Today I'll apply the same technique to put options.

#### Put Option Returns

I'll stick with the same assumptions I used in the call option example:

- The stock price returns follow a normal distribution with mean $\mu=8%$ and volatility $\sigma=16%$
- The option is European with no dividends and matures in exactly 1 year.
- The stock currently trades at \$100. (This is convenient to think about the strike price in terms of percentage away from the current stock price, e.g. a strike price of \$90 is 10% away from the current stock price.

Generally speaking, put options offer negative expected returns. This makes sense in light of the fact that put options are sort of like an insurance contract. The writer of the insurance is the one actually taking on the risk.

Oddly though, returns actually start heading positive around the \$68 strike price mark. They then appear to go up to infinity (my calculations in Excel eventually break down.)

Looking at returns versus volatility is probably less informative but here's the chart anyway:

The higher volatility correspond with the lower strike prices (further out of the money (OTM)). It reflects the fact that most of these will expire worthless but there is a small possibility of it expiring ITM in which case returns (in those scenarios) are very large relative to the low price of the put option.

Last, but not least, are the skewness and kurtosis curves.

Like the call option, the positive skew reflects the fact that most of these will expire worthless while a few will give outsized returns. The positive excess kurtosis reflects those fat tails as well.

#### Further Thoughts

I'm not entirely sure what to make out of the fact that expected returns tend toward infinity. Those events, of course, are highly unlikely under the normal distribution assumption. We'd expect about 95.5% of returns to fall within 2 standard deviations (strike price of \$76 to \$140). The probability that the stock would be at \$68 (about where returns start going positive) is only about 0.62%.

Granted, we know that stock returns don't follow a normal distribution but some of that is priced in via volatility smile.

All of this may further suggest that those writing this kind of "insurance" may be mispricing it and taking on more risk than they can swallow.

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