Arbitrage-Free Valuation Models

In term structure models, it's really hard to come up with accurate parameters that will allow the short-term interest rate to match market values over time. A solution is to allow them to change over time.

This is the idea behind __arbitrage-free models__. They also have a drift term and a stochastic term, like the Cox–Ingersoll–Ross (CIR) and Vasicek models. But unlike those models with a few static parameters, arbitrage-free models have parameters indexed with time. How do you think this will affect their accuracy?

No. The flexibility of parameter values allows for greater accuracy.

Of course.

The model prices will match market prices with this continual recalibration. That's why the models are arbitrage free. They start with observed prices and instruments rather than ending with an estimate.

The __Ho–Lee model__ is a popular arbitrage-free model. This model allows short-term interest rates to follow a pattern very similar to the CIR and Vasicek models. $$\displaystyle dr_t = \theta_t dt + \sigma dZ $$ Again, the short-term interest rate is evolving in small increments. The drift term is represented by $$\theta$$, and the random walk term, $$dZ$$, is now a binary variable, which is either +1 or -1 in each period. Aside from $$dZ$$ and $$dr_t$$, which value do you think changes each period?

That's right. It's indexed by _t_; that's the big clue. So each time period, these three variables change. The drift parameter is obtained from market prices of the debt instruments used. For example, maybe it's 0.84% in Period 1, and 0.96% in Period 2.

No. This is a parameter that's estimated just once; not everything needs to change.

No. This is a time increment that's set at the start.

Suppose the short-term rate is at 3% today, and the model assumes monthly movements. If $$\sigma$$ is set at 2.5% annually, then $$\displaystyle dt = \frac{1}{12} $$. Monthly volatility is $$\displaystyle \sigma \sqrt{\frac{1}{12}} = 0.025\sqrt{\frac{1}{12}} = 0.0072 = 0.72 \% $$. Using the model, what is the Period 1 interest rate if there is an "up" movement?

Exactly! If it's an "up" movement, then the Period 1 rate is 3.79%. $$\displaystyle r_{1,u} = 3 \% + (0.84 \%)(\frac{1}{12}) + 0.72 \% = 3.79 \% $$ If it were a "down" movement, then it would be 2.35%. $$\displaystyle r_{1,d} = 3 \% + (0.84 \%)(\frac{1}{12}) - 0.72 \% = 2.35 \% $$ Notice that the rate drifts up either way, but the stochastic factor makes the final decision as to which way it moves.

That's not quite right. Don't forget to include the stochastic factor. That's what's really causing the "up" or "down" movement.

No. The "up" movement would have to set the Period 1 rate higher than today's 3% level.

Then in Period 2, the Period 1 rates each drift up by $$\displaystyle 0.96 \% \times \frac{1}{12} = 0.08 \% $$. Then they'll adjust either up or down by the stochastic element of 0.72%. How many possible Period 2 values do you expect to find?

Well, no. Consider that a couple of paths lead to the same outcome.

Yes. The "up then down" path is the same as the "down then up" path, leading to three outcomes.  The model produces a normal distribution of future rates. Since it remains consistent with market prices, it can be used to calculate prices for zero-coupon bonds and make the spot curve. However, an arbitrage-free model like the Ho–Lee model is called a __partial equilibrium model__ since it just estimates the single factor of _r_, taking the shape of the yield curve as given. It doesn't try to explain everything, just a key thing.

Not so. There are already two rates in Period 1, and now the rates will branch out again from there.

The __Kalotay-Williams-Fabozzi (KWF) model__ is pretty similar to the Ho-Lee model. Same assumptions of constant drift and constant volatility with no mean reversion, but it models the log of the short-term interest rate instead of just the rate: $$\displaystyle d \, ln(r_t) = \theta_t dt + \sigma dZ $$ What restriction would that suggest for the interest rate?

To summarize: [[summary]]

Actually, it couldn't be zero. You can't take the natural log of zero, as it approaches negative infinity.

Right; you can't take the natural log of zero or anything lower (negative). So the KWF model assumes positive interest rates just like the CIR model. And of course many other models exist as well. The "Gauss+" model uses short-term, intermediate-term, and long-term interest rates, so it can incorporate different yield curve shapes. It's a multifactor model used a lot in hedging. As markets evolve, modeling evolves; these won't be the last models you see!

No, you can take the natural log of 10%; you'll just get a negative value, but that's fine.

Arbitrage-free models will be less accurate

Arbitrage-free models will be more accurate

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σ

_dt_

2.94%

3.06%

3.79%

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It would have to be positive

It would have to be zero

It would have to be greater than 1

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