Arbitrage, Martingales, and Pricing Kernels · Arbitrage, Martingales, and Pricing Kernels 14/ 36. 10.1: Martingales 10.2: Kernels 10.3: Alternative 10.4: ... and transforming the

10.1: Martingales 10.2: Kernels 10.3: Alternative 10.4: Applications 10.5: Summary

Arbitrage, Martingales, and Pricing Kernels

George Pennacchi

University of Illinois

George Pennacchi University of Illinois

Arbitrage, Martingales, and Pricing Kernels 1/ 36


Introduction

A contingent claim�s price process can be transformed into amartingale 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 transformedprocess�s future payo¤.

We derive the continuous-time state price de�ator thattransforms actual probabilities into risk-neutral probabilities.

Valuing a contingent claim might be simpli�ed by de�atingthe contingent claim�s price by that of another risky asset.

We consider applications: options on assets that pay acontinuous dividend; the term structure of interest rates.




Arbitrage and Martingales

Let S be the value of a risky asset that follows a generalscalar di¤usion process

dS = �Sdt + �Sdz (1)

where both � = � (S ; t) and � = � (S ; t) may be functions ofS and t and dz is a Brownian motion.Itô�s lemma gives the process for a contingent claim�s price,c(S ; t):

dc = �ccdt + �ccdz (2)

where �cc = ct + �ScS +12�

2S2cSS and �cc = �ScS , and thesubscripts on c denote partial derivatives.Consider a hedge portfolio of �1 units of the contingent claimand cS units of the risky asset.




Arbitrage and Martingales cont�d

The value of this hedge portfolio, H, satis�es

H = �c + cSS (3)

and the change in its value over the next instant is

dH = �dc + cSdS (4)

= ��ccdt � �ccdz + cS�Sdt + cS�Sdz= [cS�S � �cc] dt

In the absence of arbitrage, the riskless portfolio change mustbe H(t)r(t)dt:

dH = [cS�S � �cc] dt = rHdt = r [�c + cSS ]dt (5)




Arbitrage and Martingales cont�d

This no-arbitrage condition for dH implies:

cS�S � �cc = r [�c + cSS ] (6)

Substituting �cc = ct + �ScS +12�

2S2cSS into (6) leads tothe Black-Scholes equation:

12�2S2cSS + rScS � rc + ct = 0 (7)

However, a di¤erent interpretation of (6) results fromsubstituting cS = �c c

�S (from �cc = �ScS ):

�� r�

=�c � r�c

� � (t) (8)

No-arbitrage condition (8) requires a unique market price ofrisk, say � (t), so that �c = r + �c� (t) :




A Change in Probability

Substituting for �c in (2) gives

dc = �ccdt + �ccdz = [rc + ��cc] dt + �ccdz (9)

Next, consider a new process bzt = zt + R t0 � (s) ds, so thatdbzt = dzt + � (t) dt.Then substituting dzt = dbzt � � (t) dt in (9):

dc = [rc + ��cc] dt + �cc [dbz � �dt]= rcdt + �ccdbz (10)

If bzt were a Brownian motion, future values of c generated bydbz occur under the Q or �risk-neutral�probability measure.The actual or �physical�distribution, P, is generated by thedz Brownian motion.




Girsanov�s Theorem

Let dPT be the instantaneous change in the cumulativedistribution at date T generated by dzt (the physical pdf).dQT is the analogous risk-neutral pdf generated by dbzt .Girsanov�s theorem says that at date t < T , the twoprobability densities satisfy

dQT = exp��Z T

t� (u) dz � 1

2

Z T

t� (u)2 du

�dPT

= (�T =�t) dPT (11)

where �t is a positive random process depending on � (t) andzt :

�� = exp��Z �

0� (u) dz � 1

2

Z �

0� (u)2 du

�(12)




Girsanov�s Theorem cont�d

Thus, multiplying the physical pdf at T by �T =�t leads to therisk-neutral pdf at T .

Since �T =�t > 0, equation (11) implies that whenever dPThas positive probability, so does dQT , making them equivalentmeasures.

Rearranging (11) gives the Radon-Nikodym derivative of Qwrt P:

dQTdPT

= �T =�t (13)

Later we will relate this derivative to the continuous-timepricing kernel.




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) eR Tt r (u)du for any date T � t.

Now de�ne C (t) � c(t)=B(t) as the de�ated price process forthe contingent claim and use Itô�s lemma:

dC =1Bdc � c

B2dB (15)

=rcBdt +

�ccBdbz � r c

Bdt

= �cCdbzsince dcdB = 0 and we substitute for dc from (10).




Money Market De�ator cont�d

An implication of (15) is

C (t) = bEt [C (T )] 8T � t (16)

where bEt [�] denotes the expectation operator under theprobability measure generated by dbz .Thus, C (t) is a martingale (random walk) process.

Note that (16) holds for any de�ated non-dividend-payingcontingent claim, including C = S

B .

Later, we will consider assets that pay dividends.




Feynman-Kac Solution

Rewrite (16) in terms of the unde�ated contingent claimsprice:

c(t) = B(t)bEt �c (T ) 1B (T )

�(17)

= bEt he� R Tt r (u)duc (T )i

Equation (17) is the �Feynman-Kac� solution to theBlack-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 discountedby the risk-free rate.




Arbitrage and Pricing Kernels

Recall from the single- or multi-period consumption-portfoliochoice problem with time-separable utility:

c (t) = Et [mt ;T c (T )] (18)

= Et

�MT

Mtc (T )

�where date T � t, mt ;T � MT =Mt and Mt = Uc (Ct ; t).

Rewriting (18):

c (t)Mt = Et [c (T )MT ] (19)

which says that the de�ated price process, c (t)Mt , is amartingale under P (not Q).




Arbitrage and Pricing Kernels cont�d

Assume that the state price de�ator, Mt , follows a strictlypositive di¤usion process of the general form

dMt = �mdt + �mdz (20)

De�ne cm = cM and apply Itô�s lemma:

dcm = cdM +Mdc + (dc) (dM) (21)

= [c�m +M�cc + �cc�m ] dt + [c�m +M�cc] dz

If cm = cM satis�es (19), that is, cm is a martingale, then itsdrift in (21) must be zero, implying

�c = ��mM� �c�m

M(22)





Consider the case in which c is the instantaneously risklessinvestment 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) = ��mM

(23)

Thus, the expected rate of change of the pricing kernel mustequal minus the instantaneous risk-free interest rate.Next, consider the general case where the asset c is risky, sothat �c 6= 0. Using (22) and (23) together, we obtain

�c = r (t)��c�mM

(24)

orGeorge Pennacchi University of Illinois




�c � r�c

= ��mM

(25)

Comparing (25) to (8), we see that

��mM

= � (t) (26)

Thus, the no-arbitrage condition implies that the form of thepricing kernel must be

dM=M = �r (t) dt � � (t) dz (27)





De�ne mt � lnMt so that dm = �[r + 12�2]dt � �dz .

We can rewrite (18) as

c (t) = Et [c (T )MT =Mt ] = Et�c (T ) emT�mt

�(28)

= Ethc (T ) e�

R Tt [r (u)+

12 �2(u)]du�

R Tt �(u)dz

iSince the price under the money-market de�ator (Q measure)and the SDF (P measure) must be the same, equating (17)and (28) implies

bEt he� R Tt r (u)duc (T )i= Et [c (T )MT =Mt ] (29)

= Ethe�

R Tt r (u)duc (T ) e�

R Tt

12 �2(u)du�

R Tt �(u)dz

iGeorge Pennacchi University of Illinois



Linking Valuation Methods

Substituting the de�nition of �� from (12) leads to

bEt he� R Tt r (u)duc (T )i= Et

he�

R Tt r (u)duc (T ) (�T =�t)

ibEt [C (T )] = Et [C (T ) (�T =�t)] (30)ZC (T ) dQT =

ZC (T ) (�T =�t) dPT

where C (t) = c (t) =B (t). Thus, relating (29) to (30):

MT =Mt = e�R Tt r (u)du (�T =�t) (31)

Hence, MT =Mt provides both discounting at the risk-free rateand transforming the probability distribution to therisk-neutral one via �T =�t .




Multivariate Case

Consider a multivariate extension where asset returns dependon an n� 1 vector of independent Brownian motion processes,dZ = (dz1:::dzn)0 where dzidzj = 0 for i 6= j .A contingent claim whose payo¤ depended on these assetreturns has the price process

dc=c = �cdt +�cdZ (32)

where �c is a 1� n vector �c = (�c1:::�cn).Let the corresponding n � 1 vector of market prices of risksassociated with each of the Brownian motions be� = (�1:::�n)

0.




Multivariate Case cont�d

Then the no-arbitrage condition (the multivariate equivalentof (8)) is

�c � r = �c� (33)

Equations (16) and (17) would still hold, and now the pricingkernel�s process would be given by

dM=M = �r (t) dt ��(t)0 dZ (34)




Alternative Price De�ators

Consider an option written on the di¤erence between twosecurities�(stocks�) prices. The date t price of stock 1, S1 (t),follows the process

dS1=S1 = �1dt + �1dz1 (35)

and the date t price of stock 2, S2 (t), follows the process

dS2=S2 = �2dt + �2dz2 (36)

where �1 and �2 are assumed to be constants anddz1dz2 = �dt.

Let C (t) be the date t price of a European option written onthe di¤erence between these two stocks�prices.




Alternative Price De�ators cont�d

At this option�s maturity date, T , its value equals

C (T ) = max [0;S1 (T )� S2 (T )] (37)

Now de�ne c (t) = C (t) =S2 (t) ; s (t) � S1 (t) =S2 (t), andB (t) = S2 (t) =S2 (t) = 1 as the de�ated price processes,where the prices of the option, stock 1, and stock 2 are allnormalized 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 = �sdt + �sdz3 (39)





Here �s � �1 � �2 + �22 � ��1�2, �sdz3 � �1dz1 � �2dz2,and �2s = �

21 + �

22 � 2��1�2.

Further, when prices are measured in terms of stock 2, thede�ated price of stock 2 becomes the riskless asset withdB=B = 0dt (the de�ated price never changes).

Using Itô�s lemma on c ,

dc =�cs �s s + ct +

12css �2s s

2�dt + cs �s s dz3 (40)

The familiar Black-Scholes hedge portfolio can be createdfrom the option and stock 1. The portfolio�s value is

H = �c + cs s (41)





The instantaneous change in value of the portfolio is

dH = � dc + csds (42)

= ��cs�s s + ct +

12css �2s s

2�dt � cs �s s dz3

+ cs�s s dt + cs�s s dz3

= ��ct +

12css �2s s

2�dt

which is riskless and must earn the riskless return dB=B = 0:

dH = ��ct +

12css �2s s

2�dt = 0 (43)

which implies

ct +12css �2s s

2 = 0 (44)





This is the Black-Scholes PDE with the risk-free rate, r , set tozero. With boundary condition (38), the solution is

c(s; t) = s N(d1) � N(d2) (45)

where

d1 =ln (s(t)) + 1

2�2s (T � t)

�spT � t

(46)

d2 = d1 � � spT � t

Multiply by S2 (t) to convert back to the unde�ated pricesystem:

C ( t) = S1 N(d1) � S2 N(d2) (47)

C (t) does not depend on r(t), so that this formula holds evenfor stochastic interest rates.




Continuous Dividends

Let S (t) be the date t price per share of an asset thatcontinuously 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 rateof return.

Consider a European call option written on this asset withexercise price of X and maturity date of T > t, where wede�ne � � T � t.Let r be the constant risk-free interest rate.




Continuous Dividends cont�d

Based on (17), the date t price of this option is

c (t) = bEt �e�r�c (T )� (49)

= e�r� bEt [max [S (T )� X ; 0]]As in (10), convert from the physical measure generated by dzto the risk-neutral measure generated by dbz , which removesthe risk premium from the asset�s expected rate of return sothat:

dS = (r � �) Sdt + �Sdbz (50)

Since r � � and � are constants, S is a geometric Brownianmotion process and is lognormally distributed under Q.





Thus, the risk-neutral distribution of ln[S (T )] is normal:

ln [S (T )] � N�ln [S (t)] + (r � � � 1

2�2)� ; �2�

�(51)

Equation (49) can now be computed as

c (t) = e�r� bEt [max [S (T )� X ; 0]] (52)

= e�r �Z 1

X(S (T )� X ) g(S (T )) dS (T )

where g(ST ) is the lognormal probability density function.Consider the change in variable

Y =ln [S (T ) =S (t)]�

�r � � � 1

2�2��

�p�

(53)





Y � N (0; 1) and allows (52) to be evaluated as

c = Se��N (d1)� Xe�r �N (d2) (54)

where

d1 =ln (S=X ) +

�r � � + 1

2�2��

�p�

d2 = d1 � �p� (55)

If contingent claims have more complex payo¤s or theunderlying asset has a more complex risk-neutral process, anumeric solution to c (t) = bEt [e�r�c (S (T ))] can beobtained, perhaps by Monte Carlo simulation.





Compared to an option written on an asset that pays nodividends, the non-dividend-paying asset�s price, S (t), isreplaced with the dividend-discounted price of thedividend-paying asset, S (t) e�� (to keep the total expectedrate of return at r).

Thus, the risk-neutral expectation of S (T ) is

Et [S (T )] = S (t) e(r��)� (56)

= S (t) e��er� = S (t) er�

where we de�ne S (t) � S (t) e�� .




Foreign Currency Options

De�ne S (t) as the domestic currency value of a unit offoreign currency (spot exchange rate).Purchase of a foreign currency allows the owner to invest atthe risk-free foreign currency interest rate, rf .Thus the dividend yield will equal this foreign currency rate,� = rf and Et [S (T )] = S (t) e(r�rf )� .This expression is the no-arbitrage value of the date t forwardexchange rate having a time until maturity of � , that is,Ft ;� = Se(r�rf )� .Therefore, a European option on foreign exchange is

c (t) = e�r� [Ft ;�N (d1)� XN (d2)] (57)

where d1 =ln[Ft;�=X ]+�2

2 �

�p�

, and d2 = d1 � �p� .




Options on Futures

Consider an option written on a futures price Ft ;t� , the date tfutures 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 simplyFT ;t� � Ft ;t� .Like forward contracts, there is no initial cost for the partieswho enter into a futures contract. Hence, in a risk-neutralworld, their expected pro�ts must be zero:

Et [FT ;t� � Ft ;t� ] = 0 (58)

so under the Q measure, the futures price is a martingale:

Et [FT ;t� ] = Ft ;t� (59)




Options on Futures cont�d

Since an asset�s expected return under Q must be r , a futuresprice is like the price of an asset with a dividend yield of � = r .

The value of a futures call option that matures in � periodswhere � � (t� � t) is

c (t) = e�r� [Ft ;t�N (d1)� XN (d2)] (60)

where d1 =ln[Ft;t�=X ]+�2

2 �

�p�

, and d2 = d1 � �p� .

Note that this is similar in form to an option on a foreigncurrency written in terms of the forward exchange rate.




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; �) = bEt he� R Tt r (u)du1i

(61)

We now rederive the Vasicek (1977) model using this equationwhere recall that the physical process for r (t) is

dr(t) = � [r � r (t)] dt + �rdzr (62)

Assuming, like before, that the market price of bond risk q isa constant,

�p (r ; �) = r (t) + q�p (�) (63)

where �p (�) = �Pr�r=P.George Pennacchi University of Illinois



Term Structure cont�d

Thus, recall that the physical process for a bond�s price is

dP (r ; �) =P (r ; �) = �p (r ; �) dt � �p (�) dzr (64)

= [r (t) + q�p (�)] dt � �p (�) dzr

De�ning dbzr = dzr � qdt, equation (64) becomesdP (t; �) =P (t; �) = [r (t) + q�p (�)] dt � �p (�) [dbzr + qdt]

= r (t) dt � �p (�) dbzr (65)

which is the risk-neutral process for the bond price since allbonds have the expected rate of return r under the Qmeasure.




Term Structure cont�d

Therefore, the process for r(t) under the Q measure is foundby also substituting dbzr = dzr � qdt:

dr(t) = � [r � r (t)] dt + �r [dbzr + qdt]= �

h�r +

q�r�

�� r (t)

idt + �rdbzr (66)

which has the unconditional mean r + q�r=�.Thus, when evaluating equation (61)

P (t; �) = bEt �exp��Z T

tr (u) du

��this expectation is computed assuming r (t) follows theprocess in (66).Doing so leads to the same solution given in the previouschapter, equation (9.41) in the text.




Summary

Martingale pricing is a generalization of risk-neutral pricingthat is applicable in complete markets.

With dynamically complete markets, the continuous-timestate price de�ator has an expected growth rate equal tominus the risk-free rate and a standard deviation equal to themarket price of risk.

Contingent claims valuation often can be simpli�ed by anappropriate normalization of asset prices, de�ating either bythe price of a riskless or risky asset.

Martingale pricing can be applied to options written on assetspaying continuous, proportional dividends, as well asdefault-free bonds.



Documents

Arbitrage, Martingales, and Pricing Kernels · Arbitrage, Martingales, and Pricing Kernels 14/ 36. 10.1: Martingales 10.2: Kernels 10.3: Alternative 10.4: ... and transforming the