A closed-form solution to the problem of super-replication under

A closed-form solution to the problem of super-replication under transaction costs  Jaksa Cvitanic

Huy^en Pham

Nizar Touzi

Department of Statistics Columbia University New York, NY 10027 [email protected]

Equipe d'Analyse et de Mathematiques Appliquees Universite Marne-la-Vallee and CREST [email protected]

CEREMADE Universite Paris Dauphine and CREST [email protected]

October 13, 1997

Abstract We study the problem of nding the minimal price needed to dominate Europeantype contingent claims under proportional transaction costs in a continuous-time diusion model. The result we prove has already been known in special cases - the minimal super-replicating strategy is the least expensive buy-and-hold strategy. Our contribution consists in showing that this result remains valid for general path-independent claims, and in providing a shorter and more intuitive, nancial mathematics-type proof. It is based on a previously known representation of the minimal price as a supremum of the prices in corresponding shadow markets, and on a PDE (viscosity) characterization of that representation.

Key words: transaction costs, super-replicating strategies, viscosity solutions. JEL classi cation: G11, G12. AMS 1991 subject classi cations: Primary 90A09, 93E20, 60H30; secondary 60G44, 90A16.  Research

of the rst author partially supported by NSF grant #DMS-95-03582.

1

1 Introduction Davis and Clark (1994) conjectured that the minimal initial wealth needed to super-replicate a European Call option in a Black-Scholes market with proportional transaction costs is just the price of one share of the underlying stock; in other words, that the minimal dominating strategy is the trivial, buy-and-hold strategy - buy a share, and hold it until the maturity time of the option. This is, of course, in sharp contrast with the classical, no-frictions market models. The conjecture was proved by \brute force" approaches in Soner, Shreve and Cvitanic (1995) by analytic methods, and (independently, and for more general models and claims) in Levental and Skorohod (1995) by probabilistic methods. Both proofs are long and \ad hoc", namely very dierent in spirit from more standard approaches in nancial mathematics. Subsequently, Cvitanic and Karatzas (1996) nd a stochastic control representation for the minimal price of a claim under transaction costs, in a general diusion model, and for general European claims. It is given as a supremum of expected discounted values of the claim, over all (pairs of) probability measures under which the \discounted wealth process" is a supermartingale. In nancial jargon, it is given as a supremum of the prices of the claim in the \shadow markets" corresponding to the incompleteness caused by the presence of transaction costs. Representations of the same type have also been obtained in Jouini and Kallal (1995), Kusuoka (1995) and Koehl, Pham and Touzi (1996), in somewhat dierent models. However, Cvitanic and Karatzas (1996) leave open the problem of calculating explicitly the value of the stochastic control problem. We solve the problem in a Markovian model, proving that the buy-and-hold strategy is, indeed, the minimal dominating strategy. Equivalently, and roughly speaking, we show that the price function of a contingent claim is given as a concave envelope of an appropriately modi ed claim, thereby proving a conjecture by Avellaneda (1996). Our proof is easy to follow, not very long, and uses a \natural", PDE-martingale-stochastic control approach to calculate the price function. In particular, it should be mentioned that the methods of viscosity solutions theory that we apply, turn out to be a very powerful and a very simple to use tool in this particular problem. The approach also enables us to give another interpretation to the well known connection between hedging with a modi ed volatility and hedging under transaction costs (see Leland (1985), Avellaneda and Paras (1994), Barles and Soner (1995)). In fact, we were motivated to use our approach after having applied it successfully to a similar problem in a stochastic volatility framework in Cvitanic, Pham and Touzi (1997). Although the type of results we get are negative and theoretical, in the sense that the minimal super-replication price is too high to be used as the actual price of a claim, it is known that the knowledge of super-replicating strategies can be helpful in solving the utility maximization problems with transaction costs, as well as in nding a more realistic, 2

\utility-based" price of the claim (see, for example, Davis, Panas and Zariphopoulou (1993), Cvitanic and Karatzas (1996)). We plan to explore this connection in more detail in future work. We present the model in Section 2. In Section 3 we get the lower bound for the superreplication cost in terms of the above mentioned stochastic control problem. Then, in Section 4, we nd a variational inequality of which the value function corresponding to the problem is a viscosity supersolution. From this we get the crucial properties of the value function concavity in the price of the underlying stock, and monotonicity in the time variable. Finally, in Section 5, we show that those properties imply that the minimal super-replicating strategy is the minimal buy-and-hold strategy.

2 The model We consider a nancial market which consists of one bank account, with constant price process S (t) = 1 for all t 2 [0; T ], and one risky asset with price process evolving according to the following stochastic dierential equation : dS (t) = (t; S (t))dt + (t; S (t))dW (t) (2.1) S (t) 0

Here W = fW (t); 0  t  T g is a standard one-dimensional Brownian motion de ned on a complete probability space ( ; F ; P ). We shall denote by IF = fF (t), 0  t  T g the P augmentation of the ltration generated by W . The assumption that the interest rate of the bank account is zero could, as usual, easily be dispensed with, by discounting. Throughout this paper, we make the standing assumptions on the coecients of the last SDE in order to ensure the existence of a unique strong solution fSt, 0  t  T g, given an initial condition : all coecients are continuous in (t; s) and satisfy for all (t; s; s0) 2 [0; T ]  (IR ) , +

2

js(t; s) ? s0(t; s0)j + js(t; s) ? s0(t; s0)j + j? (t; s) ? ? (t; s0)j  C js ? s0j (2.2) 1

1

for some positive constant C . Notice that we also require Lipschitz property on ? , which will be needed for technical reasons. We also assume the uniform ellipticity condition : 1

9" > 0; 8(t; s) 2 [0; T ]  IR ;

(t; s)  ":

+

(2.3)

A trading strategy is a pair L = (L ; L ) of IF -adapted processes on [0; T ], with left continuous, nondecreasing paths and L(0) = 0. L (t) (resp. L (t)) represents the cumulative amount of funds transferred from bank account to stock (resp. from stock to bank account) up to time t. Given proportional transaction costs  2 (0; +1) and  2 (0; 1), for such 0

1

0

1

0

3

1

transfers, and initial holdings x = (x ; x ) in bank and stock respectively, the portfolio holdings (X x;L; X x;L) corresponding to a given strategy L = (L ; L ) evolve according to the equations : 0

0

1

0

1

1

X x;L(t) = x ? (1 +  )L (t) + (1 ?  )L (t) Z t x;L dS (u) x;L X (t) = x + L (t) ? L (t) + X (u) S (u) : 0

0

1

1

0

0

0

1

1

(2.4) (2.5)

1

1

0

De nition 2.1 A contingent claim is a pair (g (S (T )); g (S (T ))) where g and g are functions

mapping IR

+

0

into IR.

1

0

1

Here g (S (T )) (resp. g (S (T ))) is understood as a target position in the bank account (resp. the stock) at the terminal time T . We shall identify a contingent claim with its payo function (g ; g ). The most popular example is the European call option with strike price  > 0, de ned by the payo functions 0

1

0

1

g (s) = ?1fsg and g (s) = s1fsg: 0

1

We say that a trading strategy L = (L ; L ) dominates the contingent claim (g ; g ) starting with x = (x ; x ) as initial holdings if : 0

0

1

0

1

1

X x;L(T ) + (1 ?  )X x;L(T )  g (S (T )) + (1 ?  )g (S (T )) X x;L(T ) + (1 +  )X x;L(T )  g (S (T )) + (1 +  )g (S (T )): 0

1

1

0

1

1

0

0

1

0

0

1

(2.6) (2.7)

Hedging in the sense of (2.6)-(2.7) simply means that one is able to cover the positions (g (S (T )); g (S (T ))) at the terminal date T , net of transaction costs. In this paper, we provide an explicit solution to the super-replication cost of a given contingent claim, i.e., nd the least amount x such that there exists an admissible (in a sense to be made precise later) strategy L which dominates the contingent claim starting from x = (x ; 0). 0

1

0

0

3 The expectation representation formula for the super-replication price It is well-known that the super-replication price of a given contingent claim can be expressed as the supremum of expected values of the value of the claim under all \feasible" equivalent martingale measures (see Jouini and Kallal (1995), Kusuoka (1995), Cvitanic and Karatzas (1996)). In this section, we recall the framework of Cvitanic and Karatzas (1996) and their expectation representation formula, which is the starting point of our analysis. 4

Consider the class P consisting of all pairs (Q; ) where : (P1) Q is a probability measure equivalent to P (P2)  = f(t); 0  t  T g is a IF -adapted process satisfying : 1 ?   (t)  1 +  ; 0  t  T; a:s: 1

(3.1)

0

(P3) The process f(t)S (t); 0  t  T g is a Q-martingale. ^ r) Remark 3.1 In this diusion model, P 6= ; since P contains at least all the pairs (P; where r is a constant in [1 ?  ; 1 +  ] and : 1

0

! dP^ = exp ? Z T  (u; S (u))dW (u) ? 1 Z T  (u; S (u))du : dP  2  We also denote by P1 the subset of P whose elements (Q; ) are such that : dQ is essentially bounded: dP^ De nition 3.2 A trading strategy L is said to be admissible for the initial holdings x, and we write L 2 A(x), if the process 2

0

2

0

X x;L(t) + (t)X x;L(t); 0  t  T; 0

1

is a Q-supermartingale for all (Q; ) 2 P1 .

Remark 3.2 The condition on a strategy to be admissible in De nition 3.2 is note very

restrictive. In particular, it is weaker than the usual \no-bankruptcy" condition, i.e., the set of admissible strategies is larger. Indeed, as in Cvitanic and Karatzas (1996), by Ito's rule, and for some adapted process , we get

d(X x;L(t) + (t)X x;L(t)) = (t)dW Q(t) ? [(1 +  ) ? (t)]dL (t) ? [(t) ? (1 ?  )]dL (t); 0

0

1

0

1

1

where W Q is a Brownian motion under Q. In particular, if the left-hand side process is nonnegative (no-bankruptcy condition), then it is also a Q-supermartingale, because of condition (P2). Now, let (g ; g ) be a contingent claim and de ne its super-replication cost, by : 0

1

U = inf fx 2 IR : 9L 2 A(x ; 0); L dominates (g ; g )g : 0

0

0

1

(3.2)

It will also be convenient to denote by U (t; s) the minimal super-replication cost U , if the initial (present) time is t and the initial stock-price is S (t) = s. We have the following expectation representation formula. 5

Theorem 3.1 (Cvitanic and Karatzas (1996)). Suppose that

g + min((1 +  )g ; (1 ?  )g )  ?C; for some positive constant C and i h E P g (S (T )) + g (S (T )) < +1: 0

0

^

1

2

0

1

(3.3)

1

(3.4)

2

1

Then, we have :

U = sup E Q [g (S (T )) + (T )g (S (T ))] : 0

Q;)2P

(

(3.5)

1

Remark 3.3 We actually need only the fact that U is greater or equal to the above quantity. That inequality is quite easy to prove directly from the de nition of P , and the fact that the supremum in (3.5) remains the same if we replace P by P1. Moreover, for that inequality we do not need condition (3.4); see Cvitanic and Karatzas (1996).

As mentioned in Section 1, in the case of an European call option, Davis and Clark (1994) conjectured that the optimal dominating strategy is the trivial buy and hold strategy, and then the super-replication cost is (1 +  )S (0). This conjecture was proved by Soner, Shreve and Cvitanic (1995), as well as Levental and Skorohod (1995) using directly the variational formulation (3.2). However, Cvitanic and Karatzas (1996) left open the problem of deriving this result from the expectation representation formula (3.5). In this paper, we prove an extension of this conjecture to general path-independent contingent claims, by using the expectation representation formula (3.5). Namely, we prove, roughly speaking, that the super-replication cost is given by the concave envelope of a suitably de ned payo function. The proof relies on the usual Bellman equation of the stochastic control problem and turns out to be conceptually much simpler than those of above mentioned papers. 0

4 An auxiliary control problem In this section we assume that condition (3.3) of Theorem 3.1 is satis ed, and that the right-hand side of (3.5) is nite. Consider a strictly increasing C -function f de ned on IR, with bounded rst, second, and third derivatives, such that : 3

1 ?   f (y)  1 +  ; y 2 IR; lim f (y) = 1 ?  and y!lim1 f (y) = 1 +  : y!?1 1

0

1

+

6

0

For example, one can check easily that the function : y e?y f (y) = 1 +  ?2  +  +2  eey ? + e?y ; y 2 IR satis es the above conditions. Next, let Q be a probability measure equivalent to P characterized by its Radon Nykodim density dQ dP = Z (0; T ), where 0

0

1

1

!

ZT 1 Z (t; T ) := exp ? t (u)dW (u) ? 2 t (u) du (4.6) and consider two bounded real-valued progressively measurable processes a = fa(t), 0  t  T g and b = fb(t), 0  t  T g. We shall denote by D the set of all such pair of processes (a; b). We then introduce the controlled process Y a;b = fY a;b (t), 0  t  T g de ned by its dynamics under Q by : ZT

2

dY a;b (t) = b(t)dt + a(t)dW  (t) where W  , de ned by

W  (t)

= W (t) +

Zt 0

(4.7)

(u)du 0  t  T;

is a Brownian motion under probability measure Q = Q , by  Girsanov  theorem. a;b We now derive conditions on  in order for the pair Q; f (Y ) to be in the set P de ned in the previous section. In order to do this, we de ne the adjusted process :

X a;b(t) = f (Y a;b (t))S (t); 0  t  T:

(4.8)

By It^o's lemma, we see that the dynamics of X a;b are described by :

h i dX a;b(t) = (t; S (t))X a;b(t) ?(t) + a;b(t) dt " 0 (Y a;b ) # f a;b +X (t) (t; S (t)) + a(t) a;b dW  (t); f (Y )

where, suppressing the dependence on t, a;b (t; S )]f 0(Y a;b ) + a f 00(Y a;b ) a;b(t; S; Y a;b) = (t; S )f (Y ) + [b +(a 0  t  T:(4.9) t; S )f (Y a;b) We notice that we can now take  = a;b in (4.6), since it is easily checked that a;b as a (random) function of Y a;b is Lipschitz, so that there is a unique solution to (4.7) with  = a;b, because (4.7) can then be written as 1 2

2

dY a;b(t) = [b(t) + a;b (t; S (t); Y a;b(t))]dt + a(t)dW (t) 7

(see, for example, Protter (1990), Fleming and Soner (1993)). Moreover, since , ? , f , f ? , f 0, f 00 and (a; b) are assumed to be bounded, so is the process a;b. Therefore, taking  = a;b in (4.6), we can de ne a probability measure Qa;b equivalent to P by : ! dQa;b = exp ? Z T a;b(u)dW (u) ? 1 Z T a;b (u) du : dP 2 We shall denote by W a;b the Brownian motion under Qa;b de ned by : 1

1

2

0

0

W a;b(t) = W (t) +

Zt 0

a;b (u)du; 0  t  T:

Notice also that the diusion coecient of the adjusted process X a;b is bounded. This proves that X a;b is a martingale under the probability measure Qa;b and therefore :



 Qa;b ; f (Y a;b ) 2 P :

The consequence of this, by Theorem 3.1 and Remark 3.3, is

Proposition 4.1 U (t; s)  sup V (t; s; y); y where

V (t; s; y) := sup

a;b)2D

(

E Qa;b



g

0

(S (T )) + f (Y a;b(T ))g

1

(4.10)

   (S (T )) S (t); Y a;b(t) = (s; y) : (4.11)

In the following, we shall study the dynamic problem associated with the right-hand side term of the last proposition, in particular its value function V (t; s; y). We will provide some properties of V via the associated Bellman equation which are sucient to determine V explicitly. We will then show that the inequality in Proposition 4.1 is in fact an equality by noting that supy V (t; s; y) is the initial wealth of a dominating admissible strategy L.

5 Necessary conditions from the Bellman equation From the de nition of the adjusted process X a;b = f (Y a;b )S , we have V (t; s; y) = V (t; sf (y); y)

(5.1)

where V is the value function for the control problem : ! " a;b V (t; x; y) = sup E Qa;b g X a;b(T ) (5.2) f (Y (T )) a;b 2D # a;b (T ) ! X a;b a;b a;b +f (Y (T ))g f (Y a;b (T )) (X (t); Y (t)) = (x; y) (

)

0

1

8

for all (x; y) 2 IR  IR. Recall from the previous section that the controlled system (X a;b; Y a;b ) is governed by : " 0 (Y a;b (t)) # f a;b a;b a;b a;b dX (t) = X (t)  (t; X (t); Y (t)) + a(t) f (Y a;b (t)) dW a;b(t) dY a;b(t) = b(t)dt + a(t)dW a;b(t) +

where  (t; x; y) = (t; x=f (y)). Consider the lower semicontinuous envelope V of V , i.e., the largest lower semicontinuous function which is smaller than V , given by (5.3) V(t; x; y) = lim inf V (t0 ; x0; y0): t ;x0 ;y0 )!(t;x;y)

( 0

We then have the following property of function V (we use the de nition of viscosity solutions from Crandall, Ishii and Lions (1992); see also Fleming and Soner (1993)). Proposition 5.1 Assume that condition (3.3) holds. Then, function V(t; x; y) is a lower semicontinuous viscosity supersolution in [0; T )  IR  IR to the Bellman equation :

h

inf

a;b)2IR2

(

where

+

i

?La v ? G a;bv = 0; !

(5.4)

0 = vt + 1 x  (t; x; y) + a f (y) vxx 2 f (y) 0 (y ) ! 1 f G a;bv = bvy + 2 a vyy + ax  (t; x; y) + a f (y) vxy :

La v

2

2

2

Moreover,

"

!

!#

V(T ?; x; y)  g f (xy) + f (y)g f (xy) ; (5.5)  where [
] on the right-hand side denotes the lower semicontinuous envelope with respect to variables (x; y). 0

1

We rst prove two helpful lemmas.

Lemma 5.1 Let (tn; xn; yn) ! (t; x; y). Then, for every T  u  t and (a; b) 2 D, there exists a (relabeled) subsequence such that

a;b a;b (u); a:s: ; Y a;b Xta;b;x ;y (u) ! Xt;x;y t ;x ;y (u) ! Yt;x;y (u); a:s:; n

n

n

n

n

n

where Xt;x;y ; Yt;x;y is the notation for processes X; Y for which the initial condition is X (t) = x; Y (t) = y.

9

t a;b 0 Proof: De ne  (t) :=  t;S t;S t . Then, the dynamics of S; Y under Q are given by ( (

0

( )) ( ))

dS (t) = S (t)(t; S (t))dW 0 (t); dY a;b(t) = F a;b (t; Y a;b(t); S (t))dt + adW 0 (t); where

(5.6) (5.7)

0 00 F a;b(t; Y; S ) := b + a ff ((YY )) [b=(t; S ) + a] + 2a(t;fS ()Yf ()Y ) : 3

Because a; b; f; f 0; f 00; f 000; f ? are bounded and ? (t; s) is uniformly Lipschitz in s (see (2.2)) and bounded, it is easily checked that F a;b(t; Y; S ) is a uniformly Lipschitz (random) function of (Y; S ). Therefore, by standard estimates, using dynamics (5.6), (5.7), It^o's rule and Gronwall inequality, we get, for a given u  t, 1

1

i 0h a;b (u)j + jS ! 0; ( u ) ? S ( u ) j E Q jYta;b;s ;y (u) ? Yt;s;y t;s;y t ;s ;y 

n n

2

n

n n

2

n

as (tn; sn; yn) ! (t; s; y). But then there is a relabeled sequence such that (Stn ;sn;yn ; Yta;b n ;sn ;y n )(u) a;b converges a.s. to (St;s;y ; Yt;s;y )(u). Consequently, the analogous statement is valid for a;b a;b = f (Y a;b )S . (Xta;b n ;xn ;y n ; Ytn ;xn ;y n )(u), with X

2

Lemma 5.2 (Dynamic Programming Principle) Under condition (3.3), we have, for every 0 < t < u < T , x > 0 and every y,    V (t; x; y)  sup E Qa;b V u; X a;b(u); Y a;b (u) D

 (X a;b; Y a;b)(t) = (x; y) :

(5.8)

Proof: We rst establish the above property for V , rather than V. This result is standard in stochastic control theory but we provide a proof for completeness: Let us simplify the a;b . Moreover, denote notation by denoting the above expectation operator by Et;x;y a;b J a;b (t; x; y) := Et;x;y

"

a;b (T ) a;b (T ) # X X a;b g ( f (Y a;b (T )) ) + f (Y (T ))g ( f (Y a;b (T )) )  ?C: 0

1

Fix an arbitrary (a0; b0 ) 2 D and notice that it is sucient to show a0 ;b0 [V (u; X a0 ;b0 (u); Y a0 ;b0 (u))]: V (t; x; y)  Et;x;y

(5.9) (5.10)

Denote g(x; y) = g (x=f (y)) + f (y)g (x=f (y)). We rst notice, by Bayes rule for conditional expectations (see Karatzas and Shreve (1991)) and Markov property, recalling notation (4.6), 0

1

a;b (u; X a0;b0 (u); Y a0 ;b0 (u)) J (5.11)   = E Z  (u; T )g(X a;b(T ); Y a;b(T )) (X a;b(u); Y a;b (u)) = (X a0 ;b0 (u); Y a0 ;b0 (u)) ; (5.12) a;b

10

and therefore that the left-hand side, besides depending on (u; X a0;b0 (u); Y a0 ;b0 (u)), depends only on the values of (a; b) on [u; T ). Therefore, the same is valid for V (u; X a0;b0 (u); Y a0 ;b0 (u)). Next, let (a; b) be an arbitrary pair in D that agrees with (a0 ; b0) on [t; u]. Denote the set of all such (a; b) by Mt;u. We have then, by Markov property, a;b [g (X a;b (T ); Y a;b (T ))] V (t; x; y)  Et;x;y = = = = =





a;b E a;b g (X a;b (T ); Y a;b (T )) (X a;b (u); Y a;b (u)) Et;x;y t;x;y a;b [J a;b (u; X a;b(u); Y a;b (u))] Et;x;y h i Et;x;y Z a;b (t; u)J a;b(u; X a;b(u); Y a;b (u))  a0;b0   a;b a;b a;b Et;x;y Z (t; u)J (u; X (u); Y (u)) a0 ;b0 [J a;b (u; X a0 ;b0 (u); Y a0 ;b0 (u))]: : : : = Et;x;y



(5.13)

Moreover, the family of random variables fJ a;b(u; X a0;b0 (u); Y a0 ;b0 (u))g a;b 2Mt;u is directed upward; namely, for any (a; b) 2 Mt;u , (c; d) 2 Mt;u and with (

)

A = f! ; J a;b (u; X a0;b0 (u); Y a0 ;b0 (u))  J c;d(u; X a0;b0 (u); Y a0 ;b0 (u))g; the process (e; f ) de ned by (e; f ) := (a; b)1A + (c; d)1Ac belongs to Mt;u and we have a.s.

J e;f (u; X a0;b0 (u); Y a0 ;b0 (u)) = maxfJ a;b(u; X a0;b0 (u); Y a0 ;b0 (u)); J c;d(u; X a0;b0;(u); Y a0 ;b0 (u))g: Then from Neveu (1975), page 121, there exists a sequence (an; bn) 2 Mt;u such that an ;bn (u; X a0 ;b0 (u); Y a0 ;b0 (u)): V (u; X a0;b0 (u); Y a0;b0 (u)) = lim (5.14) n J

Now, if we substitute (an; bn) for (a; b) in (5.13) and take the limit as n ! 1, we get (5.10) by Fatou's lemma. Next we show that if the dynamic programming inequality from the statement of the lemma is true for V then it is also true for V. Let (tn; xn; yn) be a sequence such that limn V (tn; xn; yn) = V (t; x; y) and (tn; xn; yn) ! (t; x; y). By Lemma 5.1, we know that a;b to there exists a relabeled subsequence such that (Yta;b n ;xn ;y n ; Xtn ;xn ;y n )(u) converges a.s. a;b a;b (Yt;x;y ; Xt;x;y )(u), for any xed (a; b) 2 D and T  u  t: We have then, since V  V, by Fatou's lemma, and since V is lower semicontinuous, a;b a;b a;b   n n n V (t; x; y) = lim n V (t ; x ; y )  limninf E [V (u; Xtn ;xn;yn (u); Ytn ;xn;yn (u))] (5.15)  E a;b [limninf V(u; Xta;bn;xn;yn (u); Yta;b n ;xn ;y n (u))] a;b (u); Y a;b (u))];  E a;b [V(u; Xt;x;y t;x;y and we are done, after taking sup over (a; b). 11

2 Proof of Proposition 5.1: Fix (t; x; y) 2 [0; T )  IR  IR and consider a test function ' 2 C ([0; T ]  IR  IR) such that +

2

+

0 = (V ? ')(t; x; y) =

min (V ? '): ;T ]IR+IR 

[0

By substituting ' for V (and t + h for u) in the inequality of the previous lemma, we get



h 

a:b (t + h); Y a;b (t + h) ? '(t; x; y ) 0  sup E Qa;b ' t + h; Xt;x;y t;x;y

i

a;b)2D

(

and by It^o's lemma, and standard stochastic control methods, we see that

(

#) "Z t h 1 a;b a;b Q lim inf h E J ' (u; Xt;x;y (u); Yt;x;y(u)) du  0; for all (a; b) 2 D; h! 0

+

t

where J a;b is the dierential operator appearing inside the in mum of (5.4). Therefore, we obtain inf J a;b'(t; x; y)  0;

a;b)2IR2

(

and the viscosity property has been shown. Finally, the boundary condition is obtained as follows. Taking limits in (5.2) as t approaches T , we see that :

# x x lim inf V (t; x; y)  g ( f (y) ) + f (y)g ( f (y) ) ; s 2 IR t%T  "

1

0

+

by Fatou's Lemma and by Lemma 5.1. The last statement of the proposition follows now from the de nition of the lower semicontinuous envelope.

2

Notice that we only establish that V is a supersolution to the Bellman equation. Our control problem is singular (due to the non-compactness of the set of controls), and it is well known that there are examples where the Bellman equation fails to hold. In general, that is also going to be the case here. We could have used the normalized Bellman equation as in Krylov (1980) which involves stronger conditions on the model in order to de ne generalized derivatives of the value function V . The main advantage of the viscosity approach is that it requires weaker conditions on the regularity of the value function V . In fact, here, where we work directly with the lower semicontinuous envelope of V , we need no regularity of 12

V , a priori. Namely, we shall show that the characterization of function V as a viscosity supersolution of the Bellman equation is sucient for our analysis. We now derive some implications from Proposition 5.1 which will be sucient to deduce the super-replication cost U (t; s). We rst need the following result.

Lemma 5.3 Let V : A  B ! IR, A  IRm, B  IR, be a lower semicontinuous supersolution to the equation

H (x; y; Vy; Vyy ) = 0; x 2 A; y 2 B; (5.16) where H : A  B  IR ! IR is a continuous function. Assume also that V (
;
)  ?C for some constant C > 0. Then, for any xed x 2 A, V (x ;
) is a lower semicontinuous 2

0

supersolution to the equation

0

H (x ; y; Vy ; Vyy ) = 0; y 2 B:

(5.17)

0

Proof: Let  : B ! IR be a C test function such that, for some y 2 B , we have 2

0

0 = V (x ; y ) ? (y ) < V (x ; y) ? (y); 8y 2 B: 0

0

0

(5.18)

0

Since it is sucient to check the viscosity property on the strict minima of V (x ;
) ? (
), all we have to do is prove that 0

H (x ; y ; y (y ); yy (y ))  0: 0

We de ne

n

( )

for all n 2 IN , and denote

0

0

(5.19)

0

(x; y) := (y) ? njx ? x j ; x 2 A; y 2 B; 0

2

D n (x; y) := V (x; y) ? ( )

n

( )

(x; y):

Let I be a compact neighborhood of (x ; y ) in A  B . Being lower semicontinous, D n attains its minimum on I , say at a point (xn ; yn). There exists then a pair (x; y) 2 I and a (relabeled) subsequence such that 0

( )

0

(xn; yn) ! (x ; y) 2 I: Now, since V is bounded from below and  is continuous, we see that if x 6= x , then D n (xn; yn) ! 1, as n ! 1. This is a contradiction, since, for example, D n (xn; yn)  D n (x ; y ) = V (x ; y ) ? (y ). We conclude that 0

( )

( )

( )

0

0

0

0

0

x = x : 0

13

On the other hand, since V (x ; y ) ? (y ) = D n (x ; y )  D n (xn ; yn), we also have 0

0

( )

0

0

( )

0

V (x ; y ) ? (y )  limninf D n (xn; yn)  V (x ; y) ? (y); 0

0

( )

0

0

by lower semicontinuity. Consequently, (5.18) implies

y = y : 0

Now, for large enough n, (xn; yn) is a point of local minimum of D n on I , because it converges to (x ; y ) (it is a classical local minimum if (x ; y ) is in the interior of A  B ; otherwise, we can always appropriately extend our functions so that (x ; y ) becomes an interior point, and so that the local minimality is preserved). Thus, by viscosity property of V (x; y), we have H (xn; yn; yn (xn; yn); yyn (xn; yn))  0: Sending n ! 1 we get (5.19), and we are done. 2 ( )

0

0

0

0

0

0

( )

( )

Lemma 5.4 Function V does not depend on y. Proof. The proof is very similar to that of Cvitanic, Pham and Touzi (1997). Fix some (t; x; y) 2 [0; T )  IR  IR. Then by the viscosity supersolution de nition, for any function ' 2 C ([0; T )  IR  IR) such that : +

2

+

0 = (V ? ')(t; x; y) =

min (V ? '); ;T )IR+IR 

(5.20)

[0

we have (La' + G a;b')(t; x; y)  0; for all (a; b) 2 IR : 2

(5.21)

By sending b to 1, it is then clear that we must have :

'y (t; x; y) = 0

(5.22)

This implies in particular that V (t; x; y) is a lower-semicontinuous viscosity supersolution of the equation vy = 0. By Lemma 5.3, we also have that the function V (t; x;
) is a viscosity supersolution to the same equation, for any xed (t; x). We x then a pair (t; x) and omit t; x in the sequel. Fix also y ; y and consider y < y < y and a test function ' such that 0

2

0

1

2

(V ? ')(y ) = 0 = y0min (V ? ')(y): yy2  1

14

Since the viscosity property remains the same if we consider local minimums rather than global minimums, we conclude 'y (y ) = 0: Think of y as a time variable, and the function V as a viscosity supersolution to the parabolic PDE vy = 0; y 2 [y ; y ]; v(y ) = V(y ): Since the constant function v = V(y ) is also a solution, we get, by maximum principle (see Crandall, Ishii and Lions (1992), Theorems 3.3 and 8.2; notice also that we reverse the direction of the time variable in this proof, compared to that paper) : 1

0

2

2

2

2

V(y)  V (y ); y  y  y : 2

0

2

Since y ; y are arbitrary, V is non-increasing. To prove the opposite inequality, de ne 0

2

W (y) = V(y + y ? y); y  y  y : 2

0

0

2

Fix some y 2 (y ; y ) and consider a C test function such that 1

0

1

2

(W ? )(y ) =

min (W ? )(y):

y0 yy2

1

Then, de ning the C test function ' by 1

'(y) = (y + y ? y); y  y  y ; 2

0

0

2

we see that (V ? ')(y + y ? y ) = 2

0

1

min (V ? ')(y) y0 yy2 

by an obvious change of variables. Therefore, we must have y (y2 + y0 ? y1 )

= ?'y (y ) = 0: 1

It follows that W is a supersolution to the parabolic PDE

vy = 0; y 2 [y ; y ]; 0

2

v(y ) = V(y ): 2

0

By the above argument this means that W (y)  W (y ), or V(y)  V (y ) for y 2 [y ; y ]: Since y ; y are arbitrary, V is also nondecreasing, hence constant (in y). 2

0

0

0

2

2

2 15

Accordingly, we drop argument y from function V in the sequel. From the last lemma and (5.4) the function V satis es in the viscosity sense :

0 1 0 (y ) ! 1 f  A sup @(V)t + x  (t; x; y) + a 2 f (y) (V)xx  0: 2

2

a2IR

(5.23)

Remark 5.1 For xed (a; y) 2 IR , the partial dierential equation 2

!

0 (y ) f 1 vt + 2 x (t; x; y) + a f (y) vxx = 0 with a terminal condition, characterizes the Black-Scholes price when the underlying asset has the adjusted price process X a;b. This is consistent with thepresult of Leland (1985) which shows that the option price under transaction costs of order
t,
t being the time space between transaction dates, corresponds to a Black-Scholes price with an adjusted volatility. Therefore, taking the supremum over a 2 IR can be interpreted as the super-replication price in a stochastic volatility model with unbounded volatility, see Cvitanic, Pham and Touzi (1997). Hence, equation (5.23) shows an analogy between stochastic volatility model with unbounded volatility and our complete market with proportional transaction costs. 2

2

Next, we have the following result.

Proposition 5.2 Assume that condition (3.3) holds. Then: (i) Function V is concave in x for any xed t 2 [0; T ). (ii) Function V is nonincreasing in t for any xed x 2 IR . Proof. We use a similar argument as in the proof of Lemma 5.1 in Cvitanic, Pham and Touzi (1997): (i) For a xed t 2 [0; T ), by sending a to +1 in equation (5.23), we have (V)xx  0 in the viscosity sense. In other words, function V(
;
) is a lower semicontinuous viscosity supersolution of the equation ?vxx = 0. By Lemma 5.3, the function V (t;
) is +

a viscosity supersolution to the same equation, for any xed t. We x t and suppress the dependence on t in the sequel. Since V is bounded below by constant ?C , it is easily checked that V c := V + C is a viscosity supersolution of the equation "v ? vxx = 0 for all " > 0. Let [a; b] be any closed interval in IR and consider the dierential equation : +

("v ? vxx)(x) = 0; v(a) = V c(a) and v(b) = V c(b): Then, since V c(
) is a lower semicontinuous viscosity supersolution of the last equation, by maximum principle (see Crandall, Ishii and Lions (1992) Theorems 3.3 and 8.2), we see that

h i h i c(a) ep" b?x ? 1 + V c (b) ep" x?a ? 1 V p V c(x)  ; e " b?a ? 1 (

)

(

(

16

)

)

for all " > 0, since the function on the right-hand side is a solution to the same equation. Sending " to zero in the last inequality provides

x + V c(a) V c(x)  [V c(b) ? V c(a)] bb ? ?a

for all x 2 [a; b]. Applying the last inequality to x = a + (1 ? )b for some  2 [0; 1] provides :

V c (a + (1 ? )b)  V c(a) + (1 ? )V c(b) for all a; b 2 IR . Therefore, V c is concave in x for any xed t, and so is V = V c ? C . (ii) For xed (x; y) 2 IR  IR, send a to ? (t; x; y)f (y)=f 0(y) in equation (5.23). Then, we have (V)t  0 in the viscosity sense and we conclude similarly as before by applying Lemma 5.3 and the maximum principle. 2 +

+

6 Explicit solution to the super-replication problem We prove in this section that the minimal super-replicating strategy of the claims of the form (g (S (T )); g (S (T )), in the presence of transaction costs, is a buy-and-hold strategy. For any real function k(s) on IR , denote its concave envelope by kcon(s). It is well-known that 0

1

+

kcon(s) = inf fc 2 IR : 9
2 IR; 8z > 0; c +
(z ? s)  k (z)g ;

(6.1)

see Rockafellar (1970), for example. We introduce the following notation:

g(x; y) = g ( f (xy) ) + f (y)g ( f (xy) ); G(x) = sup g (x; y); y

(6.2)

con(sf (y )): g^(s) = sup G y

(6.3)

0

1

where g(x; y) is the lower semicontinuous envelope of g. Moreover, we de ne We need the following mild assumption, easily checked to be satis ed for the standard options: Assumption 6.1 For all x > 0 we have



con 

con

 g ( 1 +x  ) + (1 +  )g ( 1 +x  ) ; con  y!1 con  x x lim sup g(x; y)  g ( 1 ?  ) + (1 ?  )g ( 1 ?  ) ; y!?1 lim sup g(x; y)

0

0

0

1

0

1

1

1

where con denotes the concave envelope with respect to x variable.

17

0

1

(6.4) (6.5)

Lemma 6.1 For any (t; s; y) 2 [0; T )  IR  IR, we have : +

sup V (t; s; y)  g^(s): y

Proof. We know from Proposition 5.1 that the function V is independent of y, nonincreasing in t and V (T ?; x)  g(x; y). Therefore, for all t < T , V (t; x)  G(x):

(6.6)

Since V is also concave in x, we get V(t; x)  Gcon(x). This implies

V (t; s; y) = V (t; sf (y))  V(t; sf (y))  Gcon(sf (y));

(6.7)

and we are done.

2

In order to prove the opposite inequality, we need

Lemma 6.2 For all s > 0, under Assumption 6.1, we have g^(s) = inf fc 2 IR : either 9
> 0; 8z > 0; c ?(1 +  )
s  g (z) + (1 +  )(g (z) ?
z) and c ?(1 +  )
s  g (z) + (1 ?  )(g (z) ?
z) or 9
< 0; 8z > 0; c ?(1 ?  )
s  g (z) + (1 +  )(g (z) ?
z) and c ?(1 ?  )
s  g (z) + (1 ?  )(g (z) ?
z)g : 0

0

0

1

0

0

1

1

1

0

0

1

1

0

1

1

Proof. Denote by h(s) the right-hand side of the last equality and let rst c be such that

there exists
> 0 for which the rst two inequalities on the right-hand side are satis ed. It is then easy to see that also

c ? f (y)
s  g (z) + f (y)(g (z) ?
z); 0

1

(6.8)

for all y and for all z > 0. In particular, if we replace z by z=f (y) and let x = sf (y), we get

c ?
(x ? z)  g (z=f (y)) + f (y)(g (z=f (y)): 0

1

(6.9)

Taking a supremum over y we get c ?
(x ? z)  G(z), for all z > 0. But then, by the characterization of concave envelopes (6.1), we also get c  Gcon(x). An analogous argument 18

gives the same result in case if
< 0 and if the last two inequalities on the right-hand side of the lemma are satis ed. Therefore, h(s)  Gcon(x) and hence

h(s)  sup Gcon(sf (y)) = g^(s): y

(6.10)

We now want to prove the converse: by de nition of g^ we have

"(

sup0 g g^(s) = sup y y

(sf (y); y0)

)con#

;

(6.11)

where con denotes the concave envelope with respect to variable x = sf (y). If we now, for example, send y0 ! ?1, by Assumption 6.1 we get

"(

g ( 1sf?(y) ) + (1 ?  )g ( 1sf?(y) ) g^(s)  sup y 0

1

1

1

)con#

1

:

(6.12)

In particular, taking
to be the left-derivative of the concave envelope in the last expression, we get

g^(s) +
(z(1 ?  ) ? sf (y))  g (z) + (1 ?  )g (z); 1

0

1

1

(6.13)

for all y and all z > 0. If now, for example,
> 0, we send y ! 1 to get

g^(s) ? (1 +  )
s  g (z) + (1 ?  )g (z) ? (1 ?  )
z: 0

0

1

1

1

(6.14)

The latter implies that g^(s) is greater or equal than the in mum of all the numbers c for which the second inequality on the right-hand side in the statement of the lemma holds with a
> 0. The remaining cases are shown in a similar fashion, and we get

g^(s)  h(s):

(6.15)

2 We now state the main result of the paper.

Theorem 6.1 Assume that Assumption 6.1 is valid and g + min ((1 +  )g ; (1 ?  )g )  ?C; for some C  0: 0

0

1

1

1

Then the super-replication cost function is U (t; s) = g^(s).

19

Proof. From Proposition 4.1, we have U (t; s)  V(t; s). Using Lemma 6.1, we then get U (t; s)  g^(s). The statement is trivially true if g^(s) = 1, so we assume g^(s) < 1. To prove U (t; s)  g^(s) it suces to check that g^(s) is the initial wealth needed in the

bank account for a (buy-and-hold) strategy dominating the contingent claim (g ; g ). Indeed, let x = (^g(s); 0) be the initial holdings in the bank and the stock. It can easily be shown that the in mum in Lemma 6.2 is attained; denote by
 the corresponding
(in fact, it is not dicult to see, as in the proof of the lemma, that we can take
 := (Gcon)0?(s), the left derivative of Gcon at s; since Gcon is concave and bounded from below, we have
  0). We consider the strategy : 0

1

L (t) =
 S (0); 0 < t  T L (t) = 0; 0  t  T: 0

1

Then it is easily checked that the corresponding portfolio holdings are given by :

X x;L(t) = g^(S (0)) ? (1 +  )
 S (0); 0 < t  T X x;L(t) =
 S (t); 0  t  T: 0

0

1

By de nition of P , it is clear that L is an admissible strategy for the initial holdings x in the sense of De nition 3.2. Moreover, by Lemma 6.2, it is also clear that L dominates the contingent claim (g ; g ), starting with x as initial holdings, in the sense of (2.6)-(2.7). 2 0

1

Hence, the optimal super-replicating strategy is the trivial buy-and-hold strategy, as is the case in stochastic volatility models with unbounded volatility without transaction costs; see Cvitanic, Pham and Touzi (1997).

Remark 6.2 It is now clear that, in general, the function V does not satisfy the HJB equation (5.4) with equality.

Conclusion We show that the least expensive dominating strategy for general path-independent contingent claims under proportional transaction costs is equal to the least expensive buy-and-hold strategy. We prove the result in a Markovian continuous-time model, using a representation of the minimal super-replication price as a supremum of expectations of the claim under all \supermartingale measures", namely under the equivalent probability measures under which the appropriately discounted \wealth process" is a supermartingale. We characterize 20

the price function using viscosity solutions approach to that stochastic control problem. As a byproduct, we also indicate how the problem is essentially equivalent to the super-replication problem with a modi ed, unbounded volatility.

Acknowledgement: We are grateful to Marco Avellaneda for suggesting that the price

function should be related to the concave envelope of the payo function, to Mete Soner for some \viscosity tips", and to Yuri Kabanov for helpful comments that prompted the introduction of Lemma 5.3.

References Avellaneda, M. and Paras, A. (1994), \Dynamic hedging portfolios for derivative

securities in the presence of large transaction costs", Appl. Math. Finance 1, 165-194.

Avellaneda, M. (1996), Private communication. Barles, G. and Soner, H.M. (1995), \Option pricing with transaction costs and a

nonlinear Black-Scholes equation", to appear in Finance & Stochastics.

Broadie, M., Cvitanic, J. and Soner, M. (1996), \Optimal replication of contingent

claims under portfolio constraints", to appear in Review of Financial Studies.

Crandall, M.G., Ishii, H. and Lions, P.L. (1992), \User's guide to viscosity solutions

of second order Partial Dierential Equations", Bull. Amer. Math. Soc. 27, 1-67.

Cvitanic, J. and Karatzas, I. (1996), \Hedging and portfolio optimization under trans-

action costs: a martingale approach", Mathematical Finance, 6, 133-165.

Cvitanic, J., Pham, H. and Touzi, N. (1997) \Super-replication in stochastic volatility

models under portfolio constraints", to appear in Applied Probability journals.

Davis, M.H.A. and Clark, J.M.C. (1994) \A note on super-replicating strategies", Phil.

Trans. Roy. Soc. London A 347, 485-494.

Davis, M.H.A., Panas, V.G. and Zariphopoulou (1993) \European option pricing

with transaction costs", SIAM J. Control Optim. 31, 470-493.

Fleming, W.H. and Soner, H.M. (1993) Controlled Markov Processes and Viscosity

Solutions, Springer-Verlag, New York.

Jouini, E. and Kallal, H. (1995), \ Martingales and arbitrage in in securities markets

with transaction costs", J. Econ. Theory, 66, 178-197. 21

Karatzas, I. and Shreve, S.E. (1991), Brownian Motion and Stochastic Calculus,

Springer-Verlag, New York.

Koehl, P.F., Pham, H. and Touzi N. (1996), \ On super-replication in discrete time

under transaction costs", preprint.

Krylov, N.V. (1980) Controlled Diusion Processes, Springer-Verlag, Berlin. Kusuoka, S. (1995) \Limit theorem on option replication with transaction costs", Ann.

Appl. Probab. 5, 198-221.

Leland, H.E. (1985) \Option pricing and replication with transaction costs", J. Finance

40, 1283-1301.

Levental, S. and Skorohod, A.V. (1997) \On the possibility of hedging options in the

presence of transaction costs", Ann. Appl. Probab. 7, 410-443.

Lions, P.-L. (1983) \Optimal Control of Diusion Processes and Hamilton-Jacobi-Bellman

Equations", Parts I and II Communications in P.D.E. 8, 1101-1174, 1229-1276.

Neveu, J. (1975) Discrete-Parameter Martingales (English Translation). North-Holland,

Amsterdam.

Protter, P. (1990) Stochastic Integration and Dierential Equations. Springer-Verlag. Rockafellar, R.T. (1970) Convex Analysis, Princeton University Press, Princeton, NJ. Soner, H.M., Shreve, S.E. and Cvitanic, J. (1995), \There is no nontrivial hedging

portfolio for option pricing with transaction costs", Ann. Appl. Prob. 5, 327-355.

22