Joint Calibration to SPX and VIX Derivative Markets with Composite Change of Time Models

By the definition from CBOE, Volatility Index (VIX), as an indicator of implied volatility in the following $30$ days, is given as

where $\tau=30/365$ and $\mathbb{Q}$ is the risk neutral measure of the equity market. If continuity of the asset price process is assumed, then equivalently

where $v^{\text{VIX}}$ is the squared volatility of $S$. Even though the asset price is not continuous, formula (1) only leads to a third-order error $O\left(\left(\frac{dS_{t}}{S_{t-}}\right)^{3}\right)$, as shown in Carr and Wu (2009). Hence the analysis below is still effective for general jump models to a large degree.

Likewise, we have the formula of VVIX, the volatility of volatility computed from VIX market:

where $v^{\text{VVIX}}$ is the variance of $\ln\text{VIX}^{2}$. It is important to note that the first equality holds if we believe that measure $\mathbb{Q}$ is risk-neutral in both SPX option market and the VIX market. That is, the two markets can be consistently modeled.

The problem of joint calibration for SPX market and VIX market is equivalent to (or at least incorporates) the calibration of current VIX and VVIX, which can be approximated by $\sqrt{v^{\text{VIX}}}$ and $\sqrt{v^{\text{VVIX}}}/2$, volatility and volatility of volatility respectively, if we consider a Markov setup and that $\tau$ is small. More generally, we have $\text{VIX}_{t}^{2}=g^{M}_{1}(v^{\text{VIX}}_{t})$ and $\text{VVIX}_{t}^{2}=g^{M}_{2}(v^{\text{VVIX}}_{t})$ by equation (1) and (2), where $g_{1}^{M}(\cdot),g_{2}^{M}(\cdot)$ are functions determined by model parameters. Therefore, it is crucial to study the relationship of $v^{\text{VIX}}$ and $v^{\text{VVIX}}$ in the problem of consistent modeling.

In the past literature on consistent modeling, there have been a lot of research on such volatility relationship. One line of work is aimed at reconstructing the widely used stochastic volatility models (SVMs) by allowing for more realistic and flexible vol-of-vol functionals. One such example is the 3/2 model in Drimus (2012) and in Baldeaux and Badran (2014). Fouque and Saporito (2018) proposed a Heston vol-of-vol model, where the vol-of-vol consists of additional stochastic factors. Additionally, following the work of Gatheral et al. (2018), authors including Bayer et al. (2016), Jacquier et al. (2018) and Gatheral et al. (2020) proposed rough volatility models in the joint calibration of stock and volatility smiles. And according to Lin and Chang (2010) and Kokholm and Stisen (2015), the role of jumps were studied and highlighted in the consistent modeling. Moreover, Papanicolaou (2022) studied the consistency condition of recovering SVMs from market models of the VIX futures term structure. Another category of models suggest a multi-factor specification of volatility. The first attempt was made by Gatheral (2008) and Bayer et al. (2013), who adopted a continuous diffusion model with double mean reverting structure. Multifactor affine specification was considered in Cont and Kokholm (2013), Cheng (2019) and Pacati et al. (2018). And Papanicolaou and Sircar (2014) considered a regime-switching Heston model, where sharp volatility regime shifts captures both volatility skews. Yuan (2020) and Bardgett et al. (2019) adopted multi-factor stochastic jump intensity models. A theoretical drawback of their models is that the variance process in their models can be negative. Finally, there is some recent research that characterizes the volatility relationship using a non-parametric framework. Guo et al. (2022) introduced a time-continuous formulation of the joint calibration problem, followed by Guyon (2020), who built a non-parametric discrete-time model that achieved exact joint calibration.

While many models above achieves satisfactory consistent modeling results, our approach, apart from having good joint calibration performance, can theoretically decoupling smiles from the two markets. Such nice interpretation of our model is achieved via time change technique by representing the vol-of-vol $v^{\text{VVIX}}$ in a linear mixture form of $v^{\text{VIX}}$ and another free factor. We present the following examples to show how restrictive or implicit the volatility relationship is in certain SVMs.

To study the volatility relationship in composite time change (CTC) models, we first introduce some background knowledge on time change approaches. Time changes can be interpreted as the intensity of business activities that drives the variation in volatility of an asset. The original clock ${t,t\geq 0}$ is referred to as calendar time, and the time change process $U$ is called business time. The log-price process of an asset is originally thought to be stationary and of independent increment, i.e., a Lévy process $L$. Every time a market event happens and drives the variation in volatility, the change is reflected in the time change, either via accelerating or slowing the business clock. The real market price of the asset is then updated under the business clock, namely $L_{U}$.

Originally, Clark (1973) and Geman et al. (2001) proposed a subordinated Brownian motion model for log price. The time change introduces jumps in volatility. Carr et al. (2003) and Carr and Wu (2004) introduced time-changed Lévy models, where the time change is absolutely continuous, through which stochastic volatility is introduced. Luciano and Schoutens (2006), Luciano and Semeraro (2007) and Eberlein and Madan (2009) modeled the dependence of multi-assets via correlation of subordinated Brownian motions. Mendoza-Arriaga et al. (2010) extended the time-changed Lévy model by considering the combination of time change and a certain type of composite time change. Recently, Ballotta and Rayée (2022) established a unified TCLM structure that allows leverage via diffusion as well as jumps.

However, like SVMs, TCMs with specifications shown in Carr and Wu (2004) are restricted by the relationship $v^{\text{VVIX}}_{t}=f(v^{\text{VIX}}_{t})$ for a model-determined function $f$. When extended to composite time change, the model may generate the form of vol-of-vol as:

where $v^{I}$ is the idiosyncratic component of $v^{\text{VVIX}}$ and $a,b$ are determined by model parameters, typically considered steady in short periods. This linear combination satisfies both the need for the dependence of VVIX on VIX and the flexibility of VVIX. $v^{\text{VIX}}$ is calibrated to the SPX market, while a free factor $v^{I}$ is calibrated to the VIX market.

Here we assume time changes to be absolutely continuous and the base Lévy process to be a standard Brownian motion. Specifically, $dU_{t}=u_{t}dt$ and $dV_{t}=v_{t}dt,$ where $u$ and $v$ are two independent Itô processes. Then the instantaneous variance of the model is $v_{t}^{\text{VIX}}=\frac{dU_{V_{t}}}{dt}=u_{V_{t}}v_{t}.$ Then the product rule

gives

Equation (4) results in different linear mixture forms according to the specification of activity rate process. And the ideal form of equation (3) can be achieved by the composite 3/2 model proposed in section 3.2. In addition, the model parameters in time change $U$ and $V$ are naturally separated and linearly combined in form.

The article is organized as follows. In section2, we summarize the theory of time-changed Lévy models developed by Carr and Wu (2004) and Ballotta and Rayée (2022) and we show how the technique of leverage neutral measure change helps form the characteristic function; section 3 develops the theory of composite time change models, where a general form is considered and characteristic fuctions derived. We also discusses some useful specifications of CTC models. In Section 4, we introduce the application of the model in derivative pricing, including European options and VIX options. We show that the European option pricing in CTC models can be conducted quite efficiently. In section 5, we perform real-market joint calibration and discusses the results. And the last section concludes.

A subordinator $J=\{J^{U},t\geq 0\}$ is a Lévy process such that $t\mapsto J^{U}$ is non-decreasing.

Leverage neutral measure, first introduced in Carr and Wu (2004), is a complex-valued measure change technique that enables the explicit computation of the characteristic function (characteristic function) of time-changed Lévy process $L_{U}$, especially when $L$ and $U$ are not independent.

Assume that time change $U$ is of type 1. Denote by $X=L_{U}$. If $U$ is independent of $L$, then the characteristic function of $X_{t}$ is

Otherwise, it is shown (Appendix A) that

where $\phi_{U}(\cdot;t)$ is the characteristic function of $U_{t}$, $\mathrm{E}[\cdot]$ and $\mathrm{E}^{\mathbb{Q}}[\cdot]$ denote expectations under measures $\mathbb{P}$ and $\mathbb{Q}$, respectively. The new class of complex-valued measures $\mathbb{Q}(m)$ is absolutely continuous with respect to $\mathbb{P}$ and is defined by

with

As a result, the characteristic exponent of $L$ under the leverage neutral measure is given by

for any well-defined $\Psi_{L}^{\mathbb{Q}}(z)$. This property is useful for deriving the dynamic of $U$ under $\mathbb{Q}(m)$.

Extra condition for $U$ is required to characterize the time-changed process $X$. Specifically, if $U$ is of type 2, then semimartingale $X$ has local characteristics

where $(\alpha,\sigma^{2},\nu(dx))$ is the Lévy characteristic of $L$ (see Küchler and Sorensen (2006)).

Under the setup above, a risk-neutral CTC model is specified as follows:

where $W,Z,B$ are $(\mathbb{P},\mathcal{F})$-Brownian motions, $J$ is a pure-jump $(\mathbb{P},\mathcal{F})$-Lévy process, $J^{U},J^{V}$ are $(\mathbb{P},\mathcal{F})$-subordinators and $\Psi(m)=-\frac{m^{2}}{2}\sigma^{2}+\Psi_{J}\left(\eta m\right)$. In addition, $\alpha^{i}(\cdot),\beta^{i}(\cdot),\gamma^{i}(\cdot),i=U,V$ are unspecified functions. Finally, we require $\gamma^{i}(\cdot)\geq 0,\;i=U,V$ to guarantee the positivity of $u$ and $v$.

To introduce leverage effect, we assume that $[W,Z]_{t}=\rho_{U}t$, $[W,B]_{t}=\rho_{V}t$ but $Z$ and $B$ are independent. We also assume that $J$ and $J^{U}$ have joint distribution $F^{U}$, $J$ and $J^{V}$ have joint distribution $F^{V}$. $J^{U}$ and $J^{V}$ are independent.

Under the leverage-neutral measure $\mathbb{Q}(m)$ defined by Equation (5), we have the following result of the characteristic function of $X$.

Next, we derive the dynamics of $U$ and $V$ under the leverage-neutral measure $\mathbb{Q}$. By Equation (6), the characteristic exponent of $L$ under measure $\mathbb{Q}$ is given by

By the independence of Brownian motions and jump processes, the characteristic exponent of $W$ under measure $\mathbb{Q}$ is

Therefore $W^{\mathbb{Q}}(t):=W(t)-im\sigma t$ is a $(\mathbb{Q},\mathcal{F})$-Brownian motion. And by the correlation $[W,Z]_{t}=\rho_{U}t$, $Z^{\mathbb{Q}}(t):=Z(t)-im\rho_{U}\sigma t$ is a $(\mathbb{Q},\mathcal{F})$-Brownian motion. $B^{\mathbb{Q}}$ is defined likewise.

Next we show the characteristic function of $J^{U}$ and $J^{V}$. The single joint distribution at any $t>0$ determines the joint distribution of $J(t)$ and $J^{i}(t),\;i=U,V$ at all time $t>0$. Under the leverage neutral measure, the characteristic function of $J^{V}$ becomes

where $\Psi_{J,J^{V}}(m,z)$ is the characteristic exponent of $mJ+zJ^{V}$. It follows that

$\square$

As a computable example, we consider imposing an affine structure on time changes. We denote by

and $\phi_{U}^{\mathbb{Q}}(-i\Psi_{L}(m);t)$ for short if there is no confusion.

In this specification, we have by equation (4) that

a linear combination of factors as in equation (3).

We clearly see a separation of effects, including the effect of VIX and the idiosyncratic component. The SPX market calibrates $v_{t}^{\text{VIX}}$ and implies a general relationship of model parameters $\sigma_{V}$ and $\sigma_{T}$, while the VIX market calibrates $v$ and determines the model parameters. In other specifications likewise, we also obtain a linear combination of factors.

To derive the characteristic function under composite 3/2, we note that, according to Carr and Sun (2007), the Laplace transform of $V$ has

where

and $M(\alpha,\gamma,z)$ is the confluent hypergeometric function, defined as

and

And if follows that

where

Despite its nice interpretation, the composite 3/2 can be time-consuming in pricing, particularly in VIX pricing. A modified version of composite 3/2 model is constructed:

That is, the second time change $V$ is substituted by a CIR process. It’s shown in the section of VIX pricing that such formation is efficient in the method of exact simulation.