Chapter 11 gave us the Itô-Doeblin Formula — the new chain rule for a random world. Now we apply it to three of the most important models in quantitative finance. Each model solves a different real problem: how stocks grow, how interest rates stay bounded, and how to prevent a model from producing physically impossible results. These are not just textbook exercises. They are the building blocks of pricing engines used in financial institutions worldwide.
Three models, three problems, one mathematical engine — the Itô-Doeblin Formula.
In Chapter 3 we saw GBM with constant drift and volatility. In practice, a stock's expected return and riskiness change over time — as the company grows, market conditions shift, or volatility spikes during a crisis. Generalised GBM allows both α(t) and σ(t) to vary continuously. Think of it as upgrading from a car with a fixed speed to one with a working accelerator and a variable engine noise.
The most important feature of this formula is the correction term −½σ². It exists because of an asymmetry in how percentage returns compound. A concrete example makes this clear:
Symmetric percentage moves are not symmetric in dollar terms. The Itô correction −½σ² is the mathematical fix for this compounding asymmetry.
The −½σ² term is not an approximation or a fudge. It is the exact result of applying the Itô-Doeblin formula to the exponential function. When the formula hits the second derivative of exp(x) — which equals exp(x) itself — the ½f''dt term becomes ½σ²dt, which then subtracts from the drift. The math enforces accounting honesty.
Interest rates behave very differently from stock prices. A stock can double, triple, or go to zero. An interest rate set by a central bank tends to stay within a corridor — it is pulled back toward a long-run average whenever it strays too far. The Vasicek model captures this mean-reverting behaviour using a single elegant equation.
The rubber band pulls the rate back toward the target whenever it strays. But the normal distribution's tails extend to −∞, allowing negative rates.
Vasicek's flaw — the possibility of negative interest rates — was long considered a theoretical curiosity. Then the European Central Bank and Bank of Japan introduced negative policy rates after 2008. This shows that model flaws can sometimes turn out to be features in disguise: Vasicek accidentally predicted something economists once thought impossible. The CIR model was designed specifically to prevent this.
The Cox-Ingersoll-Ross model (CIR) makes one targeted change to Vasicek: it replaces the constant volatility σ with σ√R(t). This single modification makes the model self-correcting at zero. As the interest rate approaches zero, the volatility term shrinks toward zero too — which means the random shocks get weaker just as the rate hits bottom, allowing the positive drift to push it back up. The rate never crosses zero.
The CIR model (gold) bounces off zero as σ√R → 0 near the floor. Vasicek (red dashed) passes straight through — a mathematical impossibility in practice.
The CIR model's self-correcting mechanism is elegant: the volatility is endogenous — it grows when rates are high and shrinks when rates are low. This means that at low rates, the market is actually calmer, and the positive mean-reversion drift dominates. The model is internally consistent in a way Vasicek is not.
Both models capture mean reversion. Both are Itô Processes driven by the same rubber-band drift. The single change — from constant σ to σ√R — creates a cascade of different properties. Understanding exactly why they differ is more important than memorising which one to use.
| Feature | Vasicek | CIR |
|---|---|---|
| Mean Reversion | Yes — rubber band | Yes — rubber band |
| Volatility structure | Constant σ — same wiggle at any rate | σ√R — wiggle grows with the rate |
| Distribution of R(t) | Normal (Bell Curve) | Non-central Chi-squared (skewed) |
| Negative rates possible? | Yes — a model flaw | No — guaranteed positive |
| Mathematical tractability | Very clean — closed-form solutions | Slightly more complex — Bessel functions |
| Best used for | Quick pricing, regime analysis | Bond pricing, option pricing on rates |
There is one final insight from this chapter that connects all three models to trading strategy. If a trader uses a deterministic strategy — a position plan that is fixed in advance and does not respond to what the market actually does — then the resulting P&L has a very special property.
Fixing your strategy in advance produces a clean Bell Curve in your P&L. Adapting to the market distorts that curve — sometimes in your favour, sometimes not.
This theorem is why passive index investing (a deterministic strategy — same allocation every period) has a mathematically predictable and symmetric risk profile. Active management (an adaptive strategy — adjusting based on market signals) distorts the P&L distribution away from Normal. Whether that distortion is an improvement — positive skew, fatter right tail — depends entirely on the skill of the signal being used.
Select a model, set the parameters, and simulate multiple paths. For Vasicek and CIR, watch the mean-reversion rubber band at work. Try pushing the starting rate far from the target and observe the different speeds of convergence. Then lower the rate close to zero — and see how CIR bounces while Vasicek crosses the floor.
GBM paths grow exponentially with drift. Vasicek/CIR paths oscillate around the target — but only CIR stays above zero.
Chapter 12 — three models, three lessons:
What comes next: We now have all the ingredients — GBM for the stock, a risk-free bank account for the alternative, and the Itô-Doeblin formula as our calculator. The Black-Scholes-Merton equation combines these three into the formula that tells you the one fair price for an option. The formula that changed Wall Street forever.