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# 511 Class Note for STAT 54000 with Professor Figueroa-Lopez at Purdue

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Date Created: 02/06/15

Statedependent utility maximization in Levy markets Jos E FigueroaLopez University of California Santa Barabara Department of Statistics and Applied Probability Joint work with Jin Ma Probability and Statistics Seminar University of Southern California February 16th 2007 Program D Formulation of the problem 0 Statedependent utility maximization 0 Motivation Shortfall minimization of contingent claims D General solution for semimartingale models using convex duality D Revised solution forjumpdiffusion models driven by Levy processes D Final conclusions Formulation of the problem Setup A a bond with value process Bt defined on a stochastic basis 9 73 775 L520 P consisting of a risky assetwith price process 8 and Goal Dynamically allocate an initial endowment 10 so that to maximize the agent s expected final utility during a finite time horizon 0 T Statedependent utility U w w R x 9 gt R Fifilmer Liill5l39ig 0 Increasing and concave in the wealth w for each state of nature w E Q a Flat for wealths 10 above certain cutoff Hw with values on 0 00 Problem Maximize E U VT over all self nancing trading strategies such that the resulting agent s wealth satisfies ogth g 109 and 2 0 Budget Constraint Solvency condition Motivation Optimal Hedging of Contingent Claims Setting a The discounted asset price Bt 185 is a semimarz ingale o The class M of Equivalent Local Martingale Measures EMM is non empty The hedging problem and superhedging 0 An agent needs to deliver at time T the payoff H of a liability o The liability H can in principle be hedged away 5 Starting with a large enough initial capital 10 there is a trading strat egy that will allow the agent to cover the payoff H while maintaining solvency That is lfw 2 1D suerM EQ ET 1H there exists admissible V2th such thatVO w and VT 2 H a An initial capital 10 smallerthan ID entails certain shortfall risk sometimes VT lt H motivating the problem of minimizing the expected shortfall MinimizeE L H VT st V0 g 10 and V 2 0 fora Loss Function L R gt R Typically L0 0 increasing and convex o LEEllmer The shortfall minimization problem is equivalent to a utility maximization problem with statedependent utility Um w z LltHltwgtgt LltltHltwgt wgtgt Convex Duality Methods 0 Basic idea Upper bound a maximization problem with constraints using a convex minimization problem without constraints For instance m n 39 9 H 33 5 quot Wgi 3x M x W A 2 o 19 3 max 96 g maxm x A g g d min30x A20 ll a We say that strong duality holds if 19 d This holds if for instance the primal problem is convex and lt 0 Slater condition 0 Applications Duality plays an important role in convex optimization i Perturbation and sensibility analysis ii Interior point methods Road towards weak duality o For E F and any admissible wealth process V with V0 g 10 a TT T i h 1W7 1EIV7 0Ejb EVT7amp W EiT j T T 7 y e 39 E U llT 7 by Jim lt lit sup Lag PM va f Ema avZU39 T E I7A TBT 1w Aw where 7Q w supUZO U v w Av Gemex Dblng Hangman a Let F be a subclass of Then pw supE lif f w g I7A TBT1w Aw e J d1th Dual problem associated to F C f Weak duality UTILITY d A StateDependent Utility Function inf E 56F 14 l7A TBT1w 45 7 19 g d Aw Convex Dual Function UH Slope is H 35 7 3 l l 25 7 I Strictly Convex Part 2 7 I 1 l 15 7 U H U0 1 7 39 um I l 05 1 i 1 1 1 1 0 05 1 15 2 25 3 35 Dual domain 4 Questions 1 Does strong duality I l hold for some A w wquot 39 a l 7 E 2 Is the dual problem lll llig39glj U a a 1r attainable 3 Is the primal problem 3 attainable Answers when T F lt am 9le Yes if oo lt E U0w g E UHw lt 00 Moreover we have a dual characterization of the optimal final wealth gtxlt gtxlt gtxlt 1 VT 6T BT 7 where Iw inf z U zw lt A H I7 w Our problem For a given market model and a concrete utility function 0 can we specify further 6 a can one narrow down the dual domain F C f where to search 6 Portfolio optimization for geometric Levy models What is a L vy process A random quantity Zt evolving in time in such a way that a The increment ZtAt Zt during a time period 15 t At is both 1 independent of the past 352 0Zs S g t and 2 with distribution law depending only on the time span At a The process can exhibit sudden changes in magnitude jumps but these occur at unpredictable times no fixed jump times A natural extension of the BlackScholes model Geometric L vy model Why a L vy Market 0 Flexible modelling of return distribution eg leptokurz ic and asymmetric 0 Consistency with asset price evolution which is discontinuous defacto Market model Geometric L vy model and a constant interest rate bond Some preliminary facts 0 Z ut O39Wt Purely discontinuous part a The pointprocess H z t AZt 815 AZt Zt Zt 3e 0 is a Poisson point process with intensity measure VdZ dt for certain mea sure V on R0 R0 the socalled L vy measure Notation o Ndt dz denotes the counting random measure on R gtlt R0 associ ated to the point process H o Ndt dz Ndt dz Vdzdt denotes the compensated Pois son measure of N 0 Information processIF Fthzm where 737 ftZV P null sets Question Can we specify the dual domain F C T where to search the dual solution 6 for a geometric L vy model Key res u It l Pill rl o 5 is a positive local martingale iff St 05 X where N X 0tGsdW8 At FszNdsdz such that F gt 1 o 5 is a positive supermartingale iff St 605X A where X is as above and A is increasing predictable such that AA lt 1 Consequence lfthe utility is unbounded Uoo 00 and the dual problem is feasible then the dual solution 6 is strictly positive and thus character ized in terms of G F A However typically UOO UH lt 00 Tentative dual domain F t t s Xt GsdWs FszNdsdz F 2 1 0 0 P z 608X8 AS st S E fX E S A increasing Some useful results a 5 505X A nonnegative with X E S and A predictable increas ing belongs to T if and only if as b 7 05 zvdz 0Gt zFt zvdz g at Izlzl R for almost every t g 7a where a is the density of the absolutely con tinuous part of A and 739 is the time 6 hits 0 o F is convex and closed under Fatou convergence of processes Dual problem associated with F There exist E F and X gt 0 such that The problem iangt0 d A Aw is attainable say at X gt 0 The dual problem d X infgep E 17A TB1CU is attain able say at E F i fAgt0 l E U w Where i W IA 5B1 E 31 B w w Strategy to prove strong duality In view of the weak duality property l W above will actually be the optimal final wealth if w is replicable with an initial endowment of 10 that is if there exists an admissible trading strat egy with associated wealth V such that V0 g 10 and V11 2 Replicability 539 Recall that the density process 51 63 iS in f for eaCh EMM t By Kunita s representation 6 E F Since F is convex 8 2 86 1 0 g 8 g 1 belongs to F and thus E I7A BT1 2 E I7A BT1 It can be proved that 1 N 8 1 N gtxlt gtxlt 1 0EEUA 5TBT UA 5TBT EB1 Xi 31 3 ST 3 cl 1 SI 1 l Then suerM EQ B1W g E 5BT 1lvg w By Kramkov s theorem of superreplication WE is replicable with an ini tial wealth of w Conclusions and final remarks 0 The method developed here is more explicit in the sense that the dual do main enjoys an explicit parametrization 0 Such a parametrization could lead to certain discrete time approximations in particular cases that are numerically feasible o It can accommodate much more general jumpdiffusion models driven by Levy processes SUCh as d dsia Sip bgdt Zagjdwg ht z1ydt dz j1 Rd o What about optimal portfolio problems with consumption In that case the problem consists of maximizing the utility coming from both consumption and final wealth T E U1VT U2tctdt 0 under a budget constraint and a solvency condition The wealth is now de termined by th TWdt tdSt ctdt and C is the instantaneous rate of consumption Partial results towards consumptionwealth optimization The dual class and the dual problem The dual theorem still holds true forthe class F but replicability requires different arguments Road towards replicability By perturbating the parameters of obtain a variational equality 1 1 E eBT I m 6 YT o for any G F ii in a suitable convex set A where t t t Yt 2 GsdWSG FszNFdsdz asds 0 0 R0 0 with w Wt f Gsds and NW dt dz Ndt dz 1 Fs zvdzds Prove that A in 6 8 X A is identically zero and hence 6 39 i WG is a standard Brown is a local martingale By the ian Motion and NF is a martingale measure with respect to the finitely additive probability measure dQ 6de up to stopping times Establish the decomposition M2Q R A3 where R is the set of all M E M2Q which can be written as Mt f5 sts with th S171 dSt and S is the discounted price process A3 is the stable subspace ofVl2 generated by the integrals Aasmwf A szNFdsdz with 3 in an appropriate class A3 Show that the Qmartingale X r EQ ET 119 BT71 Vt admits the representation t A 0 Then the portfolio with initial value 10 and trading strategy BtBt is admissible and its final wealth is Vi I X B l T

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