Unchaining the Knobs: Black and Karasinski's Lognormal Short Rate

A year after Black, Derman and Toy published their tree, Fischer Black was back in the Financial Analysts Journal, this time with Piotr Karasinski, fixing the one thing about BDT that had always bothered them.

The fix is small to describe and important in practice, and the resulting model is still on trading floors today, particularly wherever people insist that interest rates cannot go below zero.

The problem: BDT's hidden wiring

The Black-Derman-Toy model has a quirk that is easy to miss and hard to live with.

In BDT, you feed the model two things from the market: the yield curve and the volatility curve. The model then bends its tree to fit both. Fine. But because of how it is constructed, the speed of mean reversion is not something you choose. It is a by-product of the shape of the volatility curve you fed in.

Specifically: if the market's quoted volatilities decline as you look at longer maturities, BDT interprets that as mean reversion, and the faster they decline, the stronger the mean reversion it assumes. If the volatility curve is flat, BDT concludes there is no mean reversion at all.

There is no economic reason those two things should be welded together. How fast rates get pulled back toward normal is a fact about the economy and central banks. How volatile long yields are relative to short ones is a fact about the option market. Sometimes they line up. Often they do not. And when they do not, a BDT user is stuck: the model has made a decision on their behalf, silently, and there is no dial to override it.

Worse, this coupling can produce perverse behaviour. Certain shapes of the input volatility curve imply negative mean reversion, meaning the model believes rates are pushed away from their long-run level over time. That is not a description of any interest rate market anyone has ever seen.

The key idea via analogy: three separate dials on the mixing desk

Black and Karasinski's move is to take the same basic setup, a lognormal short rate, so it can shrink toward zero but never cross it, and simply refuse to let the parameters be entangled.

Think of a sound engineer's mixing desk with three faders, each of which can be moved along a track through time:

1. The target. Where is the elastic lead tied? This is the level rates are being pulled toward, and Black and Karasinski let it change over time. It is what you solve for in order to make the model fit today's yield curve exactly.
2. The pull. How hard does the elastic yank the rate back toward the target? This is the mean-reversion speed, and crucially, in this model you set it yourself. It is your dial, not a by-product.
3. The jitter. How energetically does the rate jump around? This is the volatility, also time-varying, also yours to set.

Three faders, three separate tracks, no wires between them. That is the entire contribution, and it is precisely what BDT lacked.

The model works on the logarithm of the short rate. It is that logarithm, not the rate itself, that mean-reverts around a moving target with a controllable pull and a controllable jitter. Because you are working with the log, exponentiating to recover the actual rate always gives you a positive number. Rates can drop to a tenth of a percent, they can hover near zero, but they cannot become negative.

Black and Karasinski point out the payoff plainly: with all three faders free and time-varying, you can match the yield curve, the volatility curve and the cap prices simultaneously, which is exactly the calibration a real options desk needs.

Why it mattered

• It gave practitioners a properly parameterised lognormal model. Black-Karasinski is, for many purposes, the model people thought BDT was. Same positivity, same lattice-friendly structure, but with mean reversion under the user's control.
• It became a workhorse for positive-rate markets. For decades it was a common choice for pricing caps, floors, swaptions and callable bonds, and it is still used in insurance and actuarial contexts and in emerging markets where high, strictly positive rates make lognormality natural.
• It sharpened a general lesson about calibration. The BDT episode is a case study in how a model's construction can smuggle in assumptions that its user never consciously made. Black and Karasinski's fix taught quants to ask, of every model: which of these parameters are genuinely free, and which are being determined behind my back?
• The positivity is genuinely useful. In any application where the modelled quantity truly cannot be negative, a default intensity, a hazard rate, an inflation index, this construction, mean-reverting in logs, is a natural and well-behaved starting point.

The honest limitations

• No closed-form bond prices. This is the big one, and it is the price of lognormality. In Vasicek and Hull-White you can write down the price of a zero-coupon bond as an explicit formula. In Black-Karasinski you cannot. Everything, bond prices included, has to be computed numerically on a tree or a lattice. That makes calibration slower and Greeks messier, and it is a real operational cost for a desk revaluing a large book.
• Lognormal rates can run away. A lognormal process has a long right tail, and combined with the compounding involved in a money market account, this creates well-known technical unpleasantness: the continuous-time lognormal short rate model implies infinite expected returns on a money market account over any finite period. Mean reversion tames this in practice, but the model must be handled carefully at long horizons and high volatilities, where it can generate absurdly high future rates.
• It cannot represent negative rates. At all. This was a feature in 1991 and a fatal flaw in 2015. When European and Japanese rates went below zero, Black-Karasinski was structurally incapable of pricing anything, which is why desks in those markets moved to Gaussian models (Hull-White) or "shifted" versions where you displace the whole distribution downward by a fixed amount.
• Still a one-factor model. One random shock drives every rate, so the entire curve moves together. It can shift, it cannot twist. Anything whose payoff depends on the spread between two maturities is outside its natural range.
• Calibration is not unique. With three time-varying faders, many different combinations can fit the same market data. Two desks can both "use Black-Karasinski," both fit the same swaption grid, and still disagree materially on an exotic. Model risk does not disappear just because the calibration is good.

The one-line takeaway

Black and Karasinski took the lognormal, always-positive short rate of BDT and cut the hidden wire that tied mean reversion to volatility, handing traders three independent, time-varying dials that could fit the yield curve, the volatility curve and cap prices all at once.