Deriving the Reciprocal FX Rate Under Geometric Brownian Motion

Summary

The document derives the dynamics of EUR/USD from a model for USD/EUR. It assumes the USD/EUR exchange rate follows geometric Brownian motion with drift r and volatility σ, then represents the reciprocal exchange rate as 1/S. Applying Itô’s formula to that reciprocal gives its drift and diffusion terms: the drift becomes σ² − r, while the volatility coefficient remains σ.

The explanation provides a direct derivation and identifies the volatility-related adjustment to the inverse rate’s drift. It is limited to the stated continuous-time model and currency convention; it does not discuss empirical estimation, interest-rate parity, or how to interpret the drift under a particular pricing measure. The result is useful when translating a stochastic model between reciprocal exchange-rate quotes, provided the original assumptions and units are kept consistent.

Key ideas

• The reciprocal exchange rate is 1/S when the original quote is S.
• Applying Itô’s formula to 1/S produces a drift of σ² − r.
• The reciprocal process retains σ as its diffusion coefficient.
• The derivation assumes the original quote follows geometric Brownian motion.

Tags

# FX Rate dynamics


# FX Rate dynamics












Let's suppose USD/EUR price in USD follows a GBM with $$ dS_t = rS_tdt + \sigma S_tdW_t $$ What process does EUR/USD follow in EUR?

## Answer by Probilitator (score 4, accepted)

https://quant.stackexchange.com/a/11020

This will be the inverse process

$$\frac{1}{S_t}$$

Applying Itô's formula the dynamics are then given by

$$d\frac{1}{S_t}=\frac{-1}{S_t^2}dS_t+\frac{1}{S_t^3}dS_tdS_t$$ some simple algebra then leads to $$d\frac{1}{S_t}=\frac{1}{S_t}(\sigma^2 -r)dt+\frac{1}{S_t}\sigma dW_t$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.