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 29 th - July 3 rd 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 P 0, under which all discounted prices are (local) martingales with respect to the evolution of the market events (for this reason P 0 is also called martingale measure). The probability space (Ω, F, P 0 ) 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 = {F t F, t [0, T ]} (T > 0), satisfying some assumptions, there exists a martingale measure P 0 such that P 0 P, i.e. P 0 (A) = 0 P(A) = 0, A F T. 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 of 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 x st (X ) is given by x st (X ) = E 0 [ Rs R t X F s ]. Here R t, t [0, T ], represents some riskless investment always achievable and always available on the market (the numéraire). For convenience we set R t 1 for all t.
The martingale measure P 0 used in the no-arbitrage evaluation is not necessarily unique: the equivalent martingale measure is unique if and only 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 Lévy 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), Björk 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 P 0 (A) small. Note in fact that the assessment under P of the risk of these events incurring can be seriously misjugded under a P 0 only equivalent to P. This is particularly relevant for the evaluation of insurance linked securities.
Goal We study the existence of EMM P 0 with densities dp0 dp 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. lying within
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 Ràzonyi (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 with F = F T. F := {F t F, t [0, T ]} Claims. For any fixed t, the achievable claims in the market payable at t constitue the convex sub-cone of the cone L + t L + p (F t ) L + p (F t ) := {X L p (Ω, F t, P) : X 0}, p [1, ). Of course 0 L + t. Note that 1 L + t, as the numéraire is always achievable.
N.B. The case L + t = L + p (F t ) for all t [0, T ] corresponds to a complete market. Otherwise the market is incomplete, i.e. L + t L + p (F t ) for at least a t [0, T ].
N.B. The case L + t = L + p (F t ) for all t [0, T ] corresponds to a complete market. Otherwise the market is incomplete, i.e. L + t L + p (F t ) 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 L t L p (F t ) : L t := L + t L + t, i.e. X L t 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 x st (X ) is an F s -measurable random variable such that x st (X ) < P a.s.
Price operators If X L + t is available at time s : s t, then its price x st (X ) is an F s -measurable random variable such that x st (X ) < P a.s. N.B. If short-selling is admitted, the price operators are defined on the sub-space L t L p (F t ) according to x st (X ) := x st (X ) x st (X ) for any X L t : X = X X with X, X L + t.
Fix s, t : s t. The price operator x st (X ), X L + t, satisfies: It is strictly monotone, i.e. for any X, X L + t x st (X ) x st (X ), X X, x st (X ) > x st (X ), X > X available at s 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 x st (X + X ) = x st (X ) + x st (X ). It is F s -homogeneous, i.e. for any X L + t F s -multiplier λ such that λx L + t, then available at s and any x st (λx ) = λx st (X ). We set x st (1) = 1 and x tt (X ) = X, X L + t. Note that x st (0) = 0.
Definition. The price operator x st (X ), X L + t, is tame if x st (X ) L p (F s ), X L + t, i.e. x st (X ) p := ( E(x st (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, x st (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 x s t(x ) x st (X ) p 0, s s. Definition. Let T [0, T ]. The family x st, s, t T : s t, of tame discounted price operators x st (X ), X L + t, is time-consistent (in T ) if for all s, u, t T : s u t x st (X ) = x su ( xut (X ) ), for all X L + t such that x ut (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 P 0, such that the prices x st (X ), X L + t, admit the representation x st (X ) = E 0 [X F s ], X L + t. Thus, for any t [0, T ] and X L + t, the price process x st (X ), 0 s t is a martingale with respect to the measure P 0. Definition. A probability measure P 0 P is tame if, for every t [0, T ], we have that E0[X F t ] L p (F t ), X L p (F T ).
Facts. Let us consider the tame P 0 P. Then the operator conditional expectation : E 0 [ F s ] : L p (F t ) L p (F s ) is tame, strictly monotone, linear, and F s -homogeneous. Hence it has all the properties of a tame price operator. Moreover, the family of conditional expectations is time-consistent: E 0 [X F s ] = E 0 [E 0 [X F u ] F s ], X L p (F t ), 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 x su (X ), X L p (F u ) (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 x st (X ), X L p (F t ), is tame, strictly monotone, linear, and F s -homogeneous if and only if it admits representation (1) x st (X ) = E 0 st[x F s ], X L p (F t ), with respect to a tame probability measure Pst(A) 0 = f st (ω)p(dω), A F t, A where f st L + 1 q (F t ), q + 1 p = 1 with f st > 0 P-a.s. In addition, the operator (1) is bounded (continuous) if and only if [( ) q ] F s essupe essup f st E[f st F s ] f st E[f st F s ] <, p (1, ) <, 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 x su (X ), X L p (F u ), s u t, are tame price operators constituting a time-consistent family. Then, for all u [s, t], the representation (2) x su (X ) = E 0 st[x F s ], X L p (F u ), holds in terms of the measure Pst 0 defined on (Ω, F t ). Moreover Pst 0 Fu = Psu, 0 for all u [s, t].
Summary and the following steps. Whenever we have a time-consistent family of tame price operators x st (X ), 0 s t T, defined on the whole cone X L + p (F t ), 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 (F t ). 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 of operators. Need to apply this to continuous time setting.
4. EMM and extension theorems for operators Let m st, M st L + q (F t) ( 1 p + 1 q = 1) such that 0 < m st M st, P a.s. Theorem [Albeverio, DiN, and Rozanov (2005)]. Fix s, t [0, T ]: s t. The price operator x st (X ), X L + t, lying in the sandwich E[Xm st F s ] x st (X ) E[XM st F s ], X L + t, admits a tame, strictly monotone, linear, and F s -homogeneous extension x st (X ), L + p (F t ), if and only if the sandwich condition E[Y m st F s ] + x st (X ) x st (X ) + E[Y M st F s ] holds for all X, X L + t X + Y X + Y. and Y, Y L + p (F t ) such that
N.B. If existing, the extension x st (X ), X L + p (F t ), is sandwich preserving, i.e. E[Xm st F s ] x st (X ) E[XM st F s ], X L + p (F t ). Moreover (see Lemma), it admits representation x st (X ) = E 0 st[x F s ] = E[Xf st F s ] and the density f st lies in the sandwich 0 < m st f st M st P a.s. ( f st := dp0 st dp ) N.B.This is a version of the sandwich extension theorem which deals with operators. The general theorem is set in the Banach lattice framework generalizing the König 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 (F T ) ( 1 p + 1 q = 1) such that 0 < m M, P-a.s. such that For example, if m = m 0T = m 0s m st m tt, M = M 0T = M 0s M st M tt. m E[m F s ] M E[M F s ] in L + q (F), we can take m st := (E[m F 0 ]) t s T E[m F t ] E[m F s ], M st := (E[M F 0 ]) t s T E[M F t ] E[M F s ].
Theorem. Let x st (X ), X L + t, 0 s t T, be a time-consistent and right-continuous family of tame price operators. Suppose that every x st (X ), X L + t, satisfies the sandwich condition: E [ Y m st F s ] + xst (X ) x st (X ) + E [ Y M st F s ] for all X, X L + t, Y, Y L + p (F t ) such that Y + X X + Y. Then there exists a tame probability measure P 0 P: P 0 (A) = f (ω)p(dω), A F, A with f L + q (F) such that E[f F 0 ] = 1 and 0 < m f M allowing the representation [ f ] x st (X ) = E X E[f F s ] F s = E 0 [X F s ], X L + t, P a.s. for all price operators. The converse is also true.
Sketch of proof. Necessary condition. Consider the set of EMM: P := {P 0 dp0 dp = f, E[f F 0] = 1, m f M : and the approximating set: P (T ) := s, t [0, T ], s t, x st (X ) = E 0 [X F s ] X L + t {P 0 dp0 dp = f, E[f F 0] = 1, m f M : s T, t [s, T ], x st (X ) = E 0 [X F s ] X L + t where T is some partition of [0, T ] of the form T = {s 0, s 1,..., s K }, with 0 = s 0 < s 1 < < s K = T. } },
Further, we consider a sequence {T n } n=1 of increasingly refined partitions, such that T n T n+1 and mesh(t n ) 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. P 0 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 z 0 )N(T, dz), N(T, dz) Poi(T ν(dz)), z>z 0 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 z 0 in the time interval [0, T ]. Let the price of H be given by the expected value principle, i.e. x 0T (H) = (1 + δ)eh = (1 + δ)t (z z 0 )ν(dz) z>z 0 and correspondingly x(x ) = β + α(1 + δ)t z 0 (z z 0 )ν(dz), X = αh + β.
Take as bounds for the possible densities to be given by: δt ν(i0) m = e M = (1 + δ) N(T,I0) e δt ν(i0), I 0 := (z 0, ) Then P is not empty. In fact the structure preserving EMM with density: f 1 = (1 + δ) N((0,T ],I0) e δt ν(i0), I 0 = (z 0, ). belongs to P. On the other side, not all EMM belong to P. For example, the structure preserving EMM with density f 2 = (1 + γ) N((0,T ],I ) e γt ν(i ), I = (z, ). with N((0, T ], I ) = N((0, T ], I 0 ) and ν(dz) = δ a (dz) + δ b (dz) for z 0 < 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: S. Albeverio, G. Di Nunno, and Y. A. Rozanov. Price operators analysis in L p-spaces. Acta Applicandae Mathematicae, 89:85 108, 2005. T. Björk and I. Slinko. Towards a general theory of good deal bounds. Review of Finance, 10:221 260, 2006. J. H. Cochrane and J. Saá-Requejo. Beyond arbitrage: good-deal asset price bounds in incomplete markets. Journal of Political Economy, 101:79 119, 2000. F. Delbaen and J. Haezendonck. A martingale approach to premium calculation principles in an arbitrage free market. Insurance: Mathematics and Economics, 8:269 277, 1989. F. Delbaen and W. Schachermayer. A general version of the fundamental theorem of asset pricing. Math. Ann., 300:463 520, 1994. F. Delbaen and W. Schachermayer. The Mathematics of Arbitrage. Springer, 2006. G. Di Nunno. Some versions of the fundamental teorem of asset pricing. Preprint 13/02, available at www.math.uio.no/eprint, 2002. G. Di Nunno. Hölder equality for conditional expectations with applications to linear monotone operators. Theor. Probability Appl., 48:177 181, 2003. B. Fuchssteiner and W. Lusky. Convex Cones. North-Holland, 1981. H. U. Gerber and E. S. W. Shiu. Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics and Economics, 18:183 218, 1996. J.M. Harrison and S. Pliska. A stochastic calculus model of continuous trading: Complete markets. Stochastic processes and their applications, 15:313 316, 1983.
Y. Kabanov and C. Stricker. On equivalent martingale measures with bounded densities, pages 139 148. Lecture Notes in Math. 1755. Springer, 2001. O. Kreps. Arbitrage and equilibrium in economics with infinitely many commodities. Journal of Math. Econom., 8:15 35, 1981. M. Rásonyi. A note on martingale measures with bounded densities. Tr. Mat. Inst. Steklova, 237:212 216, 2002. D. Rokhlin and W. Schachermayer. A note on lower bounds of martingale measure densities. Illinois Journal of Mathematics, 50:815 824, 2006. D. Rokhlin. Lower bounds of martingale measure densities in the Dalang-Morton-Willinger theorem. arxiv: 0804.1761.v1. M. Schweizer. A guided tour through quadratic hedging approaches. In E. Jouini, J. Cvitanic, and M. Musiela, editors, Option Pricing, Interest Rates and Risk Management, pages 538 574. Cambridge University Press, 2001. J. Staum. Fundamental theorems of asset pricing for good-deal bounds. Mathematical Finance, 50:141 161, 2006. V. R. Young. Premium principles. In Encyclopedia of Actuarial Science. J. Wiley & Sons, Ltd, 2004.