Arbitrage-Free Pricing Before and Beyond Probabilities

Financial models with defaultable numéraires

“…In the case when the contingent claim C is a European call (with the first currency chosen as numéraire), this second term corresponds exactly to the ad hoc correction in Lewis (). 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.…”

“…This point of view then enables us to interpret the correction term in the valuation formula of Lewis () as the value of the contingent claim's payoff in the scenarios where the numéraire devalues. 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.…”

“…Often such models are chosen because they can be interpreted as bubbles (Protter, ) or they are analytically tractable (Carr, Fisher, & Ruf, ; Hulley & Platen, )). Both practitioners (Lewis, ; Paulot, ) and academics (Cox & Hobson, ; Madan & Yor, ) suggest nonclassical valuation formulas for contingent claims in such models in order to be consistent with market prices. In this paper, we argue that strict local martingale dynamics are consistent with the interpretation that the corresponding numéraire devalues.…”

“…We call aggregation the reverse step, namely, taking a possibly nonequivalent family of probability measures, corresponding to the different currencies as numéraires, and constructing a valuation measure for the basket. Embedding a strict local martingale model in a family of numéraire‐consistent probability measures and then aggregating this family yields the nonclassical valuation formulas of Lewis (), Madan and Yor (), Paulot (), and Carr et al. ().…”

Asymptotics of the Time-Discretized Log-Normal Sabr Model: The Implied Volatility Surface

“…Short maturity asymptotics for the asset price distribution, option prices and implied volatility were first derived at leading order by Hagan et al [27]. The expansion was extended to higher order in [28] by Paulot [43] and Lewis [38]. The asymptotics were studied using operator expansion methods in [7].…”