# On exponentially ane martingales. Johannes Muhle-Karbe

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1 On exponentially ane martingales AMAMEF 2007, Bedlewo Johannes Muhle-Karbe Joint work with Jan Kallsen HVB-Institut für Finanzmathematik, Technische Universität München 1

2 Outline 1. Semimartingale characteristics 2. Ane processes 3. Exponentially ane martingales 4. Applications HVB-Institut für Finanzmathematik, Technische Universität München 2

3 Semimartingale characteristics Idea: Local characterization of semimartingales Deterministic: Linear functions characterized by constant increments Derivative: local approximation by linear functions Stochastic analogon: Lévy processes characterized by independent, stationary increments Semimartingale characteristics: local approximation by Lévy processes HVB-Institut für Finanzmathematik, Technische Universität München 3

4 Semimartingale characteristics Linear function determined by slope b R Distribution of Lévy process (X t ) t R+ on R d determined by Lévy-Khintchine triplet (b, c, F ) (Dierential) Semimartingale characteristics: X t := (b t (ω), c t (ω), F t (ω, )) Local Lévy-Khintchine triplet, time-dependent and random Connection to (integral) characteristics from Jacod & Shiryaev: B t = t 0 b s ds, C t = t 0 c s ds, ν([0, t] G) = t 0 F s (G)ds HVB-Institut für Finanzmathematik, Technische Universität München 4

5 Semimartingale characteristics Idea: Modeling through local dynamics Deterministic: Ordinary dierential equation d dt X t = f(t, X t ) Stochastic analogon: Martingale problem, X = (b, c, F ) with β, γ, ϕ constant: X Lévy process b t (ω) = β(t, X t (ω)) c t (ω) = γ(t, X t (ω)) F t (ω, G) = ϕ(t, G, X t (ω)) HVB-Institut für Finanzmathematik, Technische Universität München 5

6 Ane processes Ane process: Characteristics X = (b, c, F ) ane in X : b t (ω) = β 0 + c t (ω) = γ 0 + d Xt (ω)β j j j=1 m Xt (ω)γ j j j=1 F t (G, ω) = ϕ 0 (G) + m Xt (ω)ϕ j j (G) j=1 (β j, γ j, ϕ j ) given Lévy-Khintchine triplets Due, Filipovi & Schachermayer (2003): Admissibility conditions for existence and uniqueness, Filipovi (2005): time-inhomogeneous triplets HVB-Institut für Finanzmathematik, Technische Universität München 6

7 Ane processes Example: Stochastic volatility model of Heston (1993) S t = E (X) t asset price, v t volatility, where (v, X) solves dv t = (κ λv t )dt + σ v t dz t dx t = (µ + δv t )dt + v t dw t W, Z Brownian motions with constant correlation ϱ Characteristics: ( v X) ane in v = v t = (( ) ( ) ) κ λvt σ, 2 v t σϱv t, 0 µ δv t σϱv t v t HVB-Institut für Finanzmathematik, Technische Universität München 7

8 Ane processes Example: Model of Carr, Geman, Madan & Yor (2001) X t = X 0 + L Vt dv t = v t dt, dv t = λv t dt + dz t L, Z independent Lévy processes with Lévy-Khintchine triplets (b L, c L, F L ) and (b Z, 0, F Z ) Characteristics: (( ) ( ) b Z λv t 0 0 b L, v t 0 c L, v t 1 G (y, 0)F Z (dy) + 1 G (0, x)f L (dx)v t ) ane in v HVB-Institut für Finanzmathematik, Technische Universität München 8

9 Exponentially ane martingales X ane process, h truncation function Goal: Conditions for E (X i ) to be a martingale Jacod & Shiryaev (2003) E (X i ) is a local martingale, i h i (x i )x i ϕ j (dx) <, 0 j d βj i + (x i h i (x i ))ϕ j (dx) = 0, 0 j d Continuous case: Local martingale, i βj i = 0, 0 j d Condition on X, easy to check in applications HVB-Institut für Finanzmathematik, Technische Universität München 9

10 Exponentially ane martingales Criterion for true martingale property? For Lévy process X: E (X) local martingale, E (X) 0 E (X)martingale Does NOT hold in general for ane processes! Some general criteria either hard to check (e.g. Wong & Heyde (2004), Jacod & Shiryaev (2003), Kallsen & Shiryaev (2002)) Others not even necessary in the Lévy case (e.g. Lepingle & Memin (1978)) On the other hand: powerful theory of ane processes HVB-Institut für Finanzmathematik, Technische Universität München 10

11 Exponentially ane martingales Theorem: X ane relative to admissible Lévy-Khintchine triplets (β j, γ j, ϕ j ), 0 j d. Then E (X i ) is a martingale, if 1. E (X i ) 0, 2. h i (x i )x i ϕ j (dx) <, 0 j d 3. βj i + (x i h i (x))ϕ j (dx) = 0, 0 j d 4. { x k >1} xk 1 + x i ϕ j (dx) <, 1 k, j d Similar criterion for exp(x i ) instead of E (X i ) Extension to time-inhomogeneous ane processes possible HVB-Institut für Finanzmathematik, Technische Universität München 11

12 Exponentially ane martingales Continuous case (e.g. Heston (1993)): E (X) positive local martingale martingale Also holds for CGMY asset price S S positive, if F L ((, 1)) = 0 S local martingale, if (yh(y))f L (dy) <, b L + (y h(y))f L (dy) = 0 For PII X: E (X) positive local martingale martingale HVB-Institut für Finanzmathematik, Technische Universität München 12

13 Application 1: Absolutely continuous change of measure X, Y semimartingales with ane characteristics Goal: Criterion for P Y loc P X Application: X model for asset price under physical, Y under risk neutral measure. Equivalence for arbitrage theory Idea: Dene appropriate candidate Z for density process Show: Z exponentially ane, local martingale martingale Dene Q loc P X via density process Z Q = P Y by Girsanov and uniqueness of ane martingale problems HVB-Institut für Finanzmathematik, Technische Universität München 13

14 Application 1: Absolutely continuous change of measure Similar results by Cheridito, Filipovi and Yor (2005) in more general setup Here, the moment conditions are often less restrictive though Example: Esscher change of measure Condition in Cheridito, Filipovi, & Yor (2005): (H x)e H x ϕ j (dx) <, 0 j d { x >1} Condition here: { x >1} e H x ϕ j (dx) <, 0 j d HVB-Institut für Finanzmathematik, Technische Universität München 14

15 Application 2: Exponential moments X R d -valued ane process Due, Filipovi and Schachermayer (2003): conditional characteristic function given by E(e iu X T F t ) = exp(φ(t t, iu) + Ψ(T t, iu) X t ), Φ and Ψ solve integro-dierential equations with initial values 0, iu If analytic extension to open set U exists, E(e p X T F t ) = exp(φ(t t, p) + Ψ(T t, p) X t ), p U Problem: construction of analytic extension is often tedious HVB-Institut für Finanzmathematik, Technische Universität München 15

16 Application 2: Exponential moments Alternative approach: Assume solutions Φ and Ψ to integro-dierential equations with initial values 0 and p R d exist Dene N t := Φ(T t) + Ψ(T t)x t Show: (X, N) is ane, exp(n) is a local martingale Results on time-inhomogeneous exponentially ane martingales, (mild) condition on the big jumps of X martingale Martingale property yields E(e p X T F t ) = E(e N T F t ) = e N t = exp(φ(t t) + Ψ(T t) X t ) HVB-Institut für Finanzmathematik, Technische Universität München 16

17 Application 3: Portfolio optimization Goal: Find trading strategy ϕ, such that E(u(V T (ϕ))) E(u(V T (ψ))), ψ u utility function Example: Power utility, i.e. u(x) = x 1 p /(1 p) ϕ, ψ admissible, i.e. V (ϕ), V (ψ) 0 Asset price modeled as an ane process, e.g. Heston (1993) or Carr, Geman, Madan & Yor (2001) HVB-Institut für Finanzmathematik, Technische Universität München 17

18 Application 3: Portfolio optimization Sucient criterion for optimality: If there exists a positive martingale Z, such that 1. (ZS) T is a local martingale 2. Z T = u (V T (ϕ)) 3. (ZV (ϕ)) T is a martingale we have E(u(V T (ϕ))) E(u(V T (ψ))), ψ Observation: S, V (ϕ) and u (V (ϕ)) exponentially ane HVB-Institut für Finanzmathematik, Technische Universität München 18

19 Application 3: Portfolio optimization Idea: Make an exponentially ane ansatz for Z as well! Computation of ansatz functions through drift conditions Verication of the candidate processes with results on exponentially ane martingales Approach works for the models proposed by Heston (1993) and Carr, Geman, Madan & Yor (2001) among others HVB-Institut für Finanzmathematik, Technische Universität München 19

20 References Carr, Geman, Madan & Yor (2001): Stochastic volatility for Lévy processes, Mathematical Finance 13(3), Cheridito, Filipovi & Yor (2005): Equivalent and absolutely continuous measure changes for jump-diusion processes, The Annals of Applied Probability 15(3), Due, Filipovi & Schachermayer (2003): Ane processes and applications in nance, The Annals of Applied Probability, 13(3), Filipovi (2005): Time-inhomogeneous ane processes, Stochastic Processes and their Applications 115(4), Heston (1993): A closed-form solution for options with stochastic volatilities with applications to bond and currency options, The Review of Financial Studies, 6(2), HVB-Institut für Finanzmathematik, Technische Universität München 20

21 References Jacod & Shiryaev (2003): Limit theorems for stochastic processes, Springer Verlag, Berlin, second edition Kallsen & Shiryaev (2002): The cumulant process and Esscher's change of measure, Finance and Stochastics, 6(4), Lépingle & Mémin (1978): Sur l'intégrabilité uniforme des martingales exponentielles, Zur Wahrscheinlichkeit Verwandte Gebiete, 42(3), Wong & Heyde (2004): On the martingale property of stochastic exponentials, Journal of Applied Probability, 41(3), HVB-Institut für Finanzmathematik, Technische Universität München 21

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