Martingale Optimal Transport and Robust Hedging
|
|
- Marsha Charles
- 6 years ago
- Views:
Transcription
1 Martingale Optimal Transport and Robust Hedging Ecole Polytechnique, Paris Angers, September 3, 2015
2 Outline Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation 1 Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation 2 Formulation and duality Optimal semi-static strategy and quasi-sure formulation 3 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem
3 The Monge-Kantorovitch optimal transport problem Financial interpretation
4 The Monge-Kantorovitch optimal transport problem Financial interpretation The problem of Déblais et Remblais (Monge 1781) Figure: Mass transport. Ref : C. Villani
5 Analytic formulation (Monge 1781) The Monge-Kantorovitch optimal transport problem Financial interpretation Initial distribution : probability measure µ on X Final distribution : probability measure ν on Y Problem : find an optimal transference plan T P M := sup T T (µ,ν) c ( x, T (x) ) µ(dx) where T (µ, ν) of all maps T : x y = T (x) such that ν = µ T 1
6 The Monge-Kantorovitch optimal transport problem Financial interpretation Randomization of mass transfert (Kantorovich 1942) Figure: Randomized mass transport
7 The Monge-Kantorovitch optimal transport problem Financial interpretation Probabilistic formulation (Kantorovich 1942) Randomization of transference plans : P K := sup E P[ c(x, Y ) ] P P(µ,ν) where Ω = X Y, X (x, y) = x, Y (x, y) = y, and P(µ, ν) := { P Prob Ω : X P µ and Y P ν }
8 Kantorovich duality The Monge-Kantorovitch optimal transport problem Financial interpretation Duality in linear programming, Legendre-Fenchel duality... D 0 { } := inf µ(ϕ) + ν(ψ) (ϕ,ψ) D 0 D 0 := { (ϕ, ψ) L 1 (µ) L 1 (ν) : ϕ ψ c } where µ(ϕ) := ϕdµ, ν(ψ) := ψdν, and ϕ ψ(x, y) := ϕ(x) + ψ(y), x X, y Y No reference probability measure on the product space
9 The duality Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation Theorem Let c 0 be measurable. Then duality : D 0 = P K existence holds for D 0 If in addition c USC, existence also holds for P K In this generality, the result is due to Kellerer 84
10 Financial interpretation The Monge-Kantorovitch optimal transport problem Financial interpretation X µ and Y ν prices of two assets at time 1 µ and ν identified from market prices of call options : C µ (K) = (x K) + µ(dx), C ν (K) = (y K) + ν(dy), K 0 Breeden-Litzenberger 1978 : µ = C µ and ν = C ν ϕ(x ) Vanilla position in X with market price µ(ϕ) ψ(y ) Y ν(ψ)
11 The Monge-Kantorovitch optimal transport problem Financial interpretation Financial interpretation : no reference probability c(x, Y ) payoff of derivative security Robust static hedging strategies for the derivative c(x, Y ) : D 0 := { (ϕ, ψ) L 1 (µ) L 1 (ν) : ϕ ψ c } Robust superhedging cost of c(x, Y ) is the Kantorovitch dual : D 0 { } = inf µ(ϕ) + ν(ψ) (ϕ,ψ) D 0 The primal Monge-Kantorovitch problem is : P K = sup E P [c(x, Y )] P P(µ,ν)
12 Outline Optimal Transport and Model-free hedging Formulation and duality Optimal semi-static strategy and quasi-sure formulation 1 Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation 2 Formulation and duality Optimal semi-static strategy and quasi-sure formulation 3 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem
13 Formulation and duality Optimal semi-static strategy and quasi-sure formulation MARTINGALE OPTIMAL TRANSPORT initiated by Pierre Henry-Labordère, Preprints Soc. Gén. Beiglböck, Henry-Labordère & Penkner Galichon, Henry-Labordère & NT
14 One asset observed at two future dates Formulation and duality Optimal semi-static strategy and quasi-sure formulation Our interest now is on the case where X = X 0 and Y = X 1 are the prices of the same asset at two future dates 0 and 1 Interest rate is reduced to zero This setting introduces a new feature : possibility of dynamic trading the asset between times 0 and 1
15 Superhedging problem Kantorovitch dual Formulation and duality Optimal semi-static strategy and quasi-sure formulation Robust super hedging problem naturally formulated as : D(µ, ν) := inf (ϕ,ψ,h) D { } µ(ϕ) + ν(ψ) where D := { (ϕ, ψ, h) L 1 (µ) L 1 (ν) L 0 : ϕ ψ + h c } where ϕ ψ(x, y) := ϕ(x) + ψ(y) and h (x, y) := h(x)(y x)
16 The Martingale Optimal Transport Problem Formulation and duality Optimal semi-static strategy and quasi-sure formulation Theorem (Beiglböck, Henry-Labordère, Penkner) Assume c USC and bounded from above. Then P = D, and existence holds for P(µ, ν). where the dual problem is : P(µ, ν) := sup E P[ c(x, Y ) ] P M(µ,ν) with M(µ, ν) := { P P(µ, ν) : E P [Y X ] = X } Strassen 65 : M(µ, ν) iff µ ν, i.e. µ(g) ν(g) g convex
17 Formulation and duality Optimal semi-static strategy and quasi-sure formulation Existence of optimal hedge does not hold in general There are easy examples where existence for the dual fails, even for bounded c, bounded support... (Beiglböck, Henry-Labordère & Penkner, Beiglböck, Nutz & NT) Let µ = ν, then M(µ, µ) = {P } where Y = X, P a.s. P(µ, µ) = E µ[ c(x, X ) ] The derivative is in fact c(x, X ), primal problem carries no information about c(x, y) outside the diagonal y = x One is only interested in hedging along the diagonal ϕ(x ) + ψ(x ) + h(x )(X X ) c(x, X ), µ a.s.
18 Duality under more general payoff functions Formulation and duality Optimal semi-static strategy and quasi-sure formulation The condition c USC is not innocent... Consider the following example of LSC payoff c(x, y) := 1I {x y}, x, y [0, 1] [0, 1] Let µ = ν = Lebesgue measure on [0, 1]. Then M(µ, µ) = {P } uniform distribution on the diagonal of the square [0, 1] 2 Then P(µ, µ) = 0 However, we may prove that D(µ, µ) = 1!
19 Quasi-sure robust superhedging Formulation and duality Optimal semi-static strategy and quasi-sure formulation Definition M(µ, ν) q.s. (quasi surely) means P a.s. for all P M(µ, ν) The quasi-sure robust superhedging cost D qs { } := inf µ(ϕ) + ν(ψ) (ϕ,ψ,h) D qs D qs := { (ϕ, ψ, h) : ϕ ψ + h c, M(µ, ν) q.s. } is more natural... Then, D(µ, ν) D qs (µ, ν) P(µ, ν) so if the duality P = D holds, it follows that D = D qs
20 Formulation and duality Optimal semi-static strategy and quasi-sure formulation Structure of polar sets in (standard) optimal transport Theorem (Kellerer) For N R R, TFAE : P[N] = 0 for all P P(µ, ν) N (N 1 R) (R N 2 ) for some N 1, N 2 R, µ(n 1 ) = ν(n 2 ) = 0 = no difference between the pointwise and the quasi-sure formulations in standard optimal transport
21 Pointwise versus Quasi-sure superhedging I Formulation and duality Optimal semi-static strategy and quasi-sure formulation Suppose Supp(µ) = [0, 2] =Supp(ν) = [0, 2], then M(µ, ν) q.s. only involves the values (x, y) [0, 2] 2 Pointwise superhedging involves all values (x, y) R 2
22 Pointwise versus Quasi-sure superhedging II Formulation and duality Optimal semi-static strategy and quasi-sure formulation Suppose Supp(µ) =Supp(ν) = [0, 2], and C µ (1) = C ν (1) E [ (X 1) +] = E [ (Y 1) +] E [ (X 1) +] by Jensen s inequality, and then {X 1} = {Y 1}
23 Formulation and duality Optimal semi-static strategy and quasi-sure formulation Structure of polar sets in martingale optimal transport Consider the partition : {C µ < C ν } = k 0 I k, I k = (a k, b k ), J k := I k {ν atoms} Theorem (Beiglböck, Nutz & NT 15) For N R R, TFAE : P[N] = 0 for all P M(µ, ν) N (N 1 R) (R N 2 ) { (I k J k ) } c for some N 1, N 2 R, µ(n 1 ) = ν(n 2 ) = 0
24 Formulation and duality Optimal semi-static strategy and quasi-sure formulation Duality and existence under quasi-sure robust superhedging Theorem (Beiglböck, Nutz & NT 15) Let µ ν and c 0 measurable. Then P(µ, ν) = D qs (µ, ν) and existence holds for D qs
25 Outline Optimal Transport and Model-free hedging Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem 1 Optimal Transport and Model-free hedging The Monge-Kantorovitch optimal transport problem Financial interpretation 2 Formulation and duality Optimal semi-static strategy and quasi-sure formulation 3 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem
26 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Geometry of optimal transport plans : cyclic monotonicity In standard optimal transport, optimality of a transport plan is a property of its support... Theorem For an optimal transport plan P, there exists Γ X Y : P [Γ] = 1, and for all finite subset (x i, y i ) i n Γ : n c(x i, y i ) i=1 n c(x i, y i+1 ) with y n+1 = y 1 i=1 (necessary and sufficient, under slight conditions)
27 Back to the original Monge formulation Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem P K P M : Kantorovitch formulation relaxation of Monge one Theorem (Rachev & Rüschendorf) µ without atoms, c C 1 with c xy > 0 (Spence-Mirrlees/Twist condition). Then there is a unique optimal transference plan : P (dx, dy) = µ(dx)δ {T (x)}(dy) with T = F 1 ν F µ Consequently P M = P K, and T solves both problems. T : monotone rearrangement, Fréchet-Hoeffding coupling Extension to R d (Brenier) : P concentrated on the graph of the gradient of some c convex function
28 On the Spence-Mirrlees condition Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem The solution of the Kantorovitch optimal transportation problem P K := sup P P(µ,ν) c(x, y)p(dx, dy) is not modified by the change of performance criterion : c(x, y) ĉ(x, y) := c(x, y)+a(x) + b(y) Spence-Mirrlees condition c xy > 0 stable by this transformation
29 Martingale monotonicity condition Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Optimality of martingale transport is also a property of its support... Theorem (Beiglböck & Juillet 12, Zaev 14) Let P be a solution of P(µ, ν). Then there exists Γ Ω, P [Γ] = 1 such that for all P 0 with Supp(P 0 ) = {ω 1,..., ω N } Γ, we have : [ ] E P 0 c(x, Y ) E P [ c(x, Y ) ] whenever P X 1 = P 0 X 1, P Y 1 = P 0 Y 1 and E P [Y X ] = E P 0 [Y X ]
30 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Necessary & sufficient Martingale monotonicity Theorem (Beiglböck, Nutz & NT 15) Let ( ˆϕ, ˆψ, ĥ) be solution of D qs (µ, ν), and set Γ := { ˆϕ ˆψ + ĥ = c} Then P solution of P(µ, ν) iff P [Γ] = 1
31 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Worst Case Financial Market Brenier Theorem Solution P M(µ, ν) exists for c USC. Is there an optimal transfert map, i.e. optimal transport of µ to ν through a map T? NO, unless µ = ν! Is there a transference plan along a minimal randomization? X Y = T u (X ) with probability q(x ) Y = Td (X ) with probability 1 q(x )
32 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Worst Case Financial Market Brenier Theorem Solution P M(µ, ν) exists for c USC. Is there an optimal transfert map, i.e. optimal transport of µ to ν through a map T? NO, unless µ = ν! Is there a transference plan along a minimal randomization? X Y = T u (X ) with probability q(x ) Y = Td (X ) with probability 1 q(x )
33 Left-monotone martingale transport Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Definition (Beiglbock & Juillet 2012) P M(µ, ν) is left-monotone if P[(X, Y ) Γ] = 1, for some Γ R R, and for all (x, y 1 ), (x, y 2 ), (x, y ) Γ : x < x = y (y 1, y 2 )
34 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Existence and uniqueness of left-monotone martingale transport Theorem (Beiglböck, Henry-Labordère & NT 15) Let P M(µ, ν) be solution of P(µ, ν). If c x strictly convex in y, µ a.e. x, then P is left-monotone Theorem (Beiglböck & Juillet 12, Beiglböck, Henry-Labordère & NT 15) Assume µ has no atoms. Then, there is a unique left-monotone P M(µ, ν) with distribution concentrated on two graphs Henry-Labordère & NT 13 provide an explicit description of the left-monotone transport plan and corresponding semi-static robust superhedging strategy
35 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem The martingale version of the Spence-Mirrlees condition... is c xyy > 0 Notice that the solution of the Martingale Transport problem is not altered by the change of payoff : c(x, y) ĉ(x, y) := c(x, y) + a(x) + b(y) + h(x)(y x) ĉ xyy = c xyy
36 Concluding remarks Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Extension to R d, i.e. µ and ν prob. meas. on R d : Duality for c USC and existence for P(µ, ν) Duality for c meas. and existence for D(µ, ν)?? Martingale version of the Brenier theorem, see Ghoussoub, Kim & Lim 2015 Extension to finite discrete-time, possibly finitely-many marginals constraints
37 Concluding remarks II Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem Continuous-Time Martingale Transport : substitute for the return from a dynamic hedge h is the stochastic integral T 0 H sdx s But without reference probability?? = Two viewpoints Pointwise definition : restrict H to have finite variation, then : T 0 H sdx s := T 0 X sdh s + H T X T H 0 X 0 Dolinsky & Soner Quasi-sure definition : under any P M(µ, ν), T 0 H sdx s is defined by the standard stochastic analysis... Guo, Tan & NT 2015
38 Back to Standard Optimal Transport Martingale Monotonicity Condition Martingale Version of the 1-dim Brenier Theorem THANK YOU FOR YOUR ATTENTION
Martingale Optimal Transport and Robust Finance
Martingale Optimal Transport and Robust Finance Marcel Nutz Columbia University (with Mathias Beiglböck and Nizar Touzi) April 2015 Marcel Nutz (Columbia) Martingale Optimal Transport and Robust Finance
More informationMartingale Transport, Skorokhod Embedding and Peacocks
Martingale Transport, Skorokhod Embedding and CEREMADE, Université Paris Dauphine Collaboration with Pierre Henry-Labordère, Nizar Touzi 08 July, 2014 Second young researchers meeting on BSDEs, Numerics
More informationA utility maximization proof of Strassen s theorem
Introduction CMAP, Ecole Polytechnique Paris Advances in Financial Mathematics, Paris January, 2014 Outline Introduction Notations Strassen s theorem 1 Introduction Notations Strassen s theorem 2 General
More informationAn Explicit Martingale Version of the one-dimensional Brenier Theorem
An Explicit Martingale Version of the one-dimensional Brenier Theorem Pierre Henry-Labordère Nizar Touzi April 10, 2015 Abstract By investigating model-independent bounds for exotic options in financial
More informationModel Free Hedging. David Hobson. Bachelier World Congress Brussels, June University of Warwick
Model Free Hedging David Hobson University of Warwick www.warwick.ac.uk/go/dhobson Bachelier World Congress Brussels, June 2014 Overview The classical model-based approach Robust or model-independent pricing
More informationOptimal martingale transport in general dimensions
Optimal martingale transport in general dimensions Young-Heon Kim University of British Columbia Based on joint work with Nassif Ghoussoub (UBC) and Tongseok Lim (Oxford) May 1, 2017 Optimal Transport
More informationRobust hedging with tradable options under price impact
- Robust hedging with tradable options under price impact Arash Fahim, Florida State University joint work with Y-J Huang, DCU, Dublin March 2016, ECFM, WPI practice is not robust - Pricing under a selected
More informationRobust Hedging of Options on a Leveraged Exchange Traded Fund
Robust Hedging of Options on a Leveraged Exchange Traded Fund Alexander M. G. Cox Sam M. Kinsley University of Bath Recent Advances in Financial Mathematics, Paris, 10th January, 2017 A. M. G. Cox, S.
More informationViability, Arbitrage and Preferences
Viability, Arbitrage and Preferences H. Mete Soner ETH Zürich and Swiss Finance Institute Joint with Matteo Burzoni, ETH Zürich Frank Riedel, University of Bielefeld Thera Stochastics in Honor of Ioannis
More informationModel-independent bounds for Asian options
Model-independent bounds for Asian options A dynamic programming approach Alexander M. G. Cox 1 Sigrid Källblad 2 1 University of Bath 2 CMAP, École Polytechnique University of Michigan, 2nd December,
More informationModel-independent bounds for Asian options
Model-independent bounds for Asian options A dynamic programming approach Alexander M. G. Cox 1 Sigrid Källblad 2 1 University of Bath 2 CMAP, École Polytechnique 7th General AMaMeF and Swissquote Conference
More informationArbitrage Theory without a Reference Probability: challenges of the model independent approach
Arbitrage Theory without a Reference Probability: challenges of the model independent approach Matteo Burzoni Marco Frittelli Marco Maggis June 30, 2015 Abstract In a model independent discrete time financial
More informationHow do Variance Swaps Shape the Smile?
How do Variance Swaps Shape the Smile? A Summary of Arbitrage Restrictions and Smile Asymptotics Vimal Raval Imperial College London & UBS Investment Bank www2.imperial.ac.uk/ vr402 Joint Work with Mark
More informationPathwise Finance: Arbitrage and Pricing-Hedging Duality
Pathwise Finance: Arbitrage and Pricing-Hedging Duality Marco Frittelli Milano University Based on joint works with Matteo Burzoni, Z. Hou, Marco Maggis and J. Obloj CFMAR 10th Anniversary Conference,
More informationMartingale Optimal Transport: A Nice Ride in Quantitative Finance
Martingale Optimal Transport: A Nice Ride in Quantitative Finance Pierre Henry-Labordère 1 1 Global markets Quantitative Research, SOCIÉTÉ GÉNÉRALE Contents Optimal transport versus Martingale optimal
More informationOptimal robust bounds for variance options and asymptotically extreme models
Optimal robust bounds for variance options and asymptotically extreme models Alexander Cox 1 Jiajie Wang 2 1 University of Bath 2 Università di Roma La Sapienza Advances in Financial Mathematics, 9th January,
More 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 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 informationarxiv: v2 [q-fin.pr] 14 Feb 2013
MODEL-INDEPENDENT BOUNDS FOR OPTION PRICES: A MASS TRANSPORT APPROACH MATHIAS BEIGLBÖCK, PIERRE HENRY-LABORDÈRE, AND FRIEDRICH PENKNER arxiv:1106.5929v2 [q-fin.pr] 14 Feb 2013 Abstract. In this paper we
More informationA New Tool For Correlation Risk Management: The Market Implied Comonotonicity Gap
A New Tool For Correlation Risk Management: The Market Implied Comonotonicity Gap Peter Michael Laurence Department of Mathematics and Facoltà di Statistica Universitá di Roma, La Sapienza A New Tool For
More informationRisk Measures and Optimal Risk Transfers
Risk Measures and Optimal Risk Transfers Université de Lyon 1, ISFA April 23 2014 Tlemcen - CIMPA Research School Motivations Study of optimal risk transfer structures, Natural question in Reinsurance.
More informationOn robust pricing and hedging and the resulting notions of weak arbitrage
On robust pricing and hedging and the resulting notions of weak arbitrage Jan Ob lój University of Oxford obloj@maths.ox.ac.uk based on joint works with Alexander Cox (University of Bath) 5 th Oxford Princeton
More informationOptimization Approaches Applied to Mathematical Finance
Optimization Approaches Applied to Mathematical Finance Tai-Ho Wang tai-ho.wang@baruch.cuny.edu Baruch-NSD Summer Camp Lecture 5 August 7, 2017 Outline Quick review of optimization problems and duality
More informationConsistency of option prices under bid-ask spreads
Consistency of option prices under bid-ask spreads Stefan Gerhold TU Wien Joint work with I. Cetin Gülüm MFO, Feb 2017 (TU Wien) MFO, Feb 2017 1 / 32 Introduction The consistency problem Overview Consistency
More informationFunctional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs.
Functional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs Andrea Cosso LPMA, Université Paris Diderot joint work with Francesco Russo ENSTA,
More informationEquity correlations implied by index options: estimation and model uncertainty analysis
1/18 : estimation and model analysis, EDHEC Business School (joint work with Rama COT) Modeling and managing financial risks Paris, 10 13 January 2011 2/18 Outline 1 2 of multi-asset models Solution to
More informationA MODEL-FREE VERSION OF THE FUNDAMENTAL THEOREM OF ASSET PRICING AND THE SUPER-REPLICATION THEOREM. 1. Introduction
A MODEL-FREE VERSION OF THE FUNDAMENTAL THEOREM OF ASSET PRICING AND THE SUPER-REPLICATION THEOREM B. ACCIAIO, M. BEIGLBÖCK, F. PENKNER, AND W. SCHACHERMAYER Abstract. We propose a Fundamental Theorem
More informationPricing and hedging in incomplete markets
Pricing and hedging in incomplete markets Chapter 10 From Chapter 9: Pricing Rules: Market complete+nonarbitrage= Asset prices The idea is based on perfect hedge: H = V 0 + T 0 φ t ds t + T 0 φ 0 t ds
More informationThe Skorokhod Embedding Problem and Model Independent Bounds for Options Prices. David Hobson University of Warwick
The Skorokhod Embedding Problem and Model Independent Bounds for Options Prices David Hobson University of Warwick www.warwick.ac.uk/go/dhobson Summer School in Financial Mathematics, Ljubljana, September
More informationAre the Azéma-Yor processes truly remarkable?
Are the Azéma-Yor processes truly remarkable? Jan Obłój j.obloj@imperial.ac.uk based on joint works with L. Carraro, N. El Karoui, A. Meziou and M. Yor Welsh Probability Seminar, 17 Jan 28 Are the Azéma-Yor
More informationAll Investors are Risk-averse Expected Utility Maximizers
All Investors are Risk-averse Expected Utility Maximizers Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel) AFFI, Lyon, May 2013. Carole Bernard All Investors are
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 informationAll Investors are Risk-averse Expected Utility Maximizers. Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel)
All Investors are Risk-averse Expected Utility Maximizers Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel) First Name: Waterloo, April 2013. Last Name: UW ID #:
More informationCOMP331/557. Chapter 6: Optimisation in Finance: Cash-Flow. (Cornuejols & Tütüncü, Chapter 3)
COMP331/557 Chapter 6: Optimisation in Finance: Cash-Flow (Cornuejols & Tütüncü, Chapter 3) 159 Cash-Flow Management Problem A company has the following net cash flow requirements (in 1000 s of ): Month
More informationWeak Reflection Principle and Static Hedging of Barrier Options
Weak Reflection Principle and Static Hedging of Barrier Options Sergey Nadtochiy Department of Mathematics University of Michigan Apr 2013 Fields Quantitative Finance Seminar Fields Institute, Toronto
More informationMartingale Optimal Transport (and Friends)
Martingale Optimal Transport (and Friends) 18 th 19 th September 2017, held at the Oxford-Man Institute, Eagle House, Walton Well Rd, Oxford OX2 6ED Organised by: Gaoyue Guo and Jan Obłój This workshop
More informationOptimal Allocation of Policy Limits and Deductibles
Optimal Allocation of Policy Limits and Deductibles Ka Chun Cheung Email: kccheung@math.ucalgary.ca Tel: +1-403-2108697 Fax: +1-403-2825150 Department of Mathematics and Statistics, University of Calgary,
More informationBOUNDS FOR VIX FUTURES GIVEN S&P 500 SMILES
BOUNDS FOR VIX FUTURES GIVEN S&P 5 SMILES JULIEN GUYON, ROMAIN MENEGAUX, AND MARCEL NUTZ Abstract. We derive sharp bounds for the prices of VIX futures using the full information of S&P 5 smiles. To that
More informationAdvanced Probability and Applications (Part II)
Advanced Probability and Applications (Part II) Olivier Lévêque, IC LTHI, EPFL (with special thanks to Simon Guilloud for the figures) July 31, 018 Contents 1 Conditional expectation Week 9 1.1 Conditioning
More informationRobust pricing and hedging under trading restrictions and the emergence of local martingale models
Robust pricing and hedging under trading restrictions and the emergence of local martingale models Alexander M. G. Cox Zhaoxu Hou Jan Ob lój June 9, 2015 arxiv:1406.0551v2 [q-fin.mf] 8 Jun 2015 Abstract
More informationOn Complexity of Multistage Stochastic Programs
On Complexity of Multistage Stochastic Programs Alexander Shapiro School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0205, USA e-mail: ashapiro@isye.gatech.edu
More informationIntroduction to Probability Theory and Stochastic Processes for Finance Lecture Notes
Introduction to Probability Theory and Stochastic Processes for Finance Lecture Notes Fabio Trojani Department of Economics, University of St. Gallen, Switzerland Correspondence address: Fabio Trojani,
More informationPortfolio Optimisation under Transaction Costs
Portfolio Optimisation under Transaction Costs W. Schachermayer University of Vienna Faculty of Mathematics joint work with Ch. Czichowsky (Univ. Vienna), J. Muhle-Karbe (ETH Zürich) June 2012 We fix a
More informationRecovering portfolio default intensities implied by CDO quotes. Rama CONT & Andreea MINCA. March 1, Premia 14
Recovering portfolio default intensities implied by CDO quotes Rama CONT & Andreea MINCA March 1, 2012 1 Introduction Premia 14 Top-down" models for portfolio credit derivatives have been introduced as
More informationOptimizing S-shaped utility and risk management
Optimizing S-shaped utility and risk management Ineffectiveness of VaR and ES constraints John Armstrong (KCL), Damiano Brigo (Imperial) Quant Summit March 2018 Are ES constraints effective against rogue
More informationDynamic Admission and Service Rate Control of a Queue
Dynamic Admission and Service Rate Control of a Queue Kranthi Mitra Adusumilli and John J. Hasenbein 1 Graduate Program in Operations Research and Industrial Engineering Department of Mechanical Engineering
More informationarxiv: v1 [q-fin.pm] 13 Mar 2014
MERTON PORTFOLIO PROBLEM WITH ONE INDIVISIBLE ASSET JAKUB TRYBU LA arxiv:143.3223v1 [q-fin.pm] 13 Mar 214 Abstract. In this paper we consider a modification of the classical Merton portfolio optimization
More information- Introduction to Mathematical Finance -
- Introduction to Mathematical Finance - Lecture Notes by Ulrich Horst The objective of this course is to give an introduction to the probabilistic techniques required to understand the most widely used
More informationArbitrage Bounds for Weighted Variance Swap Prices
Arbitrage Bounds for Weighted Variance Swap Prices Mark Davis Imperial College London Jan Ob lój University of Oxford and Vimal Raval Imperial College London January 13, 21 Abstract Consider a frictionless
More informationCONVERGENCE OF OPTION REWARDS FOR MARKOV TYPE PRICE PROCESSES MODULATED BY STOCHASTIC INDICES
CONVERGENCE OF OPTION REWARDS FOR MARKOV TYPE PRICE PROCESSES MODULATED BY STOCHASTIC INDICES D. S. SILVESTROV, H. JÖNSSON, AND F. STENBERG Abstract. A general price process represented by a two-component
More informationRegression estimation in continuous time with a view towards pricing Bermudan options
with a view towards pricing Bermudan options Tagung des SFB 649 Ökonomisches Risiko in Motzen 04.-06.06.2009 Financial engineering in times of financial crisis Derivate... süßes Gift für die Spekulanten
More informationRISK MINIMIZING PORTFOLIO OPTIMIZATION AND HEDGING WITH CONDITIONAL VALUE-AT-RISK
RISK MINIMIZING PORTFOLIO OPTIMIZATION AND HEDGING WITH CONDITIONAL VALUE-AT-RISK by Jing Li A dissertation submitted to the faculty of the University of North Carolina at Charlotte in partial fulfillment
More informationCONSISTENCY AMONG TRADING DESKS
CONSISTENCY AMONG TRADING DESKS David Heath 1 and Hyejin Ku 2 1 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA, email:heath@andrew.cmu.edu 2 Department of Mathematics
More informationAre the Azéma-Yor processes truly remarkable?
Are the Azéma-Yor processes truly remarkable? Jan Obłój j.obloj@imperial.ac.uk based on joint works with L. Carraro, N. El Karoui, A. Meziou and M. Yor Swiss Probability Seminar, 5 Dec 2007 Are the Azéma-Yor
More informationOn an optimization problem related to static superreplicating
On an optimization problem related to static superreplicating strategies Xinliang Chen, Griselda Deelstra, Jan Dhaene, Daniël Linders, Michèle Vanmaele AFI_1491 On an optimization problem related to static
More informationIn Discrete Time a Local Martingale is a Martingale under an Equivalent Probability Measure
In Discrete Time a Local Martingale is a Martingale under an Equivalent Probability Measure Yuri Kabanov 1,2 1 Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 253 Besançon,
More informationClass Notes on Financial Mathematics. No-Arbitrage Pricing Model
Class Notes on No-Arbitrage Pricing Model April 18, 2016 Dr. Riyadh Al-Mosawi Department of Mathematics, College of Education for Pure Sciences, Thiqar University References: 1. Stochastic Calculus for
More informationSample Path Large Deviations and Optimal Importance Sampling for Stochastic Volatility Models
Sample Path Large Deviations and Optimal Importance Sampling for Stochastic Volatility Models Scott Robertson Carnegie Mellon University scottrob@andrew.cmu.edu http://www.math.cmu.edu/users/scottrob June
More informationLECTURE 4: BID AND ASK HEDGING
LECTURE 4: BID AND ASK HEDGING 1. Introduction One of the consequences of incompleteness is that the price of derivatives is no longer unique. Various strategies for dealing with this exist, but a useful
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 informationRobust Pricing and Hedging of Options on Variance
Robust Pricing and Hedging of Options on Variance Alexander Cox Jiajie Wang University of Bath Bachelier 21, Toronto Financial Setting Option priced on an underlying asset S t Dynamics of S t unspecified,
More informationFinite Additivity in Dubins-Savage Gambling and Stochastic Games. Bill Sudderth University of Minnesota
Finite Additivity in Dubins-Savage Gambling and Stochastic Games Bill Sudderth University of Minnesota This talk is based on joint work with Lester Dubins, David Heath, Ashok Maitra, and Roger Purves.
More informationUNIFORM BOUNDS FOR BLACK SCHOLES IMPLIED VOLATILITY
UNIFORM BOUNDS FOR BLACK SCHOLES IMPLIED VOLATILITY MICHAEL R. TEHRANCHI UNIVERSITY OF CAMBRIDGE Abstract. The Black Scholes implied total variance function is defined by V BS (k, c) = v Φ ( k/ v + v/2
More information4 Martingales in Discrete-Time
4 Martingales in Discrete-Time Suppose that (Ω, F, P is a probability space. Definition 4.1. A sequence F = {F n, n = 0, 1,...} is called a filtration if each F n is a sub-σ-algebra of F, and F n F n+1
More information1 Shapley-Shubik Model
1 Shapley-Shubik Model There is a set of buyers B and a set of sellers S each selling one unit of a good (could be divisible or not). Let v ij 0 be the monetary value that buyer j B assigns to seller i
More informationPortfolio Management and Optimal Execution via Convex Optimization
Portfolio Management and Optimal Execution via Convex Optimization Enzo Busseti Stanford University April 9th, 2018 Problems portfolio management choose trades with optimization minimize risk, maximize
More informationRobust hedging of double touch barrier options
Robust hedging of double touch barrier options A. M. G. Cox Dept. of Mathematical Sciences University of Bath Bath BA2 7AY, UK Jan Ob lój Mathematical Institute and Oxford-Man Institute of Quantitative
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 informationMartingales. by D. Cox December 2, 2009
Martingales by D. Cox December 2, 2009 1 Stochastic Processes. Definition 1.1 Let T be an arbitrary index set. A stochastic process indexed by T is a family of random variables (X t : t T) defined on a
More informationPricing in markets modeled by general processes with independent increments
Pricing in markets modeled by general processes with independent increments Tom Hurd Financial Mathematics at McMaster www.phimac.org Thanks to Tahir Choulli and Shui Feng Financial Mathematics Seminar
More informationMarkets with convex transaction costs
1 Markets with convex transaction costs Irina Penner Humboldt University of Berlin Email: penner@math.hu-berlin.de Joint work with Teemu Pennanen Helsinki University of Technology Special Semester on Stochastics
More informationLecture 7: Linear programming, Dedicated Bond Portfolios
Optimization Methods in Finance (EPFL, Fall 2010) Lecture 7: Linear programming, Dedicated Bond Portfolios 03.11.2010 Lecturer: Prof. Friedrich Eisenbrand Scribe: Rached Hachouch Linear programming is
More informationNon-semimartingales in finance
Non-semimartingales in finance Pricing and Hedging Options with Quadratic Variation Tommi Sottinen University of Vaasa 1st Northern Triangular Seminar 9-11 March 2009, Helsinki University of Technology
More informationKim Weston (Carnegie Mellon University) Market Stability and Indifference Prices. 1st Eastern Conference on Mathematical Finance.
1st Eastern Conference on Mathematical Finance March 216 Based on Stability of Utility Maximization in Nonequivalent Markets, Finance & Stochastics (216) Basic Problem Consider a financial market consisting
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 informationOn Existence of Equilibria. Bayesian Allocation-Mechanisms
On Existence of Equilibria in Bayesian Allocation Mechanisms Northwestern University April 23, 2014 Bayesian Allocation Mechanisms In allocation mechanisms, agents choose messages. The messages determine
More informationStochastic Programming and Financial Analysis IE447. Midterm Review. Dr. Ted Ralphs
Stochastic Programming and Financial Analysis IE447 Midterm Review Dr. Ted Ralphs IE447 Midterm Review 1 Forming a Mathematical Programming Model The general form of a mathematical programming model is:
More informationComputing Bounds on Risk-Neutral Measures from the Observed Prices of Call Options
Computing Bounds on Risk-Neutral Measures from the Observed Prices of Call Options Michi NISHIHARA, Mutsunori YAGIURA, Toshihide IBARAKI Abstract This paper derives, in closed forms, upper and lower bounds
More informationLocal Volatility Dynamic Models
René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Columbia November 9, 27 Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding
More informationStochastic Proximal Algorithms with Applications to Online Image Recovery
1/24 Stochastic Proximal Algorithms with Applications to Online Image Recovery Patrick Louis Combettes 1 and Jean-Christophe Pesquet 2 1 Mathematics Department, North Carolina State University, Raleigh,
More informationOptimal Execution: II. Trade Optimal Execution
Optimal Execution: II. Trade Optimal Execution René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Purdue June 21, 212 Optimal Execution
More informationLecture 14: Examples of Martingales and Azuma s Inequality. Concentration
Lecture 14: Examples of Martingales and Azuma s Inequality A Short Summary of Bounds I Chernoff (First Bound). Let X be a random variable over {0, 1} such that P [X = 1] = p and P [X = 0] = 1 p. n P X
More informationModel-Independent Arbitrage Bounds on American Put Options
Model-Independent Arbitrage Bounds on American Put Options submitted by Christoph Hoeggerl for the degree of Doctor of Philosophy of the University of Bath Department of Mathematical Sciences December
More informationThe Uncertain Volatility Model
The Uncertain Volatility Model Claude Martini, Antoine Jacquier July 14, 008 1 Black-Scholes and realised volatility What happens when a trader uses the Black-Scholes (BS in the sequel) formula to sell
More informationinduced by the Solvency II project
Asset Les normes allocation IFRS : new en constraints assurance induced by the Solvency II project 36 th International ASTIN Colloquium Zürich September 005 Frédéric PLANCHET Pierre THÉROND ISFA Université
More informationSLE and CFT. Mitsuhiro QFT2005
SLE and CFT Mitsuhiro Kato @ QFT2005 1. Introduction Critical phenomena Conformal Field Theory (CFT) Algebraic approach, field theory, BPZ(1984) Stochastic Loewner Evolution (SLE) Geometrical approach,
More informationInformation Aggregation in Dynamic Markets with Strategic Traders. Michael Ostrovsky
Information Aggregation in Dynamic Markets with Strategic Traders Michael Ostrovsky Setup n risk-neutral players, i = 1,..., n Finite set of states of the world Ω Random variable ( security ) X : Ω R Each
More informationWeek 1 Quantitative Analysis of Financial Markets Basic Statistics A
Week 1 Quantitative Analysis of Financial Markets Basic Statistics A Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October
More informationTangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey
More informationIntroduction to game theory LECTURE 2
Introduction to game theory LECTURE 2 Jörgen Weibull February 4, 2010 Two topics today: 1. Existence of Nash equilibria (Lecture notes Chapter 10 and Appendix A) 2. Relations between equilibrium and rationality
More informationMartingale Pricing Theory in Discrete-Time and Discrete-Space Models
IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,
More informationRohini Kumar. Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque)
Small time asymptotics for fast mean-reverting stochastic volatility models Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque) March 11, 2011 Frontier Probability Days,
More informationI. Time Series and Stochastic Processes
I. Time Series and Stochastic Processes Purpose of this Module Introduce time series analysis as a method for understanding real-world dynamic phenomena Define different types of time series Explain the
More informationOptimal Stopping Rules of Discrete-Time Callable Financial Commodities with Two Stopping Boundaries
The Ninth International Symposium on Operations Research Its Applications (ISORA 10) Chengdu-Jiuzhaigou, China, August 19 23, 2010 Copyright 2010 ORSC & APORC, pp. 215 224 Optimal Stopping Rules of Discrete-Time
More informationExact replication under portfolio constraints: a viability approach
Exact replication under portfolio constraints: a viability approach CEREMADE, Université Paris-Dauphine Joint work with Jean-Francois Chassagneux & Idris Kharroubi Motivation Complete market with no interest
More informationOn the Lower Arbitrage Bound of American Contingent Claims
On the Lower Arbitrage Bound of American Contingent Claims Beatrice Acciaio Gregor Svindland December 2011 Abstract We prove that in a discrete-time market model the lower arbitrage bound of an American
More informationPortfolio selection with multiple risk measures
Portfolio selection with multiple risk measures Garud Iyengar Columbia University Industrial Engineering and Operations Research Joint work with Carlos Abad Outline Portfolio selection and risk measures
More informationStochastic Partial Differential Equations and Portfolio Choice. Crete, May Thaleia Zariphopoulou
Stochastic Partial Differential Equations and Portfolio Choice Crete, May 2011 Thaleia Zariphopoulou Oxford-Man Institute and Mathematical Institute University of Oxford and Mathematics and IROM, The University
More informationLinear-Rational Term-Structure Models
Linear-Rational Term-Structure Models Anders Trolle (joint with Damir Filipović and Martin Larsson) Ecole Polytechnique Fédérale de Lausanne Swiss Finance Institute AMaMeF and Swissquote Conference, September
More informationHedging of Contingent Claims under Incomplete Information
Projektbereich B Discussion Paper No. B 166 Hedging of Contingent Claims under Incomplete Information by Hans Föllmer ) Martin Schweizer ) October 199 ) Financial support by Deutsche Forschungsgemeinschaft,
More information