On the Pricing of Contingent Claims under Constraints I. KARATZAS † Departments of Mathematics and Statistics Columbia University New York, NY 10027
[email protected]
∗
S. G. KOU Department of Statistics Hill Center, Busch Campus Rutgers University Piscataway, NJ 08855
[email protected]
November 1994; revised, November 1995
Abstract We discuss the problem of pricing contingent claims, such as European calloptions, based on the fundamental principle of “absence of arbitrage” and in the presence of constraints on portfolio choice, e.g. incomplete markets and markets with shortselling constraints. Under such constraints, we show that there exists an arbitragefree interval which contains the celebrated BlackScholes price (corresponding to the unconstrained case); no price in the interior of this interval permits arbitrage, but every price outside the interval does. In the case of convex constraints, the endpoints of this interval are characterized in terms of auxiliary stochastic control problems, in the manner of Cvitani´c & Karatzas (1993). These characterizations lead to explicit computations, or bounds, in several interesting cases. Furthermore, a unique fair price pˆ is selected inside this interval, based on utility maximization and “marginal rate of substitution” principles; again, characterizations are provided for pˆ, and these lead to very explicit computations. All these results are also extended to treat the problem of pricing contingent claims in the presence of a higher interest rate for borrowing. In the special case of a European calloption in a market with constant coefficients, the endpoints of the arbitragefree interval are the BlackScholes prices corresponding to the two different interest rates; and the fair price coincides with that of Barron & Jensen (1990). AMS 1991 Subject Classification: Primary 90A09, 93E20, 60H30; Secondary 60G44, 90A10, 90A16, 49N15. Key words and phrases: pricing of contingent claims, constrained portfolios, incomplete markets, two different interest rates, BlackScholes formula, utility maximization, stochastic control, martingale representations, equivalent martingale measures, minimization of relative entropy. ∗
Research supported by the National Science Foundation, under Grant NSFDMS9319816. Part of this author’s work was carried out while he was visiting the Courant Institute of Mathematical Sciences, New York University, in Fall 1994; he extends his appreciation to his hosts at the Institute, for their hospitality. †
1
Contents 1 Introduction and summary
2
2 The financial market model
4
3 Portfolio, consumption and wealth processes
5
4 Contingent claims and arbitrage in the unconstrained market
8
5 Upper and lower arbitrage prices
11
6 Representations for convex constraints
15
7 A fair price
24
7.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.2
Connections with Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.3
A representation for convex constraints . . . . . . . . . . . . . . . . . . . . . . . . 31
8 European calloption in a market with constant coefficients
37
8.1
Lower and upper arbitrage prices . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8.2
Computation of the fair price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
8.3
Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9 Market with higher interest rate for borrowing
45
10 A table
48
11 Discussion
50
1
Introduction and summary
The famous Black & Scholes (1973) formula provides the unique price of a European contingent claim in an ideal, complete and unconstrained market, as laid out in sections 2 and 3 of the present paper, based on the fundamental principle of “absence of arbitrage opportunities”. In other words, this price is the unique one for which there are no arbitrage opportunities by taking either a short or a long position in the claim, and investing wisely in the market. This price coincides with the minimal initial capital, starting with which one can duplicate exactly 2
the claim at the terminal time, and also with the expectation of the claim’s discounted value under the unique, “riskneutral” equivalent probability measure (cf. Merton (1973), Cox & Ross (1976), Cox & Rubinstein (1984), Harrison & Kreps (1979), Harrison & Pliska (1981), Karatzas (1989); see also section 4 of this paper for a brief survey). However, in the presence of constraints on portfolio choice (e.g., constraints on borrowing, on shortselling of stocks, even on accessing certain stocks at all, as in the case of “incomplete markets”), there ceases to exist a unique price for a contingent claim based solely on the principle of absence of arbitrage. Instead, there appears an “arbitrage free” interval [hlow , hup ] which contains the BlackScholes price u0 ; see the following figure. Here, hup represents the least price the seller can accept without risk, and hlow the greatest price the buyer can afford to pay without risk. r
r
r
0
hlow
u0
r

hup
This interval has the following properties: (i) every pricelevel outside the interval leads to an arbitrage opportunity; (ii) there are no arbitrage opportunities for pricelevels in the interior of the interval. These facts are demonstrated, to our knowledge for the first time, in section 5 of this paper. Furthermore, if the constraints on portfolio choice are convex, it turns out that the endpoints of the arbitragefree interval can be characterized as the values of certain suitable stochastic control problems, as in Cvitani´c & Karatzas (1993), or El Karoui & Quenez (1995) for incomplete markets; see section 6 and, in particular, Theorem 6.1. Roughly speaking, the upper (resp., lower) endpoint of the interval is equal to the supremum (resp., infimum) of the BlackScholes prices of the claim over a family of auxiliary, slightly more complicated in structure but unconstrained, markets. There remains the question of how to choose then a unique price for the claim, in the presence of constraints on portfolio choice. There seems to be no definitive answer to this question, though several approaches have been suggested—most of them in the context of incomplete markets (e.g. F¨ollmer & Sondermann (1986), Foldes (1990), F¨ollmer & Schweizer (1991), Duffie & Skiadas (1991), Davis (1994), etc.), and some in different but related contexts (different interest rates for borrowing and saving, Barron & Jensen (1990); transaction costs, Hodges & Neuberger (1989)). We adopt in section 7 the approach of Davis (1994), which is based on utility maximization and on the principle of “zero marginal rate of substitution ”.
3
These considerations lead to the notion of a “fair price ” pˆ (Definition 7.3), which, under certain mild conditions (cf. Assumptions 7.1, 7.2), is shown to lie within the arbitragefree interval (Theorem 7.1). Counterexamples for which the fair price lies outside the arbitragefree interval are also given in section 8.3. In the special case of convex constraints, we show that the fair price admits a BlackScholes representation under a certain “minimal” or “leastfavorable” equivalent probability measure (Theorem 7.4). In the derivation of this latter result, we draw on the powerful results of Cvitani´c & Karatzas (1992) for utility maximization under convex portfolio constraints (cf. Karatzas, Lehoczky, Shreve & Xu (1991) for the special case of incomplete markets). The representation of Theorem 7.4 leads to explicit computations of the fair price pˆ (Examples 7.17.4) for rather general portfolio constraints, including incomplete markets, shortselling or borrowing constraints, etcetera. In particular, it is shown that pˆ is independent of both initial wealth and utility function, in a market with deterministic coefficients and in the presence of coneconstraints on portfolios; and in this case, the corresponding equivalent martingale measure is also obtained by means of relative entropy minimization. Section 8 offers a host of explicit computations for hlow , hup and pˆ in the special but important case of a European call option, for a market with constant coefficients and under various kinds of constraints; these computations are tabulated in section 10, and constitute one of the main results of this paper. Explicit computations are also possible for a pathdependent (or “lookback”) option; see Example 7.4. A most interesting result, from a practical point of view, is that the same ideas and techniques can also treat the problem of pricing contingent claims in a market with higher interest rate for borrowing than for saving. More precisely, it is shown in section 9 that in this case there also exists an arbitragefree interval, and a fair price pˆ which always lies within that interval. In the special case of European call option in a market with constant coefficients, the endpoints of the arbitragefree interval are the two BlackScholes prices corresponding to the two different interest rates; and the fair price pˆ coincides with the socalled minimax price in Barron & Jensen (1990), if a powertype utility function is employed.
2
The financial market model
We shall deal exclusively in this paper with a financial market M in which d + 1 assets (or “securities”) can be traded continuously. One of them is a nonrisky asset, called the bond (also
4
frequently called “savings account”), with price P0 (t) given by (2.1)
dP0 (t) = P0 (t)r(t)dt,
P0 (0) = 1.
The remaining d assets are risky; we shall refer to them as stocks, and assume that the price Pi (t) per share of the ith stock, is governed by the linear stochastic differential equation (2.2)
dPi (t) = Pi (t)[bi (t)dt +
d X
σij (t)dWj (t)],
Pi (0) = pi ,
i = 1, 2, . . . d.
j=1
In this model, W (t) = (W1 (t), . . . , Wd (t))∗ is a standard Brownian motion in Rd , whose components represent the external, independent sources of uncertainty in the market M; with this interpretation, the volatility coefficient σij (·) in (2.2) models the instantaneous intensity with which the j th source of uncertainty influences the price of the ith stock. As is standard in the literature, M is assumed to be an ideal market; in other words, we have infinitely divisible assets, no constraints on consumption, no transaction costs or taxes. We shall allow, however, for constraints on portfolio choice, such as limitations on borrowing (from the savings account ) or on shortselling (of stocks), and so on; see the Examples in Section 6. The probabilistic setting will be as follows: the Brownian motion W will be defined on a complete probability space (Ω, F, P), and we shall denote by {Ft } the Paugmentation of the natural filtration FtW = σ(W (s); 0 ≤ s ≤ t). The coefficients of M, that is, the interest rate process r(t), the appreciation rate vector process b(t) = (b1 (t), . . . , bd (t))∗ of the stocks, and the volatility matrixvalued process σ(t) = {σij (t)}1≤i,j≤d , will all be assumed to be progressively measurable with respect to {Ft } and bounded uniformly in (t, ω) ∈ [0, T ] × Ω. We shall also 4
impose that the following strong nondegeneracy condition on the matrix a(t) = σ(t)σ ∗ (t), (2.3)
ξ ∗ a(t)ξ ≥ ² k ξ k2 ,
∀(t, ξ) ∈ [0, T ] × Rd
holds almost surely, for a given real constant ² > 0. All processes encountered throughout the paper will be defined on the fixed, finite horizon [0, T ], and adapted to the filtration {Ft }. We shall introduce also the “ relative risk ” process 4
θ(t) = σ −1 (t)[b(t) − r(t)1], ˜
(2.4)
where 1 = (1, 1, . . . , 1)∗ . The exponential martingale ˜ (2.5)
4
Z0 (t) = exp{−
Z t 0
1 θ (s)dW (s) − 2 ∗
5
Z t 0
k θ(s) k2 ds},
the discount process 4
(2.6)
γ0 (t) = exp{−
Z t 0
r(s)ds}
and the Brownian motion with drift 4
(2.7)
W0 (t) = W (t) +
Z t 0
θ(s)ds,
0≤t≤T
will be employed quite frequently. REMARK 2.1. It is a straightforward consequence of the strong nondegeneracy condition (2.3), that the matrices σ(t), σ ∗ (t) are invertible, and that the norms of (σ(t))−1 , (σ ∗ (t))−1 are bounded above and below by δ and 1/δ, respectively, for some δ ∈ (1, ∞); compare with Karatzas & Shreve (1991) (hereafter abbreviated as [KS]), page 372. The boundedness of b(·), r(·) and (σ(·))−1 implies that of θ(·); therefore, the process Z0 (·) of (2.5) is indeed a martingale, and not just a local martingale.
3
Portfolio, consumption and wealth processes
Consider now a small economic agent, whose actions cannot affect market prices, and who can decide, at any time t ∈ [0, T ], (i) how many shares of bond φ0 (t), and how many shares of stocks, (φ1 (t), φ2 (t), . . . , φd (t))∗ to hold, and (ii) what amount of money C(t + h) − C(t) ≥ 0 to withdraw for consumption during the interval (t, t + h], h > 0. Of course, all these decisions can only be based on the current information Ft , without anticipation of the future. More precisely, we have the following. DEFINITION 3.1. A trading strategy in the market M is a progressively measurable vector process (φ0 (t), φ1 (t), . . . , φd (t)) such that
RT 0
φ2i (t)dt < ∞, 0 ≤ i ≤ d, almost surely.
The processes φ0 and φi represent the number of shares of the bond and the ith stock, respectively, 1 ≤ i ≤ d, which are held or shorted at any given time t. A short position in the bond (respectively, the ith stock), i.e., φ0 < 0 (resp., φi < 0 ), should be thought of as a loan. DEFINITION 3.2. A cumulative consumption process is a nonnegative progressively measurable process {C(t), 0 ≤ t ≤ T } with increasing, RCLL paths on (0, T ] (Right Continuous with Left Limits), and with C(0) = 0, C(T ) < ∞ a.s. A basic assumption in the market M, is that trading and consumption strategies should
6
satisfy the socalled selffinancing condition (3.1)
d X
φi (t)Pi (t) =
i=0
d X
d Z t X
φi (0)Pi (0) +
i=0 0
i=0
φi (u)dPi (u) − C(t),
0≤t≤T
almost surely. The meaning of the equation is that, starting with an initial amount x = φ0 (0) +
Pd
i=1 φi (0)pi
of wealth, all changes in wealth are due to capital gains (appreciation of
stocks, and interest from the bond), minus the amount consumed. For both economic and mathematical considerations, it is useful to introduce wealth and portfolio processes. DEFINITION 3.3. A portfolio process is a progressively measurable process π(·) = (π1 (·), . . . , πd (·)): [0, T ] × Ω → Rd . DEFINITION 3.4. For a given initial capital x, a portfolio process π(·) as in Definition 3.3, and a cumulative consumption process C(·) as in Definiton 3.1, consider the wealth equation dX(t) = X(t)[1 − (3.2)
= X(t)[1 −
d X i=1 d X
πi (t)]
d dP0 (t) X dPi (t) + X(t)πi (t) − dC(t) P0 (t) Pi (t) i=1
πi (t)]r(t)dt +
i=1
d X
X(t)πi (t)[bi (t)dt +
i=1
d X
σij (t)dWj (t)] − dC(t),
j=1
= X(t)r(t)dt + X(t)π ∗ (t)σ(t)dW0 (t) − dC(t), X(0) = x, or equivalently (3.3)
γ0 (t)X(t) = x −
Z t 0
γ0 (s)dC(s) +
Z t 0
γ0 (s)X(s)π ∗ (s)σ(s)dW0 (s), 0 ≤ t ≤ T,
in the notation of (2.1), (2.2) and (2.5)(2.7). If this equation has a unique solution X(·) ≡ X x,π,C (·), this is then called the wealth process corresponding to the triple (x, π, C). 2 The interpretation here is that π(·) represent the propotions of the wealth X(·) which are invested in the respective stocks i = 1, . . . , d. REMARK 3.1. In the setup of Definition 3.4, notice that for the stochastic integral to be well defined we must have (
φi (t) =
RT 0
X 2 (t)π(t)2 dt < ∞, a.s. Furthermore, if we define
X(t)πi (t)/Pi (t) ; i = 1, . . . , d P X(t)(1 − dj=1 πj (t))/P0 (t) ; i = 0
)
, for 0 ≤ t ≤ T,
then φ(·) = (φ0 (·), φ1 (·), . . . , φd (·))∗ constitutes a trading strategy in the sense of Definition 3.1 and we have (3.4)
X(t) =
d X
φi (t)Pi (t), 0 ≤ t ≤ T,
i=0
7
as well as the selffinancing condition (3.1), which follows then from the wealth equation (3.2). Notice that the wealth process X(·) can clearly take both positive and negative values. The equation (3.3) leads us to consider the process (3.5)
4
N0 (t) = γ0 (t)X(t) +
Z t 0
γ0 (s)dC(s) = x +
Z t 0
γ0 (s)X(s)π ∗ (s)σ(s)dW0 (s),
0 ≤ t ≤ T,
which is seen to be a continuous local martingale under the socalled “riskneutral ” probability measure (or “equivalent martingale measure”) 4
P0 (A) = E[Z0 (T )1A ],
(3.6)
A ∈ FT ,
in the notation of (2.5). DEFINITION 3.5. A portfolio/consumption process pair (π, C) is called admissible for the initial capital x ∈ R, and we write (π, C) ∈ A(x), if (i) the pair π(·), C(·) obeys the conditions of Definitions 3.23.4; (ii) the solution X x,π,C (·) ≡ X(·) of equation (3.2) satisfies, almost surely: (3.7)
X x,π,C (t) ≥ −Λ,
∀ 0 ≤ t ≤ T.
Here, Λ is a nonnegative random variable with E0 (Λp ) < ∞, for some p > 1. 2 The admissibility requirements in Definition 3.5 are imposed in order to prevent pathologies like doubling strategies (c.f. Harrison & Pliska (1981), Karatzas & Shreve (1995)); such strategies achieve arbitrarily large levels of wealth at t = T , but require X(·) to be unbounded from below on [0, T ]. If (π, C) ∈ A(x), the P0 local martingale N0 (·) of (3.5) is also bounded uniformly from below, and is thus a P0 supermartingale. Consequently (3.8)
E0 [γ0 (T )X x,π,C (T ) +
Z T 0
γ0 (t)dC(t)] ≤ x, ∀(π, C) ∈ A(x).
Here E0 denotes the expectation operator corresponding to the probability measure P0 of (3.6); under this measure the process W0 (·) of (2.7) is standard Brownian motion, by the Girsanov theorem (e.g. Karatzas & Shreve (1991), Section 3.5), and the discounted stock processes γ0 (·)Pi (·) are martingales, since (3.9)
dPi (t) = Pi (t)[r(t)dt +
d X
(j)
σij (t)dW0 (t)], Pi (0) = pi ;
i=1
8
i = 1, . . . , d,
(j)
from (2.2) and (2.7), where W0
is the j th component of W0 .
REMARK 3.2. For any x ∈ R and (π, C) ∈ A(x), let F = X x,π,C (T ). Then for any a 6= 0, we have X ax,π,aC (·) = a · X x,π,C (·) from (3.2). In particular, (i) if a > 0: (π, aC) ∈ A(ax), X ax,π,aC (T ) = aF , a.s. (ii) if a = −1, C(·) ≡ 0: X −x,π,0 (T ) = −F .
4
Contingent claims and arbitrage in the unconstrained market
The dynamics of the market M become more interesting, once we introduce contingent claims such as options. Suppose, in particular, that at time t = 0 we sign a contract which gives us the right (but not the obligation, whence the term option) to buy, at the specified time T (“expiration date”), one share of the stock i = 1 at a specified price q (“exercise price”). At expiration t = T , if the price P1 (T, ω) of the share is below the exercise price, the contract is worthless to us; on the other hand, if P1 (T, ω) > q, we can exercise our option at time t = T , which means to buy one share of the stock at the exercise price q, and then sell the share immediately in the market for P1 (T, ω). In other words, this contract entitles its holder to a payment of B(T ) ≡ B(T, ω) = (P1 (T, ω) − q)+ at time t = T ; it is called a European call option, in contradistinction with an “American call option” that can be exercised at any stopping time (with values) in [0, T ]. See Myneni (1992) for a survey on the pricing of American options with unconstrained portfolios. In this paper we shall deal primarily with the pricing problem under constraints on portfolio choice, and confine ourselves to European options; the similar problem for American options will be treated elsewhere. The following definition generalizes the concept of European call option. DEFINITION 4.1. A European Contingent Claim (ECC) is a financial instrument consisting of a payment B(T ) at maturity time T ; here, B(T ) is a nonnegative, FT measurable random variable with E[(B(T ))1+² ] < ∞ for some ² > 0. We shall denote the price at time t = 0 of the ECC by B(0). The main purpose of this paper is to find out what B(0) should be in the market M; in other words, how much an agent should charge for selling such a contractual obligation, and how much another agent could afford to pay for it. It turns out that the answer depends on the structure of the market M. In this section, we consider the simplest case: that of a complete, unconstrained market, i.e., one in which every asset can be traded, and unlimited shortselling of both the bond and stocks is also permitted 9
(subject to the admissibility requirements of Definition 3.5). More precisely, πi (·) takes values in R, for each 1 ≤ i ≤ d. In this case the answer to the pricing problem is well known. A standard approach to this problem is to utilize the concept of arbitrage in the market M with the ECC, denoted by (M, B) for short, with B standing for the pair (B(0), B(T )). DEFINITION 4.2. There is an arbitrage opportunity in (M, B), if there exist an initial wealth x ≥ 0 (respectively, x ≤ 0), an admissible pair (π, C) ∈ A(x), and a constant a = −1 (respectively, a = 1), such that x + a · B(0) = X x,π,C (0) + a · B(0) < 0 at time t = 0, and X x,π,C (T ) + a · B(T ) ≥ 0
a.s.
at time t = T . The values a = ±1 indicate long or short positions in the ECC, respectively. 2 This definition of arbitrage is standard in the literature; see, for example, Chapter 6 in Duffie (1992) and Myneni (1992). Such an arbitrage opportunity represents a riskless source of generating profit, strictly bigger than the profit from the bond, by the combination of a trading/consumption strategy and the ECC. Furthermore, from the scaling properties in Remark 3.2, we know then that the profits from such a scheme are limitless. Such opportunities should not exist in a wellbehaved, rational market. One of the most interesting “classical” results on option pricing is that by only excluding such arbitrage opportunities, the price of the ECC can be uniquely determined, namely as (4.1)
4
u0 = E0 [γ0 (T )B(T )] = E[γ0 (T )B(T )Z0 (T )].
More precisely, if the ECC has a price B(0) > u0 at time t = 0, then there is an arbitrage opportunity involving a trading/consumption strategy (φ0 , . . . , φd , 0)∗ and a short position in the ECC; conversely, for any ECC having price B(0) < u0 , there is also an arbitrage opportunity using exactly (−φ1 , . . . , −φd , 0)∗ and taking a long position in the ECC. Hence the price for the ECC has to be u0 , if no arbitrage is allowed in M. This price is called the arbitragefree price, also known as BlackScholes price. Furthermore, corresponding to the BlackScholes price u0 , there is a ”hedging portfolio” process π(·) (hence also a corresponding trading process φ(·)) and a consumption process C(·) ≡ 0, such that (4.2)
X u0 ,π,0 (T ) = B(T );
10
and with the same portfolio π(·) (hence the opposite trading strategy −φ(·)), we have X −u0 π,0 (T ) = −B(T ).
(4.3)
REMARK 4.1. We have u0 < ∞ in (4.1); indeed, with c < ∞ denoting a common upper bound on θ(·) and r(·), and with p = 1 + ², 1/p + 1/q = 1, ³
u0 ≤ e−cT E(B(T ))p
´1/p ³
E(Z0 (T ))q
´1/q
≤ e−cT +(q−1)c
2 T /2
³
E(B(T ))p
´1/p
< ∞.
If M is a market with constant coefficients b, r, σ in (2.1) and (2.2), then explicit calculations are possible for u0 of (4.1) in the following cases. EXAMPLE 4.1. European call option, B(T ) = (P1 (T ) − q)+ ; then (4.4)
u0 = p · Φ(µ+ (T, p)) − qe−rT Φ(µ− (T, p)), p = P1 (0)
where
Z
z 1 2 4 1 µ± (t, p) = √ [log(p/q) + (r ± σ 2 /2)t] and Φ(z) = √ e−u /2 du σ t 2π −∞ is the cumulative standard normal distribution function; we have set σ = σ11 > 0. Furthermore,
(4.5)
the portfolio process in (4.2) and (4.3) satisfies (4.6)
π1 (t) > 1
and
πi (t) = 0,
2 ≤ i ≤ d,
a.s.
We refer the reader to Harrison & Pliska (1981), Cox & Rubinstein (1984), Karatzas (1989), Karatzas & Shreve (1991) or Duffie (1992) for details. EXAMPLE 4.2. Pathdependent (“lookback”) option B(T ) = max P1 (t). 0≤t≤T
of Goldman et al. (1979). Then the price of (4.1) is given by u0 = pe−rT 4 r σ
where σ = σ11 > 0, ρ = (4.7)
−
σ 2
Z ∞ 0
f (T, ξ; ρ)eσξ dξ,
p = P1 (0)
and 4
f (t, ξ; ρ) = 1 − Φ
³ ξ − ρt ´
√ t
11
h
+ e2ξρ 1 − Φ
³ ξ + ρt ´i
√ t
.
Further, the portfolio π(·) of (4.2), (4.3) is given by πi (t) ≡ 0, i = 2, . . . , d and (4.8)
π1 (t) =
eσΥ(t) f (T − t, Υ(t); ρ) + σ eσΥ(t) + σ
R∞
R∞
Υ(t) f (T
Υ(t) f (T
− t, ξ; ρ)eσξ dξ
− t, ξ; ρ)eσξ dξ
for 0 ≤ t ≤ T , where 4
Υ(t) = max (W0 (s) + ρs) − (W0 (t) + ρt) = 0≤s≤t
³ max ´ 1 0≤s≤t P1 (s) log . σ P1 (t)
We refer the reader to Karatzas & Shreve (1996), section 2.4 for the details. 2 A main drawback in the above classical argument is its dependence on the assumptions of completeness and unconstrainedness for the market M. More to the point, as we have seen in the above discussion, it is critical to be able to use −φ as a trading strategy, if φ is permitted in the market, and to trade in all (d + 1) assets if necessary. However, if we are in a constrained market, for instance, a market in which shortselling of stocks is prohibited (i.e., with φi (·) ≥ 0, for each i = 1, . . . , d), then the admissibility of the strategy (φ0 , . . . , φd )∗ does not imply that of (−φ0 , . . . , −φd )∗ . Furthermore, in an incomplete market, not all the assets are accessible. A general arbitrage argument is needed to cover these cases as well as the classical unconstrained case.
5
Upper and lower arbitrage prices
Let us introduce now further constraints on portfolio choice, in addition to those of Definition 3.5. Suppose that we are given two nonempty Borel subsets K+ and K− of Rd ; for any x ∈ R, we shall consider portfolio/consumption pairs in the class (5.1)
4
A0 (x) = {(π, C) ∈ A(x) : π(t) ∈ K+ if X x,π,C (t) > 0, and π(t) ∈ K− if X x,π,C (t) < 0, ∀ t ∈ [0, T ), a.s.}.
In other words, K+ ( respectively, K− ) represents our constraint on portfolio choice when the wealth is positive (resp., negative). We shall see examples in Section 6 where such different constraints on portfolio, depending on the sign of the level of wealth, arise quite naturally. DEFINITION 5.1. Given a European contingent claim B(T ) as in Definition 4.1, introduce the lower hedging class (5.2)
4 ˇ ∈ A− (−x), such that X −x,ˇπ,Cˇ (T ) ≥ −B(T ), a.s. } L = {x ≥ 0 : ∃(ˇ π , C)
12
and the upper hedging class 4
ˆ
ˆ ∈ A+ (x), such that X x,ˆπ,C (T ) ≥ B(T ), a.s. }. U = {x ≥ 0 : ∃(ˆ π , C)
(5.3)
Here we have set 4 ˇ ˇ ∈ A(y) : π A− (y) = {(ˇ π , C) ˇ (t) ∈ K− and X y,ˇπ,C (t) ≤ 0, ∀ 0 ≤ t < T, 4
ˆ
ˆ ∈ A(z) : π A+ (z) = {(ˆ π , C) ˆ (t) ∈ K+ and X z,ˆπ,C (t) ≥ 0, ∀ 0 ≤ t < T,
a.s. }, for y ≤ 0 , a.s. }, for z ≥ 0.
ˇ (resp., (ˆ ˆ in the definitions of the classes L and U are called lower The elements (ˇ π , C) π , C)) (resp., upper) hedging strategies for the ECC. 2 Clearly, the set L contains the origin. On the other hand, it is a straightforward consequence of the Definition 5.1, that both sets L and U are (connected) intervals. More precisely, we have the following result. PROPOSITION 5.1 For any x1 ∈ L, 0 ≤ y1 ≤ x1 implies y1 ∈ L. Similarly, for any x2 ∈ U, y2 ≥ x2 implies y2 ∈ U. PROOF. Suppose (π2 , C2 ) ∈ A(x2 ) satisfies the conditions of (5.3). Then, with y2 ≥ x2 , one “just consumes immediately the amount y2 − x2 ”; in other words, with Cˆ2 (t) = C2 (t) + ˆ
(y2 − x2 ) · 1(0,T ] (t), we have X y2 ,π2 ,C2 (t) ≡ X x2 ,π2 ,C2 (t) for all 0 < t ≤ T , and thus y2 ∈ U. A similar argument works for L. 2 The purpose of this section is to show that, in the presence of constraints as in (5.1), the BlackScholes price u0 = E0 [γ0 (T )B(T )] is replaced by an interval [hlow , hup ] which contains u0 and is defined by (5.4) below, in the following sense: If B(0), the price of the ECC at time t = 0, (i) does not belong to [hlow , hup ], then there exists an arbitrage opportunity (Theorem 5.2); (ii) belongs to the interior (hlow , hup ) of this interval, or if B(0) = hlow = hup , then arbitrage opportunities do not exist (Theorem 5.3 and Corollary 5.1). DEFINITION 5.2. The lower arbitrage and the upper arbitrage prices are defined by (5.4)
4
hlow = sup{x : x ∈ L},
4
hup = inf{x : x ∈ U},
respectively. Here we adopt the convention that inf Ø = +∞. In Section 6 we shall provide characterizations of the numbers hlow , hup in terms of suitable stochastic control problems, which lead to explicit computation in several interesting special cases (cf. Section 8). 13
REMARK 5.1. Heuristically, the upper arbitrage price may be viewed as the minimal amount necessary for the seller of the ECC to set aside at time t = 0, in order to make sure that he will be able to cover his obligation at time t = T . Similarly, the lower arbitrage price can be viewed as the maximal amount that the buyer of the ECC is willing to pay at t = 0, and still be sure that he will be able to cover, at time t = T , the debt he incurred at t = 0 by purchasing the ECC. This intuition suggests that the lower arbitrage price hlow cannot be larger than the upper arbitrage price hup . The following theorem shows in fact that for general constraint sets K+ and K− , a stronger result holds. THEOREM 5.1 We have for any nonempty constraint sets K+ and K− in B(Rd ), 0 ≤ hlow ≤ u0 ≤ hup , where u0 = E0 [γ0 (T )B(T )] is the BlackScholes price of (4.1). PROOF. By (3.8) and the definition of U, we get 0
h
x ≥ E γ0 (T )X
ˆ x,ˆ π ,C
(T ) +
Z T 0
i
ˆ γ(s)dC(s) ≥ E0 [γ0 (T )B(T )] = u0 , ∀x ∈ U.
Hence, hup ≥ u0 . Similarly, h
ˇ
−y ≥ E0 γ0 (T )X −y,ˇπ,C (T ) +
Z T 0
i
ˇ γ0 (s)dC(s) ≥ E0 [γ0 (T )(−B(T ))] = −u0 , ∀y ∈ L,
whence y ≤ u0 and hlow ≤ u0 . 2 One feature of the above theorem is that it holds for any constraint sets, therefore it is applicable to many situations. For instance, in the case of a European calloption B(T ) = (P1 (T ) − q)+ on the first stock, and assuming that this stock can be traded, we have P1 (0) ∈ U, and thus: 0 ≤ hlow ≤ u0 ≤ hup ≤ P1 (0) < ∞. We define the notion of arbitrage with portfolios constrained as in (5.1), by analogy with Definition 4.2. DEFINITION 5.3. We say that there exists in (M, B) an arbitrage opportunity with constrained portfolios, if there exists an initial wealth x ≥ 0 (resp., x ≤ 0), an admissible portfolio/consumption process pair (π, C) in the class A+ (x) (respectively, A− (x)) of Definition 5.1, and a constant a = −1 (resp., a = 1) such that (5.5)
x + a · B(0) = X x,π,C (0) + a · B(0) < 0 14
and (5.6)
X x,π,C (T ) + a · B(T ) ≥ 0, a.s.
Again, the values a = ±1 represent long or short positions in the ECC, respectively. 2 THEOREM 5.2 For any ECC price B(0) > hup , there exists an arbitrage opportunity in the sense of Definition 5.3; similarly for any ECC price B(0) < hlow . PROOF. Suppose that B(0) > hup ; then for any x1 ∈ (hup , B(0)) we know that x1 ∈ U, ˆ ∈ A+ (x1 ) such that by the definition of hup . Thus there exists a (ˆ π , C) ˆ
X x1 ,ˆπ,C (0) − B(0) = x1 − B(0) < 0, and ˆ
X x1 ,ˆπ,C (T ) − B(T ) ≥ B(T ) − B(T ) = 0, a.s. Hence (5.5) and (5.6) in Definition 5.3 are satisfied with a = −1. For the case B(0) < hlow , there is an arbitrage opportunity which satisfies (5.5) and (5.6) with a = 1. The argument is similar to the first one, and we omit the details. 2 THEOREM 5.3 For any ECC price B(0) 6∈ (U
S
L), there is no arbitrage in (M, B) with
constrained portfolios. PROOF. We shall prove this by contradiction. Suppose B(0) 6∈ U, B(0) 6∈ L and that there is an arbitrage opportunity in (M, B) with constrained portfolios. Two cases may arise. Case 1: The arbitrage opportunity satisfies (5.5) and (5.6) with a = −1. In this case, there exist an initial wealth x ∈ [0, ∞) and a pair (π1 , C1 ) ∈ A+ (x), such that x = X x,π1 ,C1 (0) < B(0) and (5.7)
X x,π1 ,C1 (T ) ≥ B(T ), a.s.
From (5.7) and the definition of U we know that x = X x,π1 ,C1 (0) ∈ U, whence B(0) ∈ U, thanks to x < B(0) and Proposition 5.1; a contradiction. Case 2: The arbitrage opportunity satisfies (5.5) and (5.6) with a = 1. The proof is similar to that of Case 1, so we omit the details. 2
15
COROLLARY 5.1 If hlow < hup , then for any price B(0) ∈ (hlow , hup ) of the ECC there is no arbitrage opportunity in (M, B) with constrained portfolios. In view of Theorems 5.2 and Corollary 5.1, the interval [hlow , hup ] is the best possible interval for the ECC price that one can obtain by using only arbitrage arguments. We shall call [hlow , hup ] arbitragefree interval. REMARK 5.2. In an unconstrained market, i.e., with K+ = K− = Rd , we know from the classical results that the BlackScholes price u0 belongs to both the lower hedging class L and the upper hedging class U (see Chapter 6 in Duffie (1992)); thus we have hlow = hup = u0 according to Theorem 5.1. REMARK 5.3. If the option price is equal to one of the two endpoints hlow or hup , it may well be that in some situations there is no arbitrage, while in others there may be an arbitrage opportunity, depending on the consumption process. For example, in the unconstrained case, if B(0) = hup = u0 , there is no arbitrage, as it can be shown that the consumption process for the hedging strategy is almost surely zero (see [KS], p.378). On the other hand, if B(0) = hup , ˆ ) > 0 a.s. (for instance, as in Remark 8.1), then this consumption can be hup ∈ U and C(T viewed as a kind of arbitrage opportunity. Within the arbitragefree interval, a unique fair price might be determined by considerations based on utility maximization, or on a stochastic game between the buyer and the seller. An approach using utility maximization, originally due to Davis (1994), is discussed in detail in Section 7.
6
Representations for convex constraints
We shall concentrate in this section on the important special case where the constraint sets K+ , K− of (5.1) are nonempty closed, convex sets in Rd . For such sets, we shall obtain in this section representations of hlow , hup in terms of auxiliary stochastic control problems (cf. (6.7) and (6.8)), which will lead in turn to explicit computations in Section 8. We start by introducing the functions 4
δ(x) = sup (−π ∗ x) : Rd 7→ R
[
{+∞}
π∈K+
and 4 ˜ δ(x) = inf (−π ∗ x) : Rd 7→ R π∈K−
16
[
{−∞}.
˜ are the support functions of the convex sets In the terminology of convex analysis, δ(·) and −δ(·) −K+ and K− , respectively; they are closed, positively homogeneous, proper convex functions ˜ are finite on their on Rd (Rockafellar (1970), p.114). The support functions δ(·) and −δ(·) ˜ + and K ˜ − , respectively, where, effective domains K 4 ˜+ = K {x ∈ Rd ; ∃ β ∈ R s.t. − π ∗ x ≤ β, ∀ π ∈ K+ } = {x ∈ Rd ; δ(x) < ∞}, 4 ˜ ˜− = K {x ∈ Rd ; ∃ β ∈ R s.t. − π ∗ x ≥ β, ∀ π ∈ K− } = {x ∈ Rd ; δ(x) > −∞}.
˜ + and K ˜ − are convex cones. The following two assumptions will be imposed Notice that both K throughout this section. ˜ are continuous on K ˜ + and K ˜ − , respecASSUMPTION 6.1. The functions δ(·) and δ(·) tively. ASSUMPTION 6.2. The function δ(·) is bounded uniformly from below by some real constant. These two assumptions are satisfied by all of the examples below. In particular, Theorem ˜ +, K ˜ − are locally 10.2, p.84 in Rockafellar (1970) guarantees that Assumption 1 is satisfied, if K simplicial; and Assumption 6.2 is satisfied if and only if K+ contains the origin. The convex constraints are perhaps among the most important constraints that arise in practice. A few of them are listed below. EXAMPLE 6.1. All of the following examples satisfy the Assumptions 6.1 and 6.2. ˜+ = K ˜ − = {0} (i) Unconstrained case: φ ∈ Rd+1 . In other words, K+ = K− = Rd . Then K ˜ + and K ˜ − , respectively. and δ = δ˜ ≡ 0 on K (ii) Prohibition of shortshelling of stocks: φi ≥ 0, 1 ≤ i ≤ d. In other words, K+ = [0, ∞)d , ˜+ = K ˜ − = [0, ∞)d , and δ ≡ 0 on K ˜ + , δ˜ ≡ 0 on K ˜ −. K− = (−∞, 0]d . Then K (iii) Constraints on the shortselling of stocks: A generalization of (ii) is K+ = [−k, ∞)d ˜+ = K ˜ − = [0, ∞)d , and δ(x) = for some k ≥ 0 and K− = (−∞, l]d for some l ∈ R. Then K P P ˜ ˜ + and K ˜ − , respectively. = −l d xi on K k d xi , δ(x) i=1
i=1
(iv) Incomplete market, in which only the first m stocks can be traded:
φi = 0, ∀i =
m + 1, . . . , d for some fixed m ∈ {1, . . . , d − 1}, d ≥ 2. In other words, K+ = K− = {π ∈ ˜+ = K ˜ − = {x ∈ Rd ; xi = 0, ∀i = 1, . . . , m} and Rd ; πi = 0, ∀i = m + 1, . . . , d}. Then K ˜ + and K ˜ −. δ = δ˜ ≡ 0 on K
17
(v) Incomplete market, with prohibition of investment in the first m stocks: φi = 0, 1 ≤ i ≤ m for some 1 ≤ m ≤ d, d ≥ 2. In other words, K+ = K− = {π ∈ Rd ; πi = 0, 1 ≤ i ≤ m}. Then ˜+ = K ˜ − = {x ∈ Rd ; xm+1 = . . . = xd = 0}, and δ = δ˜ ≡ 0 on K ˜ + and K ˜ −. K ˜ + (K ˜ − ) = {x ∈ Rd ; π ∗ x ≥ (vi)Both K+ and K− are closed, convex cones in Rd . Then K ˜ ≡ 0 on K ˜ + (K ˜ − ). This clearly generalizes all the previous examples 0, ∀π ∈ K+ (K− )} and δ (δ) except (iii). P
(vii)Prohibition of borrowing: φ0 ≥ 0. In other words, K+ = {π ∈ Rd : di=1 πi ≤ 1}, P ˜+ = K ˜ − = {x ∈ Rd : x1 = x2 = · · · = xd ≤ 0}, and K− = {π ∈ Rd ; di=1 πi ≥ 1}. Then K ˜ ˜ + , δ(x) ˜ −. δ(x) = −x1 on K = −x1 on K P
(viii) Constraints on borrowing: A generalization of (vii) is K+ = {π ∈ Rd : di=1 πi ≤ k}, P ˜+ = K ˜ − = {x ∈ Rd : x1 = for some k ≥ 1 and K− = { di=1 πi ≥ l} for some l ∈ R. Then K ˜ ˜ + , δ(x) ˜ −. · · · = xd ≤ 0}, δ(x) = −kx1 on K = −lx1 on K Explicit formulae or bounds for hlow and hup , for all these examples, in the case of a European call option in a market with constant coefficients, will be presented in detail in ˜ + is equal to K ˜ − (in this Section 8. It is interesting to notice that, for all these examples, K connection, see also Proposition 7.2). In general, this will not be the case; see Example 8.8. The technique to handle such convex constraints is developed in Cvitani´c & Karatzas (1993), hereafter abbreviated as [CK2]. The basic idea is to introduce a family of auxiliary markets, in which the unconstrained (hedging) problem is relatively easy to solve, and then try to come back to the original market. This basic idea will help us here to give representations for the lower arbitrage price hlow and the upper arbitrage price hup , in terms of appropriate stochastic control problems which involve optimization with respect to “parameters” of the auxiliary markets. In order to introduce these families of auxiliary markets, the notation of sections 5 and 6 in [CK2] will be carried over here for K+ ; in addition, we shall consider the analogous notation ˜ to be the set of progressively measurable process ν = for K− . Define the class H (resp., H) R
˜ + (resp., K ˜ − ), which satisfy E T (ν(t)2 + δ(ν(t)))dt < ∞ {ν(t), 0 ≤ t ≤ T } with values in K 0 RT S ˜ 2 ˜ (resp., E (ν(t) + δ(ν(t)))dt < ∞); also introduce, for every ν ∈ H H, the analogues 0
4
θν (t) = θ(t) + σ −1 (t)ν(t), 4
γν (t) = exp[− (6.1)
4
γ˜ν (t) = exp[−
Z t 0
Z t 0
{r(s) + δ(ν(s)}ds], ˜ {r(s) + δ(ν(s))}ds], 18
4
Z t
(6.2)
Zν (t) = exp[−
(6.3)
Wν (t) = W (t) +
4
0
θν∗ (s)dW (s) −
Z t 0
1 2
Z t 0
kθν (s)k2 ds],
θν (s)ds,
of the processes in (2.4)(2.7), as well as the measure 4
Pν (A) = E[Zν (T )1A ] = Eν [1A ], A ∈ FT
(6.4)
˜ the subset consisting of the processes by analogy with (3.6). Finally, denote by D (resp., D) ˜ such that ν is bounded uniformly in (t, ω) ∈ [0, T ] × Ω: ν ∈ H (resp., H) (6.5)
sup
ν(t, ω) < ∞.
(t,ω)∈[0,T ]×Ω
Therefore, for every ν ∈ D
S ˜ D, the exponential local martingale Zν (·) of (6.2) is actually a
martingale, from which we conclude that the measure Pν of (6.4) is a probability measure and the process Wν (·) of (6.3) is a Pν Brownian motion, by the Girsanov theorem; in terms of this new Brownian motion Wν (·), the stock price equations (2.2) can be rewritten as h
(6.6)
dPi (t) = Pi (t) (r(t) − νi (t))dt +
d X
i
σij (t)dWν(j) (t) , i = 1, . . . , d.
j=1
In the special case of an incomplete market (Example 6.1 (iv)), this equation shows that the discounted prices γ0 (·)Pi (·), i = 1, . . . , d are martingales under every probability measure in the class {Pν }ν∈D of (6.4). THEOREM 6.1 With the above notation, we have: (i) the lower arbitrage price is given by (6.7)
hlow = inf Eν [˜ γν (T )B(T )] =: g, ˜ ν∈D
˜ is bounded uniformly from below by some real constant; provided that the function δ(·) (ii) the upper arbitrage price is given by (6.8)
hup = sup Eν [γν (T )B(T )], ν∈D
and if the righthand side of (6.8) is finite, then hup ∈ U. In particular, taking ν ≡ 0 in (6.7) and (6.8) we recover the result 0 ≤ hlow ≤ u0 ≤ hup ˜ observe that the number Eν [γν (T )B(T )] (resp. of Theorem 5.1. For ν ∈ D (resp. ν ∈ D), 19
Eν [˜ γν (T )B(T )]) is exactly the BlackScholes price of the contingent claim in a new auxiliary market with unconstrained portfolios. Notice that δ˜ ≥ 0 in all the cases of Example 6.1, except in (iii) when l > 0, and in (viii) when l < 0. We shall treat these two cases separately (see Example 8.1 and Example 8.2) by employing the definition of hlow directly. The representation (6.8) for hup is proved as in [CK2], although a set bigger than our D is used there, so we only need to establish (6.7). As in [CK2], the proof uses the martingale representation and DoobMeyer decomposition theorems, and relies on the construction of a submartingale with regular sample paths. Let us denote by S the set of all {Ft }stopping times τ with values in [0, T ], and by Sρ,ξ the subset of S consisting of stopping times τ such that ρ(ω) ≤ τ (ω) ≤ ξ(ω), ∀ω ∈ Ω, for any two stopping times ρ ∈ S and ξ ∈ S such that ρ ≤ ξ a.s. For every τ ∈ S, consider also the Fτ measurable random variables h
(6.9)
4 V˜ (τ ) = essinfν∈D˜ Eν B(T ) exp{−
and (6.10)
4 ˜ ˜ ν (τ ) = Q V (τ )e−
Rτ 0
Z T τ
˜ (r(u)+δ(ν(u)))du
¯
¯ ˜ (r(s) + δ(ν(s)))ds} ¯Fτ
= V˜ (τ )˜ γν (τ ),
i
˜ ν ∈ D.
˜ τ ∈ S, α ∈ Sτ,T we have the submartingale property LEMMA 6.1 For every ν ∈ D, ˜ ν (τ ) ≤ Eν [Q ˜ ν (α)Fτ ], a.s. Q
LEMMA 6.2 There exists a RCLL modification V˜ + (·) of the process V˜ (·). Furthermore, ν ˜+ ˜+ if we define Q ν (·) by analogy with (6.10 ), then {Qν (t), Ft , 0 ≤ t ≤ T } is a P submartingale with RCLL paths. The proofs of Lemma 6.1 and Lemma 6.2 are carried out in a manner similar to that of the Appendix in [CK2]. ˜ ν (·) of (6.9) and (6.10) we have LEMMA 6.3 For the processes V˜ (·), Q (6.11)
E0 [ sup (V˜ (t))p ] < ∞,
∀p ∈ (1, 1 + ²),
0≤t≤T
(6.12)
˜ ν (t)] < ∞, Eν [ sup Q
˜ ∀ν ∈ D.
0≤t≤T
˜ ν (T )) < ∞, ∀ν ∈ D. ˜ In particular, Eν [˜ γν (T )B(T )] = Eν (Q 20
PROOF. From (6.9) it follows that 0 ≤ V˜ (t) ≤ E0 [B(T )e−
(6.13)
RT t
r(s)ds
Ft ] ≤ ecT B(t),
0≤t≤T 4
holds almost surely, in the notation of Remark 4.1 and with B(t) = E0 [B(T )F(t)]. Now with 1 < p < 1 + ², r = (1 + ²)/p and 1/r + 1/s = 1, we have from the H¨older inequality and the Doob maximal inequality: E0 [ sup (B(t))p ] ≤ const · E0 (B(T ))p = const · E[Z0 (T )(B(T ))p ] 0≤t≤T
≤ const · (E[(B(T ))pr ])1/r · (E[(Z0 (T ))s ])1/s . Therefore, E0 [ sup (B(t))p ] ≤ const · (E[(B(T ))1+² ])1/r · exp(
(6.14)
0≤t≤T
s−1 2 c T ) < ∞, 2
which proves (6.11) in conjuction with (6.13). ˜ is uniformly bounded On the other hand, from (6.13), (6.10), and the assumption that δ(·) from below by some real constant, we obtain that (6.15)
˜ ν (t) = V˜ (t) exp[− 0≤Q
Z t 0
˜ (r(s) + δ(ν(s)))ds] ≤ const · B(t),
0≤t≤T
˜ also holds almost surely. With 1 < p < 1 + ², 1/p + 1/q = 1 we get then, for any fixed ν ∈ D: Eν [ sup B(t)] = E0
h Z (T ) ν
0≤t≤T
Z0 (T )
i
³
´1/p ³
· sup B(t) ≤ E0 [ sup (B(t))p ] 0≤t≤T
· E0
0≤t≤T
³ Z (T ) ´q ´1/q ν
Z0 (T )
< ∞.
We have used again the H¨older and Doob inequalities, (6.14), as well as the uniform boundedness of the process σ −1 (·)ν(·) in Zν (t) = exp{− Z0 (t)
Z t 0
(σ
−1
1 (s)ν(s)) dW0 (s) − 2 ∗
Z t 0
σ −1 (s)ν(s)2 ds},
0 ≤ t ≤ T.
In conjuction with (6.15), this leads then to (6.12). 2 PROOF OF THEOREM 6.1. The proof is similar to [CK2]. From now on we consider only ˜ ν , hence we can assume that these processes do have RCLL the RCLL modifications of V˜ and Q paths. Part I. We shall first prove the inequality hlow ≥ g. This is obvious if g = 0 so let us assume, for the remainder of this part of the proof, that g > 0, and try to show that g ∈ L. From Lemma ˜ ν (·) is a submartingale of class D[0, T ] under Pν , for every ν ∈ D. ˜ Thus from 6.2 and (6.12), Q 21
the martingale representation theorem (section 3.4 in [KS]) and the DoobMeyer decomposition ˜: for submartingales (section 1.4 in [KS]), we have for every ν ∈ D (6.16)
˜ ν (t) = V˜ (0) + Mν (t) + Aν (t) = g + Q
where Mν (t) =
Z t
ψν∗ (s)dWν (s) + Aν (t),
0
0≤t≤T
Rt ∗ ν d 0 ψν (s)dWν (s), 0 ≤ t ≤ T , is an ({Ft }, P )martingale, ψν (·) is an R valued,
{Ft }progressively measurable and almost surely square integrable process, and Aν (·) is {Ft }predictable with increasing, RCLL paths and Aν (0) = 0, Eν Aν (T ) < ∞. Introduce the negative process (6.17)
˜ ν (t) Q 4 ˇ X(t) = −V˜ (t) = − , 0 ≤ t ≤ T, γ˜ν (t)
˜ for every ν ∈ D.
Then ˇ X(0) = −V˜ (0) = −g,
ˇ ) = −B(T ). and X(T
ˇ ∈ A− (−g) such Hence, in order to show g ∈ L, it is enough to find an admissible pair (ˇ π , C) ˇ ˇ ˇ that X(·) = X −g,ˇπ,C (·); recall from (6.11) that V˜ (·) = −X(·) is dominated by the random variable Λ = sup0≤t≤T V˜ (t) ≥ 0 with E0 (Λp ) < ∞ for some p > 1. ˜ ν∈D ˜ we have from (6.10), Let us start by observing that for any µ ∈ D, hZ t
˜ µ (t) = Q ˜ ν (t) exp Q
0
˜ δ(ν(u)du −
Z t 0
i
˜ δ(µ(u))du ,
0 ≤ t ≤ T.
Thus, from the differential form of (6.16) we get Z t
˜ µ (t) = exp[ dQ
0
˜ δ(ν(s))ds −
Z t 0
˜ ˜ ˜ ˜ ν (t){δ(ν(t)) δ(µ(s))ds] · [Q − δ(µ(t))}dt +
ψν∗ (t)dWν (t) + dAν (t)]
(6.18)
Z t
= exp[
0
˜ δ(ν(s))ds −
Z t 0
˜ ˜ ˜ ˇ γν (t){δ(ν(t)) δ(µ(s))ds] · [−X(t)˜ − δ(µ(t))}dt
+ dAν (t) + ψν∗ (t)σ −1 (t)(ν(t) − µ(t))dt + ψν∗ (t)dWµ (t)], ˇ where the last equation comes from the definition of X(·) and the connection between Wµ (·) and Wν (·) (cf. (6.3)). Comparing (6.18) with the DoobMeyer decomposition ˜ µ (t) = ψ ∗ (t)dWµ (t) + dAµ (t), dQ µ
(6.19)
we conclude from the uniqueness of this decomposition that Z t
ψν (t) exp[
0
Z t
˜ δ(ν(s))ds] = ψµ (t) exp[
22
0
˜ δ(µ(s))ds],
0 ≤ t ≤ T,
so that the process Z t
4
(6.20)
h(t) = ψν (t) exp[
0
˜ δ(ν(s))ds],
0 ≤ t ≤ T,
does not depend on ν.
We claim that we also have, almost surely, Z T
(6.21)
1{X(t)=0} h(t)2 dt = 0. ˇ
0
˜ 0 (·) of (6.16). From the TanakaIndeed, consider the nonnegative P0 submartingale Q(·) ≡ Q Meyer formula (Meyer (1976), p.365, equations (12.1), (12.3)) we have Q(t) = g +
Z t 0
1{Q(s)>0} dQ(s) + Λ(t) +
X
1{Q(s−)=0} ∆Q(s),
0
where Λ(·) is the local time of Q(·) at the origin: a continuous increasing process, flat off the set {0 ≤ t ≤ T ; Q(t) = 0}, a.s. Comparing this expression with (6.16), we obtain that 4
M (t) =
Z t 0
X
1{Q(s)=0} dM0 (s) = Λ(t)+
1{Q(s−)=0} ∆Q(s)−
Z t
0
0
1{Q(s)=0} dA0 (s),
0≤t≤T
is a continuous martingale of bounded variation. Thus, its quadratic variation < M > (T ) =
Z T 0
1{Q(t)=0} d < M0 > (t) =
Z T 0
1{Q(t)=0} h(t)2 dt
is almost surely equal to zero, and (6.21) follows (recall here that M0 (t) = Rt ∗ 0 h (s)dW0 (s) from ((6.16) and (6.20)).
Rt ∗ 0 ψ0 (s)dW0 (s) =
Therefore, if we fix an arbitrary π ˇ ∈ K− and define 4
(6.22)
π ˇ (t) =
−1 (σ ∗ (t))−1 h(t) · 1{X(t)<0} +π ˇ · 1{X(t)=0} , ˇ ˇ ˇ γ0 (t)X(t)
we obtain a portfolio process that satisfies almost surely ˇ π ∗ (t)σ(t) = h∗ (t), −γ0 (t)X(t)ˇ
a.e. on
[0, T ].
¿From this and from (6.18)(6.20), we have Z t
exp[
0
˜ δ(ν(s))ds −
Z t 0
˜ δ(µ(s))ds]·
˜ ˜ ˇ γν (t){δ(ν(t)) [−X(t)˜ − δ(µ(t))}dt + dAν (t) + ψν∗ (t)σ −1 (t)(ν(t) − µ(t))dt] = dAµ (t), whence
Z t
exp[
0
˜ δ(ν(s))ds −
Z t 0
˜ δ(µ(s))ds]· 23
˜ ˜ ˇ γν (t){δ(ν(t)) [−X(t)˜ +π ˇ ∗ (t)ν(t) − δ(µ(t)) −π ˇ ∗ (t)µ(t)}dt + dAν (t)] = dAµ (t), ˇ defined as thanks to (6.22). Therefore, the process C(·) (6.23)
4 ˇ = C(t)
Z t 0
Z t
γ˜ν−1 (s)dAν (s) −
0
˜ ˇ X(s)[ δ(ν(s)) + ν ∗ (s)ˇ π (s)]ds, 0 ≤ t ≤ T,
˜ In particular, taking ν ≡ 0, we see that is independent of ν ∈ D. ˇ = C(t)
Z t 0
γ0−1 (s)dA0 (s), 0 ≤ t ≤ T
ˇ ˇ ) < ∞ almost surely. In other is an increasing, adapted, RCLL process with C(0) = 0 and C(T ˇ is a consumption process. words, C(·) The same argument as on p. 664 of [CK2] shows then that ˜ δ(ν(s)) + ν ∗ (s)ˇ π (s) ≤ 0,
0≤s≤T
˜ Therefore, the proof in [CK1], p. 782783, and Theorem holds almost surely, for every ν ∈ D. 13.1 in Rockafellar (1970), p. 112, give us π ˇ (·) ∈ K− , a.s. Notice that, for these arguments to ˜ as well as the condition that δ(·) ˜ be bounded uniformly work, we need the continuity of δ(·) from below by some real constant. Now putting the various pieces together, we obtain ˇ γν (t)) = dQ ˜ ν (t) = ψν∗ (t)dWν (t) + dAν (t) d(−X(t)˜ ˜ ˇ + X[ ˇ δ(ν(t)) ˇ π ∗ (t)σ(t)dWν (t)], = γ˜ν (t)[dC(t) + ν ∗ (t)ˇ π (t)]dt − X(t)ˇ so that, ˜ ˇ γν (t)) = −˜ ˇ − γ˜ν (x)X(t)[ ˇ d(X(t)˜ γν (t)dC(t) δ(ν(t)) + ν ∗ (t)ˇ π (t)]dt
(6.24)
ˇ π ∗ (t)σ(t)dWν (t). +˜ γν (t)X(t)ˇ Taking ν ≡ 0 in (6.24), we obtain the wealth equation (3.2) in the form ˇ ˇ + γ0 (t)X(t)ˇ ˇ π ∗ (t)σ(t)dW0 (t), X(0) ˇ d(γ0 (t)X(t)) = −γ0 (t)dC(t) = −g, ˇ = X −g,ˇπ,Cˇ (·). The proof of h whence X(·) low ≥ g is now complete. Part II. Let us consider the proof of the reverse inequality hlow ≤ g. This is obvious if hlow = 0, so we assume that hlow = 0. Thus we have L 6= Ø in (5.2), and for any x ∈ L there exists (π, C) ∈ A− (−x) such that X −x,π,C (T ) ≥ −B(T ) almost surely. It is easy to see from (3.3) and (6.1) that the analogue of (6.24) holds, and thus γ˜ν (t)X
−x,π,C
(t) +
Z t 0
γ˜ν (s)dC(s) +
Z t 0
˜ γ˜ν (s)X −x,π,C (s)[δ(ν(s)) + π ∗ (s)ν(s)]ds, 0 ≤ t ≤ T 24
is actually a Pν local martingale, whence a supermartingale. This is because γ˜ν (·)X −x,π,C (·) is bounded from below by a Pν integrable random variable, thanks to (3.7), (6.5), and the H¨older inequality. Therefore, h
−x ≥ Eν γ˜ν (T )X −x,π,C (T ) + ≥ Eν [˜ γν (T )(−B(T ))]
Z T 0
γ˜ν (s)dC(s) +
Z T 0
i
˜ γ˜ν (s)X −x,π,C (s)(δ(ν(s)) + π ∗ (s)ν(s))ds
˜ or equivalently x ≤ Eν [˜ for every x ∈ L, ν ∈ D, γν (T )B(T )], from which hlow ≤ g follows. 2
7
A fair price
We have seen that, if the upper arbitrage price hup is strictly bigger than the lower arbitrage price hlow , then the arbitrage argument alone is not enough to determine a unique price for the contingent claim. Several approaches have been proposed to get around this problem in the special case of incomplete markets (as in Example 6.1 (iv)); see, for example, F¨ollmer & Sondermann (1986), F¨ollmer & Schweizer (1991), Duffie & Skiadas (1991), Foldes (1990) and Davis (1994). There are also some approaches that have been suggested in different, but related, contexts, such as pricing in the presence of transaction costs (Hodges & Neuberger (1989)) or under different interest rates for borrowing and saving (Barron & Jensen (1990)). Although perhaps none of them is totally satisfactory, we shall try in this section to generalize one of them, the Davis (1994) approach, to the constrained setup of Section 5. The purpose of this section is not to solve the problem completely (because it might turn out that, from a practical point of view, the most convenient price to use is still the BlackScholes price u0 ; cf. Remark 11.4 in Section 11), but rather to see when the generalization works and when it does not, and hopefully to focus attention on the study of possible connections between arbitrage and utility maximization.
7.1
Definition
Davis’s “fair price” is only defined for an agent with positive wealth and involves the concept of utility function. Before presenting the definition of the fair price, we shall briefly recall that of utility function. DEFINITION 7.1. A function U : (0, ∞) → R will be called a utility function, if it is strictly increasing, strictly concave, of class C 1 and satisfies 4
4
4
U (0) = U (0+), U 0 (0+) = lim U 0 (x) = ∞, U 0 (∞) = lim U 0 (x) = 0. x→∞
x↓0
25
We shall denote by I(·) the inverse of the function U 0 (·). Notice that the function I(·) maps (0, ∞) onto itself and satisfies I(0+) = ∞, I(∞) = 0, U 0 (I(x)) = x. Consider the following “constrained portfolio” optimization problem h ³
4
(7.1)
V (x) =
´i
E U X x,π,C (T ) ,
sup
0 < x < ∞,
(π,C)∈A+ (x)
where one tries to maximize expected terminal utility over portfolio/consumption pairs in the class A+ (x) of Definition 5.1. Clearly, we have ³
Z T
V (x) ≥ EU x exp[
0
´
r(t)dt] ≥ U (xer0 T ) > −∞,
where r0 is a lowerbound on r(·). ASSUMPTION 7.1. For all x > 0, the value V (x) of (7.1) is attainable; in other words, h ³
´i
V (x) = E U X∗x (T ) ,
(7.2)
where
4
X∗x (T ) = X x,π
∗ ,C ∗
(T ),
for some (π ∗ , C ∗ ) ∈ A+ (x), and we assume that the derivative of V (·) exists and is strictly positive: V 0 (·) > 0, on (0, ∞). 2 This assumption is satisfied in many interesting cases, in particular with C ∗ (·) = 0. Indeed, it holds for all convex constraint sets K+ , subject to the rather mild Assumptions 6.1 and 6.2; see §7.3. Suppose that at time t = 0, the price of the contingent claim is p = B(0) and one diverts an amount δ, δ < x, of money into the contingent claim B (i.e, buys δ/p shares of the contingent claim). Then one chooses an optimal portfolio/consumption strategy to achieve maximal expected utility from terminal wealth. Formally, one solves the stochastic control problem (7.3)
4
W (δ, p, x) =
³ ´ δ EU X x−δ,π,C (T ) + B(T ) , p (π,C)∈A0 (x−δ)
sup
δ < x,
where we set formally U (x) = −∞ for x < 0. Notice that W (0, p, ·) coincides with the function V (·) of (7.1) for every p > 0, and that we can actually take X x−δ,π,C (T ) > 0 in (7.3) above. If the contingent claim price p is set so that this small diversion of funds has a neutral effect on W , in the sense (7.4)
∂W (0, p, x) = 0, ∂δ 26
then we tend to call this p the “fair price” of the contingent claim. Indeed, Davis (1994) uses exactly (7.4) to define the fair price. However, the differentiability of W (·, p, x) is often difficult to check directly. Here we shall use a requirement weaker than differentiability, and reminiscent of the notion of “viscosity solutions” as in Crandall and Lions (1983). DEFINITION 7.2 For a given x > 0, we call p a weak solution of (7.4) if, for every differentiable function φ(·, p, x) satisfying φ(δ, p, x) ≥ W (δ, p, x) for all δ ∈ (−x, x), and φ(0, p, x) = W (0, p, x) ≡ V (x), we have ∂φ (0, p, x) = 0. ∂δ Notice the similarity of this notion with that of “viscosity subsolution” (see, for example Definition 7.2 in Shreve and Soner (1994), or Fleming and Soner (1993) p. 66). DEFINITION 7.3 Suppose that for any given x > 0, the weak solution p = pˆ(x) > 0 of (7.4) is unique. Then we call this pˆ(x) the fair price for the contingent claim at time t = 0, corresponding to initial wealth x > 0. In economic terms, the requirement (7.4) postulates a “zero marginal rate of substitution” for W (·, pˆ(x), x) at δ = 0. Generally speaking, Davis’s fair price depends on the utility function U (·) and on the particular initial wealth x > 0. However, for convex constraint sets K+ and K− , we shall present in §7.3 conditions under which pˆ(x) can be rendered independent of the utility function U (·) and/or the initial wealth x > 0.
7.2
Connections with Arbitrage
An immediate question that we have to settle, is whether there exist any arbitrage opportunities in (M, B) if the contingent claim price B(0) is set to be pˆ(x). In other words, whether pˆ(x) belongs or not to the interval [hlow , hup ], for every initial wealth x > 0. In general, the answer can be affirmative or negative, depending on the constraint sets K+ and K− (indeed, several counterexamples are given in Section 8.3); however, if we adopt the fairly general Assumption 7.2 below, then the answer is always affirmative. ASSUMPTION 7.2. Suppose that (π (1) , C (1) ) ∈ A0 (x) and (π (2) , C (2) ) ∈ A0 (y), for arbitrary but fixed x ∈ R, y ∈ R. Then there exists a (π, C) ∈ A0 (x + y) such that the corresponding terminal wealth X x+y,π,C (T ) is obtained by superposition: X x+y,π,C (T ) = X x,π
(1) ,C (1)
(T ) + X y,π
27
(2) ,C (2)
(T ), a.s.
THEOREM 7.1 Suppose that the Assumptions 7.1 and 7.2 are satisfied, and that the fair price pˆ(x) exists for every x > 0; then (7.5)
∀x > 0,
hlow ≤ pˆ(x) ≤ hup .
The meaning of Assumption 7.2 is that, whenever an agent chooses to invest in two different accounts X1 (·) ≡ X x,π
(1) ,C (1)
(·) and X2 (·) ≡ X y,π
(1) ,C (2)
(·) separately, then this is equivalent, in
terms of terminal wealth, to investing and consuming according to some strategy (π, C) which is admissible for the initial wealth level x+y, for arbitrary real numbers x and y. This assumption holds, in particular, if the pair π = (π (1) X1 + π (2) X2 )/(X1 + X2 ),
C = C (1) + C (2)
is indeed in A0 (x + y). A sufficient condition, for Assumption 7.2 to hold in the case of convex sets K± , is given along these lines in Proposition 7.1 below. PROOF OF THEOREM 7.1: We establish the upper bound first. Suppose that pˆ(x) > 0 is the fair price of Definition 7.3 for the initial wealth x > 0. For arbitrary y ∈ U, we want to show that pˆ(x) ≤ y. Now for any δ ∈ (−xˆ p(x)/y, 0) ∩ (−x, 0), by Remark 3.2 and the definition of the class U in (5.3), there exists an admissible pair (π (1) , C (1) ) ∈ A+ (ζ) with 4
ζ = (−δy/ˆ p(x)) ∈ (0, x), such that X ζ,π
(7.6)
(1) ,C (1)
(T ) ≥ (−δ/ˆ p(x)) · B(T )
holds almost surely. On the other hand, by Assumption 7.1, there is an admissible pair (π (2) , C (2) ) ∈ A+ (w) which is optimal for the problem of (7.1) with the initial wealth w = x − δ + δy/ˆ p(x) = x − δ − ζ > x − ζ > 0 (recall that δ < 0), i.e., the resulting wealth process X w,π
(2) ,C (2)
(·) ≥ 0 satisfies h ³
V (w) = V (x − δ − ζ) = E U X w,π
(7.7)
(2) ,C (2)
´i
(T ) .
Thus, from Assumption 7.2, we know that there is an admissible pair (π (3) , C (3) ) ∈ A0 (x − δ) such that (7.8) X x−δ,π
(3) ,C (3)
(T ) = X ζ,π
(1) ,C (1)
(T ) + X w,π
(2) ,C (2)
28
(T ) ≥ X w,π
(2) ,C (2)
(T ) −
δ B(T ), pˆ(x)
by (7.6). Hence, by the definition of W in (7.3), ³
W (δ, pˆ(x), x) ≥ EU X x−δ,π
(3) ,C (3)
´
³
(T )+(δ/ˆ p(x))B(T ) ≥ EU X w,π
(2) ,C (2)
´
³
´
(T ) = V x−δ+δy/ˆ p(x) ,
thanks to (7.8) and (7.7). Therefore, for any function φ as in Definition 7.2, we have φ(δ, pˆ(x), x) − φ(0, pˆ(x), x) W (δ, pˆ(x), x) − V (x) V (x − δ + δy/ˆ p(x)) − V (x) ≤ ≤ , δ δ δ since δ < 0, and in the limit, as δ ↑ 0, 0=
³ y ´ ∂φ (0, pˆ(x), x) ≤ − 1 V 0 (x). ∂δ pˆ(x)
Since V 0 (x) > 0 by Assumption 7.1, we obtain y ≥ pˆ(x), from which the upper bound in (7.5) follows. Now consider the lower bound. For arbitrary z ∈ L, we want to show z ≤ pˆ(x). Given any δ ∈ (0, x), again by Remark 3.2 and the definition of L, we know that there exists a pair 4
(π (4) , C (4) ) ∈ A− (−ξ) with ξ = δz/ˆ p(x) > 0 such that X −ξ,π
(7.9)
(4) ,C (4)
(T ) ≥ (δ/ˆ p(x))(−B(T )),
a.s. 4
Also by Assumption 7.1, there exists a pair (π (5) , C (5) ) ∈ A+ (η) where η = x − δ + ξ = x − δ + δz/ˆ p(x) > 0, with corresponding wealth process X η,π h ³
V (η) = V (x − δ + δz/ˆ p(x)) = E U X η,π
(7.10)
(5) ,C (5)
(5) ,C (5)
(·) ≥ 0 which satisfies ´i
(T ) .
From Assumption 7.2, we know that there exists a pair (π (6) , C (6) ) ∈ A0 (x − δ) such that (7.11)
X x−δ,π
(6) ,C (6)
(T ) = X −ξ,π
(4) ,C (4)
(T ) + X η,π
(5) ,C (5)
(T ) ≥ X η,π
(5) ,C (5)
(T ) −
δ B(T ), pˆ(x)
almost surely. Therefore, ³
W (δ, pˆ(x), x) ≥ EU X x−δ,π
(6) ,C (6)
´
³
(T )+(δ/ˆ p(x))B(T ) ≥ EU X η,π
(5) ,C (5)
´
(T ) = V (x−δ+δz/ˆ p(x)),
via (7.9), (7.10) and (7.11). Thus, for any function φ as in Definition 7.2, we have φ(δ, pˆ(x), x) − φ(0, pˆ(x), x) V (x − δ + δz/ˆ p(x)) − V (x) ≥ , ∀ δ ∈ (0, x), δ δ and in the limit, as δ ↓ 0, 0=
³ z ´ ∂φ (0, pˆ(x), x) ≥ − 1 V 0 (x). ∂δ pˆ(x)
29
Again, V 0 (x) > 0 leads to the lower bound z ≤ pˆ(x) of (7.5). 2 REMARK 7.1. It is readily seen that the first part of the proof of Theorem 7.1 goes through, and thus the upper bound pˆ(x) ≤ hup of (7.5) is valid, even in the absence of Assumption 7.2, provided that the set K+ is convex. PROPOSITION 7.1 If the constraint sets K+ and K− are convex, then a sufficient condition for the validity of Assumption 7.2 is (
(7.12)
∀π+ ∈ K+ , π− ∈ K− :
λπ+ + (1 − λ)π− ∈
K+ , if λ ≥ 1 K− , if λ ≤ 0
PROOF. For xi ∈ R and (π (i) , C (i) ) ∈ A0 (xi ), let Xi (·) ≡ X xi ,π
(i) ,C (i)
)
.
(·), i = 1, 2 be the
4
4
corresponding wealth processes and define C(·) = C (1) (·) + C (2) (·), x = x1 + x2 , X(·) = X1 (·) + X2 (·). Then it is not hard to see from the wealth equation (3.2) that X(·) = X x,π,C (·), where the portfolio π(·) is given by (7.13)
4
π(t) = [λ(t)π (1) (t) + (1 − λ(t))π (2) (t)]1(X(t)6=0) ,
λ(t) = X1 (t)/X(t).
To show that (π, C) ∈ A0 (x), we have to check that (7.14)
π(t) ∈ K+ on {X(t) > 0}, and π(t) ∈ K− on {X(t) < 0}.
Now on {X1 (t) > 0, X2 (t) = 0} we have π(t) = π (1) (t) ∈ K+ in (7.13); similarly, π(t) = π (2) (t) ∈ K+ on {X1 (t) = 0, X2 (t) > 0}. By analogy, we have π(t) ∈ K− on {X(t) < 0, X1 (t)X2 (t) = 0}. It remains to see what happens on {X1 (t)X2 (t) 6= 0}. We distinguish several cases. (i) {X1 (t) > 0, X2 (t) > 0}: On this event, π (i) (t) ∈ K+ (i = 1, 2) and 0 < λ(t) < 1, so π(t) ∈ K+ by the convexity of K+ . (ii) {X1 (t) < 0, X2 (t) < 0}: By similar arguments, π(t) ∈ K− . (iii) {X1 (t) > 0 > X2 (t), X(t) > 0}: Then π1 (t) ∈ K+ , π2 (t) ∈ K− , λ(t) > 1 and π(t) ∈ K+ , by (7.12). (iv) {X1 (t) > 0 > X2 (t), X(t) < 0}: Here λ(t) < 0, and (7.12) gives π(t) ∈ K− . (v) {X2 (t) > 0 > X1 (t), X(t) > 0} and (vi) {X2 (t) > 0 > X1 (t), X(t) < 0} can be treated by analogy with (iii), (iv). 30
In all these cases, (7.14) holds.
2
The condition (7.12) is satisfied in the context of Examples 6.1, for the cases (i), (ii), (iii) with l ≤ −k, (iv), (v), (vi) with K− = −K+ , (vii), (viii) with l ≥ k. For a discussion of how things can go wrong in (7.5) if the condition (7.12) fails, see the examples of §8.3. PROPOSITION 7.2 For any two closed convex sets K+ , K− that satisfy (7.12 ), we have ˜ on ˜ on K ˜+ = K ˜ − and δ(·) ≤ δ(·) ˜ + (= K ˜ − ); futhermore, if K+ T K− 6= Ø, then δ(·) = δ(·) K ˜ + (= K ˜ − ). K ˜ + , so that δ(x) < ∞. For λ > 1 and arbitrary π+ ∈ K+ , PROOF. Fix an arbitrary x ∈ K π− ∈ K− we have x∗ (λπ+ ) + x∗ ((1 − λ)π− ) = x∗ (λπ+ + (1 − λ)π− ) ≥ inf (x∗ π) = −δ(x). π∈K+
˜ Therefore, taking infima, and recalling the positive homogeneity properties of δ(·) and −δ(·), we get ˜ −λδ(x) + (λ − 1)δ(x) ≥ −δ(x) > −∞. ˜ ˜ ˜+ ⊆ K ˜ − ) and in fact δ(x) ≤ δ(x). It follows that δ(x) > −∞ (thus K ˜ ˜ − , so that −δ(x) Now fix an arbitrary x ∈ K < ∞. For λ < 0 and arbitrary π+ ∈ K+ , π− ∈ K− we have ˜ x∗ (λπ+ ) + x∗ ((1 − λ)π− ) = x∗ (λπ+ + (1 − λ)π− ) ≤ sup (x∗ π) = −δ(x). π∈K−
Therefore, again by taking suprema, and using the same homotheticity properties, we obtain ˜ ˜ −λδ(x) − (1 − λ)δ(x) ≤ −δ(x) < ∞. ˜ ˜+ ⊇ K ˜ − ) and again δ(x) ≤ δ(x). It follows that δ(x) < ∞ (whence K T ˜ The inequality δ(x) ≤ δ(x) on Rd follows directly from K+ K− 6= Ø. 2 REMARK 7.2. If the two closed convex sets K+ , K− satisfy the conditions (7.12) and K+
T
K− 6= Ø, then the endpoints of the arbitragefree interval [hlow , hup ] are characterized solely in terms of the set K+ (recall Theorem 6.1 and the notation of section 6).
31
7.3
A representation for convex constraints
The following result will be used to obtain the representation (7.25) for the fair price pˆ(x). It was estabilished by Davis (1994), but we provide here an alternative arguments, based on our Definitions 7.3, 7.2 for the fair price. THEOREM 7.2 Under the Assumption 7.1, the fair price pˆ(x) is uniquely determined by h
(7.15)
pˆ(x) =
³
E U 0 X x,π
∗ ,C ∗
´
i
(T ) B(T )
V 0 (x)
,
∀ x > 0.
PROOF. We shall use the inequalities (7.16)
U (x) + (y − x)U 0 (x) ≥ U (y) ≥ U (x) + (y − x)U 0 (y),
∀ 0 < x < y < ∞,
which is a simple consequence of concavity. With the notation of (7.2), we have from the second inequality in (7.16), for x > δ > 0, p > 0: h ³ ´i δ W (δ, p, x) ≥ E U X∗x−δ (T ) + B(T ) p ´ i δ h ³ δ ≥ E[U (X∗x−δ (T ))] + E U 0 X∗x−δ (T ) + B(T ) · B(T ) . p p Since x 7→ X∗x (T ) is nondecreasing, we get ´ i δ δ h ³ (7.17) W (δ, p, x) ≥ V (x − δ) + E U 0 X∗x (T ) + B(T ) · B(T ) . p p Thus, from Fatou’s lemma, W (δ, p, x) − W (0, p, x) lim inf δ↓0 δ V (x − δ) − V (x) 1 δ ≥ lim + lim inf E[U 0 (X∗x (T ) + B(T )) · B(T )] δ↓0 δ p δ↓0 p 1 ≥ −V 0 (x) + E[U 0 (X∗x (T )) · B(T )]. p On the other hand, with δ < 0, p > 0, we have, from the first inequality in (7.16), that (7.17) is
valid again (with the interpretation U 0 (x) ≡ U 0 (0+) ≡ ∞ for x < 0), and thus by the monotone convergence theorem, lim sup δ↑0
W (δ, p, x) − W (0, p, x) δ
V (x − δ) − V (x) 1 δ + lim E[U 0 (X∗x (T ) + B(T )) · B(T )] δ↑0 δ p δ↑0 p 1 ≤ −V 0 (x) + E[U 0 (X∗x (T )) · B(T )]. p
≤ lim
32
Therefore, for all x > 0 and p > 0, (7.18)
lim sup δ↑0
W (δ, p, x) − W (0, p, x) δ
1 ≤ −V 0 (x) + E[U 0 (X∗x (T )) · B(T )] p ≤ lim inf δ↓0
W (δ, p, x) − W (0, p, x) . δ
Let φ denote an arbitrary function as in Definition 7.2; then (7.18) yields ∂φ 1 (0, p, x) = −V 0 (x) + E[U 0 (X∗x (T )) · B(T )], ∂δ p from which it is easy to check that pˆ(x) defined in (7.15) is the unique weak solution of (7.4) in the sense of Definition 7.2. 2 To give an explicit form of the fair price for convex constraints, we need a result from [CK1] along with some additional notation and assumptions. For each ν ∈ D, introduce the (continuous, strictly decreasing) function 4
Jν (y) = Eν [γν (T )I(yγν (T )Zν (T ))], 0 < y < ∞, along with its inverse 4
Yν (x) = Jν−1 (x), 0 < x < ∞. Furthermore, let us impose, in addition to the requirements of Definition 7.1, the following conditions on the utility function U : 4
(7.19)
U (∞) = lim U (x) = ∞,
(7.20)
U (0) > −∞, or U (x) = log x,
(7.21)
x 7→ xU 0 (x) is nondecreasing on (0, ∞),
x→∞
and (7.22)
for some α ∈ (0, 1), γ ∈ (1, ∞) we have αU 0 (x) ≥ U 0 (γx), ∀x ∈ (0, ∞).
The following result of [CK1] describes the terminal wealth corresponding to the optimal pair (π ∗ , 0) ∈ A+ (x) for the constrained portfolio optimization problem of (7.1) under the conditions (7.19)(7.22), and shows that these guarantee the validity of Assumption 7.1. THEOREM 7.3 Suppose that the constraint set K+ is closed, convex, and satisfies Assumptions 6.1 and 6.2; assume also that the conditions (7.19)(7.22) hold. Then, for every x > 0, there exists a νˆ = νˆx ∈ D and a pair (π ∗ , 0) ∈ A+ (x) with corresponding terminal wealth (7.23)
X x,π
∗ ,0
(T ) = I(Yνˆ (x)γνˆ (T )Zνˆ (T )), a.s. 33
This pair attains the supremum V (x) of (7.2), i.e., is optimal for the problem of (7.2). The value function V (·) is continuously differentiable, and its derivative can be represented as (7.24)
V 0 (x) = Yνˆ (x) > 0, ∀x > 0.
The process νˆ(·) ∈ D is optimal in a dual (minimization) stochastic control problem, whence the adjectives “minimal”, “dualoptimal” or “leastfavorable” for it. Now Theorem 7.3 leads directly to a representation for pˆ(x) in the market with convex constraints. THEOREM 7.4 We have for all x > 0, pˆ(x) = Eνˆ [γνˆ (T )B(T )].
(7.25)
PROOF. We have for any given x > 0, E[U 0 (X∗x (T ))B(T )] = E[U 0 (X x,π
∗ ,0
(T ))B(T )]
= E[U 0 (I(Yνˆ (x)γνˆ (T )Zνˆ (T )))B(T )]
(by (7.23))
= E[Yνˆ (x)γνˆ (T )Zνˆ (T )B(T )]
(using U 0 (I(x)) = x)
= V 0 (x) · E[γνˆ (T )Zνˆ (T )B(T )]
(by (7.24))
= V 0 (x) · Eνˆ [γνˆ (T )B(T )]. We can now apply Theorem 7.2, to get (7.25). 2 REMARK 7.3. Combining the representation for pˆ(x) in the above Theorem 7.4, the representations for hlow and hup in Theorem 6.1, and Proposition 7.2, we recover (7.5): namely, if the two closed convex sets K+ , K− satisfy the condition (7.12), then pˆ(x) ∈ [hlow , hup ] for all x > 0. It follows from Theorem 7.4 that pˆ(x) is the BlackScholes price Eνˆ [γνˆ (T )B(T )] of B(T ) in an auxiliary unconstrained market Mνˆ with interest rate r(·) + δ(ˆ ν (·)), appreciation rate vector b(·)+ νˆ(·)+δ(ˆ ν (·))1 and volatility matrix σ(·), corresponding to the “minimal” (“dualoptimal”) ˜ process νˆ(·) = νˆx (·) of Theorems 7.3, 7.4. Here are some examples from [CK1], in which this process can be computed explicitly.
34
EXAMPLE 7.1. Logarithmic utility function, general random adapted coefficients. If U (x) = log x, then it is shown in [CK1], p. 790 that νˆ(·) is given by νˆ(t) = argminy∈K˜ + [2δ(y) + θ(t) + σ −1 (t)y2 ], 0 ≤ t ≤ T.
(7.26)
Thus, νˆ(·) (as well as pˆ) does not depend on the initial wealth x ∈ (0, ∞). ˜ ≡ 0 on K ˜ + ), the expression of (7.26) becomes In particular, if K+ is a cone (thus δ(·) νˆ(t) = argminy∈K˜ + θ(t) + σ −1 (t)y2 , 0 ≤ t ≤ T ;
(7.26)0
this νˆ(·) also minimizes the relative entropy 4
Z
H(PPν ) = E(log
Z
T dP 1 T ∗ ) = E(− log Z (T )) = E[ θ (t)dW (t) + θν (t)2 dt] ν ν dPν 2 0 0 Z T 1 = E θ(t) + σ −1 (t)ν(t)2 dt 2 0
over ν ∈ D, answering a question of John van der Hoek. Now consider the special case K+ = {π ∈ Rd ; πi = 0, ∀ i = 1, . . . , m} of an incomplete market as in Example 6.1 (v) for some m = 1, . . . , d − 1, d ≥ 2. Then (7.26)0 becomes "
νˆ(t) =
r(t)1m − ˆb(t) 0n˜ ˜
#
, 0≤t≤T
where ˆb(t) = (b1 (t), . . . , bm (t))∗ and n = d − m; see Karatzas, Lehoczky, Shreve & Xu (1991), p. 721 and CK[1], pp. 797798 (as well as Hofmann et al. (1992), who show that Pνˆ , the “leastfavorable” equivalent martingale measure of Karatzas et al. (1991), coincides in this case with the “minimal equivalent martingale measure” in the sense of F¨ollmer & Schweizer (1991)). 2 EXAMPLE 7.2. Deterministic coefficients, utility function of powertype. Suppose that the coefficients r(·), b(·), σ(·) of the market M in (2.1), (2.2) are nonrandom (deterministic) functions, and that the utility function U (·) is of the socalled “powertype” (
(7.27)
4
Uα (x) =
xα /α; 0<α<1 α log x = limα↓0 x /α; α = 0
)
, 0 < x < ∞.
Then it is shown in [CK1], p.802 that (7.28)
νˆ(t) = argminy∈K˜ + [2(1 − α)δ(y) + θ(t) + σ −1 (t)y2 ], 0 ≤ t ≤ T
is again independent of the initial wealth; the same is thus true of pˆ. 2 35
EXAMPLE 7.3. Deterministic coefficients, cone constraints. Suppose again that r(·), b(·), σ(·) are deterministic, and that the constraint set K+ is a (closed, convex) cone in Rd (as ˜ ≡ 0 on K ˜ + . Then it is shown in [CK1], p.801 in Examples 6.1 (i), (ii), (iv)(vi)), so that δ(·) that, under certain mild conditions on the utility function U (·), the function νˆ(t) = argminy∈K˜ + θ(t) + σ −1 (t)y2 ,
(7.29)
0≤t≤T
is independent , not only of the initial wealth x > 0 , but also of the utility function U (·); these same properties are inherited by pˆ as well. Notice that νˆ(·) of (7.29) minimizes not only the relative entropy H(PPν ) as in Example 7.1, but also the relative entropy dPν ) = Eν [− dP
4
H(Pν P) = Eν (log = Eν [− =
Z T
0
0
θν∗ (t)dWν (t) +
0
Z T
1 ν E [ 2
Z T
θ(t) + σ
−1
θν∗ (t)dW (t) − 1 2
Z T 0
1 2
Z T 0
θν (t)2 dt]
θν (t)2 dt]
(t)ν(t)2 dt]
over ν ∈ D. 2 In any of these Examples 7.17.3, and with deterministic market coefficients (r(·), b(·), σ(·)), the process of (7.26), (7.28) or (7.29) is again a nonrandom (deterministic) function ˜ + . Suppose, furthermore, that νˆ : [0, T ] 7→ K
(7.30)
B(T ) = ϕ(P (T )), where P (·) = (P1 (·), . . . , Pd (·))∗ is the vector of stock price processes and ϕ(p) : (0, ∞)d 7→ [0, ∞) a continuous function that satisfies polynomial growth conditions in both p and 1/p.
Then from (7.25), (7.30), (6.6) and the FeynmanKac theorem (cf. [KS], p. 366), we see that the fair price for B(T ) is given by pˆ = e−
(7.31)
RT 0
(r(s)+δ(ˆ ν (s)))ds
· Q(0, P (0)).
Here Q(t, p) : [0, T ] × (0, ∞)d 7→ [0, ∞) is the solution of the Cauchy problem for the linear parabolic equation (7.32)
d d X d X ∂Q 1 X ∂Q ∂2Q + + (r(t) − νˆi (t))pi = 0; aij (t)pi pj ∂t 2 i=1 j=1 ∂pi ∂qj i=1 ∂pi
subject to the terminal condition (7.33)
Q(T, p) = ϕ(p),
p = (p1 , . . . , pd ) ∈ (0, ∞)d , 36
0 ≤ t < T,
where we recall that the matrix a(t) = (aij (t)) = σ(t)σ ∗ (t) is as in (2.3). The Cauchy problem of (7.32), (7.33) has a unique classical solution, subject to mild regularity conditions on the coefficients and on the terminal condition ϕ; see Chapter 1 in Friedman (1964). REMARK 7.4. In the case of constant coefficients (r(·) = r, b(·) = b, σ(·) = σ), the formulae (7.31)(7.33) become pˆ = e−(r+δ(ˆν ))T Q(0, P (0)),
(7.34)
(
(7.35)
Q(T − t, p) =
(2πt)−d/2 ϕ(p)
R Rd
ϕ(h(t, p, σz))e−z
2 /2t
dz ; 0 < t ≤ T, p ∈ (0, ∞)d ; t = 0, p ∈ (0, ∞)d
)
where νˆ = argminy∈K˜ + [2(1 − α)δ(y) + σ −1 (b − r + y)2 ] of (7.28) is now a constant vector in ˜ + , and the function h : [0, T ] × (0, ∞)d × Rd 7→ (0, ∞)d is given by K 1 4 hi (t, p, y) = pi exp[(r − νˆi − aii )t + yi ], 2
(7.36)
i = 1, . . . , d.
The Gaussian computation of (7.35) takes a very explicit form in the special case of a European call option on the first stock, where ϕ(p) = (p1 − q)+ , 0 < p1 < ∞, for some exercise price q > 0 in a market with constant coefficients (r, b, σ). Then (7.34) becomes pˆ = e−(ˆν1 +δ(ˆν1 ))T · u0 (r − νˆ1 , q; P1 (0)),
(7.37) where (7.38)
(
u0 (r, q; p) is the BlackScholes price of (4.4), (4.5) with interest rate r, exercise price q, and P1 (0) = p
)
.
EXAMPLE 7.4. “Lookback” option B(T ) = max0≤t≤T P1 (t) with constant coefficients, d = 1 and U (·) = Uα (·) as in (7.27). Again νˆ = argminy∈K˜ + [2(1 − α)δ(y) + σ −1 (b − r + y)2 ] is constant, and (7.25) becomes pˆ = P1 (0)e 4 r−ˆ ν σ
in the notation of (4.7) with ρˆ =
−(r+δ(ˆ ν ))T
Z ∞ 0
− σ2 .
37
f (T, ξ; ρˆ)eσξ dξ
8
European calloption in a market with constant coefficients
In this section, we use the general results of previous sections to study in detail the three prices hlow , hup and pˆ for a European call option on the first stock B(T ) = (P1 (T ) − q)+ , in a market with constant coefficients, i.e., when the coefficient b(·) ≡ b = (b1 , b2 , . . . , bd )∗ , r(·) ≡ r and σ(·) ≡ σ = (σij ) in (2.2) and (2.1) are all constants. All our examples involve closed, convex sets K+ , K− as in Section 6.
8.1
Lower and upper arbitrage prices EXAMPLE 8.1. Constraints on Borrowing, Example 6.1 (viii) with K+ = (−∞, k],
K− = [l, ∞) for some k ≥ 1 and l ≤ 1. It is easy to see from (4.6) that the BlackScholes price u0 belongs to L, thus (8.1)
hlow = u0 ,
by Theorem 5.1. On the other hand, we claim that hup ≤ E0 [γ0 (T )(
(8.2)
1 k−1 P1 (T ) − q)+ ] + P1 (0) =: ak . k k
PROOF OF (8.2). By the definition of hup it is enough to show that we can find for ˜ ˜ ˜ ∈ A(ak ), such that π ak an admissible pair (˜ π , C) ˜ (·) ≤ k and X ak ,˜π,C (·) ≥ 0, X ak ,˜π,C (T ) ≥ (P1 (T ) − q)+ almost surely. Actually, we can take C˜ ≡ 0. Define for 0 ≤ t ≤ T , h ³k − 1 ´+ ¯ i 1 1 ¯ E0 γ0 (T ) P1 (T ) − q ¯Ft + P1 (t) γ0 (t) k k 1 ˜ (1) (T − t, P1 (t)) + P1 (t), 0 ≤ t ≤ T, = U k 4
X (1) (t) = (8.3) where (8.4)
h ³ ´+ ¯ i 4 0 −rt k − 1 ¯ ˜ (1) (t, x) = U E e P1 (t) − q ¯P1 (0) = x , 0 ≤ t ≤ T, 0 < x < ∞. k
It is clear from (8.3) that X (1) (0) = ak ,
X (1) (T ) =
³k − 1
k
P1 (T ) − q
38
´+
1 + P1 (T ) ≥ (P1 (T ) − q)+ = B(T ). k
˜ (1) (t, x) of (8.4), we can define Using the function U π
(1)
(t) =
˜ (1) (T −t,P1 (t)) ∂U · P1 (t) + k1 P1 (t) ∂x . ˜ (1) (T − t, P1 (t)) + 1 P1 (t) U k
We shall show that X (1) (·) = X ak ,π
(1) ,0
(·), and π (1) (·) ≤ k.
˜ (1) (t, x) Notice, by the FeynmanKac formula (cf. [KS], p. 366) and (8.4), that the function U satisfies the Cauchy problem ( ˜ (1) ∂U
˜ (1) = 1 σ 2 x2 ∂ 2 U˜ 2(1) ∂t + r U 2 ∂x ˜ (1) (0, x) = ( k−1 x − q)+ . U k
(8.5)
˜ (1)
+ rx ∂ U∂x
¿From (8.5), (3.9) in the form dP1 (t) = rP1 (t)dt + σP1 (t)dW 0 (t), and Itˆo’s rule, we obtain ˜ (1) (T − t, P1 (t)) dU ˜ (1) ˜ (1) ˜ (1) ∂U ∂U 1 ∂2U = − dt + · dP1 (t) + d < P1 (t) > ∂t ∂x 2 ∂x2 ´ ³1 ˜ (1) ˜ (1) ∂2U ∂U ˜ (1) dt − r U = − σ 2 P12 (t) + rP (t) 1 2 ∂x2 ∂x (1) ³ ´ 1 ∂2U ˜ (1) ˜ ∂U · rP1 (t)dt + σP1 (t)dW 0 (t) + · σ 2 P12 (t)dt + ∂x 2 ∂x2 ˜ (1) (T − t, P( t)) ∂U ˜ (1) (T − t, P1 (t))dt + = rU σP1 (t)dW 0 (t). ∂x Therefore, dX (1) (t)
= =
= =
˜ (1) (T − t, P1 (t)) + 1 dP1 (t) dU k ˜ (1) ˜ (1) (T − t, P1 (t))dt + ∂ U (T − t, P1 (t)) σ · P1 (t)dW 0 (t) rU ∂x 1 1 0 + rP1 (t)dt + σP1 (t)dW (t) k k ³ ∂U ´ ˜ (1) (T − t, P1 (t)) 1 rX (1) (t)dt + · P1 (t)σdW 0 (t) + σP1 (t)dW 0 (t) ∂x k rX (1) (t)dt + X (1) (t)˜ π (t)σdW 0 (t).
Thus X (1) (·) satisfies the equation (3.2) with C ≡ 0, whence the pair (˜ π , 0) is indeed the one we needed, except we have to verify π ˜ (·) ≤ k. This can be checked easily from (8.4) and the inequality
1 1 x(ϕ0 (x) + ) ≤ k(ϕ(x) + x), k k 39
where ϕ(x) = (x(k − 1)/k − q)+ . 2 REMARK 8.1. The case k = 1 corresponds to the socalled “noborrowing” constraints and is discussed in [CK2], where it is also shown that hup = a1 = P1 (0) (for k = 1). In addition, these authors show that the consumption process C corresponding to the hedging strategy can be taken as C(t) = 0, for 0 ≤ t < T , and C(T ) = min(P1 (T ), q) > 0 at time t = T . REMARK 8.2. If k > 1, then we can rewrite ak as k−1 1 u0 (r, qk/(k − 1); P1 (0)) + P1 (0). k k Furthermore, if k increases (in other words, as the constraint becomes weaker) it is readily seen that the upper bound ak converges to the BlackScholes price u0 : k→∞
hup −→ u0 = u0 (r, q; P1 (0)) = BlackScholes price .
EXAMPLE 8.2. Constraints on shortselling, Example 6.1 (iii) with d = 1, K+ = [−k, ∞), K− = (−∞, l], for some k ≥ 0 and l > 1. It is easy to see from (4.6) that u0 ∈ U, whence (8.6)
hup = u0 .
We claim that in this case, (8.7)
hlow ≥ E0 [γ0 (T )(P1 (T ) − q)1[P1 (T )≥ql/(l−1)] ] =: ρl .
PROOF OF (8.7). Clearly, it is enough to show that ρl ∈ L. Define the process, 1 E0 [γ0 (T )(P1 (T ) − q)1[P1 (T )≥ql/(l−1)] Ft ] γ0 (t) ˜ (2) (T − t, P1 (t)), 0 ≤ t ≤ T, = −U 4
X (2) (t) = − (8.8) where
4 0 −rt ˜ (2) (t, x) = (8.9) U E [e (P1 (t) − q)1[P1 (t)≥ql/(l−1)] P1 (0) = x], 0 ≤ t ≤ T, 0 < x < ∞.
Then we have at time t = 0, X (2) (0) = −E0 [γ0 (T )(P1 (T ) − q)1[P1 (T )≥lq/(l−1)] ] = −ρl < 0, 40
and at time t = T , X (2) (T ) = −(P1 (T ) − q)1[P1 (T )≥ql/(l−1)] ≥ −(P1 (T ) − q)+ . ˜ (2) (t, x) ≤ E0 [e−rt P1 (t)P1 (0) = x] = x, so that, from (8.8), On the other hand, (8.9) gives 0 ≤ −U the positive process −X (2) (·) is dominated by the Pintegrable random variable max0≤t≤T P1 (t). Hence, it is enough to find a pair (π (2) , C (2) ) ∈ A(−ρl ) with X (2) (·) = X −ρl ,π π
(2)
˜ (2) (T −t,P1 (t)) ∂U · P1 (t) ∂x , ˜ (2) (T − t, P1 (t)) U
(t) =
(2) ,C (2)
(·). Introduce
0 ≤ t ≤ T.
˜ (2) (t, x) of (8.9) satisfies the Cauchy problem Again by the FeynmanKac formula, the function U ( ˜ (2) ∂U
˜ (2) = 1 σ 2 x2 ∂ 2 U˜ 2(2) + rx ∂ U˜ (2) ∂t + r U 2 ∂x ∂x ˜ (2) (0, x) = (x − q)1[x≥ql/(l−1)] , U
and from Itˆo’s rule, h ∂U ˜ (2)
i ˜ (2) ˜ (2) ∂U 1 ∂2U dP1 (t) + d < P (t) > 1 ∂t ∂x 2 ∂x2 2 (2) (2) ´ ³1 ˜ ˜ ∂ U ∂U ˜ (2) dt σ 2 P12 (t) − r U = + rP (t) 1 2 ∂x2 ∂x (2) ³ ´ 1 ∂2U ˜ (2) ˜ ∂U − rP1 (t)dt + σP1 (t)dW 0 (t) − · σ 2 P12 (t)dt ∂x 2 ∂x2 ˜ (2) ˜ (2) (T − t, P1 (t))dt − ∂ U σP1 (t)dW 0 (t) = −rU ∂x (2) (2) (2) = rX (t)dt + X (t)π (t)σdW 0 (t).
dX (2) (t) = − −
dt +
Hence the wealth equation (3.2) is satisfied with C = C (2) ≡ 0. To check that π (2) (·) ≤ l, we need only verify that
˜ (2) (T −t,P1 (t)) ∂U · P1 (t) ∂x (2) ˜ U (T − t, P1 (t))
≤ l.
This bound is not hard to derive, from (8.9) and the inequality ϕ0 (x) · x ≤ lϕ(x),
where ϕ(x) = (x − q)1[x≥ql/(l−1)] .
The proof is now complete. 2 REMARK 8.3.
Notice that we have from (2.2), P1 (t) = e(r−σ
2 /2)t+σN (t)
· p, where
p = P1 (0) > 0 and N (·) is standard Brownian motion under the probability measure P0 .
41
Therefore, ρl can be rewritten as ³
´
³
´
³
´
h
ρl = u0 r, ql/(l − 1); P1 (0) + E0 γ0 (T )(P1 (T ) − ql/(l − 1))1[P1 (T )≥ql/(l−1)]
i
= u0 r, ql/(l − 1); P1 (0) + qγ0 (T )/(l − 1) · P0 (P1 (T ) ≥ ql/(l − 1)) = u0 r, ql/(l − 1); P1 (0) +
³ ´ qe−rT · P0 (r − σ 2 /2)T + σN (T ) ≥ log(ql/(P1 (0)(l − 1))) , l−1
in the notation of (7.38). Invoking the normal distribution, we arrive after a bit of algebra at ³
´
ρl = u0 r, lq/(l − 1); P1 (0) +
³ 1 ³ √ ´´o qe−rT n . 1 − Φ √ · log ql/(P1 (0)(l − 1)) − (r − σ 2 /2) T l−1 σ T
As before, if the constraints become weaker and weaker (i.e., l → ∞), then ρl converges to the BlackScholes price u0 : hlow → u0 , as l → ∞. REMARK 8.4. If we consider the no shortselling case of Example 6.1 (ii) (or equivalently, Example 8.2 with k = l = 0) then instead of the inequality (8.7), we can actually prove hlow = 0. Indeed, we have from (6.6) that e
Rt 0
ν1 (s)ds
Z t
γ0 (t)P1 (t) = P1 (0) exp[
0
σ(s)dWν (s) −
1 2
Z t 0
σ(s)2 ds],
˜ d the subset of all nonrandom functions which is a Pν martingale. Thus, if we denote D ˜ − in the set D, ˜ we have ν : [0, T ] 7→ K (8.10)
ν
E [˜ γν (T )P1 (T )] = P1 (0)e
−
RT 0
˜ (δ(ν(s))+ν(s))ds
,
˜d. ∀ν ∈ D
By Theorem 6.1, we get the inequalities, 0 ≤ hlow = ≤
inf Eν [˜ γν (T )(P1 (T ) − q)+ ]
˜ ν∈D
inf Eν [˜ γν (T )P1 (T )]
˜ ν∈D
≤ P1 (0) inf e
−
˜d ν∈D
= P1 (0) inf e− ˜d ν∈D
RT 0
RT 0
˜ (δ(ν(s))+ν(s))ds
ν(s)ds
= 0,
as we can let ν tend to ∞. Thus we conclude that hlow = 0 in the no shortselling case. 42
EXAMPLE 8.3. Constraints on Borrowing, Example 6.1 (viii) with d = 1 and K+ = (−∞, k], K− = [l, ∞) for some k ≥ 1, l ≥ k, l > 1. Here again the upper bound hup ≤ ak on the upper arbitrage price holds, as in (8.2) of Example 8.1. Now, however, hlow = 0, so that the complete picture is 0 = hlow < u0 < hup ≤ ak < ∞. ˜ ˜ ± = (−∞, 0] and δ(x) = −kx, δ(x) ˜ ± , so that for Indeed, we have here K = −lx on K deterministic ν(·), Eν [˜ γν (T )P1 (T )] = P1 (0)e−
RT 0
˜ (δ(ν(s))+ν(s))ds
= P1 (0)e(l−1)
RT 0
ν(s)ds
as in (8.10), and we obtain hlow = 0 much like in Remark 8.4, except now letting ν(·) tend to −∞. EXAMPLE 8.4. Constraints on shortselling, Example 6.1 (iii) with d = 1 and K+ = [−k, ∞), K− = (−∞, −k] for some k ≥ 0. In this case, 0 = hlow < u0 = hup < ∞. Indeed, hup = u0 follows as in (8.6) of Example 8.2. As for hlow = 0, observe that we have ˜ ˜ ± = [0, ∞), δ(x) = δ(x) ˜ ± , and (8.10) become now K = kx on K ν
E [˜ γν (T )P1 (T )] = P1 (0)e
−(1+k)
RT 0
ν(s)ds
for deterministic ν(·); we conclude hlow = 0, by letting ν(·) becomes very large as in Remark 8.4. EXAMPLE 8.5. Incomplete market cases. (a). Only the first m stocks can be traded, with 1 ≤ m ≤ d − 1, d ≥ 2 as in Example 6.1 (iv). Then by the explicit formula in (4.6), we have that hup = hlow = u0 . (b). The first m stocks cannot be traded, 1 ≤ m ≤ d − 1 as in Example 6.1.(v). In this case, it can be shown that hup = ∞, as in [CK2]. We can show that hlow = 0. In fact, observe, once again from (6.6), that e
Rt 0
ν1 (s)ds
Z t
γ0 (t)P1 (t) = P1 (0) exp[
0
σ1 (s)dWν (s) −
1 2
Z t 0
σ1 (s)2 ds],
where σ1 = (σ11 , σ12 , . . . , σ1d )∗ . Then the same argument as for the no shortselling case will lead to the desired result.
43
8.2
Computation of the fair price
Let us compute in this subsection the fair price pˆ of (7.25) in a few examples, with closed and convex sets K± that satisfy the condition (7.12)—so that pˆ is in the interval [hlow , hup ] for all these examples. EXAMPLE 8.6. Cone constraints. Let K+ be a (closed, convex) cone in Rd , and K− = ˜ + , we have −K+ . Then from (7.29), (7.37) and the fact that δ ≡ 0 on K pˆ = e−ˆν1 T u0 (r − νˆ1 , q; P1 (0))
(8.11) in the notation of (7.38), where
νˆ = argminy∈K˜ + σ −1 (b − r + y)2 .
(8.12)
In particular, pˆ does not depend on either the utility function or the initial level of wealth. Here are some particular cases. (a) Incomplete market with only the first m stocks available, Example 6.1 (iv). Then (8.12) gives νˆ1 = 0, and (8.11) becomes pˆ = u0 (r, q; P1 (0)) = BlackScholes price. (b) Incomplete market with the first m stocks unavailable, Example 6.1 (v). We again have from (8.12) that νˆ1 = r − b1 (see also Example 7.1), and (8.11) takes the form: pˆ = e−(r−b1 )T u0 (b1 , q; P1 (0)). (c)Prohibition of shortselling, Example 6.1 (ii) with d = 1. Then it can be seen by simple algebra that in this case νˆ = (r − b)+ in (8.12), thus (8.11) becomes (
(8.13)
pˆ =
u0 (r, q; P1 (0)) ; if r ≤ b e−(r−b)T · u0 (b, q; P1 (0)) ; if r > b
)
.
EXAMPLE 8.7. Utility function of the power type (7.27). In this case, (7.28) or (7.26) give (8.14)
νˆ = argminy∈K˜ + [σ −1 θ(b + r − y)2 + 2(1 − α)δ(y)]
and pˆ is then as in (7.37); for a set K+ that is not a cone, this pˆ depends in general on the utility function through the constant α ∈ [0, 1). Here are some concrete examples.
44
(a) Prohibition of borrowing, Example 6.1 (vii) with d = 1. Then (8.14) gives νˆ = (r − b + (1 − α)σ 2 )− , and thus (7.37) becomes (
pˆ =
u0 ((b + (α − 1)σ 2 ), q; P1 (0)) ; if r ≤ b + (α − 1)σ 2 u0 (r, q; P1 (0)) ; otherwise
)
.
˜ + = (−∞, 0], for some (b) Constraints on borrowing, Example 8.3. Then δ(x) = −kx on K k ≥ 1, so (8.14) and (7.37) give νˆ = (r − b + (1 − α)σ 2 )− and (
(8.15)
pˆ =
2 u0 (r, q; P1 (0)) ³ ´ ; if b + k(α − 1)σ ≤ r 2 e−(k−1)(b+k(α−1)σ −r) u0 b + k(α − 1)σ 2 , q; P1 (0) ; otherwise
)
.
˜ + = [0, ∞) for some (c) Constraints on shortselling, Example 8.4. Then δ(x) = kx on K k ≥ 0, and (8.14), (7.37) lead respectively to νˆ = (r − b + k(α − 1)σ 2 )+ and (
(8.16)
8.3
pˆ =
2 u0 (r, q; P1 (0)) ³ ´ ; if r ≤ b + k(1 − α)σ 2 e−(1+k)(r−b+k(α−1)σ ) u0 b + k(1 − α)σ 2 , q; P1 (0) ; otherwise
)
.
Counterexamples
Finally, let us demonstrate by some examples that the lower bound of (7.5) may fail, in the absence of condition (7.12) on the sets K± . In all these examples the set K+ is convex, so the upper bound of (7.5) must hold; see Remark 7.1. EXAMPLE 8.1 (cont’d) with K+ = (−∞, k], K− = [l, ∞) and k > 1, l ≤ 1. Here it is easy to check that condition (7.12) fails; and with utility function Uα (·), 0 ≤ α < 1 as in (7.27), the fair price pˆ is given by (8.15) and satisfies pˆ → 0
as b → ∞,
for fixed (r, k, α, q, σ 2 , l), since u0 (x, q; P1 (0)) → P1 (0) as x → ∞ (See Cox & Rubinstein (1984), p.216). However, we know from (8.1) that hlow ≡ u0 (r, q; P1 (0)) > 0, whence hlow > pˆ > 0 for all sufficiently large appreciation rates b. EXAMPLE 8.2 (cont’d). Here K+ = [−k, ∞), K− = (−∞, l] for some k ≥ 0, l > 1. Again, it is verified that condition (7.12) fails; and with utility function of the type (7.27), the fair price pˆ is given by (8.16) and satisfies pˆ → 0, as r → ∞ 45
for (b, k, α, q, σ 2 ) fixed. On the other hand, we have from Remark 8.3: ³ 1 ³ √ ´´o qe−rT n hlow ≥ ρl = u0 (r, lq/(l − 1); p) + · 1 − Φ √ · log lq/(p(l − 1)) − (r − σ 2 /2) T l−1 σ T r→∞
−→ p ≡ P1 (0) > 0.
Consequently, for all sufficiently large interest rates r, hlow > pˆ > 0. EXAMPLE 8.8. Take d = 1, r > b, K+ = [0, ∞), K− = [1, ∞) (a combination of ˜ + = [0, ∞) and K ˜ − = (−∞, 0], Examples 6.1 (ii), (vii)), so that (7.12) fails again. Now K hlow = u0 (r, q; P1 (0)) as in (8.1), and from (8.13): pˆ = e−(r−b)T u0 (b, q; P1 (0)) < u0 (r, q; P1 (0)) = hlow .
9
Market with higher interest rate for borrowing
We have studied so far the pricing problem for contingent claims in a financial market with the same interest rate for borrowing and for saving. However, the techniques developed in the previous sections can be adapted to a market M∗ with interest rate R(·) for borrowing higher than the bond rate r(·) (saving rate). We consider in this section an unconstrained market M∗ with two different (bounded, {Ft }progressively measurable) interest rate processes R(·) ≥ r(·) for borrowing and saving, respectively. In this market M∗ , it is not reasonable to borrow money and to invest money in the bond, at the same time. Therefore, the relative amount borrowed at time t is equal to
³
1−
Pd
´−
i=1 πi (t)
. As shown in [CK1], the wealth process X(·) = X x,π,C (·) corresponding to
initial wealth x and a portfolio/consumption pair (π, C) as in Definition 3.5, satisfies now the analogue of the wealth equation (3.2) h
³
(9.1) dX(t) = r(t)X(t)dt − dC(t) + X(t) π ∗ (t)σ(t)dW0 (t) − (R(t) − r(t)) 1 −
d X
´−
πi (t)
i
dt ,
i=1
whence 4
N (t) = γ0 (t)X(t) +
Z t 0
γ0 (t)dC(t) +
Z t 0
³
γ0 (t)X(t)[R(t) − r(t)] 1 −
d X
´−
πi (t)
i=1
is a P0 local martingale by Itˆo’s rule, in the notation of (2.4)(2.7) and (3.6). 46
dt, 0 ≤ t ≤ T
All the arguments in Section 5 go through under slight modifications. For example, the lower and upper hedging classes are now defined to be 4 ˇ ∈ A(−x), such that X −x,ˇπ,Cˇ (·) ≤ 0 and X −x,ˇπ,Cˇ (T ) ≥ −B(T ), almost surely}, L = {x ≥ 0 : ∃(ˇ π , C) 4 ˆ ∈ A(x), such that X x,ˆπ,Cˆ (·) ≥ 0 and X x,ˆπ,Cˆ (T ) ≥ B(T ), almost surely}. U = {x ≥ 0 : ∃(ˆ π , C)
The statements of Definition 5.2, Theorem 5.1, 5.2 and 5.3 hold without change. We set δ(ν(t)) = −ν1 (t), 0 ≤ t ≤ T , for ν ∈ D, where D is the class of progressively measurable processes ν : [0, T ] × Ω 7→ Rd with r − R ≤ ν1 = · · · = νd ≤ 0, l ⊗ P − a.e. Then with this notation the theory of Section 6 also goes through with only minor changes, such as ˜ by D, etc. In particular, Theorem 6.1 now states that replacing δ˜ by δ and D hlow = inf Eν [γν (T )B(T )]
(9.2)
ν∈D
and that hup = sup Eν [γν (T )B(T )].
(9.3)
ν∈D
The proofs of (9.2) and (9.3) follow the same lines of Theorem 6.1 and [CK2], respectively. We sketch the proof of (9.2) here. SKETCH OF PROOF FOR (9.2). We first repeat the proof of Theorem 6.1, right up to ˇ (6.23). There, we change the definition of the consumption process C(·), to read 4 ˇ = C(t)
Z t 0
γν−1 (s)dAν (s) −
Z t 0
h
³
ˇ X(s) δ(ν(s)) + π ˇ ∗ (s)ν(s) + (R(s) − r(s)) 1 −
d X
´− i
π ˇi (s)
ds.
i=1 4
Taking ν(t) = λ(t) ≡ λ1 (t)1, where λ1 (t) = [r(t) − R(t)]1(Pd πˇ (t)>1) , we get i=1 i ˜ ˇ = C(t)
Z t 0
γλ−1 (s)dAλ (s)
as required. Skip the lines in which π ˇ (t) ∈ K− is shown, and observe that we have now ∗ ˇ d(−X(t)γ ν (t)) = dQν (t) = ψν (t)dWν (t) + dAν (t)
n
h
³
d X
ˇ ˇ = γν (t) −dC(t)− X(t) δ(ν(t))+(R(t)−r(t)) 1−
´−
π ˇi (t)
i
i=1
ˇ − X(t)Ψ ˇ = γ˜ν (t)[−dC(t)
ν,ˇ π
ˇ π ∗ (t)σ(t)dWν (t)], (t)dt + X(t)ˇ
47
o
ˇ π ∗ (t)σ(t)dWν (t) +ˇ π ∗ (t)ν(t) dt+X(t)ˇ
where ³
4
Ψν,ˇπ (t) = [R(t) − r(t) + ν1 (t)] 1 −
d X
´−
π ˇi (t)
³
− ν1 (t) 1 −
i=1
d X
π ˇi (t)
´+
, 0≤t≤T
i=1
ˇ = X −g,ˇπ,Cˇ (·) by comparing with the is a nonnegative process. Now taking ν ≡ 0, we get X(·) new wealth equation (9.1), where g is now the righthand side of (9.2). The rest of the proof proceeds in a clearly analogous way. 2 With the adoption of the new δ(·) and D, we can define the fair price by analogy with (7.4), and proceed in the same way as we did before, to obtain all theorems in Section 7. In particular, an encouraging phenomenon is that the fair price always lies within the arbitrage interval, as the Assumption 7.2 is always satisfied in this case. The argument in [CK2] for computing hup in a market M∗ with d = 1 and constant coefficients also works for hlow , after slight adjustments. For example, change “sup” and “max” in (9.8) and (9.9) of [CK2] to “inf” and “min” respectively; then from the HamiltonJacobiBellman (HJB) equation we can also get hlow = u0 (r, q; P1 (0)) for the European call option B(T ) = (P1 (T ) − q)+ . In other words, the lower arbitrage price is exactly the BlackScholes price with interest rate r, while, as it has be shown in [CK2], the upper arbitrage price is the BlackScholes price with interest rate R: hup = u0 (R, q; P1 (0)). For the fair price within the interval [hlow , hup ], we still use Theorem 7.4 and Remark 7.1 to get the explicit fair price pˆ for the constant coefficient market M∗ . More precisely, with utility function Uα (·) as in (7.27), it is shown in [CK1], p.816 that the νˆ in Theorem 7.4 is given by
0
; if r ≥ b1 + σ 2 (α − 1) 2 2 r − b1 − σ (α − 1) ; if r ≤ b1 + σ (α − 1) ≤ R . νˆ = r−R ; if b1 + σ 2 (α − 1) ≥ R.
Hence, (7.37) gives the fair price u0 (r, q; P1 (0))
(9.4)
: if r ≥ b1 + σ 2 (α − 1) 2 u0 (b1 + σ (α − 1), q; P1 (0)) ; if r ≤ b1 + σ 2 (α − 1) ≤ R . pˆ = u(R, q; P (0)) ; otherwise 1
48
REMARK 9.1. The expression (9.4) coincides with the socalled “minimax price” in Barron & Jensen (1990), defined to be the number p˜ = p˜(x) for which the function δ 7→ W (δ, p˜(x), x) of (7.3) is minimized at δ = 0. (Clearly, with pˆ(x) as in our Definition 7.3, δ 7→ W (δ, pˆ(x), x) is minimized at δ = 0, so we can take p˜(x) = pˆ(x), justifying this “coincidence”.) REMARK 9.2. More generally, with d ≥ 1, utility function Uα (·) of the type (7.27) and deterministic coefficients (resp., α = 0 in (7.27) and general random coefficients), the function (resp., process) νˆ(t) = νˆ1 (t)1 is given as ˜
h
νˆ1 (t) = argminr(t)−R(t)≤y≤0 σ −1 (t)(b(t) − r(t) + y12 − 2y ˜ ; ξα (t) ≤ 0 0 r(t) − R(t) ; ξα (t) ≥ R(t) − r(t) = −ξ (t) ; 0 ≤ ξα (t) ≤ R(t) − r(t) α
i
by analogy with (7.26) and (7.28), where ξα (t) = (α − 1 + θ∗ (t)σ −1 (t)1)/{tr[(σ −1 (t))∗ (σ −1 (t))]}. ˜ In the special case B(T ) = ϕ(P (T )) of (7.30) with deterministic coefficients, the computations of (7.31)(7.36) for the fair price pˆ are all still valid.
10
A table
The results of previous discussions and examples, concerning the pricing of a European call option B(T ) = (P1 (T ) − q)+ in a market with constant coefficients, can be summarized on a table as follows.
49
Unconstrained market Incomplete market, with first m stocks available Incomplete market, first m stocks unavailable No shortselling of stocks (K+ = [0, ∞), K− = (−∞, 0])
hlow u0 (r)
hup u0 (r)
pˆ u0 (r) ∗
u0 (r)
u0 (r)
u0 (r) *
0
∞
0
u0 (r)
No borrowing u0 (r) (K+ = (−∞, 1], K− = [1, ∞)) Constraints on shortselling (K+ = [−k, ∞), 0 K− = (−∞, −k], k ≥ 0) Constraints on borrowing (K+ = (−∞, k], 0 K− = [l, ∞), l ≥ k > 1) Constraints on shortselling (K+ = [−k, ∞), ≥ ρl K− = (−∞, l], k ≥ 0, l > 1) Constraints on borrowing (K+ = (−∞, k], u0 (r) K− = [l, ∞), k > 1, l ≤ 1) Market with higher interest rate R > r u0 (r) for borrowing (∗) For arbitrary utility function.
(
e−(r−b1 )T u0 (b1 ) * ) u0 (r); if r ≤ b1 ∗ e−(r−b1 )T u0 (b1 ) ; if r > b1 (
P1 (0) (
u0 (r) (
u0 (r); if r ≥ f u0 (f ); otherwise u0 (r) ; if r ≤ f ck ; otherwise
)
† )
†
≤ ak
u0 (r) ; if r ≥ b1 + k(α − 1)σ 2 dk ; otherwise
u0 (r)
not appropriate (ˆ p < hlow )
≤ ak
not appropriate (ˆ p < hlow ) u0 (r)
u0 (R)
; if r ≥ f u0 (f ) ; if r ≤ f ≤ R u (R) ; if f ≥ R 0
)
†
†
(†) For utility function Uα (·) of the form (7.27) with 0 ≤ α < 1. In the above table, r is the interest rate of the bond (savings account); b1 is the appreciation rate of the first stock, on which the option is written; σ 2 is the stock volatility; u0 (x) ≡ u0 (x, q; P1 (0)) is the BlackScholes price for interest rate x and exercise price q; and P1 (0) is the price for the first stock at time t = 0. Finally, ³ ´ 1 k−1 k→∞ · u0 r, qk/(k − 1); P1 (0) + P1 (0) −→ u0 (r), k k ³ ´ qe−rT = u0 r, lq/(l − 1); P1 (0) + · l−1 n ³ 1 √ ´´ol→∞ 1 − Φ √ · log(ql/(P1 (0)(l − 1)) − (r − σ 2 /2) T −→u0 (r) σ T
ak = ρl
2
ck = e−(1+k)(r−b1 +(α−1)kσ ) · uo (b1 + (1 − α)kσ 2 ) dk = e−(k−1)(b1 +(α−1)kσ
2 −r)
· u0 (b1 + (α − 1)kσ 2 ) 50
f
= b1 + (α − 1)σ 2 .
REMARK 10.1. It should be observed that all the exact values, as well as the bounds, for hlow and hup are independent of the appreciation rate b of the stock, which is often difficult to estimate. This makes the lower and upper arbitrage prices relatively easy to use. In contrast, a main drawback of the fair price is that it does depend on b. Heuristically, it may well be that hedging, as it is based on the arbitrage arguments, is a sort of “global” property. On the other hand, the Definition 7.3 of the fair price looks like a “local” property, as it involves a derivative; this makes the fair price pˆ more likely to depend on the local “drift” b (appreciation rate) of the price process.
11
Discussion
1. With a little additional care, the method also works for the European option with dividend rate g(t). For example, the analogue for Theorem 6.1 will be hlow = inf Eν [˜ γν (T )B(T ) + ˜ ν∈D
and hup = sup Eν [γν (T )B(T ) + ν∈D
Z T 0
Z T 0
γ˜ν (s)g(s)ds]
γν (s)g(s)ds].
2. A similar lower price hlow (for “buyers”, as opposed to the hup which refers to “sellers”) was mentioned by El Karoui & Quenez (1992) in the incomplete market case, but without justification based on considerations of arbitrage. 3. Suppose we want to consider constraints on the number of shares φ, or on the total amount of money invested in every asset, instead of on the vector π, of the proportions of wealth invested in assets. Then the general arbitrage arguments in Section 5 still hold. However, we no longer have an easy way to get all the representations of Section 6. For instance, the nice equation (6.24) is changed, as the very “helpful” term δ(ν(s)) + ν ∗ (s)π(s) disappears. 4. For practical purposes, one may recommend the use of the BlackScholes price u0 as a “roughandready” unique price, for a constrained market with the same interest rate for borrowing and saving, when the fair price pˆ is difficult to compute. The reasons are: 51
(a) The BlackScholes price u0 always lies within the arbitragefree interval [hlow , hup ]; (b) As we saw in the case of constraints on borrowing and shortselling, the arbitragefree interval will shrink to u0 as the constraints become weaker and weaker; (c) The BlackScholes price u0 does not involve the stock appreciation rate b; (d) Many numerical procedures, including software, have been developed to calculate u0 . Acknowledgements: We wish to thank Professor Jakˇsa Cvitani´c for his careful reading of the manuscript and his helpful comments, and Professors N. El Karoui and Mark Davis for their encouragement. We are indebted to Dr. Frank Oertel for a suggestion that led to our Example 7.4, to Prof. Steven Shreve for prompting us to “tighten” the argument in Theorem 7.2, and to Dr. John van der Hoek for asking us to consider the connection of the fair price with entropy minimization. Thanks also due to the anonymous referee for his very conscientious reading of the paper, and his many corrections and suggestions.
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