Minimal Variance Hedging in Large Financial Markets: random fields approach

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1 Minimal Variance Hedging in Large Financial Markets: random fields approach Giulia Di Nunno Third AMaMeF Conference: Advances in Mathematical Finance Pitesti, May based on a work in progress with Inga B. Eide

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

3 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 (2), where the concept of asymptotic free lunch is introduced.

4 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 Björk abd Näslund (1998), dedonno (24), dedonno, Guasoni and Pratelli (25) and Campi (22). We are also fitting our study in this framework.

5 Goals Our goals are: To describe a suitable model for the study of the minimal variance hedging problem for a large financial market; To embed our model in a wider frame of an idealized market model in which a continuum of assets can be traded; 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.

6 2. Markets with a countable number of traded assets Framework. Complete probability space (Ω, F, P) Fixed time horizon [, T ], T > Right continuous filtration F := { F t, t [, T ] } representing the flow of information available in the market. For semplicity, F is trivial (up to P-null events) and F = F T Market is frictionless, admitting short-selling and continuous trading Risk free asset with price (value per unit): { t } R t = exp r s ds, t [, T ] (R = 1), where r = r t, t [, T ], is the (deterministic) interest rate.

7 Risky traded assets are represented by a countable set X = {x 1, x 2,...} of different assets with prices (value per unit): (F-adapted processes). S t (x n ), t [, T ], n = 1, 2,... Correspondingly, we can define the excessive return process η t (x n ) := S t(x n ) R t S (x n ), t [, T ], n = 1, 2,...

8 We assume that the measure P is a risk-neutral probability measure such that the processes η(x n ) are square integrable orthogonal martingales with respect to F, i.e. F-adapted and E[η t (x n ) F s ] = η s (x n ), s t, E[η 2 t (x n )] <, and t-continuous in this convergence, E [ (η t (x n ) η s (x n )) (η t (x m ) η s (x m )) F s ] =, s < t, (n m) In this framework we can apply the Itô type calculus and consider: M t (x n ) :=< η(x n ) > t, t [, T ], m t (x n ) := E[η 2 t (x n )] = E[M t (x n )], t [, T ], for every n.

9 In a no-arbitrage framework prices and eccessive returns are additive, then we can define, for B = {x 1,...x N } selection of N assets, the process η t (B) := N η t (x n ) n=1 and correspondingly, thanks to orthogonality, also M t (B) := m t (B) := N M t (x n ) n=1 N m t (x n ). n=1

10 We consider X = {x 1, x 2,...} as a topological space equipped with the discrete topology and B X represents the Borel σ-algebra. We can define the set function and accordingly also µ(b (s, u]) := η u (B) η s (B) M(B (s, u]) := M u (B) M s (B) and m(b (s, u]) := m u (B) m s (B) for any s u and B = {x 1,...x N } X. Naturally, we have E[µ(B (s, u])] = E[µ 2 (B (s, u]) F s ] = E[M(B (s, u]) F s ] E[µ 2 (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 [, T ], B X B [,T ] ).

11 The class of contingent claims considered is the set of square integrable random variables. The self-financing strategies is a couple (e, ϕ) where e is an initial endowment (F -measurable) and ϕ is an element of L 2 (P), i.e. a stochastic field ϕ(ω, x n, t), ω Ω, x n X, t [, T ] n = 1, 2,... measurable with respect to the predictable σ-algebra P with [ E X T ] [ ϕ 2 (x, t)m(dxdt) = E n=1 T The σ-algebra P is generated by the sets ] ϕ 2 (x n, t)m(x n, dt) <. A B (s, u], s < u, A F s, B B X. The set L 2 (P) is a (closed) subspace in L 2 (F B X B [,T ] ).

12 The value process of the strategy (e, ϕ) satisfies dξ t = ξ t r t dt + R t = ξ t r t dt + R t n=1 X ϕ(x n, t)dη t (x n ) ϕ(x, t)µ(dxdt) = ξ t r t dt + R t dη t (ϕ), ξ = e. by means of the F-martingale t η t (ϕ) := ϕ(x, s)µ(dxds), t [, T ]. X In view of this formulae we call ϕ density of investments.

13 The solution of the equation above is ξ t = er t + R t η t (ϕ) = er t + R t X t ϕ(x, s)µ(dxds), which naturally confirms that a value process ξ t, t [, T ], of some strategy (e, ϕ) is actually a discount martingale, i.e. ξ t R t, t [, T ], is an F-martingale. Proposition. A stochastic process V t, t [, T ], is a value process for some strategy if it is a discount martingale and the random variable V T R T V admits stochastic integral representation with respect to µ(dxdt), i.e. there exists an integrand ψ L 2 (P) such that V T T V = ψ(x, s)µ(dxds). R T X Then it is the strategy (V, ψ) which yields V t, t [, T ].

14 Remarks If the strategy involves only a finite number of investments B = {x k1,..., x kn }, then we have reduced the large market to the corresponding small market and η t (ϕ) = N n=1 t In particular, η t (ϕ) = B ϕ(x kn, s)dη s (x kn ) = B t ϕ(x, s)µ(dxds). t µ(dxds) = N n=1 η t(x kn ) = η t (B). Cf. dedonno (24): naive and generalized strategies, attainable and asymptotically attainable claims. See also Campi (22). Cf. Björk and Näslund (1998): asymptotic assets, infinitely diversified portfolios (large mutual funds).

15 Remarks The claim F is replicable if there exists a self-financing strategy (e, ϕ) with value process ξ t, t [, T ], such that X = ξ T. Since any square integrable F is associated with a discount martingale V t, t [, T ], with final value V T = F, i.e. V t = R t E[R 1 T F F t], t [, 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, it is sufficient that there exits some sequence X n with m(x n ) < and n X n = X such that, for every n, the corresponding small market is complete. In case F is replicable then e = E[R 1 T F F ].

16 Minimal variance hedging problem 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 T ˆF = E[ˆF F ] + R T ϕ(x, t)µ(dxdt) X and the density of investments ϕ such that E[(F ˆF ) 2 ] = min Y E[(F Y ) 2 ] (Y replicable) Similarly, if we have some process V t, t [, T ], even if it is a discount martingale, it may not be replicable. Then the minimal variance hedging problem is to find the value process ˆV t, t [, T ]: t t ˆV t = ˆV + r s ˆVs ds + R s ϕ(x, s)µ(dxds) such that, for all t, E[(V t ˆV t ) 2 ] = min ξ E[(V t ξ) 2 ] X (ξ value process).

17 3. Martingale random fields Let X be a separable topological space equipped with its Borel σ-algebra B X which we assume to be generated by a countable semi-ring. Let m( ) be a σ-finite measure on X such that m(x {}) =. Definition. A martingale random field with respect to F is a set-function µ( ), B X B [,T ], such that µ( ) L 2 (P), for : m( ) < and (i) for any B X B [,t], the value µ( ) is an F t -measurable random variable, (ii) for any B X B (t,t ], the value µ( ) satisfies E [ µ( ) F t ] =. Moreover, we consider (1) E [ µ( ) ] =, E [( µ( ) ) 2] = m( ) and (2) E [ µ( 1 ) µ( 2 ) F t ] =, for all disjoint 1, 2 B X B (t,t ].

18 We can see that the conditional variance can be represented as (3) E [( µ( ) ) 2 Ft ] = E [ M( ) Ft ], for all B X B (t,t ]. Namely, the conditional variance is associated with a non-negative additive stochastic set-function (4) M( ), B X B [,T ], with values in L 1 (P) for : m( ) <. In fact, the set-function (4) admits a regular modification M which is a stochastic measure: for every ω Ω, M(ω, ), B X B [,T ], is a measure, for every B X B [,T ], M(ω. ), ω Ω, is a random variable in L 1 (P) with E [ M( ) ] = m( ).

19 Theorem (Doob-Meyer type decomposition) Let us define the set-function (5) PM(A ) := E [ 1 A ( µ( ) 2 ], for all B X B [,T ] of form = B (s, u] and A F s and we set PM(A B {}) =, for all B B X and A F. Then (5) defines a measure on F B X B [,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 B X B [,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, ), B X B [,T ], for P-a.a. ω.

20 Non-anticipative integration. The classical integration scheme which proceeds from simple integrands to general integrands can be applied. Necessary steps: Concept of partitions of X [, T ] The set of integrands coincides with L 2 (P), i.e. a stochastic field ϕ(ω, x, t), ω Ω, x X, t [, T ], measurable with respect to the predictable σ-algebra P, i.e. the σ-algebra generated by the sets A B (s, u], A F s, B B X, s < u, ] with ϕ 2 L 2 = E[ X [,T ] ϕ2 (x, t)m(dxdt) <.

21 Lemma. Any element ϕ L 2 (P) can be approximated, i.e. ϕ ϕ n L2, n, by simple functions ϕ n, n = 1, 2,..., in L 2 (P) of form κ n ϕ n = ϕ nk 1 nk k=1 where nk = B nk (s nk, u nk ], k = 1,..., κ n, are elements of the n th -series of the partitions of X (, T ] and [ 1 (8) ϕ nk := E E [ ] ϕ(y, z)m(dydz) ] Asnk. M( nk ) A snk nk

22 4. Markets with a continuum of traded assets Complete probability space (Ω, F, P) Fixed time horizon [, T ], T > Right continuous filtration F := { F t, t [, T ] } representing the flow of information available in the market. For semplicity, F is trivial (up to P-null events) and F = F T Market is frictionless, admitting short-selling and continuous trading Risk free asset with price (value per unit): { t } R t = exp r s ds, t [, T ] (R = 1), where r = r t, t [, T ], is the (deterministic) interest rate.

23 Let X denote the set of risky assets. Then a self-financing strategies is a couple (e, ϕ) where e is an initial endowment (F -measurable) and a density of investments ϕ L 2 (P) with value process of the strategy (e, ϕ) satisfying dξ t = ξ t r t dt + R t ϕ(x, t)µ(dxdt) X = ξ t r t dt + R t dη t (ϕ), ξ = e. by means of the F-martingale t η t (ϕ) := ϕ(x, s)µ(dxds), t [, T ]. X Here µ(dxdt), (x, t) X [, T ], is a martingale random field generated by the excessive return processes of the investments on some groups of assets B with total price S t (B), t [, T ]: η t (B) := S t(b) R t S (B), t [, T ], B B X.

24 Minimal variance hedging problem Let F be a contingent claim. Then the minimal variance hedging problem is to find the replicable claim T ˆF = E[ˆF F ] + R T ϕ(x, t)µ(dxdt) and the density of investments ϕ such that E[(F ˆF ) 2 ] = min Y E[(F Y ) 2 ] X (Y replicable) Similarly, if we have some process V t, t [, T ], even if it is a discount martingale, it may not be replicable. Then the minimal variance hedging problem is to find the value process ˆV t, t [, T ]: t t ˆV t = ˆV + r s ˆVs ds + R s ϕ(x, s)µ(dxds) such that, for all t [, T ], E[(V t ˆV t ) 2 ] = min ξ E[(V t ξ) 2 ] X (ξ value process).

25 Stochastic differentiation The non-anticipating stochastic derivative of F DF = D (x,t) F, (x, t) X [, T ], is the adjoint operator to the Itô type integral with respect to µ(dxdt). Theorem. For all F L 2 (P), the non-anticipating derivative DF is (9) DF = lim κ n n k=1 [ E F µ( nk ) E [ ] ] A µ( nk ) 2 s 1 nk A snk with limit inl 2 (P). Here the sum is taken on the elements nk = B nk (s nk, u nk ], the n th -series of partitions of X (, T ]. The non-anticipating stochastic derivative DF represents the integrand in the orthogonal projection ˆξ of ξ on the subspace of all integrals with respect to µ, i.e. (1) F = F D (x,t) F µ(dxdt) X [,T ] and F L 2 (P) : DF. Cf. dinunno (22a, 22b, 27).

26 Minimal variance hedging strategy Theorem. Let F be a contingent claim. Then the minimal variance hedge ˆF exists and the self-financing strategy to achieve it is: ϕ = R 1 T D (x,t)f e = E[F F ]. The corresponding minimal variance hedging value process is characterized by: t t ˆV t = E[F F ] + r s ˆVs ds + X R s R 1 T D (x,s)f µ(dxds)

27 Theorem. Let V t, t [, T ], be a discount martingale. Then the minimal variance hedge ˆV t, t [, T ], exists and the self-financing strategy to achieve it is: ϕ = R 1 T D (x,t)v T e = V. The corresponding minimal variance hedging value process is characterized by: t t ˆV t = V + r s ˆVs ds + R s R 1 T D (x,s)v T µ(dxds) X Corollary. Let V t, t [, T ], be a discount martingale only on some sub-set T [, T ] with T T. Then the self-financing strategy ϕ = R 1 T D (x,t)v T e = V. yields the minimal variance hedge ˆV t, t [, T ], meaning that for all t T, E[(V t ˆV t ) 2 ] = min ξ E[(V t ξ) 2 ] (ξ value process). Note that for T = {T } we have the theorem before.

28 Other commets The countable large market is embedded in the larger framework of the martingale random fields approach Campi (22) gives some results on the existence of mean-variance hedging. The approach taken is substantially different. Results on minimal variance hedging and partial information can be given via the non-anticipating derivative Some differentiation formula for the non-anticipating derivative is studied in dinunno (27) 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 (28).

29 References T. Björk, B. Näslund (1998): Diversified portfolios in continuous time. European Finance Review 1, L. Campi (22): Mean-variance hedging in large financial markets. Prépublication PMA-758, Universités Paris 6/7. To appear in Stochastic Analysis and Applications. R. Cairoli, J.B. Walsh (1975): Stochastic integrals in the plane. Acta Math. 134, M. De Donno (24): A note on completeness in large financial markets. Math. Finance 14, M. De Donno, P. Guasoni, M. Pratelli (25): Super-replication and utility maximization in large financial markets. Stochastic Processes Appl. 115, F. Delbaen and W. Schachermayer (26): The Mathematics of Arbitrage. Springer-Verlag, Berlin. G. Di Nunno (22a): Stochastic integral representations, stochastic derivatives and minimal variance hedging. Stochastics Stochastics Rep. 73, G. Di Nunno(22b): Random fields evolution: non-anticipating integration and differentiation. Theory of Probability and Math. Statistics, 66, G. Di Nunno (27): Random fields evolution: non-anticipating derivative and differentiation formulae, Inf. Dim. Anal. Quantum Prob., 1, G. Di Nunno, B. Øksendal and F. Proske (28): Malliavin Calculus for Lévy Processes with Applications to Finance, Springer - in progress. Yu.M. Kabanov, D.O. Kramkov (1994): Large financial markets: asymptotic arbitrage and contiguity. Theory Probab. Appl. 39, Yu.M. Kabanov, D.O. Kramkov (1998): Asymptotic arbitrage in large financial markets. Finance Stochast. 2, I. Klein, W. Schachermayer (1996): Asymptotic arbitrage in non-complete large financial markets. Theory Probab. Appl. 41, I. Klein (2): A fundamental theorem of asset pricing for large financial markets. Math. Finance 1, A.N. Kolmogorov, S.V. Fomin (1989): Elements of the Theory of Functions and Functional Analysis. Sixth edition. Nauka, Moscow.

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