Minimal-variance hedging in large financial markets: random field

Minimal Variance Hedging in Large Financial Markets: random fields approach Giulia Di Nunno

Third AMaMeF Conference: Advances in Mathematical Finance Pitesti, May 5-10 2008 based on a work in progress with Inga B. Eide

Outlines 1. Introduction: large financial markets 2. Markets with a countable number of traded assets B Minimal variance hedging problem 3. Martingale random fields 4. Markets with a continuum of traded assets B Minimal variance hedging problem B Stochastic differentiation B Minimal variance hedging strategy B Other comments References

1. Introduction: large financial markets Large financial markets were first introduced by Kabanov and Kramkov (1994) as a sequence of finite dimensional markets, called “small markets” by Klein and Schachermayer (1996). Each small market is defined on its own space, filtration and time horizon. With this approach the large financial market can be seen as a market where it is possible to choose a finite number of securities to trade, but a priori this number is not bounded. In this framework asymptotic arbitrage and the corresponding versions of the fundamental theorem of asset pricing is studied. Kabanov and Kramkov (1994, 1998) provide a first extension of this theorem connecting the concepts of asymptotic arbitrage with the notion of contiguity of a sequence of equivalent martingale measures. A very general version of this theorem is given by Klein (2000), where the concept of asymptotic free lunch is introduced.

If we assume that all probability spaces coincide, then we have an alternative approach. One can define a large market as a countable number of assets and, correspondingly, a sequence of price processes on one fixed probability space, filtration and time horizon. This is a model for an idealized market in which it is allowed to trade on countably many assets. This framework is more suitable for considering questions related to hedging and completeness of the market. In fact it has been chosen e.g. by Bj¨ ork abd N¨aslund (1998), deDonno (2004), deDonno, Guasoni and Pratelli (2005) and Campi (2002). We are also fitting our study in this framework.

Goals

Our goals are: I

To describe a suitable model for the study of the minimal variance hedging problem for a large financial market;

I

To embed our model in a wider frame of an idealized market model in which a continuum of assets can be traded;

I

To study a minimal variance hedging problem in this framework.

Markets with a continuum of assets can be e.g. markets were the derivatives (e.g. various options with different strike price) are traded and used for hedging purposes. Bond market.

2. Markets with a countable number of traded assets Framework. I

Complete probability space (Ω, F, P)

Fixed time horizon [0, T ], T > 0  I Right continuous filtration F := Ft , t ∈ [0, T ] representing the flow of information available in the market. For semplicity, F0 is trivial (up to P-null events) and F = FT I

I

Market is frictionless, admitting short-selling and continuous trading

I

Risk free asset with price (value per unit): Rt = exp

nZ

t

o rs ds ,

t ∈ [0, T ]

(R0 = 1),

0

where r = rt , t ∈ [0, T ], is the (deterministic) interest rate.

I

Risky traded assets are represented by a countable set X = {x1 , x2 , ....} of different assets with prices (value per unit): St (xn ),

t ∈ [0, T ],

n = 1, 2, ....

(F-adapted processes). Correspondingly, we can define the excessive return process ηt (xn ) :=

St (xn ) − S0 (xn ), Rt

t ∈ [0, T ],

n = 1, 2, ...

We assume that the measure P is a risk-neutral probability measure such that the processes η(xn ) are square integrable orthogonal martingales with respect to F, i.e. I

F-adapted and E [ηt (xn )|Fs ] = ηs (xn ), s ≤ t,

E [ηt2 (xn )] < ∞, and t-continuous in this convergence,   I E (ηt (xn ) − ηs (xn )) · (ηt (xm ) − ηs (xm ))|Fs = 0, s < t, (n 6= m) I

In this framework we can apply the Itˆ o type calculus and consider: I

Mt (xn ) :=< η(xn ) >t , t ∈ [0, T ],

I

mt (xn ) := E [ηt2 (xn )] = E [Mt (xn )], t ∈ [0, T ], for every n.

In a no-arbitrage framework prices and eccessive returns are additive, then we can define, for B = {x1 , ...xN } selection of N assets, the process ηt (B) :=

N X

ηt (xn )

n=1

and correspondingly, thanks to orthogonality, also Mt (B) :=

N X

Mt (xn )

n=1

mt (B) :=

N X n=1

mt (xn ).

We consider X = {x1 , x2 , ...} as a topological space equipped with the discrete topology and BX represents the Borel σ-algebra. We can define the set function µ(B × (s, u]) := ηu (B) − ηs (B) and accordingly also M(B × (s, u]) := Mu (B) − Ms (B) and m(B × (s, u]) := mu (B) − ms (B) for any s ≤ u and B = {x1 , ...xN } ⊆ X . Naturally, we have I

E [µ(B × (s, u])] = 0

I

E [µ2 (B × (s, u])|Fs ] = E [M(B × (s, u])|Fs ]

I

E [µ2 (B × (s, u])] = E [M(B × (s, u])] = m(B × (s, u]).

We can extend these set functions to be σ-finite random measures on (X × [0, T ], BX × B[0,T ] ).

I

The class of contingent claims considered is the set of square integrable random variables.

I

The self-financing strategies is a couple (e, ϕ) where e is an initial endowment (F0 -measurable) and ϕ is an element of L2 (P), i.e. a stochastic field ϕ(ω, xn , t),

ω ∈ Ω, xn ∈ X , t ∈ [0, T ]

n = 1, 2, ....

measurable with respect to the predictable σ-algebra P with E

hZ Z X

∞ Z i hX ϕ2 (x, t)M(dxdt) = E

T

0

n=1

T

i ϕ2 (xn , t)M(xn , dt) < ∞.

0

The σ-algebra P is generated by the sets A × B × (s, u],

s < u, A ∈ Fs , B ∈ BX .

The set L2 (P) is a (closed) subspace in L2 (F × BX × B[0,T ] ).

The value process of the strategy (e, ϕ) satisfies dξt = ξt rt dt + Rt

∞ X

ϕ(xn , t)dηt (xn )

n=1

Z = ξt rt dt + Rt

ϕ(x, t)µ(dxdt) X

= ξt rt dt + Rt dηt (ϕ), by means of the F-martingale Z Z t ηt (ϕ) := ϕ(x, s)µ(dxds), X

ξ0 = e.

t ∈ [0, T ].

0

In view of this formulae we call ϕ density of investments.

I

The solution of the equation above is Z Z ξt = eRt + Rt ηt (ϕ) = eRt + Rt

t

ϕ(x, s)µ(dxds), X

0

which naturally confirms that a value process ξt , t ∈ [0, T ], of some strategy (e, ϕ) is actually a discount martingale, i.e. ξt , Rt I

t ∈ [0, T ],

is an F-martingale.

Proposition. A stochastic process Vt , t ∈ [0, T ], is a value process for some strategy if it is a discount martingale and the random variable VRTT − V0 admits stochastic integral representation with respect to µ(dxdt), i.e. there exists an integrand ψ ∈ L2 (P) such that Z Z T VT ψ(x, s)µ(dxds). − V0 = RT X 0 Then it is the strategy (V0 , ψ) which yields Vt , t ∈ [0, T ].

Remarks – If the strategy involves only a finite number of investments B = {xk1 , ..., xkN }, then we have reduced the large market to the corresponding small market and ηt (ϕ) =

N Z X n=1

t

Z Z ϕ(xkn , s)dηs (xkn ) =

0

In particular, ηt (ϕ) =

ϕ(x, s)µ(dxds). B

R Rt B

0

µ(dxds) =

PN

n=1

t

0

ηt (xkn ) = ηt (B).

– Cf. deDonno (2004): naive and generalized strategies, attainable and asymptotically attainable claims. See also Campi (2002). – Cf. Bj¨ ork and N¨aslund (1998): asymptotic assets, infinitely diversified portfolios (large mutual funds).

Remarks – The claim F is replicable if there exists a self-financing strategy (e, ϕ) with value process ξt , t ∈ [0, T ], such that X = ξT . – Since any square integrable F is associated with a discount martingale Vt , t ∈ [0, T ], with final value VT = F , i.e. Vt = Rt E [RT−1 F |Ft ], t ∈ [0, T ], then we have: Proposition. the market is complete if and only if any contingent claim F admits integral representation with respect to µ(dxdt). Proposition. Let the information be generated by the values of µ. For a market to be complete,Sit is sufficient that there exits some sequence Xn with m(Xn ) < ∞ and n Xn = X such that, for every n, the corresponding small market is complete. – In case F is replicable then e = E [RT−1 F |F0 ].

Minimal variance hedging problem I

In an incomplete markets a claim F may not be replicable. In this case the minimal variance hedging problem is to find the replicable claim Z Z T Fˆ = E [Fˆ |F0 ] + RT ϕ(x, t)µ(dxdt) X

0

and the density of investments ϕ such that E [(F − Fˆ )2 ] = minY E [(F − Y )2 ] I

(Y replicable)

Similarly, if we have some process Vt , t ∈ [0, T ], even if it is a discount martingale, it may not be replicable. Then the minimal ˆt , t ∈ [0, T ]: variance hedging problem is to find the value process V Z t Z Z t ˆs ds + ˆt = V ˆ0 + V rs V Rs ϕ(x, s)µ(dxds) 0

X

0

such that, for all t, ˆt )2 ] = minξ E [(Vt − ξ)2 ] E [(Vt − V

(ξ value process).

3. Martingale random fields Let X be a separable topological space equipped with its Borel σ-algebra BX which we assume to be generated by a countable semi-ring. Let m(∆) be a σ-finite measure on ∆ ∈ X such that m(X × {0}) = 0. Definition. A martingale random field with respect to F is a set-function µ(∆),

∆ ∈ BX × B[0,T ] ,

such that µ(∆) ∈ L2 (P), for ∆: m(∆) < ∞ and (i) for any ∆ ∈ BX × B[0,t] , the value µ(∆) is an Ft -measurable random variable,   (ii) for any ∆ ∈ BX × B(t,T ] , the value µ(∆) satisfies E µ(∆)|Ft = 0. Moreover, we consider   (1) E µ(∆) = 0,


2  E µ(∆) = m(∆)

and (2)

  E µ(∆1 ) · µ(∆2 )|Ft = 0,

for all disjoint ∆1 , ∆2 ∈ BX × B(t,T ] .

We can see that the conditional variance can be represented as (3)


2    E µ(∆) |Ft = E M(∆)|Ft ,

for all ∆ ∈ BX × B(t,T ] . Namely, the conditional variance is associated with a non-negative additive stochastic set-function (4)

M(∆),

∆ ∈ BX × B[0,T ] ,

with values in L1 (P) for ∆: m(∆) < ∞. In fact, the set-function (4) admits a regular modification M which is a stochastic measure: I

for every ω ∈ Ω, M(ω, ∆), ∆ ∈ BX × B[0,T ] , is a measure,

I

for every ∆ ∈BX × B  [0,T ] , M(ω.∆), ω ∈ Ω, is a random variable in L1 (P) with E M(∆) = m(∆).

Theorem (Doob-Meyer type decomposition) Let us define the set-function   (5) PM(A × ∆) := E 1A µ(∆)2 , for all ∆ ∈ BX × B[0,T ] of form ∆ = B × (s, u] and A ∈ Fs and we set PM(A × B × {0}) = 0, for all B ∈ BX and A ∈ F0 . Then (5) defines a measure on F × BX × B[0,T ] which admits representation in product form, i.e. (6)

PM(dωdxdt) = P(dω) × M(ω, dxdt),

where the component (7)

M(dxdt) := M(ω, dxdt),

ω ∈ Ω,

is a σ-finite Borel measure on BX × B[0,T ] depending on ω as a parameter. The stochastic measure M is unique in the sense that any other stochastic measure satisfying (5) and (6) would have trajectories equal to M(ω, ∆), ∆ ∈ BX × B[0,T ] , for P-a.a. ω.

Non-anticipative integration. The classical integration scheme which proceeds from simple integrands to general integrands can be applied. Necessary steps: I

Concept of ”partitions” of X × [0, T ]

I

The set of integrands coincides with L2 (P), i.e. a stochastic field ϕ(ω, x, t),

ω ∈ Ω, x ∈ X , t ∈ [0, T ],

measurable with respect to the predictable σ-algebra P, i.e. the σ-algebra generated by the sets

with kϕk2L2

A × B × (s, u], A ∈ Fs , B ∈ BX , s < u, hR i = E X ×[0,T ] ϕ2 (x, t)M(dxdt) < ∞.

I

Lemma. Any element ϕ ∈ L2 (P) can be approximated, i.e. kϕ − ϕn kL2 −→ 0, n → ∞, by simple functions ϕn , n = 1, 2, ..., in L2 (P) of form κn X ϕn = ϕnk 1∆nk k=1

where ∆nk = Bnk × (snk , unk ], k = 1, ..., κn , are elements of the nth -series of the partitions of X × (0, T ] and Z i h 1  ϕ(y , z)M(dydz) Asnk . (8) ϕnk := E  E M(∆nk )|Asnk ∆nk

4. Markets with a continuum of traded assets I

Complete probability space (Ω, F, P)

Fixed time horizon [0, T ], T > 0  I Right continuous filtration F := Ft , t ∈ [0, T ] representing the flow of information available in the market. For semplicity, F0 is trivial (up to P-null events) and F = FT I

I

Market is frictionless, admitting short-selling and continuous trading

I

Risk free asset with price (value per unit): Rt = exp

nZ

t

o rs ds ,

t ∈ [0, T ]

(R0 = 1),

0

where r = rt , t ∈ [0, T ], is the (deterministic) interest rate.

Let X denote the set of risky assets. Then a self-financing strategies is a couple (e, ϕ) where e is an initial endowment (F0 -measurable) and a density of investments ϕ ∈ L2 (P) with value process of the strategy (e, ϕ) satisfying Z dξt = ξt rt dt + Rt ϕ(x, t)µ(dxdt) X

= ξt rt dt + Rt dηt (ϕ), by means of the F-martingale Z Z t ηt (ϕ) := ϕ(x, s)µ(dxds), X

ξ0 = e.

t ∈ [0, T ].

0

Here µ(dxdt), (x, t) ∈ X × [0, T ], is a martingale random field generated by the excessive return processes of the investments on some groups of assets B with total price St (B), t ∈ [0, T ]: ηt (B) :=

St (B) − S0 (B), Rt

t ∈ [0, T ], B ∈ BX .

Minimal variance hedging problem I

Let F be a contingent claim. Then the minimal variance hedging problem is to find the replicable claim Z TZ Fˆ = E [Fˆ |F0 ] + RT ϕ(x, t)µ(dxdt) 0

X

and the density of investments ϕ such that E [(F − Fˆ )2 ] = minY E [(F − Y )2 ] I

(Y replicable)

Similarly, if we have some process Vt , t ∈ [0, T ], even if it is a discount martingale, it may not be replicable. Then the minimal ˆt , t ∈ [0, T ]: variance hedging problem is to find the value process V Z t Z Z t ˆt = V ˆ0 + ˆs ds + V rs V Rs ϕ(x, s)µ(dxds) 0

X

0

such that, for all t ∈ [0, T ], ˆt )2 ] = minξ E [(Vt − ξ)2 ] E [(Vt − V

(ξ value process).

Stochastic differentiation The non-anticipating stochastic derivative of F DF = D(x,t) F ,

(x, t) ∈ X × [0, T ],

is the adjoint operator to the Itˆ o type integral with respect to µ(dxdt). Theorem. For all F ∈ L2 (P), the non-anticipating derivative DF is κn h X i µ(∆nk )  As 1∆nk (9) DF = lim E F·  n→∞ E |µ(∆nk )|2 |Asnk k=1 with limit inL2 (P). Here the sum is taken on the elements ∆nk = Bnk × (snk , unk ], the nth -series of partitions of X × (0, T ]. The non-anticipating stochastic derivative DF represents the integrand in the orthogonal projection ξˆ of ξ on the subspace of all integrals with respect to µ, i.e. Z 0 (10) F =F ⊕ D(x,t) F µ(dxdt) X ×[0,T ] 0

0

and F ∈ L2 (P) : DF ≡ 0. Cf. diNunno (2002a, 2002b, 2007).

Minimal variance hedging strategy

Theorem. Let F be a contingent claim. Then the minimal variance hedge Fˆ exists and the self-financing strategy to achieve it is: ϕ = RT−1 D(x,t) F

e = E [F |F0 ].

The corresponding minimal variance hedging value process is characterized by: Z t Z Z t ˆt = E [F |F0 ] + ˆs ds + V rs V Rs RT−1 D(x,s) F µ(dxds) 0

X

0

Theorem. Let Vt , t ∈ [0, T ], be a discount martingale. Then the minimal ˆt , t ∈ [0, T ], exists and the self-financing strategy to variance hedge V achieve it is: ϕ = RT−1 D(x,t) VT e = V0 . The corresponding minimal variance hedging value process is characterized by: Z t Z Z t ˆt = V0 + ˆs ds + V rs V Rs RT−1 D(x,s) VT µ(dxds) 0

X

0

Corollary. Let Vt , t ∈ [0, T ], be a discount martingale only on some sub-set T ⊆ [0, T ] with T ∈ T . Then the self-financing strategy ϕ = RT−1 D(x,t) VT

e = V0 .

ˆt , t ∈ [0, T ], meaning that yields the minimal variance hedge V for all t ∈ T, ˆt )2 ] = minξ E [(Vt − ξ)2 ] E [(Vt − V

(ξ value process).

• Note that for T = {T } we have the theorem before.

Other commets I

The countable large market is embedded in the larger framework of the martingale random fields approach

I

Campi (2002) gives some results on the existence of mean-variance hedging. The approach taken is substantially different.

I

Results on minimal variance hedging and partial information can be given via the non-anticipating derivative

I

Some differentiation formula for the non-anticipating derivative is studied in diNunno (2007)

I

The non-anticipating derivative has some connections with the Clark-Ocone formula in the case of common framework (Brownian, Poisson random measures, random measures with independent values). See references in diNunno, Øksendal and Proske (2008).

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