# Geometric Brownian Motion Models

Cryptocurrency:

Shown above are probability density functions that are solutions of the Garman-Kohlhagen Model, a type of geometric Brownian motion model: $\text{d}X_t = (r_d – r_f) X_t \text{d}t +\sigma X_t \text{d}W_t,$ where $$X_t$$ is the exchange rate at time $$t$$, $$r_d$$ is the domestic risk-free rate of return, $$r_f$$ is the foreign rate of return, $$\sigma$$ is the volatility, and $$W_t$$ is a standard Wiener process (i.e. the noise comes from a Gaussian distribution). Here we use the LIBOR for the risk-free rate $$r_d$$ and set $$r_f=0$$ since cryptocurrencies do not have a yield. The volatility is estimated from 90 days of historical returns as described here. The distribution shows the probability of observing an exchange rate after 7 days of trading. $$\mu$$ on the y-axis represents $$10^{-6}$$. This chart is updated daily. Informational purposes only.

BTC = Bitcoin, ETH = Ethereum, BCH = Bitcoin Cash, XRP = Ripple, LTC = Litecoin, DASH = Dash, XMR = Monero, XEM = NEM, ETC = Ethereum Classic, XLM = Stellar Lumens, ZEC = Zcash, NXT = Nxt, REP = Augur, LSK = Lisk, FCT = Factom.

Raw price data courtesy of Poloniex.