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In this study, we make use of both the specific method of Monte Carlo simulation and the spot-futures parity with the cost of carry to establish a dynamic price model of Bitcoin futures and to conduct the appraisals and numerical analyses. More specifically, the electricity fees and equipment costs are taken into account and the proposed model is thereby built. Numerical results show that various cost factors have significant effects on the Bitcoin futures price. We employ Monte Carlo simulation to approximate the Bitcoin futures price and we use Python to program the computations.

The prototype of Bitcoin was first introduced in the work of Nakamoto (2008) [

In recent years, Bitcoin has been receiving a lot of attention of investors. This cryptocurrency as an emerging alternative asset is based on the block chain technology and decentralized network. It has an inelastic money supply with a limit of 21,000,000 bitcoins, which is going to be achieved by today’s prediction in 2140. There are more than 2000 kinds of cryptocurrency in transaction markets currently; the total market value reaches 8200 billion USD in January, 2018. Because any increase may be followed by dramatic drops for Bitcoin prices, the market participants represent serious concerns for such a bumpy ride. According to the characteristics of high risk and high return on investment of volatile spot price of Bitcoin, therefore, in order to allow investors to have function of hedging and spot price forecast in Bitcoin investment, the trading service of Bitcoin futures was introduced by the Chicago Board Option Exchange (CBOE) on December 10, 2017. In addition, the Chicago Mercantile Exchange (CME) also activated such service on December 18 in the same year. However, what is comparatively special is both COBE and CME are based on contracts using cash settlement. This means on the due date there will be no actual trading of Bitcoin and instead according to the final settlement price, cash is recorded into one of the accounts and yield is realized through price difference. Bitcoin is one of emerging financial asset. Ever since the subprime mortgage crisis, currency of various countries depreciates and US dollar is no longer a strong currency. In order to seek for reliable investment financial commodity, market investors invest in Bitcoin with a decentralized characteristic one after another to prevent against the autocratic control of the government and the supply and demand is reacted by market fluctuation under the market system. Through the publicity of media, the high risk and high return on investment characteristics of Bitcoin receive attention from market investors broadly. Its futures commodity was introduced in 2017 to provide function of hedging of spot Bitcoin and price forecast for market investors. Delivery of Bitcoin futures in cash becomes a new option for Bitcoin investment, as in Hale et al. (2018) [

To accompany the hedge demand against high volatility in Bitcoin market scientifically, we dive into futures price modeling for Bitcoin. A correctly priced Bitcoin futures provides the market with price discovery, leverage, transparency, and risk transfer capacities. This study adopts GBM under the risk-neutral probability measure as the benchmark model of Bitcoin futures price. In addition, multi-factors included in the GBM according to the original cost carry model are factors that will affect the price of Bitcoin future. Moreover, consideration is also given to electricity fee and equipment cost as the price difference reference factors of spot and futures of Bitcoin and this dynamic model is thereby built. In the risk-neutral world and under the hypotheses that the change in the price of Bitcoin futures can conform to GBM, this study uses Monte Carlo simulation to simulate all possible price changes to seek for its approximated price and to find out factors that will affect the approximated price. Moreover, we study on the relationship of the amplification factor model and the futures price. The result of this study discovers that various cost factors have significant effect on the Bitcoin futures price. We provide investors with findings of expected price thereby achieving the function of evading the spot price risk of Bitcoin futures.

The remainder of this study is organized as follows. Section 2 introduces the model framework. Section 3 presents the dynamic model and numerical method. Section 4 provides the numerical results and discussions. Section 5 concludes this study.

Uncovering the Bitcoin price process is a necessary step for choosing the appropriate futures price model. Geometric Brownian motion (GBM) is the classical stochastic process under continuous-time setting, it is been used to describe the dynamics of asset price in previous studies. The characteristic is the assumption that asset price parameters are unrelated to time and its model is set as follows:

d S t = μ S t d t + σ S t d W t

The above formularepresents the dynamic path of the asset price within a short time (dt) in the real world, and in which

S t refers to the asset price at time t;

μ refers to the growth rate or return on investment;

σ refers to the fluctuation of return on investment;

W t refers to the Wiener process (i.e., Brownian motion) over time.

GBM is applied in financial mathematics to simulate underlying stock price within the framework of Black and Scholes (1973) [

d S t = r S t d t + σ S t d W t

where S t is the Bitcoin spot price at time t, r is the risk-free interest rate, and σ is the volatility.

Futures prices will change according to the spot price of the underlying asset. However, basis often appears between futures and spot that will result in different spot price and future price of the underlying asset. Apart from the arbitrage space generated from the price difference of the two assets, the function of price mining is also provided. Therefore, for the relationship between these two asset prices, the put-call futures parity proposed by Tucker (1991) [

The cost of carry model is constructed in the arbitrage portfolio demonstration under the hypothesis of perfect markets that is used to express the model of interrelation of time difference between futures prices and spot prices. This means for the futures price, apart from the spot price, consideration has to be given to costs required within the period from carrying the spot till the date of delivery including warehousing cost, transportation cost, interest cost and insurance cost, etc. However, for different assets, the costs of carry are different. In general, the cost of carry includes the disbursement of loan interest disbursement less the yield of that asset. Regarding the cost of carry of consumption asset, apart from the four aforementioned aspects, the convenience yield has to be deducted. Only for the investment asset in the traditional futures commodities, its cost of carry is formed by the sum of the four aforementioned factors. Pindyck (2001) [

The price of Bitcoin futures can be affected by some factors. However, this study is of the opinion that traditional GBM can only capture the stochastic change of prices and under the traditional GBM the factor of the cost of carry theory can no longer capture the change of Bitcoin futures effectively and as a result the effectiveness of the appraisal tool is limited by this model. As Bitcoin is not a traditional financial asset, we take a further step according to the cost required for mining Bitcoin to provide new factors in order to improve the cost factor of Bitcoin futures price, to improve the cost of carry theory of traditional futures, to revise its cost of carry factors as electricity fee and mining expense, and the rate of convenience yield is also considered that are further incorporated to the GBM for capturing the basis generated due to carry of cost so that the precision of this model can be enhanced. Consequently, the amplified GBM model is formally given by the following:

d S t = ( r + E + F − R ) S t d t + σ S t d W t

under the risk-neutral probability measure, factors of cost of carry are incorporated into the amplified geometric Brownian model, in which E is the electricity fee, F is the equipment expense, R is the rate of convenience yield. Electricity fee is the main variable cost of Bitcoin mining as extensive electricity computation has to be put in during the course of mining. Therefore, this study utilizes the weighted average of Bitcoin quantity generated from various mines in USA as the simulation factor of electricity fee of various states of US. In addition, equipment expense is based on the present high efficiency miner, the “Ant Miner S9” as the example that the price is approximately US$2000 and the rate of convenience yield is based on the quantity of Bitcoin produced by the mines multiplied by the spot price.

The famous Monte Carlo simulation is the numerical simulation method that simulation is conducted on the stochastic process of price change of derivative commodity assets so as to obtain the approximated value of the derivative commodity further. Based on the law of large numbers, if the number of times of simulation is increasing, its simulated value can be closer to the actual value, as in Lian et al. (2015) [_{t} maturing at time T is given by the proposed cost of carry relationship:

F t ( S t , T ) = S t e ( r + E + F − R ) ( T − t )

in which the 1-year US treasury bill rate of 2% is used as a proxy for the risk-free rate r, E represents the electricity fee, F indicates the equipment expense, and R is the rate of convenience yield. Electricity fee is based on the average electric fee required to mine one Bitcoin in US, the cost of ant miner S9 required to mine one Bitcoin on the average, the individual spot price cost proportion is E, F, and R. R is the rate of convenience yield of Bitcoin spot based on the quantity of Bitcoin produced multiplied by the Bitcoin spot price. Assuming there are 252 trading days in one year, and then the discretization interval is 1/252, we conduct 100,000 simulations for calculating each Bitcoin futures price. Upon the implementation of simulation of 100,000 times, the path number of 100,000 on Bitcoin futures pricesis obtained and then these are averaged to compare with the result of the Black-Scholes model and the real path of Bitcoin futures prices.

The RMSE (root means squared error) used in this study is to measure errors on the actual price change by the amplified model and in the scenario analysis, consideration is given to whether the precision of the model can be improved if the numerical values of factors are adjusted. The RMSE is given by the following:

R M S E = 1 n ∑ i = 1 n ( X o b s , i − X m o d e l , i ) 2

The RMSE of the traditional GBM and the real price is 9419.897 and the RMSE of the amplified model is 2054.559. This shows that when the cost of carry factor is included in the Bitcoin futures in the traditional GBM, the price forecast ability of the model can be improved and can provide investors with a more precise expected price trend, moreover the results are able to achieve the function of risk management and evade drastic fluctuation of Bitcoin futures price earlier.

r | σ | S_{0} | T | lnE | lnF | lnR |
---|---|---|---|---|---|---|

0.02 | 0.9997 | 19,400 | 252 | −1.43829 | −1.84831 | −0.00933 |

This study further conducts scenario analysis on various factors to check whether change in the numerical value of various factors under the identical risk-neutral probability measure will affect the precision of the model. First the fluctuation rate of σ Bitcoin spot historical prices is adjusted as plus and minus 500% to examine the effect of extreme fluctuation on the model (

When the fluctuation rate is raised (+500%), the RMSE of its model and real price will rise to 2842.335 and this can show that although the fluctuation trend of short term price can be seen due to over intensive price change, yet it will result in the rise of RMSE value and the forecast precision of the model will be lowered; when the fluctuation rate is lowered (−500%), the RMSE of its model and real price rises to 2132.267 and this can show that moderating the risk will not produce significant result on the precision of the model.

The natural logarithm of electricity fee plus or minus 20% where other conditions remain unchanged is acquired to study on whether there is effect on the amplified model due to the rise and fall of the price of electricity fee in different regions (

Compared with the original amplified model, if the natural logarithm of electricity fee is increased or decreased by 20%, the RMSE of the 20% increase of the natural logarithm of electricity fee will be 2119.451; the RMSE of 20% decrease of the natural logarithm of electricity fee will be 2092.007. Both of the RMSE values are higher than the original amplified model. Therefore, adjustment on the natural logarithm of electricity fee cannot effectively improve precision.

Under the circumstance of similarity in other conditions, the plus or minus 30% on the natural logarithm of equipment expense is acquired to explore the environment of price fluctuation of various miner models in the market and the effect on the amplified model (

Compared with the original amplified model, if the natural logarithm of equipment expense is increased or decreased by 30%, the RMSE of 30% increase in the natural logarithm of equipment expense acquired is 2262.691, the RMSE of 30% decrease in the natural logarithm of equipment expense acquired is 2278.591. The RMSE value above are both higher than the original amplified model, therefore, adjustment on the equipment cost of natural algorithm cannot effectively improve the precision of forecast.

Aiming at adjusting the natural logarithm of the rate of convenience yield by increasing or decreasing 10% to explore whether the rate of convenience yield of change in trading price in the exchange in different regions will cause effect on the amplified model of Bitcoin futures price (

In the dynamic model of Bitcoin futures prices, both the electricity fee and the equipment expense are incorporated as the factors of cost of carry and the rate of convenience yield is also considered. Upon simulation on 100,000 times by means

of Monte Carlo method, more price movement can be captured so that more precise forecast can be provided to investors. According to this empirical study, it can prove that compared with the traditional GMB, more precise dynamic price change can be captured. This model can provide investors with findings of expected price, transparency of prices can be enhanced, and the function of evading the spot price risk of Bitcoin futures can be achieved.

In scenario analyses, individual factor is adjusted to explore whether change of the factor will affect the precision of the amplified model. When it is adjusted as intensive fluctuation rate, it shows that although the short-term price change can be captured due to increase in the scale of the expected price change, yet, it will also make the error to expand simultaneously. However, reduction of the fluctuation rate will make the reaction of the price of the model not sensitive enough and will become countering to the characteristics of Bitcoin risk and also cannot improve the precision of the improved amplified model. Where the attempt is to change the parameter value of electricity fee, equipment cost and the rate of convenience yield in the amplified model, these three factors cannot reduce the RMSE value through adjustment on the preset parameter value. Therefore, this study is of the opinion that the preset parameter value presumed by the original amplified model can effectively provide an accurate simulation price of the amplified model.

The main contribution of this study is that: we present the market with price discovery, leverage, transparency, and risk transfer capacities based on correctly Bitcoin futures price modeling. In the aspect of function on regulatory, the final target is to reduce the risk of Bitcoin spot market via introducing the corresponding derivative market as an alternative place for market participants.

The authors thank the anonymous referees for helpful comments and suggestions. Yu-Min Lian is grateful for the funding support from the Ministry of Science and Technology under grant MOST107-2410-H-030-001.

The authors declare no conflicts of interest regarding the publication of this paper.

Lian, Y.-M., Cheng, C.-H., Lin, S.-H. and Lin, J.-H. (2019) A Cost of Carry-Based Framework for the Bitcoin Futures Price Modeling. Journal of Mathematical Finance, 9, 42-53. https://doi.org/10.4236/jmf.2019.91004