We propose a notion of market completeness which is invariant under
change to an equivalent probability measure. Completeness means that an
operator T acting on stopping time simple trading strategies has dense
range in the weak* topology on bounded random variables. In our setup, the
claims which can be approximated by attainable ones has codimension equal to
the dimension of the kernel of the adjoint operator $T*$ acting on signed
measures, which in most cases is equal to the "dimension of the space of
martingale measures." From this viewpoint the example of Artzner and Heath
is no longer paradoxical since all the dimensions are 1. We also illustrate how
one can check for injectivity of $T*$ and hence for completeness in the case of
price processes on a Brownian filtration (e.g., Black-Scholes,
Heath-Jarrow-Morton) and price processes driven by a multivariate point
process.
References
ANSEL, J.-P. and STRICKER, C. 1992 . Lois de martingales, densities, et decomposition de ´ ´ Follmer Schweizer, Ann. Inst. H. Poincare Probab. Statist. 28 375 392. ´ Z .
ANSEL, J.-P. and STRICKER, C. 1994 . Couverture des actifs contingents et prix maximum. Ann. Inst. H. Poincare Probab. Statist. 30 303 315. ´ Z .
ARTZNER, P. and HEATH, D. 1995 . Approximate completeness with multiple martingale measures. Math. Finance 5 1 11. Z .
BJORK, T., DI MASI, G., KABANOV, Y. and RUNGGALDIER, W. 1997 . Towards a general theory of ¨ bond markets. Finance and Stochastics. To appear. Z .
BJORK, T., KABANOV, Y. and RUNGGALDIER, W. 1996 . Bond market structure in the presence of ¨ marked point processes. Preprint. Z .
DALANG, R. C., MORTON, A. and WILLINGER, W. 1990 . Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stochastics Stochastic Rep. 29 185 201. Z .
DELBAEN, F. and SCHACHERMAy ER, W. 1994 . A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463 520. Z .
DELLACHERIE, C. and MEy ER, P.-A. 1975 . Probabilites et Potentiel, 1 3. Hermann, Paris. ´ Z .
GROTHENDIECK, A. 1973 . Topological Vector Spaces. Gordon and Breanch, Montreux.
HARRISON, J. M. and KREPS, D. M. 1979 . Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20 381 408. Z .
HARRISON, J. M. and PLISKA, S. R. 1981 . Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11 215 260. Z .
HE, S.-W., WANG, J.-G. and YAN, J.-A. 1992 . Semimartingale Theory and Stochastic Calculus. Science Press, Beijing. Z .
HEATH, D., JARROW, R. and MORTON, A. 1992 . Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60 77 105. Z .
JACOD, J. 1975 . Multivariate point processes: predictable projection, Radon Nikody m derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235 253. Z .
JACOD, J. 1979 . Calcul Stochastique et Problemes de Martingales. Lecture Notes in Math. 714. Springer, Berlin. Z .
JACOD, J. 1980 . Integrales stochastiques par rapport a une semimartingale vectorielle et changments de filtration. Seminaire de Probabilite XIV. Lecture Notes in Math. 784 ´ ´ 161 172. Springer, Berlin. Z .
JACOD, J. and SHIRy AEV, A. N. 1987 . Limit Theorems for Stochastic Processes. Springer, Berlin. Z .
JARROW, R. A. and MADAN, D. B. 1997 . Hedging contingent claims on semimartingales. Finance and Stochastics. To appear. Z . JEANBLANC-PICQUE, M. and PONTIER, M. 1990 . Optimal portfolio for a small investor in a ´ market model with discontinuous prices. Appl. Math. Optim. 22 287 310. Z .
LAKNER, P. 1993 . Martingale measures for a class of right-continuous processes. Math. Finance 3 43 53. Z .
MERCURIO, F. and RUNGGALDIER, W. J. 1993 . Option pricing for jump diffusions: approximations and their interpretation. Math. Finance 3 191 200. Z .
MERTON, R. C. 1976 . Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3 125 144. Z .
REVUZ, D. and YOR, M. 1991 . Continuous Martingales and Brownian Motion. Springer, Berlin. Z .
RUDIN, W. 1991 . Functional Analy sis, 2nd ed. McGraw-Hill, New York. Z .
SCHACHERMAy ER, W. 1994 . Martingale measures for discrete-time processes with infinite horizon. Math. Finance 4 25 55. Z .
TAQQU, M. S. and WILLINGER, W. 1987 . The analysis of finite security markets using martingales. Adv. Appl. Probab. 19 1 25.
