Describe what arbitrage-free models are good for, and evaluate the difficulty of making a model match observed market prices.
The idea
An arbitrage-free model takes today's market prices as inputs. Its parameters, such as the Ho-Lee drift at each date, are fitted so that the model reprices the benchmark bonds exactly. Its purpose is to price other securities, mainly derivatives and off-the-run bonds, consistently with those benchmarks and with no arbitrage against them. It is not a forecasting tool, and it cannot call the fitted benchmarks rich or cheap, since they match by construction. Matching is not free. Each date needs its own parameter, so many prices need many parameters. The fitted values absorb noise such as bid-ask spreads and can swing erratically. A drift fitted to bond prices also says nothing about volatility, which must be calibrated to option prices separately. Exam point: a perfect fit to today's bond prices does not validate the model's dynamics.
A practice question
What is the main use of an arbitrage-free term structure model?
- Forecasting the future level of the short rate more accurately than the forward curve.Why not: Arbitrage-free models take today's curve as given; they are pricing tools, not forecasts.
- Identifying which of the fitted benchmark bonds are rich or cheap.Why not: The fitted bonds are matched exactly by construction, so the model cannot flag them.
- Pricing other securities, such as derivatives, consistently with the observed prices of the benchmark bonds used to fit it. - correct
- Estimating the real-world long-run mean of the short rate from history.Why not: That is a parameter of an equilibrium model such as Vasicek, not what the arbitrage-free fit does.
An arbitrage-free model reproduces the market prices of the benchmark bonds exactly and then prices derivatives and other instruments relative to them without introducing arbitrage.
A harder one
A desk uses a Ho-Lee tree with semi-annual steps. The 6-month rate is observed directly. How many drift values must be fitted to match the prices of zero-coupon bonds at every semi-annual maturity out to 10 years?
- 20Why not: Counts a drift for the first period too, although the observed 6-month rate already fixes it.
- 10Why not: Counts years rather than semi-annual periods.
- 1Why not: Assumes a single constant drift can match the whole curve.
- 19 - correct
There are 20 semi-annual maturities. The 6-month bond is matched by the observed starting rate, so the 19 remaining bonds each need one drift value, one per later date.
The hardest tier
A desk calibrates a Ho-Lee model exactly to every on-the-run Treasury price each day, and the fitted drifts are noticeably erratic from day to day. Which conclusion best reflects the difficulty of making a model match observed market prices?
- An exact fit makes the model unable to call any fitted bond rich or cheap, and the drifts absorb pricing noise, so the fit says little about whether the model's dynamics are right. - correct
- An exact fit proves that the model's volatility and distribution assumptions are correct.Why not: Matching today's bond prices is a statement about the drift, not about volatility or the distribution of future rates.
- Ho-Lee cannot match a full curve because a single volatility parameter limits the fit.Why not: One drift per date is enough to match any zero curve; volatility does not restrict the bond fit.
- Erratic drifts show that the arbitrage-free condition has been violated.Why not: Exact reproduction of prices is what makes the model arbitrage-free; erratic drifts signal noise and overfitting instead.
With a free parameter per date the model can fit any set of bond prices, noise included. The fit is perfect by construction, so it carries no evidence about the dynamics and no rich/cheap signal for the fitted bonds.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in The Art of Term Structure Models: Drift
- Build a short-rate tree for normally distributed rates, with and without drift, and describe how well it works.
- Work out the short-rate change and the standard deviation of that change under a normal-rates model with zero drift.
- Describe how term structure models can cope with short rates possibly going below zero.
- Build the Ho-Lee short-rate tree, in which drift is a function of time.
- Outline how a simple, recombining short-rate tree is built under the mean-reverting Vasicek Model.
- Using the Vasicek Model, calculate the change in rate and its standard deviation, along with the rate expected T years out and the half-life.
- Characterise how well the Vasicek Model performs.