Abstract:
This paper derives a unified framework for portfolio optimization, derivative pricing,
financial modeling, and risk measurement. It is based on the natural assumption that
investors prefer more rather than less, in the sense that given two portfolios with the
same diffusion coefficient value, the one with the higher drift is preferred. Each such
investor is shown to hold an efficient portfolio in the sense of Markowitz with units
in the market portfolio and the savings account. The market portfolio of investable
wealth is shown to equal a combination of the growth optimal portfolio (GOP) and
the savings account. Tnthis setup the capital asset pricing model follows without the use
of expected utility functions, Markovianity, or equilibrium assumptions. The expected
increase of the discounted value of the GOP is shown to coincide with the expected
increase of its discounted underlying value. The discounted GOP has the dynamics of
a time transformed squared Bessel process of dimension four. The time transformation
is given by the discounted underlying value of the GOP. The squared volatility of the
GOP equals the discounted GOP drift, when expressed in units of the discounted GOP.
Risk-neutral derivative pricing and actuarial pricing are generalized by the fair pricing
concept, which uses the GOP as numeraire and the real-world probability measure as
pricing measure. An equivalent risk-neutral martingale measure does not exist under
the derived minimal market model.
