Interest Rate Modeling

Stochastic Calculus in Quantitative Finance

Overview

Welcome to QuantRates. This project explores short-rate models used in quantitative finance to simulate the evolution of interest rates over time.

Understanding the behavior of the instantaneous spot rate, \(r_t\), is crucial for pricing fixed-income derivatives, such ad bonds and swaptions.

General Diffusion Process

Most one-factor short-rate models follow a Stochastic Differential Equation (SDE) of the form:

\[ dr_t = \mu(r_t, t)dt + \sigma(r_t, t)dW_t \]

Where:

• \(r_t\): The instantaneous short rate.
• \(\mu(r_t, t)\): The drift term (deterministic trend).
• \(\sigma(r_t, t)\): The diffusion term (volatility).
• \(W_t\): A Standard Brownian Motion (Wiener Process).

Models Covered

1. The Vasicek Model: A model allowing for mean reversion, though it allows negative interest rates.
2. The CIR Model: An extension that precludes negative interest rates by scaling volatility with the square root of the rate.