# Ito Calculus **URL:** \ **Category:** Factual Questions\ **Created:** [December 16, 2017, 7:53pm UTC](https://boards.straightdope.com/t/ito-calculus/804672 "2017-12-16T19:53:51Z")\ **Posts on this page:** 3\ **Page:** 1
**Author:** ![RaftPeople](https://avatars.discourse-cdn.com/v4/letter/r/6f9a4e/32.png) [@RaftPeople](https://boards.straightdope.com/u/RaftPeople)\ **Post date:** [December 16, 2017, 7:53pm UTC](https://boards.straightdope.com/t/ito-calculus/804672/1 "2017-12-16T19:53:51Z")
Can someone explain in layman’s terms how this can provide any valuable information about a random system? I was reading about financial analysis, saw the term, googled and still unclear how you can get any valuable information out of a random system. For example, if we’re talking coin flipping (one of the examples), is there anything you can say about flip #27 other than the chance is still 50-50? The explanations showed lots of math but not much description that explained the value/info provided. ---
**Author:** ![Hellestal](https://avatars.discourse-cdn.com/v4/letter/h/3ab097/32.png) [@Hellestal](https://boards.straightdope.com/u/Hellestal)\ **Post date:** [December 16, 2017, 8:21pm UTC](https://boards.straightdope.com/t/ito-calculus/804672/2 "2017-12-16T20:21:00Z")
> [@RaftPeople](#): > > Can someone explain in layman’s terms how this can provide any valuable information about a random system? I don’t know how clear it can be made. But hey. Let’s try. > [@RaftPeople](#): > > I was reading about financial analysis, saw the term, googled and still unclear how you can get any valuable information out of a random system. For example, if we’re talking coin flipping (one of the examples), is there anything you can say about flip #27 other than the chance is still 50-50? It’s not directly about learning about flip #27, or any particular flip. In continuous-time financial contexts, the assumption is normally some variant of Brownian motion (a Wiener process) where each “flip” – each change of the asset price – is supposed to be independent of the previous flips. This is the case for the Black-Scholes option pricing model, where the price of the option is based on the underlying price of the original asset (the stock). And of course, the stock price is assumed to change with continuous small fluctuations. Essentially, the Itō calculus is about taking the “derivative” when dealing with such continuously fluctuating systems. Or to put it another way: we have a new function (such as the option price), which is dependent on the original Brownian motion process (such as the stock price). This function, based on the underlying process, might be concave. Ordinary calculus can’t handle the volatility, but this is exactly what [Itō’s lemma](https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma) is all about. It tells us how to handle dealing with a function of a random process. There are many, many, many other steps involved for something like Black-Scholes option pricing, but the idea here is understanding how _functions_ of random processes change with those random processes. Any concavity or convexity in the function is a big deal – not because we are predicting the 27th flip, but because a random series of “good” flips is not necessarily as helpful to the function as a random series of “bad” flips can be hurtful. Just because the original process is a random walk does not mean a _function_ that depends on that original process weights a bunch of ups equally with a bunch of downs. The curvature of the function is extremely important to find out how the underlying randomness will change it. I’m not sure if that’s clear, but… that’s a start, I guess. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [December 16, 2017, 11:18pm UTC](https://boards.straightdope.com/t/ito-calculus/804672/3 "2017-12-16T23:18:00Z")
My best advice is to read up on the Binomial pricing model > **[Binomial options pricing model](https://en.wikipedia.org/wiki/Binomial_options_pricing_model)** > > In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting. > The binomial model was first proposed by William Sharpe in the 1978 edition of Investments (.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-wor... This requires nothing more that simple algebra and probability. The Black-Scholes model is simply the limit of this model as the duration of each time interval goes to zero. If you know the central limit theorem, you’ll understand why the normal distribution comes into place. If you’d like even more details PM me.