# ITM Puts under negatively skewed return distribution (volatility skew)

I read Hull (2009) on implied volatilies. I understand that (given a negatively skewed return distribution) an OTM-Put is more worth than under a normal distribution and that a OTM-Call is worth less which leeds to the volatility skew for equity options.

ITM-Puts and OTM-Calls imply the same volatility due to put-call-parity.

Is there another inuitive explanation for why ITM-Puts are worth less under a negatively skewed distribution than under a normal distribution?

Thank you in advance!

I think there are 2 approaches being a bit mixed up here.

1. You can analyze the option market by looking at implied volatilities and apply Black-Scholes (BS), thus assuming that log-returns follow a Gaussian distribution. Implied volatilies are the parameters that bring together BS and market prices. Then you will observe a pattern of implied volatilies for varying moneyness. This is called the volaility smile (or skew) meaning that implied volatilities not at-the-money are generally higher than at-the-money. Keep in mind that the Gaussian distribution which is assumed in BS is not skewed. Put-Call parity holds if you look at one strike (e.g with a stock price of $100\$$put call parity holds with a Put with strike 80\$$ (OTM) and a call with strike$80\ (ITM)).

2. On the other hand you can apply another model to price options. E.g. some skewed distribution in a Lévy model. But these models usually have much more parameters than one volatility parameter - maybe even a jump measure. What people do is: calibrate a model to the data and then calculated implied volatilies (for BS) that fit these model (!) prices. These (model-) implied volatilies also have a pattern and a model is assumed to be good if it can reproduce the smile observed at the market.

In this answer I try to write down my thoughts about implied volatilities of BS and skewed distributions of alternative models.

I guess the cumulative distribution function of stock prices at maturity is an intuitive way to understand this (I'm not 100% sure).

Assume you would value an ITM Put by using simulation. If it is worth less than under BS, then the sum of simulated payoffs should be less than under normally distributed returns. This is the case if fewer stock prices below the strike of the ITM put are possible than under normally distributed returns. Hence the line of the CDF of stock prices of a normal distribution should be above the CDF of "real" stock prices.

For OTM Puts the opposite is the case. The CDF of "real" stock prices lies above the CDF of stock prices under normally distibuted returns. Stock prices below the strike of the OTM-Puts are more likely "in reality" which is why it's worth more than under BS.

I don't know if this is right - Maybe an alternative for using put call parity an OTM Calls and transferring it to ITM Puts.