Exchange Rates with Stochastic Domestic and Foreign Rates
Summary
The document formulates an exchange-rate model in which the spot rate follows geometric Brownian motion and the domestic and foreign short rates each follow a Vasicek mean-reverting process. Under the domestic risk-neutral measure, the spot drift is the difference between the two rates, while each rate is pulled toward its own long-run level. The three Brownian shocks may be correlated, so spot and rate movements can interact.
The author asks whether the resulting spot rate is normally distributed and how to compute its expectation and variance, focusing on the integrated rate differential in the solution. The setup is useful for understanding how stochastic funding rates enter an FX model, but the document does not supply the distribution or derive moments. It also does not specify parameter values or correlations, so it offers a model formulation and an analytical question rather than a complete pricing or simulation method.
Key ideas
- The exchange-rate drift under the stated measure is the domestic rate minus the foreign rate.
- Both short rates are modeled as mean-reverting Vasicek processes.
- Correlations among the spot and rate shocks can affect the joint model.
- The integrated stochastic rate differential complicates the distribution of the exchange rate.
- The document poses the distribution and moment problem without solving it.
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Full text
# Distribution and Analytical solution of a GBM with stochastic interest rate?
# Distribution and Analytical solution of a GBM with stochastic interest rate?
We model the exchange rate $S_t$ with a geometric Brownian motion and the USD and EUR interest rates $r_u$ and $r_e$ each according to the Vasicek model. Under the domestic equivalent martingale measure $Q$, our model with three state variables can be written as \begin{eqnarray*} \frac{dS_t}{S_t} &=& (r_t^u - r_t^e)dt + \sigma_S dW_t^S \\ dr_t^u &=& \kappa_u(\theta_u - r_t^u)dt + \sigma_d dW_t^u \\ dr_t^e &=& \kappa_e(\theta_e - r_t^e) dt + \sigma_e dW_t^e \end{eqnarray*} where $\kappa_u$ and $\kappa_e$ are the mean reversion rates, $\theta_u$ and $\theta_e$ are the long-term means, $\sigma_S$, $\sigma_u$, and $\sigma_e$ are the instantaneous volatilities, and $W_t^S$, $W_t^u$, and $W_t^e$ are the Brownian motions linked by the correlation matrix \begin{eqnarray*} \rho = \left(\begin{array}{cc c} 1 & \rho_{Su} & \rho_{Se} \\ - &1& \rho_{ue} \\ - & - & 1 \end{array}\right) \end{eqnarray*}
I am looking for the distribution of $S_t$. I know that the analytical solution of $S_t$ takes the following form, of course, if I am right:
\begin{eqnarray*} S_t = S_l \exp\Big\{ \int_{l}^{t}\Big((r_y^u - r_y^e) - \frac{\sigma_S^2}{2}\Big)dy + \sigma_S W_{t-l}^S \Big\}, \end{eqnarray*}
I wonder if $S_t$ is normally distributed? If so, how I can find the expectation and variance. The most difficulty goes for the integral part.
Thank you in advance for any help you guys can provide me.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.