Why Implied Volatility Varies by Strike and Includes Risk Premia

# Expected Forward Volatility vs. Different Strikes


# Expected Forward Volatility vs. Different Strikes












While theoretical options prices are derived from models, such as Black-Scholes, IV and IV skew reminds us that options prices are ultimately based on supply and demand. My question is the following: how can we claim that implied volatility measures the expected volatility of the underlying during the life of the option, when the IV varies by strike? If we are specifically talking about the ATM volatility, which I assume, how do we interpret the IVs of other strikes — especially as we move further into or away from the money? Finally, how do we interpret IV in an illiquid market, in which option prices don’t change for, say, weeks at a time? Do we say “there has been no change in the volatility expectations” — other than theoretical changes due to DvegaDtime — or do we just deem the data stale and irrelevant?

## Answer by Jan Stuller (score 4, accepted)

https://quant.stackexchange.com/a/58756

Nice question. My interpretation is via the concept of a risk premium (i.e. risk adversity of market participants).

Let me introduce the concept of a risk premium first via US corporate bonds: one can observe that the credit spread of these bonds increases as the credit quality decreases. However when looking at actual historical realized defaults of corporate companies in the US across the various credit-quality buckets, one can see that the realized default frequencies are lower than the credit premiums charged: in other words, the expected returns (under the real world probability measure) increase as you move down the credit quality. That is because investors need that extra premium to invest in these lower credit-quality bonds, to be compensated for the increase in risk, specifically tail-risks related to defaults of poor quality bonds during stressed events.

With options, the option writers demand a similar risk premium for writing options that contain a "tail risk": that's why you'd typically observe a higher IV on OTM puts on equities (because there is a tail risk related to stress events such as Covid, for which the put option writer wants to be compensated). You'd observe a similar increase in IV on OTM Call FX options written on USD vs. emerging market currencies (i.e. USDTRY): because here, it would be the OTM calls that are exposed to tail-risk events.

Last but not least, even ITM options would contain some risk premium: imagine that the IV priced into the written option would only reflect the actual expected future realized volatility; then the option writer would make zero money by delta-hedging the option. The option writers demand a premium in general for writing the options & managing the related risks via hedging. That's why I'd always argue that IV is a not a good measure of market's expectation of future volatility: because of risk premiums, the IV always overstates the expected future volatility.

## Answer by Aditya Gupta (score 0)

https://quant.stackexchange.com/a/76114

great question. Sorry I am quite late to answer this, and if it is still relevant for you. I think the use of "market implied" IVs is to extrapolate a pdf curve for the underlying. As black Scholes assumes constant IV and log normal distribution. When you trace back the IVs to create a probability distribution of the underlying, and then you can do all sorts of speculative analysis of the underlying moves.

## Answer by Arshdeep (score 0)

https://quant.stackexchange.com/a/76117

You don't claim that implied vol measures the expected realized vol. This is completely theoretical and hinges on the ability to delta hedge continuously. In reality there is a PnL leak due to discrete hedging and vol autocorrelation. In order to make up for this residual PnL variance, one might as well charge an extra premium.

This is why implied vol value contains a "gotta make up for less than perfect hedging" premium. The exact value depends on strike (because the PnL variance depends on gamma which depends on strike).

Potentially one could do a PCA of the implied vols and that factor should actually track the realized vols better in my opinion.

Movement of implied vol of the same contract is an admission that asset dynamics are not lognormal and just that. The real quantity, $Vol(dS/S)$ is naturally non constant and the expectations reflect that.