14:15
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.