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. Due, 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 Due, 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} pk,whichimplies

u0(cT ) pk = E xk (7) { }

= u0(eT +N⇤x) I Then m 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 be a continuous linear functional on Lp(µ), where p [1, ).ThenthereexistsauniqueY Lq(µ),where1 + 1 =1,s.t. 2 1 2 p q

(X)= X(!)Y (!)dµ(!) X Lp(µ) . (9) Z⌦ 8 2 If µ is a probability measure ,thiscanbewritten(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 and real number ↵ s.t.

(x) ↵ (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 . I ⌘ { 2 C } ⇤ Let B y : pm(y) 0 , the set of a↵ordable consumption plans. I ⌘ { 2 M  } I By the continuity and convexity of preferences and linearity of pm, A is open and convex, B is convex, A is nonempty since it contains xˆ,andA and B are disjoint by the viability condition. Therefore, by the S.H.T., a nontrivial continuous linear functional on all of I 9 C s.t. (x) 0 on A and (x) 0 on B. 

I It can be shown that is strictly positive, using the x>ˆ 0 and the continuity and strict monotonicity of preferences.

Finally, it can be shown that can be scaled to equal pm on .Inparticular, I M let x and let b = xpˆ m(x)/pm(ˆx)+x. Both b and b are in B since 2 M pm(b)=pm( b)=0.So(b) 0 and ( b) 0= (b)=0=   ) ) (x)=(ˆx)pm(x)/pm(ˆx).Solet

p (ˆx) p m . ⌘ (ˆx)

Then p = pm and p is a strictly positive continuous linear functional on . |M C ⇤ I Note that when the market is incomplete, i.e., packages of Arrow-Debreu state contingent claims cannot be unpacked, then distinct sdfs can exist that would price the individual claims di↵erently if they were separately tradable, but price the package the same. Securities Market with Multiple Trading Dates

I Now let’s operationalize these one-period results in a more practical setting where securities are traded on multiple dates and consumption plans are generated as payo↵s of dynamic trading strategies. Deﬁnition 5 A stochastic process on a time interval [0,T] in a ﬁltered probability space (⌦, , , F) is a mapping X(!,t) from ⌦ [0,T] to .EachX(!, ) is F P ⇥ R · a sample path and each X( ,t) is a random variable. The process is adapted to F · if each Xt is measurable w.r.t. t.Itisright-continuous if it has right-continuous F sample paths.

I Suppose there are n+1 long-lived securities traded with adapted, right-continuous p price processes S =(S0,S1,...,Sn),whereeachS L ( ) and S0 is strictly k,t 2 P positive.

Deﬁnition 6 A trading strategy N =(N0,N1,...,Nn) is an adapted n +1- dimensional row-vector-valued process where Nk,t denotes the number of shares of 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 partition

0=t0

Nk0 if t [t0,t1] Nk,t = 2 (10) ( Nkj if t (tj,tj+1] 2 security k =0, 1,...,n and trading date j =0,...,J 1. 8 Deﬁnition 8 A simple trading strategy is self-ﬁnancing if

Ntj Stj Ntj 1Stj (11)  cost of new portfolio at tj proceeds from sale of old portfolio at tj | {z } | {z } for each j =1,...,J 1. The trading strategy is tight (tightly self-ﬁnancing) if the above holds with equality, i.e.,

Ntj Stj = Ntj 1Stj j =1,...,J 1 . (12) 8 Proposition 1 A simple trading strategy is tight the following wealth evolution , equation (WEE) holds:

j NtSt = N0S0 + Nt [St t St ] (13) i i+1^ i ending ptf. value beginning value iX=0 | {z } | {z } trading gains | {z } for every t (tj,t +1] and j =0,...,J 1. 2 j Proof: Homework. Remark 1 If S has continuous sample paths, then the summation above can be t written 0 NudSu, a traditional Riemann-Stieltjes integral path by path. DefinitionR 9 A consumption plan x is marketed if there exists a tight trading 2 C strategy that generates x,i.e.,N s.t. N S = x.Let be the linear subspace of T T M consumption plans that are marketed. Definition 10 is dynamically complete if = . M M C Remark 2 The market will not generally be complete unless is finite-dimensional, C i.e., there are only a finite number of states !.

Deﬁnition 11 An arbitrage opportunity is a self-ﬁnancing trading strategy N s.t. N S 0, N S > 0 > 0,andN0S0 0. T T P{ T T }  I Assume there are no arbitrage opportunities. Proposition 2 (The Law of One Price/The Law of One Process) If two simple, tight trading strategies N and Nˆ generate the same consumption plan x,thenthe portfolio processes NtSt and NˆtSt are indistinguishable.

x x Deﬁnition 12 The price process S of a marketed consumption plan x is St = NtSt where N is any trading strategy that generates x.Theprice functional pm on is x M pm(x) S = N0S0, the start-up cost of any trading strategy that generates x. ⌘ 0 Proposition 3 pm is linear and strictly positive.

Proposition 4 If the market is dynamically complete then pm is a continuous linear p functional on all of L ( ) so there exists a unique sdf m s.t. pm(x)=E mx . P { } Theorem 2 (Harrison and Kreps) The price system S is viable, in that there exists an investor with continuous, strictly increasing, convex preferences who can ﬁnd an optimal trading strategy pm can be represented with a sdf, i.e., there exists an sdf , m s.t. pm(x)=E mx . { } The Martingale Property

Deﬁnition 13 An adapted process X with E Xt < t [0,T] is a martingale if | | 18 2

Et Xs = Xt a.s. 0 t s T. (14) { } 8   

X is a submartingale if Et Xs Xt a.s. 0 t s T . { } 8    X is a supermartingale if Et Xs Xt a.s. 0 t s T . { }  8    S S1 S2 Sn I Let S⇤ =(1, , ,..., ) be the “discounted” security price processes. ⌘ S0 S0 S0 S0 Proposition 5 A simple trading strategy is tight the wealth evolution equation , holds in discounted terms (WEE*):

NtSt⇤ = N0S0⇤ + Nti[St⇤ t St⇤ ] (15) i+1^ i iX=0 for every t (tj,t +1] and j =0,...,J 1. Proof: Homework. 2 j Deﬁnition 14 A probability measure on (⌦, ) is an equivalent martingale P⇤ F measure (emm) if is equivalent to ,andS is a vector martingale under . P⇤ P ⇤ P⇤

Deﬁnition 15 A probability measure on (⌦, ) is equivalent to ,written P⇤ F P , if they agree on which events A have zero probability. P⇤ ⇠ P 2 F I In particular, equivalent measures agree about which trading strategies are arbitrage opportunities.

Radon-Nikodym Theorem ⇤ there exists a strictly positive random variable d P ⇠ P , d P⇤ s.t. for every event A , (A)=E 1 P⇤ ,where1 (!)=1if ! A and d 2 F P⇤ { A d } A 2 zeroP otherwise. In addition, P d =1 d ⇤ 1. d P / dP . P⇤ P = d ⇤ 2. E⇤ X E X dP . { } { P } 3. Conditional expectation under :If is a coarser information set than P⇤ G ⇢ F F then d ⇤ E X dP E⇤ X = { P |G} . { |G} d ⇤ E dP { P |G} Theorem 3 (Harrison and Kreps) There is a one-to-one correspondence between sdf’s m and emm’s given by P⇤

S0,0 d m = P⇤ . (16) S0 d ,T P I The proof of involves applying the law of iterated expections to WEE* to verify S ( 0,0 d ⇤ that S dP is a sdf. 0,T P The proof of involves working with the formal deﬁnition of conditional expection I ) to show that the sdf property implies the martingale condition.

x Proposition 6 Let x and let S be its price process. If there exists an emm ⇤ Sx 2 M P then is a ⇤-martingale. S0 P Proof: Homework. Intuition: Once discounted securities prices are martingales, then so are all portfolios of them, because linear combinations of martingales are martingales, and so are portfolio value processes under tight trading strategies, because these are essentially non-forward-looking dynamic linear combinations of martingales.

I In particular, if there is an emm, we get the risk-neutral pricing equation (RNPE):

x S0,t St = Et⇤ x a.s. t [0,T] . (17) {S0,T } 8 2

Using the R.N.T, we can rewrite the RNPE above using expectation under : I P d ⇤ I Let Zt E dP t . ⌘ { P |F } S0,0 S0,T I Let the sdf process Mt Zt = E m t . ⌘ S0,t { S0,t |F } I Then

S0,t S0,T Et x m x S0,t {S0,T S0,0 } Et mx St = Et⇤ x = = { } . (18) {S0,T } Zt Mt

x It follows that MtS is a -martingale. I t P I So, for example, for given dates t and u,a(cum-dividend)priceprocessP satisﬁes

Mu S0,t Pt = Et Pu = Et⇤ Pu . (19) { Mt } {S0,u } Problem Set 1

Prove the following propositions.

1. Let N be a simple trading strategy with trading dates 0=t1 <

NtSt = N0S0 + Nti(Sti t Sti 1), a.s. (20) ^ iX=1

for all t (tj 1,tj] and j =1,...,J. 2 2. Suppose there are no free lunches. For every payo↵ x in the space of marketed M payo↵s, let the price of x, p(x), be the initial portfolio value N0S0 under a trading strategy N that ﬁnances x. Prove that p is a well-deﬁned, strictly positive, linear functional on . M 3. If the price system ( ,p) (generated by the security prices S)isviable,then M there are no free lunches.