } h(t, R ( z)ñ(dt, dz) d ξ t = E 0 G(s)dW s + ) F (t, 0 R z)ñ(dt, dz). d

16 Solution using convex duality Refinements for concrete models A natural problem For a specific market model and a given utility function, Can one narrow down the dual domain Γ Γ where to search ξ? 1 Itô incomplete market: (Karatzas et. al. 91) ds t = S t {{b t dt + σ t dw t }. t ξ t = exp 0 G(s)dW s 1 t 2 ds} 0 G(s) 2 = E( 0 G(s)dW s). 2 Lévy market: (Kunita 03) ds t = S t {b t dt + σ t dw t + } h(t, R ( z)ñ(dt, dz) d ξ t = E 0 G(s)dW s + ) F (t, 0 R z)ñ(dt, dz). d

17 Solution using convex duality What is being used and assumed? Key result Kunita-Watanabe (1967) ξ is a positive local martingale iff ξ t = ξ 0 E(X) with X t := t 0 t G(s)dW s + 0 F(s, z)ñ(ds, dz), F > 1. ξ is a positive supermartingale iff ξ t = ξ 0 E(X A) where X is as above and A is increasing predictable s.t. A < 1. Assumptions on the utility function Strictly increasing and concave. Inada conditions: U (0 + ) = and U ( ) = 0. U is unbounded.

18 An explicit" solution for Lévy-driven markets Set-up 1 The model: ds t = S t (b dt + d Z t ), where Z is a Lévy process. B t 1. 2 Primal problem: p (w) := sup V :V 0 w E {U(V T, ω)}. where U is a bounded state-dependent utility function. 3 Dual problem: } d (λ) := inf {Ũ(λξT E, ω), Γ ξ Γ where Γ is a suitable subclass of Γ to be chosen so that the dual theorem" holds.

19 An explicit" solution for Lévy-driven markets The Dual Theorem Let Γ be a convex subclass of Γ such that (i) w Γ := sup ξ Γ E {ξ T H} < (ii) Γ is closed under Fatou convergence. Then, for each 0 < w < w Γ, there exist λ > 0 and ξ Γ s.t. } 1 d (λ ) := inf Γ ξ Γ E {Ũ(λ ξ T, ω) is attainable at ξ 2 E { ξ I ( λ ξ )} T T = w, where I(z) := (U ) 1 (z) H 3 p (w) E [ U ( I ( λ ξ ))] T Furthermore, if (iii) Γ contains ξ t := E [ ] dq dp F t for any EMM Q M then I ( λ ξ T ) is super-replicable by an admissible V s.t. V 0 = w. Hence, V solves the primal problem.

20 An explicit" solution for Lévy-driven markets Construction of the dual class Γ 1 Let S be a family of loc. bounded local martingales s.t. S is predictable convex S is closed under Émery distance Then, the class Γ := {ξ := ξ0 E (X A) : X S, A increasing, and ξ 0}, is convex and closed under Fatou convergence". 2 The class S := {X t := t 0 t G(s)dW s + 0 meets the above conditions. F(s, z)ñ(ds, dz) : F 1}, 3 The class Γ := Γ Γ fulfills the conditions necessary for the Dual Theorem.

21 Conclusions The method here is more explicit in the sense that the dual domain enjoys an explicit parametrization. Such a parametrization could lead to certain discrete time approximations. It can accommodate more general jump-diffusion models driven by Lévy processes such as ds i (t) = S i (t ){bt idt + d j=1 σij t dw j t + R h(t, z)ñ(dt, dz)}. d What about optimal portfolio problems with consumption, { } sup E V :V 0 w & V 0 U 1 (V T ) + T where dv t := rv t dt + β t ds t c t dt? 0 U 2 (t, c(t))dt,

22 Graphs Bibliography Utility function and its convex dual function Ũ(z, ω) := sup {U(w, ω) z w} = U(I(z)) z I(z), 0 w H I(z) := (U ) 1 (z) H = Ũ (z). Return 1 Return 2 Return 3

23 Graphs Bibliography Bibliography Figueroa-Lopez and Ma. State-dependent utility maximization in Lévy markets, Preprint. Available at Kramkov and Schachermayer. The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Finance and Stochastics, Föllmer and Leukert. Efficient hedging: Cost versus shortfall risk. Finance and Stochastics, Karatzas, Lehoczky, Shreve, and Xu. Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control and Optimization, Kunita. Variational equality and portfolio optimization for price processes with jumps. In Stoch. Proc. and Appl. to Mathem. Fin., Kramkov. Optional decomposition of supermartingales and pricing of contigent claims in incomplete security markets. Prob. Th. and Rel. fields, 1996.

Sensitivity analysis of utility based prices and risk-tolerance wealth processes

Sensitivity analysis of utility based prices and risk-tolerance wealth processes Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,

More information

Bubbles and futures contracts in markets with short-selling constraints

Bubbles and futures contracts in markets with short-selling constraints Bubbles and futures contracts in markets with short-selling constraints Sergio Pulido, Cornell University PhD committee: Philip Protter, Robert Jarrow 3 rd WCMF, Santa Barbara, California November 13 th,

More information

Optimal Investment with Derivative Securities

Optimal Investment with Derivative Securities Noname manuscript No. (will be inserted by the editor) Optimal Investment with Derivative Securities Aytaç İlhan 1, Mattias Jonsson 2, Ronnie Sircar 3 1 Mathematical Institute, University of Oxford, Oxford,

More information

L. Campi and M. Del Vigna Weak Insider Trading and Behavioral Finance

L. Campi and M. Del Vigna Weak Insider Trading and Behavioral Finance WEAK INSIDER TRADING AND BEHAVIORAL FINANCE L. Campi 1 M. Del Vigna 2 1 Université Paris XIII 2 Department of Mathematics for Economic Decisions University of Florence 5 th Florence-Ritsumeikan Workshop

More information

arxiv:1502.06681v2 [q-fin.mf] 26 Feb 2015

arxiv:1502.06681v2 [q-fin.mf] 26 Feb 2015 ARBITRAGE, HEDGING AND UTILITY MAXIMIZATION USING SEMI-STATIC TRADING STRATEGIES WITH AMERICAN OPTIONS ERHAN BAYRAKTAR AND ZHOU ZHOU arxiv:1502.06681v2 [q-fin.mf] 26 Feb 2015 Abstract. We consider a financial

More information

Essays in Financial Mathematics

Essays in Financial Mathematics Essays in Financial Mathematics Essays in Financial Mathematics Kristoffer Lindensjö Dissertation for the Degree of Doctor of Philosophy, Ph.D. Stockholm School of Economics, 2013. Dissertation title:

More information

The Meaning of Market Efficiency

The Meaning of Market Efficiency The Meaning of Market Efficiency Robert Jarrow Martin Larsson February 23, 211 Abstract Fama (197) defined an efficient market as one in which prices always fully reflect available information. This paper

More information

Mathematical Finance

Mathematical Finance Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

More information

Non-Arbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results

Non-Arbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results Proceedings of Symposia in Applied Mathematics Volume 00, 1997 Non-Arbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results Freddy Delbaen and Walter Schachermayer Abstract. The

More information

Introduction to Arbitrage-Free Pricing: Fundamental Theorems

Introduction to Arbitrage-Free Pricing: Fundamental Theorems Introduction to Arbitrage-Free Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 8-10, 2015 1 / 24 Outline Financial market

More information

Option Pricing for a General Stock Model in Discrete Time

Option Pricing for a General Stock Model in Discrete Time University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations August 2014 Option Pricing for a General Stock Model in Discrete Time Cindy Lynn Nichols University of Wisconsin-Milwaukee

More information

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic mean-variance problems in

More information

Risk Minimizing Portfolio Optimization and Hedging with Conditional Value-at-Risk

Risk Minimizing Portfolio Optimization and Hedging with Conditional Value-at-Risk Risk Minimizing Portfolio Optimization and Hedging with Conditional Value-at-Risk Jing Li, University of North Carolina at Charlotte, Department of Mathematics and Statistics, Charlotte, NC 28223, USA.

More information

In which Financial Markets do Mutual Fund Theorems hold true?

In which Financial Markets do Mutual Fund Theorems hold true? In which Financial Markets do Mutual Fund Theorems hold true? W. Schachermayer M. Sîrbu E. Taflin Abstract The Mutual Fund Theorem (MFT) is considered in a general semimartingale financial market S with

More information

Lecture 6 Black-Scholes PDE

Lecture 6 Black-Scholes PDE Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent

More information

6.4 The Infinitely Many Alleles Model

6.4 The Infinitely Many Alleles Model 6.4. THE INFINITELY MANY ALLELES MODEL 93 NE µ [F (X N 1 ) F (µ)] = + 1 i

More information

Risk-minimization for life insurance liabilities

Risk-minimization for life insurance liabilities Risk-minimization for life insurance liabilities Francesca Biagini Mathematisches Institut Ludwig Maximilians Universität München February 24, 2014 Francesca Biagini USC 1/25 Introduction A large number

More information

OPTIMAL TIMING OF THE ANNUITY PURCHASES: A

OPTIMAL TIMING OF THE ANNUITY PURCHASES: A OPTIMAL TIMING OF THE ANNUITY PURCHASES: A COMBINED STOCHASTIC CONTROL AND OPTIMAL STOPPING PROBLEM Gabriele Stabile 1 1 Dipartimento di Matematica per le Dec. Econ. Finanz. e Assic., Facoltà di Economia

More information

Duality of linear conic problems

Duality of linear conic problems Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

More information

FINANCIAL MARKETS WITH SHORT SALES PROHIBITION

FINANCIAL MARKETS WITH SHORT SALES PROHIBITION FINANCIAL MARKETS WITH SHORT SALES PROHIBITION by Sergio Andres Pulido Nino This thesis/dissertation document has been electronically approved by the following individuals: Protter,Philip E. (Chairperson)

More information

Reaching Goals by a Deadline: Digital Options and Continuous-Time Active Portfolio Management

Reaching Goals by a Deadline: Digital Options and Continuous-Time Active Portfolio Management Reaching Goals by a Deadline: Digital Options and Continuous-Time Active Portfolio Management Sad Browne * Columbia University Original: February 18, 1996 This Version: July 9, 1996 Abstract We study a

More information

Optimisation Problems in Non-Life Insurance

Optimisation Problems in Non-Life Insurance Frankfurt, 6. Juli 2007 1 The de Finetti Problem The Optimal Strategy De Finetti s Example 2 Minimal Ruin Probabilities The Hamilton-Jacobi-Bellman Equation Two Examples 3 Optimal Dividends Dividends in

More information

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

More information

The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees

The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees Torsten Kleinow Heriot-Watt University, Edinburgh (joint work with Mark Willder) Market-consistent

More information

Invariant Option Pricing & Minimax Duality of American and Bermudan Options

Invariant Option Pricing & Minimax Duality of American and Bermudan Options Invariant Option Pricing & Minimax Duality of American and Bermudan Options Farshid Jamshidian NIB Capital Bank N.V. FELAB, Applied Math Dept., Univ. of Twente April 2005, version 1.0 Invariant Option

More information

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia Estimating the Degree of Activity of jumps in High Frequency Financial Data joint with Yacine Aït-Sahalia Aim and setting An underlying process X = (X t ) t 0, observed at equally spaced discrete times

More information

Fair Valuation and Hedging of Participating Life-Insurance Policies under Management Discretion

Fair Valuation and Hedging of Participating Life-Insurance Policies under Management Discretion Fair Valuation and Hedging of Participating Life-Insurance Policies under Management Discretion Torsten Kleinow Department of Actuarial Mathematics and Statistics and the Maxwell Institute for Mathematical

More information

Liquidity Risk and Arbitrage Pricing Theory

Liquidity Risk and Arbitrage Pricing Theory Noname manuscript No. (will be inserted by the editor) Liquidity Risk and Arbitrage Pricing Theory Umut Çetin 1, Robert A. Jarrow 2, Philip Protter 3 1 Technische Universität Wien, Institut für Finanz-

More information

Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing

Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common

More information

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t. LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

More information

Master of Mathematical Finance: Course Descriptions

Master of Mathematical Finance: Course Descriptions Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support

More information

Option Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25

Option Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25 Option Pricing Chapter 11 Options on Futures Stefan Ankirchner University of Bonn last update: 13/01/2014 at 14:25 Stefan Ankirchner Option Pricing 1 Agenda Forward contracts Definition Determining forward

More information

Option Pricing. 1 Introduction. Mrinal K. Ghosh

Option Pricing. 1 Introduction. Mrinal K. Ghosh Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified

More information

Numerical methods for American options

Numerical methods for American options Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment

More information

Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms.

Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms. Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms. Erwin Topp P. Universidad de Santiago de Chile Conference HJ2016, Rennes, France May 31th, 2016 joint work with Gonzalo Dávila

More information

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

More information

On a comparison result for Markov processes

On a comparison result for Markov processes On a comparison result for Markov processes Ludger Rüschendorf University of Freiburg Abstract A comparison theorem is stated for Markov processes in polish state spaces. We consider a general class of

More information

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for

More information

Simple Arbitrage. Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila. Christian Bender. Saarland University

Simple Arbitrage. Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila. Christian Bender. Saarland University Simple Arbitrage Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila Saarland University December, 8, 2011 Problem Setting Financial market with two assets (for simplicity) on

More information

A Martingale System Theorem for Stock Investments

A Martingale System Theorem for Stock Investments A Martingale System Theorem for Stock Investments Robert J. Vanderbei April 26, 1999 DIMACS New Market Models Workshop 1 Beginning Middle End Controversial Remarks Outline DIMACS New Market Models Workshop

More information

Who Should Sell Stocks?

Who Should Sell Stocks? Who Should Sell Stocks? Ren Liu joint work with Paolo Guasoni and Johannes Muhle-Karbe ETH Zürich Imperial-ETH Workshop on Mathematical Finance 2015 1 / 24 Merton s Problem (1969) Frictionless market consisting

More information

Numeraire-invariant option pricing

Numeraire-invariant option pricing Numeraire-invariant option pricing Farshid Jamshidian NIB Capital Bank N.V. FELAB, University of Twente Nov-04 Numeraire-invariant option pricing p.1/20 A conceptual definition of an option An Option can

More information

A SIMPLE COUNTER-EXAMPLE TO SEVERAL PROBLEMS IN THE THEORY OF ASSET PRICING.

A SIMPLE COUNTER-EXAMPLE TO SEVERAL PROBLEMS IN THE THEORY OF ASSET PRICING. A SIMPLE COUNTER-EXAMPLE TO SEVERAL PROBLEMS IN THE THEORY OF ASSET PRICING. Freddy Delbaen Walter Schachermayer Departement für Mathematik, Eidgenössische Technische Hochschule Zürich Institut für Statistik,

More information

Designing and managing unit-linked life insurance contracts with guarantees

Designing and managing unit-linked life insurance contracts with guarantees Designing and managing unit-linked life insurance contracts with guarantees Ronald Hochreiter a, Georg Pflug a, Volkert Paulsen b a Department of Statistics and Decision Support Systems, University of

More information

4: SINGLE-PERIOD MARKET MODELS

4: SINGLE-PERIOD MARKET MODELS 4: SINGLE-PERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market

More information

Hedging bounded claims with bounded outcomes

Hedging bounded claims with bounded outcomes Hedging bounded claims with bounded outcomes Freddy Delbaen ETH Zürich, Department of Mathematics, CH-892 Zurich, Switzerland Abstract. We consider a financial market with two or more separate components

More information

Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6

Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 1 Black-Scholes Equation and Replicating Portfolio 2 Static

More information

Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space

Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle

More information

Liquidity costs and market impact for derivatives

Liquidity costs and market impact for derivatives Liquidity costs and market impact for derivatives F. Abergel, G. Loeper Statistical modeling, financial data analysis and applications, Istituto Veneto di Scienze Lettere ed Arti. Abergel, G. Loeper Statistical

More information

Optimal Static-Dynamic Hedges for Exotic Options under Convex Risk Measures

Optimal Static-Dynamic Hedges for Exotic Options under Convex Risk Measures Optimal Static-Dynamic Hedges for Exotic Options under Convex Risk Measures Aytaç İlhan Mattias Jonsson Ronnie Sircar April 8, 28; revised February 6, 29 and June 15, 29 Abstract We study the problem of

More information

arxiv:math/0702413v1 [math.pr] 14 Feb 2007

arxiv:math/0702413v1 [math.pr] 14 Feb 2007 The Annals of Applied Probability 2006, Vol. 16, No. 4, 2140 2194 DOI: 10.1214/105051606000000529 c Institute of Mathematical Statistics, 2006 arxiv:math/0702413v1 [math.pr] 14 Feb 2007 SENSITIVITY ANALYSIS

More information

Real options with constant relative risk aversion

Real options with constant relative risk aversion Journal of Economic Dynamics & Control 27 22) 329 355 www.elsevier.com/locate/econbase Real options with constant relative risk aversion Vicky Henderson a;, David G. Hobson b a Financial Options Research

More information

Pricing catastrophe options in incomplete market

Pricing catastrophe options in incomplete market Pricing catastrophe options in incomplete market Arthur Charpentier arthur.charpentier@univ-rennes1.fr Actuarial and Financial Mathematics Conference Interplay between finance and insurance, February 2008

More information

ON NEUMANN TYPE BOUNDARY CONDITIONS FOR HAMILTON-JACOBI EQUATIONS IN SMOOTH DOMAINS

ON NEUMANN TYPE BOUNDARY CONDITIONS FOR HAMILTON-JACOBI EQUATIONS IN SMOOTH DOMAINS ON NEUMANN TYPE BOUNDARY CONDITIONS FOR HAMILTON-JACOBI EQUATIONS IN SMOOTH DOMAINS MARTIN V. DAY Abstract. Neumann or oblique derivative boundary conditions for viscosity solutions of Hamilton-Jacobi

More information

Stock Loans in Incomplete Markets

Stock Loans in Incomplete Markets Applied Mathematical Finance, 2013 Vol. 20, No. 2, 118 136, http://dx.doi.org/10.1080/1350486x.2012.660318 Stock Loans in Incomplete Markets MATHEUS R. GRASSELLI* & CESAR GÓMEZ** *Department of Mathematics

More information

The Discrete Binomial Model for Option Pricing

The Discrete Binomial Model for Option Pricing The Discrete Binomial Model for Option Pricing Rebecca Stockbridge Program in Applied Mathematics University of Arizona May 4, 2008 Abstract This paper introduces the notion of option pricing in the context

More information

Characterizing Option Prices by Linear Programs

Characterizing Option Prices by Linear Programs Contemporary Mathematics Characterizing Option Prices by Linear Programs Richard H. Stockbridge Abstract. The price of various options on a risky asset are characterized via a linear program involving

More information

Options pricing in discrete systems

Options pricing in discrete systems UNIVERZA V LJUBLJANI, FAKULTETA ZA MATEMATIKO IN FIZIKO Options pricing in discrete systems Seminar II Mentor: prof. Dr. Mihael Perman Author: Gorazd Gotovac //2008 Abstract This paper is a basic introduction

More information

European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

More information

Hedging of Life Insurance Liabilities

Hedging of Life Insurance Liabilities Hedging of Life Insurance Liabilities Thorsten Rheinländer, with Francesca Biagini and Irene Schreiber Vienna University of Technology and LMU Munich September 6, 2015 horsten Rheinländer, with Francesca

More information

SENSITIVITY ANALYSIS OF UTILITY-BASED PRICES AND RISK-TOLERANCE WEALTH PROCESSES

SENSITIVITY ANALYSIS OF UTILITY-BASED PRICES AND RISK-TOLERANCE WEALTH PROCESSES The Annals of Applied Probability 2006, Vol. 16, No. 4, 2140 2194 DOI: 10.1214/105051606000000529 Institute of Mathematical Statistics, 2006 SENSITIVITY ANALYSIS OF UTILITY-BASED PRICES AND RISK-TOLERANCE

More information

STOCK LOANS. XUN YU ZHOU Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong 1.

STOCK LOANS. XUN YU ZHOU Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong 1. Mathematical Finance, Vol. 17, No. 2 April 2007), 307 317 STOCK LOANS JIANMING XIA Center for Financial Engineering and Risk Management, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

More information

Proximal mapping via network optimization

Proximal mapping via network optimization L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

More information

Dynamic optimal portfolios benchmarking the stock market

Dynamic optimal portfolios benchmarking the stock market Dynamic optimal portfolios benchmarking the stock market A. Gabih a, M. Richter b, R. Wunderlich c a Martin-Luther-Universität Halle Wittenberg, Fachbereich für Mathematik und Informatik, 06099 Halle (Saale),

More information

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist

More information

The Center for Applied Probability Columbia University: Nov. 9, th Annual CAP Workshop on Derivative Securities and Risk Management

The Center for Applied Probability Columbia University: Nov. 9, th Annual CAP Workshop on Derivative Securities and Risk Management The Center for Applied Probability Columbia University: Nov. 9 00 8 th Annual CAP Workshop on Derivative Securities and Risk Management A Simple Option Formula for General Jump-Diffusion and other Exponential

More information

American Option Pricing with Transaction Costs

American Option Pricing with Transaction Costs American Option Pricing with Transaction Costs Valeri. I. Zakamouline Department of Finance and Management Science Norwegian School of Economics and Business Administration Helleveien 30, 5045 Bergen,

More information

Health Insurance and Retirement Incentives

Health Insurance and Retirement Incentives Health Insurance and Retirement Incentives Daniele Marazzina Joint work with Emilio Barucci and Enrico Biffis Emilio Barucci and Daniele Marazzina Dipartimento di Matematica F. Brioschi Politecnico di

More information

Valuation and Hedging of Participating Life-Insurance Policies under Management Discretion

Valuation and Hedging of Participating Life-Insurance Policies under Management Discretion Kleinow: Participating Life-Insurance Policies 1 Valuation and Hedging of Participating Life-Insurance Policies under Management Discretion Torsten Kleinow Department of Actuarial Mathematics and Statistics

More information

Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach

Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach Outline Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach The University of New South Wales SPOM 2013 Joint work with V. Jeyakumar, B.S. Mordukhovich and

More information

Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems

Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems Wolfgang Wagner Mohrenstrasse 39 10117 Berlin Germany Tel. +49 30 20372 0 www.wias-berlin.de WIAS workshop,

More information

Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework

Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework Quan Yuan Joint work with Shuntai Hu, Baojun Bian Email: candy5191@163.com

More information

Cross hedging with stochastic correlation

Cross hedging with stochastic correlation Gregor Heyne (joint work with Stefan Ankirchner) Humboldt Universität Berlin, QPL March 20th, 2009 Finance and Insurance, Jena Motivation Consider a call option on the DAX. How to hedge? Maybe with Futures

More information

IMPLICIT CLIENT SORTING AND BONUS MALUS CONTRACTS

IMPLICIT CLIENT SORTING AND BONUS MALUS CONTRACTS IMPLICIT CLIENT SORTING AND BONUS MALUS CONTRACTS Francisco J. Vazquez (C. U. Francisco Vitoria & Universidad Autónoma de Madrid) Richard Watt* (Universidad Autónoma de Madrid) Abstract In an earlier paper,

More information

Practical and theoretical aspects of market-consistent valuation and hedging of insurance liabilities

Practical and theoretical aspects of market-consistent valuation and hedging of insurance liabilities Practical and theoretical aspects of market-consistent valuation and hedging of insurance liabilities Łukasz Delong Institute of Econometrics, Division of Probabilistic Methods Warsaw School of Economics

More information

On exponentially ane martingales. Johannes Muhle-Karbe

On exponentially ane martingales. Johannes Muhle-Karbe On exponentially ane martingales AMAMEF 2007, Bedlewo Johannes Muhle-Karbe Joint work with Jan Kallsen HVB-Institut für Finanzmathematik, Technische Universität München 1 Outline 1. Semimartingale characteristics

More information

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-ormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last

More information

Portfolio Optimization with a Defaultable Security

Portfolio Optimization with a Defaultable Security Portfolio Optimization with a Defaultable Security Tomasz R. Bielecki Department of Applied Mathematics, Illinois Institute of Technology, 10 West 32nd Street Chicago, IL 60616, Tel:312-567-3165, Email:bielecki@iit.edu

More information

Introduction to portfolio insurance. Introduction to portfolio insurance p.1/41

Introduction to portfolio insurance. Introduction to portfolio insurance p.1/41 Introduction to portfolio insurance Introduction to portfolio insurance p.1/41 Portfolio insurance Maintain the portfolio value above a certain predetermined level (floor) while allowing some upside potential.

More information

Calibration of Stock Betas from Skews of Implied Volatilities

Calibration of Stock Betas from Skews of Implied Volatilities Calibration of Stock Betas from Skews of Implied Volatilities Jean-Pierre Fouque University of California Santa Barbara Joint work with Eli Kollman (Ph.D. student at UCSB) Joint Seminar: Department of

More information

Health Insurance, Portfolio Choices and Retirement Incentives

Health Insurance, Portfolio Choices and Retirement Incentives Health Insurance, Portfolio Choices and Retirement Incentives Daniele Marazzina Joint work with Emilio Barucci and Enrico Biffis Emilio Barucci and Daniele Marazzina Dipartimento di Matematica F. Brioschi

More information

General Equilibrium Theory: Examples

General Equilibrium Theory: Examples General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer - 1 consumer several producers and an example illustrating the limits of the partial equilibrium approach

More information

Hedging market risk in optimal liquidation

Hedging market risk in optimal liquidation Hedging market risk in optimal liquidation Phillip Monin The Office of Financial Research Washington, DC Conference on Stochastic Asymptotics & Applications, Joint with 6th Western Conference on Mathematical

More information

Exam Introduction Mathematical Finance and Insurance

Exam Introduction Mathematical Finance and Insurance Exam Introduction Mathematical Finance and Insurance Date: January 8, 2013. Duration: 3 hours. This is a closed-book exam. The exam does not use scrap cards. Simple calculators are allowed. The questions

More information

TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh

TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp -. TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*

More information

Guaranteed Annuity Options

Guaranteed Annuity Options Guaranteed Annuity Options Hansjörg Furrer Market-consistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options Contents A. Guaranteed Annuity Options B. Valuation and Risk

More information

An optimal transportation problem with import/export taxes on the boundary

An optimal transportation problem with import/export taxes on the boundary An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint

More information

American Options in incomplete Markets: Upper and lower Snell Envelopes and robust partial Hedging

American Options in incomplete Markets: Upper and lower Snell Envelopes and robust partial Hedging American Options in incomplete Markets: Upper and lower Snell Envelopes and robust partial Hedging DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Mathematik

More information

The Black-Scholes-Merton Approach to Pricing Options

The Black-Scholes-Merton Approach to Pricing Options he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

More information

Hedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15

Hedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15 Hedging Options In The Incomplete Market With Stochastic Volatility Rituparna Sen Sunday, Nov 15 1. Motivation This is a pure jump model and hence avoids the theoretical drawbacks of continuous path models.

More information

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the

More information

Operations Research and Financial Engineering. Courses

Operations Research and Financial Engineering. Courses Operations Research and Financial Engineering Courses ORF 504/FIN 504 Financial Econometrics Professor Jianqing Fan This course covers econometric and statistical methods as applied to finance. Topics

More information

Disability insurance: estimation and risk aggregation

Disability insurance: estimation and risk aggregation Disability insurance: estimation and risk aggregation B. Löfdahl Department of Mathematics KTH, Royal Institute of Technology May 2015 Introduction New upcoming regulation for insurance industry: Solvency

More information

Fuzzy Probability Distributions in Bayesian Analysis

Fuzzy Probability Distributions in Bayesian Analysis Fuzzy Probability Distributions in Bayesian Analysis Reinhard Viertl and Owat Sunanta Department of Statistics and Probability Theory Vienna University of Technology, Vienna, Austria Corresponding author:

More information

THE AMERICAN PUT IS LOG-CONCAVE IN THE LOG-PRICE

THE AMERICAN PUT IS LOG-CONCAVE IN THE LOG-PRICE THE AMERICAN PUT IS LOG-CONCAVE IN THE LOG-PRICE ERIK EKSTRÖM AND JOHAN TYSK Abstract. We show that the American put option price is log-concave as a function of the log-price of the underlying asset.

More information

Pricing American Options: A Duality Approach

Pricing American Options: A Duality Approach Pricing American Options: A Duality Approach Martin B. Haugh and Leonid Kogan December 2001 Abstract We develop a new method for pricing American options. The main practical contribution of this paper

More information

Revealed Preference. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Revealed Preference October 8, / 17

Revealed Preference. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Revealed Preference October 8, / 17 Ichiro Obara UCLA October 8, 2012 Obara (UCLA) October 8, 2012 1 / 17 Obara (UCLA) October 8, 2012 2 / 17 Suppose that we obtained data of price and consumption pairs D = { (p t, x t ) R L ++ R L +, t

More information

Dynamic Asset Allocation: a Portfolio Decomposition Formula and Applications

Dynamic Asset Allocation: a Portfolio Decomposition Formula and Applications 1 Dynamic Asset Allocation: a Portfolio Decomposition Formula and Applications Jérôme Detemple Boston University School of Management and CIRANO Marcel Rindisbacher Rotman School of Management, University

More information

A new Feynman-Kac-formula for option pricing in Lévy models

A new Feynman-Kac-formula for option pricing in Lévy models A new Feynman-Kac-formula for option pricing in Lévy models Kathrin Glau Department of Mathematical Stochastics, Universtity of Freiburg (Joint work with E. Eberlein) 6th World Congress of the Bachelier

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information