# 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%

• I think your question lacks some details. You have a stock with $\mu$ and $\sigma$ but what is your time horizon? If I understand well, you would like to determine the expected return of your stock in each probability interval? – Wiles01 May 19 '19 at 16:32
• Yes exactly! My time horizon would be one year. – Jj Blevins May 20 '19 at 10:37
• The stock over one year is distributed as $X\sim LN(\mu,\sigma^2)$. So you need to determine $E[X|a\leq X \leq b]$ where $a=exp(\mu+\sigma\Phi^{-1}(p))$ and $b=exp(\mu+\sigma\Phi^{-1}(p+0.1))$. It should be feasible to compute this expression explicitly. – Wiles01 May 21 '19 at 7:42
• So if my calculations are correct for the first decile a = 1.07768 and b = 1.12991? This seems a bit off to me. How can I now calculate the conditional expected value of the first decile? Shouldn't I use risk-neutral valuation somehow? – Jj Blevins May 21 '19 at 10:41
• $\Phi$ here represents the inverse cdf of a normal distribution. In particular $\Phi^{-1}(0)$ equals $-\infty$ and $\Phi^{-1}(0.1)=-1.281552$. Hence, $a=0$ and $b=0.8217572$. Whether you should compute the risk-neutral dynamics, it depends on your purpose. If you want to price a derivative, then you should do everything under the risk-neutral measure. If you want to determine the expected return of a certain asset portfolio then it is under the real-world measure. – Wiles01 May 21 '19 at 13:32