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Advanced model calibration on bitcoin options

  • Dilip B. Madan
  • Sofie ReynersEmail author
  • Wim Schoutens
Original Article

Abstract

In this paper, we investigate the dynamics of the bitcoin (BTC) price through the vanilla options available on the market. We calibrate a series of Markov models on the option surface. In particular, we consider the Black–Scholes model, Laplace model, five variance gamma-related models and the Heston model. We examine their pricing performance and the optimal risk-neutral model parameters over a period of 2 months. We conclude with a study of the implied liquidity of BTC call options, based on conic finance theory.

Keywords

Cryptocurrency Modelling Bitcoin Calibration 

JEL Classification

C52 C60 G10 

1 Introduction

In 2008, an anonymous (group of) author(s) posted a white paper under the name Nakamoto (2008). The paper suggests an electronic payment system, Bitcoin, that does not depend on any central authority. In particular, the authority is replaced by a distributed ledger, the so-called blockchain, that records all transactions. Only verified blocks are added to the blockchain. Verification is—roughly speaking—based on solving a complicated puzzle and hence demands a lot of CPU power. People who put their computer at work are, therefore, rewarded with bitcoins: they are mining bitcoins. This is how new bitcoins are created. For a detailed overview of the key concepts of Bitcoin, we refer to Becker et al. (2013), Dwyer (2015) and Segendorf (2014).

In October 2009, New Liberty Standard determined the first bitcoin exchange rate. Its value was based on the price of the electricity needed for mining in the United States. They concluded that one US Dollar was worth 1309.03 bitcoins, or a BTC–USD rate of approximately 0.0008. Later on, multiple exchange platforms emerged. The price of bitcoin gradually increased and reached parity with the US Dollar in 2011. At the end of 2013, the barrier of 1000 USD was reached, but this did not hold for long. The real breakthrough came in 2017, due to a combination of events, among which the announcement of bitcoin derivatives.

On 24 July 2017, the US Commodity Futures Trading Commission (CFTC) granted permission to LedgerX to clear and settle derivative contracts for digital currencies. Three months later, LedgerX became the first US exchange platform trading bitcoin options. Bitcoin derivatives were born. In December 2017, the CFTC authorized the Chicago Mercantile Exchange (CME) and the Cboe Futures Exchange (CFE) to start trading BTC futures. Some argue that the peak in BTC–USD rate mid-December (Fig. 1) was caused by the launch of their futures. Before the existence of BTC futures, investors had indeed few possibilities to bet on a decreasing bitcoin price. Futures made this easier, which could be a reason for the sharp decline in price late December 2017.
Fig. 1

Historical price chart of the BTC–USD rate

A lot of research has been carried out to gain insight into the dynamics of the bitcoin price. A substantial part of this research focuses on time series modelling, see for instance (Kristoufek 2013; Garcia et al. 2014; Kjærland et al. 2018). These authors explain the bitcoin price fluctuations using several technical and socio-economical factors, e.g. the cost of mining bitcoins. Baur et al. (2018) and Dwyer (2015) on the other hand compare historical bitcoin returns with those of other assets. However, the market of bitcoin derivatives is not taken into account yet. Literature on bitcoin derivatives usually focuses on their regulation instead of their valuation. Bitcoin derivatives and their prices are barely studied from a modelling point of view. Some articles regarding option pricing (models) are available, see for instance (Chen et al. 2018; Cretarola and Figà-Talamanca 2017), but an extensive (empirical) analysis is missing.

In this paper, we aim to fill the gap between the bitcoin price modelling and bitcoin derivative products. We use some traditional and more advanced asset pricing models to get a better understanding of the bitcoin market. In particular, we investigate how well a series of frequently used Markov–Martingale models match the bitcoin market. Bitcoin vanilla options lie at the heart of this research. Given the option surface(s), we fit the Black–Scholes model, Laplace model, variance gamma model, bilateral (double) gamma model, VG Sato model, VG-CIR model and Heston model on the bitcoin market. We start by calibrating the models on the option surface and elaborate on the different fits. We discuss and compare the implied volatility smiles according to the Laplace model and the Black–Scholes model. Finally, we investigate the Black–Scholes implied liquidity, based on the theory of conic finance.

We believe this study is useful, because it gives us for the first time a glimpse of the risk-neutral \({\mathbb {Q}}\)-measure. Time series modelling lacks this feature, and also bitcoin futures do not contain this information. The calibrated models provide us moreover with new insights in bitcoin’s volatility, for instance through the initial volatility and the vol-of-vol parameter in the Heston model. Due to the turbulent behavior of the bitcoin price, the traditional arguments against the Black–Scholes model in an equity setting do also apply here. In fact, these effects are here even more pronounced, which encourages the use of advanced jump processes and stochastic volatility to model the bitcoin market even more.

2 The data

The study is performed on vanilla options with underlying BTC price index, consisting of quotes according to six leading BTC–USD exchanges: Bitfinex, Bitstamp, GDAX, Gemini, Itbit and Kraken. Options traded are European style and cash settled in BTC, while USD acts as the numeraire. Option datasets used in this study were collected manually from various unregulated exchanges. The data are extracted each Friday approximately 2 h after expiration, to avoid extremely short-dated options. In this way, we collect ten option surfaces, starting from Friday 29 June 2018 until Friday 31 August 2018. Each surface contains options with four or five different expiries, ranging from 1 week to at most 8 months. Longer term options are not (yet) frequently traded and are hence not included in the analysis.

Before we start modelling, some additional remarks are given about the option surfaces. First of all, the options are expensive, which is directly related to bitcoin’s high volatility. More remarkable is the range of the strikes. For short-term options (up to 1 month), the strikes roughly range from 85 to 125% of the initial value of the underlying. For higher maturities, however, it is not exceptional to encounter strike ranging from 100 to 500%. Last, we checked whether the no-arbitrage assumption is roughly satisfied. Consider the two-price put-call parity, i.e.
$$\begin{aligned} {\hbox {bidEC}}(K,T) - {\hbox {askEP}}(K,T) \le S_0 - K \le {\hbox {askEC}}(K,T) - {\hbox {bidEP}}(K,T), \end{aligned}$$
(1)
where \(S_0\) is the initial bitcoin price and e.g. \({\hbox {bidEC}}(K,T)\) denotes the bid price of a European call option with strike price K and time to maturity T. The inequality is respected with only a few exceptions. Out of the 491 strike–maturity combinations, ten violate this relation. Note that we assumed here zero risk-free interest rates, both for BTC and USD. The same assumption is implicitly made in the remainder of the article.

3 Model calibration

We first examine how well the pricing models fit the atypical bitcoin option surface. To this purpose, we calibrate the models on the entire surface of one particular day, Friday 29 June 2018, for which the data are provided in “Appendix B”. Note that the choice of this day is arbitrary. Second, the stability of the risk-neutral model parameters over time is investigated based on a weekly time series of option surfaces ranging from 29 June until 31 August. The length and frequency of this time series are again arbitrarily chosen.

3.1 Calibration procedure

The option surface is cleaned by considering only out of the money options for which both a bid and an ask price are available. Since out of the money options have no intrinsic value, their prices only consist of the so-called time value. Considering only out of the money options, therefore, leads to a more robust calibration. The optimization procedure is based on the mid-prices of the remaining options. We determine optimal model parameters by minimizing the root mean squared error (rmse) of the model prices with respect to the market prices. The resulting fit is evaluated in terms of three other measures: the average absolute error (aae), the average pricing error (ape), i.e. the aae as a percentage of the average option price, and the average relative pricing error (arpe).
$$\begin{aligned} {\hbox {rmse}}&= \sqrt{\frac{1}{n} \sum _{i=1}^{n}({{\text {marketprice}}}_i - {{\text {modelprice}}}_i )^2 }, \end{aligned}$$
(2)
$$\begin{aligned} {{{\mathrm{aae}}}}&= \frac{1}{n} \sum _{i=1}^{n} |{\text {marketprice}}_i - {\text {modelprice}}_i |, \end{aligned}$$
(3)
$$\begin{aligned} {\mathrm{ape}}&= \frac{ \sum _{i=1}^{n} |{\text {marketprice}}_i - {\text {modelprice}}_i |}{ \sum _{i=1}^{n} {\text {marketprice}}_i }, \end{aligned}$$
(4)
$$\begin{aligned} {\mathrm{arpe}}&= \frac{1}{n} \sum _{i=1}^{n} \frac{|{\text {marketprice}}_i - {\text {modelprice}}_i |}{{\text {marketprice}}_i}. \end{aligned}$$
(5)
When a model does not support a closed-form pricing formula for vanilla options, model prices are calculated with the Fast Fourier-Transform (FFT) pricing method developed by Carr and Madan (1999). The price of a European call option with strike K and time to maturity T is given by
$$\begin{aligned} C(K,T) = \frac{\exp (-\alpha \log (K))}{\pi }\int _{0}^{\infty }\exp (-iv\log (K))\rho (v){{\mathrm {d}}}v, \end{aligned}$$
(6)
where
$$\begin{aligned} \rho (v) = \frac{\phi _T(v-(\alpha +1)i)}{\alpha ^2 + \alpha - v^2 + i(2\alpha +1)v} \end{aligned}$$
(7)
and \(\alpha = 1.5\). The characteristic function \(\phi _t\) of the log-price process at time t is specified below for each model.

3.2 The Black–Scholes model

The Black–Scholes model is a natural reference point to start the analysis. Let \((S_t)_{t\ge 0}\) denote the bitcoin price process. In a zero interest rate Black–Scholes world, the price process is given by
$$\begin{aligned} S_t = S_0 \exp \left( \frac{-\sigma ^2t}{2} + \sigma W_t\right), \end{aligned}$$
(8)
where \((W_t)_{t \ge 0}\) is a standard Brownian motion, \(S_0\) the initial value of the bitcoin price and \(\sigma\) denotes the volatility. Calibrating the model on the option surface of 29 June results in an overall best fitting volatility of 70.47%. In Fig. 2a, the Black–Scholes prices are plotted against the observed market prices to visualize the fit. As expected, the optimal volatility is relatively high in comparison with more common assets. The evolution of the optimal Black–Scholes volatility in the next weeks of the dataset is shown in Fig. 2b. More details on the goodness of fit are given in Table 1 and Fig. 11.
Table 1

Summary of the pricing performance on 29 June

 

rmse

\({\mathrm{ape}}\)

\({\mathrm{aae}}\)

\({\mathrm{arpe}}\)

BS

27.8368

0.1142

22.0807

0.3379

Laplace

34.2618

0.1384

26.7563

0.4990

Heston

9.2144

0.0360

6.9505

0.1207

VG

20.4969

0.0858

16.5809

0.2410

BG

16.5971

0.0691

13.3652

0.2383

BDG

9.9357

0.0322

6.2255

0.1436

VG Sato

10.5010

0.0414

8.0007

0.1011

VG-CIR

9.2275

0.0350

6.7651

0.1028

Fig. 2

Black–Scholes calibration

Alternatively, we can compute the implied volatility for each option in the surface. Implied volatility smiles based on the mid-prices of out of the money vanilla options as published on Friday 29 June 2018 are displayed in Fig. 4 for each maturity. The 1-week implied volatility smiles are the only graphs that actually resemble the shape of a smile. However, recall that the strike range for the larger maturities is highly asymmetric around the initial BTC–USD rate.

3.3 The Laplace model

Instead of modelling the log-returns with a normal distribution, the Laplace model employs the heavier tailed Laplace distribution. The model has again one parameter, \(\sigma\), corresponding to the volatility. A brief summary of the model specifications is given in “Appendix A”. Model calibration on the option surface of 29 June leads to an optimal volatility \(\sigma\) of 67.35%. The optimal volatility is similar to its Black–Scholes counterpart, but the overall price fits in Figs. 2a and 3a are not the same. While the Black–Scholes model underestimates the deep out of the money call option prices in the right tail, the Laplace model overestimates them. However, neither of these models captures the option surface accurately.
Fig. 3

Laplace calibration

We can extend the comparative analysis by calculating the implied volatilities according to the Laplace model. Across all maturities, the Laplacian implied volatility in Fig. 4 exceeds the Black–Scholes implied volatility near the money. For larger strikes, the Laplacian implied volatility decreases and stabilizes, while the Black–Scholes implied volatility behaves adversely and increases. This can be explained by the tail behavior of the log-returns in both models, being, respectively, Laplacian and Gaussian distributed. The phenomenon of differing implied (Lévy) volatilities was already discussed in Corcuera et al. (2009).
Fig. 4

Implied volatility smiles corresponding to out of the money options on 29 June 2018

3.4 Heston’s stochastic volatility model

Besides the high volatility of the bitcoin price, the high variability of this volatility should be analyzed. Incorporating a stochastic volatility process is hence a reasonable next step. We consider Heston’s stochastic volatility model (Heston 1993) with parameters \(\kappa , \rho , \theta ,\eta\) and \(v_0\).1 Formally, we model the risk-neutral bitcoin price process \((S_t)_{t\ge 0}\) as follows:
$$\begin{aligned} \frac{{{\mathrm {d}}}S_t}{S_t} = \sqrt{v_t}{{\mathrm {d}}}W_t, \end{aligned}$$
(9)
where the squared volatility process \(v_t\) is given by
$$\begin{aligned} {{\mathrm {d}}}v_t = \kappa (\eta - v_t){{\mathrm {d}}}t + \theta \sqrt{v_t}{{\mathrm {d}}}{\tilde{W}}_t \end{aligned}$$
(10)
with \((W_t)_{t\ge 0}\) and \(({\tilde{W}}_t)_{t\ge 0}\) two correlated standard Brownian motions such that
$$\begin{aligned} {\hbox {Cov}}\left( {{\mathrm {d}}}W_t{{\mathrm {d}}}{\tilde{W}}_t \right) = \rho {{\mathrm {d}}}t. \end{aligned}$$
(11)
The characteristic function \(\phi _t(u)\) of \(\log (S_t)\) is given by
$$\begin{aligned} \phi _t(u)&= \exp ( iu\log (S_0) ) \nonumber \\&\quad \times \exp \left( \eta \kappa \theta ^{-2} \left( \left( \kappa - \rho \theta ui - d\right) t - 2\log \left( (1-g\exp (-\,{\hbox {d}}t)) (1-g)^{-1} \right) \right) \right) \nonumber \\&\quad \times \exp \left( v_0\theta ^{-2}( \kappa - \rho \theta iu - d)\left( 1 - \exp \left( -\,{\hbox {d}} t\right) \right) (1-g\exp (-\,{\hbox {d}} t))^{-1} \right) \end{aligned}$$
(12)
with
$$\begin{aligned} d&= ((\rho \theta ui - \kappa )^2 - \theta ^2(-iu - u^2) )^{1/2}, \end{aligned}$$
(13)
$$\begin{aligned} g&= ( \kappa - \rho \theta ui - d )(\kappa - \rho \theta ui + d)^{-1}. \end{aligned}$$
(14)
Figure 5a shows a good fit. The optimal vol-of-vol parameter \(\theta\) takes a high value of 236%, indicating that the previous assumption of constant volatility was indeed too strong. The vol-of-vol parameter gives, moreover, new information about bitcoin’s volatility, which is often said to be high. This study indeed reflects a high volatility—as confirmed by the optimal values of \(\eta\) and \(v_0\)—but it additionally shows that bitcoin’s volatility itself is very volatile. Note that the optimal value of \(\theta\) increases sharply over time, reaching a maximum of nearly 1300% at the end of July.
Fig. 5

Heston calibration based on the FFT method

3.5 Gamma models

The turbulent bitcoin price encourages us to include jumps in the model. The presence of jumps in the bitcoin market is, moreover, extensively motivated in Scaillet et al. (2018). In gamma-based models, the market is modelled by a subtraction of two independent gamma processes, being pure jump processes that model, respectively, the upward and downward motions. We first fit the well-known variance gamma (VG) model (Madan and Milne 1991; Madan and Seneta 1990; Madan et al. 1998) on the bitcoin option surface. Second, the bilateral gamma (BG) model is considered. This model extends the variance gamma model by modelling the speed of positive and negative jumps separately. More generalizations and extensions consist in imposing the model parameters themselves to be gamma distributed, as explained in Madan et al. (2018). We consider one such model, the bilateral double gamma (BDG) model. Two alternative generalized variance gamma models, the VG Sato and the VG-CIR models, conclude our selection of gamma models.

3.5.1 The variance gamma model

We calibrate the variance gamma model in the original \((\sigma , \ \nu , \ \theta )\)-parametrization. The risk-neutral process of the bitcoin price is modelled by
$$\begin{aligned} S_t = S_0 \frac{\exp (X_{t})}{E[\exp (X_{t})]}, \end{aligned}$$
(15)
where \((X_t)_{t\ge 0}\) is a variance gamma process, i.e. it starts at zero, has independent and stationary increments and the increment \(X_{s+t} - X_s\) follows a \({\mathrm{VG}}(\sigma \sqrt{t}, \ \nu /t, \ \theta t)\) distribution over the time interval \([s, s+t]\). In this parametrization, \(\theta\) controls the skewness, \(\nu\) the kurtosis and \(\sigma\) the volatility of the risk-neutral distribution of the log-returns. Equivalently, the characteristic function of \(\log (S_t)\) is given by
$$\begin{aligned} \phi _t(u) = \exp \left( iu \left( \log (S_0) - \log (\phi _{X_t}(-i)) \right) \right) \times \phi _{X_t}(u) \end{aligned}$$
(16)
with \(\phi _{X_t}\) the characteristic function of the VG process at time t,
$$\begin{aligned} \phi _{X_t}(u) = \left( 1 - iu \theta \nu + \frac{1}{2}\sigma ^2\nu u^2 \right) ^{-t/\nu }. \end{aligned}$$
(17)
Figure 6a summarizes the calibration results of 29 June. Figure 6b again reflects the period of high volatility in the first 2 weeks of August. The value of \(\theta\) becomes strongly negative in this period, corresponding to more negative skewness in the risk-neutral distribution of log-returns.
Fig. 6

Variance gamma calibration based on the FFT method

3.5.2 The bilateral gamma model

The VG process \((X_t)_{ t\ge 0 }\) is in the CGM-parametrization2 defined by
$$\begin{aligned} X_t = G_t^{(1)} - G_t^{(2)}, \end{aligned}$$
(18)
where \((G^{(1)}_t)_{ t\ge 0 }\) and \((G^{(2)}_t)_{ t\ge 0 }\) are two independent gamma processes with, respectively, parameters (CM) and (CG). Equivalently, the characteristic function of the VG process at time t can be rewritten as
$$\begin{aligned} \phi _{X_t}(u) = \left( \frac{1}{1-\frac{iu}{M}} \right) ^{Ct} \left( \frac{1}{1+\frac{iu}{G}} \right) ^{Ct} = \left( \frac{GM}{GM + (M-G)iu + u^2} \right) ^{Ct}. \end{aligned}$$
(19)
The bilateral gamma model distinguishes between the rate of the positive and negative motions by introducing the parameters \(c_p\) and \(c_n\). When defining \(b_p = 1/M\) and \(b_n = 1/G\), i.e. the scale of respectively positive and negative jumps, the resulting characteristic function is given by
$$\begin{aligned} \phi _{X_t}^{{\mathrm{(BG)}}}(u) = \left( \frac{1}{1-iub_p} \right) ^{c_pt}\left( \frac{1}{1+iub_n} \right) ^{c_nt}. \end{aligned}$$
(20)
Calibrating this model on the bitcoin option surface of 29 June leads to the fit displayed in Fig. 7a. The optimal BG model parameters in the weeks thereafter are given in Fig. 7b. Note that the optimal values of \(c_p\) and \(c_n\) are overall significantly different. The speed at which negative jumps occur is much higher than that of positive jumps, while the scale is larger for positive jumps.
Fig. 7

Bilateral gamma calibration based on the FFT method

3.5.3 The bilateral double gamma model

The bilateral double gamma model (Madan et al. 2018) goes one step further in the generalization by allowing the speed parameters \(c_p\) and \(c_n\) to vary randomly. They are assumed to be gamma distributed, with characteristic functions:
$$\begin{aligned} \phi _{c_p}(u) = \left( \frac{1}{1-iu\beta _p} \right) ^{\eta _p} \quad {\text {and}} \quad \phi _{c_n}(u) = \left( \frac{1}{1-iu\beta _n} \right) ^{\eta _n}. \end{aligned}$$
(21)
The resulting bilateral double gamma (BDG) process depends on six parameters: \(b_p, \beta _p, \eta _p, b_n, \beta _n\) and \(\eta _n\), and is defined by
$$\begin{aligned} \phi _{X_t}^{(BDG)}(u) = \left( \frac{1}{1+\beta _pt\log (1-iub_p)} \right) ^{\eta _p}\left( \frac{1}{1+\beta _nt\log (1+iub_n)} \right) ^{\eta _n}. \end{aligned}$$
(22)
The results of calibrating this model on the bitcoin option data are displayed in Fig. 8a, b.
Fig. 8

Bilateral double gamma calibration based on the FFT method

3.6 The VG Sato model

Alternatively, the variance gamma model can be extended to a Sato model (Sato 1999). The VG Sato process \((Y_t)_{t\ge 0}\) is defined by scaling the VG process as follows:
$$\begin{aligned} Y_t \sim {\mathrm{VG}}\left( \sigma t^{\gamma }, \ \nu , \ t^\gamma \theta \right), \end{aligned}$$
(23)
or alternatively,
$$\begin{aligned} \phi _{Y_t}(u) = \left( 1 - iu t^\gamma \theta \nu + \frac{1}{2}\sigma ^2\nu u^2 t^{2\gamma } \right) ^{-1/\nu }. \end{aligned}$$
(24)
Figure 9a shows that the VG Sato model fits the observed bitcoin option prices significantly better than the ordinary VG model. An overview of the evolution of the risk-neutral parameters is given in Fig. 9b.
Fig. 9

VG Sato calibration based on the FFT method

3.7 The VG-CIR model

None of the previously considered gamma-based models took into account any notion of stochastic volatility. The calibration results of the Heston model in Sect. 3.4, however, indicate that it may be interesting to also include a similar effect here. The VG-CIR model (Carr et al. 2003) brings this in by making time stochastic. Assuming for the rate of time change \(y_t\) the classical Cox–Ingersoll–Ross (CIR) process,
$$\begin{aligned} {{\mathrm {d}}}y_t = \kappa (\eta - y_t){{\mathrm {d}}}t + \lambda y_t^{1/2}{{\mathrm {d}}}W_t, \end{aligned}$$
(25)
where \(\eta\) is the long-run rate of time change, \(\kappa\) the rate of mean-reversion and \(\lambda\) governs the volatility of the time change. The economic time elapsed in t units of calendar time is then given by the integrated CIR process \((Y_t)_{t\ge 0}\), where
$$\begin{aligned} Y_t = \int _{0}^{t} y_s {{\mathrm {d}}}s. \end{aligned}$$
(26)
The risk-neutral bitcoin price process is under the VG-CIR model hence given by
$$\begin{aligned} S_t = S_0 \frac{\exp (X_{Y_t})}{E[\exp (X_{Y_t}) \ | \ y_0]}, \end{aligned}$$
(27)
where \((X_t)_{t\ge 0}\) is a variance gamma process. The corresponding characteristic function of \(\log (S_t)\) is, therefore, given by
$$\begin{aligned} \phi _t(u) = \exp \left( iu \left( \log (S_0) \right) \right) \frac{\phi _{Y_t}( -i\phi _{X_t}(u) )}{\phi _{Y_t}( -i\phi _{X_t}(-i))^{iu}}, \end{aligned}$$
(28)
where \(\phi _{Y_t}\) is the characteristic function of \(Y_t\). Figure 10a shows that the VG-CIR model fits the bitcoin option surface fairly well. The optimal model parameters according to the other weeks in the dataset are displayed in Fig. 10b.
Fig. 10

VG-CIR calibration based on the FFT method

3.8 Comparison of fits

We summarize the pricing performance for the previously calibrated models. Table 1 displays the rmse, \({\mathrm{aae}}\), \({\mathrm{ape}}\) and \({\mathrm{arpe}}\) for the calibration on 29 June 2018, while Fig. 11 shows for each model the evolution of the rmse and \({\mathrm{ape}}\) over time.

Fig. 11

Evolution of the pricing error over time

It is clear that both one-parameter models perform poorly. The variance gamma model already does a better job than the former models, but extending it to the bilateral gamma model and the bilateral double gamma model results in slightly better fits. The VG Sato model, which generalizes the VG model in an alternative way, leads in general to lower error scores than the former models. However, the VG-CIR and the Heston model, which both incorporate stochastically changing volatility, have overall the best performance.

4 Conic finance: implied liquidity

The bitcoin derivative market is a relatively young, but growing market. In this section, we investigate the liquidity of the available derivatives. Instead of using the ordinary bid–ask spread as a measure for liquidity, we use the more advanced implied liquidity. Implied liquidity, \(\lambda\), is a unitless measure for liquidity that is introduced in Corcuera et al. (2012), arising from the theory of conic finance by Madan and Schoutens (2016).

Conic finance provides an alternative to the law of one price. Instead of focusing on one risk-neutral price, it models both the bid and ask prices of an asset. In practice, a derivative’s bid price is given by the average of the discounted distorted payoff, while the ask price is modelled by the negative average of the distorted distribution of the negative payoff. The distortion function applied in the analysis below is the Wang transform (Wang 2000), defined for \(0< u < 1\) as
$$\begin{aligned} \varPsi _{\lambda }^{{\mathrm{WANG}}}(u) = N\left( N^{-1}(u) + \lambda \right) , \quad \lambda \ge 0, \end{aligned}$$
(29)
where \(N(\cdot )\) denotes the cdf of a standard normal distribution. In the Black–Scholes model, this distortion function leads to closed-form bid and ask prices for vanilla options, e.g. for a call option we have:
$$\begin{aligned} {\mathrm {bidEC}}&= \exp (-rT)\int _{0}^{\infty } x{{\mathrm {d}}}\varPsi _{\lambda }^{{\mathrm{WANG}}}\left( F_{(S_T-K)^+}(x)\right) \nonumber \\&= {\mathrm {EC}}(K,T,S_0,r,q+\lambda \sigma /\sqrt{T},\sigma ), \end{aligned}$$
(30)
$$\begin{aligned} {\hbox {askEC}}&= -\exp (-rT)\int _{-\infty }^{0} x{{\mathrm {d}}}\varPsi _{\lambda }^{{\mathrm{WANG}}}\left( F_{-(S_T-K)^+}(x)\right) \nonumber \\&= {\mathrm {EC}}\left( K,T,S_0,r,q-\lambda \sigma /\sqrt{T},\sigma \right), \end{aligned}$$
(31)
where \({\mathrm {EC}}\) stands for the classical risk-neutral Black–Scholes price of a European call option. Conic prices are hence obtained by shifting the dividend yield in the original one-price formula. Note that the resulting option prices depend on the parameter \(\lambda\). In the special case, where \(\lambda\) is zero, bid price equals ask price and we are again in the one-price framework. In other words, a value of \(\lambda\) equal to zero corresponds to a bid–ask spread of zero.

Implied liquidity is defined as the particular value of \(\lambda\) so that conic option prices perfectly match the observed bid and ask prices in the market. In the particular case of BTC call options, we use the following procedure. First calculate mid-prices corresponding to the bid and ask prices available in the market. For each mid-price, the (Black–Scholes) implied volatility \(\sigma _{{\mathrm{impl}}}\) is computed. Implied liquidities \(\lambda _{{\mathrm{bid}}}\) and \(\lambda _{{\mathrm{ask}}}\) are then calculated by matching the market bid and ask prices with the conic bid and ask prices, respectively, where \(\sigma _{{\mathrm{impl}}}\) is plugged into formulas (30) and (31).

The procedure above leads to a surface of implied liquidities. The higher \(\lambda\) is, the less liquid the product is. However, since the implied liquidity is unitless, it is only appropriate to compare the liquidity of products. Figure 12 displays the implied liquidity smiles corresponding to the call options provided on 29 June 2018.
Fig. 12

Black–Scholes-implied liquidity (\(\lambda _{{\mathrm{bid}}}\) and \(\lambda _{{\mathrm{ask}}}\)) for call options on 29 June 2018

Implied liquidity follows an upward trend with respect to the strike price. Out of the money BTC options are hence less liquid than their near the money analogs. Figure 12 further indicates that the liquidity increases (\(\lambda\) decreases) with the maturity of the options.

5 Conclusion

With the advent of bitcoin options, a new opportunity arose to analyze the characteristics of the bitcoin market. While time series models mainly focus on historical dynamics of the bitcoin price, the option surface contains information about the risk-neutral distribution governing the bitcoin log-returns. In this paper, we performed an empirical study based on the available market prices of bitcoin vanilla options. We calibrated a series of elementary and more advanced market models on the option surface. While the classical Black–Scholes model did not capture the surface very well, more advanced models managed to produce a good fit. We observed that models including some notion of stochastic volatility, like the Heston model and the VG-CIR model, generally do a better job. In the last part, the liquidity of the bitcoin option market was examined based on the implied liquidity measure originating from conic finance theory. In the money, long-term options were found to be the most liquid options in the bitcoin market.

Footnotes

  1. 1.

    \(\kappa\) = rate of mean reversion, \(\rho\) = correlation stock—vol, \(\theta\) = vol-of-vol, \(\eta\) = long-run variance, \(v_0\) = initial variance.

  2. 2.

    \(C = 1/\nu\), \(G = \left( \sqrt{\frac{\theta ^2\nu ^2}{4} + \frac{\sigma ^2\nu }{2} } - \frac{\theta \nu }{2} \right) ^{-1}\) and \(M = \left( \sqrt{\frac{\theta ^2\nu ^2}{4} + \frac{\sigma ^2\nu }{2} } + \frac{\theta \nu }{2} \right) ^{-1}\)

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Robert H. Smith School of BusinessUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of LeuvenLeuvenBelgium

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