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Lower and upper bounds of martingale measure densities in continuous time markets

Giulia Di Nunno CMA, Univ. of Oslo Workshop on Stochastic Analysis and Finance Hong Kong, June 29th - July 3rd 2009.

Based on a joint work with Inga B. Eide

Outlines

• 1. Market modeling: EMM and no-arbitrage pricing principle
• 2. Framework: Claims and price operators
• 3. No-arbitrage pricing and representation theorems
• 4. EMM and extension theorems for operators
• 5. A version of the fundamental theorem of asset pricing

References

• 1. Market modeling

Market modeling is based on a probability space (Ω, F, P) identifying the possible future scenarios. The probability measure P is derived from DATA and/or EXPERTS’ BELIEFS of the possible scenarios and the possible dynamics of the random phenomenon. On the other hand the modeling of asset pricing is connected with the idealization of a FAIR MARKET. This is based on the principle of no-arbitrage and its relation with a risk neutral probability measure P0, under which all discounted prices are (local) martingales with respect to the evolution of the market events (for this reason P0 is also called martingale measure). The probability space (Ω, F, P0) provides an efficient mathematical framework.

Fundamental theorem of asset pricing

Naturally, we would like the models of mathematical finance to be both consistent with data analysis and to be mathematically feasible. This topic was largely investigated for quite a long time yielding the various versions of the fundamental theorem of asset pricing. The basic statement is: For a given market model on (Ω, F, P) and the flow of market events F = {Ft ⊆ F , t ∈ [0, T]} (T > 0), satisfying some assumptions, there exists a martingale measure P0 such that P0 ∼ P, i.e. P0(A) = 0 ⇐ ⇒ P(A) = 0, A ∈ FT.

• Cf. e.g. Delbaen, Harrison, Kreps, Pliska, Schachermayer.

No-arbitrage pricing principle

Mathematically the existence of an equivalent martingale measure (EMM) implies the absence of arbitrage opportunities, this embodying the economical fact that in a “fair” market there should be no possibility

• f earning riskless profit.

In fact the principle of no-arbitrage provides the basic pricing rule in mathematical finance: For any claim X, achievable at time t and purchased at time s, its “fair” price xst(X) is given by xst(X) = E0 Rs Rt X|Fs

• .

Here Rt, t ∈ [0, T], represents some riskless investment always achievable and always available on the market (the num´ eraire). For convenience we set Rt ≡ 1 for all t.

The martingale measure P0 used in the no-arbitrage evaluation is not necessarily unique: the equivalent martingale measure is unique if and

• nly if the market is complete, i.e. if all claims are attainable in the

market. In an incomplete market, if the claim X is attainable, the no-arbitrage evaluation of the price is independent of the choice of the martingale measure applied, thus the price is unique. But, if X is not-attainable, then the no-arbitrage principle does not give a unique price, but a whole range of prices that are equally valid from the no-arbitrage point of view. Many authors have been engaged in the study of how to select a martingale measure to be used. The approaches have been different.

◮ Selection of one measure that either is in some sense optimal or

whose use is justified by specific arguments in incomplete markets. Without aim or possibility to be complete we mention the minimal martingale measure and variance-optimal martingale measure which are both in some sense minimizing the distance to the physical measure (ref. e.g. Schweizer 2001). The Esscher measure is motivated by utility arguments to justify its use and it is also proved that it is also structure-preserving when applied to L´ evy driven models (ref. e.g. Delbaen et al. 1989, Gerber et al. 1996).

◮ Instead of searching for the unique ”optimal” equivalent martingale

measure, one can try to characterize probability measures that are in some sense ”reasonable”. This is the case of restrictions of the set of EMM in such a way that not only arbitrage opportunities are ruled out, but also deals that are ”too good” as in the case of bounds on the Sharpe ratio (the ratio of the risk premium to the volatility). Ref. e.g. Cochrane et al. (2000), Bj¨

• rk et al. (2006),

Staum (2006).

In various stochastic phenomena the impact of some events is crucial both for its own being and for the market effects triggered. As example, think of the devastating effects of natural catastrophies (e.g. earthquakes, hurricanes, etc.) and epidemies (e.g. SARS, bird flu, etc.). These events, though devastating, occur with small, but still positive probabilities. Having this in mind, the choice of a ”reasonable” EMM should take in to account a proper evaluation of ”small probabilities”, i.e. ”P(A) small” ⇔ ”P0(A) small.” Note in fact that the assessment under P of the risk of these events incurring can be seriously misjugded under a P0 only equivalent to P. This is particularly relevant for the evaluation of insurance linked securities.

Goal

We study the existence of EMM P0 with densities dP0

dP lying within

pre-considered lower and upper bounds: 0 < m ≤ dP0 dP ≤ M < ∞ P-a.s. We have to stress that these bounds are random variables: m = m(ω), ω ∈ Ω, M = M(ω), ω ∈ Ω.

A bit on related literature:

◮ In Rokhlin and Schachermayer (2006) and Rokhlin (2008), we find

the existence of a martingale measure with lower bounded density.

◮ In Kabanov and Stricker (2001) (see also R`

azonyi (2002)) densities are bounded from above. Actually the study shows that the set of equivalent σ-martingale measures with density in L∞(F) is dense (in total variation) in the set of equivalent σ-martingale measures. In our paper, we consider lower and upper bounds for martingale measure densities simultaneously.

• 2. Framework

We consider a continuous time market model without friction for the time interval [0, T] (T > 0) on the complete probability space (Ω, F, P). The flow of information is described by the right-continuous filtration F := {Ft ⊆ F , t ∈ [0, T]} with F = FT. Claims. For any fixed t, the achievable claims in the market payable at t constitue the convex sub-cone L+

t ⊆ L+ p (Ft)

• f the cone

L+

p (Ft) := {X ∈ Lp(Ω, Ft, P) : X ≥ 0},

p ∈ [1, ∞). Of course 0 ∈ L+

t .

Note that 1 ∈ L+

t , as the num´

eraire is always achievable.

N.B. The case L+

t = L+ p (Ft)

for all t ∈ [0, T] corresponds to a complete market. Otherwise the market is incomplete, i.e. L+

t ⊂ L+ p (Ft)

for at least a t ∈ [0, T].

N.B. The case L+

t = L+ p (Ft)

for all t ∈ [0, T] corresponds to a complete market. Otherwise the market is incomplete, i.e. L+

t ⊂ L+ p (Ft)

for at least a t ∈ [0, T]. N.B. If borrowing and short-selling are admitted, then the variety of claims is a linear sub-space Lt ⊆ Lp(Ft) : Lt := L+

t − L+ t ,

i.e. X ∈ Lt can be represented as X = X ′ − X ′′ with X ′, X ′′ ∈ L+

t .

Price operators

If X ∈ L+

t is available at time s : s ≤ t, then its price xst(X) is an

Fs-measurable random variable such that xst(X) < ∞ P − a.s.

Price operators

If X ∈ L+

t is available at time s : s ≤ t, then its price xst(X) is an

Fs-measurable random variable such that xst(X) < ∞ P − a.s. N.B. If short-selling is admitted, the price operators are defined on the sub-space Lt ⊆ Lp(Ft) according to xst(X) := xst(X ′) − xst(X ′′) for any X ∈ Lt : X = X ′ − X ′′ with X ′, X ′′ ∈ L+

t .

Fix s, t : s ≤ t. The price operator xst(X), X ∈ L+

t , satisfies:

◮ It is strictly monotone, i.e. for any X ′, X ′′ ∈ L+

t available at s

xst(X ′) ≥ xst(X ′′), X ′ ≥ X ′′, xst(X ′) > xst(X ′′), X ′ > X ′′ The notation “≥” represents the standard point-wise “≥ P-a.s.”, while “>” means that, in addition to “≥ P-a.s.”, the point-wise relation > holds on an event of positive P-measure.

◮ It is additive, i.e. for any X ′, X ′′ ∈ L+

t available at s

xst(X ′ + X ′′) = xst(X ′) + xst(X ′′).

◮ It is Fs-homogeneous, i.e. for any X ∈ L+

t available at s and any

Fs-multiplier λ such that λX ∈ L+

t , then

xst(λX) = λxst(X). We set xst(1) = 1 and xtt(X) = X, X ∈ L+

t . Note that xst(0) = 0.

• Definition. The price operator xst(X), X ∈ L+

t , is tame if

xst(X) ∈ Lp(Fs), X ∈ L+

t ,

i.e. xst(X)p :=

• E(xst(X)

p)1/p < ∞. N.B. This definition is motivated by the forthcoming arguments on time-consistency of the price operators. We consider the family of price operators of X ∈ L+

t ,

xst(X), 0 ≤ s ≤ t (t ∈ [0, T]).

• Definition. The given family of the prices is right-continuous at s if X is

available for some interval of time [s, s + δ] (δ > 0) and xs′t(X) − xst(X)p − → 0, s′ ↓ s.

• Definition. Let T ⊆ [0, T]. The family xst, s, t ∈ T : s ≤ t, of tame

discounted price operators xst(X), X ∈ L+

t , is time-consistent (in T ) if

for all s, u, t ∈ T : s ≤ u ≤ t xst(X) = xsu

• xut(X)
• ,

for all X ∈ L+

t such that xut(X) ∈ L+ u .

• Comment. This axiomatic approach to price processes is inspired by risk

measure theory. The requirements of monotonicity, additivity, and homogenuity are related to the concept of coherent risk measures. The additional assumption of strict monotonicity, is related to the concept of a relevant risk measure.

• 3. No-arbitrage pricing and representation theorems

We can reformulate the basic statement of the fundamental theorem saying that: The absence of arbitrage is ensured by the existence of an EMM P0, such that the prices xst(X), X ∈ L+

t , admit the representation

xst(X) = E 0[X|Fs], X ∈ L+

t .

Thus, for any t ∈ [0, T] and X ∈ L+

t , the price process

xst(X), 0 ≤ s ≤ t is a martingale with respect to the measure P0.

• Definition. A probability measure P0 ∼ P is tame if, for every t ∈ [0, T],

we have that E0[X|Ft] ∈ Lp(Ft), X ∈ Lp(FT).

• Facts. Let us consider the tame P0 ∼ P. Then the operator conditional

expectation : E 0[ · |Fs] : Lp(Ft) − → Lp(Fs) is tame, strictly monotone, linear, and Fs-homogeneous. Hence it has all the properties of a tame price operator. Moreover, the family of conditional expectations is time-consistent: E 0[X|Fs] = E 0[E 0[X|Fu]|Fs], X ∈ Lp(Ft), 0 ≤ s ≤ u ≤ t, and also right-continuous (thanks to the right-continuity of the filtration). Quite remarkably, it turns out that the converse is also true: All the price operators xsu(X), X ∈ Lp(Fu) (0 ≤ s ≤ u ≤ t), admit representation as conditional expectation with respect to the same EMM.

Lemma [DiN. (2003)] [Albeverio, DiN., Rozanov (2005)]. Fix s, t ∈ [0, T] : s ≤ t. The operator xst(X), X ∈ Lp(Ft), is tame, strictly monotone, linear, and Fs-homogeneous if and only if it admits representation (1) xst(X) = E 0

st[X|Fs],

X ∈ Lp(Ft), with respect to a tame probability measure P0

st(A) =

• A

fst(ω)P(dω), A ∈ Ft, where fst ∈ L+

q (Ft), 1 q + 1 p = 1 with fst > 0 P-a.s. In addition, the

• perator (1) is bounded (continuous) if and only if

   essupE

• fst

E[fst|Fs]

q Fs

• < ∞,

p ∈ (1, ∞) essup

fst E[fst|Fs] < ∞,

p = 1.

The result above is restricted to the two fixed time points s ≤ t. Now we keep s fixed and we compare the representations for different time points u ≤ t. Theorem. Let s, t ∈ [0, T]: s ≤ t. Assume that the operators xsu(X), X ∈ Lp(Fu), s ≤ u ≤ t, are tame price operators constituting a time-consistent family. Then, for all u ∈ [s, t], the representation (2) xsu(X) = E 0

st[X|Fs],

X ∈ Lp(Fu), holds in terms of the measure P0

st defined on (Ω, Ft). Moreover

P0

st|Fu = P0 su, for all u ∈ [s, t].

Summary and the following steps.

◮ Whenever we have a time-consistent family of tame price operators

xst(X), 0 ≤ s ≤ t ≤ T, defined on the whole cone X ∈ L+

p (Ft), we

have an EMM.

◮ This is always true in complete markets. However, this is not the

general situation. Usually operators are defined on the sub-cones L+

t ⊆ L+ p (Ft).

◮ Then the existence of an EMM is linked to the admissibility of an

extension of the price operator from the sub-cones to the corresponding cones.

◮ Need to give conditions (necessary and sufficient) for the extension

• f operators.

◮ Need to apply this to continuous time setting.

• 4. EMM and extension theorems for operators

Let mst, Mst ∈ L+

q (Ft) ( 1 p + 1 q = 1) such that

0 < mst ≤ Mst, P − a.s. Theorem [Albeverio, DiN, and Rozanov (2005)]. Fix s, t ∈ [0, T]: s ≤ t. The price operator xst(X), X ∈ L+

t , lying in the

sandwich E[Xmst|Fs] ≤ xst(X) ≤ E[XMst|Fs], ∀X ∈ L+

t ,

admits a tame, strictly monotone, linear, and Fs-homogeneous extension xst(X), ∈ L+

p (Ft), if and only if the sandwich condition

E[Y ′mst|Fs] + xst(X ′) ≤ xst(X ′′) + E[Y ′′Mst|Fs] holds for all X ′, X ′′ ∈ L+

t and Y ′, Y ′′ ∈ L+ p (Ft) such that

X ′ + Y ′ ≤ X ′′ + Y ′′.

N.B. If existing, the extension xst(X), X ∈ L+

p (Ft), is sandwich

preserving, i.e. E[Xmst|Fs] ≤ xst(X) ≤ E[XMst|Fs], ∀X ∈ L+

p (Ft).

Moreover (see Lemma), it admits representation xst(X) = E 0

st[X|Fs] = E[Xfst|Fs]

• fst := dP0

st

dP

• and the density fst lies in the sandwich

0 < mst ≤ fst ≤ Mst P − a.s. N.B.This is a version of the sandwich extension theorem which deals with

• perators. The general theorem is set in the Banach lattice framework

generalizing the K¨

• nig extension theorem for functionals.

Further extensions to convex operators are under construction (see [Bion-Nadal and DiN. (forthcoming)].

• 5. A version of the fundamental theorem of asset pricing

Let m, M ∈ L+

q (FT) ( 1 p + 1 q = 1) such that 0 < m ≤ M, P-a.s. such that

m = m0T = m0s mst mtT, M = M0T = M0s Mst MtT. For example, if

m E[m|Fs] ≤ M E[M|Fs] in L+ q (F), we can take

mst := (E[m|F0])

t−s T E[m|Ft]

E[m|Fs], Mst := (E[M|F0])

t−s T E[M|Ft]

E[M|Fs].

Theorem. Let xst(X), X ∈ L+

t , 0 ≤ s ≤ t ≤ T, be a time-consistent and

right-continuous family of tame price operators. Suppose that every xst(X), X ∈ L+

t , satisfies the sandwich condition:

E

• Y ′′mst|Fs
• + xst(X ′′) ≤ xst(X ′) + E
• Y ′Mst|Fs
• for all X ′, X ′′ ∈ L+

t , Y ′, Y ′′ ∈ L+ p (Ft) such that Y ′′ + X ′′ ≤ X ′ + Y ′.

Then there exists a tame probability measure P0 ∼ P: P0(A) =

• A

f (ω)P(dω), A ∈ F, with f ∈ L+

q (F) such that E[f |F0] = 1 and 0 < m ≤ f ≤ M

P − a.s. allowing the representation xst(X) = E

• X

f E[f |Fs]|Fs

• = E 0[X|Fs],

X ∈ L+

t ,

for all price operators. The converse is also true.

Sketch of proof. Necessary condition. Consider the set of EMM: P :=

• P0| dP0

dP = f , E[f |F0] = 1, m ≤ f ≤ M : ∀s, t ∈ [0, T], s ≤ t, xst(X) = E 0[X|Fs] ∀X ∈ L+

t

• and the approximating set:

P(T ) :=

• P0| dP0

dP = f , E[f |F0] = 1, m ≤ f ≤ M : ∀s ∈ T , t ∈ [s, T], xst(X) = E 0[X|Fs] ∀X ∈ L+

t

• ,

where T is some partition of [0, T] of the form T = {s0, s1, . . . , sK}, with 0 = s0 < s1 < · · · < sK = T.

Further, we consider a sequence {Tn}∞

n=1 of increasingly refined

partitions, such that Tn ⊂ Tn+1 and mesh(Tn) − → 0 as n − → ∞. Clearly P(Tn+1) ⊆ P(Tn). Then the proof proceeds with the following steps:

• A. P(T ) is non-empty for any finite partition T .

Here we have to work with the consistency of the extensions, in fact these may not be time-consistent and have to be constructed accurately.

• B. The infinite intersection ∞

n=1 P(Tn) is non-empty.

Here we have to work with arguments of weak* compactness.

• C. P0 ∈ ∞

n=1 P(Tn) is also in P. Here we have to work with the time

continuity of the family of prices and the right-continuity of the filtration.

Example: single period market model.

Let L+

T := {X = αH + β : α, β ≥ 0} be the set of claims, i.e. α

represents the fraction of the claim H =

• z>z0

(z − z0)N(T, dz), N(T, dz) ∼ Poi(Tν(dz)), and β is the amount in a money market account with zero-interest. Note that H can be interpreted as an insurance policy covering all losses exceeding the deductible z0 in the time interval [0, T]. Let the price of H be given by the expected value principle, i.e. x0T(H) = (1 + δ)EH = (1 + δ)T

• z>z0

(z − z0)ν(dz) and correspondingly x(X) = β + α(1 + δ)T ∞

z0

(z − z0)ν(dz), X = αH + β.

Take as bounds for the possible densities to be given by: m = e−δTν(I0) M = (1 + δ)N(T,I0)e−δTν(I0), I0 := (z0, ∞) Then P is not empty. In fact the structure preserving EMM with density: f1 = (1 + δ)N((0,T],I0)e−δTν(I0), I0 = (z0, ∞). belongs to P. On the other side, not all EMM belong to P. For example, the structure preserving EMM with density f2 = (1 + γ)N((0,T],I ∗)e−γTν(I ∗), I ∗ = (z∗, ∞). with N((0, T], I ∗) = N((0, T], I0) and ν(dz) = δa(dz) + δb(dz) for z0 < a < z∗ < b does not belong to P.

References

This presentation was based on:

• G. Di Nunno and Inga B. Eide. Events of small but positive probability and a fundamental theorem of asset pricing.

E-print, University of Oslo 2008. Relevant related references:

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