Geometric Brownian Motion Models


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.
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