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Optimal hedging in a dynamic futures market with a nonnegativity constraint on wealth
Journal
of Economic 20(1996)
Dynamics and Control 1101-1113
Optimal hedging in a dynamic futures market with a nonnegativity constraint on wealth Abraham Lioui”, Patrice Poncet*‘avb “University of Paris--I Sorbonne, Paris, France bESSEC, Finance Department, 95021 Cergy Pontoise Cedex, France (Received
September
1994; final version
received
September
1995)
Abstract This paper examines the issue of optimal hedging demands for futures contracts from an investor who cannot freely trade his portfolio of primitive assets in the context of lognormal, rather than normal, returns and of a constant absolute risk aversion utility function. In this context, the nonnegativity constraint on wealth is binding and the optimal hedging demands are not identical with those encountered in the bulk of the literature, which has largely overlooked this problem. Negative results concerning the derivation of equilibrium in the futures markets and the computatiorl of the open interests then follow. Key words: Hedging; Financial futures; Solvability constraint; JEL classijcation: Gl 1; G13
Martingale approach
1. Introduction The theory of dynamic hedging considers an investor holding a fixed, nontraded portfolio of spot market assets and choosing a continuously rebalanced (hedge) portfolio of futures contracts so as to maximize the expected utility of his terminal wealth. Hedging profits and losses on the futures positions being marked to market in an interest-bearing (or paying) margin account, terminal
*Corresponding
author.
We have benefited from fruitful discussions with Bernard Dumas, Roland Portait, and the participants of the XVth International Conference of Finance organized by the French Finance Association in Tunis, June 1994. We have also received very helpful comments from an Associate Editor. All remaining errors are ours.
0165-1889/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0165-1889(95)00891-X
1102
A. Lioui, P. Poncet /Journal
of Economic Dynamics and Control 20 (1996) IIOI-1113
wealth is the sum of the final value of the fixed portfolio of spot market assets and the terminal value of the margin account. Solutions to this hedging problem have been derived mostly in terms of the derivatives of the value function deemed to solve the Hamilton-Jacobi-Bellman equation for optimal control, which precludes closed-form equations for the individual’s various demand functions’ (see, for example, Anderson and Danthine, 1983; Breeden, 1984; Ho, 1984; Stultz, 1984; Adler and Detemple, 1988a, b; Poncet and Portait, 1993; Svensson and Werner, 1993). A notable exception is Duffie and Jackson (1990) who provide explicit solutions in several special cases, such as when asset prices are Gaussian, which allows negative prices with positive probability, and utility functions exhibit constant absolute risk aversion (CARA). Unfortunately in all cases, except when the utility function exhibits infinite marginal utility at zero wealth (such as the logarithmic), the nonnegativity constraint on wealth is binding and the space of admissible trading strategies is thus restricted. The task then is to specify how one could handle this problem since, in general, in the real world a state of bankruptcy will drive the investor out of the market. The purpose of this paper is to characterize the restriction and show how the constraint modifies the optimal solution of the nonconstrained problem. We will assume, like Duffie and Jackson (1990), that utility functions exhibit CARA but, unlike them, that asset price increments are lognormal rather than Gaussian. In addition to forbidding negative asset prices, this assumption also prevents hedgers’ myopia which occurs when asset prices are Gaussian, by which is meant that investors hedge only local changes in wealth, thus construct optimal dynamic hedges that are the same as the corresponding static ones. The remainder of the paper is organized as follows. Section 2 outlines our framework, while Section 3 derives the dynamics of optimal wealth and the optimal hedging demands for futures contracts. Section 4 contains conclusions, briefly discusses the issue of equilibrium, and offers a discussion of possible directions for further research. The Appendix includes a technical derivation.
2. The framework We consider an individual choosing a dynamic futures trading strategy to maximize his expected utility of wealth at a future date T, in the following environment.
‘See Adler and Detemple (1988b) for a lucid discussion of why closed-form solutions obtain (only) for the log-utility (Bernoulli) investor, when the nontraded asset(s) is (are) perfectly correlated with its (their) associated futures contract(s).
A. Lib,
P. Poncet 1 Journal of Economic Dynamics and Control 20 (1996) 1101-1113
1103
Assumption 1: There exist N primitive assets to be hedged, whose value obeys an N-dimensional diffusion process S supposed to be well-defined, with stochastic differential representation:
dst = ~sJ@,S,)dt + zs,o(t, S,)dZ,,
(1)
where Is, denotes the diagonal (N x N) matrix with the elements S:’ (j = 1, . . , N), ~1is an N-dimensional process, 0 is an (N x N) matrix, and Z denotes a standard Brownian Motion in RN, Z = (Z’, . . , ZN)r, where z indicates transpose. Hence the uncertainty is formalized by the complete filtered space (n, F, {F f >tsco,Tl, P), where P is the actual (historical) probability and the filtration is the augmented filtration generated by the Brownian Motion. The matrix u is assumed to be full rank, thus markets are complete for unconstrained investors, i.e., those who can freely trade on these assets. Note that Duffie and Jackson’s framework (1990) allows the market for primitive assets to be incomplete, but only under severe restrictions on both the utility functions and the price processes can they derive explicit solutions. To ease the exposition, we note ~(t, S,) = ~~ and a(t, S,) = (T,. Also, the primitive assets do not pay any dividends between 0 and T. Assumption 2: There exists a constant continuously compounding interest rate r 2 0 at which investors may lend or borrow. It follows immediately that forward and futures prices are equal. Assumption 3: Markets are frictionless and there are no arbitrage opportunities. This, together with the completeness assumption, implies that there exists a unique probability measure Q, associated with the riskless asset’ yielding r, that makes discounted asset prices martingales (see Harrison and Kreps, 1979; Harrison and Pliska, 1981). Q is constructed such that
dQ
dPF,
(c;l(p,-
rZ))‘dZ, -f
0
rIc;‘(C(,
- r1)12dt
.
s 0
To simplify further computations, we assume that the ‘market price of risk’ IC,E C; ’ ( pt - r I) is deterministic, which makes the diffusion process (1) only slightly more general than a Geometric Brownian Motion. Thus:
(2)
‘The riskless asset is worth
one at time 0, then is worth
e” at t > 0.
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A. Lioui, P. Poncet / Journal of Economic Dynamics and Control 20 (1996) IIOI-1113
Note that, under Q, using Girsanov’s Theorem, the process 2: such that Z: = Z, + si ~,ds is a Brownian Motion. Assumption 4: There exist N futures contracts available for trade, written on the N primitive assets, whose maturity date, T, coincides with the investor’s horizon. Given the absence of arbitrage opportunities (F, = S, erCT-‘)) and using ItB’s Lemma, the futures prices process F, is given by
dF, = I,,q,dt + ZFtvtdZt,
(3)
withq,=p(,--rlandv,=o,. Notice that the existence of these N futures contracts ensures the completeness of the market for the constrained investor who cannot freely trade the primitive assets S but can freely trade the riskless asset. A hedge is an N-dimensional stochastic process 8 3 (O’, . . . , ON)’ specified by a futures position in each futures contract.3 At time t, the position is marked to market, so that gains and losses generated by futures price changes accrue to the hedger’s margin account in a continuous manner. Assumption 5: The margin account current value, denoted by X,, is credited or debited (if negative) with interest at the constant continuously compounding rate r. The margin account then is equal to
fs
x, =
e’@-“)&dF,
0
Applying It& lemma and using (3) yields dX, = rX,dt + O;dF, = (rX, + 8:1,q,)dt
+ 8:1,,v,dZ,.
(4)
Assumption 6: The investor holds a fixed, nontraded portfolio l7 of the N assets, with terminal random value H’ST at horizon date T. Given the futures strategy 0 and denoting by B1the number (possibly negative) of units of riskless asset, his total wealth at time t is W, such that W, = FS, + X, + j$e” and4 dW, = Il’dS, + dX, + p,rerfdt = (nrI,~t ‘For
technical
conditions
+ rX, + &IF,v, + j&re”)dt + (Z7’1,o, + O:Z,,v,)dZ,. satisfied
by these strategies,
“Note that PO is equal to zero since the investor’s assumption.
see Cox and Huang
(5)
(1989).
initial wealth is equal to IIrS,, and X,, is zero by
A. Lioui, P. Poncet /Journal
of Economic Dynamics and Control 20 (1996) 1101-1113
1105
7: The investor is a Von Neumann-Morgenstern expected utility of terminal wealth maximizer and his utility function exhibits constant absolute risk aversion, i.e., U(W,) = - ee-Ywrwhere y > 0 is the coefficient of absolute risk aversion. Assumption
3. The hedging demand for futures
We first explain briefly the general ideas behind the methodology, then apply it to the specific problem at hand, namely to derive the optimal terminal wealth, the dynamics of both the constrained and unconstrained wealths, then the optimal hedging demands for futures contracts. 3.1. The methodology
The main idea of Karatzas, Lehoczky, and Shreve (1987) and Cox and Huang (1989, 1991) is to transform the dynamic optimal control problem into a static one by solving first for the terminal optimal wealth WT. Then we compute the value W, at each time t of this wealth. Identifying W, with the wealth process given by (5), it is easy to get the optimal hedging demands for futures contracts. When markets are complete, it is clear that there always exists a replicating strategy using futures that insures the choosen level of this terminal wealth. When markets are incomplete, things get complicated and we refer the interested reader to, e.g., He and Pearson (1991). Recall that our setting implies complete markets even for our constrained investor because of the existence of N futures contracts and of the riskless asset. In the process, two cases are possible. For some utility functions, the nonnegativity constraint on wealth is not binding, such as when the utility function exhibits infinite marginal utility at zero wealth (e.g., the logarithmic). For the others, the constraint is binding but Cox and Huang (1989) provide us with the following powerful equivalence result: the solution to the constrained program is equal to the solution to the unconstrained one to which is added an insurance contract (a protective put of strike zero written on the unconstrained terminal wealth) which will pay off the negative wealth if it occurs. In other words, the optimal constrained strategy will allocate the entire initial wealth between the unconstrained strategy and the protective put written on the unconstrained strategy. 3.2. The derivation of terminal wealth We derive first the solution to the unconstrained program; then we compute the optimal terminal wealth for the constrained problem, using the Cox-Huang method. We will denote I@ the solution to the unconstrained problem, with
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A. Lioui, P. Poncet /Journal
of Economic Dynamics and Control 20 (1996) 1101-1113
initial value wO, and denote W the solution to the constrained problem, with initial value Wo. Hence, W. 2 w,,. Like Cox and Huang, we now solve the following, unconstrained, program: max EP[ - e-@‘r], W, = EP[@Trzre-rT] i s.t. - EQ[@re-‘r]
(6) = @,,,
where I@e stands for the initial wealth necessary to solve the investor’s unconstrained program. It will become clear that this amount is less than (or equal to) the real Wo. The unique solution to (6) is such that
where 1 is the Lagrangian multiplier, or
@r=
-:ln
:
+i[rT--lnar].
(7)
0 Using the budget constraint EQII@re-rT] value given by (2), we have
= I@,, and substituting UT for its
Substituting in (7) the value of - (l/y)ln(A/y) given above yields 1
pT = tioerT +-
T
K; dZ:,
Ys 0
where it is readily seen that terminal wealth can be negative.
(8)
A. Lioui. P. Poncet /Journal
of Economic Dynamics and Control 20 (1996) 1101.-1113
1107
We now solve explicitly for the constrained program: max EP [ - e - Ywr], WT s.t. EQ[WTe-‘T] = WO,
(9)
Given Cox and Huang (1989, Thm. 2.4, p. 64), the solution to (9) is
w, = [PVT] +.
(10)
It is worth noting at this stage that a) knowing investor’s actual initial wealth to the unconstrained program (6), we readily get the component WI, devoted to the unconstrained strategy, and b) [ tiT]’ = I@r + max [O; - pT] which highlights the put (insurance) component of the optimal constrained strategy. WO and the solution
3.3. The dynamics of optimal wealth We first solve for the dynamics of the optimal constrained wealth. Then, by identification with Eq. (5), we solve explicitly for the optimal demands for futures contracts. The value W, at t of optimal terminal wealth given by (10) is W,e-”
= EQIWr-ePrT]F1].
Hence, using (8),
Since we have assumed that the market price of risk K~is deterministic, we can compute explicitly the expectation present in the above equation.5 Denote A the event of nonnegative terminal wealth WT. This occurs when
s T
bPoerT +
1 K:dZ: Y 0
2 0,
‘Had we assumed more generally that K, is stochastic, we would have ended up with a partial differential equation for the optimal wealth at each time t - such as Cox and Huang (1989) in their general framework - without being able to obtain an explicit solution.
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A. Lioui, P. Poncet /Journal
of Economic Dynamics and Control 20 (1996) IIOI-1113
s s T
K: dZ: 2 - y I@, erT,
0
T
rc5dZ: 2 - (yqoerT
+ ji h-:dZ:).
f
With respect to Q, the LHS of this equation is independent of F, and Gaussian with mean zero and variance JT 1rcslzds. Thus, we can write
where u is the standard (0, 1) Normal. It is important to note that d, is a function of optimal unconstrained wealth at time t, as will be shown later on. Then:
W, = e-r(T-t)
=e
-r(T-1)
w. erT EQ[l,] POerTN(d,)
+ + E Q[ ( [; IcfdZs* + lfT .:dZ:)UF,]~
-t
dN(4)
* s KS
dZ:
0
where IA is the indicator function. As shown in the Appendix, the last term EQ [ ] is equal to n(d,) [ jT )K,J* ds] 1/2where n( ) is the normal density function. Thus:
(
W, = emrcT-*) N(d,)
(_ Woe rT + -:saK:dZ:)+5n(d,)[J]l,r,,‘ds]lllj. (11)
A. Lioui. P. Poncet J Journal of Economic
Dynamics
and Control 20 (I 996) 1101-I 113
1109
Using (8) and the definition of Q, we have mtemr’ = EQII@‘Te-‘TJF,]
Note that the term under expectation in the RHS is orthogonal to F, and of expectation 0 with respect to Q. Then we get:
fs
dZ:
(12)
--n(d,)[ {;,KS,zds]‘:2. Y
(13)
KS
Y
0
Using (12) (11) becomes e-r(T-l)
IV, = W’,N(d,) +
Note for further reference that using (12) and the definition of dt, we have
(14) It remains to derive the dynamics of optimal constrained wealth at any time t. Applying Itb’s Lemma to Eq. (13) and substituting dZ, for dZ: ( = dZ, + K,dt), we get6 e-r(T-t)
dW, = g( .)dt + N(d,)--Y
6 dZt,
(15)
where g( ) is the (complicated) drift of the wealth process whose exact value is of no interest here. 3.4. The optimal hedging demands In order to derive the vector 0 of optimal positions in the futures contracts, we now identify the dynamics of wealth given by Eq. (5) with that given by Eq. (15).
6The detailed
derivation
is given in the Appendix
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A. Lioui. P. Poncet / Journal of Economic Dynamics and Control 20 (1996) IIOI-1 I I3
For that purpose, it seems at first sight necessary to identify both the drift terms and the diffusion coefficients. However, it is in fact sufficient to identify the latter, forgetting altogether the former, since, by arbitrage, it is clear that two optimal strategies of identical initial value and identical risk must have the same expected return.’ Thus, from Eqs. (5) and (15) e-r(T-t)
zITZs,a, + e:z&v, = N(d,) ___ Y
lcrt*
Then, using IF, v, = er(T-f)ZS,rr.Iand kt = o;‘(p, 0;
=
e-*u-‘)
[
- rl) = a,‘~~, we have
nz + ~N(d,)(a;‘tl,)‘(lriv,)-’ . 1
Eq. (16), which is the main result of this paper, elicits the following remarks: l It is clear that the role of (the adequate number of) futures contracts in essentially complete primitive markets is to allow investors, who cannot freely trade on the primitive assets, to achieve the same optimum they would arrive at if they were not constrained. l The first term in the RHS is the so-called ‘pure hedge’ component of the strategy. The presence of the discount factor is due to the fact that the future position is continuously marked-to-market as opposed to the cash position which is not. l The second term represents the ‘speculative’ component. It differs from the one usually encountered in the literature because of the explicit consideration of the nonnegativity constraint on wealth, which introduces the extra term N(d,). l The term N(d,) is reminiscent of the Black-&holes formula. This does not come out as a surprise since, as we have seen previously, the investor implicitly must buy a protective European put written on his unconstrained wealth. However, it is important to recall that the unconstrained terminal wealth @r, which may be negative and which is the underlying asset of the put, is not lognormally but normally distributed under both measures P and Q. Also note that the second term of the RHS of (16) is multiplied by N(d,) < 1 and, thus, the speculative demand for futures is smaller, in absolute terms, than the one which would obtain without the nonnegativity constraint on wealth. This is because
‘Note that direct identification of the drift terms is almost untractable due, in particular, to the presence of X, in the drift term of Eq. (5). This explains why we did not have to be ;xplicit about g(.) in Eq. (15).
A. Lioui, P. Poncet 1 Journal of Economic Dynamics and Control 20 (1996) I 101-l 113
1111
being long in the put is equivalent to being short in the underlying asset. The intuition behind this result is that the investor must impose limits on his trading of futures contracts without which his wealth could become negative. l Eq. (16) reveals a major difficulty ignored in the literature: even in the case of Geometric Brownian Motions, and perfect correlation between futures and spot assets, it is not possible to derive ‘closed-form’ solutions for the optimal hedging demands, in the sense that 0 depends on d, and d, depends, not on effective wealth W,, but on ‘unconstrained’ wealth J@,. It is not indeed possible to recover W, since the relation between @, and W, is not invertible. Thus, we have shown that only for those utility functions that exhibit infinite marginal utility at zero wealth (e.g., the logarithmic) can we get closed-form solutions in the aforementioned sense. One can view this as a negative result. l Note however that Eq. (16) is not worthless. Indeed it is possible to solve with numerical methods for flO, then @, and then 0,.
4. Conclusion The main thrust of the paper is that the literature relative to the use of futures contracts as hedging vehicles has largely ignored the issue of possible negative wealth. Since in general bankrupt investors are driven out of the market, we must constrain the set of admissible strategies to the set of strategies for which only nonnegative wealth can occur. Except in the case of very special utility functions such as the logarithmic, for which this constraint is not binding, this implies that the ‘speculative’ component of optimal hedging demands is smaller, in absolute terms, than is usually reported. Also, it is not possible to derive the hedging demands explicitly in terms of the investor’s actual wealth, making numerical procedures necessary. As far as equilibrium is concerned, we know that, by arbitrage, the drift term of the diffusion process for F, is Q = pL,- rl. Thus the research for an ‘equilibrium’ q1 that would cancel the global excess demand for futures contracts, a la Duffie and Jackson (1990), is at best misleading. Aggregation through investors in our setting, and thus the derivation of open interest for constrained investors in each futures contract, is untractable. Potential extensions of our framework would include the following: a) dividend-paying primitive securities, b) maturity dates of the futures contracts not equal to the investor’s horizon, c) a stochastic riskless interest rate, and d) a nonperfect correlation between the returns of the primitive assets and the futures. While a) and b) are easy to deal with, c) is a serious problem since, in particular, the simple cash-and-carry relationship does not hold any longer. As for d), it is well-known since Adler and Detemple (1988b) that it is both a crucial and a difficult task since then markets are not complete for constrained investors. This work is in progress in a companion paper.
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A. Lioui, P. Poncet / Journal of Economic Dynamics and Control 20 (1996) I IOI-I I I3
Appendix
Computation of L = EQ[(jT K3dZ:)la(F,]
in Eq. (II)
Recall that j,r ~:dZz is independent of F, and Gaussian with respect to the measure Q of mean zero and variance s,r [K,)~ ds. Then:
= EQ[ul,].
(from the definition of u and d, in the text)
[ j-:lgl’d]l’2
=[~~l~~12ds]1’2~~q~.eV.‘2du.
Let y 3 u2/2. Then: L =[JrIrcs12ds]"' Jl,2&e-Ydy
=[J:Iri,lzds]l12n(d,),
where n( ) is the normal density function. Derivation of Eq. (1.5) In fact, we are only concerned with the diffusion coefficient of the investor’s wealth. Thus, we need not specify the drift term which could be calculated explicitly. For further reference, applying Ito’s lemma to (12) yields dW’, = (.)dt + :e-‘(‘PL’K:dZ:
= (.)dt + ~e-‘(T-‘)h_~dZ,.
(A-1)
Now, using Leibnitz rule, we get
aN(d,)
=
n(d,).
Applying Ito’s Lemma to (13) leads to dW, = N(d,)dW’, + W&&)dd, + (.)dt e-r(T-t) - 4 --n(dt)[ Y
~,rla,12ds]llldd,.
64.2)
A. Lioui. P. Poncet /Journal
of Economic Dynamics and Control 20 (1996) 1101-1113
1113
Substituting for d, in (A.2), this equation becomes dW, = N(d,)d@, + (.)dt.
(A.3)
The result (15) follows by substituting for (A.l) in (A.3).
References
Adler, M. and J. Detemple, 1988a, Hedging with futures in an intertemporal portfolio context, Journal of Futures Markets 8, 2499269. Adler, M. and J. Detemple, 1988b, On the optimal hedge of a non-traded cash position, Journal of Finance 43, 143-l 53. Anderson, R.W. and J.P. Danthine, 1983, The time pattern of hedging and the volatility of futures prices, Review of Economic Studies 50, 2499266. Breeden, D.T., 1984, Futures markets and commodity options: Hedging and optimality in incomplete markets, Journal of Economic Theory 32,275-300. Cox, J. and C.F. Huang, 1989, Optimal consumption and portfolio policies when asset prices follow a diffusion process, Journal of Economic Theory 49, 33-83. Cox, J. and C.F. Huang, 1991, A variational problem arising in financial economics, Journal of Mathematical Economics 20, 4655487. Duffie, D. and M.O. Jackson, 1990, Optimal hedging and equilibrium in a dynamic futures market, Journal of Economic Dynamics and Control 14.21-33. Harrison, M. and D. Kreps, 1979, Martingales and multiperiod securities markets, Journal of Economic Theory 20, 381-408. Harrison, M. and S. Pliska, 1981, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and their Applications 11, 2155260. He, H. and N. Pearson, 1991, Consumption and portfolio policies with incomplete markets and short sale constraints, Journal of Economic Theory 54, 259-304. Ho, T.S.Y., 1984, Intertemporal commodity futures hedging and the production decision, Journal of Finance 34, 351-375. Karatzas, I., J. Lehoczky, and S. Shreve, 1987, Optimal portfolio and consumption decisions for a small investor on a finite horizon, SIAM Journal on Control and Optimization 25,1557-1587. Poncet, P. and R. Portait, 1993, Investment and hedging under a stochastic yield curve: A two-statevariable, multi-factor model, European Economic Review 37, 1127 1147. St&z, R.M., 1984, Optimal hedging policies, Journal of Financial and Quantitative Analysis 19. 1277140. Svensson, L.E.O. and I.M. Werner, 1993, Nontraded assets in incomplete markets: Pricing and portfolio choice, European Economic Review 37, 114991168.
of Economic 20(1996)
Dynamics and Control 1101-1113
Optimal hedging in a dynamic futures market with a nonnegativity constraint on wealth Abraham Lioui”, Patrice Poncet*‘avb “University of Paris--I Sorbonne, Paris, France bESSEC, Finance Department, 95021 Cergy Pontoise Cedex, France (Received
September
1994; final version
received
September
1995)
Abstract This paper examines the issue of optimal hedging demands for futures contracts from an investor who cannot freely trade his portfolio of primitive assets in the context of lognormal, rather than normal, returns and of a constant absolute risk aversion utility function. In this context, the nonnegativity constraint on wealth is binding and the optimal hedging demands are not identical with those encountered in the bulk of the literature, which has largely overlooked this problem. Negative results concerning the derivation of equilibrium in the futures markets and the computatiorl of the open interests then follow. Key words: Hedging; Financial futures; Solvability constraint; JEL classijcation: Gl 1; G13
Martingale approach
1. Introduction The theory of dynamic hedging considers an investor holding a fixed, nontraded portfolio of spot market assets and choosing a continuously rebalanced (hedge) portfolio of futures contracts so as to maximize the expected utility of his terminal wealth. Hedging profits and losses on the futures positions being marked to market in an interest-bearing (or paying) margin account, terminal
*Corresponding
author.
We have benefited from fruitful discussions with Bernard Dumas, Roland Portait, and the participants of the XVth International Conference of Finance organized by the French Finance Association in Tunis, June 1994. We have also received very helpful comments from an Associate Editor. All remaining errors are ours.
0165-1889/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0165-1889(95)00891-X
1102
A. Lioui, P. Poncet /Journal
of Economic Dynamics and Control 20 (1996) IIOI-1113
wealth is the sum of the final value of the fixed portfolio of spot market assets and the terminal value of the margin account. Solutions to this hedging problem have been derived mostly in terms of the derivatives of the value function deemed to solve the Hamilton-Jacobi-Bellman equation for optimal control, which precludes closed-form equations for the individual’s various demand functions’ (see, for example, Anderson and Danthine, 1983; Breeden, 1984; Ho, 1984; Stultz, 1984; Adler and Detemple, 1988a, b; Poncet and Portait, 1993; Svensson and Werner, 1993). A notable exception is Duffie and Jackson (1990) who provide explicit solutions in several special cases, such as when asset prices are Gaussian, which allows negative prices with positive probability, and utility functions exhibit constant absolute risk aversion (CARA). Unfortunately in all cases, except when the utility function exhibits infinite marginal utility at zero wealth (such as the logarithmic), the nonnegativity constraint on wealth is binding and the space of admissible trading strategies is thus restricted. The task then is to specify how one could handle this problem since, in general, in the real world a state of bankruptcy will drive the investor out of the market. The purpose of this paper is to characterize the restriction and show how the constraint modifies the optimal solution of the nonconstrained problem. We will assume, like Duffie and Jackson (1990), that utility functions exhibit CARA but, unlike them, that asset price increments are lognormal rather than Gaussian. In addition to forbidding negative asset prices, this assumption also prevents hedgers’ myopia which occurs when asset prices are Gaussian, by which is meant that investors hedge only local changes in wealth, thus construct optimal dynamic hedges that are the same as the corresponding static ones. The remainder of the paper is organized as follows. Section 2 outlines our framework, while Section 3 derives the dynamics of optimal wealth and the optimal hedging demands for futures contracts. Section 4 contains conclusions, briefly discusses the issue of equilibrium, and offers a discussion of possible directions for further research. The Appendix includes a technical derivation.
2. The framework We consider an individual choosing a dynamic futures trading strategy to maximize his expected utility of wealth at a future date T, in the following environment.
‘See Adler and Detemple (1988b) for a lucid discussion of why closed-form solutions obtain (only) for the log-utility (Bernoulli) investor, when the nontraded asset(s) is (are) perfectly correlated with its (their) associated futures contract(s).
A. Lib,
P. Poncet 1 Journal of Economic Dynamics and Control 20 (1996) 1101-1113
1103
Assumption 1: There exist N primitive assets to be hedged, whose value obeys an N-dimensional diffusion process S supposed to be well-defined, with stochastic differential representation:
dst = ~sJ@,S,)dt + zs,o(t, S,)dZ,,
(1)
where Is, denotes the diagonal (N x N) matrix with the elements S:’ (j = 1, . . , N), ~1is an N-dimensional process, 0 is an (N x N) matrix, and Z denotes a standard Brownian Motion in RN, Z = (Z’, . . , ZN)r, where z indicates transpose. Hence the uncertainty is formalized by the complete filtered space (n, F, {F f >tsco,Tl, P), where P is the actual (historical) probability and the filtration is the augmented filtration generated by the Brownian Motion. The matrix u is assumed to be full rank, thus markets are complete for unconstrained investors, i.e., those who can freely trade on these assets. Note that Duffie and Jackson’s framework (1990) allows the market for primitive assets to be incomplete, but only under severe restrictions on both the utility functions and the price processes can they derive explicit solutions. To ease the exposition, we note ~(t, S,) = ~~ and a(t, S,) = (T,. Also, the primitive assets do not pay any dividends between 0 and T. Assumption 2: There exists a constant continuously compounding interest rate r 2 0 at which investors may lend or borrow. It follows immediately that forward and futures prices are equal. Assumption 3: Markets are frictionless and there are no arbitrage opportunities. This, together with the completeness assumption, implies that there exists a unique probability measure Q, associated with the riskless asset’ yielding r, that makes discounted asset prices martingales (see Harrison and Kreps, 1979; Harrison and Pliska, 1981). Q is constructed such that
dQ
dPF,
(c;l(p,-
rZ))‘dZ, -f
0
rIc;‘(C(,
- r1)12dt
.
s 0
To simplify further computations, we assume that the ‘market price of risk’ IC,E C; ’ ( pt - r I) is deterministic, which makes the diffusion process (1) only slightly more general than a Geometric Brownian Motion. Thus:
(2)
‘The riskless asset is worth
one at time 0, then is worth
e” at t > 0.
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Note that, under Q, using Girsanov’s Theorem, the process 2: such that Z: = Z, + si ~,ds is a Brownian Motion. Assumption 4: There exist N futures contracts available for trade, written on the N primitive assets, whose maturity date, T, coincides with the investor’s horizon. Given the absence of arbitrage opportunities (F, = S, erCT-‘)) and using ItB’s Lemma, the futures prices process F, is given by
dF, = I,,q,dt + ZFtvtdZt,
(3)
withq,=p(,--rlandv,=o,. Notice that the existence of these N futures contracts ensures the completeness of the market for the constrained investor who cannot freely trade the primitive assets S but can freely trade the riskless asset. A hedge is an N-dimensional stochastic process 8 3 (O’, . . . , ON)’ specified by a futures position in each futures contract.3 At time t, the position is marked to market, so that gains and losses generated by futures price changes accrue to the hedger’s margin account in a continuous manner. Assumption 5: The margin account current value, denoted by X,, is credited or debited (if negative) with interest at the constant continuously compounding rate r. The margin account then is equal to
fs
x, =
e’@-“)&dF,
0
Applying It& lemma and using (3) yields dX, = rX,dt + O;dF, = (rX, + 8:1,q,)dt
+ 8:1,,v,dZ,.
(4)
Assumption 6: The investor holds a fixed, nontraded portfolio l7 of the N assets, with terminal random value H’ST at horizon date T. Given the futures strategy 0 and denoting by B1the number (possibly negative) of units of riskless asset, his total wealth at time t is W, such that W, = FS, + X, + j$e” and4 dW, = Il’dS, + dX, + p,rerfdt = (nrI,~t ‘For
technical
conditions
+ rX, + &IF,v, + j&re”)dt + (Z7’1,o, + O:Z,,v,)dZ,. satisfied
by these strategies,
“Note that PO is equal to zero since the investor’s assumption.
see Cox and Huang
(5)
(1989).
initial wealth is equal to IIrS,, and X,, is zero by
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1105
7: The investor is a Von Neumann-Morgenstern expected utility of terminal wealth maximizer and his utility function exhibits constant absolute risk aversion, i.e., U(W,) = - ee-Ywrwhere y > 0 is the coefficient of absolute risk aversion. Assumption
3. The hedging demand for futures
We first explain briefly the general ideas behind the methodology, then apply it to the specific problem at hand, namely to derive the optimal terminal wealth, the dynamics of both the constrained and unconstrained wealths, then the optimal hedging demands for futures contracts. 3.1. The methodology
The main idea of Karatzas, Lehoczky, and Shreve (1987) and Cox and Huang (1989, 1991) is to transform the dynamic optimal control problem into a static one by solving first for the terminal optimal wealth WT. Then we compute the value W, at each time t of this wealth. Identifying W, with the wealth process given by (5), it is easy to get the optimal hedging demands for futures contracts. When markets are complete, it is clear that there always exists a replicating strategy using futures that insures the choosen level of this terminal wealth. When markets are incomplete, things get complicated and we refer the interested reader to, e.g., He and Pearson (1991). Recall that our setting implies complete markets even for our constrained investor because of the existence of N futures contracts and of the riskless asset. In the process, two cases are possible. For some utility functions, the nonnegativity constraint on wealth is not binding, such as when the utility function exhibits infinite marginal utility at zero wealth (e.g., the logarithmic). For the others, the constraint is binding but Cox and Huang (1989) provide us with the following powerful equivalence result: the solution to the constrained program is equal to the solution to the unconstrained one to which is added an insurance contract (a protective put of strike zero written on the unconstrained terminal wealth) which will pay off the negative wealth if it occurs. In other words, the optimal constrained strategy will allocate the entire initial wealth between the unconstrained strategy and the protective put written on the unconstrained strategy. 3.2. The derivation of terminal wealth We derive first the solution to the unconstrained program; then we compute the optimal terminal wealth for the constrained problem, using the Cox-Huang method. We will denote I@ the solution to the unconstrained problem, with
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initial value wO, and denote W the solution to the constrained problem, with initial value Wo. Hence, W. 2 w,,. Like Cox and Huang, we now solve the following, unconstrained, program: max EP[ - e-@‘r], W, = EP[@Trzre-rT] i s.t. - EQ[@re-‘r]
(6) = @,,,
where I@e stands for the initial wealth necessary to solve the investor’s unconstrained program. It will become clear that this amount is less than (or equal to) the real Wo. The unique solution to (6) is such that
where 1 is the Lagrangian multiplier, or
@r=
-:ln
:
+i[rT--lnar].
(7)
0 Using the budget constraint EQII@re-rT] value given by (2), we have
= I@,, and substituting UT for its
Substituting in (7) the value of - (l/y)ln(A/y) given above yields 1
pT = tioerT +-
T
K; dZ:,
Ys 0
where it is readily seen that terminal wealth can be negative.
(8)
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of Economic Dynamics and Control 20 (1996) 1101.-1113
1107
We now solve explicitly for the constrained program: max EP [ - e - Ywr], WT s.t. EQ[WTe-‘T] = WO,
(9)
Given Cox and Huang (1989, Thm. 2.4, p. 64), the solution to (9) is
w, = [PVT] +.
(10)
It is worth noting at this stage that a) knowing investor’s actual initial wealth to the unconstrained program (6), we readily get the component WI, devoted to the unconstrained strategy, and b) [ tiT]’ = I@r + max [O; - pT] which highlights the put (insurance) component of the optimal constrained strategy. WO and the solution
3.3. The dynamics of optimal wealth We first solve for the dynamics of the optimal constrained wealth. Then, by identification with Eq. (5), we solve explicitly for the optimal demands for futures contracts. The value W, at t of optimal terminal wealth given by (10) is W,e-”
= EQIWr-ePrT]F1].
Hence, using (8),
Since we have assumed that the market price of risk K~is deterministic, we can compute explicitly the expectation present in the above equation.5 Denote A the event of nonnegative terminal wealth WT. This occurs when
s T
bPoerT +
1 K:dZ: Y 0
2 0,
‘Had we assumed more generally that K, is stochastic, we would have ended up with a partial differential equation for the optimal wealth at each time t - such as Cox and Huang (1989) in their general framework - without being able to obtain an explicit solution.
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s s T
K: dZ: 2 - y I@, erT,
0
T
rc5dZ: 2 - (yqoerT
+ ji h-:dZ:).
f
With respect to Q, the LHS of this equation is independent of F, and Gaussian with mean zero and variance JT 1rcslzds. Thus, we can write
where u is the standard (0, 1) Normal. It is important to note that d, is a function of optimal unconstrained wealth at time t, as will be shown later on. Then:
W, = e-r(T-t)
=e
-r(T-1)
w. erT EQ[l,] POerTN(d,)
+ + E Q[ ( [; IcfdZs* + lfT .:dZ:)UF,]~
-t
dN(4)
* s KS
dZ:
0
where IA is the indicator function. As shown in the Appendix, the last term EQ [ ] is equal to n(d,) [ jT )K,J* ds] 1/2where n( ) is the normal density function. Thus:
(
W, = emrcT-*) N(d,)
(_ Woe rT + -:saK:dZ:)+5n(d,)[J]l,r,,‘ds]lllj. (11)
A. Lioui. P. Poncet J Journal of Economic
Dynamics
and Control 20 (I 996) 1101-I 113
1109
Using (8) and the definition of Q, we have mtemr’ = EQII@‘Te-‘TJF,]
Note that the term under expectation in the RHS is orthogonal to F, and of expectation 0 with respect to Q. Then we get:
fs
dZ:
(12)
--n(d,)[ {;,KS,zds]‘:2. Y
(13)
KS
Y
0
Using (12) (11) becomes e-r(T-l)
IV, = W’,N(d,) +
Note for further reference that using (12) and the definition of dt, we have
(14) It remains to derive the dynamics of optimal constrained wealth at any time t. Applying Itb’s Lemma to Eq. (13) and substituting dZ, for dZ: ( = dZ, + K,dt), we get6 e-r(T-t)
dW, = g( .)dt + N(d,)--Y
6 dZt,
(15)
where g( ) is the (complicated) drift of the wealth process whose exact value is of no interest here. 3.4. The optimal hedging demands In order to derive the vector 0 of optimal positions in the futures contracts, we now identify the dynamics of wealth given by Eq. (5) with that given by Eq. (15).
6The detailed
derivation
is given in the Appendix
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For that purpose, it seems at first sight necessary to identify both the drift terms and the diffusion coefficients. However, it is in fact sufficient to identify the latter, forgetting altogether the former, since, by arbitrage, it is clear that two optimal strategies of identical initial value and identical risk must have the same expected return.’ Thus, from Eqs. (5) and (15) e-r(T-t)
zITZs,a, + e:z&v, = N(d,) ___ Y
lcrt*
Then, using IF, v, = er(T-f)ZS,rr.Iand kt = o;‘(p, 0;
=
e-*u-‘)
[
- rl) = a,‘~~, we have
nz + ~N(d,)(a;‘tl,)‘(lriv,)-’ . 1
Eq. (16), which is the main result of this paper, elicits the following remarks: l It is clear that the role of (the adequate number of) futures contracts in essentially complete primitive markets is to allow investors, who cannot freely trade on the primitive assets, to achieve the same optimum they would arrive at if they were not constrained. l The first term in the RHS is the so-called ‘pure hedge’ component of the strategy. The presence of the discount factor is due to the fact that the future position is continuously marked-to-market as opposed to the cash position which is not. l The second term represents the ‘speculative’ component. It differs from the one usually encountered in the literature because of the explicit consideration of the nonnegativity constraint on wealth, which introduces the extra term N(d,). l The term N(d,) is reminiscent of the Black-&holes formula. This does not come out as a surprise since, as we have seen previously, the investor implicitly must buy a protective European put written on his unconstrained wealth. However, it is important to recall that the unconstrained terminal wealth @r, which may be negative and which is the underlying asset of the put, is not lognormally but normally distributed under both measures P and Q. Also note that the second term of the RHS of (16) is multiplied by N(d,) < 1 and, thus, the speculative demand for futures is smaller, in absolute terms, than the one which would obtain without the nonnegativity constraint on wealth. This is because
‘Note that direct identification of the drift terms is almost untractable due, in particular, to the presence of X, in the drift term of Eq. (5). This explains why we did not have to be ;xplicit about g(.) in Eq. (15).
A. Lioui, P. Poncet 1 Journal of Economic Dynamics and Control 20 (1996) I 101-l 113
1111
being long in the put is equivalent to being short in the underlying asset. The intuition behind this result is that the investor must impose limits on his trading of futures contracts without which his wealth could become negative. l Eq. (16) reveals a major difficulty ignored in the literature: even in the case of Geometric Brownian Motions, and perfect correlation between futures and spot assets, it is not possible to derive ‘closed-form’ solutions for the optimal hedging demands, in the sense that 0 depends on d, and d, depends, not on effective wealth W,, but on ‘unconstrained’ wealth J@,. It is not indeed possible to recover W, since the relation between @, and W, is not invertible. Thus, we have shown that only for those utility functions that exhibit infinite marginal utility at zero wealth (e.g., the logarithmic) can we get closed-form solutions in the aforementioned sense. One can view this as a negative result. l Note however that Eq. (16) is not worthless. Indeed it is possible to solve with numerical methods for flO, then @, and then 0,.
4. Conclusion The main thrust of the paper is that the literature relative to the use of futures contracts as hedging vehicles has largely ignored the issue of possible negative wealth. Since in general bankrupt investors are driven out of the market, we must constrain the set of admissible strategies to the set of strategies for which only nonnegative wealth can occur. Except in the case of very special utility functions such as the logarithmic, for which this constraint is not binding, this implies that the ‘speculative’ component of optimal hedging demands is smaller, in absolute terms, than is usually reported. Also, it is not possible to derive the hedging demands explicitly in terms of the investor’s actual wealth, making numerical procedures necessary. As far as equilibrium is concerned, we know that, by arbitrage, the drift term of the diffusion process for F, is Q = pL,- rl. Thus the research for an ‘equilibrium’ q1 that would cancel the global excess demand for futures contracts, a la Duffie and Jackson (1990), is at best misleading. Aggregation through investors in our setting, and thus the derivation of open interest for constrained investors in each futures contract, is untractable. Potential extensions of our framework would include the following: a) dividend-paying primitive securities, b) maturity dates of the futures contracts not equal to the investor’s horizon, c) a stochastic riskless interest rate, and d) a nonperfect correlation between the returns of the primitive assets and the futures. While a) and b) are easy to deal with, c) is a serious problem since, in particular, the simple cash-and-carry relationship does not hold any longer. As for d), it is well-known since Adler and Detemple (1988b) that it is both a crucial and a difficult task since then markets are not complete for constrained investors. This work is in progress in a companion paper.
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Appendix
Computation of L = EQ[(jT K3dZ:)la(F,]
in Eq. (II)
Recall that j,r ~:dZz is independent of F, and Gaussian with respect to the measure Q of mean zero and variance s,r [K,)~ ds. Then:
= EQ[ul,].
(from the definition of u and d, in the text)
[ j-:lgl’d]l’2
=[~~l~~12ds]1’2~~q~.eV.‘2du.
Let y 3 u2/2. Then: L =[JrIrcs12ds]"' Jl,2&e-Ydy
=[J:Iri,lzds]l12n(d,),
where n( ) is the normal density function. Derivation of Eq. (1.5) In fact, we are only concerned with the diffusion coefficient of the investor’s wealth. Thus, we need not specify the drift term which could be calculated explicitly. For further reference, applying Ito’s lemma to (12) yields dW’, = (.)dt + :e-‘(‘PL’K:dZ:
= (.)dt + ~e-‘(T-‘)h_~dZ,.
(A-1)
Now, using Leibnitz rule, we get
aN(d,)
=
n(d,).
Applying Ito’s Lemma to (13) leads to dW, = N(d,)dW’, + W&&)dd, + (.)dt e-r(T-t) - 4 --n(dt)[ Y
~,rla,12ds]llldd,.
64.2)
A. Lioui. P. Poncet /Journal
of Economic Dynamics and Control 20 (1996) 1101-1113
1113
Substituting for d, in (A.2), this equation becomes dW, = N(d,)d@, + (.)dt.
(A.3)
The result (15) follows by substituting for (A.l) in (A.3).
References
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