FRM Exam Part II · The Vasicek and Gauss+ Models

Vasicek Model Option Pricing and Calibration

Updated 11 October 2026 · Fact-checked

In the Vasicek model the short rate mean-reverts with normal volatility, so a zero-coupon bond's price is lognormal-like and options on it have a Black-style closed form. You price with the bond-price volatility σ_P, then calibrate k, θ and σ to market data. Three parameters cannot match a whole curve exactly.

Understand Vasicek Model: Option Pricing and Calibration

The Vasicek model says the short rate follows dr = k(θ − r)dt + σdw under the risk-neutral measure. The rate is pulled toward a long-run level θ at speed k. Volatility σ is a constant in rate units, so rates can go negative.

Because the model is affine, a zero-coupon bond price has the form P(τ) = A(τ) × e^(−B(τ)r). The rate r is normally distributed at any future date. So the bond price at option expiry is lognormal in this setup. That is why a bond option has a Black-Scholes-style formula, usually called the Jamshidian result.

The key step is the volatility of the bond price at option expiry. It depends on σ, on the sensitivity B of the underlying bond's remaining life, and on how much the short rate can move before expiry. Mean reversion (k) dampens both effects. Higher k means lower bond-price volatility and lower option value.

Calibration means choosing k, θ and σ so model prices sit close to market prices. Yield-curve data mostly pin down k and θ. Option or rate-volatility data pin down σ and help with k. You minimise squared errors between model and market. But with only a few constant parameters, the model cannot reproduce an arbitrary market curve. Ho-Lee and Hull-White fix this by letting the drift depend on time, so the model fits today's curve exactly. Hull-White keeps mean reversion. Ho-Lee has none and constant volatility.

Key formulas to remember

Vasicek short-rate process: dr = k(θ − r)dt + σ dw; k is the speed of mean reversion, θ the long-run rate, σ the constant normal volatility. Hull-White writes the same thing as dr = a(b − r)dt + σ dz.
Zero-coupon bond price: P(τ) = A(τ) × e^(−B(τ) r), with B(τ) = (1 − e^(−kτ)) ÷ k; B is the rate sensitivity of the bond. It rises with τ but flattens out near 1/k because of mean reversion.
Bond-price volatility at option expiry: σ_P = σ × B(S − T) × √[(1 − e^(−2kT)) ÷ (2k)]; T is option expiry, S is bond maturity. Use B for the bond's remaining life S − T, not for S.
Call on a zero-coupon bond: C = P(0,S) N(h) − K P(0,T) N(h − σ_P); K is the strike on bond price at T. P(0,S) and P(0,T) are today's discount factors.
Put on a zero-coupon bond: Put = K P(0,T) N(σ_P − h) − P(0,S) N(−h); Check with put-call parity: C − Put = P(0,S) − K P(0,T).
Parameter h: h = ln[P(0,S) ÷ (P(0,T) K)] ÷ σ_P + σ_P ÷ 2; The log term compares the forward bond price with the strike.
Model comparison: Vasicek: constant drift parameters. Ho-Lee: time-dependent drift, no mean reversion. Hull-White: time-dependent drift plus mean reversion.; Only the time-dependent drift models fit today's curve exactly.

How to solve Vasicek Model: Option Pricing and Calibration questions

Use this order for any Vasicek option or calibration question. It keeps the pieces from getting mixed up.

1Identify the option expiry T, the bond maturity S, and the strike K. Confirm K is a bond price, not a yield.
2Compute B(S − T) = (1 − e^(−k(S − T))) ÷ k.
3Compute the short-rate variance factor √[(1 − e^(−2kT)) ÷ (2k)].
4Multiply σ, B and the factor to get σ_P.
5Find the forward ratio P(0,S) ÷ (P(0,T) K), take its log, and compute h.
6Look up N(h) and N(h − σ_P), then apply the call or put formula. Cross-check with put-call parity if both are asked.
7For calibration questions, count parameters (k, θ, σ) against the data points. Say whether an exact fit is possible, and name the model that fixes it.
8State the interpretation: higher k or lower σ lowers option value, and a fitted model is only as good as the instruments used to fit it.

Quickest way: Shortcut: check direction and limits first

When to use it: Use when answer options are far apart or when the question is about model properties rather than a full number.

Look at how σ_P moves. Higher σ raises it, higher k lowers it, longer option expiry raises it.
An option's value rises with σ_P. If an option says value falls with higher σ, drop it.
If the question asks about fitting today's curve exactly, the answer is a time-dependent drift model (Ho-Lee or Hull-White), never plain Vasicek.
If the call is deep out of the money, expect a small price. If the forward bond price is near the strike, price is roughly 0.4 × σ_P × the discounted forward bond price.
Only do the full N(h) calculation if the options are close.

Common mistakes in Vasicek Model: Option Pricing and Calibration

Using σ as the bond-price volatility in the option formula.
Students see σ in the process and carry it straight into the Black-style formula.
Fix: Always build σ_P = σ × B(S − T) × √[(1 − e^(−2kT)) ÷ (2k)] first.
Using B(S) instead of B(S − T) in σ_P.
B is introduced for the bond maturity, so the full maturity feels natural.
Fix: The option cares about the bond's remaining life after expiry, which is S − T.
Believing a calibrated Vasicek model matches the whole market yield curve.
Calibration sounds like fitting, and students forget how few parameters exist.
Fix: Remember k, θ and σ are constants. A curve with many points will show residual errors. Only time-dependent drift models fit exactly.
Saying Ho-Lee has mean reversion, or that Hull-White has no mean reversion.
All three are called drift models and the names blur together.
Fix: Ho-Lee: time-dependent drift, no mean reversion. Hull-White: time-dependent drift plus mean reversion. Vasicek: mean reversion with constant parameters.
Mixing the call and put d-terms, such as using N(h) for both.
Formulas are memorised without the logic.
Fix: The call uses N(h) and N(h − σ_P). The put uses the complements N(−h) and N(σ_P − h). Test with put-call parity.
Ignoring that Vasicek allows negative rates when asked about limitations.
Students focus on mean reversion as the main feature.
Fix: List limitations together: negative rates possible, constant volatility, imperfect curve fit, and a limited range of curve shapes.

Worked examples

Example 1

Vasicek model with k = 0.5 and σ = 0.01. Today's discount factors are P(0,1) = 0.96 and P(0,3) = 0.88. Price a 1-year European call on the 3-year zero-coupon bond (face 1) with strike K = 0.92. Use N(−0.35607) ≈ 0.3609 and N(−0.36612) ≈ 0.3572.

Show the solution

Bond life after expiry S − T = 2. B(2) = (1 − e^(−1)) ÷ 0.5 = 0.632121 ÷ 0.5 = 1.26424.
Rate factor = √[(1 − e^(−2×0.5×1)) ÷ (2×0.5)] = √0.632121 = 0.79506.
σ_P = 0.01 × 1.26424 × 0.79506 ≈ 0.010051.
Forward ratio = 0.88 ÷ (0.96 × 0.92) = 0.88 ÷ 0.8832 = 0.99638. Its log ≈ −0.003630.
h = −0.003630 ÷ 0.010051 + 0.010051 ÷ 2 = −0.36115 + 0.00503 ≈ −0.35607.
h − σ_P = −0.35607 − 0.01005 = −0.36612.
C = 0.88 × 0.3609 − 0.92 × 0.96 × 0.3572 = 0.31759 − 0.31546 ≈ 0.0021.

Answer: The call is worth about 0.0021 per 1 of face value, roughly 0.21 per 100 face. The option is slightly out of the money on a forward basis (forward bond price ≈ 0.9167 vs strike 0.92), so the price is small.

Example 2

A risk manager has calibrated a Vasicek model to 10 points on the USD par yield curve by least squares. Pricing errors remain at several maturities. She wants a model that keeps mean reversion and also reproduces today's curve exactly. Which model should she use? A) Vasicek with k, θ and σ re-estimated B) Ho-Lee C) Hull-White (extended Vasicek) D) Normal model with zero drift

Show the solution

Vasicek has three constant parameters. Ten curve points cannot all be matched, so re-estimating (A) will not remove the errors.
Ho-Lee has a time-dependent drift and fits the curve exactly, but it has no mean reversion. That fails the first requirement.
The normal model with zero drift has no mean reversion and no time-dependent drift. It cannot fit the curve exactly.
Hull-White adds a time-dependent drift θ(t) to Vasicek's mean reversion, so it fits today's curve exactly and keeps mean reversion.

Answer: C) Hull-White (extended Vasicek). Its time-dependent drift gives the exact fit, and it keeps mean reversion.

Exam tips

Expect conceptual comparisons of Vasicek, Ho-Lee and Hull-White more often than long calculations. Learn the table: mean reversion yes/no, time-dependent drift yes/no, exact curve fit yes/no.
If a calculation appears, it is usually a call or put on a zero-coupon bond with σ_P given or easy to build. Write σ_P first.
Watch for the word 'exactly'. It signals a time-dependent drift model.
On limitations questions, name at least: constant volatility, negative rates possible, and limited curve fit.
Check put-call parity as a quick sanity test when both options are priced.

Practice questions from The Vasicek and Gauss+ Models

Vasicek Model: Option Pricing and Calibration: frequently asked questions

How do you price an option on a zero-coupon bond in the Vasicek model?

Compute the bond-price volatility σ_P from σ, k, the option expiry and the bond's remaining life. Then use the Black-style formula with discount factors: call = P(0,S) N(h) − K P(0,T) N(h − σ_P). The put follows from the complement or from put-call parity.

How do you calibrate the Vasicek model to the yield curve?

Choose k, θ and σ to minimise the squared gap between model and market prices or yields. Curve data mainly drive k and θ, and option or volatility data help set σ. The fit will not be exact because three constants cannot match a full curve.

What is the difference between Vasicek, Ho-Lee and Hull-White?

Vasicek has mean reversion with constant parameters. Ho-Lee has a time-dependent drift but no mean reversion and constant volatility. Hull-White combines mean reversion with a time-dependent drift, so it fits today's curve exactly.

What are the main limitations of the Vasicek model?

It allows negative rates, has constant volatility, and cannot match an arbitrary market yield curve exactly. The range of curve shapes it can produce is also limited. Time-dependent drift models address the curve fit but not all the other issues.