Statedependent utility maximization in Lévy markets


 Verity Carpenter
 2 years ago
 Views:
Transcription
1 Statedependent utility maximization in Lévy markets José FigueroaLópez 1 Jin Ma 2 1 University of California at Santa Barbara 2 Purdue University KENTPURDUE 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 Lévydriven 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 Statedependent 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 Superreplication 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 nonempty. 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 Superreplication: 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 Superreplication 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 nonempty. 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 Superreplication: 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 Superreplication 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 nonempty. 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 Superreplication: 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 Superreplication 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 nonempty. 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 Superreplication: 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 Superreplication 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 nonempty. 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 Superreplication: 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 Superreplication 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 nonempty. 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 Superreplication: 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 KunitaWatanabe (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 Lévydriven markets Setup 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 statedependent 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 Lévydriven 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 superreplicable by an admissible V s.t. V 0 = w. Hence, V solves the primal problem.
20 An explicit" solution for Lévydriven 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 jumpdiffusion 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 FigueroaLopez and Ma. Statedependent 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 risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationBubbles and futures contracts in markets with shortselling constraints
Bubbles and futures contracts in markets with shortselling constraints Sergio Pulido, Cornell University PhD committee: Philip Protter, Robert Jarrow 3 rd WCMF, Santa Barbara, California November 13 th,
More informationOptimal 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 informationL. 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 FlorenceRitsumeikan Workshop
More informationarxiv:1502.06681v2 [qfin.mf] 26 Feb 2015
ARBITRAGE, HEDGING AND UTILITY MAXIMIZATION USING SEMISTATIC TRADING STRATEGIES WITH AMERICAN OPTIONS ERHAN BAYRAKTAR AND ZHOU ZHOU arxiv:1502.06681v2 [qfin.mf] 26 Feb 2015 Abstract. We consider a financial
More informationEssays 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 informationThe 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 informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationNonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results
Proceedings of Symposia in Applied Mathematics Volume 00, 1997 NonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results Freddy Delbaen and Walter Schachermayer Abstract. The
More informationIntroduction to ArbitrageFree Pricing: Fundamental Theorems
Introduction to ArbitrageFree Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 810, 2015 1 / 24 Outline Financial market
More informationOption 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 WisconsinMilwaukee
More informationCOMPLETE 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 meanvariance problems in
More informationRisk Minimizing Portfolio Optimization and Hedging with Conditional ValueatRisk
Risk Minimizing Portfolio Optimization and Hedging with Conditional ValueatRisk Jing Li, University of North Carolina at Charlotte, Department of Mathematics and Statistics, Charlotte, NC 28223, USA.
More informationIn 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 informationLecture 6 BlackScholes PDE
Lecture 6 BlackScholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the riskneutral measure Q by If the contingent
More information6.4 The Infinitely Many Alleles Model
6.4. THE INFINITELY MANY ALLELES MODEL 93 NE µ [F (X N 1 ) F (µ)] = + 1 i
More informationRiskminimization for life insurance liabilities
Riskminimization 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 informationOPTIMAL 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 informationDuality 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 informationFINANCIAL 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 informationReaching Goals by a Deadline: Digital Options and ContinuousTime Active Portfolio Management
Reaching Goals by a Deadline: Digital Options and ContinuousTime Active Portfolio Management Sad Browne * Columbia University Original: February 18, 1996 This Version: July 9, 1996 Abstract We study a
More informationOptimisation Problems in NonLife Insurance
Frankfurt, 6. Juli 2007 1 The de Finetti Problem The Optimal Strategy De Finetti s Example 2 Minimal Ruin Probabilities The HamiltonJacobiBellman Equation Two Examples 3 Optimal Dividends Dividends in
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE 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 informationThe 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 HeriotWatt University, Edinburgh (joint work with Mark Willder) Marketconsistent
More informationInvariant 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 informationEstimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine AïtSahalia
Estimating the Degree of Activity of jumps in High Frequency Financial Data joint with Yacine AïtSahalia Aim and setting An underlying process X = (X t ) t 0, observed at equally spaced discrete times
More informationFair Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion
Fair Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion Torsten Kleinow Department of Actuarial Mathematics and Statistics and the Maxwell Institute for Mathematical
More informationLiquidity 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 informationLecture 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 informationMoreover, 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 informationMaster 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 informationOption 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 informationOption 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 informationNumerical 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 informationSolvability 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 information6.254 : 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 PricingCongestion Game Example Existence of a Mixed
More informationOn 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 informationQUANTIZED 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 informationSimple 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 informationA 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 informationWho Should Sell Stocks?
Who Should Sell Stocks? Ren Liu joint work with Paolo Guasoni and Johannes MuhleKarbe ETH Zürich ImperialETH Workshop on Mathematical Finance 2015 1 / 24 Merton s Problem (1969) Frictionless market consisting
More informationNumeraireinvariant option pricing
Numeraireinvariant option pricing Farshid Jamshidian NIB Capital Bank N.V. FELAB, University of Twente Nov04 Numeraireinvariant option pricing p.1/20 A conceptual definition of an option An Option can
More informationA SIMPLE COUNTEREXAMPLE TO SEVERAL PROBLEMS IN THE THEORY OF ASSET PRICING.
A SIMPLE COUNTEREXAMPLE TO SEVERAL PROBLEMS IN THE THEORY OF ASSET PRICING. Freddy Delbaen Walter Schachermayer Departement für Mathematik, Eidgenössische Technische Hochschule Zürich Institut für Statistik,
More informationDesigning and managing unitlinked life insurance contracts with guarantees
Designing and managing unitlinked 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 information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD 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: SinglePeriod Market
More informationHedging bounded claims with bounded outcomes
Hedging bounded claims with bounded outcomes Freddy Delbaen ETH Zürich, Department of Mathematics, CH892 Zurich, Switzerland Abstract. We consider a financial market with two or more separate components
More informationLecture 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 BlackScholes Equation and Replicating Portfolio 2 Static
More informationSensitivity analysis of European options in jumpdiffusion models via the Malliavin calculus on the Wiener space
Sensitivity analysis of European options in jumpdiffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle
More informationLiquidity 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 informationOptimal StaticDynamic Hedges for Exotic Options under Convex Risk Measures
Optimal StaticDynamic 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 informationarxiv: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 informationReal 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 informationPricing catastrophe options in incomplete market
Pricing catastrophe options in incomplete market Arthur Charpentier arthur.charpentier@univrennes1.fr Actuarial and Financial Mathematics Conference Interplay between finance and insurance, February 2008
More informationON NEUMANN TYPE BOUNDARY CONDITIONS FOR HAMILTONJACOBI EQUATIONS IN SMOOTH DOMAINS
ON NEUMANN TYPE BOUNDARY CONDITIONS FOR HAMILTONJACOBI EQUATIONS IN SMOOTH DOMAINS MARTIN V. DAY Abstract. Neumann or oblique derivative boundary conditions for viscosity solutions of HamiltonJacobi
More informationStock 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 informationThe 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 informationCharacterizing 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 informationOptions 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 informationEuropean 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 informationHedging 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 informationSENSITIVITY ANALYSIS OF UTILITYBASED PRICES AND RISKTOLERANCE 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 UTILITYBASED PRICES AND RISKTOLERANCE
More informationSTOCK 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 informationProximal mapping via network optimization
L. Vandenberghe EE236C (Spring 234) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
More informationDynamic optimal portfolios benchmarking the stock market
Dynamic optimal portfolios benchmarking the stock market A. Gabih a, M. Richter b, R. Wunderlich c a MartinLutherUniversität Halle Wittenberg, Fachbereich für Mathematik und Informatik, 06099 Halle (Saale),
More informationEXIT 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 informationThe 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 JumpDiffusion and other Exponential
More informationAmerican 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 informationHealth 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 informationValuation and Hedging of Participating LifeInsurance Policies under Management Discretion
Kleinow: Participating LifeInsurance Policies 1 Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion Torsten Kleinow Department of Actuarial Mathematics and Statistics
More informationError 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 informationWeierstrass 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.wiasberlin.de WIAS workshop,
More informationValuation 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 informationCross 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 informationIMPLICIT 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 informationPractical and theoretical aspects of marketconsistent valuation and hedging of insurance liabilities
Practical and theoretical aspects of marketconsistent valuation and hedging of insurance liabilities Łukasz Delong Institute of Econometrics, Division of Probabilistic Methods Warsaw School of Economics
More informationOn exponentially ane martingales. Johannes MuhleKarbe
On exponentially ane martingales AMAMEF 2007, Bedlewo Johannes MuhleKarbe Joint work with Jan Kallsen HVBInstitut für Finanzmathematik, Technische Universität München 1 Outline 1. Semimartingale characteristics
More informationLecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. LogNormal Distribution
Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Logormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last
More informationPortfolio 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:3125673165, Email:bielecki@iit.edu
More informationIntroduction 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 informationCalibration of Stock Betas from Skews of Implied Volatilities
Calibration of Stock Betas from Skews of Implied Volatilities JeanPierre Fouque University of California Santa Barbara Joint work with Eli Kollman (Ph.D. student at UCSB) Joint Seminar: Department of
More informationHealth 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 informationGeneral 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 informationHedging 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 informationExam Introduction Mathematical Finance and Insurance
Exam Introduction Mathematical Finance and Insurance Date: January 8, 2013. Duration: 3 hours. This is a closedbook exam. The exam does not use scrap cards. Simple calculators are allowed. The questions
More informationTRIPLE 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 informationGuaranteed Annuity Options
Guaranteed Annuity Options Hansjörg Furrer Marketconsistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options Contents A. Guaranteed Annuity Options B. Valuation and Risk
More informationAn 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 informationAmerican 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 informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationHedging 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 informationFour 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 informationOperations 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 informationDisability 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 informationFuzzy 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 informationTHE AMERICAN PUT IS LOGCONCAVE IN THE LOGPRICE
THE AMERICAN PUT IS LOGCONCAVE IN THE LOGPRICE ERIK EKSTRÖM AND JOHAN TYSK Abstract. We show that the American put option price is logconcave as a function of the logprice of the underlying asset.
More informationPricing 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 informationRevealed 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 informationDynamic 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 informationA new FeynmanKacformula for option pricing in Lévy models
A new FeynmanKacformula 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 information2.3 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