Can Geometric Brownian Motion Model Currency Spot Rates Like Stock Prices?

Yes, geometric Brownian motion (GBM) can be used to model currency spot rates in a similar manner to how it is used for stock prices. The formula provided is a standard representation of GBM, which is often used in financial modeling to simulate the evolution of asset prices over time:

S_t S_0 e^{(mu-0.5sigma^2)t sigma W_t}

Key Considerations for Currency Spot Rates

While the mathematical framework remains similar, there are several important considerations specific to currency markets:

Interest Rate Differentials

Currency prices are influenced by the interest rates of the respective countries. The risk-neutral measure often incorporates the difference between the domestic and foreign interest rates, which can be represented in the GBM model as follows:

S_t S_0 e^{(r_d - r_f - 0.5sigma^2)t sigma W_t}

r_d is the domestic interest rate. r_f is the foreign interest rate. sigma is the volatility of the exchange rate. W_t is a standard Wiener process.

Volatility

Currency volatility can differ significantly from stock volatility. Exchange rates may exhibit different patterns of volatility, including time-varying volatility, which can be captured using models like GARCH (Generalized Autoregressive Conditional Heteroskedasticity).

Market Microstructure

Currency markets operate 24/5 and are influenced by various factors such as geopolitical events, economic data releases, and central bank policies. These factors can lead to jumps and non-normality in returns that GBM may not capture effectively.

Mean Reversion

Some currency pairs may exhibit mean-reverting behavior, especially in the context of long-term equilibrium levels. This behavior can be better modeled using other processes such as the Ornstein-Uhlenbeck process rather than GBM.

Liquidity and Transaction Costs

Currency markets may have different liquidity profiles compared to equity markets. Transaction costs and bid-ask spreads can also impact the simulation of currency prices, warranting a more nuanced approach to modeling.

Conclusion

In summary, while GBM can be used to model currency spot rates, it is essential to consider the unique characteristics of the currency markets, specifically interest rate differentials, volatility dynamics, and other market factors. Depending on the specific requirements of your analysis or simulation, you might want to incorporate additional models or adjustments to capture these complexities effectively.