advanced model calibration on bitcoin options · digial finance 1 3...
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Digital Financehttps://doi.org/10.1007/s42521-019-00002-1
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ORIGINAL ARTICLE
Advanced model calibration on bitcoin options
Dilip B. Madan1 · Sofie Reyners2 · Wim Schoutens2
Received: 21 September 2018 / Accepted: 13 March 2019 © Springer Nature Switzerland AG 2019
AbstractIn 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 veri-fied 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 min-ing bitcoins. This is how new bitcoins are created. For a detailed overview of the key
* Sofie Reyners [email protected]
Dilip B. Madan [email protected]
Wim Schoutens [email protected]
1 Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA2 Department of Mathematics, University of Leuven, Celestijnenlaan 200B, 3001 Leuven,
Belgium
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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 cur-rencies. Three months later, LedgerX became the first US exchange platform trading bitcoin options. Bitcoin derivatives were born. In December 2017, the CFTC author-ized 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 exist-ence 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.
A lot of research has been carried out to gain insight into the dynamics of the bit-coin 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 fac-tors, 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
1 October 2013
18 December 2017
29 June 2018
31 August 2018$ 0
$ 5000
$ 10 000
$ 15 000
$ 20 000
Fig. 1 Historical price chart of the BTC–USD rate
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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 mar-ket. We start by calibrating the models on the option surface and elaborate on the differ-ent 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 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 encour-ages 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, con-sisting 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 col-lected manually from various unregulated exchanges. The data are extracted each Fri-day 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 sur-faces. 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 underly-ing. 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.
(1)bidEC(K, T) − askEP(K, T) ≤ S0 − K ≤ askEC(K, T) − bidEP(K, T),
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where S0 is the initial bitcoin price and e.g. 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. Consid-ering 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 eval-uated 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).
(2)rmse =
√√√√1
n
n∑
i=1
(marketpricei −modelpricei)2,
(3)aae =1
n
n∑
i=1
|marketpricei −modelpricei|,
(4)ape =
∑n
i=1�marketpricei −modelpricei�∑n
i=1marketpricei
,
(5)arpe =1
n
n∑
i=1
|marketpricei −modelpricei|marketpricei
.
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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
where
and � = 1.5 . The characteristic function �t of the log-price process at time t is speci-fied below for each model.
3.2 The Black–Scholes model
The Black–Scholes model is a natural reference point to start the analysis. Let (St)t≥0 denote the bitcoin price process. In a zero interest rate Black–Scholes world, the price process is given by
where (Wt)t≥0 is a standard Brownian motion, S0 the initial value of the bitcoin price and � 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 data-set is shown in Fig. 2b. More details on the goodness of fit are given in Table 1 and Fig. 11.
Alternatively, we can compute the implied volatility for each option in the sur-face. Implied volatility smiles based on the mid-prices of out of the money vanilla
(6)C(K, T) =exp(−� log(K))
� ∫∞
0
exp(−iv log(K))�(v)dv,
(7)�(v) =�T (v − (� + 1)i)
�2 + � − v2 + i(2� + 1)v
(8)St = S0 exp
(−�2t
2+ �Wt
),
Table 1 Summary of the pricing performance on 29 June
rmse ape aae arpe
BS 27.8368 0.1142 22.0807 0.3379Laplace 34.2618 0.1384 26.7563 0.4990Heston 9.2144 0.0360 6.9505 0.1207VG 20.4969 0.0858 16.5809 0.2410BG 16.5971 0.0691 13.3652 0.2383BDG 9.9357 0.0322 6.2255 0.1436VG Sato 10.5010 0.0414 8.0007 0.1011VG-CIR 9.2275 0.0350 6.7651 0.1028
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options as published on Friday 29 June 2018 are displayed in Fig. 4 for each matu-rity. 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 param-eter, � , corresponding to the volatility. A brief summary of the model specifica-tions is given in “Appendix A”. Model calibration on the option surface of 29 June leads to an optimal volatility � 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
0.5 1 1.5 2 2.5 3 3.5 4Strike (USD) 104
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Market priceBlack-Scholes model price
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(b) Evolution over time.
Fig. 2 Black–Scholes calibration
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Laplace ( = 0.67, rmse = 34.26, ape = 0.14)
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(b) Evolution over time.
Fig. 3 Laplace calibration
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call option prices in the right tail, the Laplace model overestimates them. However, neither of these models captures the option surface accurately.
We can extend the comparative analysis by calculating the implied volatili-ties 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).
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 reason-able next step. We consider Heston’s stochastic volatility model (Heston 1993) with parameters �, �, �, � and v0.1 Formally, we model the risk-neutral bitcoin price pro-cess (St)t≥0 as follows:
5400 5600 5800 6000 6200 6400 6600 6800Strike
0.5
0.6
0.7
0.8
0.9
1
Impl
ied
vola
tility
T = 7 days, S = 5906
Black-ScholesLaplace
5500 6000 6500 7000 7500 8000 8500 9000 9500 10000Strike
0.5
0.6
0.7
0.8
0.9
1
Impl
ied
vola
tility
T = 28 days, S = 5872
Black-ScholesLaplace
0.5 1 1.5 2Strike 104
0.5
0.6
0.7
0.8
0.9
1
Impl
ied
vola
tility
T = 91 days, S = 5867
Black-ScholesLaplace
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0.5
0.6
0.7
0.8
0.9
1
Impl
ied
vola
tility
T = 182 days, S = 5906
Black-ScholesLaplace
Fig. 4 Implied volatility smiles corresponding to out of the money options on 29 June 2018
1 � = rate of mean reversion, � = correlation stock—vol, � = vol-of-vol, � = long-run variance, v0 =
initial variance.
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where the squared volatility process vt is given by
with (Wt)t≥0 and (W̃t)t≥0 two correlated standard Brownian motions such that
The characteristic function �t(u) of log(St) is given by
with
Figure 5a shows a good fit. The optimal vol-of-vol parameter � 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 bit-coin’s volatility, which is often said to be high. This study indeed reflects a high volatility—as confirmed by the optimal values of � and v0—but it additionally shows that bitcoin’s volatility itself is very volatile. Note that the optimal value of � increases sharply over time, reaching a maximum of nearly 1300% at the end of July.
(9)dSt
St=√vtdWt,
(10)dvt = 𝜅(𝜂 − vt)dt + 𝜃√vtdW̃t
(11)Cov(dWtdW̃t
)= 𝜌dt.
(12)
�t(u) = exp(iu log(S0))
× exp(���−2
((� − ��ui − d)t − 2 log
((1 − g exp(− dt))(1 − g)−1
)))
× exp(v0�
−2(� − ��iu − d)(1 − exp (− dt))(1 − g exp(− dt))−1)
(13)d = ((��ui − �)2 − �2(−iu − u2))1∕2,
(14)g = (� − ��ui − d)(� − ��ui + d)−1.
0.5 1 1.5 2 2.5 3 3.5 4Strike (USD) 104
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)Heston ( = 2.08, = -0.09, = 2.36, = 0.93, v0 = 0.41,
rmse = 9.21, ape = 0.04)
Market priceHeston model price
(a) Calibration on 29 June 2018.
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v0
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Fig. 5 Heston calibration based on the FFT method
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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 sub-traction 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 Sen-eta 1990; Madan et al. 1998) on the bitcoin option surface. Second, the bilat-eral 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 them-selves to be gamma distributed, as explained in Madan et al. (2018). We consider one such model, the bilateral double gamma (BDG) model. Two alternative gen-eralized 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 (�, �, �)-parametrization. The risk-neutral process of the bitcoin price is modelled by
where (Xt)t≥0 is a variance gamma process, i.e. it starts at zero, has independent and stationary increments and the increment Xs+t − Xs follows a VG(�
√t, �∕t, �t) dis-
tribution over the time interval [s, s + t] . In this parametrization, � controls the skew-ness, � the kurtosis and � the volatility of the risk-neutral distribution of the log-returns. Equivalently, the characteristic function of log(St) is given by
with �Xt the characteristic function of the VG process at time t,
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 � becomes strongly negative in this period, corresponding to more negative skewness in the risk-neutral distribution of log-returns.
(15)St = S0exp(Xt)
E[exp(Xt)],
(16)�t(u) = exp(iu(log(S0) − log(�Xt
(−i))))
× �Xt(u)
(17)�Xt(u) =
(1 − iu�� +
1
2�2�u2
)−t∕�
.
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3.5.2 The bilateral gamma model
The VG process (Xt)t≥0 is in the CGM-parametrization2 defined by
where (G(1)t )t≥0 and (G(2)
t )t≥0 are two independent gamma processes with, respec-tively, parameters (C, M) and (C, G). Equivalently, the characteristic function of the VG process at time t can be rewritten as
(18)Xt = G(1)t − G
(2)t ,
(19)�Xt(u) =
(1
1 −iu
M
)Ct(1
1 +iu
G
)Ct
=
(GM
GM + (M − G)iu + u2
)Ct
.
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Market priceVG model price
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Fig. 6 Variance gamma calibration based on the FFT method
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Market priceBG model price
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cn
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Fig. 7 Bilateral gamma calibration based on the FFT method
2 C = 1∕� , G =
(√�2�2
4+
�2�
2−
��
2
)−1
and M =
(√�2�2
4+
�2�
2+
��
2
)−1
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The bilateral gamma model distinguishes between the rate of the positive and nega-tive motions by introducing the parameters cp and cn . When defining bp = 1∕M and bn = 1∕G , i.e. the scale of respectively positive and negative jumps, the resulting characteristic function is given by
Calibrating this model on the bitcoin option surface of 29 June leads to the fit dis-played in Fig. 7a. The optimal BG model parameters in the weeks thereafter are given in Fig. 7b. Note that the optimal values of cp and cn are overall significantly different. The speed at which negative jumps occur is much higher than that of posi-tive jumps, while the scale is larger for positive jumps.
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 cp and cn to vary randomly. They are assumed to be gamma distributed, with characteristic functions:
The resulting bilateral double gamma (BDG) process depends on six parameters: bp, �p, �p, bn, �n and �n , and is defined by
(20)�(BG)
Xt(u) =
(1
1 − iubp
)cpt(
1
1 + iubn
)cnt
.
(21)�cp(u) =
(1
1 − iu�p
)�p
and �cn(u) =
(1
1 − iu�n
)�n
.
(22)�(BDG)
Xt(u) =
(1
1 + �pt log(1 − iubp)
)�p(
1
1 + �nt log(1 + iubn)
)�n
.
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)BDG (bp = 0.11, p = 1.33, p = 13.56, bn = 14.34, n = 54.21, n = 0.02
rmse = 9.94, ape = 0.03)
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n
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n
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Fig. 8 Bilateral double gamma calibration based on the FFT method
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The results of calibrating this model on the bitcoin option data are displayed in Fig. 8a, b.
3.6 The VG Sato model
Alternatively, the variance gamma model can be extended to a Sato model (Sato 1999). The VG Sato process (Yt)t≥0 is defined by scaling the VG process as follows:
or alternatively,
Figure 9a shows that the VG Sato model fits the observed bitcoin option prices sig-nificantly better than the ordinary VG model. An overview of the evolution of the risk-neutral parameters is given in Fig. 9b.
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, how-ever, 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 yt the classical Cox–Ingersoll–Ross (CIR) process,
where � is the long-run rate of time change, � the rate of mean-reversion and � gov-erns the volatility of the time change. The economic time elapsed in t units of calen-dar time is then given by the integrated CIR process (Yt)t≥0 , where
(23)Yt ∼ VG(�t� , �, t��),
(24)�Yt(u) =
(1 − iut��� +
1
2�2�u2t2�
)−1∕�
.
(25)dyt = �(� − yt)dt + �y1∕2t dWt,
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Market priceVG Sato model price
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imal
par
amet
ers
VG Sato model calibration
(b) Evolution over time.
Fig. 9 VG Sato calibration based on the FFT method
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The risk-neutral bitcoin price process is under the VG-CIR model hence given by
where (Xt)t≥0 is a variance gamma process. The corresponding characteristic func-tion of log(St) is, therefore, given by
where �Yt is the characteristic function of Yt . 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.
(26)Yt = ∫t
0
ysds.
(27)St = S0
exp(XYt)
E[exp(XYt) | y0]
,
(28)�t(u) = exp(iu(log(S0)
)) �Yt(−i�Xt
(u))
�Yt(−i�Xt
(−i))iu,
0.5 1 1.5 2 2.5 3 3.5 4Strike (USD) 104
0
200
400
600
800
1000
1200
Pric
e (U
SD
)VG-CIR (y0 = 1.57, G = 26.43, M = 27.78, C = 95.01, = 0.78, = 5.87, = 3.55
rmse = 9.23, ape = 0.03)
Market priceVG-CIR model price
(a) Calibration on 29 June 2018.
0
1
2
3 104 VG-CIR model calibration
CGM
29 Ju
ne 20
18
6 July
2018
13 Ju
ly 20
18
20 Ju
ly 20
18
27 Ju
ly 20
18
3 Aug
ust 2
018
10 A
ugus
t 201
8
17 A
ugus
t 201
8
24 A
ugus
t 201
8
31 A
ugus
t 201
80
5
10
15
20
Opt
imal
par
amet
ers
y0
(b) Evolution over time.
Fig. 10 VG-CIR calibration based on the FFT method
29 Ju
ne 20
18
6 July
2018
13 Ju
ly 20
18
20 Ju
ly 20
18
27 Ju
ly 20
18
3 Aug
ust 2
018
10 A
ugus
t 201
8
17 A
ugus
t 201
8
24 A
ugus
t 201
8
31 A
ugus
t 201
80
20
40
60
80
100
120
rmse
RMSEBlack-ScholesLaplaceHestonVariance GammaBilateral GammaBilateral Double GammaVG Sato
29 Ju
ne 20
18
6 July
2018
13 Ju
ly 20
18
20 Ju
ly 20
18
27 Ju
ly 20
18
3 Aug
ust 2
018
10 A
ugus
t 201
8
17 A
ugus
t 201
8
24 A
ugus
t 201
8
31 A
ugus
t 201
80
0.05
0.1
0.15
0.2
0.25
0.3
ape
APEBlack-ScholesLaplaceHestonVariance GammaBilateral GammaBilateral Double GammaVG SatoVG-CIR
Fig. 11 Evolution of the pricing error over time
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3.8 Comparison of fits
We summarize the pricing performance for the previously calibrated models. Table 1 displays the rmse, aae , ape and arpe for the calibration on 29 June 2018, while Fig. 11 shows for each model the evolution of the rmse and 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 bilat-eral 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, � , is a unitless measure for liquidity that is intro-duced 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 prac-tice, a derivative’s bid price is given by the average of the discounted distorted pay-off, while the ask price is modelled by the negative average of the distorted distribu-tion of the negative payoff. The distortion function applied in the analysis below is the Wang transform (Wang 2000), defined for 0 < u < 1 as
where N(⋅) 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:
where 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
(29)�WANG�
(u) = N(N−1(u) + �
), � ≥ 0,
(30)bidEC = exp(−rT)∫
∞
0
xd�WANG�
�F(ST−K)
+ (x)�
= EC(K, T , S0, r, q + ��∕√T , �),
(31)
askEC = − exp(−rT)∫0
−∞
xd�WANG�
�F−(ST−K)
+ (x)�
= EC�K, T , S0, r, q − ��∕
√T , �
�,
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original one-price formula. Note that the resulting option prices depend on the parameter � . In the special case, where � is zero, bid price equals ask price and we are again in the one-price framework. In other words, a value of � equal to zero cor-responds to a bid–ask spread of zero.
Implied liquidity is defined as the particular value of � so that conic option prices perfectly match the observed bid and ask prices in the market. In the par-ticular 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 �impl is computed. Implied liquidities �bid and �ask are then calculated by matching the market bid and ask prices with the conic bid and ask prices, respectively, where �impl is plugged into formulas (30) and (31).
The procedure above leads to a surface of implied liquidities. The higher � 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.
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 ana-logs. Figure 12 further indicates that the liquidity increases ( � decreases) with the maturity of the options.
0.5 1 1.5 2 2.5 3 3.5 4Strike 104
0
0.1
0.2
0.3
0.4
0.5Im
plie
d liq
uidi
tyBlack-Scholes implied liquidity
T = 7 daysT = 28 daysT = 91 daysT = 182 days
Fig. 12 Black–Scholes-implied liquidity ( �bid
and �ask
) for call options on 29 June 2018
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5 Conclusion
With the advent of bitcoin options, a new opportunity arose to analyze the charac-teristics 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 cali-brated 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, gen-erally 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.
Appendix A: The Laplace market model
A brief summary of the Laplace market model is given below. More details can be found in Madan (2016), and Madan and Wang (2017).
Laplace distribution
The density function of a Laplace distributed random variable L with mean � and vari-ance �2 is given by
and we denote L ∼ (�, �2) . We refer to the Laplace distribution with zero mean and unit variance as the standard Laplace distribution L∗ . The characteristic function of L is given by
We refer to Kotz et al. (2001) for a broad introduction to Laplace distributions and its extensions.
Market model
Assume that the log-returns of an asset S are modelled via the Laplace distribution:
(32)fL(x) =1
√2�
exp
�−√2�x − ��
�
�,
(33)�(u) = exp(iu�)
(1 +
�2u2
2
)−1
.
(34)log(St+s) − log(St) ∼ (�s, �2s),
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and hence
We apply a mean-correcting measure change, assuming zero interest rates. The dis-tribution is shifted to
Note that this equation is only valid for 𝜎2t < 2 . The characteristic function of the log-price process log(St) at time t equals:
Remark This model is not identical to the variance gamma model with � = 0 and � = 1 . However, the VG(� , 1, 0) distribution equals the Laplace distribution (0, �2) . On the other hand, in the VG(� , 1, 0) model the log-returns are distributed as
The two models hence only give the same results on a maturity of 1 year.
Vanilla option pricing
The price of a call option is given by
with
(35)St ∼ S0 exp��t + �
√tL∗
�.
(36)St ∼ S0 exp
�log
�1 −
�2t
2
�+ �
√tL∗
�.
(37)�t(u) = E[exp(iu log(St))] =exp
(iu(log S0 + log
(1 −
�2t
2
)))
1 +�2t
2u2
.
(38)log(St+s) − log(St) ∼ VG��√s, 1∕s, 0
�.
(39)C(K, T;�) = S0SL
�d,
�√T
√2
�− KSL(d, 0)
(40)SL(x, s) =
{1
2(1 + s) exp((1 − s)x) for x ≤ 0
1 −1
2(1 − s) exp(−(1 + s)x) for x > 0
,
(41)d =log(S0∕K) + log
�1 −
�2T
2
�
�√T√2
.
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Appendix B: Option data of 29 June 2018
Underlying Strike Maturity (days) Is call Bid price (USD) Ask price (USD)
5906 5500 6.92 1 434.06 472.445906 5750 6.92 1 256.88 289.365906 6000 6.92 1 132.87 147.635906 6250 6.92 1 64.96 91.535906 6500 6.92 1 29.53 50.25906 6750 6.92 1 8.86 26.575906 5500 6.92 0 70.86 91.535906 5750 6.92 0 132.87 168.35906 6000 6.92 0 256.89 301.185906 6250 6.92 0 431.09 484.245906 6500 6.92 0 634.84 699.85906 6750 6.92 0 862.21 930.125906 7000 6.92 0 1098.43 1169.295872 5500 27.92 1 607.7 648.85872 6000 27.92 1 352.29 393.395872 6500 27.92 1 193.76 211.375872 7000 27.92 1 102.74 126.235872 7500 27.92 1 58.71 76.325872 8000 27.92 1 35.23 49.95872 8500 27.92 1 17.61 29.365872 9000 27.92 1 14.68 23.495872 10,000 27.92 1 2.94 14.685872 5500 27.92 0 240.73 275.965872 6000 27.92 0 478.53 528.435872 6500 27.92 0 810.27 874.855872 7000 27.92 0 1203.66 1288.795872 7500 27.92 0 1652.83 1743.845872 8000 27.92 0 2122.55 2222.365872 8500 27.92 0 2606.95 2712.635872 9000 27.92 0 3097.22 3202.95872 10,000 27.92 0 4089.5 4195.195872 11,000 27.92 0 5087.65 5190.415872 12,000 27.92 0 6085.29 6190.975867 5500 90.92 1 920.96 973.765867 6000 90.92 1 692.19 739.125867 6500 90.92 1 513.23 530.835867 7000 90.92 1 387.12 428.185867 7500 90.92 1 287.41 328.475867 8000 90.92 1 217.02 255.155867 8500 90.92 1 164.23 199.43
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Underlying Strike Maturity (days) Is call Bid price (USD) Ask price (USD)
5867 9000 90.92 1 126.11 158.375867 10,000 90.92 1 79.18 105.585867 11,000 90.92 1 43.99 70.395867 12,000 90.92 1 29.33 52.795867 13,000 90.92 1 17.6 38.135867 14,000 90.92 1 8.8 29.335867 15,000 90.92 1 2.93 23.465867 20,000 90.92 1 2.93 8.85867 5500 90.92 0 557.27 610.065867 6000 90.92 0 824.17 882.835867 6500 90.92 0 1132.14 1196.665867 7000 90.92 0 1492.9 1572.095867 7500 90.92 0 1886.08 1965.285867 8000 90.92 0 2317.27 2428.735867 8500 90.92 0 2748.46 2877.525867 9000 90.92 0 3206.04 3340.975867 10,000 90.92 0 4147.62 4253.215867 11,000 90.92 0 5112.65 5259.325867 12,000 90.92 0 6132.9 6238.495867 13,000 90.92 0 7073.79 7226.35867 14,000 90.92 0 8065.06 8217.575867 15,000 90.92 0 9056.33 9211.775867 20,000 90.92 0 14,027.34 14,203.315867 25,000 90.92 0 19,010.09 19,203.655867 30,000 90.92 0 23,989.9 24,206.925867 35,000 90.92 0 28,972.64 29,207.265867 40,000 90.92 0 33,955.38 34,210.535906 6250 181.88 1 1012.77 1086.585906 7500 181.88 1 658.45 729.315906 8750 181.88 1 439.95 504.95906 10,000 181.88 1 310.03 363.185906 12,500 181.88 1 159.44 203.735906 15,000 181.88 1 100.39 121.065906 20,000 181.88 1 41.34 59.055906 25,000 181.88 1 20.67 29.535906 30,000 181.88 1 11.81 20.675906 35,000 181.88 1 5.91 14.765906 40,000 181.88 1 2.95 11.815906 6250 181.88 0 1328.7 1417.285906 7500 181.88 0 2199.88 2350.475906 8750 181.88 0 3215.69 3398.775906 10,000 181.88 0 4328.8 4523.685906 12,500 181.88 0 6658.24 6900.35
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Underlying Strike Maturity (days) Is call Bid price (USD) Ask price (USD)
5906 15,000 181.88 0 9064.7 9371.775906 20,000 181.88 0 13,966.13 14,391.315906 25,000 181.88 0 18,908.9 19,384.285906 30,000 181.88 0 23,863.48 24,442.25906 35,000 181.88 0 28,823.96 29,479.465906 40,000 181.88 0 33,787.4 34,519.66
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