Granular and Star-Shaped Price Systems
- 1 Accademia Nazionale Virgiliana and Università Bocconi, Mantova and Milan, Italy
- 2 Università degli Studi di Parma Dipartimento di Economia, Parma, Italy
- 3 Università degli Studi di Parma Dipartimento di Economia, Parma, Italy
- 4 Università degli Studi di Parma Dipartimento di Economia, Parma, Italy
Abstract
Linear price systems, typically used to model “perfect” markets, are widely known not to accommodate most of the typical frictions featured in “actual” ones. Since some years, “proportional” frictions (taxes, bid-ask spreads, and so on) are modeled by means of sublinear price functionals, which proved to give a more “realistic” description. In this paper, we want to introduce two more classes of functionals, not yet widely used in Mathematical Finance, which provide a further improvement and an even closer adherence to actual markets, namely the class of granular functionals, obtained when the unit prices of traded assets are increasing w.r.t. the traded amount; and the class of star-shaped functionals, obtained when the average unit prices of traded assets are increasing w.r.t. the traded amount. A characterisation of such functionals, together with their relationships with arbitrages and other (more significant) market inefficiencies, is explored.
- Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, 9, 203-228. http://dx.doi.org/10.1111/1467-9965.00068
- Björk, T. (1999). Arbitrage Theory in Continuous Time. Oxford: Oxford University Press.
- Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-654. http://dx.doi.org/10.1086/260062
- Castagnoli, E., Favero, G., & Maccheroni, F. (2009). Marchetto, il prezzatore perfetto. Elogio dell'internalità. Milan: Electronic Draft.
- Castagnoli, E., Favero, G., & Modesti, P. (in print). Price Systems for Random Amounts: A Unified Approach. In D. Jakóbczak (Ed.), Analyzing Risk through Probabilistic Modeling in Operations Research. Hershey: IGI Global.
- Castagnoli, E., Favero, G., & Tebaldi, C. (2011). One-Penny Arbitrages, or: A Free Snack without a Free Lunch. Journal of Applied Computer Science & Mathematics, 10, 20-21.
- Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., & Montrucchio, F. (2011). Risk Measures: Rationality and Diversification. Mathematical Finance, 21, 743-774.
- Chateauneuf, A., & Aouani, Z. (2008). Exact Capacities and Star-Shaped Distorted Probabilities. Mathematical Social Sciences, 56, 185-194. http://dx.doi.org/10.1016/j.mathsocsci.2008.01.006
- Cvitanic, J., Pham, H., & Touzi, N. (1999). A Closed-Form Solution for the Problem of Super-Replication under Transaction Costs. Finance and Stochastic, 3, 35-54. http://dx.doi.org/10.1007/s007800050051
- Davis, M. H. A., & Clark, J. M. C. (1994). A Note on Super-Replicating Strategies. Philosophical Transactions: Physical Sciences and Engineering, 347, 485-494.
- de Finetti, B., & Obry, S. (1933). L’optimum nella misura del riscatto. Atti del Secondo Congresso Nazionale di Scienza delle Assicurazioni, 2, 99-123.
- Delbaen, F., & Schachermayer, W. (1994). A General Version of the Fundamental Theorem of Asset Pricing. Mathematische Annalen, 300, 463-520. http://dx.doi.org/10.1007/BF01450498
- Dothan, M. U. (1990). Prices in Financial Markets. Oxford: Oxford University Press.
- El Karoui, N., & Quenez, M. C. (1995). Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market. SIAM Journal on Control and Optimization, 33, 29-66. http://dx.doi.org/10.1137/S0363012992232579