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. Figure 1 depicts the bitcoin price chart with scattered spikes.

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 [5], Heston model [8], etc.). However, empirical analysis of data showed the presence of discontinuity on many processes like stock price, exchange rate, etc. Those discontinuities are defined as jumps processes. Jump processes provide more realistic results. The importance of jump-diffusion model is its ability to capture stylized facts as skew, leptokurtosis, volatility smile/skew etc as shown by [6]. Furthermore, many studies have shown the presence of jumps on bitcoin dynamics ( [14] [15] ) and therefore they are incorporated on pricing Bitcoin derivatives.

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 Figure 2, we have plotted the daily log returns of Bitcoin for the same data sample.

The series of returns fluctuates around zero. Figure 2 depicts sustained periods of low or high volatility; it shows an effect of volatility clustering meanig that small returns are followed by small returns and large returns are followed by large returns. From 2015 to 2017, the values of returns are almost the same and slow. By 2017, we have high values. That sudden change in returns can be explained by the fact that the price was too high by that year.

Thus, the residual is conditionally heteroskedastic. The use of stochastic volatility in our work is supported by such effect of clusters. Also, [16] proved that the local and the time dependent volatility model do not necessarily capture the volatility clusters. Furthermore, the results of [17] showed that the dynamic of bitcoin is better modeled by the stochastic volatility model than the GARCH volatility.

In Figure 3, we depict the volatility of the same data frame used for the log returns. An analysis of such data gives an evidence that the volatility tends to fluctuate. Graphically, from the beginning to year 2017, we observe standard fluctuations (up and down movements). By end of 2017, there is a high move of the volatility that reverts down by 2018.

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 [6]

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 [6] has introduced the double exponential jump diffusion model, many methods for estimation of the model’s parameters were used. Among them, we have the Maximum Likelihood Estimation (MLE) method ( [18] ), the Generalized Method of Moment (GMM) ( [19] ), the Empirical Characteristic Exponent (ECE) ( [20] ), the Markov Chain Monte Carlo(MCMC) method ( [9] ) etc. In this study, we also consider the MCMC method. Our motivation is its efficiency to identify the latent variables such that the stochastic volatility, the jump sizes and the jump. First used in stochastic volatility model’s estimation by [21], MCMC method was extended in jump-diffusion models by [22] and it becomes very popular in quantitative finance.

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 [23].

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 [7] [9] [24], we consider the following prior distributions: μ ∼ N ( 0,25 ) , λ ∼ B e t a ( 2,40 ) , η 1 ∼ I W ( 10,40 ) , η 2 ∼ I W ( 10,40 ) , p ∼ B e t a ( 2,40 ) , ( α , β ) ∼ N ( 0 2 × 1 , I 2 × 2 ) , σ ∼ l n N ( 2.5,0.1 ) , η ′ 1 ∼ I W ( 10,40 ) , η ′ 2 ∼ I W ( 10,40 ) , q ∼ B e t a ( 2,40 ) , ρ ∼ U ( − 1,1 ) . IW represents the Inverted-Wishart distribution.

For the hyperparameters, we assume those in [24] and for the double exponential distribution, parameters were set under convenience to obtain the convergence of the chains.

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, Table 1 contains the estimated value of the underlying model and two others models that are later used for comparison.

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.

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.

Figure 4 contains the twelve trace plots that represent each chain for each parameter of the DEJDSVJ model. Each graph proves the convergence for one of the parameters.

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 [5], the Binomial option pricing, the Monte Carlo simulation etc. We are motivated by the later due to its efficiency to simulate price paths for high dimensional stochastic processes that have sources of uncertainty. The M.C simulation is based on the Euler-Maruyama discretization of the dynamic model. Knowing the price at t = 0 , the Strike K, the interest rate r and T the maturity in term of years, we compute the steps below to get an approximation of the European Call Option using the parameters in Table 1 for the 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 Table 2, we made a comparison between those market’s prices, the DEJDSVJ model European Call

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 [10]. We use the APE because of its ability to not skew error rate around or equal to zero ( [10] ). The formula is:

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.

Table 3 contains the Average Percentage Error for each model at a given time of maturity.

The APE for the DEJDSVJ model is smaller meaning that model outperforms the other models. Furthermore, the three graphs in Figure 5 show the prices of

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.