Vasicek Short-Rate Model & Yield Curve Lab

Simulate Oldrich Vasicek's (1977) Ornstein-Uhlenbeck stochastic interest rate framework. Compute closed-form zero-coupon bond prices, derive the complete term structure of interest rates, evaluate mean-reversion speed, and model yield curve convexity adjustments.

Vasicek (1977) SDE Ornstein-Uhlenbeck Term Structure of Yields Jensen's Convexity Effect Fixed Income Quants

Macro Regime Presets:

1. Stochastic Short-Rate Parameters

Vasicek SDE Parameters: dr = a(b - r)dt + σ dW

Current Short Rate ($r_0$, %)

Instantaneous spot rate $r(0)$.

Long-Term Mean ($b$, %)

Equilibrium mean level.

Mean Reversion Speed ($a$)

Pull rate toward $b$. Half-life: 2.77 yrs.

Volatility ($sigma$, %)

Annualized rate volatility.

Zero-Coupon Pricing Horizon

Benchmark Bond Maturity (Years)

Zero-coupon maturity for detailed pricing and duration breakdown.

2. Term Structure & Analytical Valuation

NORMAL CURVE

Benchmark Zero Yield

4.18%

10-Year Zero

Zero Price ($P)

$65.84

Par = $100

Asymptotic Yield

4.38%

-σ²/(2a²) = -11.5 bp

Yield Slope (10Y - 1Y)

+52 bps

Upward Sloping

Zero-Coupon Bond Pricing Function: $P(t,T) = A(t,T) e^{-B(t,T) r_t}$

Component	Analytical Formula	Value	Interpretation
Duration Factor $B(t,T)$	`[1 - exp(-a × τ)] / a`	3.671	Effective rate sensitivity (approaches $1/a$ as $ au o infty$)
Scale Factor $A(t,T)$	`exp{ (b - σ²/2a²)[B - τ] - σ²B²/4a }`	0.7513	Combines equilibrium drift and Jensen convexity effect
Infinite Yield ($R_infty$)	`b - σ² / (2 × a²)`	4.38%	Maximum long-term yield under complete mean reversion
Zero-Coupon Price ($P$)	`100 × A(t,T) × exp(-B(t,T) × r_0)`	$65.84	Fair value per $100 face value

Mean Reversion Half-Life: $t_{1/2} = rac{ln(2)}{a}$. At $a = 0.25$, half of any unexpected interest rate shock dissipates in 2.77 years.

3. Term Structure of Yields & Forward Curve

Yield Curve (0 to 30 Years): Blue solid curve depicts the zero-coupon spot rate $Y(0,T)$, red dashed curve shows the instantaneous forward rate $f(0,T)$, and green horizontal line indicates the asymptotic yield $R_infty$.

4. Analytical Term Structure Schedule

Continuous Compounding

Maturity	Zero Price ($P)	Zero Yield ($Y$)	Forward ($f$)	Factor B

Maturity yields: $Y(0,T) = -rac{ln P(0,T)}{T}$. Note how the forward curve leads the spot curve and asymptotes toward $R_infty$.

5. Sensitivity: Mean Reversion ($a$) vs. Volatility ($sigma$) on 10Y Yield (%)

Demonstrates how higher volatility $sigma$ pulls down long-term yields via Jensen's inequality convexity adjustment.

6. Sensitivity: Short Rate ($r_0$) vs. Equilibrium Mean ($b$) on 30Y Yield (%)

Evaluates ultra-long zero-coupon yields across different initial rate shocks and central bank terminal equilibrium levels.

7. Vasicek Term Structure Mathematics & Economic Properties

The Ornstein-Uhlenbeck Process in Fixed Income

In 1977, Czech mathematician Oldrich Vasicek introduced the first arbitrage-free continuous-time model for the term structure of interest rates in his paper "An Equilibrium Characterization of the Term Structure" (Journal of Financial Economics). The short rate $r_t$ follows an Ornstein-Uhlenbeck stochastic differential equation:

$$dr_t = a(b - r_t)dt + sigma dW_t$$

Integrating this SDE reveals that the conditional expectation and variance of the short rate at any future time $s > t$ are given by:

$$E[r_s mid r_t] = r_t e^{-a(s-t)} + b(1 - e^{-a(s-t)})$$ $$ ext{Var}(r_s mid r_t) = rac{sigma^2}{2a} [1 - e^{-2a(s-t)}]$$

Unlike geometric Brownian motion (used in Black-Scholes), interest rates do not grow without bound; they exhibit economic mean reversion due to central bank macroeconomic stabilization.

Jensen's Inequality & The Convexity Effect

Because zero-coupon bond prices are a convex function of yields ($P = e^{-yT}$), unexpected volatility in interest rates increases bond prices relative to simple expectations. Solving the term structure equation yields the closed-form zero price:

$$P(t, T) = A(t, T) e^{-B(t, T) r_t}$$

Where the asymptotic long-term yield is strictly below the equilibrium mean:

$$R_infty = lim_{T o infty} Y(t, T) = b - rac{sigma^2}{2a^2}$$

Model Limitation: Because the Vasicek distribution is Gaussian (normally distributed), the short rate can theoretically turn negative. In 1985, Cox, Ingersoll, and Ross (CIR) introduced the square-root diffusion model ($sigma sqrt{r_t} dW_t$) to guarantee non-negative rates.

8. Vasicek Model Mastery Self-Assessment Quiz

9. Related Fixed Income & Derivative Labs

Hull-White Model Lab · Cox-Ingersoll-Ross (CIR) Model Lab · Interest Rate Swap & Hedging Lab

Value plain vanilla fixed-for-floating interest rate swaps, mark-to-market NPV, and par swap rates.

Bond Portfolio Immunization Lab

Model multi-bond duration matching, convexity surplus, and Fong-Vasicek structural risk.

Bond Duration & Convexity Lab

Calculate Macaulay Duration, Modified Duration, Convexity, and DV01 for individual bonds.

10. Frequently Asked Questions (FAQ)

The Vasicek model, published by Oldrich Vasicek in 1977, describes the instantaneous short rate of interest r_t as an Ornstein-Uhlenbeck stochastic mean-reverting process: dr_t = a*(b - r_t)*dt + sigma*dW_t. It was the first model to provide a closed-form analytical solution for the entire term structure of zero-coupon bond prices and yields based on market parameters.

Parameter 'a' is the speed of mean reversion, dictating how rapidly the short rate returns to equilibrium (half-life = ln(2)/a); 'b' is the long-term equilibrium short rate level toward which rates gravitate; and 'sigma' is the annualized volatility of interest rate changes.

Because bond prices are convex functions of yields, Jensen's inequality creates an asymmetric pricing benefit known as the convexity effect. In the Vasicek model, this pulls down long-term yields by an exact term of sigma^2 / (2*a^2): R_infinity = b - sigma^2 / (2*a^2).

Under Vasicek, the yield curve can assume three distinct forms: (1) Normal upward-sloping when current short rates are below R_infinity; (2) Inverted downward-sloping when short rates exceed the long-term equilibrium b; (3) Humped when short rates are at intermediate levels, causing yields to rise initially before the convexity effect drags the long end downward.

Yes. This lab provides a fully sanitized RFC 4180 compliant CSV export containing calibrated model parameters, analytical zero-coupon bond prices, spot yields, forward rates, and 5x5 sensitivity matrices with formula injection defense.