Downward-sloping volatility skew in equity prices

Question

I’m learning the market price for FRM, and I’m having a hard time understand a question in the assessment:

From my understanding, the volatility skew for equity is the graph on the right upper corner:

So with strike price going up, the implied volatility goes down, and the equity option price goes down due to less need for “protection”.

But my question is: the explanation says “a heavy left tail puts more mass or probability on the up side or higher side of stock prices.” Why is this? Doesn’t a heavy left tail— like the graph in the first picture, show a higher probability for lower strike and lower stock price? So why is B wrong?

Thank you!

Answer 1

The question is not tricky, FRM just makes it unnecessarily complicated. The answer and hopefully understanding follows from the following steps.

Let's assume for simplicity but without loss of generality that risk-free rate is 0.

Key idea: under the risk-neutral measure, no matter the shape of the risk-neutral measure, the expected value of the asset is it's value today (because zero interest rate).

1. Suppose a simple one period model for asset prices. Let's say today the asset price is 100, and tomorrow it can either go up to 110 or down to 90. What are the risk-neutral probabilities for the asset going up to 110 and going down to 90?

2. Now let's introduce 'skew' by assigning a probability of 5% that the asset can also go down to 70. What are now the risk-neutral probabilities for 90 and 110?