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# Mathematics Of Finance STAT 54000

Purdue

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This 39 page Class Notes was uploaded by Bailey Macejkovic on Saturday September 19, 2015. The Class Notes belongs to STAT 54000 at Purdue University taught by Jose Figueroa-Lopez in Fall. Since its upload, it has received 92 views. For similar materials see /class/207930/stat-54000-purdue-university in Statistics at Purdue University.

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Date Created: 09/19/15

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 consisting ofa risky assetwith price process 8 and a bond with value process Bt defined on a stochastic basis 9 73 775 L520 P 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 Fifiillmar 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 04 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 Um empty unilallcent to Lucil quot lhfl i i MM 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 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 iljFlr Lieu 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 o 19 max gt 3 t g 0 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 Convex duality in portfolio optimization problems I i r l 73 gquot 1 ll V u 3 A u WW I SUPEUVTW such that V0 g 10 and V 2 0 Primal problem Assumption w lt ID suerM EQ ET 1H lt 00 The dual domain f Nonnegaz ive supermarz ingales hzo such that i 0 g 50 g 1 and ii tB1Vt tgt0 is a supermaringale for all admissible wealth processes W tgt0 t Consequences E 6 B lV lt ifV lt w T T T Lquotquot 39V U 2 0 39 o M C f The density process St 2 3 t E ffor each EMM Q Road towards weak duality o For E f and any admissible wealth process V with V0 g 10 EUKV EUIT QW AE TBT MT Lw W 7 P 1 7L x E M llWM A MM mm Aquot T U LJ quotr J lt1 E QgBmmAm W 1 x r lik H 3 BMW U U Lu Ag v I ALL 17 P 0339 where 7Q w supUZO U v w Av Cahvex Familth a Let F be a subclass of Then pw E511le E NEEgalaxy g I7A TBT1 w Aw d3 Dual problem associated to F C f Weak duality UTILITY d 21 El7A TB1w StateDependent Utility Function 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 l 15 7 U H U0 1 39 um I 0395 0 5 1 1 5 i 25 95 3 5 Dual domain 4 Questions 1 Does strong duality l hold for some A 3 a quot 1 1 2 Is the dual problem llillll ggfl U a attainable 39 V 71 39l lt1 l 1 J l a 1 x 1 quot Q L K quot T quot 2 J attainable Answers when T F Ff ivlllll nil llLafllWW l 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 d5t Sr b dt dZt dBt rBtdt B0 1 Some preliminary facts 0 Z ut O39Wt Purely discontinuous part a The pointprocess H z t AZt 815 AZt Zt Zt 7 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 result Will a 5 is a positive local martingale iff St 605X where X 0tGsdW8 At Fsz1vdsdz such thatF 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 0 S 605X A nonnegative with X E S and A predictable increas N 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 A gt 0 such that i j The problem iangt0 dl f A Aw is attainable say at A gt 0 The dual problem d A infgep E INJA T BT1 w is attain able say at E F iangt0 dl f A Aw E U 1ng7 where l W IWS BT71 E B1W 10 Strategy to prove strong duality In view of the weak duality property T 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 if Recall that the density process 6 dQITt legrt is in f for each EMM Q By Kunita s representation 6 E F Since F is convex 8 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 0 EEUA 513 UA 5TBT 8 1 EB1A B1NEST 631 x w 39 1x 1 I Then suerM EQ BT lwg g E BT1W 10 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 fl 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 g X A is identically zero and hence 6 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 M2 R e 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 it Show that the Qmartingale gtxlt 1 gtxlt gtxlt 1 Xt EQk BT I 5T BT m admits the representation t 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 Discretetime financial models The most important assets Jos FigueroaL pez Purdue University Math 515 Stat 540 Spring 2008 39 Bond o It is a contract where for a suitable price now the holder gets a fixed payment called the principal value or face value at a prespecified fixed time T called the maturity 39 Bond o It is a contract where for a suitable price now the holder gets a fixed payment called the principal value or face value at a prespecified fixed time T called the maturity 0 Longterm bonds usually bear coupons If no coupon is paid the bond is called a zerocoupon or pure discount bond As these assets provide the owner with fixed cash flow they are also called fixed income securities 39 Bond o It is a contract where for a suitable price now the holder gets a fixed payment called the principal value or face value at a prespecified fixed time T called the maturity 0 Longterm bonds usually bear coupons If no coupon is paid the bond is called a zerocoupon or pure discount bond As these assets provide the owner with fixed cash flow they are also called fixed income securities 0 Issued by government eg tbill and notes or many financial companies 39 Bond It is a contract where for a suitable price now the holder gets a fixed payment called the principal value or face value at a prespecified fixed time T called the maturity Longterm bonds usually bear coupons If no coupon is paid the bond is called a zerocoupon or pure discount bond As these assets provide the owner with fixed cash flow they are also called fixed income securities Issued by government eg tbill and notes or many financial companies Risks Default risk Interestrate risk 39 Bond It is a contract where for a suitable price now the holder gets a fixed payment called the principal value or face value at a prespecified fixed time T called the maturity Longterm bonds usually bear coupons If no coupon is paid the bond is called a zerocoupon or pure discount bond As these assets provide the owner with fixed cash flow they are also called fixed income securities Issued by government eg tbill and notes or many financial companies Risks Default risk Interestrate risk Similar to saving account also called moneymarket account 39 Stocks or equity 0 The holder of a share the basic unit of stock owns a fraction of a public limited company 39 Stocks or equity 0 The holder of a share the basic unit of stock owns a fraction of a public limited company 0 The shareholder is not liable for the debt of the company 39 Stocks or equity 0 The holder of a share the basic unit of stock owns a fraction of a public limited company 0 The shareholder is not liable for the debt of the company 0 Traded in very organized stock markets eg Nasdaq NYSE and Amex This is called overthecounter market 39 Stocks or equity 0 The holder of a share the basic unit of stock owns a fraction of a public limited company 0 The shareholder is not liable for the debt of the company 0 Traded in very organized stock markets eg Nasdaq NYSE and Amex This is called overthecounter market 0 Investors usually buy and sell shares from large companies called Market Makers or dealers eg Morgan Stanley Merrill Lynch etc 39 Stocks or equity The holder of a share the basic unit of stock owns a fraction of a public limited company The shareholder is not liable for the debt of the company Traded in very organized stock markets eg Nasdaq NYSE and Amex This is called overthecounter market 0 Investors usually buy and sell shares from large companies called Market Makers or dealers eg Morgan Stanley Merrill Lynch etc Nasdaq include over 300 market makers which post bid and ask prices into the Nasdaq network 39 Stocks or equity The holder of a share the basic unit of stock owns a fraction of a public limited company The shareholder is not liable for the debt of the company Traded in very organized stock markets eg Nasdaq NYSE and Amex This is called overthecounter market 0 Investors usually buy and sell shares from large companies called Market Makers or dealers eg Morgan Stanley Merrill Lynch etc Nasdaq include over 300 market makers which post bid and ask prices into the Nasdaq network 0 Find more information on Market Mechanics by James J Angel 39 Stocks or equity The holder of a share the basic unit of stock owns a fraction of a public limited company The shareholder is not liable for the debt of the company Traded in very organized stock markets eg Nasdaq NYSE and Amex This is called overthecounter market Investors usually buy and sell shares from large companies called Market Makers or dealers eg Morgan Stanley Merrill Lynch etc Nasdaq include over 300 market makers which post bid and ask prices into the Nasdaq network Find more information on Market Mechanics by James J Angel Risks Unpredictable price variability driven by the large amount of market participants with different objectives and preferences 39 Contingent claim or derivative 0 What is it Financial instrument whose payoff X at a specified time called expiration date is contingent to the value of other traded asset called the underlying Formally it is a contract in which one party who is said to take a short position or to be the writer receives payment P0 at t 0 and makes the payment X at t T The counter party who is said to take a long position or to be the buyer makes the payment Po at t 0 and receives the payment X at t T 39 Contingent claim or derivative 0 What is it Financial instrument whose payoff X at a specified time called expiration date is contingent to the value of other traded asset called the underlying Formally it is a contract in which one party who is said to take a short position or to be the writer receives payment P0 at t 0 and makes the payment X at t T The counter party who is said to take a long position or to be the buyer makes the payment Po at t 0 and receives the payment X at t T 0 Some popular derivatives Forward Agreement to buy or sell an asset at a predetermined future time for a predetermined price What is the Payoff Call option Contract that gives the not the obligation to buy a quantity of some asset during a specified period of time at a price fixed today What is the Payoff 39 Options derivatives or contingent Claims 0 Why are they useful for Risk management I Financial companies use options to increase or reduce certain type of risk This practice is sometimes called hedging I Therefore they are used as a type of insurance instruments I Speculative investment I What does a purchaser of call options hope for a What does a purchaser of put options hope for 39 Options derivatives or contingent Claims 0 Why are they useful for Risk management I Financial companies use options to increase or reduce certain type of risk This practice is sometimes called hedging I Therefore they are used as a type of insurance instruments I Speculative investment I What does a purchaser of call options hope for a What does a purchaser of put options hope for I Where are they traded Its value is more sensitive to market changes I Amounts to be gained and lost are larger and so they are much shipper than the underlying 39 Options derivatives or contingent Claims 0 Why are they useful for Risk management I Financial companies use options to increase or reduce certain type of risk This practice is sometimes called hedging I Therefore they are used as a type of insurance instruments I Speculative investment I What does a purchaser of call options hope for a What does a purchaser of put options hope for I Where are they traded Its value is more sensitive to market changes I Amounts to be gained and lost are larger and so they are much shipper than the underlying o What are potential risks of option over usage Options might limit the possible size of one s gains

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