(Ordinary differential equations for the mixed affineyield...

(Ordinary differential equations for the mixed affineyield model). In the mixed model of Subsection 10.2.3, as in the two-factor Vasicek model and the two-factor Cox-Ingersoll-Ross model, zero-coupon bond prices have the affine-yield form $$ f\left(t, y_1, y_2\right)=e^{-y_1 C_1(T-t)-y_2 C_2(T-t)-A(T-t)}, $$ where $C_1(0)=C_2(0)=A(0)=0$. (i) Find the partial differential equation satisfied by $f\left(t, y_1, y_2\right)$. (ii) Show that $C_1, C_2$, and $A$ satisfy the system of ordinary differential equations $$ \begin{aligned} C_1^{\prime} & =-\lambda_1 C_1-\frac{1}{2} C_1^2-\sigma_{21} C_1 C_2-(1+\beta) C_2^2+\delta_1, \\ C_2^{\prime} & =-\lambda_2 C_2+\delta_2, \\ A^{\prime} & =\mu C_1-\frac{1}{2} \alpha C_2^2+\delta_0 . \end{aligned} $$

(Ordinary differential equations for the mixed affineyield model). In the mixed model of Subsection 10.2.3, as in the two-factor Vasicek model and the two-factor Cox-Ingersoll-Ross model, zero-coupon bond prices have the affine-yield form
$$
f\left(t, y_1, y_2\right)=e^{-y_1 C_1(T-t)-y_2 C_2(T-t)-A(T-t)},
$$
where $C_1(0)=C_2(0)=A(0)=0$.
(i) Find the partial differential equation satisfied by $f\left(t, y_1, y_2\right)$.
(ii) Show that $C_1, C_2$, and $A$ satisfy the system of ordinary differential equations
$$
\begin{aligned}
C_1^{\prime} & =-\lambda_1 C_1-\frac{1}{2} C_1^2-\sigma_{21} C_1 C_2-(1+\beta) C_2^2+\delta_1, \\
C_2^{\prime} & =-\lambda_2 C_2+\delta_2, \\
A^{\prime} & =\mu C_1-\frac{1}{2} \alpha C_2^2+\delta_0 .
\end{aligned}
$$

Key Concepts

Ordinary Differential Equations for Coefficient Functions (Riccati Equations)

In the context of affine term structure models, the coefficients in the exponential-affine expression of the bond price often satisfy a system of ordinary differential equations, many of which are of the Riccati type. These ODEs govern the evolution of the functions that linearly combine the state variables and the constant term in the exponent. Solving these equations is essential for obtaining closed-form or semi-analytical solutions for bond prices and yields.

Affine Term Structure Models

These models represent bond prices or yields as an exponential-affine function of the state variables. That is, the logarithm of the bond price is an affine function (a linear combination plus a constant) of certain factors driving the interest rates. This approach greatly simplifies the analysis and calibration of term structure models, as it leads to tractable solutions for bond pricing and interest rate derivatives.

Zero-Coupon Bond Pricing

Zero-coupon bond pricing involves determining the price of a bond that pays no interim coupons and only a single payment at maturity. In affine term structure models, the price is typically expressed in an exponential form where the exponents are affine functions of the current state variables. This representation is crucial for deducing pricing formulas and for understanding the impact of the underlying factors on bond prices.

Partial Differential Equations in Financial Modelling

Partial differential equations (PDEs) arise naturally in financial models through methods such as the Feynman-Kac representation, which relates expectations under stochastic processes to solutions of PDEs. When pricing bonds or other derivatives under a given stochastic process for the factors, one typically derives a PDE that the pricing function must satisfy. This PDE encodes information about the time evolution and spatial (state variable) dynamics of the bond price.

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