Cost-efficiency in multivariate Lévy models

• Autorzy: Ludger Rüschendorf , Viktor Wolf
• Języki publikacji: EN

Abstrakty

EN

In this paper we determine lowest cost strategies for given payoff distributions called cost-efficient strategies in multivariate exponential Lévy models where the pricing is based on the multivariate Esscher martingale measure. This multivariate framework allows to deal with dependent price processes as arising in typical applications. Dependence of the components of the Lévy Process implies an influence even on the pricing of efficient versions of univariate payoffs.We state various relevant existence and uniqueness results for the Esscher parameter and determine cost efficient strategies in particular in the case of price processes driven by multivariate NIG- and VG-processes. From a monotonicity characterization of efficient payoffs we obtain that basket options are generally inefficient in Lévy markets when pricing is based on the Esscher measure.We determine efficient versions of the basket options in real market data and show that the proposed cost efficient strategies are also feasible from a numerical viewpoint. As a result we find that a considerable efficiency loss may arise when using the inefficient payoffs.

Słowa kluczowe

EN

cost-efficient strategies; multivariate Lévy models; multivariate Esscher transform; basket option

Kategorie tematyczne

• 91B24: Price theory and market structure
• 60G51: Processes with independent increments; L\'evy processes
• 91B16: Utility theory

• Wydawca: De Gruyter Open
• Czasopismo: Dependence Modeling
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2014-11-05

zaakceptowano

2015-04-06

online

2015-04-16

Twórcy

autor

Ludger Rüschendorf

• Department of Mathematical Stochastics, University of Freiburg,
  Eckerstraße 1, 79104 Freiburg, Germany

autor

Viktor Wolf

• Department of Mathematical Stochastics, University of Freiburg,
  Eckerstraße 1, 79104 Freiburg, Germany

Bibliografia

• ---
• [1] O. E. Barndorff-Nielsen (1977). Exponentially decreasing distributions for the logarithm of particle size. Proc. Roy. Soc. London Ser. A, 353(1674), 401–419.
• [2] O. E. Barndorff-Nielsen, J. Kent, and M. Sørensen (1982). Normal variance mean-mixtures and z-distributions. Internat. Statist. Rev., 50(2), 145–159. [Crossref]
• [3] C. Bernard, P. P. Boyle, and S. Vanduffel (2014). Explicit representation of cost-efficient strategies. Finance, 35(2), 5–55.
• [4] C. Bernard, F. Moraux, L. Rüschendorf, and S. Vanduffel (2015). Optimal payoffs under state-dependent preferences. To appear in Quant. Finance, DOI:10.1080/14697688.2014.981576. [WoS][Crossref]
• [5] C. Bernard, L. Rüschendorf, and S. Vanduffel (2014). Optimal claims with fixed payoff structure. J. Appl. Probab., 51A, 175–188. [Crossref]
• [6] P. Blæsild (1981). The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen’s bean data. Biometrika, 68(1), 251–263. [Crossref]
• [7] C. Burgert and L. Rüschendorf (2006). On the optimal risk allocation problem. Statistics & Decisions, 24(1), 153–171.
• [8] T. Chan (1999). Pricing contingent claims on stocks driven by Lévy processes. Ann. Appl. Probab., 9(2), 504–528. [Crossref]
• [9] J. E. Jr. Dennis and R. B. Schnabel (1996). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia.
• [10] P. Dybvig (1988a). Distributional analysis of portfolio choice. J. Business, 61(3), 369–393. [Crossref]
• [11] P. Dybvig (1988b). Inefficient dynamic portfolio strategies or how to throw away a million dollars in the stock market. Rev. Financ. Stud., 1(1), 67–88. [Crossref]
• [12] E. Eberlein and U. Keller (1995). Hyperbolic distributions in finance. Bernoulli, 1(3), 281–299. [Crossref]
• [13] E. Eberlein, A. Papapantoleon and A. N. Shiryaev (2009). Esscher transform and the duality principle for multidimensional semimartingales. Ann. Appl. Probab., 19(5), 1944–1971. [Crossref][WoS]
• [14] F. Esche and M. Schweizer (2005). Minimal entropy preserves the Lévy property: How and why. Stochastic Process. Appl., 115(2), 299–327.
• [15] H. Föllmer and A. Schied (2004). Stochastic Finance. An Introduction in Discrete Time. 2nd revised and extended edition, de Gruyter, Berlin.
• [16] H. U. Gerber and E. S. W. Shiu (1994). Option pricing by Esscher transforms. T. Soc. Actuaries, 46, 99–191.
• [17] T. Goll and J. Kallsen (2000). Optimal portfolios for logarithmic utility. Stochastic Process. Appl., 89(1), 31–48.
• [18] T. Goll and L. Rüschendorf (2001). Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Finance Stoch., 5(4), 557–581.
• [19] E. A. v. Hammerstein (2010). Generalized Hyperbolic Distributions: Theory and Application to CDO Pricing. PhD thesis, University of Freiburg i. Br.
• [20] E. A. v. Hammerstein, E. Lütkebohmert, L. Rüschendorf, and V. Wolf (2014). Optimality of payoffs in Lévy models. Appl. Finance, 17(6), 1-46.
• [21] F. Hubalek and C. Sgarra (2006). Comparisons of dependence for stationary Markov processes. Quant. Finance, 6(2), 125–145. [Crossref]
• [22] E. Jouini and H. Kallal (2001). Efficient trading strategies in the presence of market frictions. Rev. Financ. Stud., 14(2), 343–369. [Crossref]
• [23] J. Kallsen and A. N. Shiryaev (2002). The cumulant process and Esscher’s change of measure. Finance Stoch., 6(4), 397–428.
• [24] E. Luciano and P. Semeraro (2010). A generalized normal mean-variance mixture for return processes in finance. Int. J. Theor. Appl. Finance, 13(3), 415–440. [Crossref]
• [25] D. B. Madan and F. Milne (1991). Option pricing with VG martingale components. Math. Finance, 1(4), 39–55. [Crossref]
• [26] T. Mäkeläinen, K. Schmidt, and G. P. H. Styan (1981). On the existence and uniqueness of the maximum likelihood estimate of a vector-valued parameter in fixed-size samples. Ann. Statist., 9(4), 758–767. [Crossref]
• [27] S. Raible (2000). Lévy processes in Finance: Theory, Numerics, and Empirical Facts. PhD thesis, University of Freiburg i. Br..
• [28] L. Rüschendorf and V. Wolf (2014). On the method of optimal portfolio choice by cost-efficiency. Preprint.
• [29] K.-I. Sato (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.
• [30] A. Takahashi and K. Yamamoto (2013). Generating a target payoff distribution with the cheapest dynamic portfolio: An application to hedge fund replication. Quant. Finance, 13(10), 1559–1573. [Crossref][WoS]
• [31] P. Tankov (2010). Financial Modeling with Lévy Processes. Lecture Notes.
• [32] S. Vanduffel, A. Chernih, M. Maj, and W. Schoutens (2009). A note on the suboptimality of path-dependent payoffs in Lévy markets. Appl. Math. Finance, 16(4), 315–330. [Crossref]
• [33] S. Vanduffel, A. Ahcan, L. Henrard, and M. Maj (2012). An explicit option-based strategy that outperforms Dollar cost averaging. Int. J. Theor. Appl. Finance, 15(2), 1250013. [Crossref]
• [34] H. Witting (1985). Mathematische Statistik I: Parametrische Verfahren bei festem Stichprobenumfang. B. G. Teubner, Stuttgart..
• [35] V. Wolf (2014). Comparison of Markovian Price Processes and Optimality of Payoffs. PhD thesis, University of Freiburg i. Br..
• [36] L. Yao, G. Yang, and X. Yang (2011). A note the mean correction martingale measure for geometric Lévy processes. Appl. Math. Lett., 24(5), 593–597. [WoS][Crossref]

Identyfikatory

DOI

10.1515/demo-2015-0001