# Option with company earnings as underlying

I need to calculate the fair value of an option, with the underlying being the earnings of a listed company.

I believe the best way to achieve this is to simulate the earnings of the company and I want to do it as follows:

1. Simulate the share price using standard GBM.
2. Simulate the PE ratio using a mean reverting AR process

From these two simulated values, I can calculate the earnings and from there the payoff using a Monte Carlo simulation.

I think this a a good approach, but if there are better methods, please let me know.

My question relates to the correlation between the share price and the PE ratio. Is it fair to assume that they are not correlated, since PE can be seen as the market's rating of the company?

• While P/E ratio is obviously correlated to price, keep in mind that the future price is going to be based on future earnings, not past earnings. – amdopt Aug 2 '17 at 13:45

I think it is reasonable to assume the two are uncorrelated, since P/E ratios $R$ do not vary a lot and tend to have more to do with the fundamentals of a given industry than the earnings levels of particular companies in them.

Assuming the two are uncorrelated lets you do your pricing using trapezoid-rule quadrature of the Black-Scholes formula. (You weren't going to use Monte Carlo, I hope.)

$$V = \int_R BS(P/R, \sigma_P) p(R) dR$$

Note that if $p(R)$ is gaussian (as in the terminal distribution of a typical AR or OU process) you'll have to cut off the bounds of integration.

Personally I would not bother assuming anything other than a constant P/E ratio $R$. I would just goose the equity volatility $\sigma_P$, which after all is just a guess at future volatility, and call the job done.

• What does it mean to "goose" the volatility? – Chris Taylor Aug 2 '17 at 14:45
• Thanks. Why shouldn't I use a Monte Carlo for this? – Johan Aug 2 '17 at 18:39
• @Johan: Monte Carlo is slow and inaccurate compared to adaptive quadrature schemes for one-dimensional integrands. – Brian B Aug 2 '17 at 18:46
• @Chris: Oops, I should have watched my word choice. To "goose" is North American dialect for "boost". – Brian B Aug 2 '17 at 18:48
• So, if the PE ratio is constant, then it will just be a standard Black-Scholes formula? – Johan Aug 2 '17 at 20:00

I assume that you are talking about the changes in P, and P/E being uncorrelated, rather than the values themselves, since you are going to be simulating the increments in your GBM or AR process.

It seems reasonable to assume that increments in price and earnings are correlated (i.e. if earnings increase, the price will probably go up). And since P/E is a simple function of P and E, its relationship with P will be determined by the relationship between P and E.

The increments of P/E are given by

$$\Delta (PE) = \frac{P}{E}\left( \frac{\Delta P}{P} - \frac{\Delta E}{E}\right)$$

and so (abusing notation a bit) the covariance between P and P/E is

\begin{align} \mathrm{Cov}(\Delta P, \Delta PE) & = \mathrm{Cov}(\Delta P, \Delta P) - \mathrm{Cov}(\Delta P, \Delta E) \\ & = \sigma_P^2 - \rho \sigma_P \sigma_E \\ & = \sigma_P \left( \sigma_P - \rho \sigma_E \right) \end{align}

therefore the increments in P will be uncorrelated with the increments in P/E if you have $\sigma_P = \rho \sigma_E$.

Since price volatility is generally higher than earnings volatility ($\sigma_P > \sigma_E$) and correlation is obviously less than one, this relationship will not hold in general, and you should expect the increments of P/E to be positively correlated with the increments of P.

Edit: I confirmed this with data, using quarterly price and earnings data for the S&P 500 from 1970 to present, and observed around an 80% correlation between log changes in price, and log changes in price-to-earnings.

• Did you look at daily data or only when the earnings get updated? I would imagine that you should only look at the data when the earnings gets updated? – Johan Aug 2 '17 at 18:41
• I looked at quarterly data, assuming that most companies have quarterly earnings announcements. I didn't make any attempt to line up the dates that I sampled the price on with the most common earnings announcement, however. – Chris Taylor Aug 2 '17 at 21:12

PE is literally price / earnings. You certainly must assume that price and PE are correlated.

• My argument is that if earnings and price are highly correlated, then the two divided by each other will not necessarily be correlated as well? – Johan Aug 2 '17 at 13:15
• I saw your edit. My assumption was that you're looking at daily data. In that case changes in PE will be largely driven by changes in the numerator. But quarterly - I agree . – onlyvix.blogspot.com Aug 3 '17 at 11:11