FRM Exam Part II · The Art of Term Structure Models: Drift

Model 1: Normal Model with Zero Drift Explained

Updated 11 October 2026 · Fact-checked

Model 1 assumes the short rate follows dr = σ dw: rate changes are normal, volatility is constant and drift is zero. Zero-coupon bond prices are P(T) = exp(−r₀T + σ²T³/6). The spot rate is r₀ − σ²T²/6, so convexity pulls long-term rates below r₀.

Understand Model 1: Normal Model with Zero Drift

Start with the idea of a model. A term structure model says how the short-term rate moves randomly over time. Model 1 is the simplest. The rate changes by a random normal amount each instant, with no pull up or down. In symbols, dr = σ dw. Here σ is the constant volatility of rate changes (in rate units per √year) and dw is a normal random shock.

Because drift is zero, the expected future short rate is the same as today's rate, r₀. The rate at time t is normal with mean r₀ and variance σ²t. It can go negative. That is a known weakness, but it is acceptable when σ is small relative to the rate level.

Now the pricing. A zero-coupon bond pays 1 at T. Its price is the expected value of exp(−∫r dt) in the risk-neutral world. The integral of the rate path is normal, with mean r₀T and variance σ²T³/3. For a normal variable X, E[exp(−X)] = exp(−mean + variance ÷ 2). That gives P(T) = exp(−r₀T + σ²T³/6).

The extra term σ²T³/6 is the convexity effect. It raises the bond price above exp(−r₀T), because the bond price is a convex function of rates. Gains from falling rates are larger than losses from rising rates. The result is that the continuously compounded spot rate is r₀ − σ²T²/6, and the instantaneous forward rate is r₀ − σ²T²/2. Both fall as maturity rises, and the forward rate falls three times as fast as the spot rate.

This is why Model 1 is a teaching model. It has no risk premium and no mean reversion. Its curve is always downward sloping from convexity alone, and it cannot fit an observed curve. Later models (constant drift, Ho-Lee, Vasicek) add drift to fix that.

Key formulas to remember

Model 1 dynamics: dr = σ dw; Zero drift and constant volatility. σ is in decimals per √year (1% = 0.01).
Distribution of the short rate: r(t) ~ Normal(r₀, σ²t); Expected future short rate equals today's rate.
Zero-coupon bond price: P(T) = exp(−r₀T + σ²T³/6); Risk-neutral price of 1 paid at T, continuous compounding.
Spot rate: r̂(T) = r₀ − σ²T²/6; Equal to −ln P(T) ÷ T. The convexity term lowers the spot rate.
Instantaneous forward rate: f(T) = r₀ − σ²T²/2; Equal to −d ln P(T)/dT. Drops faster than the spot rate.
Convexity effect in the spot rate: σ²T²/6; Grows with the square of maturity and of volatility.

How to solve Model 1: Normal Model with Zero Drift questions

Use this order for any Model 1 question, whether it asks for a price, a spot rate, a forward rate or the convexity effect.

1Confirm the model: zero drift, constant σ, risk-neutral dynamics dr = σ dw.
2Convert all inputs to decimals and years. Turn 1% into 0.01 and 120 bp into 0.012. Check that σ is annual.
3Decide what is asked: price P(T), spot rate, forward rate or convexity effect.
4Compute the convexity term first: σ²T²/6 for the spot rate, σ²T²/2 for the forward rate, σ²T³/6 for the price exponent.
5Apply the formula: spot = r₀ − σ²T²/6; price = exp(−spot × T) = exp(−r₀T + σ²T³/6).
6Convert back to the requested units (percent or basis points, price per 1 or per face value).
7Sanity check: the price should exceed exp(−r₀T), the spot rate should be below r₀, and the forward rate below the spot rate.

Quickest way: Basis-point shortcut for the convexity effect

When to use it: Use when the question gives σ in basis points and asks for the rate or the convexity effect, and you want to avoid decimals.

Square σ in bp and multiply by T².
Divide by 60,000 to get the spot-rate convexity effect in bp. Use 20,000 for the forward-rate effect.
Subtract the result from r₀ in bp.
If you need a price, use exp(−spot × T).

Common mistakes in Model 1: Normal Model with Zero Drift

Using σ²T²/2 for the spot rate
The forward-rate formula has /2 and the spot-rate formula has /6, so they get mixed up.
Fix: Remember that spot is the average of forwards. Averaging r₀ − σ²t²/2 over 0 to T gives r₀ − σ²T²/6.
Putting σ in percent into the formula
Candidates plug in 1 for 1% and get a convexity effect 10,000 times too large.
Fix: Convert to decimals first: 1% = 0.01, 100 bp = 0.01.
Adding the convexity term to the rate
The price formula has a plus sign (+σ²T³/6), which gets copied into the rate formula.
Fix: Convexity raises the price and lowers the rate. Spot = r₀ − σ²T²/6.
Thinking zero drift means a flat curve
Zero drift is confused with zero expected change in prices or rates.
Fix: Zero drift only sets the expected short rate equal to r₀. The curve still slopes down because of convexity.
Using σ²T³/6 as the price instead of the exponent
The formula is memorised without the exponential.
Fix: Always write P = exp(−r₀T + σ²T³/6) and compute the exponent first.
Treating Model 1 as a realistic description of the curve
Students forget that it has no risk premium, no mean reversion and allows negative rates.
Fix: Call it the baseline model. Drift terms are added in later models to fit actual curves.

Worked examples

Example 1

In Model 1, the short rate is 5% and annual volatility is 1%. Find the 10-year zero-coupon bond price per 1 of face value and the 10-year continuously compounded spot rate.

Show the solution

Convert inputs: r₀ = 0.05, σ = 0.01, T = 10.
Convexity term for the spot rate: σ²T²/6 = 0.0001 × 100 ÷ 6 = 0.001667, or 0.1667%.
Spot rate = 5% − 0.1667% = 4.8333%.
Price = exp(−0.048333 × 10) = exp(−0.48333).
Check with the price formula: −r₀T + σ²T³/6 = −0.5 + 0.0001 × 1000 ÷ 6 = −0.5 + 0.016667 = −0.483333. Same result.
exp(−0.48333) ≈ 0.6167.

Answer: The price is about 0.6167 per 1 of face value and the spot rate is about 4.83%. Without convexity the price would be exp(−0.5) ≈ 0.6065.

Example 2

In Model 1, r₀ = 4% and σ = 1.2% per year. Find the 20-year spot rate, the 20-year instantaneous forward rate and the 20-year zero-coupon bond price.

Show the solution

Convert inputs: r₀ = 0.04, σ = 0.012, T = 20. σ² = 0.000144 and T² = 400.
σ²T² = 0.0576.
Spot convexity effect = 0.0576 ÷ 6 = 0.0096, or 0.96%. Spot rate = 4% − 0.96% = 3.04%.
Forward convexity effect = 0.0576 ÷ 2 = 0.0288, or 2.88%. Forward rate = 4% − 2.88% = 1.12%.
Price = exp(−0.0304 × 20) = exp(−0.608).
exp(−0.6) = 0.5488 and exp(−0.008) = 0.9920, so the product ≈ 0.5444.

Answer: The spot rate is 3.04%, the forward rate is 1.12% and the price is about 0.5444. The forward rate lies below the spot rate, as expected, because the forward effect is three times the spot effect.

Exam tips

Questions usually give σ in basis points. Convert it to decimals before squaring, or use the bp shortcut.
Expect a conceptual question: why spot rates fall with maturity when drift is zero. The answer is convexity, not expectations of lower rates.
Remember the ratio: the forward-rate effect is three times the spot-rate effect (σ²T²/2 versus σ²T²/6).
Check the direction of every answer: price above exp(−r₀T), spot below r₀, forward below spot.
Know the limits of the model: negative rates are possible, and there is no mean reversion and no fit to the market curve.

Practice questions from The Art of Term Structure Models: Drift

Model 1: Normal Model with Zero Drift: frequently asked questions

What does zero drift mean in Model 1?

It means the expected change in the short rate is zero, so the expected future short rate equals today's rate. The rate still moves randomly with volatility σ. Zero drift does not make the term structure flat.

Why does the spot rate fall with maturity in Model 1?

Bond prices are convex in rates, so rate volatility raises the value of long bonds. A higher price means a lower yield. The spot rate falls by σ²T²/6, which grows quickly with maturity.

How do I get the zero-coupon bond price in the zero drift model?

Use P(T) = exp(−r₀T + σ²T³/6). Convert σ to decimals and compute the exponent first. You can also compute the spot rate r₀ − σ²T²/6 and use exp(−spot × T).

Is Model 1 used in practice?

Rarely for pricing, because it cannot fit the observed curve and allows negative rates. Its value is as a baseline that isolates the convexity effect. Models with drift, such as Ho-Lee and Vasicek, build on it.