Semi‐efficient valuations and put‐call parity

Abstract

We propose an approach to the valuation of payoffs in general semimartingale models of financial markets where prices are nonnegative. Each asset price can hit 0; we only exclude that this ever happens simultaneously for all assets. 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). A valuation process for a payoff is then called semi-efficient 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 and valuing 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 three terms in general. We also show that our approach recovers all the put-call parity respecting valuation formulas in the classic theory as special cases, and we explain when and how the different terms in the put and call valuation formulas disappear or simplify. Along the way, we also define and characterize completeness for general semimartingale financial markets and connect this to the classic theory. K E Y W O R D Sabsence of arbitrage, completeness, consistent valuation, efficiency, incomplete markets, maximal strategies, NFLVR, NUPBR, option valuation, put-call parity, risk-neutral valuation, semi-efficient markets, viability Mathematical Finance. 2018;28:1061-1106.wileyonlinelibrary.com/journal/mafi

On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model

“…The prices of the related put options could be computed via the call-put parity. For details, see Eberlein et al (2008) and Herdegen and Schweizer (2018).…”

“…The obtained formulas depend on the values of special mathematical functions but can be computed over 0.5 s on modern software. The prices of forward-start put options can be calculated by exploiting the duality principle (Eberlein et al 2008;Herdegen and Schweizer 2018). Future research may relate to the computation of the prices of exotic stock options and options on bonds in the discussed models.…”

Pricing without no-arbitrage condition in discrete time

“…We say that p(Z) is instantaneous profit-free, in the spirit of [13,Definition 1.30] where the concept was introduced for the no-arbitrage condition NA and the super-replication price. This problem naturally arises in the models under NA but also, more generally, for other types of no-arbitrage condition, as in [33].…”

Financial models with defaultable numéraires

“…Thus, retrieves exactly the valuation formulas in Lewis (), Madan and Yor (), Paulot (), and Kardaras (). We also refer to section 6 in Herdegen and Schweizer () for an alternative approach based on well‐chosen no‐arbitrage principles.…”

“…Thus, the valuation formulas of Lewis (), Madan and Yor (), Paulot (), or Kardaras () arise as special cases of this paper's framework. Parallel to this paper, Herdegen and Schweizer () have developed an alternative consistent and general valuation framework that always guarantees put–call parity.…”