Correlating an Asset with Factors in a Two-Factor Gaussian Rate Model

Summary

The document describes a model combining a two-factor Gaussian short-rate process with an asset whose price follows a Black–Scholes-style stochastic differential equation. The two rate factors are mean reverting, and their Brownian drivers are specified as correlated. The asset process also includes a yield term and a time-varying volatility term, with its Brownian driver allowed to correlate with each rate factor.

The question is how to choose those two asset-factor correlations to obtain a prescribed correlation between the asset shock and the sum of the factor shocks. No answer or derivation is provided, and the stated target uses the price increment while describing a correlation with the Brownian shock. The document therefore identifies a covariance-modeling problem but does not establish a solution, feasibility conditions, or implications for pricing. Readers would need to distinguish correlations among Brownian drivers from correlations involving the asset’s price increment and account for the scaling in its stochastic term.

Key ideas

• The short rate is modeled as the sum of two mean-reverting Gaussian factors and a deterministic term.
• The asset’s stochastic driver may be correlated with both rate-factor drivers.
• The question asks how those pairwise correlations determine the relationship between the asset shock and the sum of factor shocks.
• The document provides no derivation or answer to the correlation-selection problem.

Tags

# Correlation between Two Factor Gaussian Shortrate Model and Black Scholes Model


# Correlation between Two Factor Gaussian Shortrate Model and Black Scholes Model












I want to implement a two factor Gaussian Shortrate Model \begin{align} r(t) & = x(t) + y(t) + \phi(t), \\ dx(t) & = -ax(t)dt + \sigma dB_1 (t), \\ dy(t) & = -by(t)dt + \eta dB_2(t), \end{align} with correlation $dB_2(t)dB_2(t) = \rho_{12}dt$ and do have the price process of an asset $S(t)$ given bei the Black Scholes equation \begin{align} dS(t) = S(t) \big[(r(t)-y)dt + \mu (t) dB_S (t) \big], \qquad y>0, \,\ S(0)=0, \end{align} so that you have \begin{align} S(t) = e^{\int_0^t (r(s)-y)ds - 0.5 \cdot \int_0^t \mu (s)^2 ds + \int_0^t \mu(s) dB_S(s)} . \end{align}

Thanks for your help!

My Question is: How do i have to choose the correlations \begin{align} dB_1(t)dB_S(t) &= \rho_{1S} dt, \\ dB_2(t) dB_S(t) &= \rho_{2S} dt, \end{align} so that i can get a predetermined correlation between $dB_S(t)$ and the sum $d(B_1(t) +B_2(t))$, i.e. i want to habe \begin{align} d(B_1(t) +B_2(t))dS(t) = \rho dt \end{align}.

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