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Fixed Income and Interest-Rate Models / Short-Rate Models

Volatility Structure and Lognormal Short-Rate Models: Dothan, BDT, and Salomon

A cap on the 3-month SOFR rate struck at 5% is priced very differently depending on whether you model the rate as normally distributed (Vasicek/Ho-Lee) or lognormally distributed. Normal models allow negative rates and assign a Gaussian distribution to the terminal short rate; lognormal models guarantee positive rates and assign a log-normal distribution — the same distinction as between Black-Scholes on the stock price versus on the log of the stock price. This distributional choice directly affects the skew of option payoffs: a deeply out-of-the-money floor has a much higher probability under a lognormal model than under a normal one at the same mean and variance. This lesson builds the lognormal short-rate models from the ground up — starting with the Dothan model's constant yield volatility, through the Salomon Brothers and Black-Derman-Toy trees that calibrate to the yield curve, to the Black-Karasinski model that combines lognormal rates with mean reversion.
Learning objectives. By the end of this note you should be able to:
  • distinguish basis-point volatility from yield volatility and state which grows with the rate level in each model class;
  • write down the Dothan SDE, integrate it explicitly, and identify its lognormal distribution;
  • state why lognormal models have an exploding money-market account and why this is less severe in discrete trees;
  • construct the Salomon Brothers lognormal tree in ln⁡r\ln r space and convert it to the rate tree;
  • explain how BDT uses time-dependent volatility σ(t)\sigma(t) to calibrate to the cap vol term structure;
  • articulate the Black-Karasinski model's key advantage (separate control of mean reversion and vol term structure) and cost (non-recombining tree without variable time steps).

In this lesson

    1. Volatility as a function of rate level
    1. The Dothan model
    1. The Salomon Brothers lognormal tree
    1. Black-Derman-Toy (BDT)
    1. The lognormal explosion problem
    1. Summary: the short-rate model landscape

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