Options Are Also Options on Options: How to Smile With Black-Scholes

31 Pages Posted: 9 Aug 2023

See all articles by Arianna Mingone

Arianna Mingone

Ecole Polytechnique de Paris (CMAP); Zeliade Systems

Claude Martini

Zeliade Systems

Date Written: August 4, 2023

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.

Keywords: Black-Scholes formula, Pricing function, Option pricing

Suggested Citation

Mingone, Arianna and Martini, Claude, Options Are Also Options on Options: How to Smile With Black-Scholes (August 4, 2023). Available at SSRN: https://ssrn.com/abstract=4534756 or http://dx.doi.org/10.2139/ssrn.4534756

Arianna Mingone (Contact Author)

Ecole Polytechnique de Paris (CMAP) ( email )

Paris
France

Zeliade Systems ( email )

Paris
France

Claude Martini

Zeliade Systems ( email )

Paris
France

HOME PAGE: http://www.zeliade.com

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