Chapter 32: No-Arbitrage Models of the Short Rate

Extensions of Equilibrium Models #

No-arbitrage models make the parameters time-dependent so that the model exactly fits today's term structure. The initial yield curve is an input (fitted perfectly), not an output.

The Ho–Lee Model (1986) #

The first no-arbitrage model:

$dr=\theta (t)dt+\sigma dz$

where $\theta (t)$ is chosen to fit the initial term structure. $\sigma$ is constant.

• Normal distribution of rates (can be negative)
• No mean reversion
• All rates are perfectly correlated
• $\theta (t)$ is derived analytically from the initial forward rate curve

The Hull–White Model (1990) #

An extension of Vasicek with time-dependent drift:

$dr=[\theta (t)-a\cdot r]dt+\sigma dz$

where:

• $a$ = constant mean reversion speed
• $\sigma$ = constant volatility (can also be time-dependent)
• $\theta (t)$ = chosen to fit the initial term structure

$\theta (t)=\frac{\partial f(0,t)}{\partial t}+a\cdot f(0,t)+\frac{{\sigma }^2}{2a}(1-e^{-2at})$

where $f(0,t)$ is the instantaneous forward rate at time 0 for maturity $t$.

The Black–Derman–Toy (BDT) Model (1990) #

$d\ln r=[\theta (t)-a(t)\ln r]dt+\sigma (t)dz$

• Lognormal model (rates always positive)
• Both mean reversion and volatility can be time-dependent
• Fits both the initial term structure and the initial volatility term structure
• Typically implemented as a binomial tree

Options on Bonds #

Under the Hull–White model, European options on zero-coupon bonds have closed-form solutions. The price of a call option:

$c=L\cdot P(0,T)\cdot N(h)-K\cdot P(0,s)\cdot N(h-{\sigma }_P)$

where $s$ is the option maturity, $T$ is the bond maturity, and ${\sigma }_P$ depends on model parameters. This is similar to Black's model but with ${\sigma }_P$ derived from the term structure model.

Volatility Structures #

The volatility structure describes how the volatility of different forward rates varies. Models can fit different volatility patterns:

• Decreasing volatility: Short-term rates are more volatile than long-term rates (common in practice)
• Humped volatility: Medium-term rates are most volatile
• Constant volatility: All rates equally volatile

Hull–White with constant $\sigma$ produces decreasing volatilities as maturity increases.

Interest Rate Trees #

No-arbitrage models are implemented using interest rate trees. The tree-building procedure:

General Trinomial Tree for Hull–White #

1. Build a preliminary tree for the $x$-process where $dx=-axdt+\sigma dz$ ($x$ has mean 0)
2. Compute $\alpha (t)=r(t)-x(t)$ at each time step to fit the initial term structure
3. Three branches: up (probability $p_u$), middle ($p_m$), down ($p_d$), with probabilities chosen to match the drift and variance

Calibration #

The tree is calibrated to match:

1. The initial zero curve (via $\theta (t)$ or $\alpha (t)$)
2. Possibly volatility term structure (via time-varying $\sigma$)
3. Volatility smiles in the options market (via more sophisticated calibration)

For the BDT model, calibration is performed iteratively: first for the lowest node, then each higher node, matching both bond prices and cap/floor volatilities at each maturity.

Hedging Using a One-Factor Model #

The one-factor assumption implies all rates move together, so a single hedging instrument (e.g., a bond or futures contract) can theoretically hedge all interest rate risk. In practice, multi-factor models or bucket hedging (hedging each maturity segment separately) is more common because rates don't move perfectly in parallel.

Key Contrast #