About the decomposition of pricing formulas under stochastic volatility models

TL;DR

This paper extends the decomposition of call option prices to general stochastic volatility models, including non-exponential cases, using non-anticipative techniques and functional calculus, and derives formulas for implied volatility derivatives.

Contribution

It generalizes existing decompositions to broader models and introduces new terms for non-exponential stock price dynamics, employing advanced stochastic calculus methods.

Findings

Decomposition of call prices in general stochastic volatility models.

New terms appear when stock prices are not exponential.

Derived formulas for implied volatility derivatives.

Abstract

We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model extending the decomposition obtained by E. Al\`os in [2] for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used are non anticipative. In particular, we see also that equivalent results can be obtained using Functional It\^o Calculus. Using the same generalizing ideas we also extend to non exponential models the alternative call option price decompostion formula obtained in [1] and [3] written in terms of the Malliavin derivative of the volatility process. Finally, we give a general expression for the derivative of the implied volatility under both, the anticipative and the non anticipative case.