Estimating a Stock’s Chance of Exceeding a Price with a Normal Return Model

Summary

The document shows how to estimate the probability that a stock closes above a specified price after a known market move. It assumes daily stock returns are normally distributed, uses the stock’s beta to estimate its expected return from the market change, and treats the stated stock-specific volatility as the return standard deviation. The target closing price is converted into a minimum required return, then standardized relative to the assumed mean and volatility to obtain an upper-tail probability.

In the given example, the stock starts at 100, the market rises by 1%, beta is 2, and daily volatility is 0.02. The answer calculates an expected return of 0.02 and a threshold return of 0.03, yielding an estimated probability of 0.31 under the normal model. This is an illustrative calculation, not a forecast validated against data. It assumes the beta-based conditional mean and volatility are appropriate and that returns are normal over the interval considered.

Key ideas

• Translate the target closing price into the return required from the previous close.
• Estimate the stock’s expected return by multiplying beta by the market move.
• Standardize the required return using the assumed mean and volatility, then evaluate the normal upper tail.
• The resulting probability depends on the normal-return and input assumptions.

Tags

# Probability of stock closing over a certain price


# Probability of stock closing over a certain price












A stock has beta of 2.0 and stock specific daily volatility of 0.02. Suppose that yesterday's closing price was 100 and today the market goes up by 1%. What's the probability of today's closing price being at least 103?

## Answer by emcor (score 11)

https://quant.stackexchange.com/a/14449

Usually stock returns are assumed to be normally distributed: $R\sim N(\mu,\sigma)$

If market goes up 1%, the expected stock return is $\mu = \beta\cdot 0.01 = 0.02$ (since β is the senstivity to market).

Stock price going from 100 to over 103 requires a return $R$ of at least 103/100 – 1 = 0.03.

As we have from the question σ = 0.02, we get:

$$ P(R\geq 0.03) = 1 - P(R\leq 0.03) = 1 - F(0.03) = 1 - \Phi\left( \frac{0.03-\mu}{\sigma} \right) = 1 - \Phi(0.5) = 0.31 $$

where $F$ is the generic normal cumulative distribution function and $\Phi$ is the cumulative distribution function for the standard normal distribution.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.