Option prices with call prices

TL;DR

This paper develops new formulas for pricing a broad class of options using call and digital option prices across all strikes, extending existing methods and enabling static hedging in general market models.

Contribution

It introduces a novel pricing formula for general options based on call and digital options, and extends results to barrier options and American options with convex payoffs.

Findings

Derived a new pricing formula for general options.

Established static hedging strategies in broad market models.

Provided bounds and conditions for American and European option equivalence.

Abstract

There exist several methods how more general options can be priced with call prices. In this article, we extend these results to cover a wider class of options and market models. In particular, we introduce a new pricing formula which can be used to price more general options if prices for call options and digital options are known for every strike price. Moreover, we derive similar results for barrier type options. As a consequence, we obtain a static hedging for general options in the general class of models. Our result can be utilised in several significant applications. As a simple example, we derive an upper bound for the value of a general American option with convex payoff and characterise conditions under which the value of this option equals to the value of the corresponding European option.