# Today’s spreadsheet modeling of risk neutral vs. true probabilities

My blog posting entitled “Actual versus True Probability that an option expires in the money” provides the necessary details for calculating risk neutral and true probabilities related to lognormally distributed asset prices and options which are based on such assets.  Here’s a screenshot of the spreadsheet that we worked on today:

In this spreadsheet, the Asset Price (S), Exercise Price (K), $\mu$, $\sigma$, r, the annual mean log return (${\mu - .5{\sigma ^2}}$), and T are the problem parameters.  So we are studying a European call option which is currently in-the-money, and wish to determine its price, as well as the risk neutral and true probabilities associated with expiring in-the-money.  I also provide the price of an otherwise identical put option, and quite trivially, the probabilities (risk neutral and true) of the put expiring in-the-money are just equal to 1 minus the corresponding call option probabilities.

The ${d_1}$ and ${d_2}$ values work out to  .6664 and .4164, respectively.  Using the normsdist function in Excel, these ${d_1}$ and ${d_2}$ values imply an option delta of .7474, and a risk neutral probability of expiring in-the-money of .6615. The true probability of expiring in-the-money is obtained by calculating $N\left( {\displaystyle\frac{{\ln \left( {S/K} \right) + \left( {\mu - .5{\sigma ^2}} \right)T}}{{\sigma \sqrt T }}} \right)$, returning .7569.