1

A Minimum Variance Result Under Market Conditions

C.Barnett  W.P.Liu  G.Salkin

December 30, 1999

Imperial College of Science, Technology and Medicine University of London

Abstract

In [8], 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. Our market shall consist of a single risky asset, whose returns are log-normal, and a riskless bond following the usual price process, the mathematical details are in the following section. We assume also that borrowing and lending are carried out at a single interest rate that is constant over time. In addition, short-selling of securities with full use of proceeds is permitted, there are no transaction costs, taxes, or margin requirements, no bid offer spreads and trading is continuous. As a consequence of the mathematical context that we employ there are no riskless arbitrage opportunitities in our market and all contingent claims are attainable. However this state of affairs depends, in part, upon a generous definition of admissible trading strategy. If one were to make a significant restriction on the available trading strategies then there would be contingent claims which were not attainable. Of course, in the ‘real world’ of trading on the FTSE or NYSE, say, restrictions on the available trading strategies are a part of every day life. Problems with liquidity, halts in trading caused by ‘circuit breakers’, the limitation to discrete time trading (because we are human) are all examples of “market condiions” that afflict daily trading. It was these kinds of market conditions, among many others, which suggested to us a class of mathematical problems in Finance. In the next few sections we shall state precisely the mathematical context in which we work, give a brief review of Richardsons’ problem and then introduce a (mathematical) definition of ‘market conditions’ which can capture some of those problems which are “part of every day life” for traders; to be specific, we concentrate on restictions on the available trading strategies. We will then return to Richardsons’ problem and solve it for a certain class of market conditions. As will become clear, the introduction of market conditions into the complete market described in section 2 leads to a market that is effectively incomplete. Contingent claims which were attainable may no longer be so. Given a contingent claim, one can then ask if there is a trading strategy which gives a portfolio which is, in some sense, the best available (given the market conditions) to meet the obligations implied by the contingent claim. We will take variance minimisation as our criteria of what is best, noting that there can be objections to this. It may be that certain kinds of claims coupled with certain kinds of market conditions are better served by something other than our quadratic objective but this level of detail is not explored in this paper. Related work includes D.Duffie and H.R. Richardson [2]. The problem of pricing under incompleteness has received a lot of attention, we mention Föllmer and M.Schweiser [4] who adopt a risk-minimisation approach and El-Karoui and Quenez [3]. The paper [1], by J. Cvitanić and I. Karatzas [1] is clearly related to our work.

2 The Market Model

The mathematical context is a probability space (Ω, ℱ _{T} , P) and time span [0, T], where T ∈ R^{+} denotes a fixed and finite time horizon for all economic activity. An element of Ω denoted by ω, is a state of the world which describes the exogenous uncertain environment from time 0 to time T. The flow of information is described by F = {ℱ _{t}; t ∈ [0, T]} , the augmented filtration generated by the standard Brownian Motion, W = {W_{t}; t ∈ [0, T]}. This completes the specification of the stochastic base. As we remarked previously, there are two assets that can be traded continuously. ^{1}. F is the information structure of the model; First, we have a riskless asset called the (riskless) bond, and its price process S_{t}^{0} is given by

S_{t}^{0} = S_{0}^{0} exp {rt}, t ∈ [0, T].  (1)

The other asset, the stock, (we assume that it pays no dividends) and its price process S_{t}^{1} is given by

S_{t}^{1} = S_{0}^{1} exp \{σW_{t} + (μ_{1} − \frac{σ^{2}}{2} )t\}, t ∈ [0, T]  (2)

where:

• σ ∈ R^{+} denotes the instantaneous stock volatility,

• r ∈ R^{+} denotes the instantaneous riskless interest rate,

• μ_{1} ∈ R^{+} denotes the instantaneous stock drift, and

• S_{0}^{0}, S_{0}^{1} ∈ R^{+} denote the security spot prices observed at time 0.

We will work hereafter with discounted quantities, we do so because it is convenient and consistent with Richardsons’ paper. So, we define

Z_{t} = \frac{S_{t}^{1}}{S_{t}^{0}}

1The filtration generated by the process W is F^{W} = {ℱ _{t}^{W}; t ∈ [0, T]} where ℱ _{t}^{W} ≡ σ({W (s); 0 ≤ s ≤ t}) is the sigma-field on Ω generated by the family of functions {W (s) : 0 ≤ s ≤ t} on Ω.

the discounted price of the stock at time t. We have ,

Z_{t} = Z_{0} exp \{σW_{t} − \frac{σ^{2}t}{2} + μt\}  (3)

where:

• μ = μ_{1} − r ∈ R^{+} denotes the expected continuous discounted rate of return of the stock, and

• Z_{0} ∈ R^{+}

An application of Itô’s lemma shows that,

Z_{t} = Z_{0} + ∫_{0}^{t} Z_{s}σ dW_{s} + ∫_{0}^{t} Z_{s}μ ds, t ∈ [0, T]  (4)
= Z_{0} + ∫_{0}^{t} Z_{s} d(σW_{s} + μs).  (5)

We recall that

Definition 1 A trading strategy is a pair Φ = (ϕ_{t}^{0}, ϕ_{t}^{1}), where ϕ_{t}^{0} denotes the number of units of bonds held at time t, while ϕ_{t}^{1} denotes the number of units of stock held at time t. Moreover (ϕ_{t}^{1}) is a progressively measurable process adapted to the filtration (ℱ _{t})_{t∈[0,T]} and it is such that

E [∫_{0}^{T} (ϕ_{s}^{1}Z_{s})^{2} ds] < ∞.  (6)

In order to specify ϕ^{0} we must discuss the value function of the portfolio. The value of the portfolio given by Φ at time t is

V_{t}(Φ) = ϕ_{t}^{0}S_{t}^{0} + ϕ_{t}^{1}S_{t}^{1}, t ∈ [0, T].

So the discounted value of V_{t}(Φ) is

V̂_{t}(Φ) = ϕ_{t}^{0} + ϕ_{t}^{1}Z_{t}.  (7)

The discounted gain due to Φ is the process

Ĝ_{t}(Φ) =^{△} Ĝ_{t}(ϕ^{1}) = ∫_{0}^{t} ϕ_{s}^{1} dZ_{s}  (8)

Our portfolio must be such that

V̂_{t}(Φ) = v̂_{0} + Ĝ_{t}(Φ),  (9)

where v̂_{0} is the value of the portfolio at time zero. From this the process ϕ^{0} is determined by

ϕ_{t}^{0} = v̂_{0} + ∫_{0}^{t} ϕ_{s}^{1} dZ_{s} − ϕ_{t}^{1}Z_{t}, t ∈ [0, T].  (10)

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} (Φ)] = v̂_{T} , while var [Ĝ_{T} (Φ)] is minimal,  (11)

here ĝ_{T} is the investment goal for the portfolio gain at time T which is assumed to be strictly positive. We refer to this problem (11) 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}(Φ)) for which Ĝ_{t}(Φ) = X, and finally the trading strategy ϕ^{1} which gives the gains process. In doing this the exponential martingale

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

plays a critical role . In the first place, expression (6) ensures that any gain Ĝ_{T} (Φ) belongs to L^{2} (Ω, ℱ _{T} , P) and 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). It is only right (and consistent with [8]) to write

⟨M_{T} ⟩^{⊥} = 𝒢,

and call 𝒢 the “gains subspace”.

This result follows by introducing the P-equivalent measure Q given by

Q(E) = ∫_{E} M_{T} dP.

Girsanov’s theorem tells us that, with respect to Q, the process (W_{t} + (\frac{μ}{σ}) t)_{t∈[0,T]} is a standard one-dimensional Brownian motion. However, a version of the Martingale representation theorem due to N. Vaillant [9] ^{2}, ensures that for X ∈ L^{2}(Ω, ℱ _{T} , P) there is a measurable adapted process, (ρ_{t}) satisfying

E_{P} [∫_{0}^{T} ρ_{s}^{2} ds] < ∞,  (12)

(notice it is a P expectation) such that

X = E_{Q}[X] + ∫_{0}^{T} ρ_{s} d (σW_{s} + μs) .  (13)

Writing ρ_{s} = \frac{ρ_{s}}{Z_{s}} Z_{s} we see that this amounts to,

X = E_{Q}[X] + ∫_{0}^{T} \frac{ρ_{s}}{Z_{s}} Z_{s} d (σW_{s} + μs)
= E_{Q}[X] + ∫_{0}^{T} (\frac{ρ_{s}}{Z_{s}} ) dZ_{s}.  (14)

Note that (\frac{ρ_{s}}{Z_{s}} ) is an admissible trading strategy. Now if X is an element of ⟨M_{T} ⟩^{⊥}, ie 𝒢, then

2Proposition 30

0 = ⟨X, M_{T} ⟩ = ∫_{Ω} XM_{T} dP = E_{Q}[X],  (15)

so

X = ∫_{0}^{T} (\frac{ρ_{s}}{Z_{s}} ) dZ_{s}.  (16)

The converse is very straightforward. The fact that this lemma required a modified version of the Martingale Representation theorem was not brought out fully in [8].

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)⟩.  (17)

Proof If X ∈ 𝒞_{0} then X ∈ 𝒢 and so its orthogonal projection π(X) = X. Also

0 = E [X] = E [π(X) · e] = ⟨π(X), e⟩ = ⟨X, π(e)⟩  (18)

so that 𝒞_{0} = ⟨π(e)⟩^{⊥} in the Hilbert space 𝒢. □

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),  (19)

for Y ∈ 𝒞_{0}.

Now the variance of a gain such as X in equation (19) is given by,

var[X] = var [Y + \frac{ĝ_{T}}{E[π(e)]} π(e)]
= var [Y] + \frac{ĝ_{T}^{2}}{E[π(e)]^{2}} var [π(e)] + 2 \frac{ĝ_{T}}{E[π(e)]} cov [Y, π(e)]. (20)

Since Y ∈ 𝒞_{0} is orthogonal to π(e) and the expectation of Y is zero then,

var[X] = var[Y] + \frac{ĝ_{T}^{2}}{E[π(e)]^{2}} var[π(e)].  (21)

We have expressed the variance of X as a sum of two 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] in (20) above. 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 [8]) 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 Qmartingale. 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 ^{3} ϕ^{∗} such that,

∫_{0}^{T} ϕ_{s}^{∗}dZ_{s} = \frac{ĝ_{T}}{E[π(e)]} π(e).  (22)

This 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 by expression (22) they are equal Q-almost surely and therefore Palmost surely for every t ∈ [0, T]. Using Ito’s lemma to write e − \frac{M_{t}}{∥M_{t}∥^{2}} as a sum of integrals and equating left and right sides in (22) we can use the uniqueness of semi-martingales to determine ϕ^{∗}.

3Where ϕ^{∗} denotes the optimal trading strategy.

3 Market Conditions

First some motivatiing examples. Suppose one has some investment objective within the market model described in the previous section. Given no restrictions, one proceeds to adopt whatever trading strategies one’s objectives allow. But suppose now that some agent intervenes at the outset to restrict your choice of strategies; for example, 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], the agent intervenes to halt ^{4} trading and one’s portfolio amounts ϕ_{τ}^{0} and ϕ_{τ}^{1} are fixed 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^{5}. While we would like to include a disussion of each of these cases in this paper we shall not because the first case is being investigated in [5] and the second in [7]. These considerations lead to the following definition.

Definition 2 A \underline{market condition} is a map of 𝒯 the 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. There is no metric structure on 𝒯 as yet, and despite the obvious step of defining,

4These conditions have actually arisen in practice in certain exotic currencies that belong to Emerging markets.

5Of course we require a time τ such that ϕ^{τ} ∈ 𝒯 for each ϕ ∈ 𝒯 .

∥ϕ∥ = E [∫_{0}^{t} ϕ_{s}^{2}Z_{s}^{2} ds]^{1/2} ,  (23)

and

𝒩 = {ϕ ∈ 𝒯 : ∥ϕ∥ = 0, }  (24)

then moving to \frac{𝒯}{𝒩} equipped with ∥ · ∥, we will resist taking this step now for reasons we hope to make clear later.

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 ^{6},

π_{0} (∫_{0}^{T} ϕ_{s} dZ_{s}) = (∫_{0}^{T} π̄_{0}(ϕ_{s}) dZ_{s}) .  (25)

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{M_{s}}{∥M_{s}∥^{2}Z_{s}} )⟩ , i.e. the Richardson strategies.

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

6Hereafter 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, 𝒯 , π_{0}.

Proof Since 𝒢 = 𝒞_{0} ⊕ ⟨π(e)⟩ then any X ∈ 𝒢 may be written as,

X = π_{0}(X) + λπ(e), λ ∈ R

= π_{0}(X) + \frac{⟨X, π(e)⟩}{∥π(e)∥^{2}} π(e)

indeed,

= 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}} ) . (26)

An application of Itô’s lemma to the process (\frac{M_{T}}{∥M_{T} ∥^{2}} ) shows that the righthand side above is equal to,

∫_{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])

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

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

= ∫_{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}} ) Z_{s} d(σW_{s} + μs)

= ∫_{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}.  (27)

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}}  (28)

corresponds to the projection of the trading strategy ϕ ∈ 𝒯 onto the subspace 𝒯_{0} from equation (27).

Let ℛ = ⟨(\frac{M_{s}}{∥M_{s}∥^{2}Z_{s}} )⟩ be the subspace generated by the single trading

strategy (\frac{M_{s}}{∥M_{s}∥^{2}Z_{s}} )^{7}.

Certainly

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

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

= \frac{α}{∥π(e)∥^{2}} E [e − \frac{M_{T}}{∥M_{T} ∥^{2}} ]

= \frac{α}{∥π(e)∥^{2}} E [π(e)].

7In [[8]] this strategy figures in the solution to Richardsons’ problem, so we called the subspace generated by this R

However

∥π(e)∥^{2} = ⟨π(e), π(e)⟩ = ⟨e, π(e)⟩ = 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{M_{s}}{∥M_{s}∥^{2}} ) = (\frac{σ^{2}}{μ} ∥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}] = \frac{σ^{2}}{μ} ∥π(e)∥^{2}α. But if also ϕ ∈ 𝒯_{0} then \frac{σ^{2}}{μ} ∥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.

So π̄_{0} is an idempotent. □

The following result is a direct consequence of Theorem 1.

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

Proof (⟹) If π̄_{0}(ϕ_{s}) = 0 then,

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}} = 0

that is,

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

∴ ϕ_{s} \underline{is} a Richardson strategy.

(⟸) If ϕ_{s} ∈ ℛ then,

ϕ_{s} = \frac{α}{∥π(e)∥^{2}} \frac{μ}{σ^{2}} \frac{M_{s}}{∥M_{s}∥^{2}} \frac{1}{Z_{s}} , where α ∈ R,

and E [∫_{0}^{T} ϕ_{s} dZ_{s}] = α, so by applying this to equation 28 we have π̄_{0}(ϕ_{s}) = 0. □

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}.

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

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

and by applying the market condition C, we have

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)),

since π̄_{0} and C̄ commute

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

So taking the expectation of this expression, we have

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

The variance is given by the expression,

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

+ 2 cov [∫_{0}^{T} π̄_{0}(C̄(ϕ_{s})) dZ_{s}, \frac{E[∫_{0}^{T} ϕ_{s} dZ_{s}]}{E[π(e)]} C(π(e))] .

Now, recall again that

𝒢 = 𝒞_{0} ⊕ ⟨π(e)⟩.

Suppose that Y ∈ 𝒢 is a vector orthogonal to 𝒞_{0}, so Y ∈ ⟨π(e)⟩. Then π_{0}(Y) = 0 and for all X ∈ 𝒢,

⟨π_{0}(X), C(Y)⟩ = ⟨X, π_{0} · C(Y)⟩

and as π_{0} and C commute,

= ⟨X, C · π_{0}(Y)⟩ = 0.

So C(Y) ∈ ⟨π(e)⟩ too!^{8}. As a consequence we have

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

Since the arguments are orthogonal and one has zero-expectation. The result now follows on observing that ∫_{0}^{T} π̄_{0}(C̄(ϕ_{s})) dZ_{s} has zero expectation. □

Remark We can see directly here that the effect of the market condition, C̄, upon the expected value of Ĝ_{T} (ϕ), is to scale it by a factor,

\frac{E[C(π(e))]}{E[π(e)]}

So linear market conditions which commute with π̄_{0} have a particularly nice action when their effect on 𝒢 is considered. They map 𝒞_{0} onto itself and ⟨π(e)⟩ likewise. In view of how the direct sum decomposition of 𝒯 reflects the direct sum decomposition of 𝒢, one might expect similar relations to hold for the action of C on 𝒯. Indeed we have the following results.

Proposition 3 If C̄ is a linear market condition commuting with π̄_{0} then C̄ maps 𝒯_{0} into itself and ℛ likewise.

Proof For ϕ ∈ 𝒯 we may write

C̄ (π̄_{0}(ϕ)) = π̄_{0} (C̄(π̄_{0}(ϕ))) .

So C̄ maps 𝒯_{0} into itself. On the other hand ϕ ∈ ℛ ⟺ π̄_{0}(ϕ) = 0, so if π̄_{0}(ϕ) = 0 then C̄(π̄_{0}(ϕ)) = 0 also - because C̄ is linear. Hence π̄_{0}(C̄(ϕ)) = 0 (by commutivity) so that C̄(ϕ) ∈ ℛ . □

8Functional Analysts will be wondering what all the fuss is about, but our audience may not be familiar with reducing subspaces

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 proposition 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. Since E[π(e)] ≠ 0 and ⟨π(e)⟩ is one-dimensional and C̄ maps ℛ into itself then E[C(π(e))] ≠ 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{E[π(e)]}{E[C(π(e))]} ĝ_{T} ,

then

E [Ĝ_{T} (C̄(ϕ))] = ĝ_{T}

subject to this last condition, we can then write the following expression.

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

The second term of the right hand side contains terms that must remain fixed for the problem that we are considering, while the first term of the right hand side may be rewritten as

∥π_{0} · C (∫_{0}^{T} ϕ_{s} dZ_{s})∥^{2} = ∥∫_{0}^{T} π̄_{0} · C̄(ϕ_{s}) dZ_{s}∥^{2} = ∥∫_{0}^{T} C̄ · π̄_{0}(ϕ_{s}) dZ_{s}∥^{2} .

We observe that by choosing a strategy, ϕ, in ℛ such that

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

(it exists !) then π̄_{0}(ϕ) = 0 so that,

var [Ĝ_{T} (C̄(ϕ))] = \frac{ĝ_{T}^{2}}{E[C(π(e))]^{2}} var [C(π(e))]

which is minimal. To summarise: under this class of market conditions the solution to the minimum variance problem is the strategy

\frac{E[π(e)]}{E[C(π(e))]} \frac{ĝ_{T}}{∥π(e)∥^{2}} \frac{μ}{σ^{2}} \frac{M_{s}}{∥M_{s}∥^{2}} \frac{1}{Z_{s}} = \frac{ĝ_{T}}{E[C(π(e))]} \frac{μ}{σ^{2}} \frac{M_{s}}{∥M_{s}∥^{2}} \frac{1}{Z_{s}} .

□

Theorem 3 Let C̄ be a homogeneous market condition. Suppose that there is ϕ ∈ 𝒯 such that ϕ ∈ ker (π̄_{0} · C̄) and ^{9} E[∫_{0}^{T} C̄(ϕ_{s}) dZ_{s}] ≠ 0. Then there is a strategy ψ, which solves the minimum variance problem.

Proof 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).

and arguing as in proposition 2

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

Since E[∫_{0}^{T} C̄(ϕ_{s}) dZ_{s}] ≠ 0 then,

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)]

9Where ker denotes the kernal.

which is minimal for the problem considered. □

Observe that the condition ϕ ∈ ker (π̄_{0} · C̄) means that C̄(ϕ) ∈ ℛ , while E[∫_{0}^{T} C̄(ϕ_{s}) dZ_{s}] ≠ 0 ensures that C̄(ϕ) is non-zero.

Corollary 1 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 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 .

Also,

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

= var [∫_{0}^{T} π̄_{0} · C̄(ψ_{s}) dZs] + \frac{ĝ_{T}^{2}}{E[π(e)]^{2}} var[π(e)]

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

but π̄_{0}(ψ) = 0, so

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

□

Remark By choosing an appropriate norm ^{10} 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(𝒯̄ ) : 𝒯̄ π̄_{0} = π̄_{0} 𝒯̄ }, where B(𝒯̄ ) is the set of bounded operators on 𝒯̄. 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 [6].

4 Concluding Remarks

Of course there are market conditions which do not commute with π_{0}. Indeed, consider the market condition τ introduced at the beginning of section (3), that is, there is a single stopping time τ, after which all trading halts. One can check that,

π_{0} · τ ≠ τ · π_{0}

What we shall see is the the geometric properties of 𝒯 suggest a strategy immediately. Denote also by τ the linear map induced on the gains space. The map τ is linear and so

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

We observe that, should

E[τ(π(e))] ≠ 0,

then there is a strategy, ϕ, in ℛ with

10The appropriate norm is not the one referred to in definition 2. This norm gives a different geometry from the one required.

E [∫_{0}^{T} ϕ_{s} dZ_{s}] = \frac{E[π(e)]}{E[τ(π(e))]} ĝ_{T} ,

and

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

For such a strategy,

τ · π_{0} (∫_{0}^{T} ϕ_{s} dZ_{s}) = 0.

So the gain one obtains is,

τ (∫_{0}^{T} ϕ_{s} dZ_{s}) = \frac{ĝ_{T}}{E[τ(π(e))]} τ(π(e))

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

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

So the trading strategy

(\frac{ĝ_{T}}{E[τ(π(e))]} \frac{μ}{σ^{2}} \frac{M_{s}^{τ}}{∥M_{s}^{τ}∥^{2}} \frac{1}{Z_{s}} )

gives a gain with the corrected expected value and known variance (knowing the risk is a virtue?)

\frac{ĝ_{T}^{2}}{E[τ(π(e))]^{2}} var[τ(π(e))].

This strategy does not minimises variance, that is another matter which will not be pursued in this paper ^{11}.

11Ref [6] for further detail.

Against the accusation that the paper contains too much mathematics and too little finance, we ask; for who? Adding that, the (restricted) idea of market conditions described above suggests a whole class of problems for mathematical finance in which the geometry of the space of trading strategies becomes an important feature. The solution of some of these problems will be of practical value while others ... ‘They also serve who only stand and wait’.

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] N. El Karoui and M.C. Quenez. Dynamic programming and the pricing of contingnt claims in an incomplete market. SIAM journal of control and optimisation., 33:29–66, 1995.

[4] H. Föllmer and M. Schweizer. Hedging of contingent claims under incomplete information. Applied Stochastic Analysis, 5:389–414, 1991. M.H.A. Davis and R.J. Elliott (eds.), Stochastics Monographs, Gordon and Breach.

[5] K. C. Liu. In preparation. PhD thesis, Imperial College, University of London, 1999.

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

[7] I. Milic. In preparation. PhD thesis, Imperial College, University of London, 1999.

[8] H.R. Richardson. A Minimum Variance Result in Continuous Trading Portfolio Optimization. Management Science, 35(9):1045–1055, Sep 1989.

[9] N. Vaillant. Mean-Variance Hedging and Pricing of Contingent Claims in Incomplete Markets. PhD thesis, Imperial College, University of London, May 1995.

[10] N. Young. An Introduction to Hilbert Space. Cambridge University Press, 1989.