Latest posts
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What Is the Vasicek Model? (And Why Nobody Uses It)
Vasicek is the first interest rate model almost everyone meets. It is elegant, it has closed-form bond prices, and it is the natural place to start because it is just mean reversion with Gaussian noise. It is also not used to price anything on a real desk. The usual explanation — that it produces negative
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What Is a Variance Swap?
A variance swap pays the difference between realised variance and a strike agreed at inception: It is the cleanest way to take a view on volatility without taking a view on direction. And the reason it exists in this form — variance rather than volatility — is not a convention. It is the
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What Is the Feller Condition?
The Feller condition is a single inequality that tells you whether a CIR process can reach zero: Satisfy it and the process stays strictly positive forever. Violate it and it touches zero with positive probability. That matters because CIR is the standard model for two things that cannot go negative — short rates
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What Is Itô’s Lemma?
Ordinary calculus has a chain rule. Stochastic calculus has one too, and it carries an extra term that ordinary calculus does not. Itô’s lemma is that chain rule, and the extra term is the whole point. Here is the cleanest way to see why it has to be there. The calculation that fails Take and
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Variance Reduction in Monte Carlo: Which Technique When?
Monte Carlo error falls like . To halve it you quadruple the paths, and past a certain point that stops being a strategy. Variance reduction techniques buy the same accuracy for fewer paths — sometimes by a factor of two, sometimes by a factor of five hundred. But they are not interchangeable, and one of
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What Is Girsanov’s Theorem?
Girsanov’s theorem is usually presented as a technical result about Brownian motion under a change of measure, and then applied without comment. That skips the part worth understanding. Here is the whole thing in one sentence: **you can change the drift of a diffusion to anything you like, and you cannot touch the volatility at
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What Is a Numéraire?
A numéraire is the thing you measure prices in. That is genuinely all it is — and it sounds so trivial that it is easy to miss why it matters. Prices are ratios. Saying a stock is worth 100 means 100 of something, normally currency. But there is nothing special about currency. You could quote
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What Is Quadratic Variation?
Two sums over the same Brownian path. Chop into pieces and add up the moves: The first is total variation — the distance the path travels. The second is quadratic variation. They look like minor variations on each other and they behave in opposite ways, which turns out to be the reason stochastic
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What Is the Radon-Nikodym Derivative?
If you have read anything about risk-neutral pricing, you have met the symbol and probably been told it is a “change of measure”. That is true and completely unhelpful. It looks like a derivative, it is not a derivative, and the notation actively obstructs the intuition. Here is what it actually is: a
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Euler vs Milstein: When Does the Extra Term Matter?
Every textbook on SDE discretisation gives you Euler-Maruyama first and then Milstein, with the note that Milstein is “first order” while Euler is “half order”, and moves on. What almost none of them tell you is when that upgrade is worth writing, when it does nothing at all, and when it fails to fix the