Brownian Motion and the Black-Scholes Formula Explained

[bgBlue]

Hi,

I am studying brownian motion and the Black-Scholes formula.

Our problem assumes that
1. Stock returns follow a normal distribution
2. Based on #1 the stock price follows a lognormal distribution because y = exp(X) is lognormal if X is normally distributed. Here the stock prices are continuously compounded.

3. So I would expect that the stock return would have an average return of "u" because this is the average value of a normal distribution... but it is actually "u - p^2/2".

p = standard deviation
http://www.bionicturtle.com/learn/article/lognormal_distribution_part_3_future_stock_price/

From what I am gathering this is because Ito's lemma tells us that...

St = So * Exp ( u - p^2 / 2 )*t + pWt

So why don't the two match?

I am having trouble following the proof and "repercussions" of the ito theorem. I tried wiki for ito's lemma but it is very confusing.

Are there any beginners info that I could use to get a better grip of this concept? Like books or online or can you explain any of this?

Thanks

 

[mighty2000]

Thank you for your question regarding Brownian motion and the Black-Scholes formula. I would be happy to provide some clarification and resources to help you better understand these concepts.

Firstly, it is important to note that Brownian motion is a mathematical model used to describe the random movement of particles in a fluid, such as the stock price in the stock market. The Black-Scholes formula is a pricing model used to determine the theoretical value of options contracts, which are financial instruments that give the holder the right to buy or sell an underlying asset at a predetermined price on or before a specific date.

In order to use the Black-Scholes formula, we make certain assumptions about the stock returns. One of these assumptions is that the returns follow a normal distribution, meaning that the majority of the returns fall within a certain range and the probability of extreme returns is very low. However, as you correctly pointed out, the stock price itself does not follow a normal distribution. Instead, it follows a lognormal distribution, which is a transformation of the normal distribution. This is because stock prices cannot be negative, and the lognormal distribution ensures that the stock price will always be positive.

Now, let's talk about Ito's lemma. This is a mathematical tool used to determine the change in a function of a random variable, such as the stock price. In the case of the Black-Scholes formula, we are interested in the change in the stock price over time. Ito's lemma tells us that the change in the stock price is equal to the current stock price multiplied by the average return, plus a term that takes into account the volatility (represented by the standard deviation, p) and a random component (represented by the term pWt). This is why the average return in the Black-Scholes formula is "u - p^2/2" instead of just "u."

I understand that the proof and implications of Ito's lemma can be quite complex, and I would recommend seeking out additional resources to help you better understand it. One book that I personally found helpful is "Options, Futures, and Other Derivatives" by John C. Hull. There are also many online resources available, such as Khan Academy's videos on Ito's lemma and Black-Scholes formula.

I hope this helps to clarify some of the concepts you are studying. If you have any further questions, please do not hesitate to ask.