Options are also options on options: how to smile with Black-Scholes

TL;DR

This paper explores the pricing and properties of Call on Call options, providing new formulas and insights into implied volatility smiles within the Black-Scholes framework.

Contribution

It introduces quasi-closed formulas for relative pricing functions of Call on Call options and analyzes their properties and implications for implied volatility smiles.

Findings

Derived new closed-form formulas for Call on Call pricing functions.

Analyzed the symmetry and moneyness dependence of the relative price functions.

Provided properties and behaviors of implied volatility smiles for these options.

Abstract

We observe that a European Call option with strike $L > K$ can be seen as a Call option with strike $L-K$ on a Call option with strike $K$. Under no arbitrage assumptions, this yields immediately that the prices of the two contracts are the same, in full generality. We study in detail the relative pricing function which gives the price of the Call on Call option as a function of its underlying Call option, and provide quasi-closed formula for those new pricing functions in the Carr-Pelts-Tehranchi family [Carr and Pelts, Duality, Deltas, and Derivatives Pricing, 2015] and [Tehranchi, A Black-Scholes inequality: applications and generalisations, Finance Stoch, 2020] that includes the Black-Scholes model as a particular case. We also study the properties of the function that maps the price normalized by the underlier, viewed as a function of the moneyness, to the normalized relative price, which allows us to produce several new closed formulas. In connection to the symmetry transformation of a smile, we build a lift of the relative pricing function in the case of an underlier that does not vanish. We finally provide some properties of the implied volatility smiles of Calls on Calls and lifted Calls on Calls in the Black-Scholes model.