叶明的五劈生活PDE for Finance Notes,Spring2003–Section9
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Notes by Robert V.Kohn,Courant Institute of Mathematical Sciences.For u only in connection with the NYU cour PDE for Finance,G63.2706.
About thefinal exam:As previously announced,our exam is Monday May12,8-10pm, in the usual room Silver207.Note the time-shift(8-10not7-9),intended to give students taking both Scientific Computing and PDE for Finance some breathing room.If this late start is a hardship for you,tell me–it is possible by request to take the exam7-9pm instead of8-10pm.The exam will be clod-book,but you may bring two sheets of notes(8.5×11, both sides,any font).The preparation such notes is an excellent study tool.
The exam covers the material in Sections1-8of the lecture notes,and Homeworks1-6.Sole exception:there will be no exam questions using the Fourier Transform.Thisfinal Section 9will not be on the exam.
A good exam question can be answered with very little calculation,provided you understand the relevant ideas.Most of the exam questions will be similar to(parts of)homework problems or examples discusd in the notes.Thefinal lecture will be devoted to answering questions and reviewing what we’ve achieved this mester.
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The martingale method for dynamic portfolio optimization.Sections5-7were devoted to stochastic control.We discusd the value function and the principle of dynamic programming.In the discrete-time tting dynamic programming gives an iterative scheme forfinding the value function;in the continuous-time tting it leads to the Hamilton-Jacobi-Bellman PDE.Stochastic control is a powerful technique for optimal decision-making in the prence of uncertainty.In particular it places few restrictions on the sources of randomness, and it does not require special hypothes such as market completeness.
In the continuous-time tting,a key application(due to Merton,around1970)is dynamic portfolio optimization.We examined two versions of this problem:one optimizing the utility of consumption(Section5),the other optimizing the utility offinal-time wealth(Homework 5).
This ction introduces an alternative approach to dynamic portfolio optimization.It is much more recent–the main papers were by Cox&Huang;Karatzas,Lehoczky,&Shreve; and Pliska,all in the mid-80’s.A very clear,rather elementary account is given in R.Korn and E.Korn,Option Pricing and Portfolio Optimization:Modern Methods of Financial Mathematics(American Mathematical Society,2001).My discussion is a simplifi watered-down)version of the one in Korn&Korn.
This alternative approach is called the“martingale method,”for reasons that will become clear prently.It is cloly linked to the modern understanding of option pricing via the discounted risk-neutral expected value.(Therefore this Section,unlike the rest of the cour, requires some familiarity with continous-timefinance.)The method is much less general than stochastic control;in particular,it requires that the market be complete.When it applies,however,it provides an entirely fresh viewpoint,quite different from Merton’s.To
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capture the main idea with a minimum of complexity,I shall(as in Section5and HW 5)consider just the ca of a single risky ast.Moreover I shall focus on the ca(like Problem1of HW5)where there is no consumption.So the goal is this:consider an investor who starts with wealth x at time0.He can invest in a risk-free ast(“bond”)which offers constant interest r,or a risky ast(“stock”)who price satisfies
dS=µSdt+σSdw.(1) He expects to mix the,putting fractionθ(t)of his wealth in stock and the rest in the bond;the resulting wealth process satisfies
dX=[(1−θ)r+θµ]Xdt+θσXdw(2) with initial condition X(0)=x.His goal is to maximize his expected utility
offinal-time wealth:
max
θ(t)
E[h(X(T)]
where h is his utility function and T is a specified time.Of cour his investment decisions must be non-anticipating:θ(t)can depend only on knowledge of S up to time t.
Review of risk-neutral option pricing.Recall that the time-0value of an option with payofff(S T)is its discounted risk-neutral expected payoff:
option value=e−rT E RN[f(S T)].
Moreover the risk-neutral process differs from(1)by having a different drift:it solves dS=rSdt+σSdw.By Girsanov’s theorem we can write the risk-neutral expected payoffusing the subjective probability as
E RN[f(S T)]=E[e−z(T)f(S T)](3)
where
z(t)= t
µ−r
2
t
µ−r
σ
Hdw(5)
with initial condition H(0)=1.
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This option pricing formula doesn’t come from thin air.It comes from the abnce of arbitrage,together with the fact that the option payofff(S T)can be replicated by a hedge portfolio,consisting of stock and bond in suitable weights.The option value is the value of the hedge
portfolio at he initial capital needed to establish this(time-varying but lf-financing)replicating portfolio.
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The connection with portfolio optimization.What does this have to do with portfolio optimization?Plenty.The hedge portfolio reprents a specific choice of weightsθ(t).The option pricing formula tells us the condition under which a randomfinal-time wealth of the form X(T)=f(S(T))is achievable by a suitable trading strategy:the only restriction is that f satisfy x=E[H(T)f(S T)].
In portfolio optimization we are not mainly interestedfinal-time wealths of the form f(S T). Rather,we are interested in tho achievable by a suitable trading a non-anticipating choice ofθ(t)for0≤t≤T).The resulting randomfinal-time wealth will,in general,be path-dependent;however it is certainly F T it is determined by knowledge of the entire Brownian process{w(t)}0≤t≤T.
Our discussion of option pricing extends,however,to more general(F T-measureable)final-time wealths.The crucial question is:which(random,F T-measureable)final-time wealths B are associated with nonanticipating trading strategiesθ(t)using initial capital x?The answer is simple:B has this prop
erty exactly if
x=E[H(T)B](6) and in that ca the associated wealth process X(t)satisfies
H(t)X(t)=E[H(T)B|F t].(7) We’ll prove just the easy direction:that if B=X(T)for some trading strategyθ(t)then (6)and(7)hold.(See Korn&Korn for the conver.)Let’s apply Ito’s formula in the form
d(HX)=HdX+XdH+dHdX
(there is no factor of1/2in front of the last term,becau f(h,x)=xh has∂2f/∂h∂x=∂2f/∂x∂h=1).Substituting(2)and(5)gives
d(HX)=−HX(rdt+µ−r
σ
θσdt.
The dt terms cancel,leaving only dw terms.So
H(T)X(T)−H(t)X(t)= T
课时费
t
[stuff]dw.
Setting t=0and taking the expectation of both sides gives
E[H(T)X(T)]=H(0)X(0)=x;
similarly,for any t we take the expectation of both sides conditioned on time-t data to get
E[H(T)X(T)|F t]=H(t)X(t).
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The are the desired asrtions.(Korn&Korn for the conver.)
The martingale approach to portfolio optimization.Relation(6)changes the task of dynamic portfolio optimization to a static optimization problem:the optimalfinal-time wealth B solves
max
E[H(T)B]=x
E[h(B)].
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The solution is easy.To avoid getting confud let’s pretend the list of possiblefinal-time states was discrete,with stateαhaving probability pα,1≤α≤N.Then the random variable B would be characterized by its list of values(B1,...,B N)and our task would be
to solve
max
H(T)αBαpα=x
h(Bα)pα
for the optimal B=(B1,...,B N).This is easily achieved using the method of Lagrange multipliers.Ifλis the Lagrange multiplier for the constraint then the solution satisfies ∂L/∂Bα=0and∂L/∂λ=0where
L(B,λ)=
h(Bα)pα+λ
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x−
H(T)αBαpα
.
The derivative inλrecovers the constraint
H(T)αBαpα=x
and the derivative with respect to Bαgives
h (Bα)=λH(T)α
for eachα.The N+1equations determine the values of the N+1unknowns Bαandλ. The continuous ca is no different:the optimal B satisfies
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h (B)=λH(T)(8) as random variables.Since h is concave,h is invertible,so we can solve(8)for B:
B={h }−1(λH(T))
where{h }−1is the inver function of h .The value ofλis uniquely determined by the condition that this B satisfy
E[H(T)B]=x.
An example:the Merton problem with logarithmic utility.So far we have not assumed anything about the drift and volatility:they can be functions of time and stock price(µ=µ(t,S)andσ=σ(t,S)).But to bring things down to earth let’s consider a familiar example:the constant-drift,constant-volatility ca,with utility h(x)=log x. Notice that for this utility h (x)=1/x,so{h }−1(y)=1/y.Also,since the drift and volatility are constant
H(t)=e−rt−z(t)
with
z(t)=µ−r
2
µ−r
2σ
2)t+σw(t).
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Thus both H (t )and S (t )are determined by knowledge of t and w (t ).Put differently:in this ca H (T )is a function of S (T ).
Specializing (8)to our example gives 1
λH (T ).
The value of λis fixed by (6),which gives x =E [H (T )B ]=
1x
.The wealth at time T is determined by (7);it gives
H (t )X (t )=E [H (T )B |F t ]=E λ
−1 F t =1/λ
whence
X (t )=H −1(t )x.To implement this solution practically,what we really need is the weight θ(t )associated with the optimal policy.To find it,obrve that by Ito’s formula applied to (5),d [H −1x ]=−H −2xdH +H −3xdHdH =H −1x
rdt + µ−r八一晚会
σdw .
This can be written as a wealth-process it has the form
dX =[(1−θ)r +θµ]Xdt +θσXdw
with X (t )=H −1x and
θ=µ−r
