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
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Readings and References Back, chapters 15 and 16.. Du e, 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. Du e, 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 k,t) dt + k,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 + ⇡t t dBt ct dt. •
No arbitrage if ⇡t t =0then ⇡t(µt rt1) = 0 ✓t s.t. t✓t = µt rt1 I ) )9 dXt = rtXt dt + ⇡t t(✓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 t = e 0 and sdf process Mt = tZt.ThentheWEEcanalsobewritten: R WEE*: d tXt + tct dt = t⇡t t dB • t⇤ WEE-M: dMtXt + Mtct dt = Mt[⇡t t ✓tXt] dBt • T u T T Mu MT I So Xt = Et⇤ t cu du + XT = Et t cu du + XT if ⇡ is mtgale-gen. { t t } { Mt Mt } R R I If is nonsingular, every c.plan (c, XT ) can be generated by a mtgale-gen. tr.strat.
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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.
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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 = tE P t = tZt = 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[⇡u u Xu✓u0 ] dBu Mucu du, (5) Z0 Z0 and rewrite the RNPE with true expectation and stochastic discounting (SDPE):
T x u T St = E⇤ cu du + W t (6) {Zt t t |F } T d ⇤ u T d ⇤ = E P [ cu du + W ] t /E P t (7) { d Zt t t |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 + ⇡t t dBt = rtXt dt + ⇡t t 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 .
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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 )+FY0 (µY Y ✓)+Ft + 2tr(FSSIS 0IS) 1 +2tr(FYY Y Y0 )+tr(FSY0 IS Y0 )
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 coe cient from the WEE with the di↵usion x coe cient from Itˆo’s lemma, we have that the di↵usion coe cient of St must satisfy
x ⇡ = FS0 IS + FY0 Y , (10)
which gives a way to compute the replicating trading strategy ⇡x.
8 Special Case: Complete Market with Constant Coe cients
Suppose r, µ, ,and 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 )+Ft + tr(FSSIS 0IS) (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
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Black-Scholes-Merton Call Option Model
I Assume n = d =1, r and are constant, and for ease of exposition, begin by assuming no dividends. The price of the risky asset or “stock” follows
dSt = µt dt + 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 + 2cSS S dt)+r⇡0 dt 1 2 2 = ct dt c 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 + 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).
I Alternative derivation with change of numeraire/change of measure: Now we can easily incorporate a constant dividend rate .Write • r(T t) Ct = E⇤ e (ST K)1 ST >K t { 2{ }|F } (T t) (B⇤ B⇤) (T t)/2 r(T t) = Ste E⇤ e T t 1 S >K t e K ⇤ ST >K t . { { T }|F } P { |F }
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Next, simplify the first term by introducing a new measure (s) defined by • (s) B 2T/2 P d /d e T⇤ (under which prices are martingales with the stock as P P⇤ ⌘ numeraire). Then
(T t) (s) r(T t) Ct = Ste S >K t e K ⇤ S >K t . (17) P { T |F } P { T |F }
Under the new measure, B(s) = B t is a standard Brownian motion, so • t t⇤
(T t) r(T t) Ct = c(St,t)=Ste N(d1(St,t)) e KN(d2(St,t)) (18)
where N is the cumulative normal distribution function,
(T t) r(T t) 2 ln(Ste /(Ke )) + (T t)/2 d1(St,t)= , (19) pT t
and d2(S(t),t)=d1(S(t),t) pT t. The option “delta,” or number of shares in the replicating portfolio, is • (T t) cS = e N(d1(St,t)).
12 Forward Contracts
I Consider a forward contract to buy a risky asset at time T for price F .Inthis case x =(0,S F ) and the time t value of the contract is T
T F rs ds V = E⇤ e t (S F ) t (20) t { T |F } R Ft The forward price Ft at time t is such that Vt =0,i.e.,
T rs ds E⇤ e t ST t Ft = { T |F } . (21) R rs ds E e t t ⇤{ |F } R If the risky asset dividend rates are deterministic, then
T s ds t T T Ste +y (T t) s ds Ft = = Ste t t (22) PRT t R
T T where Pt is the time t price of a zero-coupon bond maturing at time T and yt is the continuously compounded (T t)-year “zero rate” or “zero yield” at time t.
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Futures Contracts (Du e and Stanton, 1992)
I A futures contract on an underlying asset with associated futures price Gt, delivery date T,andcontinuous resettlement is a contract which produces a
cumulative cash flow of Gu Gt between any two dates t and u with 0 t u T . In addition, at time T , the contract obliges the holder to buy one share of the asset at price GT .
I Given the futures price process G,thetimet value of a futures contract is u T G T rs ds rs ds V = E e t dGu + e t (S G ) t . t ⇤{ t T T |F } R R R I Buying and selling futures contracts is costless, so the equilibrium futures price process G must be such that V G 0 and G = S . ⌘ T T t 2 rs ds Theorem 2 Suppose E S < and e 0 is bounded above and below ⇤{ T } 1 away from zero. Then there exists a uniqueR Itˆo process G with E⇤ G, G T < G h i 1 satisfying V 0 and G = S . It is given by Gt = E S t . ⌘ T T ⇤{ T |F } Sketch of Proof The conditions V f 0 and G = S imply G is a -martingale ⌘ T T P⇤ with last element S ,soGt = E S t . T ⇤{ T |F }
14 I To compare futures and forward prices, note that
T rs ds cov⇤ ST ,e t t Ft = Gt + { T |F } . (23) rRs ds E e t t ⇤{ |F } R T rs ds I In particular, they are equal if and only if ST and e t are uncorrelated. R I This will be the case if interest rates are deterministic. I But if interest rates are stochastic, they may di↵er. For example, bond futures prices are typically lower than their forward prices.
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Forward Measure vs. Futures Measure
I When the interest rate r is stochastic, it can be convenient for contingent claims pricing to work with the so-called forward measure, (T ),underwhich P prices normalized by the price of the zero-coupon bond maturing at time T are martingales. (T ) (T ) d T T The forward measure is defined by P = ,whereP = E is the I d P T 0 ⇤ T P P⇤ 0 { } time 0 price of the zero maturing at time T . I Note that the forward price F0 of a risky asset with price S satisfies
T (T ) 0=E⇤ (S F0) = P E S F0 , (24) { T T } 0 { T }
(T ) (T ) which implies E S = F0,whichiswhy is called the forward measure. { T } P By contrast, E S = G0,so is sometimes called the futures measure. I ⇤{ T } P⇤ I Both are “risk-neutral” measures in the sense that the price of time T payo↵ W can be written as
T (T ) W0 = E⇤ W = P E W (25) { T } 0 { } with discounting at a “riskless” rate. They are the same when r is nonstochastic.
16 State Prices Implicit in Options Prices
I The price of a call on a risky asset with strike K expiring at time T can be written
T (T ) + T 1 S,T C0 = P0 E (ST K) = P0 (s K)f (s) ds , (26) { } ZK
assuming S has a well-defined probability density function f S,T (s) under (T ). T P I If C0 is a twice-di↵erentiable function of the strike price K,then
2 @C0 @ C0 = P T 1( 1)f S,T (s) ds and = P T f S,T (K) , (27) 0 2 0 @K ZK (@K)
so we can recover the pdf of the future asset price with a continuum of call prices.
I Then, if a claim’s payo↵ is a path-independent function of the future risky asset price, W = '(ST ),itspricecanbewrittenas
2 @ C0 W = P T 1 '(s)f S,T (s) ds = P T 1 '(s) ds . (28) 0 0 0 2 K=s Z0 Z0 (@K) |
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I Finally, note that the call price above can be written
T (S) (T ) C0 = P [F S >K K S >K ] , (29) 0 P { T } P { T }
(S) d ST where P = , which is a generalization of the Black-Scholes formula. d (T ) F P
18 Problem Set 3
1. Derive the (Margrabe) valuation formula for a European option to exchange asset 1 for asset 2 under the assumption that each asset’s dividend rate and volatility is nonstochastic. Describe the dynamics of the replicating trading strategy. 2. Derive the (Merton) European call option valuation formula in the case that both the underlying stock and the zero-coupon bond maturing on the option expiration date have nonstochastic volatility and the stock has a nonstochastic dividend rate. Determine the replicating trading strategy. 3. Suppose the market coe cients are constant. Derive the (Black-Scholes) European call and put option valuation formulas (with dividends) and the replicating trading strategies.
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