Economically Consistent Valuations and Put-Call Parity

We propose an approach to the valuation of contingent claims in general, symmetric semimartingale models of financial markets. We start from two simple, economically motivated axioms, namely absence of arbitrage (in the sense of NUPBR) and absence of relative arbitrage among all buy-and-hold strategies (called static efficiency). We then call a valuation process for a contingent claim economically consistent if the financial market enlarged by that process still satisfies this combination of properties. It turns out that this approach lies in the middle between the extremes of valuing by risk-neutral expectation or by absence of arbitrage alone. We show that this always yields put-call parity, although put and call values themselves can be nonunique, even for complete markets. We provide general formulas for put and call values in complete markets and show that these are symmetric and that both contain in general three terms. We also show that our approach contains all the put-call parity respecting valuation formulas in the classic theory as special cases, and we explain precisely when and how the different terms in the put and call valuation formulas disappear or simplify.