Model-Free Bounds for Option Prices in Incomplete Markets

Abstract

This paper develops a methodology to quantify the uncertainty about option prices that persists in (statically) incomplete option markets due to finite strike and maturity grids as well as bid-ask price quoting. Uncertainty measures are provided for European option prices and their slopes (i.e., tail probabilities), each derived from model-free bounds that are consistent with a finite sample of observed bid and ask option prices and elementary no-arbitrage constraints in a multi-period setting. We obtain novel explicit expressions for these bounds, which are simple and efficient to compute. Moreover, we suggest a decomposition of the bounds and uncertainty measures into contributions of strike, maturity, and price quote incompleteness. In an empirical analysis of S&P 500 index options, we quantify the empirical uncertainty about option prices. We document sizable uncertainty levels for option prices and tail probabilities. Our findings challenge the common belief that such "option-implied" (tail) information can be measured at sufficiently high precision, which has implications for many applications that use this information.