Can someone please explain to me conceptually, and in english, what these things mean:
Risk Neutral Probabilities and why you'd use them
The LogNormal Distribution Model and why it works well in the short term but not the long term
I'd definitely appreciate it and give you a favor you need back
what kind of favors are we talking about here?
I can't explain risk neutral probability.
The lognormal distribution is esentially a normal distribution tailored so that it doesnt have negative values (a normal distribution is equally weighted positive to negative). It's used to model stock prices because it is a well known distribtuion with easily identifiable moments it is an essential part of brownian motion which is the major part of Black-Scholes. Look at a brownian motion path and you will see that it looks like a stock price almost.
You can do it in excel by choosing data analysis random number generator and choosing values between -1,1 and generate maybe 1000 numbers to get a good idea and select uniform distribution. Then in another column start with 0 and add to it the first number you generated divided by the square root of 1000 and do this for all 1000 numbers you generated. You are essentially saying that I have this brownian motion process that is moving over the course of 1000 units of time. if you graph the new column it will be a brownian motion process.
It's not too good in the long term because there are times when a price jumps outside of what you would ever expect using the distribution far too often (this is what ruined LTCM). Stocks appear follow more of a poisson/jump diffusion process which is where new financial math is going.
Risk neutral probability comes from the risk neutral measure and measure theory. These are grad level topics in probability that really cannot be answered in detail without spending a bit of time on them. But I'll attempt to describe it in a simple way...
For argment sake, think of "real world" probability (what we commonly work with in prob/statistics) as a ruler in measures of inches. And think of "risk neutral probability as a ruler in measures of centimeters. Both measure the same length in absolute terms, but they have their own scaling.. 1inch = 2.54cm For instance, a particular event measured in real world probability may have 20% chance, but the same event may have 30% chance in a risk neutral world.
The importance of the "skewed" risk neutral probability in finance is that in the real world, they produce a fair price to a security with multiple possible outcomes. Using "real world" probabilities actually mis-price the securities and cause arbitrage opportunities.
Anyone feel free to correct me if I'm wrong
I'll try to keep the 50-cent words to a minimum.
The term "Risk Neutral Probabilities" is kind of a misnomer. Instead of probability, think "pricing distribution." RNP's are a mathematical concept used to fulfill the requirement that each element or part of a calculation range from 0-1 and collectively sum to 1 (this is where they got probability from). Additionally, risk neutral probabilities are only used in a relative sense (i.e. you want to value x in terms of y where there is a known relationship between the two). You would use RNP's because you want to view how a derivative is priced in the context of a world with the absence of arbitrage or risk.
Simplistically stated, I create a replicating portfolio long, an option short, put all the pricing equations together and I come out with the actuarial fair value of an option using a risk neutral pricing distribution.
Now that I think about it risk neutral just makes this sound confusing. You're not taking a risk neutral position its more of a future or forward neutral. So there you have it, forward neutral pricing distribution in lieu of risk neutral probabilities.
LogNormal distributions are the least shitty option to model most financial time series data. For example, I take a time series of corn positions, take the natural log of all these random variables and I have a pretty LogNormal distribution to look at.
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