Abstract
In this paper we generalize the recent comparison results of El Karoui et al. (Math Finance 8:93–126, 1998), Bellamy and Jeanblanc (Finance Stoch 4:209–222, 2000) and Gushchin and Mordecki (Proc Steklov Inst Math 237:73–113, 2002) to d-dimensional exponential semimartingales. Our main result gives sufficient conditions for the comparison of European options with respect to martingale pricing measures. The comparison is with respect to convex and also with respect to directionally convex functions. Sufficient conditions for these orderings are formulated in terms of the predictable characteristics of the stochastic logarithm of the stock price processes. As examples we discuss the comparison of exponential semimartingales to multivariate diffusion processes, to stochastic volatility models, to Lévy processes, and to diffusions with jumps. We obtain extensions of several recent results on nontrivial price intervals. A crucial property in this approach is the propagation of convexity property. We develop a new approach to establish this property for several further examples of univariate and multivariate processes.
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References
Barndorff-Nielsen O.E. (1977) Exponentially decreasing distributions for the logarithm of particle size. Proc R Soc Lond Ser A 353, 401–419
Bellamy N., Jeanblanc M. (2000) Incompleteness of markets driven by a mixed diffusion. Finance Stoch 4, 209–222
Bergman Y.Z., Grundy B.D., Wiener Z. (1996) General properties of option prices. J Finance 51, 1573–1610
Chan T. (1999) Pricing contingent claims on stocks driven by Lévy processes. Ann Appl Probab 9, 504–528
Davis M.H.A. (1997). Option pricing in incomplete markets. In: Dempster M., Pliska S. (eds). Mathematics of derivative securities. Cambridge University Press, Cambridge, pp. 216–226
Delbaen F., Schachermayer W. (1996) The variance optimal martingale measure for continuous processes. Bernoulli 2, 81–105
Eberlein E., Jacod J. (1997) On the range of options prices. Finance Stoch 1, 131–140
El Karoui N., Jeanblanc-Picqué M., Shreve S.E. (1998) Robustness of the Black and Scholes formula. Math Finance 8, 93–126
Frey R. (1997) Derivative asset analysis in models with level-dependent and stochastic volatility. CWI Q 10, 1–34
Frey R., Sin C.A. (1999) Bounds on European option prices under stochastic volatility. Math Finance 9, 97–116
Frittelli M. (2000) The minimal entropy martingale measure and the valuation problem in incomplete markets. Math Finance 10, 39–52
Goll T., Kallsen J. (2000) Optimal portfolios for logarithmic utility. Stoch Proc Appl 89, 31–48
Goll T., Rüschendorf L. (2001) Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Finance Stoch 5, 557–581
Gushchin A.A., Mordecki E. (2002) Bounds on option prices for semimartingale market models. Proc Steklov Inst Math 237, 73–113
Henderson V. (2005) Analytical comparisons of option prices in stochastic volatility models. Math Finance 15, 49–59
Henderson V., Hobson D.G. (2003) Coupling and option price comparisons in a jump-diffusion model. Stoch Stoch Rep 75, 79–101
Henderson V., Hobson D.G., Howison S., Kluge T. (2005) A comparison of q-optimal option prices in a stochastic volatility model with correlation. Rev Derivatives 8, 5–25
Heston S.L. (1993) A closed form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6, 327–343
Hobson D.G. (1998) Volatility misspecification, option pricing and superreplication via coupling. Ann Appl Probab 8, 193–205
Hodges S.D., Neuberger A. (1989) Optimal replication of contingent claims under transaction costs. Rev Futures Mark 8, 222–239
Hofmann N., Platen E., Schweizer M. (1992) Option pricing under incompleteness and stochastic volatility. Math Finance 2, 153–187
Hull D., White A. (1987) The pricing of options on assets with stochastic volatilities. J Finance 42, 281–300
Jacod J., Shiryaev A.N. (2003). Limit theorems for stochastic processes 2nd edn. Springer, Berlin Heidelberg New York
Jakubenas P. (2002) On option pricing in certain incomplete markets. Proc Steklov Inst Math 237, 114–133
Janson S., Tysk J. (2004) Preservation of convexity of solutions to parabolic equations. J Diff Eq 206, 182–226
Kallsen, J. Duality links between portfolio optimization and derivative pricing. Universität Freiburg, Preprint 40 (1998)
Keller, U. Realistic modelling of financial derivatives. Dissertation, Universität Freiburg,http://www.mathfinance.ma.tum.de/personen/kallsen.php1997
Kloeden P., Platen E. (1992) Numerical solution of stochastic differential equations. Springer, Berlin Heidelberg New York
Liu X.Q., Li C.W. (2000) Weak approximations and extrapolations of stochastic differential equations with jumps. SIAM J Numer Anal 37, 1747–1767
Martini C. (1999) Propagation of convexity by Markovian and martingalian semigroups. Potential Anal 10, 133–175
Masuda H. (2004) On multidimensional Ornstein–Uhlenbeck processes driven by a general Lévy process. Bernoulli 10, 97–120
Møller T. (2004) Stochastic orders in dynamic reinsurance markets. Finance Stoch 8, 479–499
Müller A., Stoyan D. (2002) Comparison methods for stochastic models and risks. Wiley, Chichester
Rüschendorf L. (2002) On upper and lower prices in discrete-time models. Proc Steklov Inst Math 237, 134–139
Rüschendorf L., Rachev S.T. (1990) A characterization of random variables with minimum L 2-distance. J Mult Anal 32, 48–54
Sato K.-I. (1999) Lévy processes and infinitely divisible distributions. Studies in advanced mathematics, vol. 68, Cambridge University Press, Cambridge
Schweizer M. (1996) Approximation pricing and the variance-optimal martingale measure. Ann Probab 24, 206–236
Scott L. (1987) Option pricing when the variance changes randomly: theory, estimation and an application. J Financ Quant Anal 22, 419–438
Stein E.M., Stein J.C. (1991) Stock price distributions with stochastic volatility: an analytic . Rev Financ Stud 4, 727–752
Wiggins J.B. (1987) Option valuation under stochastic volatility: theory and empirical estimates. J Financ Econ 19, 351–372
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Bergenthum, J., Rüschendorf, L. Comparison of Option Prices in Semimartingale Models. Finance Stochast. 10, 222–249 (2006). https://doi.org/10.1007/s00780-006-0001-9
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DOI: https://doi.org/10.1007/s00780-006-0001-9