# Advanced model calibration on bitcoin options

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## 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.

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.

*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

*K*and time to maturity

*T*is given by

*t*is specified below for each model.

### 3.2 The Black–Scholes model

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 |

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

### 3.4 Heston’s stochastic volatility model

^{1}Formally, we model the risk-neutral bitcoin price process \((S_t)_{t\ge 0}\) as follows:

### 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

*t*,

#### 3.5.2 The bilateral gamma model

^{2}defined by

*C*,

*M*) and (

*C*,

*G*). Equivalently, the characteristic function of the VG process at time

*t*can be rewritten as

#### 3.5.3 The bilateral double gamma model

### 3.6 The VG Sato model

### 3.7 The VG-CIR model

*t*units of calendar time is then given by the integrated CIR process \((Y_t)_{t\ge 0}\), where

### 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.

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).

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).

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

- 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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