How a Heavy Left Tail Affects Risk-Neutral Probabilities

Summary

The question concerns the interpretation of equity option volatility skew: why a heavy left tail in the risk-neutral distribution can be described as shifting probability mass toward higher stock prices, rather than simply increasing the chance of low prices. The response starts from the risk-neutral pricing constraint that, with a zero risk-free rate, the expected future asset value equals its current value.

It illustrates the idea with a one-period stock-price model that initially has an up and a down outcome, then adds a small-probability, more severe downside outcome. To preserve the same expected value, probabilities assigned to the other outcomes must adjust. This setup is intended to clarify the counterintuitive probability shift. However, the excerpt stops before working through the adjusted probabilities or explicitly resolving the original question, so it offers only the core setup and constraint rather than a complete derivation. It does not establish that implied volatility directly equals physical probability.

Key ideas

• Under risk-neutral pricing with a zero rate, the expected future asset value equals its current value.
• Adding a severe downside outcome requires adjusting other risk-neutral probabilities to preserve that expectation.
• A heavy left tail can therefore affect probability assigned to higher outcomes as well as lower ones.
• The provided explanation is incomplete and does not show the probability calculation.

Tags

# Downward-sloping volatility skew in equity prices


# Downward-sloping volatility skew in equity prices












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 by user34971 (score 2)

https://quant.stackexchange.com/a/49755

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).

- 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?

- 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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.