Estimating Equal-Probability Stock Return Outcomes from a Random Walk

Summary

The document asks how to represent a stock’s payoff across ten equally likely return states, given an assumed drift and volatility. It frames the task as finding conditional expected values for successive deciles and asks whether to calculate physical-distribution quantiles first, then assign risk-neutral probabilities to them. The example payoff table is explicitly illustrative, with its returns described as made up.

No solution or derivation is provided, so the distinction between physical and risk-neutral probabilities remains unresolved. In particular, the text does not specify a return distribution, investment horizon, or whether “payoff in a decile” means a quantile boundary or the conditional mean within that decile. Those choices matter for calculating the requested outcomes. The document is useful as a formulation of a probability-modeling question, but it offers no empirical evidence or recommended calculation method.

Key ideas

• The question seeks ten equally probable stock return outcomes ordered from the worst decile to the best.\nIt gives drift and volatility as inputs to a random-walk model.\nIt asks whether physical quantiles should be calculated before considering risk-neutral probabilities.\nThe illustrative table does not establish a calculation method or a validated payoff distribution.

Tags

# Calculation of Conditional Expected Value and Pay-Off Diagram


# Calculation of Conditional Expected Value and Pay-Off Diagram












I have a stock with mu 6% and sigma 20% following a random walk and I would like to to calculate the Conditional expected Value of the stock in 10 states with equal probability (10%). Meaning, I would like to know how much the stock pays in the deciles from worst case to best case.

Would it be correct, to first calculate the physical quantiles of the distribution and afterwards the risk-neutral probabilities of the physical quantiles?

In the end I would like to say that:

Given a stock with these characteristics your expected payoff in a 10 state-digram would look like this: (with the returns values of course made up right now)

```
Probability       Return
10%                 -30%
10%                 -25%
10%                 -10%
10%                 -5%
10%                  15%
10%                  20%
10%                  30%
10%                  40%
10%                  50%
10%                  70%
```

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