Economic Modelling · Models of the term structure of interest rates

No-Arbitrage and Risk-Neutral Pricing of Bonds

Updated 11 October 2026 · Fact-checked

No-arbitrage pricing says a zero-coupon bond price must leave no risk-free profit. Under the risk-neutral measure Q, the discounted bond price is a martingale, so P(t,T) = E_Q[exp(−∫r ds) | F_t]. The market price of risk λ links the real-world drift to the Q drift. Solve by switching to Q and computing that expectation.

Understand No-Arbitrage and Risk-Neutral Pricing of Bonds

A zero-coupon bond pays ₹1 at time T. Its price at time t is P(t,T). If the short rate r(t) were known, P(t,T) would be exp(−∫ from t to T of r(s) ds). In practice r(t) is random, so the price depends on how you model r(t).

No arbitrage means you cannot build a portfolio with zero cost that has no chance of loss and some chance of gain. In a one-factor short rate model, all bonds are driven by the same random source. So the excess return per unit of risk must be the same for every maturity. This common value is the market price of risk, λ.

The risk-neutral measure Q is a change of probability measure that removes the risk premium. Under real-world measure P, the short rate follows dr = μ dt + σ dW. Under Q, the drift becomes μ − λσ, with a Q-Brownian motion W~ where dW~ = dW + λ dt. Sign conventions differ between texts, so check which one your question uses.

Under Q, every traded asset has expected return equal to the short rate. Equivalently, the price of any asset divided by the bank account B(t) = exp(∫ from 0 to t of r(s) ds) is a martingale. This gives the pricing rule P(t,T) = E_Q[exp(−∫ from t to T of r(s) ds) | F_t].

The price depends on Q-dynamics, not on the real-world drift. This is why you can price bonds without estimating real-world expected returns. It also means the choice of λ matters: it changes the Q-drift and so changes bond prices.

Key rules to remember

Bank account: B(t) = exp(∫₀ᵗ r(s) ds); The numeraire used for discounting under the risk-neutral measure.
Zero-coupon bond price under Q: P(t,T) = E_Q[exp(−∫ₜᵀ r(s) ds) | F_t]; Pays ₹1 at T, so P(T,T) = 1.
Martingale condition: P(t,T) ÷ B(t) = E_Q[ 1 ÷ B(T) | F_t ]; Discounted price of any traded asset is a Q-martingale.
Bond price dynamics: dP(t,T) = P(t,T)[ m(t,T) dt + S(t,T) dW ]; m is the drift and S the volatility of the bond price under the real-world measure.
Market price of risk: λ(t) = (m(t,T) − r(t)) ÷ S(t,T); Must not depend on T, or arbitrage exists between bonds.
Change of drift: dW~ = dW + λ dt, so dr = (μ − λσ) dt + σ dW~; Sign depends on how λ is defined. Follow the question's definition.
Bond price under Q: dP(t,T) = r(t) P(t,T) dt + S(t,T) P(t,T) dW~; Q-drift of every bond equals the short rate.
Constant rate check: P(t,T) = exp(−r (T − t)); If r is deterministic and constant, the general formula reduces to this.

How to solve No-Arbitrage and Risk-Neutral Pricing of Bonds questions

Use this order for any question on no-arbitrage or risk-neutral bond pricing.

1Identify the model: write the short rate SDE under the real-world measure P and note any given market price of risk λ.
2Check the sign convention for λ and the form of dW~ = dW ± λ dt as stated in the question.
3Convert the short rate to Q: replace the drift μ by μ − λσ (or as per the given convention) and replace dW by dW~.
4Write the pricing equation P(t,T) = E_Q[exp(−∫r ds) | F_t], or the bond price form such as A(t,T) exp(−B(t,T) r(t)) if the model is given.
5Compute the expectation or use the given closed form. Substitute the Q-parameters, not the real-world ones.
6Check that P(T,T) = 1, the price lies between 0 and 1 for positive rates, and that the Q-drift of P equals r.
7For no-arbitrage arguments with two bonds, form the risk-free portfolio and show that (m − r) ÷ S is the same for both.

Quickest way: Same-λ shortcut

When to use it: Use when you are given two bond drifts and volatilities, or asked for the Q-drift of a short rate.

For each bond compute (m − r) ÷ S. This is λ.
If the two values differ, there is an arbitrage. If equal, no arbitrage.
To get Q dynamics, subtract λ times the volatility from the real-world drift.
For any bond, the Q-drift is simply r. Use this to check your answer.

Common mistakes in No-Arbitrage and Risk-Neutral Pricing of Bonds

Using the real-world drift when taking the expectation for the bond price.
The short rate SDE is given under P and students forget to change measure.
Fix: Always convert to Q first. Write the Q-dynamics explicitly before computing E_Q.
Getting the sign of λ wrong when changing drift.
Textbooks define dW~ = dW + λ dt or dW − λ dt, and students mix them.
Fix: State the convention at the start. Check by confirming that the Q-drift of the bond equals r.
Thinking the bond price is the expected discounted payoff under P.
The familiar expectation formula is used without a measure.
Fix: An expectation under P uses real-world drifts, which embed the risk premium, so it does not give the no-arbitrage price. Only the Q expectation does.
Allowing λ to depend on the maturity T.
Students compute (m − r) ÷ S for each bond and accept different values.
Fix: Different values for the same risk source imply arbitrage. λ may depend on t and r, not on T.
Discounting with r(T) or the rate at time t only.
The deterministic formula exp(−r(T−t)) is overused.
Fix: With random rates, discount with the integral of the path of r. The expectation sits outside the exponential.
Saying risk-neutral means investors are actually risk-neutral.
The name is misleading.
Fix: Q is a mathematical device. Prices are the same whatever the investors' real preferences are.

Worked examples

Example 1

Two zero-coupon bonds depend on the same one-factor risk. Bond A has drift 7% and volatility 4% per year. Bond B has drift 9% and volatility 10% per year. The short rate is 5%. Is there an arbitrage?

Show the solution

Compute λ for A: (7% − 5%) ÷ 4% = 2 ÷ 4 = 0.5.
Compute λ for B: (9% − 5%) ÷ 10% = 4 ÷ 10 = 0.4.
The values differ: 0.5 ≠ 0.4.
So a portfolio of A and B, weighted to remove the random term, would earn more than r with no risk.
Build the hedge: buy ₹100 of A and short ₹40 of B. The dW terms cancel because 100 × 4% = 4 and 40 × 10% = 4. The ratio is the same as holding 1 ÷ 0.04 = 25 of A per 1 ÷ 0.10 = 10 of B.
Check the excess return over r: 100 × (7% − 5%) − 40 × (9% − 5%) = 2 − 1.6 = ₹0.40 a year. This equals (0.5 − 0.4) × 4 and carries no risk.

Answer: Yes. The market price of risk differs (0.5 for A, 0.4 for B), so there is an arbitrage. Buy A (the higher λ) and short B in amounts that cancel the dW term, for example ₹100 of A against ₹40 of B. This hedged portfolio earns ₹0.40 a year above r with no risk.

Example 2

Under the real-world measure, dr = 0.02 dt + 0.01 dW. The market price of risk is λ = 0.5 (constant) with dW~ = dW + λ dt. Find the Q-dynamics of r and the Q-expected value of r(t) if r(0) = 0.04 and t = 3.

Show the solution

Write dW = dW~ − λ dt = dW~ − 0.5 dt.
Substitute: dr = 0.02 dt + 0.01 (dW~ − 0.5 dt).
Simplify the drift: 0.02 − 0.005 = 0.015.
So dr = 0.015 dt + 0.01 dW~ under Q.
The drift is constant, so E_Q[r(3)] = 0.04 + 0.015 × 3.
Compute: 0.015 × 3 = 0.045, so E_Q[r(3)] = 0.04 + 0.045 = 0.085.

Answer: Under Q, dr = 0.015 dt + 0.01 dW~, and E_Q[r(3)] = 0.04 + 0.045 = 0.085 (8.5%).

Exam tips

Write the measure (P or Q) beside every expectation. Examiners award marks for this clarity.
State the sign convention for λ in your first line. It protects you if the sign differs from the marker's.
Written answers on no-arbitrage often ask for a risk-free portfolio argument. Show the weights and the cancelled dW term.
In MCQs, check the Q-drift of a bond equals r. It quickly rules out wrong options.
For computer-based work, state the Q-dynamics you simulate and discount along each path before averaging.

Practice questions from Models of the term structure of interest rates

No-Arbitrage and Risk-Neutral Pricing of Bonds in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

No-Arbitrage and Risk-Neutral Pricing of Bonds: frequently asked questions

What is the market price of risk in interest rate models?

It is the extra expected return per unit of volatility that investors require for bearing interest rate risk. It equals (m − r) ÷ S for a bond with drift m and volatility S. In a one-factor model it is the same for all maturities.

Why is the discounted bond price a martingale under Q?

Q is chosen so every traded asset has expected return equal to the short rate. Dividing by the bank account removes that growth. What remains has no drift, so it is a martingale.

Do I need the real-world drift to price a bond?

No. Bond prices depend only on Q-dynamics. The real-world drift matters only for forecasting or risk management, and it links to Q through λ.

Does a different λ give a different bond price?

Yes. λ changes the Q-drift of the short rate, which changes the expectation in the pricing formula. This is why λ is often fixed by calibrating to the observed yield curve.