Describing Ito’s Lemma.

I need help understanding Ito’s Lemma.

What’s a Lemma?

A lemma is known as a helping therom. In other words, it’s a mini therom in which a bigger therom is based off of.

Who is Ito?

Kiyoshi Ito is a mathematician from Hokusei, Japan. He studied stochastic processes while working at Kyoto University and has his own branch of calculus named after him. He is very nice looking, isn’t he?

What’s a Stochastic Process?

There are two types of stochastic processes: discrete and continuous. Discrete values can be observed by measuring values at certain intervals of time, like every week, for a certain amount of time, like a year. Continuous means that the time in the intervals have been shrunk down to very very small and there are an infinite amount of intervals.

Say your probability space is all the values a stock maybe in the next year, your stochastic process would be a collection of all those random stock prices indexed by time. Which could be anything because it is assumed stock prices move randomly up or down with a 50% chance either way.

What Does Ito’s Lemma Look Like?

Ito’s Lemma describes the change in stock price from one time to the next. It can be decribed like this: “the change in stock price at time t equals, the average rate of return of the stock (over a certain amount of time), minus the continuous dividend rate of the stock, times the change in time, plus the volatitily of the stock, times the change of some sort of randomness factor.”

How Does Ito’s Lemma Help?

Because the value of a derivative is based on an underlying asset such as a stock, Ito’s lemma can be used to show the relationship between the two.

By using the two variable Taylor-series expansion formula (only the first three terms are usable because afterwards, the number becomes so small it’s not worth calculating by hand), the value of an option can be expressed in terms of the derivatives of the option value and the Ito proccess.

For example, if the value of an option is expressed in term of a fuction “ln(x)”, VS would equal “1/x” and the second derivative Vss would be “-1/x^2.” The derivative Vt in respect to time would equal 0. Because the Greeks delta, gamma, and theta are all partial derivatives of the option value, they can replace the Vs, Vss, and Vt (Reminder: delta relates to change in price, gamma relates to change in delta, and theta is time decay).

What Next?

We can use Ito’s lemma to help define Brownian motion, which is a more specific category under the Ito’s lemma formula’s umbrella. And because of the previous option valuation formula, we now can apply Ito’s lemma to the Black-Scholes Equation (which follows a geometric Brownian motion) to help us value any derivative with a stock as its underlying asset.

If you are reading this, please comment and help me learn!