Is (independent) subordination relevant in option pricing?

TL;DR

This paper investigates whether local semimartingales can be represented as time-changed Brownian motions with independent time-changes, finding that such independence is generally incompatible with market data and calibration performance.

Contribution

It proves that local semimartingales cannot be represented as independent time-changed BMs and introduces the additive normal tempered stable process as a superior calibration model.

Findings

Independent additive subordination performs poorly in calibration.

ATS models accurately fit equity volatility surfaces.

Market data contradicts the independence assumption in subordination.

Abstract

Monroe (1978) demonstrates that any local semimartingale can be represented as a time-changed Brownian Motion (BM). A natural question arises: does this representation theorem hold when the BM and the time-change are independent? We prove that a local semimartingale is not equivalent to a BM with a time-change that is independent from the BM. Our result is obtained utilizing a class of additive processes: the additive normal tempered stable (ATS). This class of processes exhibits an exceptional ability to accurately calibrate the equity volatility surface. We notice that the sub-class of additive processes that can be obtained with an independent additive subordination is incompatible with market data and shows significantly worse calibration performances than the ATS, especially on short time maturities. These results have been observed every business day in a semester on a dataset of S&P 500 and EURO STOXX 50 options.