MaMaMoMaMa: BTC options

Dilip B. Madan∗, Sofie Reyners†, Wim Schoutens‡

September 15, 2018

Abstract
In this paper, we investigate the behavior 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 stability of the optimal risk-neutral parameters
over a period of two months. The analysis proceeds with the construction of Black-
Scholes and Laplace implied volatity smiles. We conclude with a study of the implied
liquidity of BTC call options, based on conic finance theory.

1 Introduction
In 2008, an anonymous (group of) author(s) posted a white paper under the name
Satoshi Nakamoto, [11]. The paper suggests an electronic payment system, Bitcoin,
that does not depend on a 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.
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 U.S. 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 U.S. 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 U.S. 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 U.S. 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. This had a significant impact on the BTC-USD rate: the peak
∗
Robert H. Smith School of Business University of Maryland College Park, MD. 20742
†
University of Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium
‡
University of Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium

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in Figure 1 coincides exactly with the launch of their futures mid December. Before
the existence of BTC futures, investors had very few possibilities to bet on a decreasing
bitcoin price. The rate was mainly pushed up by optimistic investors who bought the
currency. Futures on the other hand made it possible for investors to bet on a decreasing
rate. In December 2017, pessimistic investors thus entered the BTC market, leading to
a sharp decline in price.

Figure 1: Historical price chart of the BTC index.

Bitcoin is an atypical asset, not a real currency, nor equity. In this paper, we try
to get a better understanding of the bitcoin market with the help of some traditional
market models. 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 match the Black-Scholes model, Laplace model, Variance Gamma model, Bilateral
(Double) Gamma model, VG Sato model, VG-CIR model and Heston model with 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.

2 The Data
The study is performed on options with as underlying a BTC index. The index consists
of quotes according to six leading BTC-USD exchanges: Bitfinex, Bitstamp, GDAX,
Gemini, Itbit and Kraken. Every six seconds, averages of bid and ask price are calculated
for each exchange. Both the highest and lowest price are left out and the index price is
calculated as the average over the remaining four mid prices. The historical evolution
of this BTC index is displayed in Figure 1. Options traded are European-style, cash-
settled and they are priced and settled in BTC. The corresponding prices in USD are
computed according to the latest future prices and will be used in our calculations. The
options expire each Friday at 08.00 UTC. Option pricing datasets used in this study
were extracted each Friday, two hours after expiration. The shortest maturity in the
dataset is therefore approximately seven days. There are usually four or five maturities

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available, ranging from one week to at most eight months. Longer term options are not
yet traded and are hence not included in the study.
Before we start modelling, some additional remarks about the option surface. 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 one 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 ranges from 100% to
500%. Finally note that we assume zero risk-free interest rates, both for BTC and USD.

3 Model Calibration
We first examine how well the typical 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. Secondly, 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.

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. The optimization procedure is then based
on the mid price 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).
v
u n
u1 X
rmse = t (market pricei − model pricei )2 (1)
n
i=1

n
1X
aae = |market pricei − model pricei | (2)
n
i=1
Pn
i=1 |market pricei − model pricei |
ape = Pn (3)
i=1 market pricei
n
1 X |market pricei − model pricei |
arpe = (4)
n market pricei
i=1

The options available on 29 June 2018 have maturities of 7, 28, 91 and 182 days.
Due to the extensive strike ranges for options with maturities of 91 and 182 days, the
majority of the considered options are call options.

3.2 The Black-Scholes Model

The Black-Scholes model is a natural reference point to start with. Calibrating the model
on the option surface of 29 June results in an overall best fitting volatility of 70.47%. In
Figure 2a, the Black-Scholes prices are plotted against the market prices to visualize the

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fit. 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 Figure 2b.

(a) Calibration on 29 June 2018. (b) Evolution over time.

Figure 2: Black-Scholes calibration.

More details on the goodness of fit are given in Table 1 and Figure 10. In section 4
the Black-Scholes implied volatility is studied in more detail.

3.3 The Laplace Model

Instead of modelling the log-returns with a normal distribution, the Laplace model em-
ploys the heavier-tailed Laplace distribution. The model has again one parameter, σ,
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 σ of 67.35%. The optimal volatility is comparable with its Black-Scholes coun-

(a) Calibration on 29 June 2018. (b) Evolution over time.

Figure 3: Laplace calibration.

terpart, but the overall price fits in Figures 2a and 3a are not. While the Black-Scholes
model underestimates the deep out of the money call option prices in the right tail, the

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Laplace model overestimates them. However, neither of the models does capture the
option surface accurately.

3.4 Heston’s Stochastic Volatility Model

Besides the high volatility of the bitcoin price, the high variability of this volatility should
be taken into account. Incorporating a stochastic volatility process is hence a reasonable
next step. We consider Heston’s stochastic volatility model [4] with parameters κ, ρ, θ, η
and v0 1 . The calibration procedure is based on the generic Fast Fourier Transform (FFT)
method [1].

(a) Calibration on 29 June 2018. (b) Evolution over time.

Figure 4: Heston calibration based on the FFT method.

Figure 4a shows a good fit. The optimal vol-of-vol parameter θ is relatively high,
showing that the previous assumption of constant volatility was too strong. Its value
moreover increases sharply over time, reaching a maximum value of nearly 1300% at the
end of July.

3.5 Gamma Models

The high volatility in the bitcoin market encourages us to include jumps in the model.
In Gamma-based models, the market is modelled by a difference of two independent
Gamma processes, being pure jump processes that model respectively the upward and
downward motion. We first fit the well-known Variance Gamma (VG) model [6], [8], [9]
on the bitcoin option surface. Secondly, 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 [10]. We
consider one such model, the Bilateral Double Gamma (BDG) model. Two alternative
generalized Variance Gamma models, the VG Sato and the VG-CIR model, conclude our
selection of Gamma models. The calibration procedures again rely on the FFT pricing
method.
1
κ = rate of mean reversion, ρ = correlation stock - vol, θ = vol-of-vol, η = long run variance, v0 =
initial variance.

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3.5.1 The Variance Gamma Model
We calibrate the Variance Gamma model in the original (σ, ν, θ)-parametrization.

(a) Calibration on 29 June 2018. (b) Evolution over time.

Figure 5: Variance Gamma calibration based on the FFT method.

Figure 5a summarizes the calibration results of 29 June. In Figure 5b the period of

high volatility in the first two weeks of August is again visible. The value of θ becomes
strongly negative in this period, corresponding to more negative skewness in the (risk-
neutral) distribution of log-returns.

3.5.2 The Bilateral Gamma Model

(V G)
In the Variance Gamma model, log-returns were modelled via a VG process (Xt )t≥0 ,
which is (in the CGM-parametrization) defined as
(V G) (1) (2)
Xt = Gt − G t (5)
(1) (2)
where Gt ∼ Gamma(Ct, M ) and Gt ∼ Gamma(Ct, G). Equivalently, the character-
istic function of the VG process at time t is given by
!Ct !Ct  Ct
(V G) 1 1 GM
φt (u) = iu
= . (6)
1− M 1 + iu
G
GM + (M − G)iu + u2

Instead of taking the same rate parameter C in both processes, the Bilateral Gamma
model distinguishes between the rate of the positive and negative motion by introducing
the parameters cp and cn . When defining bp = 1/M and bn = 1/G, the resulting
characteristic function is given by
 cp t  cn t
(BG) 1 1
φt (u) = . (7)
1 − iubp 1 + iubn

Calibrating this model on the bitcoin option surface of 29 June leads to the fit displayed
in Figure 6a. The optimal BG model parameters in the weeks thereafter are given in
Figure 6b. Note that the optimal values of cp and cn are overall significantly different.

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 (a) Calibration on 29 June 2018. (b) Evolution over time.

Figure 6: Bilateral Gamma calibration based on the FFT method.

3.5.3 The Bilateral Double Gamma Model

The Bilateral Double Gamma model [10] 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
 ηp  ηn
1 1
φcp (u) = and φcn (u) = . (8)
1 − iuβp 1 − iuβn
The resulting Bilateral Double Gamma (BDG) process depends on six parameters,
bp , βp , ηp , bn , βn and ηn and is defined by
 ηp  ηn
(BDG) 1 1
φt (u) = . (9)
1 + βp t log(1 − iubp ) 1 + βn t log(1 + iubn )
The results of calibrating this model on the bitcoin option data are displayed in Figure
7a and Figure 7b.

(a) Calibration on 29 June 2018.

(b) Evolution over time.

Figure 7: Bilateral Double Gamma calibration based on the FFT method.

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3.6 The VG Sato Model
Alternatively, the Variance Gamma model can be extended to a Sato model [12]. The
VG Sato process (Yt )t≥0 is defined by scaling the VG process as follows:

Yt ∼ V G(σtγ , ν, tγ θ) (10)

Figure 8a 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 Figure 8b.

(a) Calibration on 29 June 2018. (b) Evolution over time.

Figure 8: 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 section 3.4 however
indicate that it may be interesting to also include a similar effect here. The VG-CIR
model [2] brings this in by making time stochastic. Assuming for the rate of time change
the classical Cox-Ingersoll-Ross (CIR) process,
1/2
dyt = κ(η − yt )dt + λyt dWt (11)

where (Wt )t≥0 is a standard Brownian motion, the economic time elapsed in t units of
calendar time is then given by the integrated CIR process (Yt )t≥0 , where
Z t
Yt = ys ds. (12)
0

The risk-neutral price process of an asset (St )t≥0 is under the VG-CIR model hence
given by
exp((r − q)t)
St = S0 exp(XYt ), (13)
E[exp(XYt ) | y0 ]
where (Xt )t≥0 is a Variance Gamma process. Figure 9a 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 Figure 9b.

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 (a) Calibration on 29 June 2018.
(b) Evolution over time.

Figure 9: 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, aae, ape and arpe for the calibration on 29 June 2018, while Figure
10 shows for each model the evolution of the rmse and ape over time.

rmse ape aae 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

Table 1: Summary of the pricing performance on 29 June.

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 previous models. However, the VG-CIR and the Heston
model, which both incorporate stochastically changing volatility, are overall performing
the best.

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 Figure 10: Evolution of the pricing error over time.

4 Implied Volatility: Laplace vs. Black-Scholes

Calibration of the Black-Scholes model and the Laplace model led in both cases to an
optimal volatility σ, corresponding to the entire option surface. Here we construct the
implied volatility smiles for each maturity, based on the mid prices of out of the money
vanilla options as published on Friday 29 June 2018.
The results are displayed in Figure 11. The one 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.
In particular, the strikes of one week options range from 93% to 114% of the initial value
of the underlying, while the six month options range in moneyness from 105% to 677%.
Across all maturities, the Laplace implied volatility exceeds the Black-Scholes implied
volatility near the money. For larger strikes, the Laplace implied volatility decreases and
stabilizes, while the Black-Scholes implied volatility behaves adversely and increases.

5 Conic Finance: Implied Liquidity

The bitcoin derivatives 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, λ, is a unitless measure for liquidity that is introduced in [3], arising
from the theory of conic finance by [7].
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 price 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 [13],
defined for 0 < u < 1 as

ΨW AN G
(u) = N N −1 (u) + λ ,


λ λ≥0 (14)

where N (·) denotes the cdf of a standard normal distribution. Under the Black-Scholes
model, this distortion function leads to closed-form bid and ask prices for vanilla options,

10

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Figure 11: Implied volatility smiles corresponding to out of the money option prices on
29 June 2018.

e.g. for a call option we have:

Z ∞
xdΨW AN G


bidEC = exp(−rT ) λ F(ST −K)+ (x) (15)
0
√
= EC(K, T, S0 , r, q + λσ/ T , σ)
Z 0
xdΨW AN G


askEC = − exp(−rT ) λ F−(ST −K)+ (x) (16)
−∞
√
= EC(K, T, S0 , r, q − λσ/ T , σ)

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 original
one-price formula. Note that the resulting option prices depend on the parameter λ.
In the special case where λ is zero, bid price equals ask price and we are again in the
one-price framework. In other words, a value of λ equal to zero corresponds to a bid-ask
spread of zero.
Implied liquidity is defined as the particular value of λ 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 corre-
sponding to the bid and ask prices available in the market. For each mid price, the
(Black-Scholes) implied volatility σimpl is computed. Implied liquidities λbid and λask

11

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are then calculated by matching the market bid and ask price with the conic bid and
ask price respectively, where σimpl is plugged into formulas (15) and (16).
The procedure above leads to a surface of implied liquidities. The higher λ, the less
liquid the product. 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

Figure 12: Black-Scholes implied liquidity (λbid and λask ) for call options on 29 June
2018.

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

6 Conclusion
An empirical study was performed to get insight in the behavior of the bitcoin market.
Market prices of bitcoin vanilla options formed the starting point of the analysis. 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.
We constructed and compared implied volatility smiles according to the Black-Scholes
and the Laplace model. Finally, the liquidity of the bitcoin option market was examined
using the implied liquidity measure, originating from conic finance. In the money, long
term options were found to be the most liquid options in the market.

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A The Laplace Market Model
A brief summary of the Laplace market model is given below.

A.1 Laplace Distribution

The density function of a Laplace distributed random variable L with mean µ and
variance σ 2 is given by
√ |x − µ|
 
1
fL (x) = √ exp − 2 (17)
2σ σ

and we denote L ∼ L(µ, σ 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
−1
σ 2 u2

φ(u) = exp(iuµ) 1 + . (18)
2
See for instance [5] for a broad introduction to Laplace distributions and its extensions.

A.2 Market Model

Assume that the log-returns of an asset S are modelled via the Laplace distribution,

log(St+s ) − log(St ) ∼ L(µs, σ 2 s) (19)

and hence √
St = S0 exp(µt + σ tL∗ ). (20)
Its risk-neutral counterpart is found by applying a mean-correcting measure change. The
process is shifted to

σ2t √ ∗
   
St = S0 exp rt + log 1 − + σ tL (21)
2

where r is the risk-free interest rate. Note that this equation is only valid for σ 2 t < 2.
The characteristic function of the log-price process log(ST ) at maturity T equals
   2

exp iu log S0 + rT + log 1 − σ 2T
φ(u; T ) = EQ [exp(iu log(ST ))] = 2 (22)
1 + σ 2T u2

Remark. This model is not the same as the Variance Gamma model with θ = 0 and
ν = 1. The VG(σ, 1, 0) distribution equals the Laplace distribution L(0, σ 2 ). On the
other hand, in the VG(σ, 1, 0) model the log-returns are distributed as
√
log(St+s ) − log(St ) ∼ V G(σ s, 1/s, 0). (23)

The two models hence only give the same results on a maturity of one year.

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A.3 Vanilla Option Pricing
The price of a call option is given by
√ !
σ T
C(K, T ; σ) = S0 SL d, − K exp(−rT )SL(d, 0) (24)
2

with
(1
2 (1 + s) exp((1 − s)x) for x ≤ 0
SL(x, s) = (25)
1 − 12 (1 − s) exp(−(1 + s)x) for x > 0

σ2 T
log(S0 /K) + rT + log(1 − 2 )
d= √ (26)
σ T
2

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References
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