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Stochastic Discount Factors and Martingales NYU Stern Financial Theory IV Continuous-Time Finance Professor Jennifer N. Carpenter Spring 2020 Course Outline 1. The continuous-time financial market, stochastic discount factors, martingales 2. European contingent claims pricing, options, futures 3. Term structure models 4. American options and dynamic corporate finance 5. Optimal consumption and portfolio choice 6. Equilibrium in a pure exchange economy, consumption CAPM 7. Exam - in class - closed-note, closed-book Recommended Books and References Back, K., Asset Pricing and Portfolio Choice Theory,OxfordUniversityPress, 2010. Duffie, D., Dynamic Asset Pricing Theory, Princeton University Press, 2001. Karatzas, I. and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1991. Karatzas, I. and S. E. Shreve, Methods of Mathematical Finance, Springer, 1998. Merton, R., Continuous-Time Finance, Blackwell, 1990. Shreve, S. E., Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004. Arbitrage, martingales, and stochastic discount factors 1. Consumption space – random variables in Lp( ) P 2. Preferences – strictly monotone, convex, lower semi-continuous 3. One-period market for payo↵s (a) Marketed payo↵s (b) Prices – positive linear functionals (c) Arbitrage opportunities (d) Viability of the price system (e) Stochastic discount factors 4. Securities market with multiple trading dates (a) Security prices – right-continuous stochastic processes (b) Trading strategies – simple, self-financing, tight (c) Equivalent martingale measures (d) Dynamic market completeness Readings and References Duffie, chapter 6. Harrison, J., and D. Kreps, 1979, Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 20, 381-408. Harrison, J., and S. Pliska, 1981, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and Their Applications, 11, 215-260. Dybvig, P, and C. Huang, 1989, Nonnegative wealth, absence of arbitrage, and feasible consumption plans, Review of Financial Studies 1, 377-401. Back, K. and S. Pliska, 1991, On the fundamental theorem of asset pricing with an infinite state space, Journal of Mathematical Economics 20, 1-18. Cox, J., and S. Ross, 1976, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3, 145-166. Vasicek, O., 1977, An equilibrium characterization of the term structure, Journal of Financial Economics, 5, 177-188. Overview – When Can Prices be Represented with a SDF/EMM? I A stochastic discount factor (sdf) is a random variable m such that for every marketed payo↵ PT with price P0,thepriceP0 can be represented as P0 = E mP . (1) { T } I A risk-neutral pricing measure, or equivalent martingale measure (emm), associated with a “riskless” numeraire asset with price S is a probability measure such P⇤ that for every marketed payo↵ PT with price P0, S0 P0 = E⇤ PT . (2) {ST } I In a given asset market, there is typically a one-to-one correspondence between sdf’s and emm’s given by S0 d m = P⇤ , (3) S d T P d where P⇤ is the so-called Radon-Nikodym derivative of the emm w.r.t. the d P⇤ true probabilityP measure . P I In a securities market with multiple trading dates, these relations expand to Mt+1 St Pt = Et Pt+1 = Et⇤ Pt+1 , (4) { Mt } {St+1 } (ignoring dividends) where it turns out this sdf process Mt can be written as d ⇤ S0 Mt = Et P = Et ST m /St . (5) { d } St { } P In other words, MtPt is a -martingale and Pt/St is a -martingale. I P P⇤ I The sdf and emm representations of prices provide important insight into the structure of asset prices and powerful computational machinery for modeling. I This lecture illustrates the generality of this representation, sets up the basic financial market model with discrete trading dates, and develops the key results. Simple Construction of a SDF from FOC of Portfolio Optimization I Suppose the space of marketed claims is spanned by a finite number of securities with time 0 prices p =(p1,...,pn) and time T payo↵s x =(x1,...,xn),and an investor with preferences described by a strictly increasing, strictly concave, di↵erentiable utility function u is able to find an optimal portfolio, i.e., a row vector of security holdings N ⇤ =(N1⇤,...,Nn⇤). I In particular, the investor chooses security holdings N =(N1,...,Nn) to max Eu(cT )=Eu(e + Nx) s.t. Np 0. (6) N The first-order condition for the investor’s optimal holding N is E u (c )x = k { 0 T k} λpk,whichimplies u0(cT ) pk = E xk (7) { λ } = u0(eT +N⇤x) I Then m λ is a sdf for this market. I If the market is incomplete, there could be di↵erent investors with di↵erent utility functions, which would produce di↵erent sdfs, but the di↵erent sdfs would generate the same prices for the marketed securities. I Introducing a previously unspanned payo↵ to the market would in general lead to re-optimization and change the prices of all assets. Formal Building Blocks of the Economy for More General Construction I Probabilistic Setting Finite time horizon [0,T]. • T Probability space (⌦, , ) with filtration F = t . • F P {F }t=0 Each ! ⌦ is a complete description of what happens from time 0 to T . • 2 is the subjective probability measure believed by people in the economy. • P Each t represents information set at time t. Formally, it is a special collection • F of events, or subsets A of ⌦, called a sigma-field. Each A t represents an event that is measurable, or distinguishable, at time • 2 F t, i.e., you know whether or not ! A at time t. 2 t +1 = , i.e., people don’t forget, and all there is to know is • F ✓ Ft ✓ FT F revealed by time T . A random variable X is a real-valued function on ⌦ that is measurable w.r.t • F (its value is known by time T ). A reason to be precise about “measurability” w.r.t., say, t, is, e.g., to be • F precise about statements like ”the trading strategy doesn’t look ahead.” Notational short-hand: Et X means E X t • { } { |F } I Consumption There is a single consumption good consumed only at time T . • A consumption plan x(!) is an -measurable random variable. • F The consumption space is Lp( ) for some p [1, ), the complete, • C P 2 1 normed vector space (Banach space) of random variables with finite norm p 1/p X p (E( X ) || || ⌘ || || I Preferences People are represented by their preferences on . Without loss of generality, • ⌫ C represents preferences for time T net trades x (net of endowment x¯). ⌫ Assume these preferences satisfy lower semi-continuity, strict monotonicity, • and convexity,whichmeansthesets x : x xˆ are convex xˆ . { 2 C ⌫ } 8 2 C For example, a preference relation defined by x y EU(x) EU(y) • ⌫ () ≥ where U is concave, strictly increasing, and grows at no more than quadratic rate satisfies these conditions. The One-Period Market Let be the subspace of marketed consumption plans. I M ⇢ C Let pm be the price functional on . I M Assume is a linear subspace and pm is a linear functional. I.e., the price of a I M portfolio is the sum of the prices of its pieces, there are no transaction costs, short sale constraints, etc. Suppose there exists x>ˆ 0 a.s. in . I M Definition 1 An arbitrage opportunity is an x s.t. x 0, x>0 > 0,and 2 M ≥ P{ } pm(x) 0. I No arbitrage implies pm is a strictly positive linear functional. Definition 2 The price system ( ,pm) is viable if preferences and net trade M 9 ⌫ x such that pm(x ) 0 and x x for every x such that pm(x) 0. ⇤ 2 M ⇤ ⇤ ⌫ 2 M Definition 3 A stochastic discount factor is a random variable m>0 such that pm(x)=E mx x . (8) { } 8 2 M Definition 4 The market is complete if = . M C Lemma 1 A positive linear functional on Lp(µ) is continuous. Riesz Representation Theorem Let φ be a continuous linear functional on Lp(µ), where p [1, ).ThenthereexistsauniqueY Lq(µ),where1 + 1 =1,s.t. 2 1 2 p q φ(X)= X(!)Y (!)dµ(!) X Lp(µ) . (9) Z⌦ 8 2 If µ is a probability measure ,thiscanbewrittenφ(X)=E XY X Lp( ). P { } 8 2 P Proposition 1 In a complete market with no arbitrage, there exists a unique sdf. Proof of Proposition 1: No arbitrage pm is a strictly positive linear functional. ) p Complete markets pm is defined on all of = L ( ).SobytheR.R.T., ! sdf. ) C P 9 Theorem 1 (Harrison and Kreps) The price system ( ,pm) is viable there M , exists a strictly positive continuous linear extension p of pm to all of . C Corollary The price system ( ,pm) is viable there exists a sdf. M , Sketch of Proof of Harrison and Kreps Theorem 1 Separating Hyperplane Theorem Let A and B be two convex subsets of a topological vector space X and assume that A has an interior point. If Int(A) B =ø \ then there exists a nontrivial continuous linear functional φ and real number ↵ s.t. φ(x) ↵ φ(y) x A, y B. ≥ ≥ 8 2 2 Theorem 1 (Harrison and Kreps) The price system ( ,pm) is viable there M , exists a strictly positive continuous linear extension p of pm to all of . C : Suppose such a p exists. Define on by x y p(x) p(y).Thenthese ( ⌫ C ⌫ , ≥ preferences together with x⇤ =0satisfy the viability condition (vc) in Definition 2. : Suppose ( ,pm) is viable. Let and x satisfy the vc, and w.l.o.g. set x =0. ) M ⌫ ⇤ ⇤ Let A x : x 0 , the set of consumption plans strictly preferred to x .
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