Arbitrage, Martingales, and Pricing Kernels

Arbitrage, Martingales, and Pricing Kernels Arbitrage, Martingales, and Pricing Kernels 1/ 36

Introduction A contingent claim s price process can be transformed into a martingale process by 1 Adjusting its Brownian motion by the market price of risk. 2 De ating by a riskless asset price. The claim s value equals the expectation of the transformed process s future payo. We derive the continuous-time state price de ator that transforms actual probabilities into risk-neutral probabilities. Valuing a contingent claim might be simpli ed by de ating the contingent claim s price by that of another risky asset. We consider applications: options on assets that pay a continuous dividend; the term structure of interest rates. Arbitrage, Martingales, and Pricing Kernels 2/ 36

Arbitrage and Martingales Let S be the value of a risky asset that follows a general scalar di usion process ds = Sdt + Sdz (1) where both = (S; t) and = (S; t) may be functions of S and t and dz is a Brownian motion. Itô s lemma gives the process for a contingent claim s price, c(s; t): dc = c cdt + c cdz (2) where c c = c t + Sc S + 1 2 2 S 2 c SS and c c = Sc S, and the subscripts on c denote partial derivatives. Consider a hedge portfolio of 1 units of the contingent claim and c S units of the risky asset. Arbitrage, Martingales, and Pricing Kernels 3/ 36

Arbitrage and Martingales cont d The value of this hedge portfolio, H, satis es H = c + c S S (3) and the change in its value over the next instant is dh = dc + c S ds (4) = c cdt c cdz + c S Sdt + c S Sdz = [c S S c c] dt In the absence of arbitrage, the riskless portfolio change must be H(t)r(t)dt: dh = [c S S c c] dt = rhdt = r[ c + c S S]dt (5) Arbitrage, Martingales, and Pricing Kernels 4/ 36

Arbitrage and Martingales cont d This no-arbitrage condition for dh implies: c S S c c = r[ c + c S S] (6) Substituting c c = c t + Sc S + 1 2 2 S 2 c SS into (6) leads to the Black-Scholes equation: 1 2 2 S 2 c SS + rsc S rc + c t = 0 (7) However, a di erent interpretation of (6) results from substituting c S = c c S (from c c = Sc S ): r = c r c (t) (8) No-arbitrage condition (8) requires a unique market price of risk, say (t), so that c = r + c (t) : Arbitrage, Martingales, and Pricing Kernels 5/ 36

A Change in Probability Substituting for c in (2) gives dc = c cdt + c cdz = [rc + c c] dt + c cdz (9) Next, consider a new process bz t = z t + R t 0 (s) ds, so that dbz t = dz t + (t) dt. Then substituting dz t = dbz t (t) dt in (9): dc = [rc + c c] dt + c c [dbz dt] = rcdt + c cdbz (10) If bz t were a Brownian motion, future values of c generated by dbz occur under the Q or risk-neutral probability measure. The actual or physical distribution, P, is generated by the dz Brownian motion. Arbitrage, Martingales, and Pricing Kernels 6/ 36

Girsanov s Theorem Let dp T be the instantaneous change in the cumulative distribution at date T generated by dz t (the physical pdf). dq T is the analogous risk-neutral pdf generated by dbz t. Girsanov s theorem says that at date t < T, the two probability densities satisfy Z T Z 1 T dq T = exp (u) dz (u) 2 du dp T t 2 t = ( T = t ) dp T (11) where t is a positive random process depending on (t) and z t : Z Z 1 = exp (u) dz (u) 2 du (12) 0 2 0 Arbitrage, Martingales, and Pricing Kernels 7/ 36

Girsanov s Theorem cont d Thus, multiplying the physical pdf at T by T = t leads to the risk-neutral pdf at T. Since T = t > 0, equation (11) implies that whenever dp T has positive probability, so does dq T, making them equivalent measures. Rearranging (11) gives the Radon-Nikodym derivative of Q wrt P: dq T dp T = T = t (13) Later we will relate this derivative to the continuous-time pricing kernel. Arbitrage, Martingales, and Pricing Kernels 8/ 36

Money Market De ator Let B (t) be the value of an instantaneous-maturity riskless money market fund investment: db=b = r(t)dt (14) Note that B (T ) = B (t) e R T t r (u)du for any date T t. Now de ne C(t) c(t)=b(t) as the de ated price process for the contingent claim and use Itô s lemma: dc = 1 B dc c db (15) B2 = rc B dt + c c B dbz r c B dt = c Cdbz since dcdb = 0 and we substitute for dc from (10). Arbitrage, Martingales, and Pricing Kernels 9/ 36

Money Market De ator cont d An implication of (15) is C (t) = b E t [C (T )] 8T t (16) where b E t [] denotes the expectation operator under the probability measure generated by dbz. Thus, C(t) is a martingale (random walk) process. Note that (16) holds for any de ated non-dividend-paying contingent claim, including C = S B. Later, we will consider assets that pay dividends. Arbitrage, Martingales, and Pricing Kernels 10/ 36

Feynman-Kac Solution Rewrite (16) in terms of the unde ated contingent claims price: c(t) = B(t) E b 1 t c (T ) (17) B (T ) = E b R T i t he t r (u)du c (T ) Equation (17) is the Feynman-Kac solution to the Black-Scholes PDE and does not require knowledge of (t). This is the continuous-time formulation of risk-neutral pricing: risk-neutral (or Q measure) expected payo s are discounted by the risk-free rate. Arbitrage, Martingales, and Pricing Kernels 11/ 36

Arbitrage and Pricing Kernels Recall from the single- or multi-period consumption-portfolio choice problem with time-separable utility: c (t) = E t [m t;t c (T )] (18) MT = E t c (T ) M t where date T t, m t;t M T =M t and M t = U c (C t ; t). Rewriting (18): c (t) M t = E t [c (T ) M T ] (19) which says that the de ated price process, c (t) M t, is a martingale under P (not Q). Arbitrage, Martingales, and Pricing Kernels 12/ 36

Arbitrage and Pricing Kernels cont d Assume that the state price de ator, M t, follows a strictly positive di usion process of the general form De ne c m = cm and apply Itô s lemma: dm t = m dt + m dz (20) dc m = cdm + Mdc + (dc) (dm) (21) = [c m + M c c + c c m ] dt + [c m + M c c] dz If c m = cm satis es (19), that is, c m is a martingale, then its drift in (21) must be zero, implying c = m M c m M (22) Arbitrage, Martingales, and Pricing Kernels 13/ 36

Arbitrage and Pricing Kernels cont d Consider the case in which c is the instantaneously riskless investment B (t); that is, dc (t) = db (t) = r (t) Bdt so that c = 0 and c = r (t). From (22), this requires r (t) = m M (23) Thus, the expected rate of change of the pricing kernel must equal minus the instantaneous risk-free interest rate. Next, consider the general case where the asset c is risky, so that c 6= 0. Using (22) and (23) together, we obtain or c = r (t) c m M (24) Arbitrage, Martingales, and Pricing Kernels 14/ 36

Arbitrage and Pricing Kernels cont d c r = m c M (25) Comparing (25) to (8), we see that m M = (t) (26) Thus, the no-arbitrage condition implies that the form of the pricing kernel must be dm=m = r (t) dt (t) dz (27) Arbitrage, Martingales, and Pricing Kernels 15/ 36

Arbitrage and Pricing Kernels cont d De ne m t ln M t so that dm = [r + 1 2 2 ]dt dz. We can rewrite (18) as c (t) = E t [c (T ) M T =M t ] = E t c (T ) e m T m t = E t hc (T ) e R T t [r (u)+ 1 2 2 (u)]du R i T t (u)dz (28) Since the price under the money-market de ator (Q measure) and the SDF (P measure) must be the same, equating (17) and (28) implies R T i be t he t r (u)du c (T ) = E t [c (T ) M T =M t ] (29) = E t he R T t r (u)du c (T ) e R T t 1 2 2 (u)du R i T t (u)dz Arbitrage, Martingales, and Pricing Kernels 16/ 36

Linking Valuation Methods Substituting the de nition of from (12) leads to R T i be t he t r (u)du R T i c (T ) = E t he t r (u)du c (T ) ( T = t ) Z be t [C (T )] = E t [C (T ) ( T = t )] Z (30) C (T ) dq T = C (T ) ( T = t ) dp T where C (t) = c (t) =B (t). Thus, relating (29) to (30): M T =M t = e R T t r (u)du ( T = t ) (31) Hence, M T =M t provides both discounting at the risk-free rate and transforming the probability distribution to the risk-neutral one via T = t. Arbitrage, Martingales, and Pricing Kernels 17/ 36

Multivariate Case Consider a multivariate extension where asset returns depend on an n 1 vector of independent Brownian motion processes, dz = (dz 1 :::dz n ) 0 where dz i dz j = 0 for i 6= j. A contingent claim whose payo depended on these asset returns has the price process dc=c = c dt + c dz (32) where c is a 1 n vector c = ( c1 ::: cn ). Let the corresponding n 1 vector of market prices of risks associated with each of the Brownian motions be = ( 1 ::: n ) 0. Arbitrage, Martingales, and Pricing Kernels 18/ 36

Multivariate Case cont d Then the no-arbitrage condition (the multivariate equivalent of (8)) is c r = c (33) Equations (16) and (17) would still hold, and now the pricing kernel s process would be given by dm=m = r (t) dt (t) 0 dz (34) Arbitrage, Martingales, and Pricing Kernels 19/ 36

Alternative Price De ators Consider an option written on the di erence between two securities (stocks ) prices. The date t price of stock 1, S 1 (t), follows the process ds 1 =S 1 = 1 dt + 1 dz 1 (35) and the date t price of stock 2, S 2 (t), follows the process ds 2 =S 2 = 2 dt + 2 dz 2 (36) where 1 and 2 are assumed to be constants and dz 1 dz 2 = dt. Let C (t) be the date t price of a European option written on the di erence between these two stocks prices. Arbitrage, Martingales, and Pricing Kernels 20/ 36

Alternative Price De ators cont d At this option s maturity date, T, its value equals C (T ) = max [0; S 1 (T ) S 2 (T )] (37) Now de ne c (t) = C (t) =S 2 (t) ; s (t) S 1 (t) =S 2 (t), and B (t) = S 2 (t) =S 2 (t) = 1 as the de ated price processes, where the prices of the option, stock 1, and stock 2 are all normalized by the price of stock 2. Under this normalized price system, the payo (37) is c (T ) = max [0; s (T ) 1] (38) Applying Itô s lemma, the process for s (t) is ds=s = s dt + s dz 3 (39) Arbitrage, Martingales, and Pricing Kernels 21/ 36

Alternative Price De ators cont d Here s 1 2 + 2 2 1 2, s dz 3 1 dz 1 2 dz 2, and 2 s = 2 1 + 2 2 2 1 2. Further, when prices are measured in terms of stock 2, the de ated price of stock 2 becomes the riskless asset with db=b = 0dt (the de ated price never changes). Using Itô s lemma on c, dc = c s s s + c t + 12 c ss 2s s 2 dt + c s s s dz 3 (40) The familiar Black-Scholes hedge portfolio can be created from the option and stock 1. The portfolio s value is H = c + c s s (41) Arbitrage, Martingales, and Pricing Kernels 22/ 36

Alternative Price De ators cont d The instantaneous change in value of the portfolio is dh = dc + c s ds (42) = c s s s + c t + 12 c ss 2s s 2 dt c s s s dz 3 = + c s s s dt + c s s s dz 3 c t + 12 c ss 2s s 2 dt which is riskless and must earn the riskless return db=b = 0: dh = c t + 12 c ss 2s s 2 dt = 0 (43) which implies c t + 1 2 c ss 2 s s 2 = 0 (44) Arbitrage, Martingales, and Pricing Kernels 23/ 36

Alternative Price De ators cont d This is the Black-Scholes PDE with the risk-free rate, r, set to zero. With boundary condition (38), the solution is c(s; t) = s N(d 1 ) N(d 2 ) (45) where d 1 = ln (s(t)) + 1 2 2 s (T t) s p T t (46) d 2 = d 1 s p T t Multiply by S 2 (t) to convert back to the unde ated price system: C( t) = S 1 N(d 1 ) S 2 N(d 2 ) (47) C(t) does not depend on r(t), so that this formula holds even for stochastic interest rates. Arbitrage, Martingales, and Pricing Kernels 24/ 36

Continuous Dividends Let S (t) be the date t price per share of an asset that continuously pays a dividend of S (t) per unit time. Thus, ds = ( ) Sdt + Sdz (48) where and are assumed to be constants. Note that the asset s total rate of return is ds=s+ dt = dt+ dz, so that is its instantaneous expected rate of return. Consider a European call option written on this asset with exercise price of X and maturity date of T > t, where we de ne T t. Let r be the constant risk-free interest rate. Arbitrage, Martingales, and Pricing Kernels 25/ 36

Continuous Dividends cont d Based on (17), the date t price of this option is c (t) = b E t e r c (T ) (49) = e r b Et [max [S (T ) X ; 0]] As in (10), convert from the physical measure generated by dz to the risk-neutral measure generated by dbz, which removes the risk premium from the asset s expected rate of return so that: ds = (r ) Sdt + Sdbz (50) Since r and are constants, S is a geometric Brownian motion process and is lognormally distributed under Q. Arbitrage, Martingales, and Pricing Kernels 26/ 36

Continuous Dividends cont d Thus, the risk-neutral distribution of ln[s (T )] is normal: 1 ln [S (T )] N ln [S (t)] + (r 2 2 ); 2 (51) Equation (49) can now be computed as c (t) = e r Et b [max [S (T ) X ; 0]] (52) Z 1 = e r (S (T ) X ) g(s (T )) ds (T ) X where g(s T ) is the lognormal probability density function. Consider the change in variable Y = ln [S (T ) =S (t)] r 1 2 2 p (53) Arbitrage, Martingales, and Pricing Kernels 27/ 36

Continuous Dividends cont d Y N (0; 1) and allows (52) to be evaluated as where c = Se N (d 1 ) Xe r N (d 2 ) (54) d 1 = ln (S=X ) + r + 1 2 2 p d 2 = d 1 p (55) If contingent claims have more complex payo s or the underlying asset has a more complex risk-neutral process, a numeric solution to c (t) = b E t [e r c (S (T ))] can be obtained, perhaps by Monte Carlo simulation. Arbitrage, Martingales, and Pricing Kernels 28/ 36

Continuous Dividends cont d Compared to an option written on an asset that pays no dividends, the non-dividend-paying asset s price, S (t), is replaced with the dividend-discounted price of the dividend-paying asset, S (t) e (to keep the total expected rate of return at r). Thus, the risk-neutral expectation of S (T ) is ^E t [S (T )] = S (t) e (r ) (56) where we de ne S (t) S (t) e. = S (t) e e r = S (t) e r Arbitrage, Martingales, and Pricing Kernels 29/ 36

Foreign Currency Options De ne S (t) as the domestic currency value of a unit of foreign currency (spot exchange rate). Purchase of a foreign currency allows the owner to invest at the risk-free foreign currency interest rate, r f. Thus the dividend yield will equal this foreign currency rate, = r f and ^E t [S (T )] = S (t) e (r r f ). This expression is the no-arbitrage value of the date t forward exchange rate having a time until maturity of, that is, F t; = Se (r r f ). Therefore, a European option on foreign exchange is c (t) = e r [F t; N (d 1 ) XN (d 2 )] (57) where d 1 = 2 ln[ft; =X ]+ 2 p, and d 2 = d 1 p. Arbitrage, Martingales, and Pricing Kernels 30/ 36

Options on Futures Consider an option written on a futures price F t;t, the date t futures price for a contract maturing at date t. The undiscounted pro t (loss) earned by the long (short) party over the period from date t to date T t is simply F T ;t F t;t. Like forward contracts, there is no initial cost for the parties who enter into a futures contract. Hence, in a risk-neutral world, their expected pro ts must be zero: ^E t [F T ;t F t;t ] = 0 (58) so under the Q measure, the futures price is a martingale: ^E t [F T ;t ] = F t;t (59) Arbitrage, Martingales, and Pricing Kernels 31/ 36

Options on Futures cont d Since an asset s expected return under Q must be r, a futures price is like the price of an asset with a dividend yield of = r. The value of a futures call option that matures in periods where (t t) is where d 1 = ln[f t;t c (t) = e r [F t;t N (d 1 ) XN (d 2 )] (60) 2 =X ]+ 2 p, and d 2 = d 1 p. Note that this is similar in form to an option on a foreign currency written in terms of the forward exchange rate. Arbitrage, Martingales, and Pricing Kernels 32/ 36

Term Structure Revisited Let P (t; ) be the date t price of a default-free bond paying $1 at maturity T = t +. Interpreting c (T ) = P (T ; 0) = 1, equation (17) is P (t; ) = b E t he R T t r (u)du 1 i (61) We now rederive the Vasicek (1977) model using this equation where recall that the physical process for r (t) is dr(t) = [r r (t)] dt + r dz r (62) Assuming, like before, that the market price of bond risk q is a constant, p (r; ) = r (t) + q p () (63) where p () = P r r =P. Arbitrage, Martingales, and Pricing Kernels 33/ 36

Term Structure cont d Thus, recall that the physical process for a bond s price is dp (r; ) =P (r; ) = p (r; ) dt p () dz r (64) = [r (t) + q p ()] dt p () dz r De ning dbz r = dz r qdt, equation (64) becomes dp (t; ) =P (t; ) = [r (t) + q p ()] dt p () [dbz r + qdt] = r (t) dt p () dbz r (65) which is the risk-neutral process for the bond price since all bonds have the expected rate of return r under the Q measure. Arbitrage, Martingales, and Pricing Kernels 34/ 36

Term Structure cont d Therefore, the process for r(t) under the Q measure is found by also substituting dbz r = dz r qdt: dr(t) = [r r (t)] dt + r [dbz r + qdt] h = r + q i r r (t) dt + r dbz r (66) which has the unconditional mean r + q r =. Thus, when evaluating equation (61) Z T P (t; ) = E b t exp r (u) du this expectation is computed assuming r (t) follows the process in (66). Doing so leads to the same solution given in the previous chapter, equation (9.41) in the text. t Arbitrage, Martingales, and Pricing Kernels 35/ 36

Summary Martingale pricing is a generalization of risk-neutral pricing that is applicable in complete markets. With dynamically complete markets, the continuous-time state price de ator has an expected growth rate equal to minus the risk-free rate and a standard deviation equal to the market price of risk. Contingent claims valuation often can be simpli ed by an appropriate normalization of asset prices, de ating either by the price of a riskless or risky asset. Martingale pricing can be applied to options written on assets paying continuous, proportional dividends, as well as default-free bonds. Arbitrage, Martingales, and Pricing Kernels 36/ 36