Super-replication in stochastic volatility models under portfolio constraints 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 a nancial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on borrowing and short-selling of a \hedger" in this market, and described by a closed convex set K . We nd explicit characterizations of the minimal price needed to superreplicate European type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or in nity or not, and also, on whether we have linear dynamics for the stock price process and whether volatility process depends on the stock price or not. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.
Kew words: stochastic volatility, portfolio constraints, hedging options, viscosity solutions. 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. He is also grateful to CREST for its hospitality in July 1996, when this research was initiated.
1
1 Introduction We consider a Markovian model of a nancial market in which the stock price is a solution of a (possibly nonlinear) stochastic di erential equation (SDE), with the volatility coecient being driven by another di usion process. There are two Brownian motions driving the corresponding SDEs, therefore the market is incomplete. This is a standard way of modeling volatility risk which is not hedgeable by investing in the underlying only, and was used in Hull and White (1987) and Wiggins (1987), among others. We are primarily interested in the minimal super-replication cost in this model, of a European type option with payo g(s) - namely the cost of the least expensive dominating strategy for the option. Additionally, we also analyze what happens if we impose convex constraints on the portfolio weights of the hedging wealth process. The latter problem is solved in Broadie, Cvitanic and Soner (1996), in the Black-Scholes, constant volatility framework. They show that the minimal cost for super-replication of g under constraints is given by pricing another option g^(s) g(s) without the constraints, with g^ appropriately de ned, to make the super-replicating portfolio satisfy the constraints. We extend their results to the case of stochastic volatility. In Section 2 we introduce the model and the constraints. In Section 3 we de ne the shadow prices corresponding to the incompleteness of the market, and we re-derive a result of Cvitanic and Karatzas (1993), Jouini and Kallal (1995) and El Karoui and Quenez (1995), which gives a lower bound on the minimal super-replication cost as a supremum of BlackScholes prices over all the associated shadow markets. In Section 4 we state the main technical result (proved in Appendix): the cost (price) function is a viscosity supersolution to the Bellman equation associated to the control problem (the use of viscosity supersolutions approach turns out to be very powerful and elegant in our problem). Next, using those results, in Section 5 we show that in the case that the volatility can reach (in the limit) both zero and in nity, or it can reach in nity and the payo g is convex, the minimal super-replication cost (without constraints) is the cost of the minimal buy-and-hold strategy - namely, equal to the concave envelope of g. If there are constraints, but such that one is allowed to put all the money in the stock, then it is the concave envelope of g^ that is equal to the minimal cost. In Section 6 we consider the case of bounded volatility, and re-derive the so-called Barenblatt PDE, suggested as the PDE for obtaining the super-replicating price under stochastic volatility in Avellaneda, Levy and Paras (1995). Our analysis shows that this gives rise not only to a super-replicating strategy, but also to the least expensive one. Moreover, in the case of linear dynamics, we extend the result to the case with portfolio constraints, simply by substituting g^ instead of g for the terminal condition. Similar results are shown to be valid in the mixed case (unbounded volatility bounded away from zero) in Section 7. Finally, in Section 8 we provide some 2
examples. Let us also mention that while we were nishing the paper, we learned that, in the case of unbounded volatility and no constraints, Frey and Sin (1997) obtained results similar to ours; however, they use completely di erent methods and deal with less general payo s, but more general price processes. Related work also includes El Karoui, Jeanblanc-Picque and Shreve (1996), and Bergman, Grundy and Wiener (1996).
2 The model We consider a nancial market which consists of one bank account, with constant price process B (t) = 1 for all t 2 [0; T ], and one risky asset with price process evolving according to the following stochastic di erential equation : dS (t) = (t; S (t); Y (t))dt + (t; S (t); Y (t))dW (t) + (t; S (t); Y (t))dW (t) (2.1) 1 1 2 2 S (t) dY (t) = (t; S (t); Y (t))dt + (t; S (t); Y (t))dW2(t): (2.2) Here W = (W1 ; W2) is a standard Brownian motion in IR2 de ned on a complete probability space ( ; F ; P ). We shall denote by fF (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 usual standing assumptions on the coecients of the last SDE in order to ensure the existence of a unique strong solution fSt ; Yt), 0 t T g, given an initial condition : all coecients are continuous in (t; s; y) and satisfy for all t 2 [0; T ] and (s; y), (s0 ; y0) 2 IR+ IR, i=1;2 (jsi (t; s; y ) ? s i (t; s ; y )j + j (t; s; y ) ? (t; s ; y )j
P
0
0 0
0 0
+ js(t; s; y) ? s0(t; s0; y0)j + j(t; s; y) ? (t; s0; y0)j C (js ? s0j + jy ? y0j);
(2.3)
for some positive constant C . We also assume
8(t; s; y) 2 [0; T ] IR+ IR; 1 (t; s; y) > 0 and (t; s; y) > 0:
(2.4)
Moreover, for simplicity, we assume that for any y 2 IR, there is a positive probability that process Y will reach y before time T .
Remark 2.1 Because our goal is mostly to illustrate the extreme behavior of the minimal super-replicating price that can happen in stochastic volatility models, we don't always aim at the most general assumptions possible. For example, we could, alternatively, assume Lipschitz conditions in log s on the coecients in (2.1) and (2.2) and the results would remain 3
the same. Moreover, we don't deal with markets with more than one risky asset. It does not seem likely that one could get as explicit results in that case, as we get in the case of one stock only (although Broadie, Cvitanic and Soner (1996) get explicit results for the multidimensional case, when volatility is constant). Nevertheless, the viscosity supersolution characterization of the minimal super-replication price would remain the same (a related viscosity characterization is obtained in Buckdahn and Hu (1997) in a di erent context). Consider now an economic agent, endowed with an initial capital x, who invests at each time t 2 [0; T ] a proportion (t) of his wealth in the risky asset and the remaining wealth in the bank account. Here = f(t), 0 t T g is an fF (t)g progressively measurable R process with 0T (t)2(12 + 22)(t; S (t); Y (t))dt < 1, a.s.. Then the wealth process X x; (t) satis es the linear stochastic di erential equation (for t < T ) : (t) (2.5) dX x;(t) = (t)X x; (t) dS S (t) X x; (0) = x: (2.6) Let K = [l; u], ?1 l 0 u 1 be an arbitrary closed convex set in IR containing 0. We say that a portfolio is K -admissible if
(t) 2 K; 0 t T
(2.7)
hold P -a.s. The set of K -admissible portfolios will be denoted by AK .
Remark 2.2 Set K speci es the constraints on borrowing and short-selling that our agent
has to adhere to: he/she cannot borrow more than u times what he/she owns at the moment, and cannot sell short more than ?l times her current wealth. Notice also that since the wealth process fX x; (t), 0 t T g is a continuous process, it is clear from (2.5) that, given an initial wealth x > 0, we have X x; (t) 0 for all t 2 [0; T ], a.s. Therefore, in this model where the portfolio is de ned as proportions of the wealth process, the no-bankruptcy condition is automatically satis ed. In this paper, we consider European contingent claims de ned by a terminal payo g(S (T )), where g is a nonzero and nonnegative function. Given such a contingent claim, we then consider the in mum U (0) of initial capitals x which induce a wealth process X x; through some admissible portfolio 2 AK such that X x; hedges g(S (T )), i.e.
U (0) = inf fx > 0 : 9 2 AK ; X x; (T ) g(S (T )) P ? a.s.g :
(2.8)
It should be pointed out that we could also allow consumption (withdrawal of funds) in the above de nitions, and the results would not change. 4
The main result of this paper is an explicit solution to the problem of calculating U (0), as in Broadie, Cvitanic and Soner (1996), who deal with the Black and Scholes model with constraints.
3 Shadow prices We introduce in this section the shadow state-price densities relevant to our incomplete market, following Cvitanic and Karatzas (1993) or Karatzas and Kou (1996). We denote by the set of all adapted, IR?valued processes . An appropriately de ned subset of will correspond to the shadow prices associated with the incompleteness of the market due to the stochastic volatility. We also introduce the shadow prices relevant to the incompleteness of the market induced by the constraints: consider the support function of the convex set ?K :
(z) = sup f?xzg = uz? ? lz+ x2[l;u]
(3.1)
and the associated e ective domain K~ = fz 2 IR : (z) < 1g = fz 2 IR : z 0 if u = 1 and z 0 if l = ?1g : (3.2) Notice that 0 on K~ since 0 2 [l; u]. Then, let D be the set of all fF (t)g adapted processes = f (t), 0 t T g such that (t) 2 K~ for all 0 t T P -a.s. Given 2 and 2 D, we introduce the shadow state-price density process M ; by ! dM ; (t) = ? ? (2 = ) + dW (t) ? dW (t) (3.3) 1 M ; (t) 1
2 M ; (0) = 1; (3.4) provided that the stochastic integrals can be de ned. This happens to be a somewhat too large class of shadow state-price densities, and we will nd it convenient to reduce it by considering subsets and D of and D (resp.), consisting only of a.s. bounded processes. The following standing assumption is a natural generalization of the standard assumption of the risk-neutral shadow state-price density being a martingale:
Assumption 3.1 The coecients functions ; 1; 2 ; ; de ning the dynamics of (S; Y )
are such that
and M ;
2
2 + ? (2 = ) +
1 0 are martingales, for all (; ) 2 D. Z
T
!
4
5
! 3
2
5
du < 1; a:s:;
For example, the latter will be satis ed if Novikov condition (see Karatzas and Shreve (1991)) is satis ed for all M ; , with and bounded a.s. Therefore, given any 2 and 2 D, we can de ne a probability measure Q; equivalent to P , by dQ; = M ; (T )dP . Then, by Girsanov's theorem,
? (2 = ) + (u)du 1 0 t W2; (t) = W2(t) + 0 (u)du W1; (t) = W1(t) +
Z
t
(3.5)
Z
(3.6)
is a Q; standard Brownian motion in IR2 and the dynamics of the process (S; Y ) are driven by the SDE : dS (t) = ? (t)dt + (t; S (t); Y (t))dW ; (t) + (t; S (t); Y (t))dW ; (t) (3.7) 1 2 1 2 S (t) dY (t) = [(t; S (t); Y (t)) ? (t)] dt + (t; S (t); Y (t))dW2; (t): (3.8) R
t Therefore, under measure, the \discounted" wealth process X~ (t) := X (t)e? 0 ((u))du , corresponding to a strategy satis es (suppressing the dependence on t in coecients),
Q;
dX~ (t) = X~ (t)[?( + ( ))dt + 1 dW1; (t) + 2 dW2; (t)]:
(3.9)
Recalling the de nition of , we see that the discounted wealth process is a (non-negative) Q; -supermartingale. In particular, for a super-replicating process X we obtain
X (0) E Q; X~ (T ) E Q; g(S (T ))e? h
i
RT
0 ( (u))du
:
The consequence of this is
Proposition 3.1 U (0) sup(;)2D E Q; g(S (T ))e?
RT
0 ( (u))du
:
Remark 3.1 A more general result of this kind, with equality, was obtained in Cvitanic and Karatzas (1993) and Karatzas and Kou (1996), in a (possibly) non-Markovian framework, but with somewhat stronger, boundedness assumptions on the volatility coecients. We will prove indirectly that the equality holds here, too (under some assumptions), by characterizing quite explicitely the right-hand side in Proposition 3.1, and showing that one can super-replicate g(S (T )) starting with the right-hand side as the initial investment. From now on we also impose the following standing assumption.
Assumption 3.2 The payo function g() is lower semicontinuous. 6
4 Necessary conditions from the Bellman equation As explained in the previous section, the super-replication cost at time t has a lower bound V (t; St; Yt) given by the following control problem :
V (t; s; y) =
sup
(; )2D
E Q;
g (S (T )) e?
RT
t
( (u))du (S (t); Y (t)) = (s; y )
:
(4.1)
In the following, we assume that function V (t; s; y) is nite. We have the following property of function V (we use the de nition of Crandall, Ishii and Lions (1992) for viscosity supersolutions).
Proposition 4.1 The value function V (t; s; y) is a lower-semicontinuous viscosity superso-
lution to the Bellman equation :
inf
(; )2IRK~
where
Moreover,
[Lv + G v + svs + vy + ( )v] = 0;
(4.2)
Lv = ?vt ? 21 s22 vss with 2 = 12 + 22 G v = ?vy ? 12 2 vyy ? s2 vsy :
(4.3)
V (T ?; s; y) g(s):
(4.5)
Proof. see Appendix 1.
(4.4)
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 many examples where the Bellman equation then fails to hold. In general, that is also going to be the case in this paper. 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, we will show that the characterization of function V as a lower-semicontinuous viscosity supersolution of the Bellman equation is sucient for our analysis. We now derive some implications from Proposition 4.1 which will be sucient to deduce the super-replication cost U (0). Fix some (t; s; y) 2 [0; T ) IR+ IR. Then by the viscosity supersolution de nition, for any function ' 2 C 2 ([0; T ) IR+ IR) such that 0 = (V ? ')(t; s; y) = 7
min (V [0;T )IR+ IR
? ');
(4.6)
we have ~ (L' + G ' + s's + 'y + ( )')(t; s; y) 0; for all (; ) 2 IR K:
(4.7)
By sending to 1 it is then clear that we must have
'y (t; s; y) = 0
(4.8)
for any test function ' satisfying (4.6). We now need the following result.
Lemma 4.1 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; (4.9) where H : A B IR2 ! IR is a continuous function. Assume also that V (; ) ?C for some constant C > 0. Then, for any xed x0 2 A, V (x0 ; ) is a lower semicontinuous supersolution to the equation
H (x0; y; Vy ; Vyy ) = 0; y 2 B:
(4.10)
Proof: Let : B ! IR be a C 2 test function such that, for some y0 2 B we have 0 = V (x0 ; y0) ? (y0) < V (x0; y) ? (y); 8y 2 B:
(4.11)
Since it is sucient to check the viscosity property on the strict minima of V (x0; ) ? (), all we have to do is prove that
H (x0; y0; y (y0); yy (y0)) 0: We de ne
(n) (x; y ) := (y )
for all n 2 IN , and denote
(4.12)
? njx ? x0 j2; x 2 A; y 2 B;
D(n)(x; y) := V (x; y) ?
(n) (x; y ):
Let I be a compact neighborhood of (x0 ; y0) 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 (xn; yn) ! (x ; y) 2 I: 8
Now, since V is bounded from below and is continuous, we see that if x 6= x0 , then D(n) (xn; yn) ! 1, as n ! 1. This is a contradiction, since, for example, D(n) (xn; yn) D(n) (x0; y0) = V (x0 ; y0) ? (y0). We conclude that
x = x0 : On the other hand, since V (x0 ; y0) ? (y0) = D(n) (x0; y0) D(n)(xn ; yn), we also have
V (x0 ; y0) ? (y0) limninf D(n) (xn; yn) V (x0 ; y) ? (y); by lower semicontinuity. Consequently, (4.11) implies
y = y0 : Now, for large enough n, (xn; yn) is a point of local minimum of D(n) on I , because it converges to (x0 ; y0) (it is a classical local minimum if (x0 ; y0) is in the interior of A B ; otherwise, we can always appropriately extend our functions so that (x0 ; y0) 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; y(n) (xn; yn); yy(n) (xn; yn)) 0: Sending n ! 1 we get (4.12), and we are done. 2 From this we get the following result.
Lemma 4.2 Function V does not depend on y. Proof: From (4.8) we conclude that V (t; s; y) is a lower semicontinuous viscosity supersolution to the equation vy = 0. By Lemma 4.1, we also have that the function V (t; s; ) is a
viscosity supersolution to the same equation, for any xed (t; s). We x then a pair (t; s) and omit t; s in the sequel. Fix also y0; y2 and consider y0 < y1 < y2 and a test function ' such that (V ? ')(y1) = 0 = y0min (V ? ')(y): yy2
Since the viscosity property remains the same if we consider local minimums rather than global minimums, we conclude 'y (y1) = 0: Think of y as a time variable, and the function V as a viscosity supersolution to the parabolic PDE vy = 0; y 2 [y0; y2]; v(y2) = V (y2): 9
Since the constant function v = V (y2) 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) : V (y) V (y2); y0 y y2: Since y0; y2 are arbitrary, V is non-increasing. To prove the opposite inequality, de ne W (y) = V (y2 + y0 ? y); y0 y y2: Fix some y1 2 (y0; y2) and consider a C 1 test function such that (W ? )(y1) = y0min (W ? )(y): yy2 Then, de ning the C 1 test function ' by '(y) = (y2 + y0 ? y); y0 y y2; we see that (V ? ')(y2 + y0 ? y1) = y0min (V ? ')(y) yy2
by an obvious change of variables. Therefore, we must have y (y2 + y0 ? y1 ) = ?'y (y1 ) = 0: It follows that W is a supersolution to the parabolic PDE vy = 0; y 2 [y0; y2]; v(y2) = V (y0): By the above argument this means that W (y) W (y2), or V (y) V (y0) for y 2 [y0; y2]: Since y0; y2 are arbitrary, V is nondecreasing, hence constant (in y). 2 Accordingly, in the sequel, we omit argument y in function V . It then follows from (4.7) that, in viscosity sense, ~ LV + sVs + ( )V 0 for all 2 K: (4.13) Writing condition (4.13) for = 0 2 K~ , we get : LV (t; s) 0; (t; s) 2 [0; T ) IR+ : (4.14) Next, suppose that there exists some 0 2 K~ and (t0; s0 ) 2 [0; T ) IR+ such that [0 sVs + (0 )V ](t0 ; s0) < 0. Then by the cone property of K~ , the left hand side of (4.13) can be sent to ?1, a contradiction. Therefore, we must have, in viscosity sense, (sVs + ( )V )(t; s) 0; for all 2 K~ and (t; s) 2 [0; T ) IR+ : (4.15) Combining (4.14) and (4.15), we get 10
Proposition 4.2 Function V does not depend on y and is a viscosity supersolution to the equation
(
min Lv ; inf~ (svs + ( )v)
)
2K
= 0:
(4.16)
In the following sections, we use the last proposition in order to characterize explicitly the solution U (0) to the hedging problem under di erent assumptions on the volatility function.
5 Unbounded Volatility In this section we provide an explicit solution to the hedging problem under the following conditions.
Assumption 5.1 sup (t; s; y) = 1 for all (t; s) 2 [0; T ) IR+ . y2IR
Assumption 5.2 yinf (t; s; y) = 0 for all (t; s) 2 [0; T ) IR+ . 2IR Then we have the following preliminary result on function V .
Lemma 5.1 (i) under Assumption 5.1, function V is concave in s for any xed t 2 [0; T ); (ii) under Assumption 5.2, function V is nonincreasing in t for any xed s 2 IR+ . Proof. From equation (4.14), we know that V is a viscosity supersolution to
?vt (t; s) ? 21 2(t; s; y)s2vss(t; s) = 0; (t; s; y) 2 [0; T ) IR+ IR: (i) By considering a maximizing sequence of (t; s; y) for any xed (t; s) 2 [0; T ) IR+ , we see that the function V (; ) is a lower semicontinuous viscosity supersolution of the equation ?vss = 0. By Lemma 4.1 the function V (t; ) is a supersolution to the same equation, for
any xed t. We x t and suppress the dependence on t in the sequel. Since V is nonnegative, it is easily checked that V is also a viscosity supersolution of the equation "v ? vss = 0 for all " > 0. Let [a; b] be any closed interval in IR+ and consider the di erential equation : ("v ? vss)(s) = 0; v(a) = V (a) and v(b) = V (b): Then, since V () 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
V (s)
p"(b?s)
V (a) e
?p1 + V (b) e e "(b?a) ? 1 i
11
h
p"(s?a)
?1
i
;
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
? s + V (a) V (s) [V (b) ? V (a)] bb ? a
for all s 2 [a; b]. Applying the last inequality to s = a +(1 ? )b for some 2 [0; 1] provides :
V (a + (1 ? )b) V (a) + (1 ? )V (b) for all a; b 2 IR+ . (ii) By considering a minimizing sequence of (t; s; y) for any xed (t; s) 2 [0; T ) IR+ , we see that the function V (; ) is a lower-semicontinuous viscosity supersolution of the equation ?vt = 0. The result follows similarly as before by an application of Lemma 4.1 and the maximum principle. 2 Now, let H be the set of all functions mapping IR+ into IR [f1g. As in Broadie, Cvitanic and Soner (1996), we introduce the operator H de ned on H by :
H (h)(s) = h^ (s) = sup h(se? )e?() ; s 2 IR+ : 2K~
(5.1)
Then, we have the following result on function V .
Lemma 5.2 For any t 2 [0; T ), the function V (t; ) is invariant by H , i.e. V^ (t; ) = V (t; ); t 2 [0; T ): Proof. From equation (4.16), we have for all 2 K~ , V (t; ) is a lower semicontinuous viscosity supersolution of
svs(t; s) + ( )v(t; s) = 0;
v(t; s0) = V (t; s0 );
Since is nonnegative, it follows from the maximum principle in the viscosity sense (see Crandall, Ishii and Lions (1992)) that
V (t; s) V (t; s0) ss 0
?( )=
~ ; 2 K;
since the right-hand side solves the same PDE. By taking s0 = se? , we see that : ~ V (t; s) V (t; se? )e?() ; 2 K; which proves that V (t; ) V^ (t; ) for all t 2 [0; T ). The reverse inequality is trivial since 0 2 K~ . 2 12
Next, we consider the following optimal stopping problem :
g~(s) = sup E [^g(Z s( ))] ; s 2 IR+ ;
(5.2)
2T
where T is the set of all nonnegative stopping times and dZ s(t) = Z s(t)dW (t), Z s(0) = s. We provide the following characterizations of the functions g^ and g~.
Lemma 5.3 g^ is the smallest invariant function by H such that g^ g. Proof. (i) From the de nition of the operator H , it is clear that g^ g, since 0 2 K~ , and H (^g) = g^. (iii) Let h be any invariant function by H such that h g. Then we have : h(s) = sup h(se? )e?() sup g(se? )e?() 2K~ 2K~ = g^(s)
2
which ends the proof.
Lemma 5.4 g~ is the smallest concave function such that g~ g and H (~g) = g~. Proof. (i) From the de nition of g~ in (5.2), g~ is a superharmonic function, hence concave. (ii) It is also clear that g~ g. (iii) We now prove that H (~g) = g~. Since Z se? (t) = e? Z s(t), we have for all 2 K~ : i
h
g~(se? )e?() = sup E g^(Z s( )e? )e?() : 2T
This implies that : h
i
H (~g)(s) = sup sup E g^(Z s( )e? )e?() : 2K~ 2T
(5.3)
By taking = 0 in the right-hand-side of (5.3), we obviously have H (~g) g~. By applying Jensen's inequality to the right-hand-side of (5.3), we obtain :
H (~g)(s) =
"
sup E sup g^(Z s( )e? )e?() 2T 2K~ sup E [H (^g)(Z s( ))] : 2T
#
(5.4)
Since H (^g) = g^, (5.4) implies that H (~g) g~ and nally H (~g) = g~. (iv) Let h be a concave function such that h g and H (h) = h. By Lemma 5.3, h g^ and so
E [h(Z s( ))] E [^g(Z s( ))] 13
for all s > 0 and all stopping time 2 T . By Jensen's inequality and since Z s is a martingale, we have for all n 2 IN :
h(s) E [^g(Z s( ))] for all stopping times n. This implies that :
h(s) sup E [^g(Z s( ))] n
and nally by sending n to in nity, h g~.
2
Lemma 5.5 For all s 2 IR+ , we have : g~(s) = inf fc > 0 : 9 2 IR;
s=c 2 K; and 8z 2 IR+ ; c + (z ? s) g^(z) : o
Proof. Let us de ne the function f by : f (s) = inf fc > 0 : 9 2 IR;
s=c 2 K; and 8z 2 IR+ ; c + (z ? s) g^(z) : o
f is a concave function since it is de ned as in mum of ane functions. It is also clear that f g^ g. Let us prove that f^ = f or equivalently that f^ f since the reverse inequality is always true. By de nition of f , we have for all 2 K~ : f (se? )e?() = inf ce?() > 0 : 9 2 IR; se? =c 2 K; and 8z 2 IR+ ; c + (z ? se? ) g^(z) = inf fc0 > 0 : 90 2 IR; s0=c0 2 K; and 8z0 2 IR+ ; c0 + 0 (z0 ? s) g^(z0 e? )e?() : n
o
o
Taking the supremum of this last relation over 2 K~ , we obtain : )
(
f^(s) inf c > 0 : 9 2 IR; s=c 2 K; 8z 2 IR+ ; c + (z ? s) sup g^(ze? )e?() : 2K~
Since H (^g) = g^, the last inequality means that f^ f and so f^ = f . Let h be a concave function such that h f and h^ = h for all s > 0. By Lemma 5.3, h g^. Since h is concave, we have for all s; z 2 IR+ :
h(s) + h0? (s)(z ? s) h(z) g^(z); 14
where h0? is the left derivative of the concave function h. Since g is a nonzero and nonnegative function, so is h; moreover, since h is concave, it is in fact positive on IR+ . Let us now prove that = h0?(s) satis es s=h(s) 2 K . Since h^ h, we have h(s) h(se? )e?() and therefore ~ log h(s) ? log h(se? ) ?( ); s > 0 and 2 K: Since K~ is a cone and is positively homogeneous, this implies for any s > 0 and 2 K~ \ IR+ :
log h(s) ? log h(se?" ) ?"( ) ; " > 0 s(1 ? e?" ) s(1 ? e?" ) and by sending " to zero, we get : 0 shh?(s()s) + ( ) 0; 2 K~ \ IR+: By a similar argument (use the change of variable s0 = se? ), one shows that the last inequality also holds for 2 K~ \ IR?. Then, by Theorem 13.1 of Rockafellar (1970), this proves that s=h(s) 2 K . Therefore, it follows by the de nition of f that h f . By Lemma 5.4, we conclude that f = g~. 2 From the last lemma, the amount g~(S (0)) is the in mum of initial capitals of buy-andhold strategies which dominate the contingent claim de ned by the payo function g^. It is easily checked that the in mum is attained and that the associated portfolio can be taken as : 0 S (t) ; ~ (t) = g~(S (0)) +g~g~?0 ((SS(0)) ? (0)) [S (t) ? S (0)] the denominator of the right-hand-side term of the last equation is equal to X g~(S(0));~ (t), the time t wealth associated to the strategy ~ and the initial capital g~(S (0)). Notice that, since g~ is concave and dominates the nonzero and nonnegative function g, we have X g~(S(0));~ (t) > 0 and
~ (t) 0 0 t T;
(5.5)
Furthermore, as in the proof of Lemma 5.5, we see that that the portfolio strategy ~ is K -admissible at time zero, i.e. ~ (0) 2 K .
Remark 5.1 In the particular case g~?0 (S (0)) = 0, it is clear that ~ = 0 and therefore the portfolio strategy ~ is K -admissible.
15
In the general case, we have the following result. Lemma 5.6 Suppose ~ (0) 6= 0. Then the portfolio strategy ~ is K -admissible if and only if u 1. Proof. From (5.5), we have ~ (t) 0 l for all t 2 [0; T ]. Therefore, the portfolio strategy ~ is K -admissible if and only if z c + (z ? s) u; for all z > 0; where s = S (0), c = g~(S (0)) and = g~?0 (S (0)) 0. Since is assumed to be nonzero, this can be written equivalently as : (5.6) z(1 ? u) u (c ? s); for all z > 0: Now, by the concavity of g~, we have c + (z ? s) g~(z) 0 for all z > 0. By sending z to zero, we see that the right-hand-side of (5.6) is nonnegative. Therefore (5.6) holds if and only if u 1.
2
Remark 5.2 The restriction u 1 means that any admissible portfolio strategy is such that the corresponding wealth proportion invested in the nonrisky asset satis es 1 ? (t) 1 ? u with 1 ? u 0. In other words, we can keep all the money in the stock, if we want to. Remark 5.3 In the case u 1, we have shown that g~(s) is the minimal initial capital in order to perform a K -admissible buy-and-hold strategy which allows to dominate the contingent claim de ned by the payo function g^. The last assertion always holds (without assuming u 1) whenever g~?0 (S (0)) = 0, see Remark 5.1. Therefore, by de nition of U (see Section 2), since g^ g, it follows that : U (0) g~(S (0)) (5.7) whenever u 1 or g~?0 (S (0)) = 0. We now state the main result of this section which holds for lower semicontinuous, nonnegative and nonzero payo functions g and under the assumptions of Sections 2 and 3. Theorem 5.1 Let Assumptions 5.1 and 5.2 hold and suppose - either g~?0 (S (0)) = 0 (5.8) - or u 1: Then we have : U (0) = g~(S (0)): 16
Proof. From Lemma 5.1 (i), the function s 7! V (t; s) is concave for all 0 t < T . Moreover, from (4.5) and Lemma 5.1 (ii), we have :
V (t; s) g(s); (t; s) 2 [0; T ) IR+ : Furthermore, from Lemma 5.2 V (t; ) is invariant by H for all t 2 [0; T ). From Lemma 5.4, this implies that V (t; s) g~(s) for all (t; s) 2 [0; T ) IR+ . Therefore, from Proposition 3.1, we have :
U (0) g~(S (0)):
2
The required result follows from (5.7). The conditions of Theorem 5.1 can be weakened for convex payo functions g.
Theorem 5.2 Suppose g is convex and let Assumption 5.1 and restriction (5.8) hold. Then, we have :
U (0) = g~(S (0)):
Proof. The inequality U (0) g~(S (0)) follows from (5.7). To see that the reverse inequality holds, notice that, by considering the controls (; ) = 0, (4.1) implies :
V (t; s) E Q0;0 [g(S (T ))j(S (t); Y (t)) = (s; y)] g(s); (t; s) 2 [0; T ) IR+ ; where we used Jensen inequality and the martingale property of fS (t), 0 t T g under Q0;0 . Now, from Lemma 5.2, V (t; ) is invariant by H for all t 2 [0; T ) and, by Lemma 5.1 (i), V is concave. Therefore, the required inequality follows from Lemma 5.4 and Proposition 3.1. 2
Remark 5.4 From the proof of Theorems 5.1 and 5.2, it is clear that without the restriction (5.8), we still have :
U (0) g~(S (0)):
Remark 5.5 It is now clear that the HJB equation (4.16) for V is not necessarily satis ed as equality. In particular, that is the case if the concave envelope g~() is strictly concave.
17
6 Bounded volatility Assume rst that there are no constraints and
Assumption 6.1 (bounded volatility) inf (t; s; y) = (t; s) and sup (t; s; y) = (t; s); for all (t; s) 2 [0; T ) IR+ ;
y2IR
y2IR
where and satisfy the same assumptions as .
This is the framework of Avellaneda, Levy and Paras (1995), and we rst want to show that we can recover their result that the minimal super-replication price in this market satis es U (0) V~ (0; S (0)) where V~ (t; s) is a solution to the Barenblatt PDE (6.1) ?vt + 21 s2 2(vss)? ? 12 s22 (vss)+ = 0; v(T; s) = g(s); (6.2) if a unique classical solution exists, which we assume. (In Appendix 2, we provide sucient conditions on the coecients , and the terminal payo function g which ensure existence of a unique solution to the Barenblatt PDE (6.1).) In fact, we shall prove here that U (0) = V~ (0; S (0)). Let, then, V~ be the solution to (6.1)-(6.2). We rst show that V V~ , hence U (0) V~ (0; S (0)). We already know that V does not depend on y. Moreover, from (4.14) we conclude that V is a lower semicontinuous viscosity supersolution of (6.1) with V (T ?; s) g(s) for all s > 0. Therefore, the maximum principle ensures that V V~ . The opposite inequality V~ (0; S (0)) U (0) is quite straightforward : It is sucient to prove that one can super-replicate g(S (T )) if one starts with the initial capital V~ (0; S (0)) at time 0. From (6.1), it is easily seen that ?LV~ 0. By It^o's rule we then get :
g(S (T )) = V~ (T; S (T )) V~ (0; S (0)) +
Z
0
T
V~s(t; S (t))S (t)[1 dW10(t) + 2dW20(t)]:
(6.3)
However, the right-hand side is the value at T of a wealth process starting with V~ (0; S (0)) at time 0, holding V~s(t; S (t)) shares of stock at time t, and it super-replicates g(S (T )).
2
Remark 6.1 The proof of V V~ given above remains the same in the case we assume that V~ is only a viscosity solution to the Barenblatt PDE. 18
We now extend the previous result to the case of constraints, in the case of linear dynamics for S :
Assumption 6.2 Functions (t; s; y), (t; s; y) and i(t; s; y), i = 1; 2, do not depend on s. The following result holds for lower semicontinuous, nonnegative and nonzero payo functions g and under the assumptions of Sections 2 and 3.
Theorem 6.1 Under Assumptions 6.1 and 6.2, and in the presence of constraints described by set K = [l; u], the minimal super-replicating cost is given by U (0) = V~ (0; S (0)) where V~ is the unique classical solution (if exists) to the Barenblatt PDE (6.1), with the terminal condition
V~ (T; s) = g^(s);
(6.4)
where g^ = H (g) is de ned in (5.1). The corresponding super-replicating portfolio is given by ~ (t) = S (t~)Vs(t; S (t)) : (6.5) V (t; S (t) Moreover, if there are no constraints (K = IR), the same is valid without assuming Assumption 6.2.
Proof. We have already shown the last statement. For the rest, we use representation
(4.1) for V (t; s). We follow the probabilistic argument of Broadie, Cvitanic and Soner (1996). Ru ( s ) ds 0 Denote, for a xed , S (u) = S (u)e t for t u T and notice that the distribution 0 ; of fS (u), t u T g under Q does not depend on . Then, by Jensen's inequality and R convexity of , and using the fact that for () 2 D we have tT (s)ds 2 K~ , a.s., we get for t < T,
V (t; s; y)
=
sup E Q; g S 0(T )e?
(; )2D
R
T t
(s)ds
R
h
T
e?( t
(s)ds)
sup E Q; g^(S 0(T )) (S 0; Y )(t) = (s; y)
(; )2D h sup E Q;0 g^(S 0(T )) 2
(S 0; Y )(t) = (s; y)
i
(6.6)
i
(S 0 ; Y )(t) = (s; y) :
We also want to prove the reverse inequality: Let fk g be a maximizing sequence such that g(se?k )e?(k ) ! g^(s), and choose the constant controls k (u) = T?k t to get (for t < T )
V (t; s; y) sup E Q;k g S 0(T )e?k e?(k ) (S 0 ; Y )(t) = (s; y) : h
2
19
i
(6.7)
By standard estimates on di usion processes, one concludes that S 0(t) converges in L2 to S 0(T ), as t ! T , therefore also almost surely along a subsequence. From this, by Fatou's lemma and lower-semicontinuity of g, we get lim inf V (t; s; y) g(se?k )e?(k ) : t!T
(6.8)
V (T ?; s; y) g^(s):
(6.9)
Taking the limit as k ! 1 we get
Therefore, in order to super-replicate g(S (T )), we have to super-replicate at least g^(S (T )), which implies that we can replace g with g^ in (4.1). But then, repeating the arguments of (6.6) with g replaced by g^, and using the fact that g^ = g^ we see that, for t < T ,
V (t; s; y) = =
sup
E Q;
g^(S (T ))e?
(; )2D h sup E Q;0 g^(S 0(T )) 2
RT
t
( (s))ds
(S; Y )(t) = (s; y) i
(S 0; Y )(t) = (s; y) :
(6.10) (6.11)
In other words the supremum over the processes 2 D is obtained at () 0. But now we are back in the case without constraints, with the payo g^, and it follows that V satis es the Barenblatt PDE (6.1) with the terminal condition (6.4). In particular, V = V~ is a classical solution to the HJB equation (4.16), hence (sVs + ( )V )(t; s) 0, for all 2 K~ , (t; s) 2 [0; T ) IR+ . By Theorem 13.1 in Rockafellar (1970), this implies that the portfolio (6.5) satis es the constraints. Consequently, V (0; S (0)) U (0), hence V (0; S (0)) = U (0).
2
7 Mixed case In this section we get similar results for the mixed case, in which Assumption 5.1 on the volatility holds, but its in mum is not necessarily zero:
Assumption 7.1 inf (t; s; y) = (t; s) for all (t; s) 2 [0; T ) IR+ ;
y2IR
and satis es the same assumptions as .
The following result holds for lower semicontinuous, nonnegative and nonzero payo functions g and under the assumptions of Sections 2 and 3. 20
Theorem 7.1 Under Assumptions 5.1, 6.2 and 7.1, and in the presence of constraints de-
scribed by set K = [l; u], the minimal super-replicating cost for the contingent claim g(S (T )), is given by U (0) = V~ (0; S (0)) where V~ is the classical solution (if exists) to the PDE (7.1) vt + 21 s22 vss = 0; v(T; s) = g~(s); (7.2)
The corresponding super-replicating portfolio is given by ~ (t) = S (t~)Vs(t; S (t)) : (7.3) V (t; S (t) If there are no constraints (K = IR), then the same is valid even if Assumption 6.2 is not satis ed.
Proof: We only sketch the proof, since it is similar to the ones above. Again, we rst
assume that there are no constraints. By Lemma 5.1 we know that function V is concave. It also has to satisfy V (T ?; )) g(), therefore V (T ?; ) g~(). Moreover, from (4.14) we see that V is a viscosity supersolution to : (7.4) ?vt + 21 s2 2vss = 0: By the maximum principle we get then V V~ , hence U (0) V~ (0; S (0)). The reverse inequality is easy to show using It^o's rule on V~ . The case with constraints is then proved as in the previous section. 2
8 Examples 8.1 European put options European put options are de ned by the terminal payo function
g(s) = ( ? s)+; s 2 IR+ ;
for some 0. Using, the characterization of g~ given in Lemma 5.5 and recalling that K contains 0, we see that g~(s) = and therefore g~0(s) = 0. It follows that restriction (5.8) is satis ed. By Theorem 5.2 we have that, under Assumption 5.1, the super-replication cost of the European put option is given by :
U (0) = and the corresponding super-replicating portfolio is ~ (t) = 0 for all t 2 [0; T ]. 21
8.2 European call options European call options are characterized by the terminal payo function
g(s) = (s ? )+; s 2 IR+ : We could get the super-replication cost directly, but we choose to use the put-call parity s ? = (s ? )+ ? ( ? s)+, to transform the problem to the case of European put options. Assume rst that there are no constraints, i.e. l = ?1 and u = +1 (> 1). Then from Proposition 3.1 :
U (0) sup E Q;0 (S (T ) ? )+ h
2
i
= sup E Q;0 ( ? S (T ))+ + (S (T ) ? ) h
2
= S (0) ? + sup E Q;0 ( ? S (T ))+ h
i
i
2
since the process fS (t), 0 t T g is a martingale under Q;0 for any 2 . Therefore, applying the results of the previous section concerning European put options, we see that, under Assumption 5.1, the super-replication cost of the European call option in the absence of portfolio constraints satis es U (0) S (0). The reverse inequality is trivial (see also (5.7)). Therefore
U (0) = S (0): Now, in the presence of constraints, this is, obviously, also the cost if u 1, since then one can still buy one share of the stock. If u < 1, g^ g~ 1, and it is impossible to super-replicate the call, see Remark 5.4.
Remark 8.1 The above examples are extreme, in the sense that the super-replicating strategies are expensive and not very interesting from the practical point of view. The example below is less extreme because the volatility process is bounded. For more examples as well as numerical calculations see Broadie, Cvitanic and Soner (1996) and Avellaneda, Levy and Paras (1995).
8.3 A butter y spread option
We consider here an example in which the volatility is bounded between 2 (t; s) :1 and 2 (t; s) :2. To make things more interesting, we look at a payo that is neither convex nor concave, given by
g(s) = (s ? 1)+ if s 2; g(s) = (3 ? s)+ if s 2: 22
(8.1)
Suppose that the constraints are given by the interval K = [?2; 2]. We will calculate the price using Theorem 6.1. It is easy to see that
g^(s) = s2=4 if s 2; g^(s) = 4=s2 if s 2: Numerical results are given in the table below, for a range of initial stock prices and times to expiration. The values of the portfolio are called \hedge ratio". There is a rounding error, that sometimes gives the portfolio values slightly outside of set K . For comparison, the second table gives the Black-Scholes price of g(S (T )) without constraints, and with volatility assumed to be constant and equal to 2 = :15.
23
Appendix 1 : proof of Proposition 4.1 We now show that V is a supersolution in the viscosity sense of the Bellman equation in Proposition 4.1. This proof is standard and is essentially based on the dynamic programming principle. We refer to Crandall, Ishii and Lions (1992) or Fleming and Soner (1993) for de nitions and standard estimates of the theory of viscosity solutions.
Lemma A.1 (Dynamic Programming Principle) We have, for all 0 < t < u < T , V (t; s; y)
sup
(; )2D
E Q;
V (u; St;s;y(u); Yt;s;y
(u)) e?
Ru
t
( (r))dr
;
where f(St;s;y (u); Yt;s;y (u)); t u T g is the process (S; Y ) satisfying (2.1)-(2.2) with the initial condition (S (t); Y (t)) = (s; y).
Proof: See Cvitanic and Karatzas (1993), Proposition 6.2 (notice that, although Cvitanic
and Karatzas (1993) assume bounded coe ecients for the di usion driving (S; Y ), their proof of the dynamic programming principle does not require such assumptions). 2 n n n We rst show that V is lower semi-continuous. Let (t ; x ; y ) be a sequence such that (tn; sn; yn) ! (t; s; y). By standard estimates of di usion processes (using the Lipschitz property of the parameters and Gronwall inequality), we get that Stn ;sn;yn (T ) and Ytn ;sn;yn (T ) converge in L2 to St;s;y (T ) and Yt;s;y (T ), respectively. So, there exists a (relabeled) subsequence (tn; sn; yn) so that the above convergences are a.s. convergences. We have then, for a xed (; ) 2 D, by Fatou's lemma, and since g is lower semicontinuous, limninf V (tn; sn; yn)
R
T limninf E ; [g(Stn ;sn;yn (T )e? tn ((r))dr ] RT E ; [limninf g(Stn ;sn;yn (T ))e? tn ((r))dr ] RT E ; [g(St;s;y (T ))e? t ((r))dr ];
(A.1)
and we are done, after taking sup over (; ). Next, x (t; s; y) 2 [0; T ) IR+ IR and consider a test function ' 2 C 2([0; T ] IR+ IR) such that 0 = (V ? ')(t; s; y) =
min (V [0;T ]IR+IR
? '):
By substituting ' for V (and t + h for u) in the inequality of the previous lemma, we get 0
sup
(; )2D
E Q;
' (t + h; St;s;y (t + h); Yt;s;y 24
R t+h ( ( u )) du ? ? '(t; s; y) (t + h)) e t
and by It^o's lemma, and standard estimates on the state process (S; Y ) under the controlled probability measure Q; , we see that (
1 E Q; lim inf h!0
" Z
h
t+h t
J ; ' (u; S
t;s;y (u); Yt;s;y (u)) du
#)
0; for all (; ) 2 D;
where J ; is the di erential operator appearing inside the in mum of (4.2). Therefore, we obtain inf
(; )2IRK~
J ; '(t; s; y) 0:
Finally, the boundary condition is obtained as follows. Considering = 0 as control in (4.1) and taking limits as t approaches T , we see that : lim inf V (t; s; y) g(s); s 2 IR+ t%T by Fatou's Lemma.
2
Appendix 2 In this Appendix, we provide conditions on the coecients , and the payo function g which guaranty the existence of a unique solution V~ to the Barenblatt PDE (6.1). Consider the auxiliary optimal control problem :
V~ (t; s) = sup E [ g(S (T ))j S (t) = s] ; 2C
where C is the set of all fF (t)g adapted processes valued in [0; 1] and S is the controlled process de ned by the stochastic di erential equation : dS (t) = ~ (t; S (t); (t))dW (t) S (t) with
~ (t; s; a) = (1 ? a)(t; s) + a (t; s): The Bellman equation associated to this optimal control problem is given by : n o vt (t; s) + 12 sup s2~ 2 (t; s; a)vss(t; s) = 0 0a1 25
which is exactly the Barenblatt PDE (6.1). Furthermore, at the terminal date T , we have V~ (T; s) = g(s). Since the controls set is bounded we can apply standard regularity results for the HJB equation (6.1). We denote by Cbk (IR) the space of real-valued functions de ned on IR such that and its derivatives of order k are continuous and bounded. The space Cbk;m([0; T ] IR) is de ned similarly. The following assumptions are made.
A1. There exists some " > 0 such that (t; s) " for all (t; s) 2 [0; T ] IR+ . A2. Let ^ (t; x; a) = ~ (t; ex; a). Then, for xed a 2 [0; 1], ^ 2 C 1;2 ([0; T ]; IR) and the functions ^ , ^t , ^x and ^xx are bounded in [0; T ] IR [0; 1]. A3. The function g^ : x 7?! g(ex) lies in Cb3 (IR).
Then, under Assumptions A1, A2 and A3, the value function V~ is the unique solution of (6.1)-(6.2) in the class of functions v such that (t; x) 7?! v(t; ex) lies in Cb1;2([0; T ]; IR), see Theorems 4.2 and 4.4 in Fleming and Soner (1993).
ACKNOWLEDGEMENT: We are grateful to Victoria Pikovsky for the numerical calculations in section 8.3, to H. Mete Soner for providing us with some \viscosity tips", and to Yuri Kabanov for helpful comments that prompted the introduction of Lemma 4.1.
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27
