#### Conclusion, c**omments, extension and further research.**

I believe this paper provides a simple (also empirically accurate enough) quantitative framework for price forecasting, also easily extendable from bitcoin to other type of assets. As early as we are in this stage, be it currency, utility token or protocol value carriers(Ether or EOS) most information is still contained in the total number of addresses, translated into prices by the newly derived form of Metcalfe’s law, if it still can be called Metcalfe’s law..

Hence by extension, if one can predict how fast and how far the count of total addresses can grow, one can in some degree estimate the price of crypto assets in short -mid term.

I’m sure most practitioners already agree with me that some factors, for instance founders track record,strength of assembled team, public interests, market sentiments etc. are very important factors evaluating a project, since these factors directly translates into the velocity of address growth. Extending this framework, VCs,fund managers will be able to built customised valuation and risk model.

The proposed modification to Metcalfe’s law is also discussed in detail and in my limited knowledge, this is the first time ever such effects is discussed, and most of time, overlooked.

Second order derivative of the model is beta itself, which can also be interpreted as volatility, explaining the observation that ,compared to traditional equity assets, the conditional volatility of this market is reversed to that of classic equity assets, and will be so in the foreseeable future. This is a very important observation for option traders on crypto assets. As volatility calibration will be theoretically and empirically reversed to that of equity assets.

Mean reverting setup of beta captures the momentum effects in price dynamics, momentum exists! and it can go to zero on a high way, nonstop.

*Flaws:*

First of all, Simulation is based on the assumption that there will always be people using the network. To some, it’s non rigorous, as abandonment would effectively drive network value to zero. I agree on this point and acknowledge my method may only be good for simulating the short to mid periods,on the networks with a group of devoted supporters.**Or, perhaps using the account who actually owns some bitcoin is a better regressor.**

Also the maths and modelling framework ** can and need to be** improved,(the background maths only works under very strict and non-robust assumptions) I modelled address growth as a deterministic curve,while it should also be dynamic and with stochastic components. I ignored most of the stochastic calculus assumptions of return and volatility.

*And the arguments on volatility and beta is a mixture of intuition and empirical evidence.*Thirdly I ignored the correlation between the velocity of address growth and beta, and modelled each as if they were independent, which is obviously untrue. Since empirical data shows that betas are correlated to the velocity of address increments.

Finally, the model requires a lot of historical data, and can only be applied individually, as different projects will have very different beta curve. I did not conduct a full test on another asset type other than bitcoin.

*Further research:*

For academics, adding a copula dynamic between beta and velocity of address growth is the goto step to enhance model accuracy. We may also need a more rigorous model specification, for example I manually decided the best fits in the ARMA process without doing any BICs or AICs tests.

For practitioners, the model can be extended to two directions, first and most apparently, pricing forecast. Secondly, this model can also be used as risk management tool, to calculate the VAR of asset under management.

Last but not least,As I failed to account for the correlation between address growth and beta, that will be the focal point of my future research.I believe hidden within it lies the answer for how the volatility and return will evolve in the future. For bitcoin, as well as other crypto assets, if mass adoption occurs, I suspect the volatility as well as the expected return will reach a plateau on par with current existing major fiats, therefore “stable coins” in the long run is only a temporary solution and not really needed. The idea was inspired by the research from Catalini et,al.(2018) on how the crypto token value will grow and evolve, as well as the discussions I saw on Twitter. Building on top of this story, I may try to verify that idea from an econometrical and quantitative prospective. The result maybe useful for crypto start-ups drafting token economics, in a way that they no longer need to hire “market makers” to manipulate price,focus on product and market will handle the rest.

Also here’s a list of less articulated but ambitious place I may want to explore.

1,A Risk controlling tool.

2, an option pricing model built on crypto currencies.

3, a mechanism to curb volatility.

4,a statistical arbitrage model that actually works.

5,continue fine-tuning this. maybe a way to estimate the proportion of cryptoassets that’s being hodl or lost.(newest update Jul-10th, improved background math a bit)

#### References:

Athey, S., Parashkevov, I., Sarukkai, V., and Xia, J. (2016), “Bitcoin pricing, adoption, and usage: Theory and evidence,” mimeo., Stanford.

Bouri, E., Azzi, G., & Dyhrberg, A. H. (2016). On the return-volatility relationship in the Bitcoin market around the price crash of 2013.

Catalini, C., & Gans, J. S. (2018). Initial coin offerings and the value of crypto tokens (No. w24418). National Bureau of Economic Research.

Creal, D., Koopman, S. J., & Lucas, A. (2013). Generalized autoregressive score models with applications. *Journal of Applied Econometrics*, *28*(5), 777–795.

Wheatley, Spencer, et al.(2018) “Are Bitcoin Bubbles Predictable? Combining a Generalized Metcalfe’s Law and the LPPLS Model.” .