Quantum Bounds for Option Prices

TL;DR

This paper develops a mathematical framework using operator algebras to establish bounds on option prices, providing a unified approach to various complex derivatives and market constraints.

Contribution

It introduces a novel operator algebra approach to derive bounds on option prices, extending traditional methods to a more general and geometric setting.

Findings

Derived upper bounds for basket option prices

Created converging families of bounds for vanilla options

Interpolated volatility smile and analyzed cross FX rate options

Abstract

Option pricing is the most elemental challenge of mathematical finance. Knowledge of the prices of options at every strike is equivalent to knowing the entire pricing distribution for a security, as derivatives contingent on the security can be replicated using options. The available data may be insufficient to determine this distribution precisely, however, and the question arises: What are the bounds for the option price at a specified strike, given the market-implied constraints? Positivity of the price map imposed by the principle of no-arbitrage is here utilised, via the Gelfand-Naimark-Segal construction, to transform the problem into the domain of operator algebras. Optimisation in this larger context is essentially geometric, and the outcome is simultaneously super-optimal for all commutative subalgebras. This generates an upper bound for the price of a basket option. With innovative decomposition of the assets in the basket, the result is used to create converging families of price bounds for vanilla options, interpolate the volatility smile, price options on cross FX rates, and analyse the relationships between swaption and caplet prices.