AdvancedQuantitative MethodsPython

Run this module

cd "Quantitative Methods - Interest Rate Models"
python "interest_rate_models.py"

Short Rate Interest Rate Models¶

Continuous-time models for the evolution of the short (instantaneous) interest rate. Used for bond pricing, interest rate derivatives, and yield curve modeling.

Functions¶

Function	Description
`vasicek_simulate(r0, kappa, theta, sigma, T, n_steps, n_paths)`	Simulate Vasicek paths
`vasicek_bond_price(r0, kappa, theta, sigma, T)`	Closed-form ZCB price
`vasicek_yield(r0, kappa, theta, sigma, T)`	Zero-coupon yield
`cir_simulate(r0, kappa, theta, sigma, T, n_steps, n_paths)`	Simulate CIR paths
`cir_bond_price(r0, kappa, theta, sigma, T)`	Closed-form ZCB price
`cir_yield(r0, kappa, theta, sigma, T)`	Zero-coupon yield
`term_structure(r0, kappa, theta, sigma, maturities, model)`	Full yield curve

Models¶

Vasicek (1977)¶

dr = kappa*(theta - r)*dt + sigma*dW - Mean-reverting: rate pulled toward theta at speed kappa - Rates can go negative (unrealistic but analytically convenient) - Closed-form bond prices

Cox-Ingersoll-Ross (1985)¶

dr = kappa*(theta - r)*dt + sigma*sqrt(r)*dW - Mean-reverting + non-negative rates (when 2*kappa*theta >= sigma²) - Volatility scales with rate level (more realistic) - Closed-form bond prices

Parameters¶

Param	Typical Range	Meaning
kappa	0.1–1.0	Mean reversion speed (0.3 = ~3yr half-life)
theta	0.03–0.07	Long-run mean rate
sigma	0.005–0.02	Rate volatility

Example¶

from interest_rate_models import term_structure

yields = term_structure(r0=0.03, kappa=0.3, theta=0.05, sigma=0.01,
                        maturities=[1, 2, 5, 10, 30], model="cir")

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