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The Vasicek Model: Unraveling Interest Rate Dynamics

A comprehensive guide to the math and magic behind interest rate modeling

What is the Vasicek Model?

The Vasicek model is a cornerstone of financial mathematics, used to model the stochastic evolution of interest rates. Whether you're a finance enthusiast, a quant, or just curious about how markets work, this guide breaks down the model's mathematical foundation in a clear and engaging way. Ready to dive in? Let's explore the math that powers bond pricing and more!

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1. Model Definition

The Vasicek model assumes the instantaneous short rate \( r(t) \) follows an Ornstein-Uhlenbeck process, described by the stochastic differential equation (SDE):

\[ dr(t) = \kappa (\theta - r(t)) dt + \sigma dW(t) \]

Where:

• \( r(t) \): Short-term interest rate at time \( t \).
• \( \kappa \): Speed of mean reversion (positive constant).
• \( \theta \): Long-term mean interest rate.
• \( \sigma \): Volatility of the interest rate (positive constant).
• \( W(t) \): Standard Wiener process (Brownian motion) under the risk-neutral measure.
• \( dt \): Infinitesimal time increment.
• \( dW(t) \): Increment of the Wiener process, with \( dW(t) \sim \mathcal{N}(0, dt) \).

This model captures mean-reverting behavior, ensuring interest rates oscillate around a long-term mean \( \theta \), with random fluctuations driven by \( \sigma dW(t) \).

2. Solving the SDE

To derive the solution, we solve the SDE:

\[ dr(t) = \kappa (\theta - r(t)) dt + \sigma dW(t) \]

This is a linear SDE, solvable using an integrating factor. Rewrite it as:

\[ dr(t) + \kappa r(t) dt = \kappa \theta dt + \sigma dW(t) \]

Step 1: Apply the Integrating Factor

The integrating factor is \( e^{\int \kappa dt} = e^{\kappa t} \). Multiply both sides by \( e^{\kappa t} \):

\[ e^{\kappa t} dr(t) + \kappa e^{\kappa t} r(t) dt = e^{\kappa t} \kappa \theta dt + e^{\kappa t} \sigma dW(t) \]

The left-hand side is the differential of a product:

\[ d(e^{\kappa t} r(t)) = e^{\kappa t} dr(t) + \kappa e^{\kappa t} r(t) dt \]

Thus, the equation becomes:

\[ d(e^{\kappa t} r(t)) = e^{\kappa t} \kappa \theta dt + e^{\kappa t} \sigma dW(t) \]

Step 2: Integrate Both Sides

Integrate from initial time \( s \) to \( t \):

\[ e^{\kappa t} r(t) - e^{\kappa s} r(s) = \int_s^t e^{\kappa u} \kappa \theta du + \int_s^t e^{\kappa u} \sigma dW(u) \]

Deterministic Integral:

\[ \int_s^t e^{\kappa u} \kappa \theta du = \kappa \theta \int_s^t e^{\kappa u} du = \theta \left[ e^{\kappa u} \right]_s^t = \theta (e^{\kappa t} - e^{\kappa s}) \]

Stochastic Integral: The term \( \int_s^t e^{\kappa u} \sigma dW(u) \) is normally distributed with mean zero and variance:

\[ \text{Var}\left( \int_s^t e^{\kappa u} \sigma dW(u) \right) = \sigma^2 \int_s^t e^{2\kappa u} du = \sigma^2 \left[ \frac{e^{2\kappa u}}{2\kappa} \right]_s^t = \frac{\sigma^2}{2\kappa} (e^{2\kappa t} - e^{2\kappa s}) \]

Step 3: Solve for \( r(t) \)

Divide through by \( e^{\kappa t} \):

\[ r(t) = e^{-\kappa (t-s)} r(s) + \theta (1 - e^{-\kappa (t-s)}) + \sigma \int_s^t e^{-\kappa (t-u)} dW(u) \]

This is the solution to the Vasicek SDE, combining the initial rate, a mean-reverting component, and a stochastic term.

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3. Distribution of \( r(t) \)

The solution \( r(t) \) is normally distributed because the stochastic integral is a linear combination of Wiener process increments. Let’s compute its mean and variance:

Mean

\[ E[r(t) | r(s)] = E\left[ e^{-\kappa (t-s)} r(s) + \theta (1 - e^{-\kappa (t-s)}) + \sigma \int_s^t e^{-\kappa (t-u)} dW(u) \right] \]

Since \( E\left[ \int_s^t e^{-\kappa (t-u)} dW(u) \right] = 0 \), we get:

\[ E[r(t) | r(s)] = e^{-\kappa (t-s)} r(s) + \theta (1 - e^{-\kappa (t-s)}) \]

Variance

\[ \text{Var}(r(t) | r(s)) = \text{Var}\left( \sigma \int_s^t e^{-\kappa (t-u)} dW(u) \right) = \sigma^2 \int_s^t e^{-2\kappa (t-u)} du \]

Compute the integral:

\[ \int_s^t e^{-2\kappa (t-u)} du = \int_0^{t-s} e^{-2\kappa v} dv = \left[ -\frac{1}{2\kappa} e^{-2\kappa v} \right]_0^{t-s} = \frac{1}{2\kappa} (1 - e^{-2\kappa (t-s)}) \]

Thus:

\[ \text{Var}(r(t) | r(s)) = \frac{\sigma^2}{2\kappa} (1 - e^{-2\kappa (t-s)}) \]

So, \( r(t) \sim \mathcal{N}\left( e^{-\kappa (t-s)} r(s) + \theta (1 - e^{-\kappa (t-s)}), \frac{\sigma^2}{2\kappa} (1 - e^{-2\kappa (t-s)}) \right) \).

4. Bond Pricing in the Vasicek Model

A key application is pricing zero-coupon bonds. The bond price at time \( t \) with maturity \( T \) is:

\[ P(t, T) = E\left[ e^{-\int_t^T r(u) du} | \mathcal{F}_t \right] \]

Using the affine term structure, the bond price is:

\[ P(t, T) = e^{A(t,T) - B(t,T) r(t)} \]

Where:

• \( B(t, T) = \frac{1 - e^{-\kappa (T-t)}}{\kappa} \)
• \( A(t, T) = \left( \theta - \frac{\sigma^2}{2\kappa^2} \right) (B(t, T) - (T - t)) - \frac{\sigma^2 B(t, T)^2}{4\kappa} \)

Derivation of Bond Price

The bond price satisfies the PDE:

\[ \frac{\partial P}{\partial t} + (\kappa (\theta - r)) \frac{\partial P}{\partial r} + \frac{\sigma^2}{2} \frac{\partial^2 P}{\partial r^2} - r P = 0 \]

With boundary condition \( P(T, T) = 1 \). Assuming \( P(t, T) = e^{A(t,T) - B(t,T) r} \), we solve for \( A(t, T) \) and \( B(t, T) \).

5. Properties and Implications

• Mean Reversion: The term \( \kappa (\theta - r(t)) \) ensures rates revert to \( \theta \).
• Normal Distribution: \( r(t) \) can be negative, realistic in some markets.
• Affine Term Structure: Enables efficient bond and derivative pricing.

6. Verification

The solution is verified using Ito’s lemma to ensure it satisfies the SDE. The bond pricing formula satisfies the PDE and boundary conditions.

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