The emergence of probability-type properties of price paths
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Fri, 28/10/2011 14:15 |
Vladmir Vovk (Royal Holloway University of London) |
Nomura Seminar  |
DH 1st floor SR |
| The standard approach to continuous-time finance starts from postulating a statistical model for the prices of securities (such as the Black-Scholes model). Since such models are often difficult to justify, it is interesting to explore what can be done without any stochastic assumptions. There are quite a few results of this kind (starting from Cover 1991 and Hobson 1998), but in this talk I will discuss probability-type properties emerging without a statistical model. I will only consider the simplest case of one security, and instead of stochastic assumptions will make some analytic assumptions. If the price path is known to be cadlag without huge jumps, its quadratic variation exists unless a predefined trading strategy earns infinite capital without risking more than one monetary unit. This makes it possible to apply the known results of Ito calculus without probability (Follmer 1981, Norvaisa) in the context of idealized financial markets. If, moreover, the price path is known to be continuous, it becomes Brownian motion when physical time is replaced by quadratic variation; this is a probability-free version of the Dubins-Schwarz theorem. | |||