BOOK-Soner-Stochastic Optimal Control in Finance

7/31/2019 BOOK-Soner-Stochastic Optimal Control in Finance

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Stochastic Optimal ControlinFinance

H. Mete Soner

Koc UniversityIstanbul, Turkey

[email protected]


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.

for my son,MehmetAliye.


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Preface

These are the extended version of the Cattedra Galileiana I gave in April2003 in Scuola Normale, Pisa. I am grateful to the Society of Amici dellaScuola Normale for the funding and to Professors Maurizio Pratelli, MarziaDe Donno and Paulo Guasoni for organizing these lectures and their hospi-tality.

In these notes, I give a very quick introduction to stochastic optimalcontrol and the dynamic programming approach to control. This is donethrough several important examples that arise in mathematical finance andeconomics. The theory of viscosity solutions of Crandall and Lions is alsodemonstrated in one example. The choice of problems is driven by my ownresearch and the desire to illustrate the use of dynamic programming andviscosity solutions. In particular, a great emphasis is given to the problemof super-replication as it provides an usual application of these methods.

Of course there are a number of other very important examples of optimalcontrol problems arising in mathematical finance, such as passport options,American options. Omission of these examples and different methods insolving them do not reflect in any way on the importance of these problemsand techniques.

Most of the original work presented here is obtained in collaboration withProfessor Nizar Touzi of Paris. I would like to thank him for the fruitfulcollaboration, his support and friendship.

Oxford, March 2004.

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Contents

1 Examples and Dynamic Programming 11.1 Optimal Control. . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Deterministic minimal time problem . . . . . . . . . . 31.2.2 Mertons optimal investment-consumption problem . . 31.2.3 Finite time utility maximization . . . . . . . . . . . . . 51.2.4 Merton problem with transaction costs . . . . . . . . . 51.2.5 Super-replication with portfolio constraints . . . . . . . 71.2.6 Buyers price and the no-arbitrage interval . . . . . . . 71.2.7 Super-replication with gamma constraints . . . . . . . 8

1.3 Dynamic Programming Principle . . . . . . . . . . . . . . . . 91.3.1 Formal Proof of DPP . . . . . . . . . . . . . . . . . . . 10

1.3.2 Examples for the DPP . . . . . . . . . . . . . . . . . . 111.4 Dynamic Programming Equation . . . . . . . . . . . . . . . . 131.4.1 Formal Derivation of the DPE . . . . . . . . . . . . . . 141.4.2 Infinite horizon . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Examples for the DPE . . . . . . . . . . . . . . . . . . . . . . 161.5.1 Merton Problem . . . . . . . . . . . . . . . . . . . . . 161.5.2 Minimal Time Problem . . . . . . . . . . . . . . . . . . 191.5.3 Transaction costs . . . . . . . . . . . . . . . . . . . . . 201.5.4 Super-replication with portfolio constraints . . . . . . . 221.5.5 Target Reachability Problem . . . . . . . . . . . . . . . 22

2 Super-Replication under portfolio constraints 252.1 Solution by Duality . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Black-Scholes Case . . . . . . . . . . . . . . . . . . . . 262.1.2 General Case . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Direct Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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2.2.1 Viscosity Solutions . . . . . . . . . . . . . . . . . . . . 29

2.2.2 Supersolution . . . . . . . . . . . . . . . . . . . . . . . 302.2.3 Subsolution . . . . . . . . . . . . . . . . . . . . . . . . 322.2.4 Terminal Condition or Face-Lifting . . . . . . . . . . 34

3 Super-Replication with Gamma Constraints 383.1 Pure Upper Bound Case . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 Super solution . . . . . . . . . . . . . . . . . . . . . . . 393.1.2 Subsolution . . . . . . . . . . . . . . . . . . . . . . . . 413.1.3 Terminal Condition . . . . . . . . . . . . . . . . . . . . 41

3.2 Double Stochastic Integrals . . . . . . . . . . . . . . . . . . . 443.3 General Gamma Constraint . . . . . . . . . . . . . . . . . . . 51

3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.1 European Call Option . . . . . . . . . . . . . . . . . . 533.4.2 European Put Option: . . . . . . . . . . . . . . . . . . 543.4.3 European Digital Option . . . . . . . . . . . . . . . . . 553.4.4 Up and Out European Call Option . . . . . . . . . . . 56

3.5 Guess for The Dual Formulation . . . . . . . . . . . . . . . . . 58

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Chapter 1

Examples and DynamicProgramming

In this Chapter, we will outline the basic structure of an optimal control prob-lem. Then, this structure will be explained through several examples mainlyfrom mathematical finance. Analysis and the solution to these problems willbe provided later.

1.1 Optimal Control.

In very general terms, an optimal control problem consists of the followingelements:

State process Z(). This process must capture of the minimal neces-sary information needed to describe the problem. Typically, Z(t) dis influenced by the control and given the control process it has a Marko-vian structure. Usually its time dynamics is prescribed through anequation. We will consider only the state processes whose dynamics isdescribed through an ordinary or a stochastic differential equation. Dy-namics given by partial differential equations yield infinite dimensionalproblems and we will not consider those in these lecture notes.

Control process (). We need to describe the control set, U, inwhich (t) takes values in for every t. Applications dictate the choiceof U. In addition to this simple restriction (t) U, there could beadditional constraints placed on control process. For instance, in the

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stochastic setting, we will require to be adapted to a certain filtration,

to model the flow of information. Also we may require the state processto take values in a certain region (i.e., state constraint). This also placesrestrictions on the process ().

Admissible controls A. A control process satisfying the constraintsis called an admissible control. The set of all admissible controls willbe denoted by A and it may depend on the initial value of the stateprocess.

Objective functional J(Z(), ()). This is the functional to be max-imized (or minimized). In all of our applications, J has an additivestructure, or in other words J is given as an integral over time.

Then, the goal is to minimize (or maximize) the objective functional Jover all admissible controls. The minimum value plays an important role inour analysis

Value function: = v = infA

J .

The main problem in optimal control is to find the minimizing controlprocess. In our approach, we will exploit the Markovian structure of theproblem and use dynamic programming. This approach yields a certainpartial differential equation satisfied by the value function v. However, in

solving this equation we also obtain the optimal control in a feedback form.This means that is the optimal process (t) is given as (Z(t)), where isthe optimal feedback control given as a function of the state and Z is thecorresponding optimal state process. Both Z and the optimal control arecomputed simultaneously by solving the state dynamics with feedback control. Although a powerful method, it also has its technical drawbacks. Thisprocess and the technical issues will be explained by examples throughoutthese notes.

1.2 Examples

In this section, we formulate several important examples of optimal controlproblems. Their solutions will be given in later sections after the necessarytechniques are developed.

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1.2.1 Deterministic minimal time problem

The state dynamics is given by

d

dtZ(t) = f(Z(t), (t)) , t > 0 ,

Z(0) = z,

where f is a given vector field and : [0, ) U is the control process. Wealways assume that f is regular enough so that for a given control process ,the above equation has a unique solution Zx ().

For a given target set T d, the objective functional is

J(Z(), ()) := inf {t 0 : Zz (t) T } (or + if set is empty),:= Tz .

Letv(z) = inf

ATz ,

where A := L([0, ); U), and U is a subset of a Euclidean space.Note that additional constraints typically placed on controls. In robotics,

for instance, control set U can be discrete and the state Z() may not beallowed to enter into certain a region, called obstacles.

1.2.2 Mertons optimal investment-consumption prob-lem

This is a financial market with two assets: one risky asset, called stock, andone riskless asset, called bond. We model that price of the stock S(t) asthe solution of

dS(t) = S(t)[dt + dW] , (1.2.1)

where W is the standard one-dimensional Brownian motion, and and are given constants. We also assume a constant interest rate r for the Bondprice, B(t), i.e.,

dB(t) = B(t)[rdt] .

At time t, let X(t) be the money invested in the bond, Y(t) be the invest-ments at the stock, l(t) be the rate of transfer from the bond holdings to

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the stock, m(t) be the rate of opposite transfers and c(t) be the rate of con-

sumption. So we have the following equations for X(t), Y(t) assuming notransaction costs.

dX(t) = rX(t)dt l(t)dt + m(t)dt c(t)dt , (1.2.2)dY(t) = Y(t)[dt + dW] + l(t)dt m(t)dt . (1.2.3)

Set

Z(t) = X(t) + Y(t) = wealth of the investor at time t,

(t) =Y(t)

Z(t).

Then,

dZ(t) = Z(t)[(r + (t)( r))dt + (t)dW] c(t)dt . (1.2.4)In this example, the state process is Z = Z,cz and the controls are (t) 1and c(t) 0. Since we can transfer funds between the stock holdings andthe bond holdings instantaneously and without a loss, it is not necessary tokeep track of the holdings in each asset separately.

We have an additional restriction that Z(t) 0. Thus the set of admis-sible controls Az is given by:

Az:=

{((

), c(

))

|bounded, adapted processes so that Z,c

Z 0 a.s.

}.

The objective functional is the expected discounted utility derived from con-sumption:

J = E

0

etU(c(t))dt

,

where U : [0, ) 1 is the utility function. The function U(c) = cpp

with0 < p < 1, provides an interesting class of examples. In this case,

v(z) := sup(,c)Az

E

0

et1

p(c(t))pdt | Z,cz (0) = z

. (1.2.5)

The simplifying nature of this utility is that there is a certain homotethy.Note that due to the linear structure of the state equation, for any > 0,(,c) Az if and only if (, c) Az. Therefore,

v(z) = pv(z) v(z) = v(1)zp . (1.2.6)

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Thus, we only need to compute v(1) and the optimal strategy associated to

it. By dynamic programming, we will see thatc(t) = (pv(1))

1p1 Z(t), (1.2.7)

(t) = r2(1 p) . (1.2.8)

For v(1) to be finite and thus for the problem to have a solution, needs tobe sufficiently large. An exact condition is known and will be calculated bydynamic programming.

1.2.3 Finite time utility maximization

The following variant of the Mertons problem often arises in finance. LetZt,z() be the solution of (1.2.4) with c 0 and the initial condition:

Zt,z(t) = z . (1.2.9)

Then, for all t < T and z +, considerJ = E

U(Zt,z(T))

,

v(z, t) = supAt,z

E[U(Zt,z(T)|Ft],

where Ft is the filtration generated by the Brownian motion. Mathematically,the main difference between this and the classical Merton problem is thatthe value function here depends not only on the initial value of z but also ont. In fact, one may think the pair (t, Z(t)) as the state variables, but in theliterature this is understood only implicitly. In the classical Merton problem,the dependence on t is trivial and thus omitted.

1.2.4 Merton problem with transaction costs

This is an interesting modification of the Mertons problem due to Constan-tinides [9] and Davis & Norman [12]. We assume that whenever we movefunds from bond to stock we pay, or loose,

(0, 1) fraction to the transac-

tion fee ,and similarly, we loose (0, 1) fraction in the opposite transfers.Then, equations (1.2.2),(1.2.3) become

dX(t) = rX(t)dt l(t)dt + (1 )m(t)dt c(t)dt , (1.2.10)dY(t) = Y(t)[dt + dW] + (1 )l(t)dt m(t)dt . (1.2.11)

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In this model, it is intuitively clear that the variable Z = X + Y, is not

sufficient to describe the state of the model. So, it is now necessary toconsider the pair Z := (X, Y) as the state process. The controls are theprocesses l, m and c, and all are assumed to be non-negative. Again,

v(x, y) := sup=(l,m,c)Ax,y

E

0

et1

p(c(t))pdt

.

The set of admissible controls are such that the solutions (Xx , Y

x ) Lfor all t 0. The liquidity set L 2 is the collection of all (x, y) thatcan be transferred to a non-negative position both in bond and stock by anappropriate transaction, i.e.,

L = {(x, y) 2 : (L, M) 0 s.t.(x + (1 )M L, Y M + (1 )L) + +}

= {(x, y) 2 : (1 )x + y 0 and x + (1 )y 0} .

An important feature of this problem is that it is possibly singular, i.e.,the optimal (l(), m()) process can be unbounded. On the other hand, thenonlinear penalization c(t)p does not allow c(t) to be unbounded.

The singular problems share this common feature that the control enterslinearly in the state equation and either is not included in the objectivefunctional or included only in a linear manner.

So, it is convenient to introduce processes:

L(t) :=

t0

l(s)ds, M (t) :=

t0

m(s)ds .

Then, (L(), M()) are nondecreasing adopted processes and (dL(t), dM(t))can be defined as random measures on [0, ). With this notation, we rewrite(1.2.10), (1.2.11) as

dX = rXdt dL + (1 )dM c(t)dt ,dY = Y[dt + dW] + (1

)dL

dM ,

and = (L,M,c) Ax,y is admissible if they are adapted (L, M) nonde-creasing, c 0 and

(Xx (t), Y

y (t)) L t 0 . (1.2.12)

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1.2.5 Super-replication with portfolio constraints

Let Zt,z() be the solution of (1.2.4) with c 0 and (1.2.9), and let St,s()be the solution of (1.2.1) with St,s(t) = s. Given a deterministic functionG : 1 1 we wish to findv(t, s) := inf{z | () adapted, (t) K and Zt,z(T) G(St,s(T)) a.s. } ,

where T is maturity, K is an interval containing 0, i.e., K = [a, b]. Herea is related to a short-sell constraint and b to a borrowing constraint (orequivalently a constraint on short-selling the bond).

This is clearly not in the form of the previous problems, but it can betransferred into that form. Indeed, set

X(z, s) :=

0, z G(s),+, z < G(s).

Consider an objective functional,

J(t,s,s; ()) := EX(Zt,z(T), St,s(T)) | Ft ,u(t ,z,s) := inf

AJ(t,s,s; ()) ,

and A if and only if is adapted with values in K. Then, observe that

u(t,z,s) = 0, z > v(t, s)

+, z < v(t, s) .and at z = v(t,z,s) is a subtle question. In other words,

v(t, s) = inf{z| u(t,z,s) = 0} .

1.2.6 Buyers price and the no-arbitrage interval

In the previous subsection, we considered the problem from the perspectiveof the writer of the option. For a potential buyer, if the quoted price z of acertain claim is low, there is a different possibility of arbitrage. She would

take advantage of a low price by buying the option for a price z. She wouldfinance this purchase by using the instruments in the market. Then she triesto maximize her wealth (or minimize her debt) with initial wealth of z. Ifat maturity,

Zt,z(T) + G(St,s(T)) 0, a.s.,

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then this provides arbitrage. Hence the largest of these initial data provides

the lower bound of all prices that do not allow arbitrage. So we define (afterobserving that Zt,z(T) = Zt,z(T)),

v(t, s) := sup{z | () adapted, (t) K and Zt,z(T) G(St,s(T)) a.s. } .

Then, the no-arbitrage interval is given by

[v(t, s), v(t, s)] .

In the presence of friction, there are many approaches to pricing. How-ever, the above above interval must contain all the prices obtained by anymethod.

1.2.7 Super-replication with gamma constraints

To simplify, we take r = 0, = 0. We rewrite (1.2.4) as

dZ(t) = n(t)dS(t) ,

dS(t) = S(t)dW(t) .

Then, n(t) = (t)Z(t)/S(t) is the number of stocks held at a given time.Previously, we placed no restrictions on the time change of rate of n() andassumed only that it is bounded and adapted. The gamma constraint, re-stricts n() to be a semimartingale,

dn(t) = dA(t) + (t)dS(t) ,

where A is an adapted BV process, () is an adapted process with values inan internal [, ].

Then, the super-replication problem is

v(t, s) := inf{z| = (n(t), A(), ()) At,s,z s.t. Zt,z(T) G(St,s(T))} .

The important new feature here is the singular form of the equation for then() process. Notice the dA term in that equation.

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1.3 Dynamic Programming Principle

In this section, we formulate an abstract dynamic programming following therecent manuscript of Soner & Touzi [20]. This principle holds for all dynamicoptimization problems with a certain structure. Thus, the structure of theproblem is of critical importance. We formulate this in the following mainassumptions.

Assumption 1 We assume that for every control and initial data (t, z),the corresponding state process starts afresh at every stopping time > t,i.e.,

Zt,z(s) = Z,Zt,z()

(s) , s .

Assumption 2 The affect of is causal, i.e., if 1(s) = 2(s) for all s ,where is a stopping time, then

Z1

t,z(s) = Z2

t,z(s) , s .Moreover, we assume that if is admissible at (t, z) then, is restricted

to the stochastic interval [, T] is also admissible starting at (, Zt,z()).

Assumption 3 We also assume that the concatenation of admissible con-trols yield another admissible control. Mathematically, for a stopping time

and At,z, set

= (, Zt,z()). Suppose A and define

(s) =

(s), s ,(s), s .

Then, we assume At,z. Precise formulation is in Soner & Touzi [20].

Assumption 4 Finally, we assume an additive structure for J, i.e.,

J =

t

L(s, (s), Zt,z(s))ds + G(Zt,z()) .

The above list of assumptions need to be verified in each example. Underthese structural assumptions, we have the following result which is called thedynamic programming principle or DPP in short.

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Theorem 1.3.1 (Dynamic Programming Principle) For any stopping

time tv(t, z) = inf

At,zE

t

Lds + v(, Zt,z()) | Ft

.

We refer to Fleming & Soner [14] and Soner & Touzi [20]for precise state-ments and proofs.

1.3.1 Formal Proof of DPP

By the additive structure of the cost functional,

v(t, z) = inf At,z

E(

T

t

Lds + G(Zt,z(T)) | Ft)

= inf At,z

E(

t

Lds + E[

T

Lds + G(Zt,z(T)) | F] | Ft) .(1.3.13)

By Assumption 2, restricted to the interval [, T] is in A,Zt,z(). Hence,

E[

Tt

Lds + G | F] v(),

where := t,z() = (, Zt,z()) .

Substitute the above inequality into (1.3.13) to obtain

v(t, z) inf

E[

t

Lds + v() | Ft] .

To prove the reverse inequality, for > 0 and , choose , A sothat

E(T

L(s, ,(s), Z, (s))ds + G(Z

, (T))

|F)

v() + .

For a given At,z, set

(s) :=

(s), s [t, ],(s), s T

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Here there are serious measurability questions (c.f. Soner & Touzi), but it

can be shown that At,z. Then, with Z = Z

t,z,

v(t, z) E(T

t

L(s, (s), Z(s))ds + G(Z(T)) | Ft)

E(

t

L(s, (s), Z(s))ds + v() + | Ft) .

Since this holds for any At,z and > 0,

v(t, z) infAt,z

E(

T

Lds + v() + |Ft) .

The above calculation is the main trust of a rigorous proof. But there aretechnical details that need to provided. We refer to the book of Fleming &Soner and the manuscript by Soner & Touzi.

1.3.2 Examples for the DPP

Our assumptions include all the examples given above. In this section we lookat the super-replication, and more generally a target reachability problemand deduce a geometric DPP from the above DPP. Then, we will outline an

example for theoretical economics for which our method does that alwaysapply.

Target Reachability.Let Zt,z, At,z be as before. Given a target set T d. Consider

V(t) := {z d : At,z s.t. Zt,z(T) T a.s.} .

This is the generalization of the super-replication problems considered before.So as before, for A d, set

XA(z) := 0, z T ,+ z T ,and

v(t, z) := infAz,t

E[XT(Zt,z(T)) | Ft] .

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Then,

v(t, z) = XV(t)(z) = 0, z V(t) ,+, z V(t) .Since, at least formally, DPP applies to v,

v(t, z) = XV(t)(z) = inf At,z

E(v(, Zt,z()) | Ft)= inf

At,zE(XV()(Zt,z()) | Ft) .

Therefore, V(t) also satisfies a geometric DPP:

V(t) = {z d : At,z s.t. Zt,z() V() a.s. } . (1.3.14)

In conclusion, this is a nonstandard example of dynamic programming,in which the principle has the above geometric form. Later in these notes,we will show that this yields a geometric equation for the time evolution ofthe reachability sets.

Incentive Controls.Here we describe a problem in which the dynamic programming does not

always hold. The original problem of Benhabib [4] is a resource allocationproblem. Two players are using y(t) by consuming ci(t), i = 1, 2 . Theequation for the resource is

dydt = (y(t)) c1(t) c2(t) ,with the constraint

y(t) 0, ci 0 .If at some time t0, y(t0) = 0, after this point we require y(t), ci(t) = 0 for

t t0. Each player is trying to maximize

vi(y) := supci

0

etci(t)dt .

Set

v(y) := supc1+c2=c

0

etc(t)dt ,

so that, clearly, v1(y) + v2(y) v(y). However, each player may bring thestate to a bankruptcy by consuming a large amount to the detriment of theother player and possibly to herself as well.

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To avoid this problem Rustichini [17] proposed a variation in which the

state equation is d

dtX(t) = f(X(t), c(t)) ,

with initial conditionX(0) = x .

Then, the pay-off is

J(x, c()) =0

etL(t, X(t), c(t))dt ,

and c(

)

Ax if

t

es L(t, X(t), c(t))ds esD(Xcx(t), c(t)), t 0 ,

where D is a given function. Note that this condition, in general, violatesthe concatenation property of the set of admissible controls. Hence, dynamicprogramming does not always hold. However, Barucci-Gozzi-Swiech [3] over-come this in certain cases.

1.4 Dynamic Programming Equation

This equation is the infinitesimal version of the dynamic programming prin-ciple. It is used, generally, in the following two ways:

Derive the DPE formallyas we will do later in these notes. Obtain a smooth solution, or show that there is a smooth solution via

PDE techniques.

Show that the smooth solution is the value function by the use of Itosformula. This step is called the verification.

As a by product, an optimal policy is obtained in the verification.This is the classical use of the DPE and details are given in the book

Fleming & Soner and we will outline it in detail for the Merton problem.The second approach is this:

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Derive the DPE rigorously using the theory of viscosity solutions ofCrandall and Lions.

Show uniqueness or more generally a comparison result between suband super viscosity solutions.

This provides a unique characterization of the value function which canthen be used to obtain further results.

This approach, which become available by the theory of viscosity solu-tions, avoids showing the smoothness of the value function. This is verydesirable as the value function is often not smooth.

1.4.1 Formal Derivation of the DPE

To simplify the presentation, we only consider the state processes which arediffusions. Let the state variable X be the unique solution

dX = (t, X(t), (t))dt + (t, X(t), (t)))dW ,

and a usual pay-off functional

J(t,x,) = E[

Tt

L(s, Xt,x(s), (s))ds + G(Xt,x(T)) | Ft] ,

v(t, x) := infAt,x J(t,x,) .

We assume enough so that the DPP holds. Use = t + h in the DPP toobtain

v(t, x) = infAt,x

E[

t+ht

Lds + v(t + h, Xt,x(t + h)) | Ft] .

We assume, without justification, that v is sufficiently smooth. This part ofthe derivation is formal and can not be made rigorous unless viscosity theoryis revoked. Then, by the Itos formula,

v(t + h, Xt,x(t + h)) = v(t, x) +

t+h

t

( t

v + L(s)v)ds + martingale ,

where

Lv := (t,x,) v + 12

tr a(t,x,)D2v ,

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with the notation,

a(t,x,) := (t,x,)(t,x,)t and tr a :=d

i=1

aii .

In view of the DPP,

supAt,x

E[t+h

t

(

tv + L(s)v + L)ds] = 0 .

We assume that the coefficients ,a,L are continuous. Divide the aboveequation by h and let h go to zero to obtain

t

v(t, x) + H(x,t, v(t, x), D2v(t, x)) = 0 , (1.4.15)

where

H(x,t,,A) := sup{. 12

traA l : (,a,l) A(t, x)} ,

and (,a,l) A(t, x) iff there exists At,x such that

(,a,l) = limh0

1

h

t+h

t

((x, (s), t), a(x, (s), t), L(x, (s), t))ds .

We should emphasize that we assume that the functions ,a,L are suffi-ciently regular and we also made an unjustified assumption that v is smooth.All these assumptions are not needed in the theory of viscosity solutions aswe will see later or we refer to the book by Fleming & Soner.

For a large class of problems, At,x = L((0, ) ; U) for some set U.Then,

A(t, x) = {((x,,t), a(x,,t), L(x,,t) : U} ,and

H(x,t,,A) = supU{(x,,t) 12tr a(x,,t)A L(x,,t)} .

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1.4.2 Infinite horizon

A important class of problems are known as the discounted infinite horizonproblems. In these problems the state equation for X is time homogenousand the time horizon T = . However, to ensure the finiteness of the costfunctional, the running cost is exponentially discounted, i.e.,

J(x, ) := E

0

etL(s, Xx (s), (s))ds .

Then, following the same calculation as in the finite horizon case, we derivethe dynamic programming equation to be

v(x) + H(x, v(x), D2

v(x)) = 0 , (1.4.16)

where for At,x = L((0, ) ; U),

H(x,,A) = supU

{(x, ) 12

tr a(x, )A L(x, )} .

1.5 Examples for the DPE

In this section, we will obtain the corresponding dynamic programming equa-tion for the examples given earlier.

1.5.1 Merton Problem

We already showed that, in the case with no transaction costs,

v(z) = v(1)zp .

This is a rare example of an interesting stochastic optimal control problemwith a smooth and an explicit solution. Hence, we will employ the first ofthe two approaches mentioned earlier for using the DPE.

We start with the DPE (1.4.16), which takes the following form for this

equation (accounting for sup instead of inf),

v(z) + inf 1,c0

{(r + ( r))zvz(z) 12

22z2vzz + cvz 1p

cp} = 0 .

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We write this as,

v(z) rzvz(z) sup1

{( r)zvz + 12

22z2vzz} supc0

{cvz + 1p

cp} = 0 .

We directly calculate that (for vz(z) > 0 > vzz (z))

v(z) rzvz(z) 12

(( r)zvz(z))22z2vzz (z)

H(vz(z)) = 0 ,

where

H(vz(z)) =1 p

p(vz(z))

pp1 ,

with maximizers

= ( r)zvz(z)2z2vzz (z)

, c = (vz(z))1

p1 .

We plug the form v(z) = v(1)zp in the above equations. The result is theequation (1.2.7) and (1.2.8) and

v(1)

rp p( r)

2

2(1 p)2

1 pp

(p v(1))p

p1 = 0 .

The solution is

v(1) = :=(1 p)1p

p

rp p( r)

2

2(1 p)2p1

,

and we require that

> rp +p( r)2

2(1 p)2 .

Although the above calculations look to be complete, we recall that thederivation of the DPE is formal. For that reason, we need to complete thesecalculations with a verification step.

Theorem 1.5.1 (Verification) The function zp, with as above, is thevalue function. Moreover, the optimal feedback policies are given by the equa-tions (1.2.7) and (1.2.8).

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Proof. Set u(z) := zp.

For z > 0 and T > 0, let = ((), c()) Az be any admissible con-sumption, investment strategy. Set Z := Zz . Apply the Itos rule to thefunction e t u(Z(t)). The result is

eTE[u(Z(T))] = u(z) +T0

et E[ u(Z(t)) + L(t),c(t)u(Z(t))] dt,

where L,c is the infinitesimal generator of the wealth process. By the factthat u solves the DPE, we have, for any and c,

u(z) L,cu(z) 1p

cp 0.

Hence,

u(z) E[e T u(Z(T)) +T0

e t1

p(c(t))p dt] .

By direct calculations, we can show that

limT

E[e T u(Z(T))] = limT

E[e T (Z(T))p] = 0 .

Also by the Fatous Lemma,

limT

E[T

0

e t1

p

(c(t)p dt] = J(z; (

), c(

)).

Since this holds for any control, we proved that

u(z) = zp v(z) = value function .To prove the opposite inequality we use the controls (, c) given by theequations (1.2.7) and (1.2.8). Let Z be the corresponding state process.Then,

u(z) L,cu(z) 1p

(c)p = 0 .

Therefore,

EeTu(Z(T)) = u(z) +T0

et E[ u(Z(t)) + L(t),c(t)u(Z(t))]dt

= u(z) + E[

T0

e t1

p(c(t))p dt] .

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Again we let T tend to infinity. Since Z and the other quantities can be

calculated explicitly, it is straightforward to pass to the limit in the aboveequation. The result isu(z) = J(z; , c).

Hence, u(z) = v(z) = J(z; , c).

1.5.2 Minimal Time Problem

For Ax = L((0, ); U), the dynamic programming equation has a simpleform,

supU

{f(x, ) v(x) 1} = 0 x T .

This follows from our results and the simple observation that

J = x =

x0

1ds .

So we may think of this problem an infinite horizon problem with zero dis-count, = 0.

In the special example, U = B1, f(x, ) = , the above equation simplifiesto the Eikonal equation,

|v(x)

|= 1 x

T,

together with the boundary condition,

v(x) = 0, x T .The solution is the distance function,

v(x) = infyT

{|x y|} = |x y| ,

and the optimal control is

=y

x

|y x| t .As in the Merton problem, the solution is again explicitly available. How-

ever, v(x) is not a smooth function, and the above assertions have to beproved by the viscosity theory.

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1.5.3 Transaction costs

Using the formulation (1.2.10) and (1.2.11), we formally obtain,

v(x, y) + inf (l,m,c)0

{rxvx yvy 12

2y2vyy

l[(1 )vy vx] m[(1 )vx vy] + cvx 1p

cp} = 0 .

We see that, since l and m can be arbitrarily large,

(1 )vy vx 0 and (1 )vx vy 0 . (1.5.17)Also dropping the l and m terms we have,

v rxvx yvy 12

2y2vyy := v Lv H(vx) ,

where

H(vx) := supc0

1

pcp cvx

.

Moreover, if both inequalities are strict in (1.5.17), then the above is anequality. Hence,

min{v Lv H(vx), vx (1 )vy, vy (1 )vx} = 0 . (1.5.18)

This derivation is extremely formal, but can be verified by the theory ofviscosity solutions, cf. Fleming & Soner [14], Shreve & Soner [18]. We alsorefer to Davis & Norman [12] who was first to study this problem using thefirst approach described earlier.

Notice also the singular character of the problem resulted in a quasi vari-ational inequality, instead of a more standard second order elliptic equation.

We again use homothety pv(x, y) = v(x,y), to represent v(x, y) so

v(x, y) = (x + y)pf(y

x + y), (x, y) L ,

where

f(u) := v(1 u, u), 1 u 1 .The DPE for v, namely (1.5.18), turns into an equation for f the coefficientfunction; a one dimensional problem which can be solved numerically bystandard methods.

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-

6

r

rrrrrrrrrrrrrrrrr

eeeeeeeeeeeeee

x

y

Region I

Consume

Region III

Sell Stock

Region II

Sell Bond

yx+y =

1

yx+y = b

yx+y

= a

yx+y

= (1)

eeu

ee

Figure 1.1:

Further, we know that v is concave. Using the concavity of v, we showthat there are points

(1)

a < mert < b

1

so that

In Region I: v Lv H(vx) = 0 for a yx + y

b ,

In Region II: vx (1 )vy = 0 for (1 )

yx + y

a ,

In Region III: vy (1 )vx = 0 for b yx + y

1

.

So we formally expect that in Region 1, no transactions are made and con-

sumption is according to , c = (vx(x, y))1

p1 . In Region 2, we sell bonds and

buy stocks, and in Region 3 we sell stock and buy bonds. Finally, the process(X(t), Y(t)) is kept in Region 1 through reflection.Constant b, a are not explicitly available but can be computed numeri-

cally.

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1.5.4 Super-replication with portfolio constraints

In the next Chapter, we will show that the DPE is (with K = (a, b) )

min{vt

12

s22vss rsvs + rv ; bv svs; svs + av} = 0

with the final conditionu(T, s) = G(s) .

Then, we will show that

u(t, s) = E[G(St,s(T)) | Ft] ,where E is the risk neutral expectation and G is the minimal function sat-isfying

(i). G G ,(ii) a sGs(s)

Gs b .

Examples of G will be computed in the last Chapter.

1.5.5 Target Reachability Problem

ConsiderdX = (t,z,(t))dt + (t,z,(t))dW ,

as before,A

= L((0,

); U). The reachability set is given by,

V(t) := {x| A such that Xt,x(T) T a.s} ,where T d is a given set. Then, at least formally,

v(t, x) := XV(t)(x) = lim0

H(w(t, x)) ,

wherew(t, x) = inf

AE[G(Xt,x(T)) | Ft] ,

H(w) =tanh(w/) + 1

2,

and G : d [0, ) any smooth function which vanishes only on T, i.e.,G(x) = 0 if and only if x T. Then, the standard DPE yields,

wt

+ supU

{(t,x,) w 12

tr(a(t,x,)D2w)} = 0 .

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We calculate that, with w := H(w),

w

t= (H)

w

t, w = (H)w, D2w = (H)D2w+(H)ww .

Hence,

(H)D2w = D2w (H)

[(H)]2w w .

This yields,

w

t+ sup

U{ w 1

2traD2w +

1

2

(H)

[(H)]2aw w} = 0 .

Here, very formally, we conjecture the following limiting equation for v =lim w:

wt

supK(w)

{ w 12

traD2w} = 0 . (1.5.19)

K() := { : t(t,x,) = 0} .Notice that for K(w), aw w = 0. And if K(w) then,aw w = |tw|2 > 0 and (H)/[(H)]2 1/H will cause the non-linear term to blow-up.

The above calculation is very formal. A rigorous derivation using theviscosity solution and different methods is available in Soner & Touzi [20, 21].

Mean Curvature Flow.This is an interesting example of a target reachability problem, which

provides a stochastic representation for a geometric flow. Indeed, consider atarget reachability problem with a general target set and state dynamics

dX =

2(I )dW ,

where () B1 is a unit vector in d. In our previous notation, the controlset U is set of all unit vectors, and

Ais the collection of all adapted pro-

cess with values in U. Then, the geometric dynamic programming equation(1.5.19) takes the form

vt

+ supK(v)

{tr(I )D2v} = 0 ,

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and

K(v) = { B1 : (I )v = 0} = { v

|v|} .So the equation is

vt

v + D2vv v|v|2 = 0 .

This is exactly the level set equation for the mean curvature flow as in thework of Evans-Spruck [13] and Chen-Giga-Goto [6].

If we usedX =

2(t)dW ,

where (t) is a projection matrix on d onto (dk) dimensional planes thenwe obtain the co-dimension k mean curvature flow equation as in Ambrosio& Soner [1].

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Chapter 2

Super-Replication underportfolio constraints

In this Chapter, we will provide all the technical details for this specificproblem as an interesting example of a stochastic optimal control problem.

For this problem, two approaches are available. In the first, after a cleverduality argument, this problem is transformed into a standard optimal con-trol problem and then solved by dynamic programming, we refer to Karatzas& Shreve [15] for details of this method. In the second approach, dynamicprogramming is used directly. Although, when available the first approachprovides more insight, it is not always possible to apply the dual method.The second approach is a direct one and applicable to all super-replicationproblems. The problem with Gamma constraint is an example for which thedual method is not yet known.

2.1 Solution by Duality

Let us recall the problem briefly. We consider a market with one stock andone bond. By multiplying er(Tt) we may take r = 0, (or equivalently takingthe bond as the numeriare). Also by a Girsanov transformation, we maytake = 0. So the resulting simpler equations for the stock price and wealthprocesses are

dS(t) = S(t)dW(t) ,

dZ(t) = (t)Z(t)dW(t) .

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A contingent claim with payoff G : [0, ) 1 is given. The minimalsuper-replication cost is

v(t, s) = inf{z : () A s.t. Zt,z(T) G(St,s(T)) a.s. } ,

where A is the set of all essentially bounded, adapted processes () withvalues in a convex set K.

This restriction of () K, corresponds to proportional borrowing (orequivalently short-selling of bond) and short-selling of stock constraints thatthe investors typically face.

The above is the so-called writers price. The buyers point of view isslightly different. The appropriate target problem for this case is

v(t, s) := sup{z : () A s.t. Zt,z(T) + G(St,s(T)) 0 a.s. } .

Then, the interval [v(t, s), v(t, s)] gives the no-arbitrage interval. That is,if the initial price of this claim is in this interval, then, there is no admissibleportfolio process () which will result in a positive position with probabilityone.

2.1.1 Black-Scholes Case

Let us start with the unconstrained case, K = 1. Since Zt,z() is a martin-gale, if there is z and () A which is super-replicating, then

z = Zt,z(t) = E[Zt,z(T) | Ft] E[G(St,s(T) | Ft] .

Our claim is, indeed the above inequality is an equality for z = v(t, s). Set

Yu := E[G(St,s(T)) | Fu] .

By the martingale representation theorem, Y() is a stochastic integral. Wechoose to write it as

Y(u) = E[G(St,s(T))|Ft] + u

t ()Y()dW() ,

with an appropriate () A. Then,

Y() = Zt,z0(), z0 = E[G(St,s(T)) | Ft] .

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Hence, v(t, s) z0. But we have already shown that if an initial capitalsupports a super-replicating portfolio then, it must be larger than z0. Hence,

v(t, s) = z0 = E[G(St,s(T)) | Ft] := vBS(t, s) ,

which is the Black-Scholes price. Note that in this case, starting with z0,there always exists a replicating portfolio.

In this example, it can be shown that the buyers price is also equal tothe Black-Scholes price vBS. Hence, the no-arbitrage interval defined in theprevious subsection is the singleton {vBS}. Thus, that is the only fair price.

2.1.2 General Case

Let us first introduce several elementary facts from convex analysis. Set

K() := supK

v, K := { : K() < } .

In the convex analysis literature, K is the support function of the convex setK. In one dimension, we may directly calculate these functions. However,we use this notation, as it is suggestive of the multidimensional case. Then,it is a classical fact that

K

+ K()

0

K .

Let z, () be an initial capital, and respectively, a super-replicating portfolio.For any () with values in K, let P be such that

W(u) := W(u) +

ut

()1

d

is a P martingale. This measure exists under integrability conditions on (),by the Girsanov theorem. Here we assume essentially bounded processes, soP exits. Set

Z(u) := Zt,z(u) exp(

u

t K(())d) .

By calculus,

dZ(u) = Z(u)[(K((u)) + (u)(u))du + dW(u)] .

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Since (u) K and (u) K , K((u) + (u)(u) 0 for all u. Therefore,

Z(u) is a super-martingale and

E[Z(T) | Ft] Z(t) = Zt,z(t) = z .

Also Zt,z(T) G(St,s(T)) Pa.s, and therefore, P-a.s. as well. Hence,

Z(T) = exp(T

t

K((v))du) Zt,z(T)

exp(T

t

K((v))du) G(St,s(T)) P a.s. .

All of these together yield,

v(t, s) z := E[exp(T

t

K((v))du)G(St,s(T)) | Ft] .

Since this holds for any () K,

v(t, s) supK

z .

The equality is obtained through a super-martingale representation for theright hand side, c.f. Karatzas & Shreve [15]. The final result is

Theorem 2.1.1 (Cvitanic & Karatzas [11]) The minimal super replicat-ing cost v(t, s) is the value function of the standard optimal control problem,

v(t, s) = E

exp(

Tt

K((v))du) G(St,s(T)) | Ft

,

where St,s solvedSt,s = S

t,s(T) [dt + dW] .

Now this problem can be solved by dynamic programming. Indeed, an

explicit solution was obtained by Broadie, Cvitanic & Soner [5]. We willobtain this solution by the direct approach in the next section.

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2.2 Direct Solution

In this section, we will use dynamic programming directly to obtain a solu-tion. The dynamic programming principle for this problem is

v(t, s) = inf{Z : () A s.t. Zt,z() v(, St,s()) a.s. } .We will use this to derive first the dynamic programming equation and

then the solution.

2.2.1 Viscosity Solutions

We refer to the books by Barles [2], Fleming & Soner [14], the Users Guide[10] for an introduction to viscosity solutions and for more references to thesubject.

Here we briefly introduce the definition. For a locally bounded functionv, set

v(t, s) := lim sup(t,s)(t,s)

v(t, s), v(t, s) := lim inf(t,s)(t,s)

v(t, s) .

Consider the partial differential equation,

F(t,s,v,vt, vs, vss) = 0 .

We say that v is a viscosity supersolution if for any C1,2 and any mini-mizer (t0, s0) of (u

),

F(t0, s0, u(t0, s0), t(t0, s0), s(t0, s0), ss(t0, s0)) 0 . (2.2.1)A subsolution satisfies

F(t0, s0, u(t0, s0), t, s, ss) 0 , (2.2.2)

at any maximizer of (u ).Note that a viscosity solution of F = 0 is not a viscosity solution of

F = 0.It can be checked that the distance function introduced in the minimal

time problem is a viscosity solution of the Eikonal equation.

Theorem 2.2.1 The minimal super-replicating cost v is a viscosity solutionof the DPE,

min{vt

12

s22vss rsvs + rv ; bv svs; svs + av} = 0 . (2.2.3)The proof will be given in the next two subsections.

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2.2.2 Supersolution

Assume G 0. Let C1,2 andv(t0, s0) (t0, s0) = 0 (v )(t, s) (t, s).

Choose tn, sn, zn such that

(tn, sn) (t0, s0), v(tn, sn) v(t0, s0),

v(tn, sn) zn (tn, sn) + 1n2

.

Then, by the dynamic programming principle, there is n() A so that

Zn(tn +1

n) v(tn + 1

n, Sn(tn +

1

n)) a.s ,

whereZn := Z

n

tn,zn, Sn := Stn,sn .

Since v v ,

Zn(tn +1

n) (tn + 1

n, Sn(tn +

1

n)) a.s .

We use the Itos rule and the dynamics of Zn(

) to obtain,

zn +

tn+ 1ntn

n(u)Zn(u)dW(u) (tn, sn)

+

tn+ 1ntn

(t + L)(u, Sn(u))du

+

tn+ 1ntn

s(u, Sn(u))Sn(u)dW(u) .

We rewrite this as

cn +tn+ 1

n

tn

an(u)du +tn+ 1

n

tn

bn(u)dW(u) 0 a.s. , n ,

where

cn := zn (tn, sn) [0, 1n2

],

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an(u) =

(t +

L)(u, Sn(u)), [

L =

1

22s2ss],

bn(u) = [n(u)Zn(u) s(u, Sn(u))Sn(u)] .

For a real number > 0, let P,n be so that

Wn (u) := W(u) +

ut

bn()d

is a P,n martingale. Set

Mn(u) := cn +

utn

an()d +

utn

bn()dW()

= cn +u

tn

(an() b2n())d +u

tn

bn()dW,n() .

Since Mn(u) 0,

0 E,nMn(tn + 1n

)

= cn + E,n[

tn+ 1ntn

(an() b2n()) d] .

Note that an()

(t

L)(t0, s0) as

t0. We multiply the above

inequality n and let n tend to infinity. The result is for every > 0,

(tL)(t0, s0) lim infn

E,nn

tn+ 1ntn

2(n(u)v(t0, s0)s0s(t0, s0))2du.

Hence,(t L)(t0, s0) 0 .

liminfn

E,nn

tn+ 1ntn

2(n(u)v(t0, s0) s0s(t0, s0))2du = 0 .

Moreover, since v

0 and n(

)

(

a, b],

b v(t0, s0) s0s(t0, s0) 0,

s0s(t0, s0) + a v(t0, s0) 0. (2.2.4)

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In conclusion,

F(t0, s0, u(t0, s0), t(t0, s0), s(t0, s0), ss(t0, s0)) 0, (2.2.5)where

F(t,a,v,q,,A) = min{q 12

2s2A; bv s; av + s}.

Here q stands for t, for s and A for ss.

Thus, we proved that

Theorem 2.2.2 v is a viscosity super solution of

F(t,s,v,vt, vs, vss) = min{vt 2

2s2vss; bv svs; av + svs} 0,

on (0, T) (0, ).

2.2.3 Subsolution

Assume that G 0 and G 0. Let C1,2, (t0, s0) be such that(v

)(t0, s0) = 0

(v

)(t, s)

(t, s) .

By considering = + (t t0)2 + (s s0)4 we may assume the abovemaximum of (v ) is strict. (Note t = t, s = s, ss = ss at (t0, s0).)We need to show that

F(t0, s0, v(t0, s0), t(t0, s0), s(t0, s0), ss(t0, s0) 0 . (2.2.6)

Suppose to the contrary. Since v = at (t0, s0), and since F and aresmooth, there exists a neighborhood of (t0, s0), say N, and > 0 so that

F(t,s,,t, s, ss) (t, s) N . (2.2.7)

Since G and G 0, and s0 = 0 then v(t0, s0) > 0. So > 0 in aneighborhood of (t0, s0). Also since (v

) has a strict maximum at (t0, s0),there is a subset ofN, denoted by Nagain so that

> 0 on N, and v(t, s) e(t, s) (t, s) N .

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Set

(t, s) =ss(t, s)

(t, s) , (t, s) N .Then by (2.2.7), K. Fix (t, s) N near (t0, s0) and set S(u) :=St,s(u),

:= inf{u t : (u, S(u)) N} .Let

dZ(u) = Z(u)(u, S(u))dW(u), u [t, ] ,Z(t) = (t, s) .

By the Itos rule, for t < u < ,

d[Z(u)

(u, S(u))] =L

(u, S(u)) du

+(u, S(u))[Z(u) (u, S(u))]dW(u) .Since Z(t) (t, S(t)) = 0, and L 0,

Z(u) (u, S(u)) , u [t, ] .In particular,

Z() (, S() .Also, (, S()) N, and therefore

v(, S()) e (, S()) .We combine all these to arrive at,

Z() (, S() e v(, S()) e v(, S()) .Then,

Z

t,e(t,s)() = eZ() v(, S()) .

By the dynamic programming principle, (also since () K)),e(t, s) v(t, s) .

Since the above inequality holds for every (t, s), by letting (t, s) tend to(t0, s0), we arrive at

e(t0, s0) v(t0, s0) .But we assumed that (t0, s0) = v(t0, s0).

Hence the inequality (2.2.6) must hold and v is a viscosity subsolution.

In the next subsection, we will study the terminal condition and thenprovide an explicit solution.

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2.2.4 Terminal Condition or Face-Lifting

We showed that the value function v(t, s) is a viscosity solution of

F(t,s,v(t, s), vt(t, s), vs(t, s), vss(t, s)) = 0, s > 0, t < T .

In particular,

b v(t, s)svs(t, s) 0, a v(t, s)svs(t, s) 0, s > 0, t < T, (2.2.8)

in the viscosity sense. Set

V(s) := lim sups

s,t

T

v(t, s), V(s) := liminfs

s,t

T

v(t, s) .

Formally, since v(t, .) satisfies (2.2.8) for every t < T, we also expect Vand V to satisfy (2.2.8) as well. However, given contingent claim G may notsatisfy (2.2.8).

Example.Consider a call option:

G(s) = (s K)+,

with K = (

, b) for some b > 1. Then, for s > K,

bG(s) sGs(s) = b(s K)+ s .

This expression is negative for s near K. Note that the Black-Scholes repli-cating portfolio requires almost one share of the stock when time to maturity(T t) near zero and when S(t) > K but close to K. Again at these pointsthe price of the option is near zero. Hence to be able finance this replicatingportfolio, the investor has to borrow an amount which is an arbitrarily largemultiple of her wealth. So any borrowing constraints (i.e. any b < +)makes the replicating portfolio inadmissible.

We formally proceed and assume V = V = V. Formally, we expect V tosatisfy (2.2.8) and also V 0. Hence,

bV(s) sVs(s) 0 aV(s) sVs(s), s 0, (2.2.9)V(s) G(s), s 0.

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Since we are looking for the minimal super replicating cost, it is natural to

guess that V() is the minimal function satisfying (2.2.9). So we defineG(s) := inf{H(s) : H satisfies (2.2.9) } . (2.2.10)

Example.Here we compute G(s) corresponding to G(s) = (s K)+ and K =

(, b] for any b > 1. Then, G satisfiesG(s)(s) (K s)+ ,

bG(s)(s) sGs(s) 0 . (2.2.11)Assume G() is smooth we integrate the second inequality to conclude

G(s0) ( s0s1

)bG(s1), s0 s1 > 0 .

Again assuming that (2.2.11) holds on [0, s] for some s we get

G(s) = (s

s)bG(s), s s .

Assume that G(s) = G(s) for s s. Then, we have the following claim

G(s) = h(s) :=

(s K)+, s s,

( ss )b(s K)+ s < s,

where s > K the unique point for which h C1, i.e., hs(s) = Gs(s) = 1.Then,

s =K

b 1It is easy to show that h (with s as above) satisfies (2.2.9). It remains toshow that h is the minimal function satisfying (2.2.9). This will be donethrough a general formula below.

Theorem 2.2.3 Assume that G is nonnegative, continuous and grows atmost linearly. Let G be given by (2.2.10). Then,

V(s) = V(s) = G(s) . (2.2.12)

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In particular, v(t, ) converges to G uniformly on compact sets. Moreover,G(s) = sup

KeK()G(s, e),

andv(t, s) = E[G(St,s(T))|Ft],

i.e., v is the unconstrained (Black-Scholes) price of the modified claim G.

This is proved in Broadie-Cvitanic-Soner [5]. A presentation is also given inKaratzas & Shreve [15] (Chapter 5.7). A proof based on the dual formulationis simpler and is given in both of the above references.

A PDE based proof of formulation G() is also available. A complete

proof in the case of a gamma constraint is given in the next Chapter. Herewe only prove the final statement through a PDE argument. Set

u(t, s) = E[G(St,s(T)) | Ft] .Then,

ut =1

22s2uss t < T, s > 0,

andu(T, s) = G(s) .

It is clear that u is smooth. Set

w(t, s) := bu(t, s) sus(t, s) .Then, since u(T, s) = G(s) satisfies the constraints,

w(T, s) 0 s 0 .Also using the equation satisfied by u, we calculate that

wt = but sust = 12

2s2[buss] s2

2[s2uss]s

= 12

2s2[buss 2uss s2usss] = 12

2s2wss .

So by the Feynman-Kac formula,

bu(t, s) sus(t, s) = w(t, s) = E[w(t, s)|Ft] 0, t T, s 0 .

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Similarly, we can show that

au(t, s) + sus(t, s) 0, s T, s 0 .

Therefore,

F(t,s,u(t, s), ut(t, s), us(t, s), uss(t, s)) = 0, t < T, s > 0,

where F is as in the previous section. Since v, the value function, also solvesthis equation, and also v(T, s) = u(T, s) = G(s), by a comparison result forthe above equation, we can conclude that u = v, the value function.

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Chapter 3

Super-Replication with GammaConstraints

This chapter is taken from Cheridito-Soner-Touzi [7, 8], and Soner-Touzi[19]. For the brevity of the presentation, again we consider a market withonly one stock, and assume that the mean return rate = 0, by a Girsanovtransformation, and that the interest rate r = 0, by appropriately discountingall the prices. Then, the stock price S() follows

dS(t) = S(t)dW(t),

and the wealth process Z() solvesdZ(t) = Y(t)dS(t),

where the portfolio process Y() is a semi-martingaledY(t) = dA(t) + (t)dS(t) .

The control processes A(t) and () are required to satisfy(S(t)) (t) (S(t)) t, a.s. ,

for given functions and . The triplet (Y(t) = y, A(), ()) = is thecontrol. Further restriction on A() will be replaced later. Then, the minimalsuper-replicating cost v(t, s) is

v(t, s) = inf{z : A s.t. Zt,z G(St,s(T)) a.s. } .

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3.1 Pure Upper Bound Case

We consider the case with no lower bound, i.e. = . Since formallywe expect vs(t, S(t)) = Y

(t) to be the optimal portfolio, we also expect(t) := vss(t, S(t)). Hence, the gamma constraint translates into a differen-tial inequality

vss(t, s) (s) .In view of the portfolio constraint example, the expected DPE is

min{vt 12

2s2vss; s2vss + s2(s)} = 0 . (3.1.1)

We could eliminate s2

term in the second part, but we choose to write theequation this way for reasons that will be clear later.

Theorem 3.1.1 Assume G is non-negative, lower semi-continuous and grow-ing at most linearly. Then, v is a viscosity solution of (3.1.1)

We will prove the sub and super solution parts separately.

3.1.1 Super solution

Consider a test function and (t0, s0) (0, T) (0, ) so that

(v )(t0, s0) = min(v ) = 0 .Choose (tn, sn) (t0, s0) so that v(tn, sn) v(t0, s0). Further choosevn = (yn, An(), n()) so that with zn := v(tn, sn) + 1/n2,

Zntn,sn(tn +1

n) v(tn + 1

n, Stn,sn(tn +

1

n)) a.s.

The existence of n follows from the dynamic programming principle withstopping time = tn +

1n

. Set := tn +1n

, Sn() = Stn,sn(). Since v v ,Itos rule yield,

zn +

ntn

Yntn,yn(u)dS(u) (tn, sn) (3.1.2)

+

ntn

[L(u, Sn(u))du + s(u, Sn(u))dS(u)] .

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Taking the expected value implies

n

n

tn

E(L(u, Sn(u)))du n[(tn, sn) zn]

= n[ v](tn, sn) 1n

.

We could choose (tn, sn) so that n[ v](tn, sn) 0. Hence, by taking thelimit we obtain,

L(t0, s0) 0To obtain the second inequality, we return to (3.1.2) and use the Itos ruleon s and the dynamics of Y(). The result is

n +

ntn

an(u)du +

ntn

nu

bn(t)dtdS(u)

+

ntn

nu

cn(t)dS(t)dS(u) +

ntn

dndS(u) 0 ,

where

n = zn (tn, sn),an(u) = L(u, Sn(u)),bn(u) = dAn(u)

Ls(u, Sn(u)),

cn(u) = n(u) ss(u, Sn(u)),dn = yn s(tn, sn) .

We now need to define the set of admissible controls in a precise way andthen use our results on double stochastic integrals. We refer to the paperby Cheridito-Soner-Touzi for the definition of the set of admissible controls.Then, we can use the results of the next subsection to conclude that

L(t0, s0) = lim suput

an(u) 0 ,

andlim sup

utcn(u) 0 .

In particular, this implies that

ss(t0, s0) (s0) .

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Hence, v is a viscosity super solution of

min{vt 12

2s2vss; s2vss + s2(s)} 0 .

3.1.2 Subsolution

Consider a smooth and (t0, s0) so that

(v )(t0, s0) = max(v ) = 0Suppose that

min{t(t0, s0) 1

2

2

s

2

0ss(t0, s0) ; s2

0ss(t0, s0) + s

2

0(s0)} > 0 .We will obtain a contradiction to prove the subsolution property. We proceedexactly as in the constrained portfolio case using the controls

y = s(t0, s0), dA = Ls(u, S(u))du, = ss(u, S(u)) .Then, in a neighborhood of (t0, s0), Y(u) = s(u, S(u)) and () satisfies theconstraint. Since L > 0 in this neighborhood, we can proceed exactly asin the constrained portfolio problem.

3.1.3 Terminal ConditionIn this subsection, we will show that the terminal condition is satisfied with amodified function G. This is a similar result as in the portfolio constraint caseand the chief reason for modifying the terminal data is to make it compatiblewith the constraints. Assume

0 s2(s) c(s2 + 1) . (3.1.3)Theorem 3.1.2 Under the previous assumptions on G,

limt

T,s

s

v(t, s) = G(s),

where G is smallest function satisfying (i) G G (ii) Gss(s) (s). Let(s) be a smooth function satisfying

ss = (s) .

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Then,

G(s) = hconc(s) + (s) ,

h(s) := G(s) (s) ,and hconc(s) is the concave envelope of h, i.e., the smallest concave functionabove h.

Proof: Consider a sequence (tn, sn) (T, s0). Choose n A(tn) so that

Zntn,v(tn,sn)+

1n

(T) G(Stn,sn(T)) .

Take the expected value to arrive at

v(tn, sn) +1

n E(Gtn,sn(T) | Ftn) .

Hence, by the Fatous lemma,

V(s0) := lim inf (tn,sn)(T,s0)

v(tn, sn) G(s0) .

Moreover, since v is a super solution of (3.1.1), for every t

vss(t, s)

(s) in the viscosity sense .

By the stability of the viscosity property,

Vss(s) (s) in the viscosity sense .

Hence, we proved that V is a viscosity supersolution of

min{V(s) G(s); Vss(s) + (s)} 0 .

Set w(s) = V(s) (s). Then, w is a viscosity super solution of

min{

w(s)

h(s);

wss(s)}

0 .

Therefore, w is concave and w h. Hence, w hconc and consequently

V(s) = w(s) + (s) hconc(s) + (s) = G(s) .

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To prove the reverse inequality fix (t, s). For > 0 set

(t, s) := (1 )(s) + c(T t)s2/2 .

Set

(u) = ss(u, S(u)), dA(u) = Ls (u, S(u)), y0 = s (t, s) + p0,

where p0 is any point in the subdifferential of h,conc, h = G (1 ).Then,

p0(s s) + h,conc(s) h,conc(s), s .

For any z, consider Zt,z with = (y0, A(), ()). Note for t near T, A(t),

Zt,z(T) = z +

Tt

(s (t, s) + p0)dS(u)

+

Tt

[

ut

Ls (r, S(r))dr + ss(r, S(r))dS(r)]dS

= z + p0 (S(T) s) +T

t

s (u, S(u))dS(u)

z + h,conc(S(T)) h,conc(s)+(S(T), T) (t, s)

T

t

L(u, S(u))du .

We directly calculate that with c as in (3.1.3),

L(t, s) = cs2 + 12

2s2((1 )ss + c(T t))

= s2{c 12

2[(1 )(s) + +c(T t)]}

s2{c 12

2[(1 )c + c(T t)]} + 12

2c

12

2c ,

provided that t is sufficiently near T and c is sufficiently large. Hence, with

z0 = h,conc(s) + (t, s) +

1

22c(T t),

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we have

Zt,z0 h,conc(S(T)) + (S(T), T) G(S(T)) (1 )(S(T)) + (1 )(S(T))= G(S(T)) .

Therefore,

z0 = h,conc(s) + (t, s) +

1

22c(T t) v(t, s) ,

for all > 0, c large and t sufficiently close to T. Hence,

limsup(t,s)(T,s0)

v(t, s)

h,conc(s0) +

(T, s0)

> 0

= h,conc(s0) + (1 )(s0)= (G (1 ))conc(s0) + (1 )(s0)= G(s0) .

It is easy to prove thatlim0

G(s0) = G(s0) .

3.2 Double Stochastic IntegralsIn this section, we study the asymptotic behavior of certain stochastic inte-grals, as t 0. These properties were used in the derivation of the viscosityproperty of the value function. Here we only give some of the results and theproofs. A detailed discussion is given in the recent manuscript of Cheridito,Soner and Touzi.

For a predictable, bounded process b, let

Vb(t) :=

t0

u0

b()dW()dW(u) ,

h(t) := t lnln1/t .

For another predictable, bounded process m, let

M(t) :=

t0

m(u)dW(u),

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Vbm := t

0

u

0

b()dM()dM(u) .

In the easy case when b(t) = , t 0, for some constant R, we have

Vb(t) =

2

W2(t) t , t 0 .

If 0, it follows from the law of the iterated logarithm for Brownianmotion that,

limsupt0

2V(t)

h(t)= , (3.2.4)

where h(t) := 2t log log 1t

, t > 0 ,

and the equality in (3.2.4) is understood in the almost sure sense. On theother hand, if < 0, it can be deduced from the fact that almost all paths ofa one-dimensional standard Brownian motion cross zero on all time intervals(0, ], > 0, that

limsupt0

2V(t)

t= . (3.2.5)

The purpose of this section is to derive formulae similar to (3.2.4) and

(3.2.5) when b = {b(t) , t 0} a predictable matrix-valued stochastic process.Lemma 3.2.1 Let and T be two positive parameters with 2T < 1 and{b(t) , t 0} an Md-valued, F-predictable process such that |b(t)| 1, forall t 0. Then

E

exp

2Vb(T) E exp 2VId(T) .

Proof. We prove this lemma with a standard argument from the theory ofstochastic control. We define the processes

Yb

(r) := Y(0)+r0 b(u)dW(u) and Z

b

(t) := Z(0)+t0 (Y

b

(r))T

dW(r) , t 0 ,where Y(0) d and Z(0) are some given initial data. Observe thatVb(t) = Zb(t) when Y(0) = 0 and Z(0) = 0. We split the argument intothree steps.

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Step 1: It can easily be checked that

E

exp

2ZId(T)Ft = ft, YId(t), ZId(t) , (3.2.6)

where, for t [0, T], y d and z , the function f is given by

f(t,y,z) := E

exp

2

z +

Tt

(y + W(r) W(t))T dW(r)

= exp( 2z) E

exp

{2yTW(T t) + |W(T t)|2 d(T t)}= d/2 exp

2z d(T t) + 22(T t)|y|2 ,

and := [1

2(T

t)]1. Observe that the function f is strictly convex in

y and

D2yz f(t,y,z) :=2f

yz(t,y,z) = g2(t,y,z) y (3.2.7)

where g2 := 83(T t) f is a positive function of (t,y,z).Step 2: For a matrix Md, we denote by L the Dynkin operator associ-ated to the process

Yb, Zb

, that is,

L := Dt + 12

tr TD2yy +

1

2|y|2D2zz + (y)TD2yz ,

where D. and D2.. denote the gradient and the Hessian operators with respect

to the indexed variables. In this step, we intend to prove that for all t [0, T],y d and z ,

maxMd , ||1

Lf(t,y,z) = LIdf(t,y,z) = 0 . (3.2.8)

The second equality can be derived from the fact that the process

f

t, YId(t), ZId(t)

, t [0, T] ,

is a martingale, which can easily be seen from (3.2.6). The first equalityfollows from the following two observations: First, note that for each Mdsuch that || 1, the matrix Id T is in Sd+. Therefore, there exists a Sd+ such that

Id T = 2 .

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Since f is convex in y, the Hessian matrix D2yyf is also in Sd+. It follows thatD

2

yyf(t,x,y) Sd

+, and therefore,

tr[D2yy f(t,x,y)] tr[TD2yyf(t,x,y)] = tr[(Id T)D2yyf(t,x,y)]= tr[D2yy f(t,x,y)] 0 . (3.2.9)

Secondly, it follows from (3.2.7) and the Cauchy-Schwartz inequality that,for all Md such that || 1,

(y)TD2yz f(t,y,z) = g2(t,y,z)(y)Ty g2(t,y,z)|y|2

= yTD2yz f(t,y,z) . (3.2.10)

Together, (3.2.9) and (3.2.10) imply the first equality in (3.2.8).

Step 3: Let {b(t) , t 0} be an Md-valued, F-predictable process such that|b(t)| 1 for all t 0. We define the sequence of stopping times

k := T inf

t 0 : |Yb(t)| + |Zb(t)| k , k N .It follows from Itos lemma and (3.2.8) that

f(0, Y(0), Z(0)) = f

k, Y

b(k), Zb(k)

k0

Lb(t)f

t, Yb(t), Zb(t)

dt

k0

[(Dyf)Tb + (Dzf)yT]

t, Yb(t), Zb(t)

dW(t)

fk, Yb(k), Zb(k)k0

[(Dyf)Tb + (Dzf)y

T]

t, Yb(t), Zb(t)

dW(t) .

Taking expected values and sending k to infinity, we get by Fatous lemma,

E

exp

2ZId(T)

= f(0, Y(0), Z(0))

lim infk

E

f

k, Y

b(k), Zb(k) E fT, Yb(T), Zb(T) = E exp 2Zb(T) ,

which proves the lemma.

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Theorem 3.2.2

a) Let{b(t) , t 0} be anMd

-valued,F

-predictable process such that|b(t)| 1 for all t 0. Thenlimsup

t0

|2Vb(t)|h(t)

1 .

b) Let Sd with largest eigenvalue () 0. If {b(t) , t 0} is abounded, Sd-valued, F-predictable process such that b(t) for all t 0,then

limsupt0

2Vb(t)

h(t) () .

Proof.

a) Let T > 0 and > 0 be such that 2T < 1. It follows from Doobsmaximal inequality for submartingales and Lemma 3.2.1 that for all 0,

P

sup

0tT2Vb(t)

= P

sup

0tTexp(2Vb(t)) exp()

exp() E exp 2Vb(T) exp() E exp 2VId(T)= exp ()exp(dT) (1 2T) d2 .

(3.2.11)

Now, take ,

(0, 1), and set for all k

N,

k := (1 + )2h(k) and k := [2

k(1 + )]1 .

It follows from (3.2.11) that for all k N,

P

sup

0tk2Vb(t) (1 + )2h(k)

exp

d

2(1 + )

1 +

1

d/2(k log

1

)(1+) .

Since

k=1 k(1+) < ,

it follows from the Borel-Cantelli lemma that, for almost all , thereexists a natural number K, () such that for all k K, (),

sup0tk

2Vb(t, ) < (1 + )2h(k) .

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In particular, for all t (k+1, k],

2Vb(t, ) < (1 + )2h(k) (1 + )2 h(t)

.

Hence,

lim supt0

2Vb(t)

h(t) (1 + )

2

.

By letting tend to one and to zero along the rationals, we conclude that

lim supt0

2Vb(t)

h(t) 1 .

On the other hand,

lim inft0

2Vb(t)

h(t)= lim sup

t0

2Vb(t)h(t)

1 ,

and the proof of part a) is complete.b) There exists a constant c > 0 such that for all t 0,

cId b(t) cId , (3.2.12)and there exists an orthogonal d d-matrix U such that

:= UUT

= diag[(), 2, . . . , d] ,

where () 2 d are the ordered eigenvalues of the matrix .Let

:= diag[3c , c , . . . , c] and := UTU .

It follows from (3.2.12) that for all t 0, b(t) 0 ,

which implies that

| b(t)| | | = ( ) = () = 3c () .

Hence, by part a),

limsupt0

2Vb(t)h(t)

3c () . (3.2.13)

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Note that the transformed Brownian motion,

W(t) := UW(t) , t 0 ,is again a d-dimensional standard Brownian motion and

lim supt0

2V(t)

h(t)= lim sup

t0

W(t)TW(t) tr()th(t)

= lim supt0

W(t)TW(t) tr()th(t)

= lim supt0

W(t)TW(t)

h(t)

limsupt0

3c(W1(t))2

h(t)= 3c .

(3.2.14)

It follows from (3.2.14) and (3.2.13) that

lim supt0

2Vb(t)

h(t) limsup

t0

2V(t)

h(t)limsup

t0

2Vb(t)h(t)

3c(3c()) = () ,

which proves part b) of the theorem.

Using the above results one can prove the following lower bound for Vbm.

Theorem 3.2.3 Assume that

supu

E|m(u) m(0)|4u4

.

Then,

lim supt0

Vbm(t)

h(t) a ,

for every b L satisfying m(0)b(0)m(u) a > 0.Theorem 3.2.4 Suppose that c L and E|m(u)|4 c. Then, for all > 0,

lim supt0

| t0 (u0 a()d)dM(u)|

t3/2+= 0 .

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Proofs of these results can be found in Cheridito-Soner-Touzi.

Lemma 3.2.5 Suppose that b L and for some > 0

supu0

E|b(u) b(0)|2u2

< .

Then,

liminft0

Vb(t)

t= 1

2b(0) .

Proof: Set b(u) := b(u) b(0). Then, E|b(u)|2/u2 < and

Vb(t) = 12

b(0)W2(t) 12

b(0)t + Vb(t) .

It can be shown that |Vb(t)| = 0(t). Hence,

liminft0

Vb(t)

t= 1

2b(0) + lim inf

t01

2b(0)

W2(t)

t= 1

2b(0) .

3.3 General Gamma Constraint

Here we consider the constraint

(S(u)) (u) (S(u)) .Let

H(vt, vs, vss, s) = min{vt 12

2s2vss; vss + (s), vss (s)} ,

and H be the parabolic majorant of H, i.e.,

H(vt, vs, vss, s) := sup{H(vt, vs, vss + Q, s) : Q 0} .Theorem 3.3.1 v is a viscosity solution of

H(vt, vs, vss, s) = 0, (t, s) (0, T) (0, )

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The super solution property is already proved. The subsolution property is

proved almost exactly as in the upper gamma constraint case; cf [Cheridito-Soner-Touzi].Terminal condition is the same in the pure upper bound case. As before

G(s) = (G )conc(s) + (s) ,

where satisfies, ss(s) = (s). Note that the lower gamma bound does

not effect G.

Proposition 3.3.2lim

(t,s)(T,s)v(t, s) = G(s)

Proof is exactly as in the upper gamma bound case.

Theorem 3.3.3 Assume 0 G(s) c[1 + s]. Then, v is the unique viscos-ity solution of H = 0 together with the boundary condition v(T, s) = G(s).

Proof is given in Cheridito-Soner-Touzi.

Example. Consider the problem with (s) 0, (s) +, G(s) = s 1.For s 1, z = s, y0 = 1, A 0, 0 yield,

Y(u) 1, Z

t,s(u) = s +u

t dS() = St,s(u) .

Hence, Zt,s(T) = St,s(T) St,s(T) 1 and for s < 1, v(t, s) s. For s 1,let z = 1, y0 = 0, A 0, 0. Then,

Y(u) 0, Zt,1(u) = 1 ,

and Zt,1(T) = 1 St,s(T) 1. So for s 1 v(t, s) 1.Hence, v(t, s) G(s). We can check that

H(Gt, Gs, Gss) = supQ0{

1

2

2s2(Gss + Q); (Gss + Q)

}= 0 .

By the uniqueness, we conclude that v = G, and the buy and hold strategydescribed earlier is optimal.

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3.4 Examples

In some cases it is possible to compute the solution. In this section, we giveseveral examples to illustrate the theory.

3.4.1 European Call Option

G(s) = (s K)+ .(a). Let us first consider the portfolio constraint:

Lv svs Uv .where (U 1) is the fraction of our wealth we are allowed to borrow (i.e.,shortsell the money market) and L is the shortsell constraint. According toour results,

v(t, s) = E[G(St,s(T))] ,

where G is the smallest function satisfying i)G G, ii) LG sGs UG.By observation, we see that only the upper bound is relevant and that thereexists s > K so that

G(s) = G(s) = (s K) s s

Moreover, for s s the constraint sG(s) = UG(s) saturates, i.e.sG(s) = UG(s) s s

HenceG(s) = (

s

s)UG(s) = (s K)( s

s)U

Now we compute s, by imposing that G C1. So

Gs(s) = Gs(s) = 1 U(s K)( 1

s) = 1 .

The result is

s = U(v 1) K .

Therefore,

G(s) =

(s K), if s s

( U1K )U1UUsU, if 0 s s .

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Set,

Gpremium(s) :=

G(s) G(s) .Then,

v(t, s) = vBS(t, s) + vpr(t, s) ,

where vBS(t, s) := EG(t, s) is the Black Scholes price, and

vpr(t, s) = E[G(s) G(S(T))]Moreover, vpr can be explicitly calculated (and its derivative) in terms of theerror function, as in the Black-Scholes Case.(b). Now consider the Gamma constraint

v

s2vss

v .

Again, since G is convex, the lower bound is irrelevant and the modified Gis given as in part (a) with

U(U 1) = U = 1 +

1 + 4

2.

So, it is interesting to note that the minimal super-replication cost of bothconstraints agree provided that U and are related as above.

3.4.2 European Put Option:

G(s) = (K s)+ .(a). Again let us consider the portfolio constraint first. In this case, since Gis decreasing, only lower bound is active. We compute as in the Call OptionCase:

G(s) =

(K s), if 0 s s ,

LL( KL+1)L+1sL, if s s ,

where

s =L

L + 1K .

(b). Consider the gamma constraint

v s2vss v .Again lower bound inactive, and the solution is as in the portfolio case with

L = ( + 1) .

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3.4.3 European Digital Option

G(s) =

1, s K ,0, s K .

(a). Again, we first consider the portfolio constraint. It can be verified that

G(s) =

( s

K)U, 0 s K ,

1, s K .If we split the price as before:

v(t, s) = vBS(t, s) + vpr(t, s) .

We have

vBS(t, s) = P(St,s(T) 1) = N(d2) := 12

d2

ex2/2dx ,

d2 =ln( s

k) 1

22(T t)

T t ,

vpr(t, s) =12

d2(

s

K)UeU[

Ttx 1

22(Tt)]e

12

x2dx

= (s

K)Ue

u(u1)2

2(Tt)[1

2 N(d2)] .

We can also compute the corresponding deltas.(b). Once again, the gamma constraint

s2vss vyields the same G as in the portfolio constraint with

= U(U

1)

U =

1 +

1 + 4

2

.

If there is no lower gamma bound, v = vBS + vpr.(c). Suppose now that there is no upper bound,

s2vss v .

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Then the dynamic programming equation is

supQ0

min{vt 2

2(s2vss + Q); (s

2vss + Q) + v} = 0 s > 0, t < T .

v(T, s) = G(s) .

By calculus, we see that this is equivalent to

vt 2

2(s2vss + v)+ +

2

2v = 0 .

Since, for any real number ,

()+ = sup0a1

{ a2 } ,

vt + inf0a1

{2a2

2s2vss +

2

(1 a2)v} = 0 .We recognize the above equation as the dynamic programming equation forthe following stochastic optimal control problem:

v(t, s) = sup0a()1

E[exp(T

t

22

(1 a2(u))du)G(Sat,s(T))] ,

wheredSa(u) = a(u)Sa(u)dW(u) .

3.4.4 Up and Out European Call Option

This is a path-dependent option

G(St,s()) =

(St,s(T) K)+, if maxtuT St,s(u) := M(T) < B ,0, M(T) B .

We can also handle this with PDE techniques. Consider the bound

v s2vss v .

The upper bound describes the modified final data G:

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(i) If

s := UU 1K < B (with U = 1 + 1 + 42 ) ,then G is as in the usual Call Option case.

(ii) If, however, s B, then

G(s) = (K B)( sB

)U .

We claim that at the lateral boundary s = B, the lower gamma boundsaturates and

s

2

vss(t, B) = v(t, B) .Using the equation, we formally guess that

vt(t, B) = 2

2s2vss(t, B) =

2

2v(t, B) .

We solve this ODE to obtain

v(t, B) = (K B) exp( 2

2(T t)) t T . (3.4.15)

Lemma 3.4.1 The minimal super-replicating cost for the up and out call

option is the unique (smooth) solution of

vt 2

2s2vss = 0, t < T, 0 < s < B ,

v(T, s) = G(s), 0 s B .together with (3.4.15). In particular,

v(t, s) = E{G(St,s(T)){t,sT} + (K B)e

2

2(tt,s){t,s


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Proof: It is clear that

s2vss(t, B) = v(t, B), and s2vss(T, s) 0 .

Setw(t, s) := s2vss(t, s) + v(t, s) .

Then,

wt 2

2s2wss = 0, t < T, 0 < s < B ,

w(T, s) > 0 ,

w(t, B) = 0 .

Hence, w 0. Similarly setz(t, s) := s2vss(t, s) v(t, s) .

Then,z(T, s) , z(t, B) < 0 .

Also,

zt 2

2s2zss = 0 .

Hence, z

0. Therefore, v solves H(vt, s

2vss, v) = 0. So by the uniqueness

v is the super-replication cost.

3.5 Guess for The Dual Formulation

As it was done for the portfolio constraint, using duality is another possibleapproach to super-replication is also available. We refer to the lecture notesof Rogers [16] for this method and the relevant references. However, the dualapproach has not yet been successfully applied to the gamma problem. Here

we describe a possible dual problem based on the results obtained throughdynamic programming.

Let us first consider the upper bound case

s2vss .

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Then the dynamic programming equation is

min{vt 22

s2vss; s2vss + } = 0 .We rewrite this as

vt + infb1

{2

2b2s2vss +

2

2(b2 1)} = 0 .

The above equation is the dynamic programming equation of the followingoptimal control problem,

v(t, s) := supb(

)

1

E 2

2

T

t

(b2(u) 1)du + G(Sbt,s(T)) ,dSbt,s(u) = b(u)S

bt,s(u)dW(u) .

Note the change in the diffusion coefficient of the stock price process.If we consider,

s2vss ,same argument yields

v(t, s) = sup0a()1

E

2

2

Tt

(1 a2(u))du + G(Sat,s(T))

.

Now consider the full constraint,

s2vss .The equation is

supQ0

min{vt 2

2(s2vss + Q); (s2vss + Q) + ; (s2vss + Q) + } = 0 .

We rewrite it as

vt + inf b1a0

{2

2s2a2b2vss +

2

2(b2 1) +

2

2(1 a2)} = 0 .

Hence,

v(t, s) = supb()1,0a()1

E

2

2

Tt

[(b(u)2 1) + (1 a2(u))]du

+G(Sa,bt,s (T))

,

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dSa,b(u) = a(u)b(u)Sa,b(u)dW(u) .

It is open to prove this by direct convex analysis methods. We finish byobserving that if

v s2vss v ,then

v(t, s) = supb()1,0a()1

E

eTt2

2[(b(u)21)+(1a2(u))] du G(Sa,bt,s (T))

.

All approaches to duality (see Rogers lecture notes [16]) yield expressions ofthe form

v(t, s) = sup E[ B Y ] ,

where B = G(S(T)) in our examples. However, in above examples S() needsto be modified in a way that is not absolutely continuous with respect to P.

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