NYU Stern Financial Theory IV Continuous-Time Finance

NYU SternFinancial Theory IVContinuous-Time FinanceProfessor Jennifer N. CarpenterSpring 2020Course Outline1. The continuous-time financial market, stochastic discount factors, martingales2. European contingent claims pricing, options, futures3. Term structure models4. American options and dynamic corporate finance5. Optimal consumption and portfolio choice6. Equilibrium in a pure exchange economy, consumption CAPM7. Exam - in class - closed-note, closed-book

Recommended Books and ReferencesBack, K., Asset Pricing and Portfolio Choice Theory, Oxford University Press,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 factors1. Consumption space – random variables in Lp(P)2. Preferences – strictly monotone, convex, lower semi-continuous3. 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 factors4. 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 completenessReadings and ReferencesDuffie, chapter 6.Harrison, J., and D. Kreps, 1979, Martingales and arbitrage in multiperiod securitiesmarkets, Journal of Economic Theory, 20, 381-408.Harrison, J., and S. Pliska, 1981, Martingales and stochastic integrals in the theoryof continuous trading, Stochastic Processes and Their Applications, 11, 215-260.Dybvig, P, and C. Huang, 1989, Nonnegative wealth, absence of arbitrage, andfeasible consumption plans, Review of Financial Studies 1, 377-401.Back, K. and S. Pliska, 1991, On the fundamental theorem of asset pricing with aninfinite state space, Journal of Mathematical Economics 20, 1-18.Cox, J., and S. Ross, 1976, The valuation of options for alternative stochasticprocesses, Journal of Financial Economics, 3, 145-166.Vasicek, O., 1977, An equilibrium characterization of the term structure, Journal ofFinancial 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 everymarketed payo PT with price P0, the price P0 can be represented asP0 E{mPT } .(1)I A risk-neutral pricing measure, or equivalent martingale measure (emm), associatedwith a “riskless” numeraire asset with price S is a probability measure P suchthat for every marketed payo PT with price P0,P0 E {S0PT } .ST(2)I In a given asset market, there is typically a one-to-one correspondence betweensdf’s and emm’s given bym S0 dP ,ST dP(3) where dPis the so-called Radon-Nikodym derivative of the emm P w.r.t. thedPtrue probability measure P .I In a securities market with multiple trading dates, these relations expand toPt Et {Mt 1StPt 1} E t {Pt 1} ,MtSt 1(4)(ignoring dividends) where it turns out this sdf process Mt can be written asMt E t {dP S0} Et{ST m}/St .dP St(5)I In other words, MtPt is a P -martingale and Pt/St is a P -martingale.I The sdf and emm representations of prices provide important insight into thestructure of asset prices and powerful computational machinery for modeling.I This lecture illustrates the generality of this representation, sets up the basicfinancial market model with discrete trading dates, and develops the key results.

Simple Construction of a SDF from FOC of Portfolio OptimizationI Suppose the space of marketed claims is spanned by a finite number of securitieswith time 0 prices p (p1, . . . , pn) and time T payo s x (x1, . . . , xn), andan 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 rowvector of security holdings N (N1 , . . . , Nn ).I In particular, the investor chooses security holdings N (N1, . . . , Nn) tomax Eu(cT ) Eu(e N x) s.t. N p 0.N(6)The first-order condition for the investor’s optimal holding Nk is E{u0(cT )xk } pk , which impliesp k E{I Then m u0 (eT N x)u0(cT )xk }(7)is a sdf for this market.I If the market is incomplete, there could be di erent investors with di erent utilityfunctions, which would produce di erent sdfs, but the di erent sdfs would generatethe same prices for the marketed securities.I Introducing a previously unspanned payo to the market would in general lead tore-optimization and change the prices of all assets.

Formal Building Blocks of the Economy for More General ConstructionI Probabilistic Setting Finite time horizon [0, T ]. Probability space ( , F , P) with filtration F {Ft}Tt 0. Each ! 2 is a complete description of what happens from time 0 to T . P is the subjective probability measure believed by people in the economy. Each Ft represents information set at time t. Formally, it is a special collectionof events, or subsets A of , called a sigma-field. Each A 2 Ft represents an event that is measurable, or distinguishable, at timet, i.e., you know whether or not ! 2 A at time t. Ft Ft 1 FT F , i.e., people don’t forget, and all there is to know isrevealed 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, Ft, is, e.g., to beprecise about statements like ”the trading strategy doesn’t look ahead.” Notational short-hand: Et{X} means E{X Ft}I Consumption There is a single consumption good consumed only at time T . A consumption plan x(!) is an F -measurable random variable. The consumption space C is Lp(P) for some p 2 [1, 1), the complete,normed vector space (Banach space) of random variables with finite norm X p (E( X p)1/pI Preferences People are represented by their preferences on C . Without loss of generality, represents preferences for time T net trades x (net of endowment x̄). Assume these preferences satisfy lower semi-continuity, strict monotonicity,and convexity, which means the sets {x 2 C : x x̂} are convex 8x̂ 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 quadraticrate satisfies these conditions.

The One-Period MarketI Let M C be the subspace of marketed consumption plans.I Let pm be the price functional on M.I Assume M is a linear subspace and pm is a linear functional. I.e., the price of aportfolio is the sum of the prices of its pieces, there are no transaction costs, shortsale constraints, etc.I Suppose there exists x̂ 0 a.s. in M.Definition 1 An arbitrage opportunity is an x 2 M s.t. x 0, P{x 0} 0, andpm(x) 0.I No arbitrage implies pm is a strictly positive linear functional.Definition 2 The price system (M, pm) is viable if 9 preferences and net tradex 2 M such that pm(x ) 0 and x x for every x 2 M such that pm(x) 0.Definition 3 A stochastic discount factor is a random variable m 0 such thatpm(x) E{mx} 8 x 2 M .(8)Definition 4 The market is complete if M C .Lemma 1 A positive linear functional on Lp(µ) is continuous.Riesz Representation Theorem Let be a continuous linear functional on Lp(µ),where p 2 [1, 1). Then there exists a unique Y 2 Lq (µ), where 1p 1q 1, s.t.(X) Z X(!)Y (!)dµ(!) 8X 2 Lp(µ) .(9)If µ is a probability measure P , this can be written (X) E{XY } 8X 2 Lp(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.Complete markets ) pm is defined on all of C Lp(P). So by the R.R.T., 9! sdf.Theorem 1 (Harrison and Kreps) The price system (M, pm) is viable , thereexists a strictly positive continuous linear extension p of pm to all of C .Corollary The price system (M, pm) is viable , there exists a sdf.

Sketch of Proof of Harrison and Kreps Theorem 1Separating Hyperplane Theorem Let A and B be two convex subsets of atopological vector space X and assume that A has an interior point. If Int(A)\B øthen there exists a nontrivial continuous linear functional and real number s.t.(x) (y) 8x 2 A, y 2 B .Theorem 1 (Harrison and Kreps) The price system (M, pm) is viable , thereexists a strictly positive continuous linear extension p of pm to all of C .(: Suppose such a p exists. Define on C by x y , p(x) p(y). Then thesepreferences together with x 0 satisfy the viability condition (vc) in Definition 2.): Suppose (M, pm) is viable. Let and x satisfy the vc, and w.l.o.g. set x 0.I Let A {x 2 C : x 0}, the set of consumption plans strictly preferred to x .I Let B {y 2 M : pm(y) 0}, the set of a ordable consumption plans.I By the continuity and convexity of preferences and linearity of pm, A is open andconvex, B is convex, A is nonempty since it contains x̂, and A and B are disjointby the viability condition.I Therefore, by the S.H.T., 9 a nontrivial continuous linear functional on all of Cs.t. (x) 0 on A and (x) 0 on B .I It can be shown that is strictly positive, using the x̂ 0 and the continuity andstrict monotonicity of preferences.I Finally, it can be shown that can be scaled to equal pm on M. In particular,let x 2 M and let b x̂pm(x)/pm(x̂) x. Both b and b are in B sincepm(b) pm( b) 0. So (b) 0 and ( b) 0 ) (b) 0 )(x) (x̂)pm(x)/pm(x̂). So letp pm(x̂).(x̂)Then p M pm and p is a strictly positive continuous linear functional on C . I Note that when the market is incomplete, i.e., packages of Arrow-Debreu statecontingent claims cannot be unpacked, then distinct sdfs can exist that wouldprice the individual claims di erently if they were separately tradable, but pricethe package the same.

Securities Market with Multiple Trading DatesI Now let’s operationalize these one-period results in a more practical setting wheresecurities are traded on multiple dates and consumption plans are generated aspayo s of dynamic trading strategies.Definition 5 A stochastic process on a time interval [0, T ] in a filtered probabilityspace ( , F , P, F) is a mapping X(!, t) from [0, T ] to R. Each X(!, ·) isa sample path and each X(·, t) is a random variable. The process is adapted to Fif each Xt is measurable w.r.t. Ft. It is right-continuous if it has right-continuoussample paths.I Suppose there are n 1 long-lived securities traded with adapted, right-continuousprice processes S (S0, S1, . . . , Sn), where each Sk,t 2 Lp(P) and S0 is strictlypositive.Definition 6 A trading strategy N (N0, N1, . . . , Nn) is an adapted n 1dimensional row-vector-valued process where Nk,t denotes the number of sharesof security k held at time t.I We’ll restrict attention to trading strategies with a finite number of trading dates:Definition 7 A trading strategy is simple if there exists a finite partition0 t0 t1 · · · tJ T of [0, T ] and random variables Nkj 2 Ftj such thatNk,t (Nk0Nkjifift 2 [t0, t1]t 2 (tj , tj 1]8 security k 0, 1, . . . , n and trading date j 0, . . . , J(10)1.Definition 8 A simple trading strategy is self-financing ifN t j St j {z }cost of new portfolio at tj (11)N t j 1 St j {z}proceeds from sale of old portfolio at tjfor each j 1, . . . , J 1. The trading strategy is tight (tightly self-financing) if theabove holds with equality, i.e.,Ntj Stj Ntj 1 Stj 8j 1, . . . , J1.(12)