A Minimum Variance Result Under Market Conditions

C.Barnett ^{∗} and W.P.Liu Department of Mathematics Imperial College of Science, Technology and Medicine University of London

August 2, 2005

Abstract In [7], H. Richardson solved the problem of achieving a specified level of expected wealth with minimum risk (variance). We introduce the idea of Market Conditions and susequently solve a problem more general than Richardsons’.

1 Introduction and Summary

We adopt the context of Richardsons’ paper. So our market is the complete frictionless structure consisting of a single risky asset, S^{1} - a stock, whose returns are log-normal, and a simple riskless bond, S^{0}, following the usual price processes. The symbols μ_{1} σ, r, denote positive constants, respectively the drift and volatility of S^{1} and the interest rate for S^{0}. On the probability space (Ω, F_{T} , P), W is a standard Brownian Motion. Trading, long or short, may be continuous and there are no transaction costs. If X ∈ L^{2}(Ω, F_{T} , P) and X is a non-negative contingent claim then there is a self-financing trading strategy, Φ = (ϕ^{0}, ϕ^{1}) which achieves X and has value V (Φ); that is,

X = ϕ_{T}^{0} S_{T}^{0} + ϕ_{T}^{1} S_{T}^{1} and E(X|F_{t}) = V_{0} + ∫_{0}^{t} ϕ_{s}^{1}dZ_{s}^{1},

for 0 ≤ t ≤ T, where V_{0} is the time zero value of the portfolio and Z is the discounted stock value,

Z_{t} = \frac{S_{t}^{1}}{S_{t}^{0}} = Z_{0} exp \{σW_{t} − \frac{σ^{2}t}{2} + μt\} = Z_{0} + ∫_{0}^{t} Z_{s}d(σW_{s} + μs)

here μ = μ_{1} − r and

∗c.barnett@imperial.ac.uk

E(∫_{0}^{T} (S_{s}^{1}ϕ_{s}^{1})^{2}ds) < ∞ and E(∫_{0}^{T} |ϕ_{s}^{0}S_{s}^{0}|ds) < ∞.

The first of these conditions keeps portfolio gains in L^{2}(Ω, F_{T} , P). We will work with discounted quantities. The discounted gain is the process

Ĝ_{t} =^{△} Ĝ_{t}(ϕ^{1}) = ∫_{0}^{t} ϕ_{s}^{1} dZ_{s}  (1)

Our portfolio must be such that

V̂_{t} = v̂_{0} + Ĝ_{t},  (2)

where v̂_{0} = V_{0} is the value of the portfolio at time zero. We can now state Richardsons’ problem; An investor with initial endowment v̂_{0} and (expected) investment goal at time T of v̂_{T} wishes to achieve this goal with minimal risk. Equating risk with variance, we want to find the portfolio, Φ, such that

E [v̂_{0} + Ĝ_{T} ] = v̂_{T} , while var \left[{ \atop Ĝ_{T}}\right] is minimal,  (3)

and we write ĝ_{T} = v̂_{T} − v̂_{0} for the (expected) investment goal for the portfolio gain at time T which is assumed to be strictly positive. We refer to this problem (3) as the minimum variance problem. In order to solve this problem, Richardson first determines the random variable X, that corresponds to the solution of this problem, then the discounted gain process (Ĝ_{t}) for which Ĝ_{t} = X, and finally the trading strategy ϕ^{1} which gives the gains process. The remaining results of this section were proved in Richardson’s paper. The exponential martingale

M_{t} = exp \{− (\frac{μ}{σ}) W_{t} − (\frac{μ}{σ})^{2} \frac{t}{2}\}, t ∈ [0, T]

turns out to be significant. We have the following result.

Lemma 1 A random variable X is the gain from an admissible trading strategy if and only if X ∈ ⟨M_{T} ⟩^{⊥}. Here the orthogonal complement occurs in L^{2} (Ω, F_{T} , P). We write

⟨M_{T} ⟩^{⊥} = G,

and call G the “gains subspace”.

So L^{2} (Ω, F, P) decomposes as an orthogonal direct sum,

G ⊕ ⟨M_{T} ⟩.

We observe that G is closed because it is an orthogonal complement. It turns out that G decomposes similarly into a direct sum in which one of the components is one-dimensional. Indeed, let π :L^{2} (Ω, F_{T} , P)⟶ G be the orthogonal projection of L^{2} (Ω, F_{T} , P) onto G. Then we have the following lemma.

Lemma 2 Let C_{0} be the closed subspace of G consisting of those gains with zero-expectation. Let e denote the function Ω ∋ ω ⟼ 1 ∈ R then,

G = C_{0} ⊕ ⟨π(e)⟩.  (4)

As a consequence of this decomposition there are gains X, with E[X] =ĝ_{T}; indeed every such gain has the form,

X = Y + \frac{ĝ_{T}}{E[π(e)]} π(e),  (5)

for Y ∈ C_{0}. Now the variance of a gain such as X in equation (5) is given by the sum of the variances of the two right hand terms. The second of these consists of terms which we cannot change. So for var[X] to be minimised subject to E[X] = ĝ_{T} we can only minimise var [Y]. So we set Y = 0. This identifies the random variable that corresponds to the solution of this problem, and it is exactly \frac{ĝ_{T}}{E[π(e)]} π(e). Since G is ⟨M_{T} ⟩^{⊥}, one can write down an explicit and elementary formula for the projection π (Lemma 3 in [7]) and consequently for π(e). Indeed π(e) = e − \frac{M_{T}}{∥M_{T} ∥^{2}} , and an application of Itô’s lemma shows that

the process (\frac{M_{t}}{∥M_{t}∥^{2}} ) is a Q-martingale. Having done this then one knows π(e)

as the time T value of a P-semi-martingale. Since \frac{ĝ_{T}}{E[π(e)]} π(e) ∈ G there is a trading strategy ϕ^{∗} in stock which defines a self-financing trading strategy Φ^{∗} such that Ĝ_{T} (Φ^{∗}) = \frac{ĝ_{T}}{E[π(e)]} π(e). Since each of the process (G_{t}(Φ^{∗})) and

(\frac{ĝ_{T}}{E[π(e)]} (e − \frac{M_{t}}{∥M_{t}∥^{2}} )) are Q-martingales then they are equal Q-almost surely andtherefore P-almost surely for every t ∈ [0, T]. Using Ito’s lemma to write e − \frac{M_{t}}{∥M_{t}∥^{2}} as a sum of integrals we can use the uniqueness of semi-martingales to determine ϕ^{∗}.

2 Market Conditions

As we indicated in the introduction, a ‘market condition’. a restriction on our ability to trade, will amount to our only being able to use trading strategies which are in the range of some map of the space of trading strategies into itself. Since we deal only with self-financing portfolios for which the bond component is determined by

ϕ_{t}^{0} = v̂_{0} + ∫_{0}^{t} ϕ_{s}^{1} dZ_{s} − ϕ_{t}^{1}Z_{t}, t ∈ [0, T].  (6)

then we consider only maps of the space of ‘stock strategies’ into itself; that is, the vector space of progressively measurable processes ϕ^{1}, on [0, T] for which E ∫_{0}^{T} (ϕ_{s}^{1}S_{s}^{1})^{2}ds) < ∞. Possible examples: a volume constraint is imposed so that for some K > 0,

−K ≤ ϕ_{t}^{1}(w) ≤ K.

There may be values of K which for a given initial endowment would prevent you from achieving your investment goal. Another example is the situation where at some random time, τ ∈ [0, T),an agent intervenes to halt trading and one’s portfolio amounts ϕ_{τ}^{0} and ϕ_{τ}^{1} are fixed from time τ until time T. In each of these cases we see that our “choice” of trading strategies are limited to the range of a map on the space of trading strategies. For the first case it is,

ϕ ⟼ −K ∨ ϕ ∧ K ≡ ϕ_{K},

while for the second case it is,

(ϕ_{s}) ⟼ (ϕ_{τ∧s}) ≡ ϕ^{τ} .

We note that the first map is non-linear with convex range, while the second is a linear mapping ^{1}. These cases are discussed in [5] and [6]. These considerations lead to the following definition.

Definition 1 A \underline{market condition} is a map of T the (stock) trading strategies, into itself. We note that T is a linear space under pointwise operations. Further we shall regard elements of T which are indistinguishable as identical with one another.

Our first result for this section concerns the structure of T - it arises from the following question; Let π_{0} : G ⟶ C_{0} denote the orthogonal projection. Since the elements of C_{0} are gains then for each ϕ ∈ T , there is π̄_{0}(ϕ) ∈ T such that

π_{0} (∫_{0}^{T} ϕ_{s} dZ_{s}) = (∫_{0}^{T} π̄_{0}(ϕ_{s}) dZ_{s}) .  (7)

Hereafter the maps of the space of trading strategies will be denoted with a bar, for example, π̄_{0} or C̄ and the corresponding maps on the space of gains by C, π_{0}. Can we express (π̄_{0}(ϕ_{s})) in terms of (ϕ_{s}) ? The answer is yes:

Theorem 1 The linear space T may be expressed as a direct sum

T = T_{0} ⊕ R,

where

ϕ ∈ T_{0} ⟺ E [Ĝ_{T} (ϕ)] = 0 and R = ⟨( \frac{μM_{s}}{σ^{2}∥M_{s}∥^{2}Z_{s}} )⟩ ,

The map ϕ ⟼ π̄_{0}(ϕ), defined in the preamble, is an idempotent of T onto T_{0}.

1Of course we require a time τ such that ϕ^{τ} ∈ T for each ϕ ∈ T .

Proof 1 Since G = C_{0} ⊕ ⟨π(e)⟩ then for any X ∈ G there is λ ∈ R such that,

X = π_{0}(X) + λπ(e) = π_{0}(X) + \frac{⟨X, π(e)⟩}{∥π(e)∥^{2}} π(e) = π_{0}(X) + \frac{E [X]}{∥π(e)∥^{2}} π(e).

So that π_{0}(X) = X − \frac{E [X]}{∥π(e)∥^{2}} π(e). If X is the gain from the strategy ϕ then

π_{0} (∫_{0}^{T} ϕ_{s} dZ_{s}) = ∫_{0}^{T} ϕ_{s} dZ_{s} − \frac{E [∫_{0}^{T} ϕ_{s} dZ_{s}]}{∥π(e)∥^{2}} (e − \frac{M_{T}}{∥M_{T} ∥^{2}} ) .  (8)

An application of Itô’s lemma to the process (\frac{M_{T}}{∥M_{T} ∥^{2}} ) shows that

∫_{0}^{T} ϕ_{s} dZ_{s}−\frac{E [∫_{0}^{T} ϕ_{s} dZ_{s}]}{∥π(e)∥^{2}}(e−[e−(\frac{μ}{σ}) ∫_{0}^{T} \frac{M_{s}}{∥M_{s}∥^{2}} dW_{s}−(\frac{μ}{σ})^{2} ∫_{0}^{T} \frac{M_{s}}{∥M_{s}∥^{2}} ds])

which simplifies to

∫_{0}^{T} (ϕ_{s} − \frac{E [∫_{0}^{T} ϕ_{s} dZ_{s}]}{∥π(e)∥^{2}} \frac{μ}{σ^{2}} \frac{M_{s}}{∥M_{s}∥^{2}} \frac{1}{Z_{s}} ) dZ_{s}.

So

π̄_{0}(ϕ_{s}) = ϕ_{s} − \frac{E [∫_{0}^{T} ϕ_{s} dZ_{s}]}{∥π(e)∥^{2}} \frac{μ}{σ^{2}} \frac{M_{s}}{∥M_{s}∥^{2}} \frac{1}{Z_{s}}  (9)

corresponds to the projection of the trading strategy ϕ ∈ T onto the subspace T_{0} from equation (1). Certainly, for any ϕ,

E [π_{0} (∫_{0}^{T} ϕ_{s} dZ_{s})] = E [∫_{0}^{T} π̄_{0}(ϕ_{s}) dZ_{s}] = 0.

On the other hand, if E[Ĝ_{T} (ϕ)] = E [∫_{0}^{T} ϕ_{s} dZ_{s}] = 0 then, π̄_{0}(ϕ_{s}) = ϕ_{s} so let T_{0} = π̄_{0}(T ). We note that if ϕ_{s} = \frac{α}{∥π(e)∥^{2}} \frac{μ}{σ^{2}} \frac{M_{s}}{∥M_{s}∥^{2}} \frac{1}{Z_{s}} , α ∈ R then,

E [∫_{0}^{T} ϕ_{s} dZ_{s}] = E [∫_{0}^{T} \frac{α}{∥π(e)∥^{2}} \frac{μ}{σ^{2}} \frac{M_{s}}{∥M_{s}∥^{2}} d (σW_{s} + μs)] = \frac{α}{∥π(e)∥^{2}} E [π(e)].

However ∥π(e)∥^{2} = E [π(e)], so E [∫_{0}^{T} ϕ_{s} dZ_{s}] = α.. Since π̄_{0} is clearly linear, T_{0} is a subspace of T . If ϕ ∈ T_{0} ∩ R, then there is α ∈ R such that,

ϕ = (\frac{α}{Z_{s}} \frac{μM_{s}}{σ^{2}∥M_{s}∥^{2}} ) = (∥π(e)∥^{2} \frac{α}{∥π(e)∥^{2}} \frac{μ}{σ^{2}} \frac{M_{s}}{∥M_{s}∥^{2}} \frac{1}{Z_{s}} ) .

So that, E [∫_{0}^{T} ϕ_{s} dZ_{s}] = ∥π(e)∥^{2}α. But if also ϕ ∈ T_{0} then ∥π(e)∥^{2}α = 0, so that α = 0. This shows that R ∩ T_{0} = {0}, so the sum is direct and equal to T . Consider now,

π̄_{0} (π̄_{0}(ϕ_{s})) = π̄_{0}(ϕ_{s}) − \frac{E [∫_{0}^{T} π̄_{0}(ϕ_{s}) dZ_{s}]}{∥π(e)∥^{2}} \frac{μ}{σ^{2}} \frac{M_{s}}{∥M_{s}∥^{2}} \frac{1}{Z_{s}}

= π̄_{0}(ϕ_{s}) since E [∫_{0}^{T} π̄_{0}(ϕ_{s}) dZ_{s}] = 0.  (10)

So π̄_{0} is an idempotent. □

Corollary 1 If ϕ ∈ T then π̄_{0}(ϕ) = 0 ⟺ ϕ ∈ R.

We turn now to consider the problem of achieving a given level of expected wealth under some market condition C̄. Let C denote the map of G into itself given by

C (∫_{0}^{T} ϕ_{s} dZ_{s}) = ∫_{0}^{T} C̄(ϕ_{s}) dZ_{s}.

We will assume that C̄ is such that C a bounded operator with closed range, the projection of G onto the range of C is written l(C). Let C(G)_{0} be the subspace of the range of C for which E(C(X)) = 0 for X ∈ G. C(G)_{0} is a closed subspace because

0 = E(C(X)) =< C(X), e >=< l(C)C(X), π(e) >=< C(X), l(C)π(e) >

so it is the subspace orthogonal to < l(C)π(e) >. Now, C(G) is a Hilbert space and E(·) a continuous linear functional on this space, E(·) implemented by the vector l(C)π(e) in C(G). So its kernel has co-dimension 1. Consequently any gain, C(X), in the range of C can be written

C(X) = Y + λl(C)π(e),  (11)

with Y ∈ C(G)_{0} and λ ∈ R. As a consequence, so long as l(C)π(e) is non zero there will be gains in the range of C for which E(C(X)) = ĝ_{T}. Every such gain has the form

C(X) = Y + \frac{ĝ_{T}}{E(l(C)π(e))}l(C)π(e).

The variance of this gain is, because the parts are orthogonal,

V ar(C(X)) = V ar(Y) + (\frac{ĝ_{T}}{E(l(C)π(e))})^{2}V ar(l(C)π(e))
= V ar(Y) + ĝ^{2}(\frac{(1 − E(l(C)π(e))}{E(l(C)π(e)}

We have “no choice” about the second of these variance terms so to minimise the variance we set Y = 0. This identifies the random variable which minimises the variance. Now we turn to cases. We start with a market condition, C̄, that commutes with the projection π̄_{0} of T onto T_{0}.

Lemma 3 Let C̄ be a linear market condition such that C̄ π̄_{0} = π̄_{0} C̄. If the strategy C̄(ϕ) where ϕ ∈ T is adopted, then the expectation of the gain is given by

E [Ĝ_{T} (C̄(ϕ))] = \frac{E[∫_{0}^{T} ϕ_{s} dZ_{s}]}{E[π(e)]} E[C(π(e))],

and the variance of the gain is given by

var [Ĝ_{T} (C̄(ϕ))] = ∥π_{0} · C (∫_{0}^{T} ϕ_{s} dZ_{s})∥^{2} + \frac{E [∫_{0}^{T} ϕ_{s} dZ_{s}]^{2}}{E[π(e)]^{2}} var [C(π(e))].

Proof 2 G is the orthogonal direct sum C_{0} ⊕ ⟨π(e)⟩, so

C (∫_{0}^{T} ϕ_{s} dZ_{s}) = C · π_{0} (∫_{0}^{T} ϕ_{s} dZ_{s}) + \frac{E[∫_{0}^{T} ϕ_{s} dZ_{s}]}{E[π(e)]} C(π(e)),

= ∫_{0}^{T} π̄_{0}(C̄(ϕ_{s})) dZ_{s} + \frac{E[∫_{0}^{T} ϕ_{s} dZ_{s}]}{E[π(e)]} C(π(e)), (12)

since π̄_{0} and C̄ commute. This means that C maps C_{0} into itself and the subspace < π(e) > likewise. Should C be non-zero on R =< π(e) > then π(e) is an

eigenvector of C, the eigenvalue is given by λ_{C} = \frac{E(∫_{0}^{T} C̄(ϕ_{s})dZ_{s})}{E(∫_{0}^{T} ϕ_{s}dZ_{s})} for any ϕ ∈ T

for which π_{0}(ϕ) ≠ ϕ, and π(e) lies in the range C. Therefore l(C)π(e) = π(e) while π̄_{0}(C̄(ϕ_{s})) produces a gain in the range of C with zero expectation. We have recovered equation 11. The calculations are now routine. □

Theorem 2 Let C̄ be a linear market condition commuting with π̄_{0}, such that C̄ is non-zero on R. Under this market condition the problem of achieving a specified expected terminal wealth with minimum variance has a solution. The solution lies in R.

Proof From the previous Lemma we have

E [Ĝ_{T} (C̄(ϕ))] = \frac{E[∫_{0}^{T} ϕ_{s} dZ_{s}]}{E[π(e)]} E[C(π(e))],

and by hypothesis C(π(e)) ≠ 0. So λ_{C} ≠ 0. Suppose that the desired expected terminal gain is ĝ_{T}. We note that if we choose ϕ ∈ T so that

E [∫_{0}^{T} ϕ_{s} dZ_{s}] = \frac{ĝ_{T}}{λ_{C}}

then

E [Ĝ_{T} (C̄(ϕ))] = ĝ_{T}

So the random variable which solves our problem is \frac{ĝ_{T}}{λ_{C}E(π(e))}π(e)and we know the strategy that will achieves this. □ Theorem 3 Let C̄ be a homogeneous market condition. Suppose that there is ϕ ∈ T such that ϕ ∈ ker (π̄_{0} · C̄) and E[∫_{0}^{T} C̄(ϕ_{s}) dZ_{s}] ≠ 0. Then there is a strategy ψ, which solves the minimum variance problem. Proof 3 Since ϕ ∈ ker (π̄_{0} · C̄) then C̄(ϕ) ∈ R and we have

C (∫_{0}^{T} ϕ_{s} dZ_{s}) = π_{0} (C (∫_{0}^{T} ϕ_{s} dZ_{s})) + \frac{E [∫_{0}^{T} C̄(ϕ_{s}) dZ_{s}]}{E[π(e)]} π(e).

Since E[∫_{0}^{T} C̄(ϕ_{s}) dZ_{s}] ≠ 0 we can set

ψ = \frac{ĝ_{T}}{E[∫_{0}^{T} C̄(ϕ_{s}) dZ_{s}]} ϕ ∈ ker (π̄_{0} · C̄) and E [∫_{0}^{T} C̄(ψ_{s}) dZ_{s}] = ĝ_{T} .

On the other hand,

var [C (∫_{0}^{T} ψ_{s} dZ_{s})] = \frac{ĝ_{T}^{2}}{E[π(e)]^{2}} var [π(e)]

which is minimal for the problem considered. □ Corollary 2 Let C̄ be a homogeneous market condition. Suppose that ∃ ϕ ∈ T such that

0 ≠ ϕ ∈ ker [π̄_{0}, C̄] ∩ R and E [∫_{0}^{T} C̄(ϕ_{s}) dZ_{s}] = α ≠ 0

where [π̄_{0}, C̄] denotes the commutator of π̄_{0} and C̄. Then a solution to the minimum variance problem exists, it lies in ker [π̄_{0}, C̄] ∩ R. Proof 4 We have

E [∫_{0}^{T} C̄ (\frac{ĝ_{T}}{α} ϕ_{s}) dZ_{s}] = ĝ_{T} .

Let ψ = \frac{ĝ_{T}}{α} ϕ then ψ ∈ ker [π̄_{0}, C̄] ∩ R. And ψ minimises the variance of the gain.

□

By defining the norm ϕ ∈ T to be the L^{2} norm of the corresponding gain we can turn T into a Hilbert space in which π̄_{0} becomes and orthogonal projection - that is, a self-adjoint idempotent. We can then recast theorem 2, in the following form.

Theorem 4 Let C̄ ∈ {T̄ ∈ B(T ) : T̄ π̄_{0} = π̄_{0} T̄ }, where B(T ) is the set of bounded operators on T . If C̄ π̄_{0}^{⊥} = 0 then all expected gains are zero and the minimum variance problem does not have a solution. If C̄ π̄_{0}^{⊥} ≠ 0 then a solution to the minimum variance problem exists.

The details of this belong to [5].

References

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[2] D. Duffie and H.R. Richardson. Mean-variance hedging in continuous time. Annals of Applied Probability, 1:1-15, Jan 1989.

[3] H. Follmer and M. Schweizer. Hedging of contingent claims under in- complete information. Applied Stochastic Analysis, 5:389-414, 1991. M.H.A. Davis and R.J. Elliott (eds.), Stochastics Monographs, Gordon and Breach.

[4] W.P. Liu. The Problem of Portfolio Optimisation Under Market Conditions with Minimum Variance Constraints. PhD thesis, Imperial College, University of London, 1997.

[5] I. Milic. PhD thesis, Imperial College, University of London, 2000.

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