Deriving the Reverse Exchange Rate Dynamics with Itô’s Lemma

Summary

The document derives the stochastic dynamics of a reverse exchange rate. If the quoted rate gives domestic currency per unit of foreign currency and follows geometric Brownian motion, the reverse rate is its reciprocal, representing foreign currency per unit of domestic currency.

Applying Itô’s lemma to the reciprocal yields a drift of volatility squared minus the original drift, and a diffusion term with the opposite sign. In relative-return form, the reverse rate therefore has drift σ² − μ and volatility −σ. This result follows from the curvature term in Itô’s lemma; simply negating the original drift would omit that correction. The derivation assumes the stated continuous-time diffusion model and does not discuss alternative rate conventions, jumps, or empirical estimation.

Key ideas

• The reverse exchange rate is the reciprocal of the original quoted rate.
• Itô’s lemma adds a volatility-squared correction to the reverse rate’s drift.
• The diffusion term changes sign when taking the reciprocal.
• The result assumes geometric Brownian motion for the original rate.

Tags

# Deriving $dR(t)$ For Reverse Exchange Rate


# Deriving $dR(t)$ For Reverse Exchange Rate












Say $Q(t)$ is the exchange rate at time $t$. It's the price in domestic currency of one unit of foreign currency and converts foreign currency into domestic currency.

The model for the dynamics of this exchange rate is:

$$\frac{dQ(t)}{Q(t)}=\mu_Q dt+\sigma_Q dB(t)$$

Then the reverse exchange rate, $R(t)$, would be the price in foreign currency of one unit of domestic currency modeled by:

$$R(t)=\frac{1}{Q(t)}$$

My question is, how would I derive $dR(t)$?

## Answer by Raskolnikov (score 2, accepted)

https://quant.stackexchange.com/a/37341

With the help of Itô's lemma, you can show that

$$df(Q)=f'(Q)dQ+\frac{1}{2}f''(Q)dQ^2 \; .$$

Putting $R=f(Q)=1/Q$ and using

$$\frac{dQ(t)}{Q(t)}=\mu_Q dt+\sigma_Q dB(t)$$

you should get

$$dR = \frac{\sigma_Q^2-\mu_Q}{Q}dt-\frac{\sigma_Q}{Q}dB$$

or equivalently

$$\frac{dR(t)}{R(t)} = (\sigma_Q^2-\mu_Q)dt-\sigma_Q dB(t) \; .$$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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