Function to find the probability distribution of a stock price

In summary, the conversation discusses the use of a log-normal distribution to calculate the probability distribution of a stock price after a certain number of days. The suggestion is to transform the normal distribution of percentage growth into a log-normal distribution, and then rescale it to get the final distribution. The use of a log-normal distribution is based on the assumption that many natural growth processes are driven by the accumulation of many small percentage changes. However, there is some disagreement on whether stock prices can be accurately modeled using this distribution.
  • #1
beamthegreat
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Hi all. I'm trying to find a formula that will calculate the probability distribution of a stock price after X days, using the assumption that the price change follows a normal distribution. In the spreadsheet, you can see the simulation I've made of the probability distribution of the price of a stock that is initially at $100 after 252 days (1 trading year, using the assumption that the price moves with an SD of 3.5% per day)

Can someone please direct me to the simplified formula I can use to calculate this? I'm certain this has been done before.

Thanks!

Link: https://docs.google.com/spreadsheets/d/1beooSijC0OJ4uBfC_a-dxveeplFmfTBTiy4QoIhbC28/edit?usp=sharing
 
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  • #2
beamthegreat said:
Hi all. I'm trying to find a formula that will calculate the probability distribution of a stock price after X days, using the assumption that the price change follows a normal distribution. In the spreadsheet, you can see the simulation I've made of the probability distribution of the price of a stock that is initially at $100 after 252 days (1 trading year, using the assumption that the price moves with an SD of 3.5% per day)
So the idea is that the probability distribution at the end of one day is a normal distribution with mean equal to the morning's value and standard deviation equal to 3.5 percent of the morning's value? And that the distribution at the end of 252 days is the result of iterating this procedure 252 times?

The obvious difficulty is that the standard deviation for the second day's normal distribution will depend on the value of the sample drawn from the first day's distribution. The obvious remediation would be to use a log-normal distribution. Sum up 252 of those and take the anti-log.

https://en.wikipedia.org/wiki/Log-normal_distribution

[As an bonus, the log normal distribution is not obviously impossible. The normal distribution is obviously impossible since the left hand tail has a non-zero probability for negative prices]
 
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  • #3
jbriggs444 said:
So the idea is that the probability distribution at the end of one day is a normal distribution with mean equal to the morning's value and standard deviation equal to 3.5 percent of the morning's value?

That is correct.

jbriggs444 said:
The obvious difficulty is that the standard deviation for the second day's normal distribution will depend on the value of the sample drawn from the first day's distribution. The obvious remediation would be to use a log-normal distribution. Sum up 252 of those and take the anti-log.

I got the gist of it, but I don't have sufficient knowledge of mathematics to derive the expression myself. I tried approximating it with a normal distribution using the following equation to find the final SD after n days, which is pretty accurate if n is small, but the distribution becomes increasingly skewed as the number of days increases: ##SD_{final} = \sqrt {nSD_{initial}^2}##

I have no idea how to use the log-normal distribution to remediate this. Any help will be appreciated.
 
  • #4
beamthegreat said:
That is correct.
I got the gist of it, but I don't have sufficient knowledge of mathematics to derive the expression myself. I tried approximating it with a normal distribution using the following equation to find the final SD after n days, which is pretty accurate if n is small, but the distribution becomes increasingly skewed as the number of days increases: ##SD_{final} = \sqrt {nSD_{initial}^2}##

I have no idea how to use the log-normal distribution to remediate this. Any help will be appreciated.
I've never used one either, but the principle seems simple enough.

With a normal distribution, you'd be looking at a distribution of percentage growth which is centered on 0% growth with standard deviation 3.5%. So we are talking about a bell shaped curve roughly from 96.5% to 103.5% of the morning's starting value.

If we represent those values as natural logs, that's ln(96.5%) to ln(103.5%). Using the small log approximation, that's about -0.035 to +0.035. If precision is important, you could transform the normal distribution more carefully, taking the natural log of every value.

Pretend for the moment that the resulting distribution is approximately normal. So we have an estimated distribution with mean 0 and standard deviation approximately 0.035. But after summing with itself 252 times, the result will be a lot closer to normal. So we estimate the true distribution after 252 days as a normal distribution with mean = 252 times 0 and standard deviation = ##\sqrt{252}## times 0.035. The standard deviation of this will be around 0.55 -- corresponding to a rise or fall by a factor of ##e^{0.55}## ~= 1.73 in price.

Rescale this normal distribution by taking the exponential of each point and you have your expected final distribution. I do not expect that this will be a normal distribution.

Edit: It is gratifying to see that Wiki makes essentially the same points:

Occurrence and applications[edit]

The log-normal distribution is important in the description of natural phenomena. This follows, because many natural growth processes are driven by the accumulation of many small percentage changes. These become additive on a log scale. If the effect of anyone change is negligible, the central limit theorem says that the distribution of their sum is more nearly normal than that of the summands. When back-transformed onto the original scale, it makes the distribution of sizes approximately log-normal (though if the standard deviation is sufficiently small, the normal distribution can be an adequate approximation).
 
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  • #5
beamthegreat said:
That is correct.
I got the gist of it, but I don't have sufficient knowledge of mathematics to derive the expression myself. I tried approximating it with a normal distribution using the following equation to find the final SD after n days, which is pretty accurate if n is small, but the distribution becomes increasingly skewed as the number of days increases: ##SD_{final} = \sqrt {nSD_{initial}^2}##

I have no idea how to use the log-normal distribution to remediate this. Any help will be appreciated.
Google "lognormal distribution".
 
  • #6
Stock prices are not lognormal, returns are not normal. What are you trying to do with this?
 

Related to Function to find the probability distribution of a stock price

1. What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of different outcomes occurring in a given event. It shows all the possible outcomes and the probability of each outcome happening.

2. Why is it important to find the probability distribution of a stock price?

Finding the probability distribution of a stock price allows us to understand the potential risks and returns associated with investing in a particular stock. It also helps in making informed decisions about buying, selling, or holding a stock.

3. How is the probability distribution of a stock price calculated?

The probability distribution of a stock price is calculated using historical data and statistical methods such as the normal distribution or the lognormal distribution. These methods take into account factors such as past performance, market trends, and volatility to determine the likelihood of future stock prices.

4. Can the probability distribution of a stock price be used to predict future stock prices?

No, the probability distribution of a stock price cannot accurately predict future stock prices. It only provides a range of possible outcomes based on historical data and statistical analysis. Other factors such as market conditions, company performance, and external events can also impact stock prices.

5. How can the probability distribution of a stock price be used in risk management?

The probability distribution of a stock price can be used in risk management by helping investors understand the potential risks associated with a particular stock. It can also be used to calculate the expected return and determine the level of risk that an investor is comfortable with. This information can then be used to make informed decisions about portfolio diversification and risk management strategies.

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