Models of the term structure of interest rates: formula sheet

Key formulas

Discount function from spot rate

v(t) = (1 + y_t)^(-t)

y_t is the annual effective spot rate for term t. Price today of ₹1 at time t.

Forward rate from spot rates (general)

(1 + y_{t+r})^(t+r) = (1 + y_t)^t × (1 + f_{t,r})^r

Rearrange for f_{t,r}. Rates are annual effective and the same compounding basis is used throughout.

One-year forward rate

1 + f_t = (1 + y_{t+1})^(t+1) ÷ (1 + y_t)^t = v(t) ÷ v(t+1)

f_t is the rate from time t to t+1. Here f_0 = y_1.

Spot rate from forward rates

(1 + y_t)^t = (1 + f_0)(1 + f_1)...(1 + f_{t-1})

The spot rate is a geometric average of the one-year forward rates.

Discount factor from forward rates

v(t) = [(1 + f_0)(1 + f_1)...(1 + f_{t-1})]^(-1)

Same relation as above, written as a price.

Par yield

c_n = (1 − v(n)) ÷ (v(1) + v(2) + ... + v(n))

Coupons paid annually, bond redeemed at par, priced at par. Denominator is the sum of discount factors.

Price of a coupon bond from spot rates

P = Σ C_t × v(t)

Discount each cash flow at the spot rate for its own term.

Forward rate from spot rates (annual compounding)

(1 + y_t)^t = (1 + y_{t-1})^(t-1) × (1 + f_{t-1,t})

y_t is the t-year spot rate and f_{t-1,t} is the one-year forward rate from t-1 to t. Needs a consistent compounding basis.

Pure expectations theory

f_{t-1,t} = E[ one-year spot rate at time t-1 ]

Forward rates are unbiased predictors of future spot rates. No risk premium.

Liquidity preference theory

f_{t-1,t} = E[ spot rate at t-1 ] + L_t, with L_t > 0

L_t is the liquidity premium, usually increasing with term. Forwards overstate expected future spot rates.

Preferred habitat theory

f_{t-1,t} = E[ spot rate at t-1 ] + P_t

P_t can be positive or negative, depending on supply and demand in each maturity habitat.

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.

Vasicek SDE

dr = α(μ − r)dt + σ dW

α > 0 is the speed of mean reversion, μ the long-run mean, σ constant volatility. Treat the parameters as risk-neutral when pricing bonds.

Vasicek distribution of r(t)

r(t) ~ Normal with mean μ + (r(0) − μ)e^(−αt) and variance σ²(1 − e^(−2αt)) ÷ (2α)

As t → ∞, the mean tends to μ and the variance tends to σ² ÷ (2α). Negative rates have positive probability.

Bond price (affine form)

P(t,T) = exp(A(τ) − B(τ)r(t)), τ = T − t

Vasicek and Hull-White take this form. CIR is written as A(τ)exp(−B(τ)r) with a different A and B.

Vasicek B(τ)

B(τ) = (1 − e^(−ατ)) ÷ α

The same B is used in Hull-White. It measures the sensitivity of ln P to r.

Vasicek A(τ)

A(τ) = (μ − σ²/(2α²))(B(τ) − τ) − σ²B(τ)² ÷ (4α)

Here μ is the risk-neutral long-run mean. If a market price of risk is used, adjust μ accordingly.

Vasicek long-run yield

R(∞) = μ − σ² ÷ (2α²)

Yield is R(τ) = −ln P ÷ τ. The long-run yield is below μ because of the convexity term.

CIR SDE and Feller condition

dr = α(μ − r)dt + σ√r dW; r stays above 0 if 2αμ ≥ σ²

If the condition fails, r can touch zero but is reflected, so it remains non-negative.

CIR bond price

P = A(τ)e^(−B(τ)r), γ = √(α² + 2σ²), B = 2(e^(γτ) − 1) ÷ [(γ + α)(e^(γτ) − 1) + 2γ], A = [2γe^((α+γ)τ/2) ÷ ((γ + α)(e^(γτ) − 1) + 2γ)]^(2αμ/σ²)

These are the risk-neutral forms with no extra market price of risk term. Check which version your question gives.

Hull-White SDE

dr = (θ(t) − αr)dt + σ dW

θ(t) is chosen to match the initial forward curve f(0,t): θ(t) = ∂f(0,t)/∂t + αf(0,t) + σ²(1 − e^(−2αt)) ÷ (2α).

Instantaneous forward rate

f(t,T) = −∂ ln P(t,T) / ∂T

P(t,T) is the price at time t of a zero-coupon bond paying 1 at T.

Bond price from forward rates

P(t,T) = exp(−∫ f(t,s) ds), integrated from s = t to s = T

Gives the full price curve from the forward curve.

Short rate from forward rate

r(t) = f(t,t)

The short rate is the forward rate with zero time to maturity.

HJM forward rate dynamics

df(t,T) = α(t,T) dt + σ(t,T) dW(t)

One-factor form. With several factors σ dW becomes a sum over the factors.

HJM no-arbitrage drift (risk-neutral measure, one factor)

α(t,T) = σ(t,T) × ∫ σ(t,s) ds, integrated from s = t to s = T

The drift is fully determined by the volatility. This is the key HJM result.

Simple forward LIBOR rate

L(t; T, T+δ) = [P(t,T) ÷ P(t,T+δ) − 1] ÷ δ

δ is the accrual period, for example 0.25 years.

Vasicek short rate model

dr = a(b − r)dt + σ dW

a is the speed of mean reversion, b the long-run mean, σ the volatility. Rates can be negative. Equilibrium model with constant parameters.

Hull-White (extended Vasicek) model

dr = [θ(t) − a r]dt + σ dW

θ(t) is chosen to fit the initial yield curve exactly. This is what makes it arbitrage-free.

Ho-Lee model

dr = θ(t)dt + σ dW

No mean reversion. θ(t) fits the initial curve.

CIR model

dr = a(b − r)dt + σ√r dW

Volatility rises with the rate. Rates stay non-negative if 2ab ≥ σ².

Zero-coupon bond price

P(0,T) = exp(−∫₀ᵀ f(0,s) ds) = exp(−T y(0,T))

f is the instantaneous forward rate and y the continuously compounded spot rate. This is the target the calibrated model must match.

Risk-neutral price of a derivative

V(0) = E_Q[ exp(−∫₀ᵀ r(s) ds) × payoff ]

Expectation is under the risk-neutral measure Q, using the model calibrated to market prices.

Calibration objective

minimise Σ (model price − market price)²

Least squares over chosen instruments. Weighting by vega or using relative errors is also common.

Quick revision

• Spot rate: yield on a zero-coupon bond from now to time t.
• Forward rate: rate agreed today for borrowing over a future period, implied by spot rates.
• Discount factor from spot rates: v(t) = (1 + s_t)^(−t), when rates are annual effective.
• Expectations theory: long rates reflect the average of expected future short rates.
• Liquidity preference theory: investors want a premium for longer maturities, so the curve tends to slope up.
• Market segmentation: different maturities are driven by separate supply and demand.
• No-arbitrage: a model must not allow a riskless profit, so prices are consistent across bonds.
• Risk-neutral pricing of a zero-coupon bond: P(t,T) = E_Q[exp(−∫ r_s ds from t to T) | F_t]. The expectation is under the risk-neutral measure Q, and the discount factor depends on the whole short rate path.
• If your syllabus covers them: Vasicek has mean reversion and normally distributed rates, so negative rates are possible.
• If your syllabus covers them: CIR has mean reversion and a square-root term that keeps rates non-negative under its usual parameter condition.
• Calibration means choosing parameters so model prices match observed market prices.
• Always state your assumptions and notation before calculating.

Common mistakes

• Averaging spot rates to get a forward rate or taking the difference y_{t+1} − y_t. Fix: Always equate accumulation factors: 1 + f_t = (1 + y_{t+1})^(t+1) ÷ (1 + y_t)^t.
• Using the wrong exponent, for example (1 + y_2)^1 instead of (1 + y_2)^2. Fix: Write the accumulation factor as (1 + y_t)^t for each term before dividing.
• Treating liquidity preference as a theory that the curve must always slope upward. Fix: Say the premium biases the curve upward. If expected short rates fall enough, the curve can still be flat or inverted.
• Saying expectations play a role in market segmentation theory. Fix: State that in pure market segmentation, yields come from supply and demand within each maturity segment, not from expected future rates.
• Using the real-world drift when taking the expectation for the bond price. Fix: Always convert to Q first. Write the Q-dynamics explicitly before computing E_Q.
• Getting the sign of λ wrong when changing drift. Fix: State the convention at the start. Check by confirming that the Q-drift of the bond equals r.
• Saying Vasicek rates can never be negative, or that CIR rates can be negative. Fix: Vasicek r(t) is normal, so P(r < 0) > 0. CIR has σ√r, which removes noise at zero, so r ≥ 0. Mean reversion alone does not stop negative values.
• Forgetting the convexity term σ²/(2α²) in the Vasicek A(τ) formula. Fix: Write the formula down in full before substituting. Remember that the long-run yield is μ − σ²/(2α²), not μ.
• Treating the HJM drift as a free parameter to be chosen. Fix: Under the risk-neutral measure the HJM drift is fixed by the volatility through no-arbitrage. Only the volatility structure is chosen.
• Saying more factors always give a better model. Fix: More factors add realism but also parameters, calibration difficulty and computation time. State both sides.

Exam tips

• Show the accumulation equation first, then the rearranged result. Method marks are given for the setup even if arithmetic slips.
• Keep at least 5 or 6 decimal places in discount factors. Rounding early is a common way to lose accuracy in the final rate.
• Read the question for the time convention. f_{t,r} is a rate from t to t+r. Make sure you give it per annum, not as a total over r years.
• In written answers, interpret the result. Say whether the curve is rising, flat or inverted, and link it to the forward rates.
• In the computer-based paper, build discount factors in a column, then compute forward rates and par yields from that column so you can check each step.
• For a discussion question, name the theory, say what drives yields, state the curve shape it predicts, and give one weakness.
• In calculations, state the compounding basis and show the forward rate formula before substituting.
• Link each theory to a real investor, such as a life insurer with long liabilities, to support market segmentation and preferred habitat.