Difficulties can arise with the valuation of American securities in incom- plete markets.For example,the exercise policy may play a role in determin- ing the marketed subspace,and therefore a role in pricing securities.If the state-price density depends on the exercise policy,it could even turn out that the notion of a rational exercise policy is not well defined. 3 Continuous-Time Modeling Many problems are more tractable,or have solutions appearing in a more natural form,when treated in a continuous-time setting.We first introduce the Brownian model of uncertainty and continuous security trading,and then derive partial differential equations for the arbitrage-free prices of derivative securities.The classic example is the Black-Scholes option-pricing formula. We then examine the connection between equivalent martingale measures and the "market price of risk"that arises from Girsanov's Theorem.Finally,we briefly connect the theory of security valuation with that of optimal portfolio and consumption choice,using the elegant martingale approach of Cox and Huang 1989]. 3.1 Trading Gains for Brownian Prices We fix a probability space (F,P).A process is a measurablel0 function on x 0,oo)into R.The value of a process X at time t is the random variable variously written as Xt,X(t),or X(.,t):-R.A standard Brownian motion is a process B defined by the properties: (a)Bo=0 almost surely; (b)Normality:for any times t and s >t,Bs-B:is normally distributed with mean zero and variance s-t; (c)Independent increments:for any times to,...,tn such that 0<to <t< ...tn <oo,the random variables B(to),B(t1)-B(to),...,B(tn)- B(tn-1)are independently distributed;and (d)Continuity:for each w in n,the sample path t B(w,t)is continuous. 10See Duffie [2001]for technical definitions not provided here. 26
Difficulties can arise with the valuation of American securities in incomplete markets. For example, the exercise policy may play a role in determining the marketed subspace, and therefore a role in pricing securities. If the state-price density depends on the exercise policy, it could even turn out that the notion of a rational exercise policy is not well defined. 3 Continuous-Time Modeling Many problems are more tractable, or have solutions appearing in a more natural form, when treated in a continuous-time setting. We first introduce the Brownian model of uncertainty and continuous security trading, and then derive partial differential equations for the arbitrage-free prices of derivative securities. The classic example is the Black-Scholes option-pricing formula. We then examine the connection between equivalent martingale measures and the “market price of risk” that arises from Girsanov’s Theorem. Finally, we briefly connect the theory of security valuation with that of optimal portfolio and consumption choice, using the elegant martingale approach of Cox and Huang [1989]. 3.1 Trading Gains for Brownian Prices We fix a probability space (Ω, F, P). A process is a measurable10 function on Ω ×[0, ∞) into R. The value of a process X at time t is the random variable variously written as Xt, X(t), or X(· , t):Ω → R. A standard Brownian motion is a process B defined by the properties: (a) B0 = 0 almost surely; (b) Normality: for any times t and s>t, Bs − Bt is normally distributed with mean zero and variance s − t; (c) Independent increments: for any times t0,...,tn such that 0 ≤ t0 < t1 < ··· < tn < ∞, the random variables B(t0), B(t1) − B(t0), ...,B(tn) − B(tn−1) are independently distributed; and (d) Continuity: for each ω in Ω, the sample path t 7→ B(ω, t) is continuous. 10See Duffie [2001] for technical definitions not provided here. 26
It is a nontrivial fact,whose proof has a colorful history,that (F,P)can be constructed so that there exist standard Brownian motions.In perhaps the first scientific work involving Brownian motion,Bachelier [1900]proposed Brownian motion as a model of stock prices.We will follow his lead for the time being and suppose that a given standard Brownian motion B is the price process of a security.Later we consider more general classes of price processes. We fix the standard filtration F=F:t>0 of B,defined for example in Protter [1990].Roughly speaking,1F is the set of events that can be distinguished as true or false by observation of B until time t. Our first task is to build a model of trading gains based on the possibility of continual adjustment of the position held.A trading strategy is an adapted process 6 specifying at each state w and time t the number 6(w)of units of the security to hold.If a strategy 0 is a constant,say 0,between two dates t and s>t,then the total gain between those two dates is 0(Bs-B), the quantity held multiplied by the price change.So long as the trading strategy 0 is piecewise constant,we would have no difficulty in defining the total gain between any two times.For example,suppose,for some stopping times To,...,TN with 0=To <T<...<TN=T,and for any n,we have (t)=0(Tn-1)for all t [Tn-1,Tn).Then we define the total gain from trade as N 0dB=∑9Tn-B(T）-BT-ll (16) n-1 More generally,in order to make for a good model of trading gains for trading strategies that are not necessarily piecewise constant,a trading strat- egyis required to satisfy the technical condition thatdalmost surely for each T.We let C2 denote the space of adapted processes satisfying this integrability restriction.For each in C2 there is an adapted process with continuous sample paths,denoted fdB,that is called the stochastic integral of 0 with respect to B.A full definition of fedB is outlined in a standard source such as Karatzas and Shreve [1988. The value of the stochastic integral fdB at time T is usually denoted Band represents the total gain generated up to timeTby trading the security with price process B according to the trading strategy 0.The stochastic integral fdB has the properties that one would expect from a 11The standard filtation is augmented,so that F contains all null sets of F. 27
It is a nontrivial fact, whose proof has a colorful history, that (Ω, F, P) can be constructed so that there exist standard Brownian motions. In perhaps the first scientific work involving Brownian motion, Bachelier [1900] proposed Brownian motion as a model of stock prices. We will follow his lead for the time being and suppose that a given standard Brownian motion B is the price process of a security. Later we consider more general classes of price processes. We fix the standard filtration F = {Ft : t ≥ 0} of B, defined for example in Protter [1990]. Roughly speaking,11 Ft is the set of events that can be distinguished as true or false by observation of B until time t. Our first task is to build a model of trading gains based on the possibility of continual adjustment of the position held. A trading strategy is an adapted process θ specifying at each state ω and time t the number θt(ω) of units of the security to hold. If a strategy θ is a constant, say θ, between two dates t and s>t, then the total gain between those two dates is θ(Bs − Bt), the quantity held multiplied by the price change. So long as the trading strategy θ is piecewise constant, we would have no difficulty in defining the total gain between any two times. For example, suppose, for some stopping times T0,...,TN with 0 = T0 < T1 < ··· < TN = T, and for any n, we have θ(t) = θ(Tn−1) for all t ∈ [Tn−1, Tn). Then we define the total gain from trade as Z T 0 θt dBt = X N n=1 θ(Tn−1)[B(Tn) − B(Tn−1)]. (16) More generally, in order to make for a good model of trading gains for trading strategies that are not necessarily piecewise constant, a trading strategy θ is required to satisfy the technical condition that R T 0 θ2 t dt < ∞ almost surely for each T. We let L2 denote the space of adapted processes satisfying this integrability restriction. For each θ in L2 there is an adapted process with continuous sample paths, denoted R θ dB, that is called the stochastic integral of θ with respect to B. A full definition of R θ dB is outlined in a standard source such as Karatzas and Shreve [1988]. The value of the stochastic integral R θ dB at time T is usually denoted R T 0 θt dBt, and represents the total gain generated up to time T by trading the security with price process B according to the trading strategy θ. The stochastic integral R θ dB has the properties that one would expect from a 11The standard filtation is augmented, so that Ft contains all null sets of F. 27
good model of trading gains.In particular,(16)is satisfied for piece-wise constant 6,and in general the stochastic integral is linear,in that,for any 0 and o in C2 and any scalars a and b,the process a0+bo is also in C2,and, for any time T>0, 。 (17) 3.2 Martingale Trading Gains The properties of standard Brownian motion imply that B is a martingale. (This follows basically from the property that its increments are independent and of zero expectation.)One must impose technical conditions on 6,how- ever,in order to ensure that fdB is also a martingale.This is natural;it should be impossible to generate an expected profit by trading a security that never experiences an expected price change.The following basic proposition can be found,for example,in Protter [1990. Proposition. <oo for all T>0,then f0dB is a martingale As a model of security-price processes,standard Brownian motion is too restrictive for most purposes.Consider,more generally,an Ito process,mean- ing a process S of the form St=x+ us ds+ (17) where x is a real number,o is in C2,and u is in Cl,meaning that u is an adapted process such thatds<o almost surely for all t.It is common to write (17)in the informal "differential"form dSt=u dt +ot dBt. One often thinks intuitively of dSt as the "increment"of S at time t,made up of two parts,the "locally riskless"part ut dt,and the "locally uncertain" part ot dBt. 28
good model of trading gains. In particular, (16) is satisfied for piece-wise constant θ, and in general the stochastic integral is linear, in that, for any θ and ϕ in L2 and any scalars a and b, the process aθ + bϕ is also in L2, and, for any time T > 0, Z T 0 (aθt + bϕt) dBt = a Z T 0 θt dBt + b Z T 0 ϕt dBt. (17) 3.2 Martingale Trading Gains The properties of standard Brownian motion imply that B is a martingale. (This follows basically from the property that its increments are independent and of zero expectation.) One must impose technical conditions on θ, however, in order to ensure that R θ dB is also a martingale. This is natural; it should be impossible to generate an expected profit by trading a security that never experiences an expected price change. The following basic proposition can be found, for example, in Protter [1990]. Proposition. If E R T 0 θ2 t dt1/2 < ∞ for all T > 0, then R θ dB is a martingale. As a model of security-price processes, standard Brownian motion is too restrictive for most purposes. Consider, more generally, an Ito process, meaning a process S of the form St = x + Z t 0 µs ds + Z t 0 σs dBs, (17) where x is a real number, σ is in L2, and µ is in L1, meaning that µ is an adapted process such that R t 0 |µs| ds < ∞ almost surely for all t. It is common to write (17) in the informal “differential” form dSt = µt dt + σt dBt. One often thinks intuitively of dSt as the “increment” of S at time t, made up of two parts, the “locally riskless” part µt dt, and the “locally uncertain” part σt dBt. 28
In order to further interpret this differential representation of an Ito pro- cess,suppose that o and u have continuous sample paths and are bounded. It is then literally the case that for any time t, =此 almost surely (18) and d a,S,),=听 almost surely, (19) where the derivatives are taken from the right,and where,for any random variable X with finite variance,vart(X)=E(X2)-[E(X)is the F- conditional variance of X.In this sense of(18)and (19),we can interpret ut as the rate of change of the expectation of S,conditional on information available at time t,and likewise interpret o?as the rate of change of the conditional variance of S at time t.One sometimes reads the associated abuses of notation "E(dst)=ut dt"and "vart(dst)=o2 dt."Of course,dSt is not even a random variable,so this sort of characterization is not rigorously justified and is used purely for its intuitive content.We will refer to u and o as the drift and diffusion processes of S,respectively. For an Ito process S of the form (17),let C(S)be the set whose elements are processes 0 with :to}in Cl and (:t>0}in C2.For 0 in C(S),we define the stochastic integral feds as the Ito process fods given by 0ed5= Oiudt+o:dB, T≥0 Assuming no dividends,we also refer to feds as the gain process generated by the trading stragegy 0,given the price process S. We will have occasion to refer to adapted processes 0 and that are equal almost everywhere,by which we mean that E(dt)=0.In fact, we shall write "=o"whenever 6=o almost everywhere.This is a natural convention,for suppose that X and Y are Ito processes with Xo=Yo and with dX=u dt+ot dB:and dyi=at dt+b dBt.Since stochastic integrals are defined for our purposes as continuous-sample-path processes,it turns out that X =Yi for all t almost surely if and only if u=a almost everywhere and o =b almost everywhere.We call this the unigue decomposition property of Ito processes. Ito's Formula is the basis for explicit solutions to asset-pricing problems in a continuous-time setting. 29
In order to further interpret this differential representation of an Ito process, suppose that σ and µ have continuous sample paths and are bounded. It is then literally the case that for any time t, d dτ Et (Sτ ) τ=t = µt almost surely (18) and d dτ vart (Sτ ) τ=t = σ2 t almost surely, (19) where the derivatives are taken from the right, and where, for any random variable X with finite variance, vart(X) ≡ Et(X2) − [Et(X)]2 is the Ftconditional variance of X. In this sense of (18) and (19), we can interpret µt as the rate of change of the expectation of S, conditional on information available at time t, and likewise interpret σ2 t as the rate of change of the conditional variance of S at time t. One sometimes reads the associated abuses of notation “Et(dSt) = µt dt” and “vart(dSt) = σ2 t dt.” Of course, dSt is not even a random variable, so this sort of characterization is not rigorously justified and is used purely for its intuitive content. We will refer to µ and σ as the drift and diffusion processes of S, respectively. For an Ito process S of the form (17), let L(S) be the set whose elements are processes θ with {θt µt : t ≥ 0} in L1 and {θt σt : t ≥ 0} in L2. For θ in L(S), we define the stochastic integral R θ dS as the Ito process R θ dS given by Z T 0 θt dSt = Z T 0 θtµt dt + Z T 0 θtσt dBt, T ≥ 0. Assuming no dividends, we also refer to R θ dS as the gain process generated by the trading stragegy θ, given the price process S. We will have occasion to refer to adapted processes θ and ϕ that are equal almost everywhere, by which we mean that E( R ∞ 0 |θt − ϕt| dt) = 0. In fact, we shall write “θ = ϕ” whenever θ = ϕ almost everywhere. This is a natural convention, for suppose that X and Y are Ito processes with X0 = Y0 and with dXt = µt dt + σt dBt and dYt = at dt + bt dBt. Since stochastic integrals are defined for our purposes as continuous-sample-path processes, it turns out that Xt = Yt for all t almost surely if and only if µ = a almost everywhere and σ = b almost everywhere. We call this the unique decomposition property of Ito processes. Ito’s Formula is the basis for explicit solutions to asset-pricing problems in a continuous-time setting. 29
Ito's Formula.Suppose X is an Ito process with dX:=udt+o:dBt and f:R2->R is twice continuously differentiable.Then the process Y,defined by Yt=f(Xt,t),is an Ito process with dY-f(Xi)+fiXi.)+jfz(X.)oi dt+fr(X:,t)o:dBi. A generalization of Ito's Formula appears later in this section. 3.3 The Black-Scholes Option-Pricing Formula We turn to one of the most important ideas in finance theory,the model of Black and Scholes [1973 for pricing options.Together with the method of proof provided by Robert Merton,this model revolutionized the practice of derivative pricing and risk management,and has changed the entire path of asset-pricing theory. Consider a security,to be called a stock,with price process S=xeat+oB0,t≥0， where x >0,a,and o are constants.Such a process,called a geometric Brownian motion,is often called log-normal because,for any t,log(St)= log()+at +oB:is normally distributed.Moreover,since X:at+oB= ads+dB,defines an Ito process with constant drift a and diffusion o,Ito's Formula implies that S is an Ito process and that dSt=uSt dt+St dBt;So=x, where u=a+o2/2.From (18)and (19),at any time t,the rate of change of the conditional mean of St is uSt,and the rate of change of the conditional variance is o292,so that,per dollar invested in this security at time t,one may think of u as the "instantaneous"expected rate of return,and o as the "instantaneous"standard deviation of the rate of return.The coefficient o is also known as the volatility of S.A geometric Brownian motion is a natural two-parameter model of a security-price process because of these simple interpretations of u and o. Consider a second security,to be called a bond,with the price process B defined by 3:=f0et,t≥0， 30
Ito’s Formula. Suppose X is an Ito process with dXt = µt dt + σt dBt and f : R2 → R is twice continuously differentiable. Then the process Y , defined by Yt = f(Xt, t), is an Ito process with dYt = fx(Xt, t)µt + ft(Xt, t) + 1 2 fxx(Xt, t)σ2 t dt + fx(Xt, t)σt dBt. A generalization of Ito’s Formula appears later in this section. 3.3 The Black-Scholes Option-Pricing Formula We turn to one of the most important ideas in finance theory, the model of Black and Scholes [1973] for pricing options. Together with the method of proof provided by Robert Merton, this model revolutionized the practice of derivative pricing and risk management, and has changed the entire path of asset-pricing theory. Consider a security, to be called a stock, with price process St = x eαt+σB(t) , t ≥ 0, where x > 0, α, and σ are constants. Such a process, called a geometric Brownian motion, is often called log-normal because, for any t, log(St) = log(x) + αt + σBt is normally distributed. Moreover, since Xt ≡ αt + σBt = R t 0 α ds+R t 0 σ dBs defines an Ito process X with constant drift α and diffusion σ, Ito’s Formula implies that S is an Ito process and that dSt = µSt dt + σSt dBt; S0 = x, where µ = α+σ2/2. From (18) and (19), at any time t, the rate of change of the conditional mean of St is µSt, and the rate of change of the conditional variance is σ2 S2 t , so that, per dollar invested in this security at time t, one may think of µ as the “instantaneous” expected rate of return, and σ as the “instantaneous” standard deviation of the rate of return. The coefficient σ is also known as the volatility of S. A geometric Brownian motion is a natural two-parameter model of a security-price process because of these simple interpretations of µ and σ. Consider a second security, to be called a bond, with the price process β defined by βt = β0 ert, t ≥ 0, 30