On Minimal Entropy Martingale Measures Andrii Andrusiv Friedrich Schiller University of Jena, Germany Marie Curie ITN
Outline 1. One Period Model Definitions and Notations Minimal Entropy Martingale Measure Esscher Transformation Alternative Characterization 2. Exponential Compound Poisson Model Definitions and Notations Some Facts about Lévy Processes and MEM Connection with One Period Model
Definitions and Notations (Ω, F, P) = (R, B(R), µ) probability space ξ(x) = x, x R F 0 := {, R} F 1 := F ξ = B(R) F = (F t ) t=0,1 Then (R, B(R), F, µ) is a filtered probability space. Price Process The price process is defined by X = (X t ) t=0,1 with X 0 = ξ 0 and X 1 = ξ. By definition X should be F-adapted and hence ξ 0 R.
Definitions and Notations The expectation of the random variable η under measure ν is denoted by E ν η := η(x)ν(dx). Let us define the set of all random variables on (R, B(R), ν) by R L 0 (ν) = {η : η is finite}. Similar notation is used for the set of all integrable random variables on (R, B(R), ν): L 1 (ν) = {η : E ν η < }. Finally, let us introduce the linear space generated by the change of price and by the random variables that are equal to 0 µ-a.s.: K = {η : η = k(ξ ξ 0 ) µ-a.s. for some k R}.
Definitions and Notations Definition A probability measure ν on (R, B(R)) is called a martingale measure, if E ν ξ = ξ 0. The set of all absolutely continuous martingale measures is denoted by M = {ν is a martingale measure, ν µ}. The set of all equivalent martingale measures is denoted by M e = {ν is a martingale measure, ν µ}. Assume µ({ξ ξ 0 }) > 0. Then there exists at least one equivalent martingale measure if and only if µ({ξ ξ 0 > 0}) > 0 and µ({ξ ξ 0 < 0}) > 0.
I (M, µ) = inf I (ν, µ). ν M Minimal Entropy Martingale Measure Definition Let ν be a probability measure on (R, B(R)). The relative entropy I (ν, µ) of ν with respect to µ is defined by [ ] dν dν E I (ν, µ) = µ log if ν µ. dµ dµ + otherwise. Definition A probability measure ν 0 M will be called the minimal entropy martingale measure (MEM) if it satisfies [ ] dν I (ν 0, µ) = min I (ν, µ) = min E dν µ log. ν M ν M dµ dµ The distance between µ and M is defined by
Minimal Entropy Martingale Measure The class of all absolutely continuous with respect to µ martingale measures with finite relative entropy is denoted by M 0 = {ν M : I (ν, µ) < + }. Let us define the class of all integrable random variables with respect to all absolutely continuous (with respect to µ) martingale measures with finite entropy L = ν M 0 L 1 (ν), and the set of random variables with non-positive expectation under every absolutely continuous (with respect to µ) martingale measures with finite entropy C 0 = { f L : E ν f 0, ν M 0}. If I (M, µ) < + and if ξ is bounded then there exists a unique MEM.
Characterisation of MEM Assume that I (M e, µ) < +. A probability ν 0 is the MEM if and only if (i) ν 0 M; (ii) dν 0 dµ = c exp( f 0) µ-a.s., where f 0 L 1 (ν 0 ), E ν0 f 0 = 0 and c > 0; (iii) f 0 C 0.
The Esscher Martingale Measure Definition Given a probability space (Ω, F, µ), assume ξ is a r.v. and k R such that E exp[k(ξ ξ 0 ] < +. Then ζ defined by is called an Esscher density. ζ := exp[k(ξ ξ 0)] E exp[k(ξ ξ 0 )] The random variable ζ is positive µ-a.s., integrable and normalized (Eζ = 1), hence ν defined by ζ = dµ dν is a probability measure, ν µ. This probability measure ν is called an Esscher measure. If Eζξ = E ν ξ = ξ 0, then ν is a martingale measure for ξ and is called an Esscher martingale measure.
Alternative Characterisation Let µ be an arbitrary probability measure on R, B(R), ξ is the random variable defined by ξ(x) = x, x R, and ξ 0 R. A probability measure ν 0 on R, B(R) is the MEM if and only if ν 0 is the Esscher martingale measure. Corollary Assume I (M e, µ) < +. Then the following conditions are equivalent: (a) There exists the MEM. (b) There exists the (unique) Esscher martingale measure. (c) There exists k R such that R (x ξ 0) exp[k(x ξ 0 )]µ(dx) = 0.
More Detailed Analysis The set of all c such that ξ exp(cξ) is finite, is denoted by E := {c R : E ξ exp(cξ) < + }. If there exists the MEM, then set E is not empty. The set E could have 4 different structures: (a, b), (a, b], [a, b), [a, b], where a b +. If E = (a, b) there always exists the MEM.
Examples Example 1. Let ξ be a Cauchy distributed symmetric random variable: µ(dx) = 1 π(1 + x 2 ) dx In this case there does not exist the MEM. Example 2. Let ξ is shifted one-side Cauchy distributed random variable. µ(dx) = 1 {x [ a,+ )} 2 π(1 + (x + a) 2 ) dx where a > 0. In this case there exists the MEM.
Definitions and Notations (Ω, F, P) probability space X t is compound Poisson process with drift N t X t = µt + ξ i, X 0 = 0 where N is Poisson with λ = ν(r), (ξ i ) iid, independent of N F = F X filtration generated by process X (R, B(R), F, µ) is filtered probability space Price process Let us define the price process S t = exp(x t ) for t [0, T ]. i=1
Stochastic Exponential Let Y be a real-valued semi-martingale and consider the stochastic differential equation dz = Z dy, Z 0 = 1. This equation has a unique (up to indistinguishability) cádlág adapted solution, called the stochastic exponential of Y, which is a semi-martingale and is denoted by E (Y ). Explicitly, E (Y ) t = exp(y t 1 2 Y c t ) s t(1 + Y s ) exp( Y s ). The mapping Y E (Y ) can be inverted. The following theorem shows this fact.
Stochastic Logarithm Let Z be a semimartingale such that Z, Z are strictly positive. Then there exists an up to indistinguishability unique semimartingale Y with Y 0 = 0 and Z = Z 0 E (Y ). It is given by Y t = t 0 1 Z u dz u, t 0. The process Y from the previous theorem is called the stochastic logarithm of Z and is written L (Z) := Y.
Stochastic Logarithm Stochastic logarithm of S is equal L (S) t = = t 0 N t i=1 [exp( X u ) 1]dN u + µt [exp(ξ i ) 1] + µt =: L t, where is Lévy process with drift µ and Lévy measure ν L = ν F 1, F (x) = e x 1 for x R.
Definitions and Notations Let us define the following sets of martingale measures: M a := {Q : (S, F) is a Q-local martingle, Q P on F T }, M e := {Q M a : Q P on bf T }, M f := {Q M a : I (Q, P) < + }, M l := {Q M a : (L, F) is a Q-Lévy process, }. Assumption: M l M f M e
Some facts about MEM Results of Esche and Schweizer(2005): (1) inf Q Ma I (Q, P) = inf Q Ml M f M e I (Q, P). (2) If the MEM Q exists (in M a ) then Q M l ( (L, F) is a Q -Lévy process). (3) If the MEM Q in M l exists. Then Q l = Q and Q P.
Some facts The process Y is a local martingale if and only if E (Y ) is a local martingale. Assume that Q P on F T and (L, F) is a Q-Lévy process. Then X (resp., L) is compound Poisson with drift µ and Lévy measure ν Q ν (resp., ν L Q = ν Q F 1 ).
Density Let (Y, F) be a Lévy process. Then (Y, F) is a local martingale if and only if (Y, F) is a martingale. If (L, F) is compound Poisson w.r.t. Q and Q P on F T. Then ν L Q νl and dq dp (F T ) = exp T (ν L (R) νq(r)) L + 0<u T log ρ( L u ), where ρ := dνl Q dν L.
Entropy Let us assume that there exists a martingale measure Q such that it preserves the Lévy property and has finite entropy: Q M e M f. Then the relative entropy I (Q, P) has the form [ ( )] I (Q, P) = T λ λ Q + λ Q log λq ν L λ + λq I Q λ Q, νl, λ where λ = ν L (R) and λ Q = ν L Q (R). I (Q, P) < + if and only if I ( ν L ) Q, νl λ Q λ < +.
Form of Density (L, F) is a Q-martingale if and only if E Q [exp(ξ 1 )] = µ λ + 1. If there is no drift µ = 0, the martingale condition can be simplified. The optimization ( problem splits into 2 independent problems: minimization of ν L ) Q I, νl λ Q λ and minimization of I (Q, P) with respect to λ Q. Hence, the optimal solution is [ ( )] ν L λ Q = λ exp I Q νl,, λq λ where νl Q λ Q is a solution of one-dimensional problem.
Form of Density And now we can calculate the density: dq dp (F T ) = exp(kl T ) E exp(kl T ) where k is the Esscher coefficient for one-dimensional distribution.
References Esche,F. Two Essays on Incomplete Markets, PhD-thesis(2004). Esche,F., Schweizer, M. Minimal entropy preserves the Lévy property: how and why., Stoch. Proc. Appl. 115, (2005), 299-327. Frittelli, M. The minimal entropy martingale measures and the valuation problem in incomplete markets., Math. Finance 10, (2000) 39-52. Goll,T., Ruschendorf, L. Minimax and minimal distance martingale measures and their relationship to portfolio optimization., Finance Stochast. 5, (2001) 557-581. Kabanov, Y., Stricker, C. On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper., Math. Finance 12, (2002) 125-134.