Market attention and Bitcoin price modeling: theory, estimation and option pricing

Abstract

The goal of this paper is to provide a novel quantitative framework to describe the Bitcoin price behavior, estimate model parameters and study the pricing problem for Bitcoin derivatives. To this end, we propose a continuous time model for Bitcoin price motivated by the findings in recent literature on Bitcoin, showing that price changes are affected by sentiment and attention of investors, see e.g., (Kristoufek in Sci Rep 3:3415, 2013, PLoS ONE 10(4):e0123923, 2015; Bukovina and Marticek in Sentiment and bitcoin volatility. Technical report, Mendel University in Brno, Faculty of Business and Economics 2016). Economic studies, such as Yermack (Handbook of Digital Currency, chapter second. Elsevier, Amsterdam, pp 31–43, 2015), have also classified Bitcoin as a speculative asset rather than a currency due to its high volatility. Building on these outcomes, the price dynamics in our suggestion is indeed affected by an exogenous factor which represents market attention in the Bitcoin system. We prove the model to be arbitrage-free under a mild condition and we fit the model to historical data for the Bitcoin price; after obtaining a approximate formula for the likelihood, parameter values are estimated by means of the profile likelihood method. In addition, we derive a closed pricing formula for European-style derivatives on Bitcoin, the performance of which is assessed on a panel of market prices for Plain Vanilla options quoted on www.deribit.com.

Appendices

Appendix A: Levy approximation

In Levy (1992) the author proves that the distribution of the mean integrated Brownian motion $\frac{1}{s} \int _0^s P_u \mathrm {d}u$ can be approximated with a log-normal distribution, at least for suitable values of the model parameters $\mu _P, \sigma _P$; the parameters of the approximating log-normal distribution are obtained by applying a moment matching technique. Set

$$\begin{aligned} IP(s):=\int _0^s P_u \mathrm {d}u,\quad s>0. \end{aligned}$$

(A.1)

Of course, the distribution of IP(s) can also be approximated by a log-normal for $s >0$. By applying the moment matching technique the parameters of the corresponding log-Normal distribution for IP(s) are given by

$$\begin{aligned} \alpha (s)= & {} \log \left( \frac{\mathbb {E}\left[ IP(s)\right] ^2}{\sqrt{\mathbb {E}\left[ IP(s)^2\right] }}\right) , \\ \nu ^2(s)= & {} \log \left( \frac{\mathbb {E}\left[ IP(s)^2\right] }{\mathbb {E}\left[ IP(s)\right] ^2}\right) . \end{aligned}$$

The approximate distribution density function $f_{IP(s)}$ of IP(s) is thus given by

$$\begin{aligned} f_{IP(s)}(x)= {\mathcal {LN}} pdf_{\alpha (s),\nu ^2(s)} \left( x\right) , \quad \text{ if } s > 0, \end{aligned}$$

where ${\mathcal {LN}} pdf_{m,v}$ denotes the probability distribution function of a log-normal distribution with parameters m and v, defined as

$$\begin{aligned} {\mathcal {LN}} pdf_{m,v}(y)=\frac{1}{y\sqrt{2\pi v}} e^{-\frac{(log (y)-m)^2}{2v}}, \quad \forall \ y \in \mathbb {R}^+. \end{aligned}$$

In the paper the above approximation is applied twice with completely different purposes. In Sect. 4, once $\tau <\Delta$ is assigned, the Levy approximation is applied to derive the distribution of $A_1$ and of $A_i$ given $A_{i-1}$ where $A_1=X_\tau ^\tau + IP(\Delta -\tau )$ and, for $i\ge 2$, $A_i=\int _{(i-1)\Delta -\tau }^{i\Delta -\tau } P_u du=\int _{-\tau }^{\Delta -\tau } P_{u+(i-1)\Delta } \mathrm {d}u =P_{(i-1)\Delta }\int _{-\tau }^{\Delta -\tau } P_{u} \mathrm {d}u =P_{(i-1)\Delta } \left( X_\tau ^\tau +IP(\Delta -\tau ) \right)$.

In Sect. 5 it is applied to derive an approximate distribution for the integrated attention process starting at $t=0$, i.e., to $X_{0,T}^\tau =X_\tau ^\tau +IP(T-\tau )$. Note that $X_{0,T}^\tau -X_\tau ^\tau =IP(T-\tau )$ hence the derivations of its distribution is trivial once that of $IP(T-\tau )$ is known.

Appendix B: Technical proofs

The following result provides basic statistical properties for the integrated attention process $X^\tau$ defined in (2.3), as well as for its variation in case they are not fully deterministic.

Lemma B.1

In the market model outlined in Eqs. (2.1)–(2.2), we have:

• (i) For $t > \tau$,

  $$\begin{aligned} \mathbb {E}\left[ X_t^\tau \right]&= X_\tau ^\tau +\frac{\phi (0)}{\mu _{P}}\left( e^{\mu _{P}(t-\tau )} -1\right) ;\\ \mathbb V\mathrm{ar}[X_t^\tau ]&= \frac{2\phi ^{2}(0)}{\left( \mu _{P}+\sigma _{P}^{2}\right) \left( 2\mu _{P}+\sigma _{P}^{2}\right) }\left( e^{\left( 2\mu _{P}+\sigma _{P}^{2}\right) (t-\tau )} -1\right) \\&\quad -\frac{2\phi ^{2}(0)}{\mu _{P}\left( \mu _{P}+\sigma _{P}^{2}\right) }\left( e^{\mu _{P}(t-\tau )}-1\right) -\left( \frac{\phi (0)}{\mu _{P}}\left( e^{\mu _{P}(t-\tau )}-1\right) \right) ^2. \end{aligned}$$

• (ii) For $\tau \le t < T$,

  $$\begin{aligned} \mathbb {E}\left[ X_{t,T}^\tau \right]&= \frac{\phi (0)e^{\mu _{P}(t-\tau )}}{\mu _{P}}\left( e^{\mu _{P}(T-t)}-1\right) ;\\ \mathbb V\mathrm{ar}[X_{t,T}^\tau ]&= \frac{2\phi ^{2}(0)e^{\left( 2\mu _{P}+\sigma _{P}^{2}\right) (t-\tau )}}{\left( \mu _{P}+\sigma _{P}^{2}\right) \left( 2\mu _{P}+\sigma _{P}^{2}\right) }\left( e^{\left( 2\mu _{P}+\sigma _{P}^{2}\right) (T-t)} -1\right) \\&\quad -\frac{2\phi ^{2}(0)e^{\mu _{P}(t-\tau )}}{\mu _{P}\left( \mu _{P}+\sigma _{P}^{2}\right) }\left( e^{\mu _{P}(T-t)} -1\right) \\&\quad -\left( \frac{\phi (0)e^{\mu _{P}(t-\tau )}}{\mu _{P}}\left( e^{\mu _{P}(T-t)} -1\right) \right) ^2. \end{aligned}$$

• (iii) For $t \le \tau < T$,

  $$\begin{aligned} \mathbb {E}\left[ X_{t,T}^\tau \right]&= \int _{t-\tau }^0 \phi \left( u\right) \mathrm {d}u + \frac{\phi (0)}{\mu _{P}}\left( e^{\mu _{P}(T-\tau )}-1\right) ;\\ \mathbb V\mathrm{ar}[X_{t,T}^\tau ]&= \frac{2\phi ^{2}(0)}{\left( \mu _{P}+\sigma _{P}^{2}\right) \left( 2\mu _{P}+\sigma _{P}^{2}\right) }\left( e^{\left( 2\mu _{P}+\sigma _{P}^{2}\right) (T-\tau )} -1\right) \\&\quad -\frac{2\phi ^{2}(0)}{\mu _{P}\left( \mu _{P}+\sigma _{P}^{2}\right) }\left( e^{\mu _{P}(T-\tau )} -1\right) -\left( \frac{\phi (0)}{\mu _{P}}\left( e^{\mu _{P}(T-\tau )} -1\right) \right) ^2. \end{aligned}$$

In Hull and White (1987) similar outcomes are claimed for $\tau =0$ without providing a proof; for the sake of clarity, we give here a self-contained proof.

Proof

In order to prove the Lemma let us first compute the mean and the variance of IP(s) given in (A.1) for each $s>0$.

Fix $s >0$. Since $P_u > 0$ for each $u \in (0, s]$, by applying the Fubini theorem we get

$$\begin{aligned} \mathbb {E}\left[ IP(s)\right] = \mathbb {E}\left[ \int _{0}^{s}P_{u}\mathrm {d}u\right] =\int _{0}^{s}\mathbb {E}\left[ P_{u}\right] \mathrm {d}u, \end{aligned}$$

where, for each $u \ge 0$, we have

$$\begin{aligned} \mathbb {E}\left[ P_{u}\right]&= \phi (0) e^{\left( \mu _{P}-\frac{\sigma _{P}^{2}}{2}\right) u} \mathbb {E}\left[ e^{\sigma _{P}Z_{u}}\right] = \phi (0)e^{\mu _{P}u}, \end{aligned}$$

since P is a geometric Brownian motion with $P_0=\phi (0)$. Hence

$$\begin{aligned} \mathbb {E}\left[ IP(s)\right] =\phi (0)\int _{0}^{s} e^{\mu _{P}u} \mathrm {d}u=\frac{\phi (0)}{\mu _{P}}\left( e^{\mu _{P}(s)}-1\right) . \end{aligned}$$

As for the variance of IP(s), we have

$$\begin{aligned} \mathbb V\mathrm{ar}[IP(s)]&= \mathbb V\mathrm{ar}\left[ \int _0^{s} P_u \mathrm {d}u\right] = \mathbb {E}\left[ \left( \int _0^{s} P_u \mathrm {d}u\right) ^2\right] - \mathbb {E}\left[ \int _0^{s} P_u \mathrm {d}u\right] ^2, \end{aligned}$$

with

$$\begin{aligned} \mathbb {E}\left[ \left( \int _0^{s} P_u \mathrm {d}u\right) ^{2}\right]&= 2\mathbb {E}\left[ \int _{0}^{s}P_{v}dv\int _{0}^{v}P_{u}\mathrm {d}u\right] =2\mathbb {E}\left[ \int _{0}^{s}\int _{0}^{v}P_{u}P_{v}\mathrm {d}v\mathrm {d}u\right] \nonumber \\&= 2\int _{0}^{s}\int _{0}^{v}\mathbb {E}\left[ P_{u}P_{v}\right] \mathrm {d}v\mathrm {d}u, \end{aligned}$$

(B.1)

where the last equality again holds thanks to Fubini’s theorem. Moreover, by the independence property of the increments of Brownian motion, for $0< u < v \le s$, we get

$$\begin{aligned} \mathbb {E}\left[ P_{u}P_{v}\right]&= \mathbb {E}\left[ P_{u}^{2}e^{\left( \mu _{P}-\frac{\sigma _{P}^{2}}{2}\right) (v-u)+\sigma _{P}\left( Z_{v}-Z_{u}\right) }\right] \\&=e^{\left( \mu _{P}-\frac{\sigma _{P}^{2}}{2}\right) (v-u)} \mathbb {E}\left[ P_{u}^{2}\mathbb {E}\left[ e^{\sigma _{P}\left( Z_{v}-Z_{u}\right) }|\mathcal {F}_u^P\right] \right] \\&=e^{\left( \mu _{P}-\frac{\sigma _{P}^{2}}{2}\right) (v-u)} \mathbb {E}\left[ P_{u}^{2}\right] \mathbb {E}\left[ e^{\frac{\sigma _{P}^{2}\left( v-u)\right) }{2}}\right] =e^{\mu _{P}(v-u)}\mathbb {E}\left[ P_{u}^{2}\right] . \end{aligned}$$

Further,

$$\begin{aligned} \mathbb {E}\left[ P_{u}^{2}\right]&=\phi ^2(0)e^{2\left( \mu _{P}-\frac{\sigma _{P}^{2}}{2}\right) u}\mathbb {E}\left[ e^{2\sigma _{P}Z_{u}}\right] =\phi ^2(0)e^{\left( 2\mu _{P}+\sigma _{P}^{2}\right) u}. \end{aligned}$$

Hence

$$\begin{aligned} \mathbb {E}\left[ P_{u}P_{v}\right] =\phi ^2(0)e^{\mu _{P}(v-u)}e^{\left( 2\mu _{P}+\sigma _{P}^{2}\right) u}, \end{aligned}$$

(B.2)

and by plugging (B.2) into (B.1), we have

$$\begin{aligned} \mathbb {E}\left[ IP(s)^{2}\right]&= 2\phi ^2(0)\int _{0}^{s}e^{\mu _{P}v} \int _{0}^{v}e^{\left( \mu _{P}+\sigma _{P}^{2}\right) u} \mathrm {d}u\mathrm {d}v\\&= \frac{2\phi ^{2}(0)}{\left( \mu _{P}+\sigma _{P}^{2}\right) \left( 2\mu _{P}+\sigma _{P}^{2}\right) }\left( e^{\left( 2\mu _{P}+\sigma _{P}^{2}\right) s} -1\right) \\&\quad -\frac{2\phi ^{2}(0)}{\mu _{P}\left( \mu _{P}+\sigma _{P}^{2}\right) }\left( e^{\mu _{P}s} -1\right) . \end{aligned}$$

Finally, gathering the results we get

$$\begin{aligned} \mathbb V\mathrm{ar}[IP(s)]&= \frac{2\phi ^{2}(0)}{\left( \mu _{P}+\sigma _{P}^{2}\right) \left( 2\mu _{P}+\sigma _{P}^{2}\right) }\left[ e^{\left( 2\mu _{P}+\sigma _{P}^{2}\right) s} -1\right] \\&\qquad - \frac{2\phi ^{2}(0)}{\mu _{P}\left( \mu _{P}+\sigma _{P}^{2}\right) }\left( e^{\mu _{P}s} -1\right) -\left( \frac{\phi (0)}{\mu _{P}}\left( e^{\mu _{P} s} -1\right) \right) ^2. \end{aligned}$$

Note that $X_t^\tau$, with $t\in [0,\tau ]$, and $X_{t,T}^\tau$, with $t <T \le \tau$, are fully deterministic and the computation is trivial.

To prove points (i)–(iii), it suffices to observe that

$$\begin{aligned} X_t^\tau= & {} X_\tau ^\tau +IP(t-\tau ),\quad t > \tau , \\ X_{t,T}^\tau= & {} \int _{t-\tau }^0 \phi (u)\mathrm {d}u + IP(T-\tau ),\quad t \le \tau < T, \end{aligned}$$

and the computation easily follows once those of IP(s) are known for $s>0$.

To prove point (ii), it is worth noticing that given $0 \le v<s$

$$\begin{aligned} IP(s)-IP(v)&=\int _v^s P_u \mathrm {d}u = \int _v^{s} P_v e^{\left( \mu _P-\frac{\sigma _P^2}{2}\right) (u-v)+\sigma _P(Z_u-Z_v)} \mathrm {d}u\nonumber \\&{\mathop {=}\limits ^{(\mathrm{law})}} P_v\int _v^{s} e^{\left( \mu _P-\frac{\sigma _P^2}{2}\right) (u-v)+\sigma _P Z_{u-v} } \mathrm {d}u \nonumber \\&= P_v\int _0^{s-v} e^{\left( \mu _P-\frac{\sigma _P^2}{2}\right) r+\sigma _P Z_r} \mathrm {d}r=P_v IP(s-v), \end{aligned}$$

(B.3)

where $r=u-v$. To obtain the desired result, it suffices to note that, for $\tau \le t<T$,

$$\begin{aligned} X_{t,T}^\tau&=IP(T-\tau )-IP(t-\tau ) \end{aligned}$$

and apply (B.3). The computation of the mean and variance of the above difference is straightforward given the independence of Brownian increments. $\square$

The system given by Eqs. (2.1)–(2.2) is well-defined in $\mathbb {R}_+$, as stated in the following theorem, which also provides its explicit solution.

Theorem B.2

In the market model outlined in (2.1)–(2.2), the followings hold:

• (i) the bivariate stochastic delayed differential equation

  $$\begin{aligned} \left\{ \begin{array}{ll} \mathrm {d}S_t = \mu _S P_{t-\tau } S_t \mathrm {d}t+\sigma _S \sqrt{P_{t-\tau }}S_t\mathrm {d}W_t, \quad S_0=s_0 \in \mathbb {R}_+, \\ \mathrm {d}P_t =\mu _P P_t\mathrm {d}t+\sigma _P P_t\mathrm {d}Z_t, \quad P_t = \phi (t),\ t \in [-L,0], \\ \end{array} \right. \end{aligned}$$

  (B.4)

  has a continuous, $\mathbb {F}$-adapted, unique solution $(S,P)=\{(S_t,P_t),\ t \ge 0\}$ given by

  $$\begin{aligned} S_t&=s_0e^{\left( \mu _{S}-\frac{\sigma _{S}^{2}}{2}\right) \int _0^t P_{u-\tau } \mathrm {d}u+\sigma _S \int _0^t \sqrt{P_{u-\tau }}\mathrm {d}W_u},\quad t \ge 0, \end{aligned}$$

  (B.5)
  $$\begin{aligned} P_t&= \phi (0)e^{\left( \mu _{P}-\frac{\sigma _{P}^{2}}{2}\right) t+\sigma _P Z_t}, \quad t \ge 0. \end{aligned}$$

  (B.6)

  More precisely, S can be computed step by step as follows: for $k=0,1,2,\ldots$ and $t \in [k\tau ,(k+1)\tau ]$,

  $$\begin{aligned} S_t = S_{k\tau }e^{\left( \mu _{S}-\frac{\sigma _{S}^{2}}{2}\right) \int _{k\tau }^t P_{u-\tau } \mathrm {d}u+\sigma _S \int _ {k\tau }^t \sqrt{P_{u-\tau }}\mathrm {d}W_u}. \end{aligned}$$

  (B.7)

  In particular, $P_t \ge 0$$\mathbf {P}$-a.s. for all $t \ge 0$. If in addition, $\phi (0) > 0$, then $P_t > 0$$\mathbf {P}$-a.s. for all $t \ge 0$.

• (ii) Further, for every $t \ge 0$, the conditional distribution of $S_{t}$, given the integrated attention $X_t^\tau$, is log-Normal with mean $\log \left( s_0\right) + \left( \mu _{S}-\frac{\sigma _{S}^{2}}{2}\right) X_t^\tau$ and variance $\sigma _{S}^{2}X_t^\tau$.

• (iii) Finally, for every $t \in [0,\tau ]$, the random variable $\log \left( S_t \right)$ has mean $\log \left( s_0\right) + \left( \mu _{S}-\frac{\sigma _{S}^{2}}{2}\right) X_t^\tau$ and variance $\sigma _{S}^{2}X_t^\tau$; for every $t > \tau$, $\log \left( S_t\right)$ has mean and variance, respectively, given by

  $$\begin{aligned} \mathbb {E}\left[ \log \left( S_t \right) \right]&= \log \left( s_0\right) + \left( \mu _S-\frac{\sigma _S^2}{2}\right) \mathbb {E}\left[ X_t^\tau \right] ;\\ \mathbb V\mathrm{ar}\left[ {\log \left( S_t \right) }\right]&= \left( \mu _S-\frac{\sigma _S^2}{2}\right) ^2\mathbb V\mathrm{ar}[X_t^\tau ]+ \sigma _S^2 \mathbb {E}\left[ X_t^\tau \right] , \end{aligned}$$

  where $\mathbb {E}\left[ X_t^\tau \right]$ and $\mathbb V\mathrm{ar}[X_t^\tau ]$ are both provided by Lemma B.1, point (i).

Proof

Point (i). Clearly, S and P, given in (B.5) and (B.6), respectively, are $\mathbb {F}$-adapted processes with continuous trajectories. Similarly to (Mao and Sabanis 2013, Theorem 2.1), we provide existence and uniqueness of a strong solution to the pair of stochastic differential equations in system (B.4) by using forward induction steps of length $\tau$, without the need of checking any assumptions on the coefficients, e.g., the local Lipschitz condition and the linear growth condition.

First, note that the second equation in the system (B.4) does not depend on S, and its solution is well known for all $t \ge 0$. Clearly, Eq. (B.6) says that $P_t \ge 0$$\mathbf {P}$-a.s. for all $t \ge 0$ and that $\phi (0) > 0$ implies that the solution P remains strictly greater than 0 over $[0,+\infty )$, i.e., $P_t > 0$, $\mathbf {P}$-a.s. for all $t \ge 0$.

Next, by the first Eq. (B.4) and applying Itô’s formula to $\log \left( S_t\right)$, we get

$$\begin{aligned} \mathrm {d}\log \left( S_t\right) =\left( \mu _S-\frac{\sigma _S^2}{2}\right) P_{t-\tau } \mathrm {d}t + \sigma _S\sqrt{P_{t-\tau }} \mathrm {d}W_t, \end{aligned}$$

(B.8)

or equivalently, in integral form

$$\begin{aligned} \log \left( \frac{S_{t}}{s_{0}}\right) = \left( \mu _S-\frac{\sigma _S^2}{2}\right) \int _0^t P_{u-\tau } \mathrm {d}u + \sigma _S\int _0^t\sqrt{P_{u-\tau }} \mathrm {d}W_u,\quad t \ge 0. \end{aligned}$$

(B.9)

For $t \in [0,\tau ]$, (B.9) can be written as

$$\begin{aligned} \log \left( \frac{S_{t}}{s_{0}}\right) = \left( \mu _S-\frac{\sigma _S^2}{2}\right) \int _0^t \phi \left( u-\tau \right) \mathrm {d}u + \sigma _S\int _0^t\sqrt{\phi \left( u-\tau \right) } \mathrm {d}W_u, \end{aligned}$$

(B.10)

that is, (B.7) holds for $k = 0$.

Given that $S_t$ is now known for $t \in [0,\tau ]$, we may restrict the first Eq. (B.4) on $t \in [\tau , 2\tau ]$, so that it corresponds to consider (B.8) for $t \in [\tau , 2\tau ]$. Equivalently, in integral form,

$$\begin{aligned} \log \left( \frac{S_{t}}{S_{\tau }}\right) = \left( \mu _S-\frac{\sigma _S^2}{2}\right) \int _\tau ^t P_{u-\tau } \mathrm {d}u + \sigma _S\int _\tau ^t\sqrt{P_{u-\tau }} \mathrm {d}W_u. \end{aligned}$$

This shows that (B.7) holds for $k = 1$. Similar computations for $k=2,3,\ldots$, give the final result.

Point (ii). Set $Y_t:=\int _0^t\sqrt{\phi \left( t-\tau \right) } \mathrm {d}W_u$, for $t \in [0,\tau ]$ and $Y_t:= Y_{k\tau }+ \int _{k\tau }^t\sqrt{P_{u-\tau }} \mathrm {d}W_u$, for $t \in [k\tau ,(k+1)\tau ]$, with $k=1,2,\ldots$. Then, by applying the outcomes in Point (i) and the decomposition

$$\begin{aligned} \log \left( \frac{S_{t}}{s_{0}}\right) =\log \left( \frac{S_{t}}{S_{k\tau }}\right) +\sum _{j=0}^{k-1}\log \left( \frac{S_{(j+1)\tau }}{S_{j\tau }}\right) , \end{aligned}$$

for $t\in [k\tau ,(k+1)\tau ]$, with $k=1,2,\ldots$, we can write

$$\begin{aligned} \log \left( S_{t}\right) = \log (s_0)+\left( \mu _S-\frac{\sigma _S^2}{2}\right) X_t^\tau +\sigma _S Y_t,\quad t \ge 0. \end{aligned}$$

(B.11)

To complete the proof, it suffices to show that, for each $t\ge 0$ the random variable $Y_t$, conditional on $X_t^\tau$, is Normally distributed with mean 0 and variance $X_t^\tau$. This is straightforward from (B.10) if $t \in [0,\tau ]$. Otherwise, we first observe that since $Z_{u-\tau }$ is independent of $W_u$ for every $\tau < u \le t$, the distribution of $Y_t$, conditional on $\{Z_{u-\tau }:\ \tau< u \le t-\tau \}=\{P_{u-\tau }:\ \tau < u \le t-\tau \} =\mathcal {F}_{t-\tau }^P$, is Normal with mean 0 and variance $\sigma _S^2X_t^\tau$.

Now, for each $t > \tau$, the moment-generating function of $Y_t$, conditioned on the history of the process P up to time $t-\tau$, is given by

$$\begin{aligned} \mathbb {E}\left[ e^{aY_t}\Big |\mathcal {F}_{t-\tau }^{P}\right]&= e^{\int _0^t \frac{a^2}{2} P_{u-\tau } \mathrm {d}u} = e^{\frac{a^2}{2} \int _0^t P_{u-\tau } \mathrm {d}u}\\&= e^{\frac{a^2}{2}\left( \sqrt{X_t^\tau }\right) ^2}, \quad a \in \mathbb {R}, \end{aligned}$$

that only depends on $X_t^\tau$ up to time t, that is,

$$\begin{aligned} \mathbb {E}\left[ e^{aY_t}\Big |\mathcal {F}_{t-\tau }^{P}\right] =\mathbb {E}\left[ e^{aY_t}\Big |X_t^\tau \right] , \quad t > \tau . \end{aligned}$$

Point (iii). The proof is trivial for $t \in [0,\tau ]$. If $t >\tau$, (B.11) and Lemma B.1, point (i), together with the null-expectation property of the Itô integral, give

$$\begin{aligned} \mathbb {E}\left[ \log \left( S_t\right) \right]&= \log \left( s_0\right) + \left( \mu _S-\frac{\sigma _S^2}{2}\right) \left( X_\tau ^\tau +\frac{\phi (0)}{\mu _{P}}\left( e^{\mu _{P}(t-\tau )} -1\right) \right) . \end{aligned}$$

Now, we compute the variance of $\log \left( S_t\right)$. Since for each $t > \tau$ the random variable $Y_t$ has mean 0 conditional on $\mathcal {F}_{t-\tau }^P$, we have

$$\begin{aligned} \mathbb V\mathrm{ar}\left[ \log \left( S_t\right) \right]&= \mathbb {E}\left[ \log ^2\left( S_t\right) \right] - \left( \mathbb {E}\left[ \log \left( S_t\right) \right] \right) ^2\\&= \left( \mu _S-\frac{\sigma _S^2}{2}\right) ^2\mathbb {E}\left[ (X_t^\tau )^2\right] + 2\left( \mu _S-\frac{\sigma _S^2}{2}\right) \mathbb {E}\left[ X_t^\tau \mathbb {E}\left[ Y_t\Big |\mathcal {F}_{t-\tau }^P\right] \right] \\&\quad + \sigma _S^2 \mathbb {E}\left[ X_t^\tau \right] - \left( \mu _S-\frac{\sigma _S^2}{2}\right) ^2 \mathbb {E}\left[ X_t^\tau \right] ^2\\&= \left( \mu _S-\frac{\sigma _S^2}{2}\right) ^2\mathbb V\mathrm{ar}[X_t^\tau ]+ \sigma _S^2 \mathbb {E}\left[ X_t^\tau \right] . \end{aligned}$$

Thus, the proof is complete. $\square$

Let $T > 0$ be a fixed and finite time horizon. The following result ensures that the model given in (2.1)–(2.2) is arbitrage-free.

Lemma B.3

Let $\phi (t) > 0$, for each $t \in [-L,0]$, in (2.2). Then, every equivalent martingale measure $\mathbf {Q}$ for S defined on $(\Omega ,\mathcal {F}_T)$ has the following density

$$\begin{aligned} \frac{\mathrm {d}\mathbf {Q}}{\mathrm {d}\mathbf {P}}\bigg |_{\mathcal {F}_T}=:L_T^\mathbf {Q}, \quad \mathbf {P}-\text{ a.s. }, \end{aligned}$$

where $L_T^\mathbf {Q}$ is the terminal value of the $(\mathbb {F},\mathbf {P})$-martingale $L^\mathbf {Q}=\{L_t^\mathbf {Q},\ t \in [0,T]\}$ given by

$$\begin{aligned} L_t^\mathbf {Q}:=\mathcal {E}\left( -\int _0^\cdot \frac{\mu _S P_{s-\tau }-r(s)}{\sigma _S\sqrt{P_{s-\tau }}} \mathrm {d}W_s - \int _0^\cdot \gamma _s\mathrm {d}Z_s\right) _t, \quad t \in [0,T], \end{aligned}$$

(B.12)

for a suitable $\mathbb {F}$-progressively measurable process $\gamma =\{\gamma _t,\ t \in [0,T]\}$.

Proof

Firstly, we prove that formula (B.12) defines a probability measure $\mathbf {Q}$ equivalent to $\mathbf {P}$ on $(\Omega ,\mathcal {F}_T)$. This means we need to show that $L^\mathbf {Q}$ is an $(\mathbb {F},\mathbf {P})$-martingale, that is, $\mathbb {E}\left[ L_T^\mathbf {Q}\right] =1$. Since the $\mathbb {F}$-progressively measurable process $\gamma$ can be suitably chosen, to prove this relation we can assume $\gamma \equiv 0$, without loss of generality. Set

$$\begin{aligned} \alpha _t := -\frac{\mu _S P_{t-\tau }-r(t)}{\sigma _S \sqrt{P_{t-\tau }}}, \quad t \in [0,T]. \end{aligned}$$

(B.13)

We observe that since $\phi (t) > 0$, for each $t \in [-L,0]$, in (2.2), by Theorem B.2, point (i), we have that $P_{t-\tau } > 0$, $\mathbf {P}$-a.s. for all $t \in [0,T]$, so that the process $\alpha =\{\alpha _t,\ t \in [0,T]\}$ given in (B.13) is well-defined, as well as the random variable $L_T^\mathbf {Q}$. Clearly, $\alpha$ is an $\mathbb {F}$-progressively measurable process. Moreover, since the trajectories of the process P are continuous, then P is almost surely bounded on [0, T] and this implies that $\int _0^T|\alpha _u|^2 \mathrm {d}u < \infty$$\mathbf {P}$-a.s.; on the other hand, the condition $\phi (t)>0$, for every $t \in [-L,0]$, implies that almost every path of $\left\{ \frac{1}{\sigma _S \sqrt{P_{t-\tau }}},\ t \in [0,T]\right\}$ is bounded on the compact interval [0, T]. Set $\mathcal {F}_t^P:=\mathcal {F}_0^P=\{\Omega ,\emptyset \}$, for $t \le 0$. Then, $\alpha _u$, for every $u \in [0,T]$, is $\mathcal {F}_{T-\tau }^P$-measurable. Since $Z_{u-\tau }$ is independent of $W_u$, for every $u \in [\tau , T]$, the stochastic integral $\int _0^T \alpha _u \mathrm {d}W_u$ conditioned on $\mathcal {F}_{T-\tau }^P$ has a normal distribution with mean zero and variance $\int _0^T |\alpha _u|^2 \mathrm {d}u$. Consequently, the formula for the moment-generating function of a normal distribution implies

$$\begin{aligned} \mathbb {E}\left[ e^{\int _0^T \alpha _u \mathrm {d}W_u}\bigg |\mathcal {F}_{T-\tau }^P\right] =e^{\frac{1}{2}\int _0^T |\alpha _u|^2 \mathrm {d}u}, \end{aligned}$$

or equivalently

$$\begin{aligned} \mathbb {E}\left[ e^{\int _0^T \alpha _u \mathrm {d}W_u-\frac{1}{2}\int _0^T |\alpha _u|^2 \mathrm {d}u}\bigg |\mathcal {F}_{T-\tau }^P\right] =1. \end{aligned}$$

(B.14)

Taking the expectation of both sides of (B.14) immediately yields $\mathbb {E}\left[ L_T^\mathbf {Q}\right] =1$. Now, set $\widetilde{S}_t:= \displaystyle \frac{S_t}{B_t}$, for each $t \in [0,T]$. It remains to verify that the discounted Bitcoin price process $\widetilde{S}=\{\widetilde{S}_t,\ t \in [0,T]\}$ is an $(\mathbb {F},\mathbf {Q})$-martingale. By Girsanov’s theorem, under the change of measure from $\mathbf {P}$ to $\mathbf {Q}$, we have two independent $(\mathbb {F},\mathbf {Q})$-Brownian motions $W^\mathbf {Q}=\{W_t^\mathbf {Q},\ t \in [0,T]\}$ and $Z^\mathbf {Q}=\{Z_t^\mathbf {Q},\ t \in [0,T]\}$ defined, respectively, by

$$\begin{aligned} W_t^\mathbf {Q}:= W_t - \int _0^t\alpha _u \mathrm {d}u, \quad Z_t^\mathbf {Q}:= Z_t + \int _0^t\gamma _u \mathrm {d}u, \quad t \in [0,T]. \end{aligned}$$

Under the martingale measure $\mathbf {Q}$, the discounted Bitcoin price process $\widetilde{S}$ satisfies the following dynamics

$$\begin{aligned} \mathrm {d}\widetilde{S}_t&= \widetilde{S}_t\sigma _S\sqrt{P_{t-\tau }}\mathrm {d}W_t^\mathbf {Q}, \quad \widetilde{S}_0=s_0 \in \mathbb {R}_+, \end{aligned}$$

which implies that $\widetilde{S}$ is an $(\mathbb {F},\mathbf {Q})$-local martingale. Finally, proceeding as above it is easy to check that $\widetilde{S}$ is a true $(\mathbb {F},\mathbf {Q})$-martingale. $\square$

Here $\mathcal {E}(Y)$ denotes the Doléans-Dade exponential of an $(\mathbb {F}, \mathbf {P})$-semimartingale Y. Then, Lemma B.3 ensures that the space of equivalent martingale measures for S is described by (B.12). Note that the attention factor dynamics under $\mathbf {Q}$ in the Bitcoin market is given by

$$\begin{aligned} \mathrm {d}P_t = (\mu _P - \sigma _P \gamma _t)P_t\mathrm {d}t + \sigma _P P_t \mathrm {d}Z_t^\mathbf {Q}, \quad P_t = \phi (t),\ t \in [-L,0]. \end{aligned}$$

The process $\gamma$ can be interpreted as the price for attention risk, i.e., the perception associated to the future direction of investor attention on the Bitcoin market. The probability measure which corresponds to the choice $\gamma \equiv 0$ in (B.12) is the so-called minimal martingale measure. Let us focus on the special case of a constant process $\gamma$; in this setting the equivalent martingale measure, denoted by $\mathbf {Q}^\gamma$, is parameterized by the constant $\gamma$ which governs the change of drift of the $(\mathbb {F},\mathbf {P})$-Brownian motion Z. As a special case of the above dynamics, the two independent $(\mathbb {F},\mathbf {Q}^\gamma )$-Brownian motions ${\widehat{W}}=\{{\widehat{W}}_t,\ t \in [0,T]\}$ and ${\widehat{Z}}= \{{\widehat{Z}}_t,\ t \in [0,T]\}$ are defined, respectively, by

$$\begin{aligned} {\widehat{W}}_t&:= W_t + \int _0^t\frac{\mu _S P_{s-\tau }-r(s)}{\sigma _S\sqrt{P_{s-\tau }}} \mathrm {d}s,\quad t \in [0,T],\\ {\widehat{Z}}_t&:= Z_t+\gamma \, t, \quad t \in [0,T]. \end{aligned}$$

Denote by $\widetilde{S}_t=\{\widetilde{S}_t,\ t \in [0,T]\}$ the discounted Bitcoin price process defined as $\displaystyle \widetilde{S}_t:=\frac{S_t}{B_t}$, for each $t \in [0,T]$. Then, on the probability space $(\Omega ,\mathcal {F},\mathbf {Q}^\gamma )$, the pair $(\widetilde{S},P)$ satisfies the following system of stochastic delayed differential equations:

$$\begin{aligned} \left\{ \begin{array}{ll} \mathrm {d}{\widetilde{S}}_t = \sigma _S \sqrt{P_{t-\tau }}{\widetilde{S}}_t\mathrm {d}{\widehat{W}}_t, \quad {\widetilde{S}}_0=s_0 \in \mathbb {R}_+, \\ \mathrm {d}P_t =(\mu _P-\gamma \,\sigma _P) P_t\mathrm {d}t+\sigma _P P_t\mathrm {d}{\widehat{Z}}_t, \quad P_t = \phi (t),\ t \in [-L,0], \\ \end{array} \right. \end{aligned}$$

(B.15)

By Theorem B.2, point (i), the explicit expression of the solution to (B.15), which provides the discounted Bitcoin price $\widetilde{S}_t$, at any time $t \in [0,T]$, is given by

$$\begin{aligned} {\widetilde{S}}_t=s_0e^{\sigma _S\int _0^t\sqrt{P_{u-\tau }}\mathrm {d}{\widehat{W}}_u - \frac{\sigma _S^2}{2}\int _0^t P_{u-\tau } \mathrm {d}u},\quad t \in [0,T], \end{aligned}$$

(B.16)

with the representation of the attention factor P still provided by (B.6) where the parameter $\mu _P$ is replaced by ${\tilde{\mu }}_P=\mu _P -\gamma \,\sigma _P$. All the outcomes on the integrated attention variable $X_{t,T}$ hold under the transformed probability measure $\mathbf {Q}^\gamma$ by replacing parameter $\mu _P$ with ${\tilde{\mu }}_P$.

In what follows we prove Lemma 3.1.

Proof of Lemma 3.1

By applying (Levy 1992) we have (see “Appendix A”) that the distribution of $A_1-X_\tau ^\tau$ can be approximated by a log-normal with parameters

$$\begin{aligned} \alpha _1&=\log \left( \frac{{\mathbb {E}}[IP(\Delta -\tau )]^2}{\sqrt{{\mathbb {E}}[IP(\Delta -\tau )^2]}} \right) ,\qquad \nu _1^2 = \log \left( \frac{{\mathbb {E}}[IP(\Delta -\tau )^2]}{{\mathbb {E}}[IP(\Delta -\tau )]^2} \right) . \end{aligned}$$

By applying the outcomes of Lemma B.1, we have

$$\begin{aligned} {\mathbb {E}}^{\mathbf {Q}^\gamma }[A_1]&= {\mathbb {E}} [P_0] \frac{e^{{\tilde{\mu }}_P (\Delta -\tau )}-1}{{\tilde{\mu }}_P} = P_0 \frac{e^{{\tilde{\mu }}_P (\Delta -\tau )}-1}{{\tilde{\mu }}_P}\\ {\mathbb {E}}^{\mathbf {Q}^\gamma }[A_1^2]&= P_0^2 \left( \frac{2}{{\tilde{\mu }}_P+\sigma _P^2} \left[ \frac{e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) (\Delta -\tau )}-1}{2{\tilde{\mu }}_P+\sigma _P^2}-\frac{e^{{\tilde{\mu }}_P(\Delta -\tau )}-1}{{\tilde{\mu }}_P} \right] \right) . \end{aligned}$$

Hence

$$\begin{aligned} \alpha _1&= \log \left( \frac{P_0\left( \frac{e^{{\tilde{\mu }}_P \Delta }-1}{{\tilde{\mu }}_P} \right) ^2}{\sqrt{ \frac{2}{{\tilde{\mu }}_P+\sigma _P^2} \left[ \frac{e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta }-1}{2{\tilde{\mu }}_P+\sigma _P^2}-\frac{e^{{\tilde{\mu }}_P\Delta }-1}{{\tilde{\mu }}_P} \right] }}\right) \\ \nu _1^2&= \log \left( \frac{ \frac{2}{{\tilde{\mu }}_P+\sigma _P^2} \left[ \frac{e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta }-1}{2{\tilde{\mu }}_P+\sigma _P^2}-\frac{e^{{\tilde{\mu }}_P\Delta }-1}{{\tilde{\mu }}_P} \right] }{\left( \frac{e^{{\tilde{\mu }}_P \Delta }-1}{{\tilde{\mu }}_P} \right) ^2}\right) . \end{aligned}$$

By applying simple computation, we get the outcomes for part (i). Moreover,

$$\begin{aligned} {\mathbb {E}}^{\mathbf {Q}^\gamma }[A_i]&= {\mathbb {E}}^{\mathbf {Q}^\gamma } [P_{i-1}] \frac{e^{{\tilde{\mu }}_P \Delta }-1}{{\tilde{\mu }}_P} = {\mathbb {E}}^{\mathbf {Q}^\gamma }[P_{i-2}] \; e^{{\tilde{\mu }}_P\Delta } \; \frac{e^{{\tilde{\mu }}_P \Delta }-1}{{\tilde{\mu }}_P} = {\mathbb {E}}[A_{i-1}]\; e^{{\tilde{\mu }}_P\Delta }\\ {\mathbb {E}}^{\mathbf {Q}^\gamma }[A_i^2]&= {\mathbb {E}}^{\mathbf {Q}^\gamma } [P_{i-1}^2] \left( \frac{2}{{\tilde{\mu }}_P+\sigma _P^2} \left[ \frac{e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta }-1}{2{\tilde{\mu }}_P+\sigma _P^2}-\frac{e^{{\tilde{\mu }}_P\Delta }-1}{{\tilde{\mu }}_P} \right] \right) \\&= {\mathbb {E}}^{\mathbf {Q}^\gamma }[P_{i-2}^2] e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta } \left( \frac{2}{{\tilde{\mu }}_P+\sigma _P^2} \left[ \frac{e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta }-1}{2{\tilde{\mu }}_P+\sigma _P^2}-\frac{e^{{\tilde{\mu }}_P\Delta }-1}{{\tilde{\mu }}_P} \right] \right) \\&= {\mathbb {E}}^{\mathbf {Q}^\gamma }[A_{i-1}^2]\; e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta }. \end{aligned}$$

Then

$$\begin{aligned} \alpha _i&= \log \left( \frac{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_i]^2}{\sqrt{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_i^2]}} \right) = \log \left( \frac{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_{i-1}]^2 \left( e^{{\tilde{\mu }}_P\Delta }\right) ^2}{\sqrt{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_{i-1}^2] \; e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta }}} \right) \\&= \log \left( \frac{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_{i-1}]^2}{\sqrt{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_{i-1}^2]}} \right) + \log \left( \frac{\left( e^{{\tilde{\mu }}_P\Delta }\right) ^2}{\sqrt{e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta }}}\right) = \alpha _{i-1} + \left( {\tilde{\mu }}_P-\frac{\sigma _P^2}{2} \right) \Delta , \\ \nu _i^2&= \log \left( \frac{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_i^2]}{\sqrt{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_i]^2}} \right) = \log \left( \frac{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_{i-1}^2] \; e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta }}{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_{i-1}]^2 \; e^{({\tilde{\mu }}_P\Delta )^2}} \right) \\&=\log \left( \frac{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_{i-1}^2]}{\sqrt{{\mathbb {E}}^{\mathbf {Q}^\gamma }[A_{i-1}]^2}} \right) + \log \left( \frac{e^{\left( 2{\tilde{\mu }}_P+\sigma _P^2\right) \Delta }}{e^{({\tilde{\mu }}_P\Delta )^2}} \right) = \nu _{i-1}^2 + \sigma _P^2\Delta . \end{aligned}$$

Conditioning to $A_{i-1}$, we get

$$\begin{aligned} \alpha _i&= \log \left( A_{i-1}\right) + \left( {\tilde{\mu }}_P-\frac{\sigma _P^2}{2}\right) \Delta , \\ \nu _i^2&= \sigma _P^2\Delta , \end{aligned}$$

which gives part (ii). $\square$

Appendix C: Finite sample behavior of QML estimates

In order to check the goodness of the log-likelihood approximation introduced in Theorem 3.2, we apply the proposed estimation method to simulated data and assume, for the sake of simplicity, $\tau =0$. We simulate m samples of length n for the processes in (2.1) and (2.2) assuming a constant finer observation step $\delta$; we extract corresponding samples for ${\mathbf {R,A}}$ at a lower frequency, with observation step $\Delta =r\delta$. In the numerical exercise we choose $n=730$, $m=1000$, $\delta =\frac{1}{365}$ (daily observations), $\Delta =7\delta$ (weekly observations); parameters values are set as $\mu _P=2$, $\sigma _P=0.5$$\mu _S=0.05$, $\sigma _S=0.3$. We end with 1000 samples of 104 observations for $\left( {\mathbf {R,A}}\right)$; for each sample we estimate the parameters by means of the quasi-maximum likelihood as suggested in previous subsection. The results are summed up in Table 11.

Table 11 Parameter fit with simulated data of QML method

We also performed a t test in order to check for estimation bias. The fitted values of $\mu _P,\mu _S,\sigma _S$ are close in mean to their theoretical value and with a reasonable standard deviation; the p-values of the t test confirm that estimated are not biased. Different conclusions are in order as for parameter $\sigma _P$ which estimations is by no doubt biased. In Fig. 2 we plot the histograms of the estimated as well as the fitted normal distribution and the expected mean of the asymptotic distribution. Pictures confirm the biasedness of the estimator for $\sigma _P$ but all other estimates perform well and outcomes may become better by increasing the sample length. The simulation exercise have been repeated by letting the parameters values, the number and the sample length vary obtaining analogous qualitative results.

Fig. 2

Histogram of parameter fit with simulated data of QML method

In order to disentangle the contribution of the Levy approximation (Levy 1992) to the estimation bias, we suggest to apply the method of moments to estimate $\mu _P$ and $\sigma _P$ considering the whole sample for P generated at the finer observation step $\delta$ to compute the sample mean and sample variance of the attention realizations. If we then we plug the estimated values in the likelihood (3.2) in order to estimate $\lbrace \mu _S, \sigma _S \rbrace$ these two estimates remain unchanged; in fact the likelihood may be maximized separately with respect to $\mu _P,\sigma _P$ and $\mu _S,\sigma _S$ since each of the two addend in the likelihood expression depends on just one of this pairs. The results of this alternative estimation method are reported in Table 12.

Table 12 Parameter fit with simulated data of Moments method

Fig. 3

Histogram of parameter $\lbrace \mu _P, \sigma _P \rbrace$ fit with simulated data of QML method and Moments method at finer step $\delta$

To visualize the bias of $\lbrace \sigma _P \rbrace$ we plot in Fig. 3 the histogram of parameter $\lbrace \mu _P, \sigma _P \rbrace$ fit with simulated data using the two methods. As we can see using the two step procedure we obtain better estimates of $\lbrace \sigma _P \rbrace$ both in terms of expected value and standard deviation. It is evident from Table 12 and Fig. 3 that the estimation of $\sigma _P$ is not biased in this case hence the estimation bias may essentially be attributed to the aggregation of the attention over time intervals and to the corresponding approximating distribution. Hence, whether the attention factor is observed at a finer step than the price, the above separate estimation is more reliable.