Price Operators: Extensions, Potentials, and the Markov Valuation of Securities

This paper provides properties of price operators, functions that map the payoff of a contingent claim to its market value as a function of the state of the economy. The principal objective is the construction of a Markov process under which the market value of a security is the expected sum of future dividends of the security, where dividends are a function of the Markov process. First we provide conditions for a norm-preserving arbitrage-free (positive) extension of an arbitrage-free price operator from the space of actually marketed assets to the space of all possible assets. This can be useful for the characterization of equilibrium in settings of asymmetric information or sequential trade. Then we show, in the multiperiod setting, that the market value of a security may be treated as the Markov potential of its dividend, and show several properties that derive from this characterization. Finally, we demonstrate the existence of an invariant measure for the corresponding Markov process under which the mean current value of any security is its discounted mean future payoff. The fixed discount factor is the spectral radius of the valuation operator, the largest possible fixed discount rate on any security.