Bitcoin Volatility and Intrinsic Time Using Double-Subordinated Lévy Processes

Abstract

We propose a doubly subordinated Lévy process, the normal double inverse Gaussian (NDIG), to model the time series properties of the cryptocurrency bitcoin. By using two subordinated processes, NDIG captures both the skew and fat-tailed properties of, as well as the intrinsic time driving, bitcoin returns and gives rise to an arbitrage-free option pricing model. In this framework, we derive two bitcoin volatility measures. The first combines NDIG option pricing with the Chicago Board Options Exchange VIX model to compute an implied volatility; the second uses the volatility of the unit time increment of the NDIG model. Both volatility measures are compared to the volatility based on the historical standard deviation. With appropriate linear scaling, the NDIG process perfectly captures the observed in-sample volatility.

1. Introduction

Cryptocurrencies have emerged as an alternative asset class in their own right. By now, institutional as well as private investors have taken a closer look at this asset class. Many view digital currencies—particularly, the leading cryptocurrency, bitcoin—as an alternative to gold for providing long-term protection against inflation1. In contrast, a major argument against investing in cryptocurrencies is their high and often erratic volatility compared to conventional speculative assets.

Data on bitcoin volatility, in terms of the Bitcoin Volatility Index (BVI), have been available since 2010. The BVI is based on historical volatility; specifically, it is based on the standard deviation of past daily log-returns. Two versions of this index are provided based on 30- and 60-day windows of past observations. The problem with historical volatility is that it is only meaningful if the returns are approximately independent and identically distributed (iid). In the presence of temporal dependence in the form of volatility clustering—a phenomenon common to virtually all speculative assets, including cryptocurrencies—historical volatility fails to adequately capture the current risk of an investment in a speculative asset2. It simply provides an average of the past risk.

Implied volatility measures attempt to tackle this shortcoming by capturing the current market view of the future risk implied by the observed transaction prices of option contracts. By assuming an option pricing method and given the price at which an option has been traded, one can back out the expected volatility of the underlying asset that is implied by the observed price. Several approaches have been developed to price options contracts, including continuum models such as the Black–Scholes–Merton (BSM) formula (Black and Scholes 1973; Merton 1973), as well as discrete models based on binomial or trinomial trees, Monte Carlo simulations, discrete stochastic volatility, and finite-differencing methods.

Alexander and Imeraj (2021) used bitcoin options data from the Deribit exchange to create a term structure for bitcoin implied volatility indices using the same variance swap fair-value formula (geometric variance swap) that the Chicago Board Options Exchange (Cboe) uses to construct the Cboe Volatility Index (VIX) (Chicago Board Options Exchange 2023), which reflects the volatility of the S&P 500 stock index3. Furthermore, Alexander and Imeraj point out that bitcoin prices appear to fluctuate excessively, so this approach tends to underestimate the fair value of geometric variance swaps. In a recent report, Kim et al. (2021) proposed a volatility index based on the cryptocurrency index CRIX4. It proxies the next month’s mean annualized volatility. Venter and Maré (2020) used the symmetric Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) option pricing model to construct various implied volatility indices, including the CRIX index. More recently, an alternative volatility index, BitVol (T3I Pty Ltd. 2019), was introduced. It reflects the expected 30-day implied volatility derived from tradable bitcoin option prices using the BSM model.

The BSM model rests upon the assumption that the returns of the underlying asset follow a normal distribution. There is, however, overwhelming empirical evidence that return distributions exhibit heavy tails and asymmetry. Osterrieder (2016) demonstrated that these properties hold for bitcoin returns. The omnipresence of extreme return observations indicates that a heavy-tailed distribution should provide a more realistic description of the behavior of bitcoin returns. Heavy tails and asymmetry contradict the normality assumption and present a challenge for BSM option pricing. The Student’s t-distribution is an obvious candidate for heavy-tailed data. However, in the context of option pricing, the use of the t-distribution in the BSM setting results in a divergent integral (Cassidy et al. 2010). Recently, Näf et al. (2019) proposed a mean-variance heterogeneous-tails mixture distribution for modeling financial asset returns. They showed that the new model captures, along with the obligatory leptokurtosis, different tail behaviors among the assets.

Mandelbrot and Taylor (1967) coined the term “intrinsic time” in finance to emphasize that market information is not continuously available but instead arrives in discrete events occurring at varying intervals. As noted by Clark (1973), “The different evolution of price series on different days is due to the fact that information is available to traders at a varying rate. On days when no new information is available, trading is slow, and the price process evolves slowly. On days when new information violates old expectations, trading is brisk, and the price process evolves much faster”. Market orders for an asset, which can differ significantly in both timing and frequency throughout the trading day, are a prime example of such events. These events provide information on the asset value (price), and their occurrence, magnitude, and sign characterize intrinsic time. Price change information may also differ in its level of informativeness, with larger changes providing more information than smaller ones and with consecutive price changes of the same sign providing more information than those with opposite signs. Researchers have explored the concept of intrinsic time and applied it to financial time series in various works, including those of Guillaume et al. (1997); Tsang (2010), and Aloud et al. (2012). According to the intrinsic time perspective, no information is available between events, and therefore, no time has elapsed. The microstructural details induced by intrinsic time can influence the tail behavior of returns (Chakraborti et al. 2011).

The method of subordination offers an alternative approach to addressing the non-normal nature of true asset returns (Bochner 1995; Sato 1999; Schoutens 2003). In single subordination models, the continuous (and deterministic) time dependence of the Brownian motion is replaced by a stochastic time process—specifically, a non-decreasing Lévy process. In general, this stochastic time dependence consists of both a continuous and a jump process, mimicking the nature of the “arrival time” (intrinsic time) of new information in the financial markets. In addition to modifying the time dependence of the Brownian motion, subordinated models add the non-decreasing Lévy process as an additional stochastic driver to the return process. This allows for additional control of skewness and kurtosis to the return process. Under double subordination, the stochastic time process is represented as a further subordinated Lévy process, allowing for a more complicated intrinsic time model and the addition of further stochastic drivers to the return process. In addition to capturing skewedness and kurtosis behaviors, Shirvani et al. (2021) have shown that the use of double subordination can be used to incorporate the views of investors in asset and option pricing models.

In this paper, we develop a bitcoin volatility index based on the Cboe variance swap fair-value formula but with log-prices following a double-subordinated process, with one of the subordinated processes modeling intrinsic time. We briefly introduce the idea of double subordination to model the dynamics of Lévy processes. We show that our model implies a heavy-tailed distribution that provides a better description of bitcoin log-prices than the Student’s t model.

Rather than using Monte Carlo option pricing in non-Gaussian settings, as put forth in Allen et al. (2011), we employ the arbitrage theorem and mean-correction martingale measure (MCMM) of Yao et al. (2011) to price options when the returns of the underlying assets are doubly subordinated. Specifically, we price European contingent claims when bitcoin returns are assumed to be driven by a double-subordinated Lévy process. We determine the equivalent martingale measure to price options using the MCMM approach and demonstrate that our proposed pricing model is arbitrage-free. Based on the generated bitcoin option data, we derive implied volatilities using the Cboe approach and the double-subordination approach.

The remainder of the paper is organized as follows. Section 2 describes the double-subordination process and applies it to modeling bitcoin log-prices. Section 3 details the option pricing framework under double subordination; Section 4 applies it to modeling vanilla European bitcoin call and put options. In Section 5, we assess bitcoin volatility using the Cboe approach, which is based on option pricing, and our intrinsic time approach based on spot pricing. Both methods are compared with the historical volatility. The paper concludes with a discussion of our results.

2. Double-Subordination Model for Bitcoin Log-Prices

The double-subordination framework adopted here involves a Lévy subordinator process. A stochastic process $\mathbb{X}=\left(X\right(t),t\ge 0)$ defined on a stochastic basis $(\mathsf{\Omega },\mathcal{F},\mathbb{F},\mathbb{P})$ is said to be a Lévy process if the following conditions hold:

• $X\left(0\right)=0,$ $\mathbb{P}$-almost surely;
• $\mathbb{X}$ is a process with independent increments; for any partition $0=t_0<t_1<\cdots <t_n<\infty$, the increments $X\left(t_i\right)-X\left(t_{i-1}\right),i=1,2,\cdots ,n$ are independent;
• $\mathbb{X}$ is a process with stationary increments; for any $0\le s<t$, the increment $X\left(t\right)-X\left(s\right)$ has the same distribution as $X\left(t-s\right)$, that is, $X\left(t\right)-X\left(s\right)\sim X\left(t-s\right)$5;
• $\mathbb{X}$ is continuous in probability; for every $\epsilon >0$ and $t\ge 0$, there exists $h_{\epsilon ,t}>0$ such that $\mathbb{P}\left(\left|X\left(t+h_{\epsilon ,t}\right)-X\left(t\right)\right|>\epsilon \right)<\epsilon$.

In dynamic asset-pricing theory, a risky financial asset is defined by its price dynamics $S_t$, $t\in [0,\tau ]$, where $\tau <\infty$ is the time horizon; that is, $\tau$ is the maturity date of a financial contract. A Lévy process $\mathbb{T}=\left(T\right(t),t\ge 0,T(0)=0)$ with non-decreasing trajectories (i.e., non-decreasing sample paths) is called a Lévy subordinator. Since $T\left(0\right)=0$, the trajectories of $\mathbb{T}$ take only non-negative values. In the BSM option pricing model, the price dynamics of the underlying asset are defined by

$$S_t^{\left(\mathrm{BSM}\right)}=e^{X_t^{\left(\mathrm{BSM}\right)}},t\in [0,\tau ],$$

(1)

where the log-process is

$$X_t^{\left(\mathrm{BSM}\right)}=X_0+{\mu }_{1}t+{\sigma }_1{B}_t,{\mu }_1\in R,{\sigma }_1>0,X_0=ln\left(S_0\right),S_0>0,$$

(2)

and $\mathbb{B}=(B_t,t\ge 0)$ is a standard Brownian motion. To allow for the non-normality of asset returns, Mandelbrot and Taylor (1967) and Clark (1973) suggested the use of a subordinated Brownian motion, where the price process $S_t^{\left(\mathrm{ss}\right)}$ and the log-price process are defined as

$$S_t^{\left(\mathrm{ss}\right)}=e^{X_t^{\left(\mathrm{ss}\right)}},t\in [0,\tau ],$$

(3)

$$X_t^{\left(\mathrm{ss}\right)}=X_0+{\mu }_{2}t+{\sigma }_2{B}_{T\left(t\right)},{\mu }_2\in R,{\sigma }_2>0,$$

(4)

where $\mathbb{T}=\left(T\right(t),t\ge 0,T(0)=0)$ is a Lévy subordinator.

Process (4) describes the well-known single-subordinated log-price process. Various studies have demonstrated that single-subordinated log-price models commonly fail to capture the heavy-tailedness observed in financial return data. For example, Lundtofte and Wilhelmsson (2013) and Shirvani et al. (2021) showed that the normal-inverse Gaussian (NIG) distribution, which is a single-subordinated Lévy process, fails to explain the equity premium puzzle. This is partly due to the fact that the tails produced by the NIG distribution are not heavy enough. However, Shirvani et al. (2021) demonstrated that the high value for the risk aversion coefficient, which gives rise to the equity premium puzzle, is compatible with a return process driven by a double-subordinator model. Shirvani et al. (2021) defined and investigated the properties of various multiple-subordinated log-return processes designed to model leptokurtic asset returns. They showed that multiple-subordinated log-return processes can imply heavier tails than single-subordinated models and that they are capable of capturing skewness and kurtosis. Therefore, a double-subordination framework may be a more appropriate candidate for modeling the rather extreme behavior of bitcoin.

To apply double subordination to modeling the bitcoin price process, let $S_t$ denote the price process, with the dynamics

$$\begin{array}{cc}\hfill S_t& =e^{X_t},t\in [0,\tau ],\hfill \\ \hfill X_t& =X_0+{\mu }_{3}t+\gamma U\left(t\right)+\rho T\left(U\left(t\right)\right)+{\sigma }_3{B}_{T\left(U\left(t\right)\right)},t\ge 0,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\mu }_3\in R,{\sigma }_3>0,X_0=ln\left(S_0\right),S_0>0,\hfill \end{array}$$

(6)

where the triplet members $\{B_s,T\left(s\right),U\left(s\right)\},s\ge 0$ are independent processes generating the stochastic basis $(\mathsf{\Omega },\mathcal{F},\mathbb{F}=\left({\mathcal{F}}_t,t\ge 0\right),\mathbb{P})$, which represents the real world. Here, $\{B_s,s\ge 0\}$ is a standard Brownian motion, while $\left\{T\right(s),s\ge 0,T(0)=0\}$ and $\left\{U\right(s),s\ge 0,U(0)=0\}$ are Lévy subordinators. $B_t$, $T\left(t\right)$, and $U\left(t\right)$ are ${\mathcal{F}}_t$-adapted processes whose trajectories are right-continuous with left limits. Shirvani et al. (2021) referred to $T\left(U\right(t\left)\right),t\ge 0$, as the double-subordinator process; hence, the process modeled by (6) is a double-subordinated log-price process.

Consider the case where the subordinators $T\left(t\right)$ and $U\left(t\right)$ are inverse Gaussian (IG) Lévy processes; that is, $T\left(1\right)\sim IG({\lambda }_T,{\mu }_T)$ has the probability density function (pdf)

$$f_{T\left(1\right)}\left(x\right)=\sqrt{\frac{{\lambda }_T}{2\pi x^3}}\exp \left(-\frac{{\lambda }_T{\left(x-{\mu }_T\right)}^2}{2{\mu }_T^{2}x}\right),x\ge 0,{\lambda }_T>0,{\mu }_T>0.$$

(7)

Similarly, $U\left(1\right)\sim IG\left({\lambda }_U,{\mu }_U\right)$. In this case, Shirvani et al. (2021) referred to $T\left(U\right(t\left)\right)$ as the double-inverse Gaussian subordinator and to $X_t$ as a normal double-inverse Gaussian (NDIG) log-price process. The characteristic function (chf) of $X_1$ is given by

$$\begin{array}{c}{\phi }_{X_1}\left(v\right)=\mathrm{E}\left[e^{iv{X}_1}\right]\hfill \\ \hfill =exp\left\{iv{\mu }_3+\frac{{\lambda }_U}{{\mu }_U}\left[1-\sqrt{1-\frac{2{\mu }_U^2}{{\lambda }_U}\left(\frac{{\lambda }_T}{{\mu }_T}\left(1-\sqrt{1-\frac{{\mu }_T^2}{{\lambda }_T}(2iv\rho -{\sigma }_3^2{v}^2)}\right)+iv\lambda \right)}\right]\right\},\end{array}$$

(8)

with $v\in \mathbb{R}$. The moment-generating function (MGF) of $X_1$ is $M_{X_1}\left(w\right)=\mathrm{E}\left[e^{w{X}_1}\right]$, $w\in \mathbb{R}$.

The NDIG model, (6) and (8), has eight parameters: namely, ${\mu }_3$, ${\sigma }_3$, $\gamma$, $\rho$, ${\mu }_{\tau }$, ${\lambda }_{\tau }$, ${\mu }_T$, and ${\lambda }_T$, which can make fitting the model to data a challenging task. However, only six of these parameters are identifiable within the model (Lindquist et al. 2022). To set the remaining two parameters, consider the expectation

$$\mathbb{E}\left[X_1\right]={\mu }_3+\gamma \mathbb{E}\left[U\left(1\right)\right]+\rho \mathbb{E}\left[X\left(v\right)\right]\mathbb{E}\left[T\left(U\right(1\left)\right)\right].$$

(9)

As processes U and T are independent IG processes, we can uniquely identify $\gamma$ and $\rho$ by requiring

$$\mathbb{E}\left[U\left(1\right)\right]={\mu }_U=1\mathrm{and}\mathbb{E}\left[T\left(U\right(1\left)\right)\right]={\mu }_U{\mu }_T=1\to {\mu }_T=1.$$

(10)

By utilizing (10), the set of model parameters that can be identified becomes ${\mu }_3$, ${\sigma }_3$, $\gamma$, $\rho$, ${\lambda }_T$, and ${\lambda }_U$.

The central moments of the NDIG can be used to fit these six parameters. The MGF $M_{X_1}\left(w\right)$ for $X_1$, which generates the moments of its probability distribution, is obtained by evaluating $\mathbb{E}\left[e^{w{X}_1}\right]$ for $w\in \mathbb{R}$. This can be obtained directly from (8) by setting $w=iv$. The MGF is written in terms of the cumulant-generating function $K_{X_1}\left(w\right)$:

$$M_{X_1}\left(w\right)=\mathbb{E}\left[e^{w{X}_1}\right]=exp\left(K_{X_1}\left(w\right)\right).$$

(11)

Using the identity

$${\phi }_{X_t}\left(v\right)=\mathbb{E}\left[e^{iv{X}_t}\right]={\left(\mathbb{E}\left[e^{iv{X}_1}\right]\right)}^t={\left[\phi \left(X_1\right)\right]}^t,$$

(12)

we can express $M_{X_t}\left(w\right)$ as $M_{X_t}\left(w\right)=exp\left(K_{X_1}\left(w\right)t\right)$, where $K_{X_1}\left(w\right)$ can be written using (11) as

$$\begin{array}{cc}\hfill K_{X_1}\left(w\right)& ={\mu }_{3}w+{\lambda }_U\left(1-\sqrt{1-g\left(w\right)}\right),\hfill \\ \hfill g\left(w\right)& =1-2\frac{{\lambda }_T}{{\lambda }_U}\left(1-\sqrt{h\left(w\right)}\right)-2\frac{\gamma }{{\lambda }_U}w,\hfill \\ \hfill h\left(w\right)& =1-\frac{2\rho w}{{\lambda }_T}-\frac{{\sigma }_3^2{w}^2}{{\lambda }_T}.\hfill \end{array}$$

(13)

From the first four central derivatives of $K_{X_1}\left(w\right)$, the first four centered moments of $X_1$ are

$$\begin{array}{cc}\hfill \mathbb{E}\left[X_1\right]& ={\mu }_3+s,\hfill \\ \hfill \mathrm{Var}\left[X_1\right]& =\sigma +\frac{s^2}{{\lambda }_U},\hfill \\ \hfill \mathrm{Skew}\left[X_1\right]& =\frac{3\left(\sigma c+\frac{s^3}{{\lambda }_U^2}\right)}{{\left(\sigma +\frac{s^2}{{\lambda }_U}\right)}^{3/2}},\hfill \\ \hfill \mathrm{Kurt}\left[X_1\right]& =3\frac{\left[{\sigma }^2\left(1/{\lambda }_T+1/{\lambda }_U\right)+2\sigma \left(c^2+{(\rho /{\lambda }_T)}^2+2{(s/{\lambda }_U)}^2\right)+5s^4/{\lambda }_U^3\right]}{{\left(\sigma +\frac{s^2}{{\lambda }_U}\right)}^2},\hfill \end{array}$$

(14)

where $s=\gamma +\rho$, $\sigma ={\rho }^2/{\lambda }_T+{\sigma }_3^2$, and $c=\rho /{\lambda }_T+s/{\lambda }_U$. Equation (32) presents four criteria for fitting the six model parameters. Section 4 discusses the additional requirements needed to complete the parameter estimation.

3. Bitcoin Option Pricing under Double Subordination

To price European contingent claims, we assume that $X_t$ follows an NDIG log-price process. We need to derive an equivalent martingale measure (EMM) $\mathbb{Q}$ of $\mathbb{P}$ on $(\mathsf{\Omega },\mathcal{F},\mathbb{F}=\left({\mathcal{F}}_t,t\ge 0\right),\mathbb{P})$ such that the discounted price process $e^{-rt}{S}_t$, with $r\ge 0$ denoting the riskless rate, is a martingale (Duffie 2010, Chapter 6). To do this, we use the MCMM approach (as the proposed pricing model is arbitrage-free under the MCMM) and estimate the parameters specifying the process. We then add the appropriate drift term to the process to ensure that the discounted price process becomes a martingale.

Yao et al. (2011) constructed a martingale measure using the MCMM approach for the geometric Lévy process model and showed that this measure is an EMM if there is a continuous Gaussian part in the Lévy process. In the case in which $X_t$ is a pure-jump Lévy process, they pointed out that this measure cannot be equivalent to a physical probability. However, pricing European options under this measure is still arbitrage-free.

Let $\mathcal{C}$ be a European call option with an underlying risky asset $\mathcal{S}$, with price and log-price processes as in (5) and (6), respectively. Let $\mathcal{B}$ be a riskless asset with a price $b_t=e^{rt},t\ge 0$, where $r\ge 0$ is the riskless rate. Then the price of $\mathcal{C}$ is

$$C(S_0,r,K,\tau )=e^{-r\tau }{\mathbb{E}}_{\mathbb{Q}}\left[\max (S_{\tau }^{\left(\mathbb{Q}\right)}-K,0)\right],$$

(15)

where $\tau >0$ denotes the maturity, $K>0$ is the strike price, and $S_t^{\left(\mathbb{Q}\right)}$ is the price dynamics of $\mathcal{S}$ on $\mathbb{Q}$ (an EMM of $\mathbb{P}$). Using the MCMM approach, we obtain the dynamics of $\mathcal{S}$ on $\mathbb{Q}$:

$$S_t^{\left(\mathbb{Q}\right)}=\frac{b_t{S}_t}{M_{X_t}\left(1\right)}=S_0{e}^{\left(r-K_{X_1}\left(1\right)\right)t+X_t},t\in [0,\tau ].$$

(16)

The chf of the log-price, $ln{S}_t^{\left(\mathbb{Q}\right)}$, is

$${\phi }_{ln{S}_t^{\left(\mathbb{Q}\right)}}\left(v\right)=S_0^{iv}\exp \left\{\left[iv(r-K_{X_1}\left(1\right))+{\psi }_{X_1}\left(v\right)\right]t\right\},t\in [0,\tau ].$$

(17)

Carr and Madan (1998) showed how to use the fast Fourier transform (FFT) to value options in the case in which the chf of the log-price of the underlying asset is known analytically. They considered the modified option price $c_a(S_0,r,k,\tau )=e^{ak}C(S_0,r,k,\tau )$, where $k=ln\left(K\right)$, which, for a range of values $a>0$, guarantees that $c_a(S_0,r,k,\tau )$ is square-integrable over $k\in (-\infty ,\infty )$. The access point for applying the FFT is the relationship

$$C(S_0,r,k,\tau )=\frac{e^{-r\tau -ak}}{\pi }{\int }_0^{\infty }{e}^{-ivk}\frac{{\phi }_{ln{S}_{\tau }^{\left(\mathbb{Q}\right)}}(v-i(a+1))}{a^2+a-v^2+i(2a+2)v}dv.$$

(18)

The numerical solution of this integral involves two fundamental concerns: an “optimum” value for a and control over the error produced by truncating the integral over a finite domain $[0,v_{\max }]$. Addressing the first of these two concerns, Carr and Madan note that a positive value for a guarantees the square-integrability of $c_a(S_0,r,K,\tau )$ over the negative k-axis but aggravates the square-integrability over the positive k-axis. They derived a sufficient condition, ${\mathbb{E}}_{\mathbb{Q}}\left[{\left(S_{\tau }^{\left(\mathbb{Q}\right)}\right)}^{a+1}\right]<\infty$, which, combined with the analytic expression for the chf, can be used to determine an upper bound on a. Addressing the second of these concerns, they developed the bound

$$v_{\max }>\frac{exp(-ak)}{\pi }\frac{\sqrt{A}}{ϵ}$$

on the error involved in truncating the integration range of (18). Here, $ϵ$ is a desired level of the truncation error and $\sqrt{A}/v^2$ is an upper bound on the magnitude of the integrand in (18).

Constructing the implied volatility surface for call prices requires the estimation of (18) over a discrete mesh of $(k,T)$ values. Put options can then be valued, assuming that put–call parity holds. Implementing an FFT requires the numerical discretization of (18) into the form

$${\widehat{Z}}_p=\sum _{j=1}^{M}exp\left[-i\frac{2\pi }{M}(j-1)(p-1)\right]Z_j,p=1,2,\cdots M,$$

(19)

which the FFT can solve in $O\left(M{ln}_2\left(M\right)\right)$ operations. Writing (18) as

$$C(·,\tau ,k)=\frac{e^{-r\tau -ak}}{\pi }\Psi \left(k\right),\mathrm{where}\Psi \left(k\right)={\int }_0^{\infty }exp(-ivk)h\left(v\right)dv,$$

(20)

we can obtain the discretization of $\Psi \left(k\right)$ over a finite range $[0,v_{\max }]$ using the left-hand rectangle rule6:

$$\Psi \left(k\right)=\sum _{j=1}^{N}exp(-i{v}_{j}k)h\left(v_j\right)\Delta v,$$

(21)

where $v_j=(j-1)\Delta v$, $j=1,\dots ,N$, $v_1=0$, and $v_N=v_{\max }-\Delta v$. For each fixed value of $\tau$, we discretize $k=ln\left(K\right)$ over a range $[-\overline{k},\overline{k}]$ with N equally spaced points to obtain $k_p=-\overline{k}+(p-1)\Delta k$, $p=1,\dots ,N$, $\Delta k=2\overline{k}/N$. On this grid, (21) becomes

$$\Psi \left(k_p\right)=\sum _{j=1}^{N}exp[-i(j-1)\Delta v(p-1)\Delta k]exp\left(i\overline{k}{v}_j\right)h\left(v_j\right)\Delta v,p=1,\dots ,N,$$

(22)

which is identical to (19) with the identifications ${\widehat{Z}}_p=\Psi \left(k_p\right)$, $Z_j=exp\left(i\overline{k}{v}_j\right)h\left(vj\right)\Delta v$, $M=N$, and $2\pi /M=\Delta v\Delta k$. This last identification gives the familiar FFT tradeoff between the span covered in the “space” domain and the span covered in the “frequency” domain:

$$v_{\max }\overline{k}=\pi N.$$

(23)

In our computations in Section 4, we used $N=2^{10}=1024$.

As noted above, Carr and Madan (1998) introduced the parameter a to ensure that the call pricing function (15) is square-integrable as $K\to 0$ (i.e., as $k=lnK\to -\infty$). They note that a sufficient condition for square-integrability is provided by the requirement that ${\varphi }_{ln\left(S_t^{\mathbb{Q}}\right)}(-i(1+a))\le \infty$. From (17),

$${\phi }_{ln{S}_t^{\left(\mathbb{Q}\right)}}(-i(1+a))=S_t^{1+a}\exp \left\{(1+a)\left(r_f-K_{X_1}\left(1\right)\right)T\right\}{\left\{\exp \left(K_{X_1}(1+a)\right)\right\}}^T\le \infty .$$

(24)

From (8) and (11), note that $\psi \left(iw\right)\le K\psi \left(w\right)$ for $w\in \mathbb{R}$. Hence, (24) and ${\varphi }_{ln\left(S_t^{\mathbb{Q}}\right)}(-i(1+a))\le \infty$ can be combined to create the requirement

$$\exp \left\{(1+a)\left(r_f-K_{X_1}\left(1\right)\right)T\right\}{\left\{\exp \left(K_{X_1}(1+a)\right)\right\}}^T\le \infty .$$

(25)

To ensure that the cumulant-generating function remains real-valued, requirement (25) can be reduced to positive argument requirements for the square-root evaluations in (13) for $K_{X_1}\in [1,a]$, where $a\in (0,\infty )$ (Lindquist et al. 2022). From (11), under the assumption that $\gamma =0$, this can be further reduced to the requirements

$$\begin{array}{c}\hfill h\left(w\right)=1-\frac{{\sigma }_3^2{w}^2}{{\lambda }_T}-\frac{2\rho w}{{\lambda }_T}\ge 0\mathrm{and}g\left(w\right)=1-2\frac{{\lambda }_T}{{\lambda }_U}\left(1-\sqrt{h\left(w\right)}\right)\ge 0.\end{array}$$

(26)

Solving the latter equation for $h\left(w\right)$ yields

$$h\left(w\right)\ge 1-\frac{{\lambda }_U}{2{\lambda }_T}\stackrel{\mathrm{def}}{=}{d}^2.$$

(27)

Combining (27) with the first equation in (26) gives

$$1-\frac{{\sigma }_3^2{w}^2}{{\lambda }_T}-2\frac{\rho w}{{\lambda }_T}\ge d^2.$$

(28)

The quadratic equation has two roots, and (28) is satisfied for $w=1+a$ when

$$-\frac{\rho }{{\sigma }_3^2}-\frac{\sqrt{{\rho }^2+{\lambda }_t{\sigma }_3^2(1-d^2)}}{{\sigma }_3^2}\le 1+a\le -\frac{\rho }{{\sigma }_3^2}+\frac{\sqrt{{\rho }^2+{\lambda }_t{\sigma }_3^2(1-d^2)}}{{\sigma }_3^2}.$$

(29)

Since $a>0$, it follows that

$$a\le a_{max}=-1-\frac{\rho }{{\sigma }_3^2}+\frac{\sqrt{{\rho }^2+{\lambda }_t{\sigma }_3^2(1-d^2)}}{{\sigma }_3^2}.$$

(30)

By utilizing (28), (30) can be rewritten as

$$a_{max}=\frac{1}{{\sigma }_3^2}\sqrt{{\rho }^2+{\lambda }_U\left(1-\frac{{\lambda }_U}{4{\lambda }_T}\right){\sigma }_3^2}-\frac{\rho }{{\sigma }_3^2}-1.$$

(31)

As shown by Lindquist et al. (2022), while the modified option price $c_a(S_0,r,k,\tau )$ is square integrable over the range $k\in (-\infty ,\infty )$ for $a\in (0,a_{max})$, this does not guarantee that $c_a(S_0,r,k,\tau )$ obeys the appropriate price bounds; for example, for a European call option, we require

$$max(S_0-K{e}^{-r\tau },0)\le c_a(S_0,r,k,\tau )\le S_0.$$

Their work demonstrates the existence of an interval $0<a_l<a_h<<a_{max}$ such that if $a\in (0,a_l)$, then $c_a(S_0,r,k,\tau )>S_0$ for a range of values $(k,\tau )$, while if $a\in (a_h,a_{max})$, then $c_a(S_0,r,k,\tau )<max(S_0-K{e}^{-r\tau },0)$ for a range of values $(k,\tau )$. Thus, to ensure that $c_a(S_0,r,k,\tau )$ remains appropriately bounded, the limit $a_l<a<a_h$ must be imposed. These limiting values are dependent on data (although the extent to which they are dependent is unexplored). For their data, Lindquist et al. (2022) established the values $a_l=0.40$, $a_h=0.90$ to two decimal digits of numerical accuracy.

4. Numerical Computation

In this section, we illustrate the double-subordinated method described in Section 2 and Section 3 and fit the NDIG model to the log-return bitcoin time series. Figure 1 shows the daily bitcoin price and log-return ($r_t$) time series covering the period from 19 July 2010 to 28 July 2023.

The return series consists of 4026 observations, to which we fit the NDIG model specified by (5)–(8). The NDIG model is characterized by six parameters: $\theta =({\mu }_3,{\lambda }_U,{\mu }_U,{\lambda }_T,{\mu }_T,{\sigma }_3)$. To further simplify the parameter set, we assume that the subordinator $U\left(t\right)$ is used to model the intrinsic time of the return process, while the subordinator $T\left(t\right)$ is used to model the return skewness and heavy-tailed behavior. It is reasonable in this model to demand that there be no $\gamma U\left(t\right)$ term in (6), i.e., $\gamma =0$. With this requirement, the first four moments of $X_1$ become

$$\begin{array}{cc}\hfill \mathbb{E}\left[X_1\right]& ={\mu }_3+\rho ,\hfill \\ \hfill \mathrm{Var}\left[X_1\right]& ={\sigma }_3^2+d{\rho }^2,\hfill \\ \hfill \mathrm{Skew}\left[X_1\right]& =\frac{3d\rho \left({\sigma }_3^2+{\rho }^2\right)}{{\left({\sigma }_3^2+d{\rho }^2\right)}^{3/2}},\hfill \\ \hfill \mathrm{Kurt}\left[X_1\right]& =\frac{3{\sigma }^{2}d+6{\rho }^2\sigma \left(d^2+1/{\lambda }_T^2+2/{\lambda }_U^2\right)+15{\rho }^4/{\lambda }_U^3}{{\left({\sigma }_3^2+d{\rho }^2\right)}^2},\hfill \end{array}$$

(32)

where $\sigma ={\rho }^2/{\lambda }_T+{\sigma }_3^2$ and $d=1/{\lambda }_T+1/{\lambda }_U$.

As there is no analytical expression for the pdf, we employ the method of moments and the empirical chf to estimate the parameters of the model. We follow Paulson et al. (1975) and Yu (2003) and use the fact that the pdf is the Fourier transform of the chf. Thus, the five parameters ${\mu }_3$, ${\sigma }_3$, $\rho$, ${\lambda }_U$, and ${\lambda }_T$ can be estimated via minimization:

$$\begin{array}{cc}\underset{{\mu }_3,{\sigma }_3,\rho ,{\lambda }_T,{\lambda }_U}{\min }& \{{(\Delta M_1)}^2+{(\Delta M_2)}^2+{(\Delta M_3)}^2+{(\Delta M_4)}^2+{(\Delta CF)}^2\}\hfill \\ s.t.& {(\Delta M_1)}^2=1-\frac{\mathbb{E}\left[X_1\right]}{\mathbb{E}\left[r_t\right]},\hfill \\ & {(\Delta M_2)}^2=1-\frac{\mathrm{Var}\left[X_1\right]}{\mathrm{Var}\left[r_t\right]},\hfill \\ & {(\Delta M_3)}^2=1-\frac{\mathrm{Skew}\left[X_1\right]}{\mathrm{Skew}\left[r_t\right]},\hfill \\ & {(\Delta M_4)}^2=1-\frac{\mathrm{Kurt}\left[X_1\right]}{\mathrm{Kurt}\left[r_t\right]},\hfill \\ & {(\Delta CF)}^2={\int }_{-\infty }^{\infty }{(\frac{1}{n}\sum _{j=1}^n{e}^{iv{x}_j}-\phi x_t(v,\theta \left)\right)}^{2}dv,\hfill \end{array}$$

(33)

where $r_t$ denotes the observed daily bitcoin return time series. The inclusion of the term $\Delta CF$ relies on the one-to-one correspondence between the cumulative distribution function and the characteristic function (since the pdf is the inverse Fourier transform of the characteristic function). The integral for $\Delta CF$ can be estimated using the method described by Yu (2003).

The parameters are estimated using the method of moments and empirical chf fitting, which is a method that is as efficient as likelihood-based methods (Yu 2003). The resulting parameter estimates are reported in

Table 1. NDIG parameter estimates for the bitcoin log-returns.

${\mathit{\mu }}_3$	${\mathit{\lambda }}_{\mathit{U}}$	${\mathit{\mu }}_{\mathit{U}}$	${\mathit{\lambda }}_{\mathit{T}}$	${\mathit{\mu }}_{\mathit{T}}$	${\mathit{\sigma }}_3$	$\mathit{\rho }$
0.004	0.145	1.00	9.9293	1.00	0.0551	−0.0008

.

For comparison, we also fit the normal and Student’s t distributions to the bitcoin daily log-return values and estimated the distribution parameters using the maximum likelihood. The empirical density and the densities for the fitted NDIG model, the best-fit Student’s t distribution (giving $df=1.60$), and best-fit normal distribution (giving $\mu =0.0032$, $\sigma =0.0551$) are compared in Figure 2.

The results confirm, as was noted in the introduction, that the normal distribution is not well suited for modeling asset returns. The graphs indicate that both the Student’s t and NDIG models match the tails of the empirical density rather well, while the NDIG matches the skewness in the empirical distribution more evenly than Student’s t-distribution. The low estimate of $df=1.60$ for the Student’s t-distribution implies that even second moments do not exist. Thus, the fat tails of Student’s t-distribution would cause divergence of the BSM integral needed to evaluate option prices.

NDIG Model for European Call Option Pricing

We apply the NDIG Lévy model to the pricing of plain vanilla European bitcoin options. Let $\mathcal{C}$ be a European call option, where the underlying risky asset $\mathcal{S}$ follows the log-price process given in (6). Put option prices are computed using put–call parity. The dynamics of $\mathcal{S}$ on $\mathbb{Q}$ are given by (16), and the chf of the log-price is given by (17). We evaluate the integral (18) using the FFT for a range of strike levels and maturity horizons, with the parameter estimates reported in Table 1. When the global parameter values provided in Table 1 are used, the upper bound of a in (32) is $a_{max}\lesssim 21.93$. As noted above, the interval $a_l\le a\le a_h$ that determines a suitably price-bounded value for the option is dependent on the data set. Following the procedure of Lindquist et al. (2022), we set $a=0.40$ for the computations here and in the remainder of the paper.

Figure 3a,b show the resulting prices for call and put options plotted against the time to maturity $\tau$ and the strike price K. Figure 3c shows the implied volatility surface, i.e., the market’s view of future volatility, versus the time to maturity and relative moneyness, which is defined as $K/S$, where S is the asset price obtained by the BSM model. Figure 3c investigates the behavior of the implied volatility as a function of moneyness and time to maturity. It is observed that as moneyness increases, moving towards out-of-the-money values, the implied volatility also increases. Conversely, as moneyness decreases, a distinct pattern emerges, referred to as the “smirk” in the implied volatility curve. Initially, as moneyness decreases and options move towards in-the-money values, the implied volatility rises, but subsequently, it starts to decrease for further in-the-money options, eventually approaching near-zero values. As we extend the time to maturity (denoted as $\tau$) of the options, we notice a noteworthy trend. The sensitivity of the implied volatility to changes in moneyness diminishes for call options, and this effect appears to converge to a constant level irrespective of moneyness. In other words, as the options approach their expiration dates, their implied volatility becomes less responsive to changes in moneyness.

5. Bitcoin Volatility Measures

Understanding the volatility of speculative assets is critical for investment decisions. Given that bitcoin is considered, at least by some, a potential alternative to fiat money, volatility characteristics are of special concern. It is, therefore, of interest to understand and adequately model the process that governs the volatility of bitcoin.

Model parameters reflecting the volatility of asset prices are known to vary stochastically and exhibit clustering. To allow for these phenomena in option pricing, Hull and White (1987) and Heston (1993) suitably randomized the volatility parameter in the Black–Scholes model, where the volatility process is governed by Brownian motion. An alternative strategy was adopted by Carr et al. (2003) based on the ideas of Geman et al. (2001) and Clark (1973). Clark conjectured that price processes are controlled by a random clock, which is a cumulative measure of economic activity, and used the transaction volume as a proxy for this measure. Geman et al. suggested the price process must have a jump component; thus, the price process can be regarded as Brownian motion subordinated to this random clock. Carr et al. developed this subordinated model using the NIG and variance gamma examples of pure-jump Lévy processes. Hurst et al. (1997) considered various subordinated processes to model the leptokurtic characteristics of stock-index returns. In the option pricing literature, Carr and Wu (2004) extended the approach of Carr et al. (2003) by providing an efficient way to allow for correlation between the stock price process and random time changes. Klingler et al. (2013) introduced two six-parameter processes based on time-varying, tempered stable distributions and developed an option pricing model based on these processes. Shirvani et al. (2021) introduced the volatility intrinsic time, or volatility subordinator model, to reflect the heavy-tail phenomena present in asset returns. They studied the question of whether the VIX is a volatility index that adequately reflects intrinsic time. They showed that this index fails to properly capture the intrinsic time for the SPDR S&P 500 Trust ETF. Apparently, the VIX, as a measure of change over time, does not reflect all the information needed to correctly capture the skewness and the fat-tailedness of the S&P 500 index. A model with a suitable volatility subordinator should adequately account for such empirical phenomena.

In this section, we apply three perspectives to analyze bitcoin volatility. The first measures the historical realized volatility through the sample standard deviation. As noted in the introduction, this is the method employed in the BVI. The second measures the implied (future) volatility in the risk-neutral derivatives world, $\mathbb{Q}$, reflecting the views of option traders. The third measures the implied volatility in the real world, $\mathbb{P}$, reflecting the views of spot traders. We employ an intrinsic time formulation in this last case. In the following, we empirically compare these three modeling strategies using daily bitcoin data from 5 May 2014 through 28 July 2023.

Historical Volatility. Historical volatility was computed based on the average standard deviation for a 1008-day rolling window of bitcoin returns. The historical volatility varies between 3.56% and 8.53% over the sample period.

Implied Volatility in $\mathbb{Q}$. Option traders commonly use the implied volatility as a proxy for the future volatility. The VIX, the index reflecting volatility expectations for the S&P 500 stock index (SPX), is based on SPX options estimated on a real-time basis by the Cboe. The VIX can be regarded as an efficient volatility forecast for the SPX provided that option markets are efficient. The value of the VIX is

$$VIX=100\times \sqrt{W_1{\sigma }_1^2+W_2{\sigma }_2^2}.$$

(34)

The subscript “1” denotes near-term options, while “2” denotes next-term options. As SPX options expire on Fridays, depending on the trading day, near-term options have 23 to 30 days until expiration, while next-term options have 31 to 37 days until expiration. The weights $W_j$ $(j=1,2)$ express these expiration times in a normalized form that is accurate to the minute:

$$W_1=\frac{M_{T_1}}{M_{30}}\left(\frac{M_{T_2}-M_{30}}{M_{T_2}-M_{T_1}}\right),W_2=\frac{M_{T_2}}{M_{30}}\left(\frac{M_{30}-M_{T_1}}{M_{T_2}-M_{T_1}}\right),$$

(35)

where $M_j$ represents the number of minutes until the settlement of the j-th term options, while $M_{30}$ represents the total number of minutes in a 30-day period. As a result, $W_1$ and $W_2$ are constrained to the range $0\le W_1,W_2\le 1$, with the additional requirement that $W_1+W_2=1$. The equation below, provided by Demeterfi et al. (1999), is used to calculate the near-term and next-term volatilities:

$${\sigma }_j^2=\frac{2e^{r_f{T}_j}}{T_j}\sum _i\frac{\Delta K_i}{K_i^2}Q\left(K_i\right)\frac{1}{T_j}{\left(\frac{F_j}{K_{0j}}-1\right)}^2.$$

(36)

Here, $T_j$ is the time to expiration measured in fractions of a year, $r_f$ is the annualized risk-free rate, $F_j$ is the forward index level derived from index option prices, $K_{0j}$ is the first strike price below $F_j$, $K_i$ is the strike price of the ith out-of-the-money option (a call if $K_i>K_0$, a put if $K_i<K_0$, and both a call and put if $K_i=K_0$), $\Delta K_i$ is the interval between strike prices, and $Q_{K_i}$ is the midpoint of the bid–ask spread for each option with a strike price $K_i$. For more information on the computation of each of these parameters, refer to the Cboe methodology paper (Chicago Board Options Exchange 2023).

To calculate a “VIX-like” volatility, which we refer to as the BVIX, the NDIG model is used to compute prices for European put and call options that have between 23 and 37 days to expiration. To compare this volatility with the historical volatility, BVIX values are calculated using historical data. For each moving 1008-day (four-year) moving window of bitcoin log-return data, BVIX values are calculated as follows.

• The NDIG model is fitted to the bitcoin log-return data in the window, and the model parameters are estimated using the minimization defined in (33).
• The dynamics of the bitcoin price $S_T^{\mathbb{Q}}$ in the risk-neutral world and the chf of the log-price $ln{S}_T^{\mathbb{Q}}$ are determined using (17).
• Call option prices with appropriate expiration dates are computed using the FFT formulation (18), and put option prices are computed using put–call parity based on the portfolio spot price $S_T^{\mathbb{Q}}$ for the last date of the moving window. In the context of option price calculations based on (18), the numerical values for the parameters a, $a_{\max }$, and $v_{\max }$ are determined through the utilization of the conditions outlined in (30) and (31), respectively. Choosing values of a that are too small results in option prices that exceed the maximum threshold, while choosing values of a that are too large produces prices that go below the minimum threshold and may be negative (Lindquist et al. 2022).
• BVIX values are then computed using (34)–(36). As there are no traded options in the bitcoin data, the following minor modifications are made to the VIX formulation:

  (a)

  The risk-free interest rate used is the annualized bond-equivalent (coupon-equivalent) yield for 3-month U.S. treasury bills published for day t.

  (b)

  The closing time for option evaluation is 4:00 PM on day t.

  (c)

  Options expire on near- and next-term Fridays at 4:00 PM.

  (d)

  As there is no bid–ask spread in the NDIG option price computations, $Q_{K_i}$ is computed directly as the NDIG option price.

  (e)

  The range of strike prices considered in the BVIX computation is from $0.75S_t$ to $1.5S$, as the NDIG computed option prices do not go to zero.

  As mentioned in step 4(d), since the underlying asset portfolio is not traded, there is no market sentiment setting bid–ask prices for the call and put options used in the BVIX calculation. Therefore, the BVIX volatility directly reflects the NDIG option price computations, and there is no risk premium or market price of risk, which is typically captured by the VIX.

The parameter values computed in step 1 are displayed in Figure 4. The parameter ${\lambda }_T$ displays the smallest, while $\rho$ displays the greatest, temporal variability. The estimated values for ${\mu }_3$ and ${\sigma }_3$ show a declining trend leading up to April 2017, followed by a subsequent abrupt change, with no discernible long-term drift thereafter.

Figure 5 compares the first four empirical moments computed from the time series in Figure 1 with the moments computed from the fitted parameters using (32). The agreement between the empirical and fitted parameter moments is very good.

Noticeable in certain plots in Figure 4 and Figure 5 are “spikes” in the results. The most discernible occurs on 23 March 2017. The reason for this (and other smaller spikes) can be traced to the following. Bitcoin is traded 24/7. However, the Bloomberg data utilized did not provide price data for weekends and imposed a “close of trade time” in order to record a final daily price. As a consequence, volatile weekend activity in bitcoin can result in large swings in Thursday–Friday–Monday or Friday–Monday–Tuesday pricing. The consequent “daily” log-returns over the period 19 July 2010 through 28 July 2023 (with weekend price data missing) are shown in Figure 6. The data set displays a $-0.584$ log-return in BTC between 5 December 2013 and 6 December 2013 and a corresponding $0.517$ log-return in BTC between 6 December 2013 and 9 December 2013. This large swing in the BTC return disappears from the moving window in the computation of parameters between 23 March 2017 and 24 March 2017. The result is a spike in the fitted parameters on the date when the weekend “discontinuity” no longer contributes to the moving window. As is evident from Figure 6, the volatility in bitcoin returns was most pronounced prior to 2015.

Addressing the restriction on a mentioned in step 3, Figure 7 shows the computed values for the upper bound $a_{max}$ over the period from 8 May 2014 to 28 July 2023.

Figure 8 shows that the values computed for the BVIX are higher than historical volatility values. This is due to the fact that the BVIX formulation is a direct application of the Cboe VIX formulation, which is designed specifically to predict volatility magnitudes for SPX and must be appropriately re-normalized to apply to options based upon other risky assets. To account for the different scales observed, the historical and BVIX volatility time series were normalized separately (by subtracting the respective series mean and dividing by the series standard deviation). The normalized plots, shown in Figure 9, are in excellent agreement. This emphasizes the fact that the sole source of variance in this numerical example is the daily returns of the underlying risky asset—the bitcoin data; this variance is directly quantified by the historical volatility. Since all option prices in this example are calculated using the Carr–Madan formulation and the NDIG model, no additional volatility is introduced into these option prices through, for instance, trader sentiment (bid–ask pricing). Therefore, the BVIX volatility formulation only captures the original asset return volatility.

Implied Volatility in $\mathbb{P}$: Intrinsic Time. By utilizing the double-subordinated method, a novel measure of volatility can be derived under the $\mathcal{P}$ measure. From (5), we view $X_t$ as the process governing the intrinsic time of the double-subordinated price process. From (32), with $\gamma =0$, the standard deviation (i.e., the volatility) of the unit increment of $X_t$ when $X_t$ follows a NDIG log-price process is given by

$$\mathrm{Vol}\left(X_1\right)\equiv \sqrt{\mathrm{Var}\left(X_1\right)}={{\sigma }_3}^2+{\rho }^2\left(\frac{1}{{\lambda }_T}+\frac{1}{{\lambda }_U}\right),$$

(37)

where ${\lambda }_T$, ${\lambda }_U$, and ${\sigma }_3$ are model parameters defined in Table 1. We refer to the volatility $\sqrt{\mathrm{Var}\left(X_1\right)}$ as the NDIG volatility, which takes into account both the Brownian motion (Gaussian) and Lévy subordinator components of the model. It is important to note that this measure differs from other measures of volatility in the literature.

We compute rolling NDIG parameter estimates from the most recent nine years of daily data using overlapping windows with a length of 1008 days and use (37) to derive the annualized volatility implied by the NDIG model. The volatility computed by the NDIG intrinsic time (NDIG IT) model is also shown in Figure 8. Plots of the NDIG intrinsic time and historical volatility are visually indistinguishable, confirming that (37) captures realistic bitcoin dispersion patterns.

We noted above that the NDIG IT volatility predictions are consistently larger than the historical volatility but otherwise demonstrate very strong synchronicity. Normalized versions of the three series are plotted in Figure 97. All three volatility time series are in very strong agreement.

For a quantitative analysis of their agreement, we assessed the cointegration of the three un-normalized time series. To apply cointegration, we first determined whether each volatility series was integrated. The results of both the augmented Dickey–Fuller and KPSS tests (Kwiatkowski et al. 1992) shown in

Table 2. The p-values from augmented Dickey–Fuller (ADF), KPSS, and Johansen tests.

Test	BVIX	NDIG IT	STD	Test	$\mathit{r}=0$	$\mathit{r}=1$
ADF	0.009	0.007	0.001	Johansen	0.001	0.001
KPSS	0.010	0.010	0.010

support the hypothesis that each series is integrated of order one (has a unit root). Performing a Johansen (1988) test involving all three (un-normalized) time series indicates a cointegrating relationship of both orders 1 and 2 among the three time series (see Table 2).

The results from Figure 9 and Table 2 demonstrate that the dynamic NDIG IT model, which is consistent with dynamic asset pricing theory and has predictive capabilities, provides excellent agreement when it is applied “in-sample”. The implied volatility model, i.e., the BVIX model, which is also dynamic and predictive, also provides very good agreement with the in-sample data.

We investigate the question of whether there is long-range dependency (LRD) in bitcoin volatility, as has been found in (see, for example, Asai et al. 2012). We fit a fractionally integrated, autoregressive, conditional heteroskedastic (FIGARCH) model to investigate both short- and long-range dependency in the bitcoin variance series. Specifically, we fit an AR(1)-FIGARCH(1,d,1) model with Student’s t innovations. (See, e.g., Bollerslev and Mikkelsen 1996; Engle 1982; Granger and Joyeux 1980). Fractional integration in the GARCH part, i.e., $d>0$, indicates an LRD in variance and non-stationarity, while $d<0$ indicates anti-persistency and short-range memory. The AR(1)-FIGARCH(1,d,1) model parameters are estimated using a rolling window of 1008 days. Figure 10 plots the sequence of d estimates for the conditional heteroskedastic equation for the variance.

The estimates fall almost exclusively in the interval $(0.9,1.0)$, indicating non-stationarity and the presence of an LRD in bitcoin volatility. There appears to be a weak secondary range dependence corresponding to a value of $d\sim 0.4$.

6. Discussion

Dealing with heavy-tailed asset returns remains a challenge in theoretical and empirical finance. Implicit in our approach is the idea that there is a relationship between the “intrinsic time” of finance and the price volatility of an asset. Whether this intrinsic time is measured in terms of the rate (ticks), volumes of transactions, or another proxy is irrelevant; we take as a stylized fact that the intrinsic time distribution of information arrival is skewed and extremely heavy-tailed. (See, for example, Chakraborti et al. (2011) for information on the microstructure of the time between successive transactions.)

The philosophy behind our approach has been as follows. The log-price process is additive Brownian motion, but the time variable that drives the Brownian motion is not the physical time t, as suggested by (2), but is intrinsic time. For an asset such as bitcoin, whose volatility is especially pronounced, this relationship should be “extractable”. We utilize subordinated Lévy processes to extract the relationship between intrinsic time and the price processes observed in physical time. As has been demonstrated by Shirvani et al. (2021), it is necessary to utilize two levels of subordination (6) to capture both the stylized facts of the operational time scale and the stylized facts (skewness, heavy tails, and clustering) observed in asset returns. In this paper, we have utilized a doubly subordinated process (6) using an IG process at each level of subordination, where the IG process $U\left(t\right)$ is used to capture the effects of intrinsic time. The results in Figure 2 demonstrate that our NDIG model is superior to Student’s t for capturing the tails of the bitcoin log-return distribution. Using the NDIG parameter estimates from Table 1, Figure 11 plots the densities $U\left(1\right)$ and $T\left(1\right)$ governing, respectively, the intrinsic time and heavy-tailed effects of the bitcoin log-return.

By extending our double-subordination approach to option pricing, we are able to show convincingly that the double-subordination model not only captures the distributional properties of the price dynamics but also very accurately captures (Figure 8) the trend of the historical (in-sample) bitcoin volatility (as measured by the historical standard deviation). Furthermore, Figure 9 demonstrates that we can derive the correct linear scaling to be applied to the double-subordinated model to capture the actual (in-sample) volatility.

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