In this opening section, we provide a general introduction to the class of subordinated market models; we also present the key points investigated in the paper, as well as the work’s overall structure.

#### 1.1. Time Subordination in Financial Modelling

Among the most striking patterns that are observable in financial time series are the phenomena of regime switching, clustering, and long memory or autocorrelation (see e.g.,

Cont (

2007) and references therein). Such stylized facts have been evidenced for several decades, Mandelbrot famously remarking in

Mandelbrot (

1963) that large price changes tend to cluster together (“

large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes”), thus creating periods of market turbulence (high volatility) alternating with periods of relative calm (low volatility). These empirical observations can be described, among other approaches, by agent based models focusing on economic interpretation, such as

Lux and Marchesi (

2000);

Niu and Wang (

2013), by tools from statistical mechanics and econophysics (

Krawiecki et al. 2002), or by the introduction of multifractals (

Calvet and Fischer 2008).

Another prominent approach to describe this subtle volatility behavior consists of introducing a time change in the stochastic process driving the market prices. Besides stochastic volatility, time changed market models also capture several stylized facts, like non-Normality of returns (the presence of jumps, asymmetry) and negative correlation between the returns and their volatility (see a complete overview in

Carr and Wu (

2004)). They are motivated by the observation that market participants do not operate uniformly through a trading period, but, on the contrary, the volume, and frequency of transactions greatly vary over time. Following the terminology of

Geman (

2009), the time process is called the stochastic clock, or business time, while the stochastic process for the underlying market (a Brownian motion, or a more general Lévy process) is said to evolve in operational time.

Historically, the first introduction of a time change in a diffusion process goes back to

Bochner (

1949); it was first applied to financial modeling in

Clark (

1973) in the context of the cotton futures market and for a continuous-time change. During the late 1990s and early 2000s, the approach was extended to discontinuous time changes, with the introduction of subordinators (i.e., non-negative Lévy processes, see the theoretical details in

Bertoin (

1999)). In other words, the business time now admits increasing staircase-like realizations, describing peak periods of activity (following, for instance, earning announcements, central bank reports, or major political events) alternating with less busy periods. Perhaps the best-known subordinators are the Gamma process, like in the Variance Gamma (VG) model by

Madan et al. (

1998), and the inverse Gaussian process, like in the Normal inverse Gaussian (NIG) model by

Barndorff-Nielsen (

1997). Let us also mention that subordination has been successfully applied in many other fields of applied science. For instance, Gamma subordination has been employed for modeling the deterioration of production equipment in order to optimize their maintenance (see

de Jonge et al. (

2017) and references therein), and inverse Gaussian subordination was originally introduced in

Barndorff-Nielsen (

1977) to model the influence of wind on dunes and beach sands.

Recently, a new type of time subordination, based on fractional calculus, has emerged. Indeed, Lévy processes are closely related to fractional calculus because, for many of them (including stable and tempered stable processes), their probability densities satisfy a space fractional diffusion equation (see details and applications to option pricing in

Cartea and del-Castillo-Negrete (

2007) and in

Luchko et al. (

2019)). By also allowing the time derivative to be fractional, as, e.g., in

Jizba et al. (

2018);

Kleinert and Korbel (

2016);

Korbel and Luchko (

2016);

Tarasov (

2019);

Tomovski et al. (

2020), it provides a new type of subordinated models: while the order of the space fractional derivative controls the heavy tail behavior of the distribution of returns, the order of the time fractional derivative acts as a temporal subordination parameter whose purpose is to capture time-related phenomena, such as temporal risk redistribution. This model, which we shall refer to as the fractional diffusion (FD) model, is an alternative to time-change models, or to subordinated random walks (

Gorenflo et al. 2006).

Regarding the practical implementation and valuation of financial derivatives within subordinated market models, the literature is dominated by numerical techniques. In time changed models notably, tools from Fourier transform (

Lewis 2001) or Fast Fourier transform (

Carr and Madan 1999), and their many refinements, such as the COS method by

Fang and Osterlee (

2008) or the PROJ method by

Kirkby (

2015). These methods are popular, notably because such models’ characteristic functions are known in relatively simple closed-forms. Similarly,

Cui et al. (

2019) provides a numerical pricing framework for a general time changed Markov processes, and

Li and Linetsky (

2014) employs eigenfunction expansion techniques. However, recently, closed-form pricing formulas have been derived, for the VG model in

Aguilar (

2020a) and for the NIG model in

Aguilar (

2020b). The technique has also been employed in the FD model, for vanilla payoffs in

Aguilar et al. (

2018) and for more exotic options in

Aguilar (

2020c).

In this paper, we extend these pricing tools to the calculation of risk sensitivities and to profit-and-loss (P&L) explanation, and we provide comparisons between time changed models (such as the NIG and the VG models) and the FD model. Like for the pricing case, risk sensitivities in the context of time changed market models (and of Lévy market models in general) are traditionally evaluated by means of numerical methods based on Fourier inversion (

Eberlein et al. 2010;

Takahashi and Yamazaki 2008); in the present paper, we will therefore show that they can be expressed in a tractable way, under the form of fast convergent series whose terms explicitly depend on the model parameters. This will allow for us to construct and compare the performance of option based portfolios, and discuss, both quantitatively and qualitatively, the impact on the parameters on risks and P&L. Related topics, such as volatility modeling, will also be discussed.