Cox-Ingersoll-Ross CIR

Tags: #Financial #Economics

Equation

$$\mathrm{d} r(t) = a[b - r(t)] \mathrm{d} t + \sigma \sqrt{r(t)} \mathrm{d} Z(t) \\ P(r, t, T) = A(T-t)e^{-rB(T-t)} \\ \gamma = \sqrt{(a-\bar{\phi})^{2} + 2 \sigma^{2}} \\ q(r, t, T) = \sigma \sqrt{r} B(T-t) \\ \text{yield to maturity} \\ \frac{2ab}{ a - \bar{\phi} + \gamma}$$

Latex Code

                                 \mathrm{d} r(t) = a[b - r(t)]  \mathrm{d} t + \sigma \sqrt{r(t)} \mathrm{d} Z(t) \\
            P(r, t, T) = A(T-t)e^{-rB(T-t)} \\
            \gamma = \sqrt{(a-\bar{\phi})^{2} + 2 \sigma^{2}} \\
            q(r, t, T) = \sigma \sqrt{r} B(T-t) \\
            \text{yield to maturity} \\
            \frac{2ab}{ a - \bar{\phi} + \gamma}

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Introduction

$\mathrm{d} r(t) = a[b - r(t)] \mathrm{d} t + \sigma \sqrt{r(t)} \mathrm{d} Z(t) \\ P(r, t, T) = A(T-t)e^{-rB(T-t)} \\ \gamma = \sqrt{(a-\bar{\phi})^{2} + 2 \sigma^{2}} \\ q(r, t, T) = \sigma \sqrt{r} B(T-t) \\ \text{yield to maturity} \\ \frac{2ab}{ a - \bar{\phi} + \gamma}$

Latex Code

            \mathrm{d} r(t) = a[b - r(t)]  \mathrm{d} t + \sigma \sqrt{r(t)} \mathrm{d} Z(t) \\
            P(r, t, T) = A(T-t)e^{-rB(T-t)} \\
            \gamma = \sqrt{(a-\bar{\phi})^{2} + 2 \sigma^{2}} \\
            q(r, t, T) = \sigma \sqrt{r} B(T-t) \\
            \text{yield to maturity} \\
            \frac{2ab}{ a - \bar{\phi} + \gamma}

Explanation

Latex code for the Cox-Ingersoll-Ross model.

$R_{0}$ : 1-year interest rate.
$P_{0}$ : 1-year bond price
$P(1, 2, r_{u}), P(1, 2, r_{d})$ : year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.
$P_{1}$ : Observed year-1 price of a 1-year bond