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This study considers a market-based economy that is composed of two asset classes: one is a digital, cryptocurrency, and the other is real, gold. We demonstrated that coins like (BTC, LTC, and DASH) can substitute a traditional safe haven “gold” in an intertemporal investment portfolio to become a new form of safe haven. The cryptocurrency follows a Jump-diffusion process. However, gold prices follow an Ornstein-Uhlenbek process to characterize the stochastic nature of the market. The stochastic optimal control approach, combined with the strategic asset allocation and the intertemporal utility theory, are used through the derivation of a Hamilton-Jacobi-Bellman (HJB) equation to determine an explicit solution of the optimal allocation problem for investors with CRRA utility function. We considered the Gamma Lévy process to solve the optimization problem. By using the secant method, we determined numerically the optimal percentage invested in the two asset classes at each time over the holding period. Our results showed that an investor can substitute gold by coins (BTC, LTC, DASH) from an investment portfolio perspective. Although Gold is supposed to be the traditional safe-haven asset, the digital currency seems to emerge as a new form of safe-haven value in a risky environment.

It is commonly agreed that cryptocurrencies are very volatile and cannot be modeled like ordinary assets or indices. The diffusion process, which seems to correspond to the modeling of the yields of cryptocurrencies, is a Brownian geometric diffusion process with a jump and a drift adjusted by the risk premium. Gold, on the other hand, presents a return whose variations follow an Ornstein-Uhlenbek process. We considered an economy where there are two types of assets. The first type is very risky, cryptocurrency. Whereas, the other type with low risk is represented by gold. The investment strategies will be organized between these two types of assets belonging to two completely different asset classes: one is digital and the other is real. Investors will define their optimal strategies based on how they perceive these two worlds, digital and real. The opposite is also true because one can, from optimal investment strategies, infer the perception of investors of these two types of assets.

In recent years, cryptocurrency has been a growing decentralized payment system. There are many types of alternative currencies such as Bitcoin, Litcoin, Dash, Ripple, Etherium, etc. Among these coins, Bitcoin is the most used digital currency with the largest market capitalization [

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The present paper aims at answering the following questions: can coins (BTC, LTC, and DASH) replace gold in an investment portfolio? Can these coins represent a current and future safe-haven value according to their characteristics? The parameters of BTC, LTC, DASH, and gold are estimated using US data for the period 2014-2019.

In the upcoming section, a survey of the relevant literature will be exposed. However, in section 3 we focus on the model formulation of the dynamic portfolio selection problem. Section 4 is devoted to presenting the stochastic control problem solution within a power utility framework. As for Section 5, it illustrates the optimal solution with a Gamma process. Section 6 presents the problem discretization, and the parameter estimates. The penultimate section serves to discuss the optimal portfolio choices and behavioral implications. Whereas, Section 8 encompasses the results’ synthesis, comments, and the concluding remarks.

In recent decades, considerable attention has been paid to cryptocurrency. Investors are drawn to the potential for high returns, the benefits of diversification, the increase rates in market capitalization, volumes traded as well as the speed of transactions [

According to the market assets’ views, cryptocurrencies are very volatile in their pricing compared to fiducial currency. The professionals who follow the stock charts have seen how the price of Bitcoin often fluctuates very violently, even at the level of ten percent. Indeed, unforeseen changes in market sentiment can result in large and sudden price movements. Generally speaking, this volatility is very frightening for investors and the markets, especially for unexperienced investors and speculators who speculate on the price of Bitcoin for short-term gain. This volatility is justified by several factors. One of the most certain causes of Bitcoin volatility is speculation. The cycle of buying and selling Bitcoins, often due to investor emotions, news, political or international decisions, creates sharp price fluctuations and generates high volatility. [

The more volatile is the financial value, the riskier it is, the more it will offer interesting trading opportunities. Therefore, price volatility allows investors to be able to make profits on the markets. Also, cryptocurrencies can offer significant trading opportunities. Indeed, the volatility of cryptocurrency increases the probability of gains. Bitcoin is very volatile and its inclusion in a diversified portfolio is very profitable [

Traditionally, gold is the ultimate asset used as a safety net for economies, a reserve of currency and the most robust safe haven for investors. Recently, cryptocurrencies have sparked controversy as a new safe haven. Recent studies have found that Bitcoin may show signs of safe haven. Bitcoin is sometimes referred to as digital gold as Bitcoin is seen as part of an alternative economy and investors could resort to Bitcoin if they lose faith in current currencies [

Much previous research shows that Bitcoin can be used as a hedge, a safe-heaven, or a diversiﬁer against traditional ﬁnancial assets (e.g., stock, bond, commodity and USD) for investors. Our work further investigates in more detail the properties of cryptocurrencies as a safe haven. The strong point of this work is the right choice of model and the methodology adopted. Indeed, in the present study, the diffusion process, which seems to correspond to the modeling of the yields of cryptocurrencies, is a Brownian geometric diffusion process with a jump and a drift adjusted by the risk premium. Gold, on the other hand, presents a return whose variations follow an Ornstein-Uhlenbek process. The advantage of applying such an approach is to take into account the stochastic nature of the market, and the high volatility of cryptocurrency and the low volatility of Gold. This helps us a lot to find reliable results.

Let ( Ω , F , P ) be a complete probability space and { F t } t ≥ 0 a filtration satisfying the usual conditions. We set up the portfolio selection model in a continuous-time framework. The two-asset model involves two different classes of assets: one is very risky (cryptocurrency) and the other is less risky (gold). So, we consider a financial market consisting of a cryptocurrency and gold. Let the Gold dynamics B ( t ) be given by the Ornstein-Uhlenbeck process to model the evolution of the spot rate and, P ( t ) be the price evolution of the cryptocurrency risky asset which is modeled by the process.

Accordingly, the dynamics of the gold spot rate of return and the prices of the coins are as follows:

d B ( t ) B ( t ) = r ( t ) d t (1)

d r ( t ) = α [ β − r ( t ) ] d t + σ r d X r ( t ) (2)

where parameters α , β and σ r are strictly positive constants and correspond to the degree of mean reversion, long-return mean, and volatility of the gold.

d P ( t ) P ( t ) = [ r ( t ) + λ σ p ] d t + σ p d X p ( t ) + ∫ ℜ γ ( t , z ) N ˜ ( d t , d z )

We will suppose that γ ( t , z ) = z . To obtain a unique solution to this stochastic differential equation, we must have 1 + γ ( t , z ) > 0 which means that z > − 1 , this implies that we may only suppose jump sizes higher than −1. As we have jumps in both directions (positive and negative), but negative jumps are not too large, we assume that:

d P ( t ) P ( t ) = [ r ( t ) + λ σ p ] d t + σ p d X p ( t ) + ∫ − 1 ∞ z N ˜ ( d t , d z ) (3)

where: μ ( t ) = r ( t ) + λ σ p with d μ ( t ) = d r ( t ) and N ˜ is a martingale,

d X p ( t ) and d X r ( t ) are standard Brownian motions with d X p ( t ) d X r ( t ) = ρ d t . ρ is the correlation between the random change in the gold spot rate and the random return on the coin asset.

We assume that the instantaneous rate of gold is the short rate a r ( t ) and the expected return μ ( t ) on the cryptocurrency equals the spot rate plus a risk premium. If y ( t ) represents the optimal fraction of the wealth invested in coins, then 1 − y ( t ) is the optimal fraction of the wealth invested in gold.

The dynamics of wealth W ( t ) is given by:

d W ( t ) = y ( t ) W ( t ) d P ( t ) P ( t ) + ( 1 − y ( t ) ) W ( t ) d B ( t ) B ( t ) (4)

Substituting (1) and (2) into (4) and simplifying the equation, we have

d W ( t ) = [ r ( t ) + y ( t ) λ σ p ] d t + y ( t ) W ( t ) σ p d X p ( t ) + y ( t ) W ( t ) ∫ − 1 ∞ z N ˜ ( d t , d z ) (5)

where W ( 0 ) = W 0 stands the initial wealth.

Letting U [ W ( t ) , t ] be a concave, additively separable utility defined over wealth, and let’s assume that the investor allocates his wealth between the two assets to maximize his expected utility of the terminal wealth. We suppose that the investor has a power utility function.

We obtain the maximized utility function J [ W ( t ) , r , t ] at the time t ∈ [ 0 , T ] such that

J [ W ( t ) , r , t ] = sup y E t ∫ t T U [ W ( s ) , s ] d s (6)

Subject to the continuous-time budget constraint of Equation (5).

where, U [ w ] = w 1 − γ 1 − γ ( 0 < γ < 1 ) is the risk averseness parameter.

Using power utility has two advantages. The first advantage is that an explicit solution can be found for our portfolio selection problem with Power utility but not with other utility functions. The second advantage is that Power utility allows an optimal solution independent of wealth and thus it simplifies the derivation. [

Equation (6) can be simplified to

J [ W ( t ) , r , t ] = sup y E t { ∫ t T J [ W + d W , r + d r , t + d t ] } (7)

We need to calculate the infinitesimal generator of an Ito-Lévy process (geometric Lévy), more specifically the generator of Ito-Lévy diffusion to find the Hamilton-Jacobi-Bellman equation. By simplifying the general version of the Ito process, assuming that E [ P ( t ) ] < ∞ , we find the form that we will use throughout the paper. The infinitesimal generator for a Lévy process with jumps is found using the Ito-Lévy theorem with the fact that N ˜ is a martingale. Letting y ( t ) ∈ A and W ( t ) the asset portfolio value process, the HJB equation associated with the dynamic portfolio problem is [

sup y ( J t + J w [ r + y λ σ p ] W + J r α ( β − r ) + 1 2 J w w W 2 σ p 2 y 2 + 1 2 J r r σ r 2 + J r w σ r σ p ρ y W + ∫ − 1 ∞ [ J ( r , s , W ( 1 + y z ) ) − J ( r , s , W ) − J w y W z ] υ ( d z ) ) = 0 (8)

Here, J t , J W , J r , J W W , J r W and J r r denote the first and second order partial derivatives with respect to t , r and W in the normal way. To solve the HJB equation, let the value function J ( t , r , W ) = W γ f ( t , r ) , where f ( T , r ) = 1 for all r. We have J t = W γ f t , J W = γ W γ − 1 f , J r = W γ f r , J r W = γ W γ − 1 f r , J W W = γ ( γ − 1 ) W γ − 2 f , J r r = W γ f r r , J ( s , r , W ( 1 + y z ) ) = W γ ( 1 + y z ) γ f .

Now replacing J-related terms by f-related terms in Equation (8) and simplifying to obtain a second order PDE for f which is written as follows:

W γ f t + γ W γ − 1 f r W + W γ f r α ( β − r ) + 1 2 W γ f r r σ r 2 + sup y { γ λ σ p W W γ − 1 y f + 1 2 γ ( γ − 1 ) W γ − 2 W 2 σ P 2 y 2 f + γ W γ − 1 W σ r σ p ρ y f r + ∫ − 1 ∞ [ W γ ( 1 + y Z ) γ f − W γ f − γ W γ − 1 f W y z ] υ ( d z ) } = 0

Dividing each side by W γ , we obtain the following Hamilton-Jacobi-Bellman (HJB) equation.

f t + γ r f + α ( β − r ) f r + 1 2 f r r σ r 2 + sup y { γ λ σ p y f + 1 2 γ ( γ − 1 ) f σ p 2 y 2 + γ f r σ r σ p ρ y + ∫ − 1 ∞ [ ( 1 + y z ) γ f − f − γ f y z ] υ ( d z ) } = 0

The term W γ is eliminated from the HJB equation because power utility is used.

By applying the first-order conditions, we obtain

λ σ p f + ( γ − 1 ) f σ p 2 y + f r σ r σ p ρ + f ∫ − 1 ∞ [ ( 1 + y z ) γ − 1 − 1 ] z υ ( d z ) } = 0 (9)

With: y = y ( f , f r , υ )

Consequently, we conjecture a solution with the following form, f ( t , r ) = g ( t ) exp ( A ( t ) r ) with terminal conditions g ( T ) = 1 , A ( T ) = 0 , [

The form of f ( t , r ) is the solution of PDE, with g and A being regular functions.

The partial derivative with respect to r is

f r = A ( t ) g ( t ) exp ( A ( t ) r ) .

After replacing f and f r terms in (9) and simplifying the equation, we obtain

λ σ p g ( t ) exp ( A ( t ) r ) + ( γ − 1 ) σ p 2 y g ( t ) exp ( A ( t ) r ) + A ( t ) g ( t ) exp ( A ( t ) r ) σ r σ p ρ + g ( t ) exp ( A ( t ) r ) ∫ − 1 ∞ [ ( 1 + y z ) γ − 1 − 1 ] z υ ( d z ) } = 0

Dividing by g ( t ) exp ( A ( t ) r ) :

λ σ p + ( γ − 1 ) σ p 2 y + A ( t ) σ r σ p ρ + ∫ − 1 ∞ [ ( 1 + y z ) γ − 1 − 1 ] z υ ( d z ) } = 0 (10)

where: A ( t ) = γ α [ 1 − exp ( α ( T − t ) ) ]

After replacing A ( t ) in Equation (10), we obtain an optimal portfolio strategy for the given portfolio problem.

λ σ p + ( γ − 1 ) σ p 2 y + σ r σ p ρ γ α [ 1 − exp ( α ( T − t ) ) ] − ∫ − 1 ∞ [ 1 − ( 1 + y z ) γ − 1 ] z ν ( d z ) = 0 (11)

This equation is non-linear in y and the difficulty comes from the integral term. Theoretically, we can solve this equation through any Lévy measure. We can provide a numerical result for the portfolio selection problem studied. We can compare the value of the optimal portfolio weights via different Lévy measures. General Lévy processes may include a complicated Lévy measurement that makes it difficult to find an optimal or even an impossible solution. However, if we limit our study to the case where the Lévy measure is continuous for the Lebesgue measure, we can provide tangible results.

So we solve the Equation (11) to determine the proportion of risky investment

Theorem 1:

We call a Lévy measure a non-negative measure ν ( d z ) on ℜ satisfying ν ( { 0 } ) = 0 and

∫ ℜ ( Z 2 ∧ 1 ) ν ( d z ) ≺ ∞

Now, we can solve the problem of portfolio selection for an investor, who is confronted with different investment opportunities described above, with a finite time horizon.

We consider the case of the Γ ( 1 , 1 ) -Lévy process whose Lévy measure is given by:

υ ( d z ) = z − 1 e − z I ( z > 0 )

This is a Lévy measure since it satisfies the inverse of Lévy-Khintchine theorem for the existence of a Lévy process, it suffices to show that

∫ ℜ min ( 1 , z 2 ) ν ( d z ) = ∫ 0 ∞ min ( 1 , z 2 ) z − 1 e − z d z = ∫ 0 1 z e − z d z + ∫ 1 ∞ z − 1 e − z d z ≺ ∞

The first term of equation is finite because the integrand is continuous on the finite closed interval [ 0 , 1 ] . Also the finiteness of the term ∫ 1 ∞ z − 1 e − z d z is immediate because of the inequality z − 1 e − z ≤ e − z , z ≥ 1 .

The finiteness of this expression guarantees the existence of a process corresponding to this measure, this process is Γ ( 1 , 1 ) . Replacing this measure, the integral equation becomes:

∫ ℜ [ 1 − ( 1 + y z ) γ − 1 ] z ν ( d z ) = ∫ 0 ∞ [ 1 − ( 1 + y z ) γ − 1 ] e − z d z = ∫ 0 ∞ e − z d z − ∫ 0 ∞ ( 1 + y z ) γ − 1 e − z d z = 1 − ∫ 0 ∞ ( 1 + y z ) γ − 1 e − z d z

Now, we can rewrite the remaining integral by using a variable change U = 1 + y z , where z = U − 1 y , we find

∫ 1 ∞ U γ − 1 e − U y e 1 y d U = 1 y e 1 y ∫ 1 ∞ U γ − 1 e − U y d U = e 1 y y [ ∫ 0 ∞ U γ − 1 e − U y d U − ∫ 0 1 U γ − 1 e − U y d U ] = e 1 y y [ y γ Γ ( γ ) − ∫ 0 1 U γ − 1 e − U y d U ]

If we take t = U y ,

∫ ℜ [ 1 − ( 1 + y z ) γ − 1 ] z ν ( d z ) = y γ − 1 e 1 y Γ ( γ ) [ 1 − P ( 1 y , γ ) ]

Definition

We have used a predefined statistical function from the program Matlab to rewrite the integral, the function is called the gamma function which is defined by

P ( a , x ) = 1 Γ ( a ) ∫ 1 x t a − 1 e − t d t

By combining these results, Equation (11) becomes:

λ σ p + ( γ − 1 ) σ p 2 y + σ r σ p ρ γ α [ 1 − exp ( α ( T − t ) ) ] − y γ − 1 e 1 y Γ ( γ ) [ 1 − P ( 1 y , γ ) ] = 0 (12)

This equation is highly nonlinear. Therefore, this equation cannot be solved explicitly. With the implementation of a numerical algorithm, we can solve this equation.

In this section, we first estimate the parameters in Equations (2) and (3) of Section 3 by maximum likelihood (ML) method. Then, we show how to implement numerically the portfolio selection model to determine the optimal proportions invested in the cryptocurrencies and gold at different times over the holding period.

Skewness and Kurtosis coefficients are calculated to measure the distribution deviation from the symmetry and to measure whether the shape of the distribution deviates from the flattening of the normal distribution. The studied samples reflect two types of asymmetry: A positive asymmetry is estimated for BTC, DASH, and LTC (0.2866), (1.4536), (2.0184) respectively. The three distributions are spread to the right. A negative asymmetry is estimated for gold (−0.0385). Gold distribution is spread to the left. A positive kurtosis is estimated for gold (4.6696), BTC (8.2070), LTC (21.6283), and DASH (12.1697). These coefficients indicate distributions with a flatter peak and thicker ends compared to the normal distribution (leptokurtic distribution).

In

The return series fluctuates around zero.

Graphically, from 2014 to 2017, the return values are almost the same (low fluctuation). In 2017, we have high values. This sudden change in returns can be explained by the fact that the price was too high during this period.

GLD | BTC | LTC | DASH | |
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Mean | 0.00003 | −0.0015 | 0.0030 | 0.0033 |

Stdev | 0.0379 | 0.0388 | 0.0596 | 0.0601 |

Skewness | −0.0385 | 0.2866 | 2.0184 | 1.4536 |

Kurtosis | 4.6696 | 8.2070 | 21.6283 | 12.1697 |

As displayed in

To estimate parameters of the Geometric Lévy process, we consider that the return of the cryptocurrencies is generated by a Geometric Brownian motion. Since the integral term does not contain the parameters that must be estimated, this term will disappear by calculating the derivatives for the two parameters ( λ , σ p ) . We estimate the parameters in Equations (2) and (3) by using the maximum likelihood (ML) method examined by [

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We use the gold as a proxy for the short rate r ( t ) and the BTC, LTC, and DASH for the price P ( t ) . The diffusion process in Equation (2) can be rewritten in discrete form as follows:

r ( t + Δ t ) − r ( t ) = α [ β − r ( t ) ] Δ t + σ r ε ( r ) Δ t (13)

where:

ε ( r ) is a standard normal deviate, r ( t + Δ t ) − r ( t ) − α [ β − r ( t ) ] Δ t is distributed as N ( 0 , σ r 2 Δ t )

The algorithm of the likelihood function L ( α , β , σ r ) , considered as a function of α , β and σ r , can be written as

L ( α , β , σ r ) = − n 2 ln ( σ r 2 ) Δ t − n 2 ln ( 2 π ) − ∑ i = 1 n [ r ( t + Δ t ) − r ( t ) − α [ β − r ( t ) ] Δ t ] 2 2 σ r 2 Δ t (14)

The diffusion process in Equation (3) can be rewritten in discrete form as follows:

Δ P ( t ) P ( t ) = ( r + λ σ p ) Δ t + σ p ε ( p ) Δ t (15)

where: ε ( p ) is a standard normal deviate.

ln ( Δ P ( t ) P ( t ) ) is distributed as N ( r ( t ) + λ σ p − σ p 2 2 , σ p )

The logarithm of the likelihood function L ( λ , σ p ) , considered as a function of λ and σ p , can be written as

L ( λ , σ p ) = − n 2 ln ( σ p 2 ) Δ t − n 2 ln ( 2 π ) − ∑ i = 1 n [ x ( t ) − [ r ( t ) + λ σ p − σ p 2 2 ] Δ t ] 2 2 σ p 2 Δ t (16)

ML estimates are obtained by maximizing Equation (14) for α , β and σ r , Equation (16) for ( λ B , σ B ) , ( λ L , σ L ) and ( λ D , σ D ) .

The estimates for the nine parameters are as follows:

The results of gold’s estimate show that the mean reversion speed and volatility are very low. This indicates that it is a less risky asset. BTC volatility is higher than that of LTC and DASH. The high volatility of BTC explains the high transaction volume and the very high return.

We apply the secant method to determine numerically the optimal percentage invested in both assets. The highly risky asset (cryptocurrency) and the less risky asset (gold). The secant method is to approach the function derivative of the first order by a deviated difference. We assume a 5-year holding period to determine the optimal fraction invested in the two asset classes at each time t ∈ [ 0 , T ] to maximize the expected utility of the terminal wealth. The optimal weights are obtained numerically using Equation (12). The estimated parameters in Equations (2) and (3) using the maximum likelihood estimation method are summarized in

Figures 2(a)-(c) show that optimal weights are more important in cryptocurrencies not only as time goes by but also as the risk aversion coefficient decreases. For Five-year investment horizon, the optimal weight in BTC (LTC) (DASH) goes from 40.96% to 42.96% to 50.91% (40.93%, 41.97%, 45.95%) (40.94%, 42.11%, 46.73%) when gamma varies from 0.1 to 0.5 to 0.9 respectively.

GOLD | ||
---|---|---|

α ^ = 0.0666 | β ^ = 0.5985 | σ ^ r = 0.0390 |

BTC | LTC | DASH |

σ ^ B = 0.0388 | σ ^ L = 0.0183 | σ ^ D = 0.0217 |

λ ^ B = 0.2188 | λ ^ L = 0.2504 | λ ^ D = 0.3278 |

These analyses would tend to validate the idea that coins are gradually emerging as a new form of a safe-haven. Gold is a traditional safe-haven. Our analyzes show that BTC, LTC, and DASH could be considered as a substitute for gold since their weights that are taken individually in an optimal investment strategy represent almost half of the proportion invested in Gold. Besides, with the high volatility of coins, there is more chance of generating a better return. An investor could have higher returns in BTC, LTC, and DASH than in Gold. There are more risks, but it is worth for investors (less risk-averse) to generate higher profits. Coins seem to be an attractive investment that can significantly increase the portfolio return. A safe-haven asset is uncorrelated with other asset classes and with the stock market indices. Evan Kuo said, “There have been almost 10 years of data suggesting that the BTC has virtually no exposure to the risk of precious metals, commodities, stocks, bonds, currencies…”. Having an uncorrelated asset can be a good way to balance an investment portfolio against other safe assets. User confidence in BTC is growing day by day and allows us to confirm better this opinion and to build BTC as a refuge in all circumstances. Our results show that the investor’s wealth is optimally distributed in equal proportions in the two asset classes and thus BTC, LTC, and Dash seem to be attractive investments for investors and are emerging as a new form of a safe investment.

In recent decades, cryptocurrencies have been one of the most important financial innovations. They have drawn a growing number of critics and supporters. To analyze cryptocurrencies as assets, we took a stochastic and dynamic modeling approach. In this study, we have dealt with an optimal portfolio model in continuous time over an infinite horizon and in a risky environment to show whether cryptocurrencies such as BTC, LTC, and DASH can replace GOLD in a portfolio of investment or not. Due to the high volatility of cryptocurrency prices, investment portfolios must be reviewed, evaluated in real time and regularly adjusted. For this reason, we worked in a dynamic and stochastic context. The closed form solution of the optimal portfolio is obtained for an investor with a CRRA utility function. Indeed, using daily data, we concluded that cryptocurrencies represent a good substitute for gold in an intertemporal investment portfolio and therefore they represent a new form of safe haven. However, the oldest and the most classic safe haven was gold. BTC, LTC and DASH are emerging as a new form of safe haven that plays an important role in the investment portfolio. Moreover, an investor can replace gold with BTC, LTC or DASH in an investment portfolio and enjoys a high return. This justifies the fact that crypto currencies are becoming a more attractive investment for investors. In this case, a cryptocurrency can also be a safe haven. You can even escape with your codes to the other side of the world, so without losing your Bitcoins or the like. In the long run, like GOLD, the rating of your cryptocurrency portfolio will tend to go up, even if you need to sell it at any time. This has led us to say that BTC, LTC and DASH become like digital gold. This analysis provides a clear answer to the question “Can Bitcoin replace gold in an investment portfolio?” that was posed by [

The results presented in this paper concern only the coin data under Jump-Diffusion model in which we only assume jump sizes that are larger than −1, so we may have jumps in both directions but negative jumps cannot be too large. However, for further works, we can change the model of the risky asset to include larger negative jumps.

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

Maghrebi, A. and Abid, F. (2021) The Investors’ Behavior towards the Relationship between Bitcoin, Litcoin, Dash Coins, and Gold: A Portfolio Modeling Approach. Journal of Mathematical Finance, 11, 495-511. https://doi.org/10.4236/jmf.2021.113028