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Contingent Claims Pricing Contingent Claims Pricing 1. European contingent claims 2. SDF process in complete markets 3. Replication cost as -expected risk-neutrally discounted payo↵ or -expected P⇤ P stochastically discounted payo↵ 4. The Markovian financial market (a) The fundamental valuation PDE (Feynman-Kac Theorem) (b) Hedging and replicating trading strategies 5. Examples and computational methods (a) Options - Black-Scholes formula (b) Change of numeraire (c) Forward contracts (d) Futures contracts (e) Forward measure vs. futures measure (f) State prices implicit in option prices 1 Readings and References Back, chapters 15 and 16.. Duffie, chapters 5 and 6. Karatzas and Shreve, 1998, chapter 2 Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy 81, 637-654. Merton, R., 1973, Theory of rational option pricing, Bell Journal of Economics and Management Science 4, 141-183. Breeden, D., and R. Litzenberger, 1978, Prices of state-contingent claims implicit in option prices, Journal of Business 51, 621-51. Duffie, D. and R. Stanton, 1992, Pricing continuously resettled contingent claims, Journal of Economic Dynamics and Control 16, 561-573. 2 Summary of the Continuous-Time Financial Market dS0,t dSk,t I Security prices satisfy = rt dt and =(µk,t δk,t) dt + σk,tdBt. S0,t Sk,t − I Given tight tr. strat. ⇡t and consumption ct, portfolio value Xt satisfies the WEE: dXt = rtXt dt + ⇡t(µt rt1) dt + ⇡tσt dBt ct dt. • − − No arbitrage if ⇡tσt =0then ⇡t(µt rt1) = 0 ✓t s.t. σt✓t = µt rt1 I ) − )9 − dXt = rtXt dt + ⇡tσt(✓t dt + dBt) ct dt. ) − t 1 t 2 d ✓ dBs ✓s ds ⇤ = = − 0 s0 −2 0 | | I Under emm ⇤ given by dP ZT where Zt e , P P t R R B = Bt + ✓s ds is Brownian motion. • t⇤ 0 t rsRds Let βt = e− 0 and sdf process Mt = βtZt.ThentheWEEcanalsobewritten: R WEE*: dβtXt + βtct dt = βt⇡tσt dB • t⇤ WEE-M: dMtXt + Mtct dt = Mt[⇡tσt ✓tXt] dBt • − T βu βT T Mu MT I So Xt = Et⇤ t cu du + XT = Et t cu du + XT if ⇡ is mtgale-gen. { βt βt } { Mt Mt } R R I If σ is nonsingular, every c.plan (c, XT ) can be generated by a mtgale-gen. tr.strat. 3 European Contingent Claims Definition 1 A European contingent claim (ecc) is a payo↵ (c, W ) . 2 C I In an incomplete market, it’s hard to nail down a unique replication cost of a ecc because there is so much flexibility in the choice of trading strategies that wealth may be only a local martingale or supermartingale. x Definition 2 The price of a ecc x =(c, W ) is S =min ⇡¯t1:¯⇡ is a tame, tight t { trading strategy that generates x , provided the minimum exists. } Proposition 1 If the market is complete then we have the RNPE T u T x rs ds rs ds St = E⇤ e− t cu du + e− t W t t [0,T]. (1) { t |F } 8 2 Z R R Proof In a complete market, there exists a tight, tame, martingale-generating strategy ⇡ that finances x and must satisfy u T rs ds rs ds Xt = E⇤ e− t cu du + e− t W t . (2) { |F } R R 4 For any other tame trading strategy that finances x, risklessly discounted cum- consumption portfolio value is a -supermartingale, so P⇤ u T rs ds rs ds Xt E⇤ e− t cu du + e− t W t . (3) ≥ { |F } ⇤ R R Definition 3 A replicating portfolio for an ecc x is a tame, martingale-generating trading strategy that tightly finances x. I In what sense is Sx an equilibrium price for an ecc x? I If a replicating portfolio of x exists, we would like to argue that the equilibrium price of x must be the price of its replicating portfolio. I But when doubling strategies are possible, the replication cost may not be unique. I And, if we are restricted to tame trading strategies, we may not be able turn an apparent mispricing into an arbitrage, because the natural long-short position to implement might get closed out if its value gets marked below an institutionally imposed lower bound before the end of the trading horizon. I Nevertheless, the minimum replication cost is a pretty economically natural notion of the price of x. 5 SDF Process and Pricing Equation I If the market is complete, we can define the unique SDF process Mt by d t t 1 t 2 ⇤ ru du ✓u0 dBu ✓u du Mt = βtE P t = βtZt = e− 0 − 0 −2 0 | | , (4) { d |F } P R R R I see that stochastically discounted wealth process MX satisfies WEE-M: t t MtXt = x0 + Mu[⇡uσu Xu✓u0 ] dBu Mucu du, (5) Z0 − − Z0 and rewrite the RNPE with true expectation and stochastic discounting (SDPE): T x βu βT St = E⇤ cu du + W t (6) {Zt βt βt |F } T d ⇤ βu βT d ⇤ = E P [ cu du + W ] t /E P t (7) { d Zt βt βt |F } { d |F } PT P Mu MT = E cu du + W t t [0,T]. (8) {Zt Mt Mt |F } 8 2 6 Fundamental PDE for Path-Independent Claims From the WEE in undiscounted terms, but with B⇤ instead of B, cum-consumption or cum-dividend portfolio value always appreciates at the riskless rate r under : P⇤ dXt + ct dt = rtXt dt + ⇡t(µt rt) dt + ⇡tσt dBt = rtXt dt + ⇡tσt dB⇤ . (9) − t Intuitively, once B⇤ absorbs the Sharpe ratios of all the securities, it also absorbs the Sharpe ratios of all the portfolio processes. 1 Definition 4 Aeccx =(c, W ) b is path-independent if ct = '1(S1,t,...,Sn,t,t) 2 C n and W = '2(S1,T ,...Sn,T ) for continuous functions '1 : + [0,T] and n R ⇥ ! R '2 : . R+ ! R I In the Markovian model, if x is path-independent then u T x rs ds rs ds S = E⇤ e− t cu du + e− t W t = F (S1,t,...,Sn,t,Yt,t) t { |F } R R for some real-valued function F . 7 I If F is smooth enough for an application of Itˆo’s lemma, then it must satisfy the following PDE, which says the drift of F from the WEE must equal the drift of F from Itˆo’s lemma: 1 rF '1 = FS0 IS(r1 δ)+FY0 (µY σY ✓)+Ft + 2tr(FSSISσσ0IS) 1 − − − +2tr(FYYσY σY0 )+tr(FSY0 ISσσY0 ) I where IS is a diagonal matrix with diagonal elements S1,S2,...,Sn and the derivatives FS and FSS are with respect to the vector (S1,S2,...,Sn). I This is subject to the boundary condition F (S1,T ,...Sn,T ,YT ,T)='2(S1,T ,...Sn,T ). I Furthermore, by matching the di↵usion coefficient from the WEE with the di↵usion x coefficient from Itˆo’s lemma, we have that the di↵usion coefficient of St must satisfy x ⇡ σ = FS0 ISσ + FY0 σY , (10) which gives a way to compute the replicating trading strategy ⇡x. 8 Special Case: Complete Market with Constant Coefficients Suppose r, µ, σ,andδ are constant. Definition 5 A function f : d [0,T] satisfies a polynomial growth condi- R ⇥ ! R tion (pgc) if there exist positive constants k1 and k2 such that k2 d f(x, t) k1(1 + x ) (x, t) [0,T] . (11) | | | | 8 2 R ⇥ Theorem 1 (Feynman-Kac) Suppose the functions '1, '2,and T r(u t) F (S1,t,...,Sn,t,t) E⇤ t e− − '1(S1,u,...,Sn,u,u) du r(T t) ⌘ { +e '2(S1 ,...S ) t − − ,T n,TR |F } each satisfy a pgc. Then F satisfies the PDE 1 rF '1 = F 0 Is(r1 δ)+Ft + tr(FSSISσσ0IS) (12) − S − 2 subject to F (S1,T ,...Sn,T ,T)='2(S1,T ,...Sn,T ). There is no other solution to this PDE that satisfies a pgc. x In addition, the trading strategy ⇡ that generates x =('1, '2) is given by ⇡x = S @F/@S ,i.e.,N = @F/@S ,fork =1, 2,...,n,and⇡x = F ⇡x1. k k k k k 0 − 9 Black-Scholes-Merton Call Option Model I Assume n = d =1, r and σ are constant, and for ease of exposition, begin by assuming no dividends. The price of the risky asset or “stock” follows dSt = µt dt + σ dBt . (13) St + Consider a call on the stock with time T payo↵ '2(S )=(S K) for some I T T − positive constant K. I The Black-Scholes argument (assume µ is constant): The time t call price Ct = c(St,t) because S is Markov. • Consider a portfolio short one call and long c shares of the stock. The • S portfolio value is X = c S c + ⇡0 where ⇡0 adjusts to make the portfolio S − self-financing. On one hand, from the wealth evolution equation dX = NdSand Itˆo’s lemma, • dX = cS dS dc + r⇡0 dt − 1 2 2 = cS dS (cS dS + ct dt + 2cSSσ S dt)+r⇡0 dt − 1 2 2 = ct dt c σ S dt + r⇡0 dt. − − 2 SS 10 On the other hand, by no arbitrage, since the portfolio is locally riskless, i.e., • has zero di↵usion, it must appreciate at the riskless rate: dX = rX dt = r(c S c + ⇡0) dt . (14) S − Therefore, it must be that 1 2 2 rScS + σ S cSS + ct = rc . (15) 2 They recognized this as the heat equation from physics, with solution • r(T t) + c(St,t)=E⇤ e− − (S K) St , (16) { T − | } which is the same as our result from equation (1).
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