If each option's strike price has different implied volatility what does IV means?

Question

After spending 2 days researching I realized not many people know how to answer this, and most answers going around and not direct.

I know that 20% implied volatility means 1 standard deviation probability that the underlying asset will move 20% from the current price.

Now, if you look at the chain of options with the same expiry, each strike price of the same option has a different implied volatility next to it on any platform.

I know that this volatility is an output from the Black Scholes Model, but if one strike has 17% volatility, and another has 26% (skew), then what does it mean about the stock?

Does it mean the stock can move 17% within 1SD? Or does it mean 26%? If not, why then does delta matter? (Delta is a product of IV).

I can't find a direct explanation of what this number means, instead I can read many theories on how we calculate it and why it is different. But what does 17% volatility for a certain strike mean? 17% of what?

EDIT:

This is a screenshot from yesterday:

The left red mark shows that each strike has a different IV. The right red mark shows some sort of generic IV which is not clear how all platforms calculate (can't be BSM because it requires a strike and this one is generic IV).

1. If each strike has an IV, and IV depends on the strike price, then what is the general IV of an option with an expiry at x?
2. If each stock has options with literally all types of strikes (and IV's) with the same expiry, then what conclusion can we draw based on that? It is always true that many people think many things. Can we make some average?

Answer 1

Implied Volatility is, basically, a signal what an option's price means, in terms of the market's opinion [ie: the most recent traded price] of the likely magnitude of future price changes that would reach a given strike price.

Simple terms:

Assume a rock-steady stock trades at $100, that most people expect will continue at the same price forever. For such a stock, buying a put option expiring this month with a $90 strike price is almost worthless - because there is barely any chance that the stock's price drops 10% in the next few weeks. So, we should expect that the price for this option should be tiny. Another way to say this same fact: "Because the price for this option is tiny, we can assume that 'the market' thinks it is very unlikely to end up in-the-money".

Mathematically, this is shown as a low IV. Another way to say this same fact: "Since this option was last traded for a tiny price, the buyer and seller of that trade are each implying that the volatility of the stock is low".

The actual IV listed for a stock option is a reflection of its most recently traded price. It is not some analyst's opinion of how the stock is doing, it is purely a mathematical function using the Black Scholes model, given all current factors (strike price, option price, time to expiry, and the current stock price). If someone overpays for an option, it will drive the calculated IV upward, basically saying "The last person who bought this option thinks the stock is incredibly volatile".

So why is IV listed as if it is an indisputable fact on an options chain? Because it creates something closer to an 'apples to apples' comparison of the price of one strike price vs the next. If you look at the pricing on an options chain, you will typically see IV's quite close to eachother, implying that buyers and sellers of all those options generally agree on the likely volatility, until you get far out of the money, when there is likely to be a gap / jump in IV. This often occurs because liquidity on those options is incredibly low, so there is a bigger spread between what a buyer is willing to pay, and what a seller is trying to demand.

So what does it mean if one option has IV of 17% and the next one has IV of 26%? Basically, it means that either:

(a) one of those options was most recently traded for a bad price [either too high or too low, relative to what similar options traded for]; OR

(b) the change in strike price is enough to fairly imply that the bigger change is far less likely to occur. As an example, assume a real estate company owns land that it could immediately sell for $1B. Based on the current revenues earned by its rents, assume the market values the company at $1.1B. This could indicate that it would be hard for the company to ever drop below $1B in value, because the company has the option to sell its land for $1B and liquidate funds to its shareholders. In a scenario like this, the IV might be relatively high for options with a strike price of $1.05B [saying this is a more likely price drop that could occur], and lower for options with a strike price of $950M [saying a drop below $1B in total share value is very unlikely to occur].

Final, critical note - options are far higher risk than simple diversified index purchases, or even manually selecting stocks to buy - please do not trade options if you don't know what you're doing, it is often closer to gambling than investing for a naïve trader.

Answer 2

20% implied volatility means 1 standard deviation probability that the underlying will move 20% from the current price

That interpretation is correct if you assume that returns are normally distributed. A non-constant vol means that the market disagrees with that aspect of the B-S model. The market sees the probability distribution of prices at expiry as "fat-tailed", meaning a slightly lower probability than "normal" at the current price and higher probabilities than "normal" at the tails.

One interpretation of the vol skew (or smile) is that the market puts higher probability on "big" moves than a "normal" distribution would imply, so the "implied vol" is higher when you get farther than at-the-money. So options that are far out of the money have a higher change of getting struck, thus a higher implied vol.

As far as delta, it just measures the sensitivity of the price of the option to movements in the underlying. If you have a delta of .40 and the underlying moves up 0.2, then the price of the option should move roughly 0.08 (0.4 * 0.2). However, delta is not constant either, so the actual price of the option may move more or less than 0.08 depending on higher-order effects (like Gamma, which is the sensitivity of delta to changes in the underlying).

So what does an "18% IV" mean? well you can still think of it as a measure of how much the option is expected to change, and it can be used comparatively, meaning a stock with a higher IV is expected to be less stable, and will have options that are "more expensive" that stocks with lower IV.

Answer 3

Researching for two days is not really that long. It is an interesting topic though for sure. @D Stanley's answer is spot on. I just like to add (likely TL;DR) details to suit the comment

Now, someone could maybe come up with a more accurate representation of that probability distribution, but the calculus might not be as neat as it is with a "normal" distribution.

There is no general IV for an option. Quoting from Just What You Need To Know About Variance Swaps - JP Morgan Equity Derivatives

For each strike and maturity there is a different implied volatility which can be interpreted as the market’s expectation of future volatility between today and the maturity date in the scenario implied by the strike. For instance, out-of-the money puts are natural hedges against a market dislocation (such as caused by the 9/11 attacks on the World Trade Center) which entail a spike in volatility; the implied volatility of out-of-the money puts is thus higher than in-the-money puts.

What most people refer to if they look at a single vol is the ATM volatility. Sometimes also a VIX like measure. The VIX is the square root of the variance swap strike, computed via SPX options that are listed for trading on the Cboe. This answer explains what this computation looks like. The gif from the answer looks like this:

Lacking far OTM quotes for single stocks means that this is usually ruled out I would say. Either way, the VIX and a 1m ATM IV is very similar as can be seen here. How IV percentile is computed can only be guessed without a documentation. I suspect it will be simply the ATM vol for the selected expiry compared to some set number of past days. So the 19% in your example means that ~20% of days had lower, ~80% of days had higher ATM IV for that expiry in the observed time span. That it is close to current IV is a coincidence in this case, because it could be any number between 0 and 100.

Some people interpret IV as a forward looking measure of standard deviation, just like the commonly used definition of historical / realized vol which is computed as the sample standard deviation of log return as shown here. However, one should be cautious when comparing IV to historical vol (HV) - also called realized volatility (RV) - because it is not necessarily useful for at least two reasons:

1 ) Empirically, IV tends to overestimate RV, commonly referred to as Volatility Risk Premium

2 ) IV is the only free parameter in the Black-Scholes-Merton (BSM) model. Higher IV can be a result of compensation for tail risk.

A simple explanation is that market participants tend to overestimate the likelihood of a significant market crash, which results in an increased demand for options as protection against an equity portfolio. This can be exploited, as for example demonstrated in Sullivan, R., Israelov, R., Ang, I., & Tummala, H. Understanding the Volatility Risk Premium. The authors show that the returns of an investor who sells the same 5% out-of-the money put option every month, delta hedges it and holds it to expiration generated 1.5% annualized returns with a Sharpe ratio of 0.68. Compared to the S&P Sharpe Ratio of 0.32 over the same observation period (1996-2016), this is an attractive strategy.

What is IVOL?
IVOL is turning an option price into a comparable number (it’s also annualized). The theory to construct IVOL is based on the world of Black Scholes (its assumptions). 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

The intuition is the same for all sorts of markets. However, FX is very helpful in getting an understanding of it. Ignoring all details, FX is quoted in IVOL, the quotes come as ATM DNS (delta neutral straddle), RR (Risk Reversals) and BF (Butterflies). 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 puts for RUB and GBP in the examples below) and
• BF the kurtosis (how pronounced the general wings are).

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. Using the vol quotes from above, one can compute strikes (for simplicity I assumed delta premium excluded to avoid using root solvers), back out option prices, and compute risk neutral implied probabilities for the underlying. I use the method shown by Malz in the Fed Staff Report No. 677 on June 2014 A Simple and Reliable Way to Compute Option-Based Risk-Neutral Distributions. I modified it a bit because the strikes derived from delta quotes do not lie on a uniform grid (they do not have constant spacing), in which case a more general formula for weighting is needed. All computed prices are monotonically decreasing, showing that the results are free from vertical spread arbitrage opportunities.

That way, it is easy to show how the quotes indeed affect the implied return distribution of the underlying.

A few observations:

• increasing ATM vol moves the vol surface up, and spreads out the RN probability distribution
• increasing BF quotes moves the tails out significantly
• a negative RR quote increases the left tail, a positive the right tail

Since the Russian invasion of the Ukraine, it is more likely for the RUB to depreciate compared to the USD. If you look at a vol surface skew now, compared to a date prior to the invasion (where no one was expecting it yet), you see that the skew is a lot more pronounced now. It always existed, because the USD is generally more stable. Also, the overall level of IVOL increases too, but the main take-away is that the skew got a lot larger. To the right, we have OTM calls (on USD, which is a put on RUB, hence buying protection against a RUB depreciation is more expensive now).

Similarly, if you look at GBP, you have Brexit as a major event. Uncertainty meant that IVOL not only anticipated the higher realized / historical vol (shown on the left screen below which is from Bloomberg's VOLC, comparing ATM 1M IVOL with realized 1M vol), but also meant that it was heavily skewed towards OTM Puts (on GBP) as displayed by the risk reversal quote over time. Realized vol takes time (1m in the example) to be computed, whereas IVOL is forward looking. That is why the spikes of the white line are before the red.

You can see that Brexit, Covid and the Russian invasion all elevated ATM IVOL (left), but the effect on the skew was a lot more pronounced during Brexit because that was a GBP specific risk. Below is a screenshot of the smile on the day of Brexit and during normal times.

Long story short, different IVOL for different strikes accounts for the possibility of larger outliers and skewness in the return distribution, thereby accounting for a short coming of the Black Scholes model.

How to compute a vol surface

With regards to your comment to @Grade 'Eh' Bacon where you wrote

you still did not answer how the total IV surface is calculated

What you look at is not a IV surface. It is simply an isolated computation for the given quoted options. You have the inputs into Black Scholes, and solve for IV, nothing more. Although doing this for an American option is not that simple after all.

If you wanted to get a vol surface, you would do something along the line of:

Filter out stale prices, unreasonable small or large bid/ask spreads, figure out a way to handle that the underlying asset and options on the asset can trade during different times on different exchanges, that there exist erratic prices, especially close to the opening and closing times of the trading session... Sorting all these difficulties out, still leaves a major problem. There is no consensus on how to model (cash) dividends, and you have borrow costs/funding costs. You can use a (continuous) dividend yield, cash dividends (implying that the observed stock price cannot follow geometric Brownian motion...), discrete proportional (discrete stock) dividends and so forth. Typically, a blend is used that uses different types for different maturities (due to lack of better data mainly).

Put another way, the problem is that the forward price is not directly quoted for listed options markets. Yet is an important quantity impacting option prices and implied volatility. Futures may be quoted but maturities frequently do not coincide with (all) option maturities. Interpolation is not trivial because future dividends (even times of payments) are usually unknown. Therefore, most practitioners (in my experience) use vanilla equity options to back out (implied) dividends. As stated in the question, the problem hereby is that put-call parity for American-Style options does not hold. Even for European-Style options, there are frequently issues with different trading times and erratic option prices etc. Since dividend payments are discrete in nature, you require a dividend schedule with dates and amounts to derive an implied dividend curve. This curve is usually noisier than one would hope for and commonly smoothed (via Kalman filter of the like).

Ignoring all these problems, de-americanization is a major building block and can be done like this:

1. Estimate the forward using spot, interest rates, dividends (forecasts) and ideally borrow costs/funding costs.

2. Use a local volatility model consistent with options of all strikes at this maturity. Alternatively, and a lot faster and simpler, compute the Black implied volatilities from put and call prices by using the estimated forward (implied forward in future iterations as explained below). This usually requires a PDE solver (or you use a binomial/ trinomial tree again). Lastly, use the estimated forward and the implied volatilities in the Black model to compute the corresponding European option prices.

3. Imply the forward and dividends using the two nearest strikes on either side of the estimated forward from step 1 (make sure you quotes are reasonable for both puts and calls). The implied forward is the average of the forward obtained at each of the two strikes by applying put-call parity to the European option mid prices computed in 2), and back out a corresponding implied dividend.

4. Evaluate the results:

• Does the implied forward lie in between the two strikes? This is a minimum requirement.
• How different is the forward from the estimated forwarded (or implied if several steps are needed)? You will need some criteria to determine the optimal stopping.

5. If the result in step 4) is not satisfying, use the current implied forward in step 2) and iterate until you are satisfied with the result.

These steps will give you Pseudo European option prices, option implied forwards and option implied dividends that are consistent with the observed American prices.

The calibration of the actual vol surface should result in the chosen (Pseudo) European option prices (only OTM, sufficiently liquid...) at any maturity to match the prices from the vol surface.

Afterwards, you will have a set of implied vols for moneyness levels, which you can fit using various techniques like SVI, SABR, or a mixture of lognormals to get a curve that best fits the existing IV and allows you to get full fledged surface for any strike and expiry.

Below is a quick SVI implementation in Python example:

spot = 1.34
forward = 1.35
t = 30 / 365.0
vols = np.array([ 12, 10, 9.5, 9, 10.5, 8, 10.24, 9.6, 11.2, 9.4, 11.9, 9.7, 20, 23,  27]) / 100
strikes = np.array([1.21, 1.3, 1.4, 1.3, 1.3, 1.32, 1.38, 1.3,
                    1.4, 1.3, 1.45, 1.25, 1.5 , 1.6,  1.8])
total_implied_variance = t * vols ** 2

def svi(k, param):
    a = param[0];
    b = param[1];
    m = param[2];
    rho = param[3];
    sigma = param[4];

    totalvariance = a + b * (rho * (k - m) + np.sqrt((k - m)** 2 + sigma**2));
    return totalvariance



def targetfunction(x):
    value=0
    for i in range(11):
        model_total_implied_variance = svi(np.log(strikes[i] / forward), x);
        value =value+(total_implied_variance[i]  - model_total_implied_variance) ** 2;
    return value**0.5

bound = [(1e-5, max(total_implied_variance)),(1e-3, 0.99),(min(strikes), max(strikes)),(-0.99, 0.99),(1e-3, 0.99)]
result = optimize.minimize(targetfunction, bound, tol=1e-8, method="BFGS")
x=result.x

K = np.linspace(-0.4, 0.4, 60)

newVols = [np.sqrt(svi(logmoneyness, x)/t) for logmoneyness in K]
plt.plot(np.log(strikes / forward), vols, marker='o', linestyle='none', label='market')
plt.plot(K, newVols, label='SVI')
plt.title("vol curve")

plt.grid()
plt.legend()
plt.show()

SABR calibration would involve fitting beta, alpha, rho and nu so that it resembles the shape of IV in the market. You can see details and the below illustration here.

Lastly, I am not convinced about the real estate example and liquidation. Real estate is usually trading far from book value, and even the companies in the S&P500 have illiquid options markets. However, there are 14 companies in the S&P500 that trade below book value according to Bloomberg's API and the field PX_TO_BOOK_RATIO. All these have IV of deep OTM puts significantly higher than ATM, and increasing all the way.

After all, the scenario implied by such low strikes will almost surely see massive haircuts for its liquidation value as well.