# Implied volatilities for different options that track the same stock

I have a somewhat basic question regarding option prices.

Suppose we have an underlying stock and two different options (that have different strike prices, maturities, etc.) that track this stock. What factors can cause these options to have different implied volatilities?

One intuitive reason is I've come up with is maturity$$-$$a longer maturity option should price in longer-term volatility than the shorter maturity option.

What are some other factors?

• They moneyness of the option - a way out of the money put might be very different from an in the money call. Feb 1 at 12:50
• For a given underlying and option type, implied volatility is indexed (and quoted) by time to maturity TTM and strike / moneyness M (as @DimitriVulis pointed out); at least in the stock / index option space that you asked about. Feb 1 at 12:56
• stock options have volatility surfaces; If the underlyings are swaps rather than stocks, then the volatility cube has dimensions: moneyness, time to option's expiry, time to underlying swap's maturity. Feb 1 at 16:35
• If all options had the same implied volatility that would mean that the volatility of the stock is constant. This is not the case as numerous empirical studies have confirmed. The volatility of the stock depends on many factors such as news, sentiment, macro and micro drivers, market activity etc. A non-constant (i.e. unpredictable) volatility for the stock leads at the very least to a vol term structure and a smile/skew. Feb 1 at 19:48
• @DimitriVulis Can you elaborate more on why/how different moneyness leads to different implied volatilities? castella08 outlined one reason below, but I'm interested if there are others. Feb 1 at 21:05

It is not simply supply and demand. There was supply and demand for options before October 1987, yet, options traded largely without a skew. Supply and demand plays a role, but its a secondary consideration.

Implied vol is a way of turning an option price into a comparable number (its annualized for example). The theory to construct vol is based on the world of Black Scholes (its assumptions). That holds for all sorts of markets. If you construct vol surfaces from equity markets (predominantly American and price quoted), one usually de-Americanises the option before building a vol surface. Many OTC markets (e.g. rates, FX) mostly quote directly in IVOL.

Black Scholes implies normally distributed stock returns, whereas real (stock) returns are negatively skewed and have fatter tails because:

• stocks (or other underlyings) tend to move down faster than they move up, so the left side has a fatter tail than the right side - known as skewness

• extreme price movements in both directions ( called outliers) are more common than the normal distribution suggests, so both tails are fatter than a normal distribution would suggest - known as kurtosis

Traders use different vols to price options (on stocks) where returns tend to move differently than the normal distribution suggests, to more accurately represent the stocks movement.

FX is the "cleanest" way volatilities are quoted in my opinion. The quotes come as ATM DNS (delta neutral straddle), RR (Risk Reversals) and BF (Butterflies). The screenshots below will ignore quite a few technicalities; ATMD will not be 50D as the stylized example suggests, and is be explained here. IVOL itself affects the value of delta and many fx pairs are quoted as delta premium included. BF itself can be quoted differently.

In a nutshell,

• ATM determines the level (you can think of it as the Black Scholes IVOL for a specific tenor),
• RR the skew (how its tilted, towards OTM pits in the example below) and
• BF the kurtosis (how pronounced the general wings are).

Using a somewhat simplified method from Malz, one can quickly demonstrate this with a few lines of code in Julia.

Black Scholes looks like this (RR and BF set to 0).

A 25D RR is quoted as 25D Call - 25D Put vol; $$RR_{25D}=C-P$$. If OTM puts are more expensive (higher vol), it means that the RR is negative and the VOL surface is higher for OTM puts (to the left).

A 25D BF is quoted as $$BF_{25D}=((C+P))/2-ATM$$ in vol quotation. If the average of Call and Put vols is higher than ATM (smaller is very rare), you account for kurtosis (fatter tails on both sides). I set RR to zero for convenience here.

Putting it all together creates the VOL surface.

Hence, the vol surface exists mainly because there are fat tails, skewness, heteroscedasticity, jumps (crashes), and so forth. None of these real world phenomena are featured in the Black Scholes formula. The market just developed ways to account for many of the shortcomings of Black Scholes.

All these factors (as well as supply and demand) increase the market price of OTM puts (and calls, depending a bit on markets though) and therefore translates into higher implied volatilities.

Maturity is the "cousin" of volatility, it is almost everything. Other item to think about include is it an American or European exercise? But its all about volatility and time, all else being equal.

The causes are simply the market feeling and the way options are used in the market, and nothing more than just supply and demand.

For example, it is usually assumed that low-strike options are more expensive (have a higher implied volatility) than those with higher strikes. The reason is that many big players in the market use out-of-the-money puts to hedge against big drops in the market. In other words, many people willing to buy OTM puts push their prices higher and that translates into higher implied volatilities (note that the volatilities quoted by the market are just a way of quoting the prices of the options).

Something similar could be claimed for different maturities, usually the shorter term ones are traded with a higher volume than long-term ones and therefore the difference in price (and therefore volatility).

• Supply and demand is a secondary consideration for IVOL. Feb 2 at 22:13