The returns previously presented were used to calculate the SVCJ model. The main code for the calculation of the SVCJ was the one used by (Chen et al., 2018) in their estimations, I did some minor changes to translate it from Matlab into R (both codes are available via www.quanlet.de). For the SVCJ estimation, a total of 5000 iterations were done in each case with a burn-in of 1000 to minimize initial value influence. Figure 5 presents the trace plot for the parameter µ (XRP not shown), which represents the long term return. It is interesting to see how the parameter fluctuates over a value near to zero in a stable interval and also not changing considerably in relation with the initial values, which supports the idea of using an
Figure 4: Returns of Bitcoin Cash (BCH), Ethereum Classic (ETH), Ontology (ONT) and Theter (USDT)
initial value ofµ∼N(0,25) as originally proposed by (Chen et al., 2018). The case of USDT is interesting since, as shown in figure 2, the returns do not present important variations and that could explain the almost null variation of the parameter after iteration 3500.
Another interesting parameter isλ, that represents the jump arrival rate, and whose trace plot is presented in figure 6. As can be seen from the figure, λdiffers between cryptos, some with higher values than others. The case of ETH seems to be interesting in the way that let us see the Metropolis-Hastings in action when the parameters is back on its path around iteration 2300. Going forward, Figure 7 presents the trace plot for the parameter ρ, correla-tion between Brownian mocorrela-tions of returns and volatility, reaching to similar conclusions as in the previous two trace plots. The remaining trace plots can be found on the code. The shiny application will not show any trace plot since it is oriented to option price estimation
Table 2 shows the estimated parameters for the different cryptos. We can see a low MSE indicating an overall good fitting. One potential pitfall of the MCMC is that results depend on the initial values of the parameters. As already mentioned one possible way to solve that
Figure 5: Trace Plot of parameter µ(MCMC iteration on x-axis)
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Figure 6: Trace Plot of parameterλ(MCMC iteration on x-axis)
SVCJOptionApp
Figure 7: Trace Plot of parameter ρ (MCMC iteration on x-axis)
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period. Still, even using a considerable burn-in there could be changes in the results depend-ing on the initial values. An additional way to prevent this is to use initial values that one extract observing the data. Unfortunately this is not easy and, eventually, only applies for few parameters such asµorσv. For the rest of the parameters, to infer the initial values by observing the data could be more complicated.
Table 3, shows a simple sensitivity analysis for the parameter mu of BTC where the es-timates are presented for different initial values. The decision to take an initial value for µ following a Normal distribution with mean a= 0 and standard deviation A = 25 is coming for the posterior equation presented in section A.1 in one hand, and from the observed long term trend of the return which oscillates around 0 for every crypto, on the other hand. The discussion could be centered in the initial value of the standard deviationA and that is what is presented in table 3.
Lets take now the example of the parameter σy, again for BTC, which represents the standard deviation of the jump size Zty. The parameterσy follows an Inverse Gaussian
dis-Crypto µ µy σy λ α β ρ σv ρj µv MSE ADA mean -0.115 0.006 52.5 0.043 0.364 -0.418 -0.022 0.045 0.039 6.425 0.897
sd 0.057 0.740 23.176 0.012 0.097 0.106 0.054 0.027 0.085 2.141
BCH mean -0.066 -0.046 59.484 0.035 0.257 -0.273 0.013 0.090 0.027 5.532 0.816 sd 0.052 0.713 27.868 0.011 0.069 0.055 0.048 0.032 0.095 2.307
BTC mean 0.029 0.004 3.974 0.025 0.009 -0.066 0.008 0.011 -0.003 0.706 0.84 sd 0.008 0.093 1.015 0.005 0.002 0.012 0.026 0.002 0.086 0.121
CRIX mean 0.032 -0.002 3.145 0.030 0.010 -0.097 0.012 0.013 -0.007 0.819 0.854 sd 0.008 0.082 0.755 0.007 0.003 0.018 0.024 0.002 0.069 0.181
DASH mean -0.023 0.100 17.345 0.049 0.049 -0.161 -0.019 0.032 -0.003 2.129 0.866 sd 0.015 0.187 4.008 0.007 0.008 0.017 0.025 0.005 0.059 0.265
EOS mean -0.082 0.081 8.201 0.164 0.161 -0.450 -0.002 0.032 0.013 2.203 0.743 sd 0.044 0.248 1.680 0.027 0.030 0.087 0.048 0.019 0.075 0.425
ETC mean -0.031 -0.479 99.434 0.024 0.221 -0.326 0.018 0.118 0.052 12.541 0.779 sd 0.030 0.620 32.063 0.007 0.094 0.088 0.037 0.028 0.037 3.609
ETH mean 0.007 0.011 4.650 0.047 0.029 -0.094 -0.018 0.032 0.000 1.594 0.886 sd 0.019 0.120 1.501 0.014 0.008 0.011 0.027 0.007 0.051 0.393
LTC mean -0.007 0.005 8.177 0.048 0.011 -0.123 -0.012 0.013 0.026 1.391 0.852 sd 0.009 0.136 1.694 0.008 0.003 0.010 0.025 0.004 0.065 0.211
ONT mean -0.067 0.079 10.554 0.085 0.505 -0.565 -0.029 0.033 0.013 3.228 1.026 sd 0.093 0.458 4.391 0.028 0.164 0.111 0.080 0.020 0.099 1.038
USDT mean -0.001 -0.032 7.536 0.048 0.003 -0.386 0.024 0.004 0.024 1.278 0.389 sd 0.006 0.395 3.775 0.016 0.001 0.036 0.069 0.001 0.232 0.405
XMR mean 0.003 0.038 29.024 0.025 0.063 -0.098 0.004 0.048 0.007 2.032 0.869 sd 0.021 0.247 9.691 0.006 0.012 0.018 0.022 0.008 0.081 0.369
XRP mean -0.043 0.054 20.358 0.043 0.026 -0.156 -0.018 0.025 0.002 3.095 0.834 sd 0.012 0.185 4.127 0.006 0.005 0.011 0.022 0.006 0.040 0.564
Table 2: SVCJ estimated parameters
Parameter
µ Init. Val: a = 0, A = 10 a = 0, A = 25 a = 0, A = 100
Estimate: 0.030 0.029 0.033
σ
yInit. Val: f = 5, F = 40 f = 100, F = 40 f = 5, F = 200
Estimate: 3.974 0.540 20.466
Table 3: BTCµ andσy parameters for different initial values
tribution. A couple of different initial values were used in table 3, but unlike µ, their choice is more difficult to support and without previous information they are completely arbitrary.
We can see from the table how the parameter estimates differs considerably. Figure 8 also facilitates the visualization while showing different trace plots for the parameter σy, accord-ing to different initial values. We can see on the Y-axis how the values differ, even after the burn-in period.
Table 3 is by no means a complete sensitivity analysis because the range of initial val-ues to be tested needs to be enlarged considerably. Even, expanding the grid of potential initial values, results could be misleading if we omit important prior knowledge. A more sophisticated numerical technique should be applied, requiring considerable computing time, but is beyond the scope of this paper. The objective of the table is only to warn about the shortcomings of the MCMC in the context of SVCJ estimation, by presenting a very simple example. A good reference to improve the results can be found in (Kristensen and Shin, 2012), where a non parametric simulated maximum likelihood is used for dynamics models where no closed-form representation of the likelihood function is available providing hints when considering an alternative way to set up the initial parameters.
Nontheless, an interesting property of the MCMC is that, as its name indicates, it works as a chain, with the advantage of estimating not just parameters but also volatility and co-variates. Figure 9 shows the estimated volatility under the SVCJ. It is interesting to see the increased volatility period at the end of year 2017. We can see how the estimated volatility is
Figure 8: Trace plots for parameterσyof BTC using different initial valuesf andF (MCMC iteration on x-axis)
SVCJOptionApp
trace plots, the estimated SVCJ volatility plot is not part of the app results. Fortunately for the users all the following plots can be also accessed using the app.
To start with the app results, another interesting plots coming from the SVCJ model are the estimated jumps in returns and volatility, presented in figure 10 where we can see the jumps in returns on the left column and the jumps in volatility on the right column. We can see how, compared to BTC, BCH exhibits a lower frequency of jumps. On the contrary, DASH highlights for having a high frequency of jumps. XMR is also interesting since the size of positive jumps in returns is higher compared to the negative jumps. In neither case the jumps in volatility are negative due to the model definition which set them up following an exponential distribution. A more detailed plot of jumps can be seen using the app since only one crypto at a time is plotted there.
The second result users can find on the app is the QQ-plot of the SVCJ residuals, such as those plotted in figure 11. It is clear from the figure how the SVCJ model residuals seems to follow a normal distribution which speaks about a good model fitting. The residuals follow
Figure 9: SVCJ in-sample fitted volatility
SVCJOptionApp
the pattern mentioned by almost all the cryptos. One exception, which is shown on purpose on the lower-right panel of figure 11, is USDT. We can see how the residuals deviate from the red line, indicating that they possibly do not follow a normal distribution. Additional comparisons can be done by users on the app, such as comparing the SCVJ residuals with a GARCH (2,2) model residuals. For a more deep analysis of the econometrics of CRIX and BTC please refer to (Chen et al., 2016).
Once we have the SVCJ parameters next step consist in simulate 5000 returns paths for each crypto. Figure 12 shows two simulated returns paths for ETH. The simulated paths show jumps with the desired frequency and size, also one can see how the returns oscillate around the zero line. If we use a reference or initial price, we can transform the simulated returns of figure 12 into simulated prices. Such is the case shown in figure 13, were five different price paths are shown, again for the case of ETH, using an initial price of 215 USD.
We can observe, as well as in the previous figure, the well defined jumps, driving prices up or down depending on the path and the observation. For every crypto 5000 price paths are
Figure 10: Estimated jumps in returns (left column) and volatility (right column)
Figure 11: SVCJ residuals
SVCJOptionApp
Figure 12: Two simulated return paths for ETH (one blue, one black) using SVCJ param-eters
SVCJOptionApp
Having the simulated price paths, what remains is to compute option prices. The way to do that is defining one strike price K and one time to maturity t and then computing the option pay-off according to formulas from section 3.3, depending if we want a call or a put option. Then to take expectations of the pay-off and discount that value. The result, again for the case of ETH, can be seen in table 4. Users can get similar tables with the app even with the possibility of download them into a csv file.
Last but not least, the final result of the app is the Implied Volatility computed using the Black Scholes formula as the one depicted in figure 14
Figure 13: Five simulated price paths for ETH using an initial price of 215 USD
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Figure 14: Implied Volatility for ETH Call Option using Black-Scholes formula
SVCJOptionApp
K/t 1 7 30 60 90 180 360 183 33.17 39.05 54.82 69.85 83.39 112.68 147.72 186 30.32 36.66 52.90 68.21 81.87 111.46 146.78 189 27.51 34.34 51.01 66.60 80.38 110.25 145.86 192 24.76 32.10 49.17 65.02 78.92 109.05 144.95 196 21.23 29.24 46.79 62.97 77.01 107.49 143.75 199 18.70 27.20 45.07 61.47 75.62 106.33 142.86 202 16.30 25.26 43.41 60.01 74.25 105.19 141.99 205 14.05 23.40 41.80 58.58 72.91 104.07 141.12 209 11.34 21.07 39.74 56.71 71.16 102.59 139.97 212 9.52 19.45 38.26 55.35 69.87 101.50 139.12 215 7.89 17.92 36.83 54.02 68.61 100.43 138.28 218 6.47 16.49 35.46 52.73 67.37 99.37 137.45 221 5.25 15.16 34.14 51.47 66.16 98.33 136.63 225 3.92 13.53 32.45 49.84 64.58 96.96 135.54 228 3.12 12.41 31.25 48.65 63.42 95.94 134.75 231 2.46 11.37 30.08 47.49 62.28 94.94 133.96 234 1.93 10.42 28.97 46.37 61.17 93.96 133.18 238 1.41 9.28 27.55 44.92 59.71 92.68 132.16 241 1.13 8.52 26.54 43.87 58.63 91.74 131.41 244 0.91 7.82 25.57 42.84 57.57 90.80 130.66 247 0.74 7.20 24.64 41.85 56.53 89.88 129.92
Table 4: Call Option prices for ETH for different strike prices K and time to maturity t
5 Conclusions
Cryptocurrencies have become an important object of study for different disciplines ranging from computer science to economics, also including mathematics, statistics and law. The reasons behind that include the revolutionary technology the have brought and also the im-portance the cryptos started having as financial assets. We should expect more developments in the crypto market that will include the participation of more agents, that so far have seem to be reluctant, such as central and commercial banks, and also the idea of the development of a derivatives market for cryptos.
An important aspect, parallel to the development of the derivatives market, is to under-stand cryptos price behavior which is a challenging task since traditional econometric models, such as ARIMA and GARCH, are not necessarily the initial option to chose and more sophis-ticated models such as the SVCJ seem to fit the data better. Even though the SVCJ model better fit the data, there is still space for incorporating additional techniques, specially those oriented to identify the prior distributions of the parameters with their initial values.
For the case of the option price estimation, additional assumptions facilitate the estima-tion since there are no real opestima-tion prices to compare. Those assumpestima-tions, such a zero risk premium, could be controversial but they facilitate initial exercises of option price estimation.
Once real cryptos options start to be traded, more realistic assumptions can be taken in order to improve the price estimation. I hope this text could help someone interested in the crypto market to get some additional insights in how to approach the problem of returns and option price estimation. Further analysis will be required considering that cryptocurrencies came to stay.
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A Appendix
A.1 Posterior Distributions
To derive the exact formulas of the posterior distribution is beyond the scope of this paper, since they require extensive algebra and the use of conjugates distributions, nevertheless, for a complete treatment of the posterior probabilities please refer to (Numatsi, 2010). Here the only the final formulas are presented.