^{1}

^{2}

^{3}

The objective of this study is, to show the importance of incorporating jumps in both returns and volatility dynamics for Bitcoin. For that purpose, we introduce the Double Exponential Jump-Diffusion model with Stochastic Volatility (DEJDSVJ) that contains asymmetric jumps. The use of the Markov Chain Monte Carlo methods for estimation has proved the meaningful presence of jumps in Bitcoin price and volatility. Moreover, based on the Bitcoin options market, a comparison between the underlying model, the Double Exponential Jump Diffusion model (DEJD) with Stochastic Volatility (no Jumps) and the Stochastic Volatility (SV) shows the goodness of the DEJDSVJ model’s calibration over others for pricing Bitcoin options.

Bitcoin is a digital currency that satisfies the technical properties of money. Contrary to fiat currencies, there is no central authority acting as a bank for Bitcoin. Its system is based on solving computational algorithms (cryptographic puzzles) known as mining process through a network called blockchain whose protocol was released by a pseudonymous Satoshi Nakamoto on 2009 [

But despite those critics, some studies proved that there will be a future role of Bitcoin in financial markets. [

Up to now, few academic researches were done in this direction because of some challenges that concern the characteristics of Bitcoin: price discontinuity, high level of speculation, high volatility.

To fulfill such challenges, it is important to model the dynamic of BTC including the occurrence of rare or extreme events as jump process in returns. In literature about options pricing model, many studies were done using diffusion

models with the geometric Brownian motion process [

In addition, the characteristic of the volatility is important for option pricing models. Volatility plays an important role in pricing of derivatives. After the financial crisis on 1987, many studies have demonstrated that the inefficiency of the Black-Scholes model [

Although few studies are done about bitcoin options pricing, we find many relevant considerations in some articles. [

In this article, we contribute to the growing academic literature about Bitcoin derivative markets by considering the jump diffusion model of [

Our difference with [

The paper is presented as follows. In Section 2, we analyze the dynamic of Bitcoin and introduce the model. In Section 3, we estimate the parameters and interpreted the results. We implement in Section 4 the option pricing exercises and compare the Double Exponential Jump-Diffusion with asymtric jump Stochastic Volatility (DEJDSVJ) model with the the Double exponential Jump Diffusion model with Mean Reverting Stochastic Volatility (DEJD) (i.e. no jumps in volatility) and the Stochastic Volatility (SV) model meaning the absence of jumps in both returns and volatility processes and finally, we graph and analyze the Implied Volatility surface in Section 5.

Currently, Bitcoin knows a huge emergence. Traders and investors give it more attention because it is not controlled by a central authority. The technology behind Bitcoin is an open source. Its properties are transparency, anonymous, fast transactions and cheap. Also, there is not prerequisites or interest rate for governments or banks. All transactions made are registered by the blockchain system whose functionnement depends on mining process. Since it was established, the price in USD dollar (exchange rate) of Bitcoin fluctuates by going either up or down. Thus, it generates a fear sentiment for some people to invest on it. Some studies have tried to investigate about the factors that drive the price ( [

To analyze such dynamic, we use a daily time series data exhibited from the database of COINBASE exchange platform one of the popular largest market of bitcoin based at USA. The data cover the time period from February 2015 to July 2020. We use a big sample of data in order to avoid bias.

It directly shows that the time series of the underlying is not stationary; there are discontinuities. The graph depicts many phases:

· From February 2015 to April 2017, the graph fit to a very flat curve line; that period is considered as normal because there is not high fluctuations of the BTC-USD values.

· From May 2017 to November 2017, data exhibit some steeply fluctuations. Regular ups and downs spikes are observed.

· However the end of 2017 is marked by unregular increasing spike of the exchange rate that reaches for the first time a value around $20,000. Such change can be defined as an irregular component because it was unpredictable. During that period, the interest of media and governments about cryptocurrencies increased.

· After that, we notice a mildly decreasing curve up to half of 2019 again followed by some fluctuations of BTC-USD exchange rate.

The aspect of the Geometric Brownian Motion (GBM) process is directly observed. It represents the continuity part of a process. Generally, GBM assumes that a constant drift is accompanied by random shocks.

Many options pricing models consider a given underlying asset as a continuous process (Black Scholes model [

As the price is nonstationary, we take the log returns as a response variable. Log returns are the first difference of the logarithm of the prices (see Equation (1)).

r t = ln ( P t ) − ln ( P t − 1 ) . (1)

where:

r t is the log returns at time t.

P t is the price of Bitcoin in USD at time t. Here, we took the log returns as percentage meaning that we multiply r t by 100. In

The series of returns fluctuates around zero.

Thus, the residual is conditionally heteroskedastic. The use of stochastic volatility in our work is supported by such effect of clusters. Also, [

In

Here, we introduce the jump diffusion process with Stochastic Volatility Jump to

model the dynamic of bitcoin.

Let define S t = e X t with S the underling process of BTC-USD price and X is the basic state process such that:

d X t = μ d t + V t d W t 1 + ( e y t 1 − 1 ) d N t . (2)

where μ is the constant drift term, V is the Stochastic Volatility with Jump defined as:

d V t = κ ( ϕ − V t ) d t + σ V t d W t 2 + ( e y t 2 − 1 ) d N t . (3)

d W 1 , d W 2 are the Brownian motion processes that are correlated. y t 1 , y t 2 are independent double exponentially distributed random variables such that:

f ( y 1 ) = p η 1 e − η 1 y 1 I { y 1 ≥ 0 } + ( 1 − p ) η 2 e − η 2 y 1 I { y 1 < 0 } (4)

f ( y 2 ) = q η ′ 1 e − η ′ 1 y 2 I { y 2 ≥ 0 } + ( 1 − q ) η ′ 2 e − η ′ 2 y 2 I { y 2 < 0 } (5)

where η 1 > 1 , η 2 > 0 , η ′ 1 > 0 and η ′ 2 > 0 .

d N is a pure jump process following a poisson distribution with constant mean jump arrival rate λ .

Let ( Ω , F , ℙ ) be the filtered probability space. Then using Itô’s formula under the probability measure ℚ , we introduce the diffusive price process d S t of [

log ( S t ) = log ( e X t )

d ( log ( S t ) ) = d ( X t )

d S t = S t − ( μ d t + V t d W t 1 + ( e y t 1 − 1 ) d N t ) (6)

By combining the above process with the stochastic volatility, we obtain the following dynamic model:

{ d S t S t − = μ d t + V t d W t 1 + Z t 1 d N t d V t = κ ( ϕ − V t ) d t + σ V t d W t 2 + Z t 2 d N t c o v ( d W t 1 , d W t 2 ) = ρ d t P ( d N t = 1 ) = λ d t Z t 1 = Y t 1 − 1 Y t 1 = e y t 1 Z t 2 = Y t 2 − 1 Y t 2 = e y t 2 (7)

S is the exchange rate (BTC-USD price), V is the stochastic variance with asymmetric jumps. κ is the mean reversion rate or the degree of volatility clustering, ϕ is the long-run mean of V; the process reverts to that level with a spread governed by κ .

W t 1 and W t 2 are two Brownian motion ρ -correlated; they represent respectively the diffusion processes for the S and V,

N is a pure jump process following a poisson distribution with constant mean

jump arrival rate λ (i.e. ℙ k ( λ d t ) = ℙ ( d N t = k ) = e − λ d t ( λ d t ) k k ! ).

σ is the volatility of volatility that controls the kurtosis.

ρ the correlation coefficient between the log returns and the volatility that affects the asymmetry or skewness. As there is not dividend yields on Bitcoin, we will not consider such parameter in the drift term of the log return process.

Under the square root variance process, the model allows for systematic volatility risk.

For the above model to be applicable in order to get option prices in following section, we have to estimate the values of parameters and the latent variables. Since [

Markov Chain Monte Carlo methods are a class of algorithms that allows to find a posterior distribution and to sample from it a Markov chain for the set of parameters. Primarly, such methods were used in Bayesian statistics, computational physics, computational linguistics in order to address multi-dimensional problems.

MCMC becomes popular in field of quantitative finance this year because of its ability to estimate the parameters of complex dynamic models that are developed to solve the problems in finance such as risk management and options pricing. The most meaningful part of the MCMC methods is the approximation of the posterior. Also called the conditional joint distribution of random variables (parameters and latent variables) given the data, the posterior is derived from Baye’s formula such that:

P ( θ , L | Y ) ∝ P ( Y | θ , L ) P ( θ , L ) . (8)

where Y is the data, θ is the set of parameters of a given model and L represents the latent variables.

After setting the initial values and the prior for each parameter, there are two approaches to get the posterior for each parameter: the Gibbs sampling method and the Metropolis-Hastings M-H algorithm. For the former, it is used when the posterior distribution derived can be approximated to a known probability distribution or in case of conjugate posterior that means the prior and the posterior for a given parameter have the same distributions with different input hyperparameters. The M-H algorithms is considered when there is not a standard distribution that links with the posterior. A good overview of the MCMC methods can be found in [

In what follows, we highlight the different steps to consider before implementing the MCMC algorithm in Matlab software.

First of all, we discretize both the log returns and the stochastic volatility under Euler-Maruyama discretization method.

Δ S t = S t − 1 ( μ Δ t + V t W 1 ( Δ t ) + Z t 1 J t )

⇒ r t = μ Δ t + V t B t 1 + Z t 1 J t

r t = Δ S t S t − 1 .

Δ V t = V t − V t − Δ t

⇒ V t = ( α + β V t − 1 ) Δ t + σ V t − 1 W 2 ( Δ t ) + Z t 2 J t

⇒ V t = ( α + β V t − 1 ) Δ t + σ V t − 1 B t 2 + Z t 2 J t

where t ∈ 1,2, ⋯ , T is discrete daily time ( Δ t = 1 ), α = κ ϕ and β = 1 − κ , P ( J t = 1 ) = λ Δ t .

B 1 and B 2 are respectively the standardized residuals of the diffusive processes W 1 and W 2 . Regarding the properties of a geometric brownian motion, W ( t ) follows a gaussian distribution ℕ ( 0, t ) . Hence, it has the same distribution than t B i with B i ∼ ℕ ( 0,1 ) , i ∈ { 1,2 } .

The discretized model becomes:

{ r t = μ + V t B t 1 + Z t 1 J t V t = α + β V t − 1 + σ V t − 1 B t 2 + Z t 2 J t (9)

After the discretization, we assume θ 1 = { μ , λ , η 1 , η 2 , p } the set of parameters in returns, θ 2 = { α , β , σ , η ′ 1 , η ′ 2 , q } contains the parameters of the stochastic volatility and Σ = { V , Z 1 , Z 2 , J } the set for the latent variables.

Thus, θ = θ 1 ∪ θ 2 ∪ { ρ } = { μ , λ , η 1 , η 2 , η ′ 1 , η ′ 2 , p , q , α , β , σ , ρ } represents the set of all parameters to estimate.

Thus, we define the posterior distribution based on Baye’s formula to respect the principle of MCMC method. We obtain:

P ( θ , V , Z 1 , Z 2 , J | r ) = P ( r | θ , V , Z 1 , Z 2 , J ) × P ( θ , V , Z 1 , Z 2 , J ) (10)

P ( θ , V , Z 1 , Z 2 , J | r ) = P ( r | θ , V , Z 1 , Z 2 , J ) × P ( θ ) × P ( V , Z 1 , Z 2 , J | θ ) (11)

P ( r | θ , V , J , Z 1 , Z 2 ) represents the likelihood of the data,

P ( V , J , Z 1 , Z 2 | θ ) is the prior of the latent variables given their parameters respectively.

P ( θ ) is the prior of the parameters.

Furthermore, we assume that all parameters have mutually independent prior distributions. That implies

P ( θ ) = P ( θ 1 ) P ( θ 2 ) P ( ρ ) (12)

where

P ( θ 1 ) = P ( μ ) P ( η 1 ) P ( η 2 ) P ( λ ) P ( p )

P ( θ 2 ) = P ( α ) P ( β ) P ( σ ) P ( η ′ 1 ) P ( η ′ 2 ) P ( q )

Based on relevant studies [

For the hyperparameters, we assume those in [

Further, we use conjugate posterior for all parameters except ρ and σ for which we consider the M-H algorithm. For the latent variables, the posteriors of the jump sizes Z 1 and Z 2 follow a double exponential distribution. The jump J has a Bernouilli distribution and the volatility V has a non stantard posterior; so we use the M-H algorithm to approximate it.

To estimate the above parameters under MCMC method, we consider a set of 2000 daily data points from the Coinbase market. We use large sample of data to avoid bias. We run N = 5000 iterations with a burn-in of n = 500 iterations. We have chosen such burn-in period to remove the effects of initial values that affect only five hundred instead of one of the different chains. The value of each parameter x is obtained by averaging the sum of the posterior’s values over N − n iterations:

x = 1 N − n ∑ i = n + 1 N x i (13)

Hence,

We have the leverage effect with a negative value of ρ . It means that bad news about the market of BTC increase the price of such asset. Thus, we can explain the huge fluctuations of the price this last year by the attention about Bitcoin. Although, some investors defend the importance of using Bitcoin, there are a lot of critics about it supported by governments and banks that do not trust the digital currency. The values η 1 and η 2 show the interest of incorporating the jumps in returns. In addition, the values of α and β exhibit an important value of spread κ and mean ϕ of revesion. So, the Stochastic Volatility is very useful for BTC price.

Parameters | DEJD | DEJDSVJ | SV |
---|---|---|---|

μ | 0.1839 | 0.1542 | 0.1843 |

η 1 | 52.9287 | 49.2762 | - |

η 2 | 42.0980 | 33.0683 | - |

η ′ 1 | - | 43.9765 | - |

η ′ 2 | - | 39.6936 | - |

λ | 0.0011 | 0.0028 | - |

α | 12.1648 | 12.0071 | 12.1641 |

β | −0.7824 | −0.7736 | −0.7825 |

ρ | −0.0871 | −0.1084 | −0.0878 |

σ | 0.3532 | 0.4 | 0.353 |

p | 0.0009 | 0.0025 | - |

q | - | 0.0024 | - |

After estimation, we checked the sensitivity of the posterior regarding the prior distributions assumption. It concerns the convergence of each chain. For that purpose, we graph the trace plot of each chain with respect to the simulation index.

We obtain quick convergence for each parameter (after 500 iterations over 5000). Thus, the prior distributions and the likelihood function of the data given the parameters are well chosen. The data too are well sampled.

In the following section, the parameters values of the models are used to price European call options.

An option pricing model involves a probabilistic approach to assign a fair value for an option. An option is a contract that gives the holder the right but not the obligation to buy (call option) or to sell (put option) an underlying security at a pre-determined price at (European option) or before (American option) a maturity time. The variables considered to price options are the current market price, the strike price, the volatility, the interest rate and time to maturity.

Let ℚ be the probabilty measure under which options are priced.

Let S ( t ) be the price a time t, r the interest rate, K the strike and T the expiry time.

The European Call Option price is defined as follow:

C ( t ) = E ℚ ( e − r ( T − t ) ( S ( T ) − K ) + ) (14)

where, ( S ( T ) − K ) + = max ( S ( T ) − K , 0 ) represents the payoff ϕ ( T ) of the option at T. For the call option to have a value, S ( T ) should be greater than K; otherwise, it is called a worthless Call Option. Many models are developed to value options such as the benchmark Black-Scholes Merton model [

· We choose two large numbers M the number of M.C simulations for price and N the number of time steps.

· Δ t = T N is the time steps.

For i = 1 , ⋯ , M :

· We generate the sources of uncertainty (randomness) B i 1 and B i 2 that are the standardized residuals of the discrete model. B 1 and B 2 are ρ -correlated. Then using the Cholesky decompositon, we construct two functions B 1 and B 2 with two independent standard normal random variables ε 1 and ε 2 such that:

{ B i 1 = ε i 1 B i 2 = ρ ε i 1 + 1 − ρ 2 ε i 2

· We simulate the stochastic volatility V i , the log of the price X i and the price S i

V i = ( α + β V i − 1 ) Δ t + σ V i − 1 ( ρ ε i 1 + 1 − ρ 2 ε i 2 ) Δ t + Z i 2 J i (15)

X i = X i − 1 + μ Δ t + V i Δ t ε i 1 + Z i 1 J i (16)

S i = exp ( X i ) (17)

· We calculate the payoff ϕ i = ( S i − K ) + and we compute the expected payoff as follow:

E ( ϕ ) = ∑ i = 1 M ϕ i M (18)

· Finally, we obtain the Call Price by discounting the expected payoff:

C ( T ) = e − r T E ( ϕ ) (19)

In what following, we consider the Monte Carlo Simulation described above to obtain the European Call Option Prices for the three models in order to compare them.

Calibration is very important because it shows the impact of the model over the prices. To calibrate the model’s call prices with the Bitcoin Options market, we consider the European call prices from the Deribit Options market on February 22, 2021 for 18 days, 32 days and 65 days to maturity. In

Expiry Time | Stock Prices | Strikes | Market Calls | DEJD Calls | DEJDSVJ Calls | SV Calls |
---|---|---|---|---|---|---|

18 days | 56,901.94 | 54,000 | 6629.46 | 6903.9 | 6798.8 | 5835.5 |

18 days | 56,901.94 | 56,000 | 5605.1 | 5689.7 | 5633 | 4753.4 |

18 days | 56,907.94 | 58,000 | 4723.36 | 4651.1 | 4555.3 | 3826 |

18 days | 56,907.94 | 60,000 | 3955.47 | 3600.6 | 3664.1 | 3039.5 |

18 days | 56,907.94 | 62,000 | 3272.22 | 3023 | 2901.2 | 2386.1 |

32 days | 57,716.26 | 48,000 | 11,606.25 | 13,259 | 12,767 | 10,680 |

32 days | 57,716.26 | 52,000 | 9754.54 | 10,191 | 10,074 | 7671.1 |

32 days | 57,716.26 | 56,000 | 7676.18 | 7577.6 | 7485.9 | 5236.4 |

32 days | 57,738.26 | 60,000 | 6004.57 | 5410.6 | 5296 | 3410.8 |

32 days | 57,738.26 | 64,000 | 4676.25 | 3897.5 | 3746.2 | 2114.3 |

65 days | 55,934.21 | 52,000 | 11,327.08 | 12,994 | 12,487 | 10,459 |

65 days | 55,934.21 | 56,000 | 9593.51 | 10,596 | 10,746 | 8596.9 |

65 days | 55,955.65 | 60,000 | 8140.86 | 8871.5 | 8432.1 | 7037.7 |

65 days | 55,955.65 | 64,000 | 6910.42 | 6989.7 | 6829.5 | 5726 |

Prices, the DEJD model European Call Prices, the SV model European Call Prices. Such comparison is to check the impact of each model and its efficiency regarding how it approaches the real market’s prices. For each model, we run M = 20000 simulations.

The prices of the DEJDSVJ model approach well the market’s prices followed by the prices of the DEJD model for all three expiry times over different strikes. The results show that the SV model is not a good model for the BTC market options pricing.

An error measure method is used to check the goodness of fit. For that purpose, there are many methods as the Root Mean Square Error, the Average Absolute Error, the Average Percentage Error (APE) used by [

APE = 1 N ⋅ ∑ i = 1 N | C ˜ i − C i | ω (20)

where,

ω = 1 N ∑ i = 1 N C i

N is the number of options used, C i , the market price and C ˜ i the price for a given model.

The APE for the DEJDSVJ model is smaller meaning that model outperforms the other models. Furthermore, the three graphs in

Model | APE for 18 days | APE for 32 days | APE for 65 days |
---|---|---|---|

DEJDSVJ | 4.2% | 8.3% | 7.4% |

DEJD | 4.3% | 8.9% | 9.6% |

SV | 17.9% | 26.7% | 11% |

the three models and the market prices against the strikes for each expiry. The curves of the DEJDSVJ’s prices and the DEJD’s prices are much closer to the market’s prices. It confirms that the DEJDSVJ and DEJD models are better than the SV model when pricing options in BTC market.

We explain such performance of the DEJDSVJ model over other models by the incorporation of the jumps in both returns and volatility. That shows the recent importance of jumps when modelling financial assets.

Implied Volatility captures the future expectations of a security’s price. Forecasting the Implied Volatility is very important in trading and investments. It helps to buy cheap and to sell expensive in order to make a profit.

We analyze such volatility in this paper by drawing the Implied Volatility surface over Strikes and Maturity times. In

· The long-dated options have higher Implied Volatility than the short-dated options;

· As the strike price increases, the Implied Volatility decreases.

As a result, it is profitable for traders and investors to buy strategies such that calls, puts with short expiry date (low Implied Volatility) and to sell strategies that have long maturity time.

Strike K Maturity T | 1 | 7 | 30 | 60 | 90 | 180 |
---|---|---|---|---|---|---|

8000 | 1015.1 | 1160.5 | 1720.1 | 2304.9 | 2873.3 | 4703 |

8100 | 910.3 | 1072.3 | 1635.3 | 2272.7 | 2846.5 | 4666.6 |

8200 | 815.4 | 993.3 | 1576.6 | 2179.1 | 2734.1 | 4503.1 |

8300 | 718.2 | 913.5 | 1502.3 | 2183 | 2725 | 4432.9 |

8400 | 619.5 | 838.6 | 1448.4 | 2029.4 | 2649.4 | 4388.5 |

8500 | 530.7 | 774.9 | 1378 | 2013 | 2613.6 | 4192.7 |

8600 | 441.7 | 682 | 1321.7 | 1905.3 | 2583.2 | 4269.8 |

8700 | 359.3 | 640.6 | 1235.7 | 1900.2 | 2485.2 | 4159.1 |

8800 | 282.9 | 584.8 | 1193.7 | 1806.1 | 2435.8 | 4212.9 |

8900 | 214.2 | 516.6 | 1140.1 | 1768.5 | 2340.1 | 4196 |

9000 | 161.2 | 460.2 | 1071.7 | 1743.8 | 2268.2 | 4127.5 |

9100 | 115.9 | 411.8 | 1038.1 | 1687.7 | 2272.2 | 3923.1 |

9200 | 79.1 | 366.3 | 960.8 | 1645.9 | 2189.8 | 4013.6 |

9300 | 53 | 311.2 | 954.7 | 1560.6 | 2117.5 | 3825.4 |

9400 | 34.3 | 283.2 | 883.7 | 1489.4 | 2074.9 | 3783.3 |

Recently, digital currencies gain a wide attention from economists, investors, traders and academicians, etc. Bitcoin, the first released cryptocurrency is the most popular because of its price’s fluctuations (USD price) and the high level of volatility. Such interest challenges people to think about the future of Bitcoin and to improve more studies about it that will be helpful for traders and miners to avoid losses.

Following that wave, this paper extends the literature about jump-diffusion models for Bitcoin market by introducing a Double Exponential Jump-Diffusion model combined with a Stochastic Volatility process that incorporates asymmetric jumps. Theoretically, the choice of such model is supported by the important features captured by jumps in both returns and volatility as Bitcoin is too volatile. The importance of the double exponential distribution is its memoryless property and it allows to see easily the difference between the magnitudes (upward and downward) and intensities of jumps via the parameters η 1 , η 2 , η ′ 1 , η ′ 2 , p , q .

As pricing options can help for portfolios management, the mechanism of call options pricing using the underlying DEJDSVJ model shows much better performance than the DEJD model (no jumps in volatility) that outperforms the Stochastic Volatility where jumps are missing in both returns and volatility. Such outcomes also support the model.

A final conclusion is our results that meet the expectations.

The results presented in this paper concern only the Bitcoin data under Jump-Diffusion model in which the jumps are asymmetric Double exponentially distributed. However, for further works, one can consider the Pareto-Beta Jump-Diffusion that also allows for two approaches towards jumps’ magnitudes in order to explore the Bitcoin market.

We acknowledge the African Union through the Pan African University, Institute of Basic Sciences, Technology and Innovation for its considerable support.

All authors have approved that there is not conflict of interest regarding the publication of this article.

Sene, N.F., Konte, M.A. and Aduda, J. (2021) Pricing Bitcoin under Double Exponential Jump-Diffusion Model with Asymmetric Jumps Stochastic Volatility. Journal of Mathematical Finance, 11, 313-330. https://doi.org/10.4236/jmf.2021.112018