What is implied volatility

The volatility is often used as a measure of risk for a given instrument: if instrument A has a volatility of 10% and instrument B has a volatility of 5%, we consider A riskier than B.

But volatility is a very generic term and can only be estimated. So, different ways to estimate it would likely give a different level of volatility for that same instrument. A market-agreed volatility, instead, is a measure that can be implied from option prices, i.e. from prices that are established by supply and demand in the market.

Options are traded instrument that are directly affected by the volatility of its underlying. To understand why this is the case, it is useful to remember that when we buy an option, we pay a premium and this is the maximum we can lose.

Let’s consider a call option. Once we buy a call, we pay a premium and when the option expires (or is exercised) we receive a positive payoff if the underlying price S is above the strike price K. In the example below, we assume that the strike price is 450 and the premium we paid is 100 USD. The orange continuous line shows our P&L profile at expiry while the dotted line is the T+0 line, i.e. our current P&L profile. Here we can see that while losses are limited to the premium we paid, we start making profit as the market moves above the strike price. At some point, the difference between the underlying price S and the strike K is large enough to compensate the premium we paid and the position will be at a profit.

A similar logic would apply to put options. The final P&L for a call and a put can be summarised below:

Clearly if volatility is very high, we have a higher chance to hit a large P&L on the option because the underlying S might move well beyond the strike K. At the same time, the losses would still be limited to the premium we pay.

The option market compensates for this higher chance of profit by increasing the premium we pay for the options. This mechanism creates a direct relationship between option price and volatilities.

Since option prices are traded in the market, we can use a pricing model to imply the value of volatility that would justify the option price we see. This measure of volatility is called implied volatility.

 

Volatility smile/skew and implied distribution

Differently to all other volatility estimates, this measure is a market-agreed estimate of the expected volatility that the underlying will experience until expiry date. An important point is that this measure is not unique and different strikes will typically have a different implied volatility, even for the same expiry date. As a result, when these implied volatilities are plotted together, they form a curve that is typically not flat. In the equity markets, we usually observe a curve that is skewed on one side, this is why we often talk about volatility skew instead of volatility smile.

But why should options at different strikes imply a different expected volatility for the same underlying over the exact same period of time?

To understand this point, let’s say that we are inverting the Black-Scholes formula (the de facto standard for the pricing of European-style options) to infer the level of implied volatility on each strike for a given expiry date. This model is built on the assumption that the log-returns (a measure for the change in the underlying price) are normally distributed (so the price is log-normally distributed). Under this assumption, we know the probability associated to all possible changes in the underlying price over the period covered by the option and, in particular, we also know the probability associated to extreme movements.

The critical point is this: the normal distribution assumes that extreme movements occur with a relatively low probability if compared to how often these movements are actually observed in the market. In other words, if the model we use to price the option assumes that the underlying movements are normally distributed it is usually under-estimating the risk of potential large movements that might occur.

Traders in the market are aware of this and don’t believe in the assumption of normal distribution. The way they correct for this is by changing the prices of those options that would be mispriced under the assumptions of the Black-Scholes model.

These are typically the OTM options, i.e. those strikes that would be touched only if the underlying makes a large movements. Exactly those kinds of large movements that, under the normal distribution, have very low probability of occurring, but that the market participants expect to happen with a relatively higher probability.

This should help you understand why there is a connection between the volatility smile/skew we observe in the market and the probability distribution of potential future movements in the underlying. The fact that the pricing model makes the wrong assumption about this distribution means that the distribution that is implied from option prices has usually fatter tails, i.e. larger area in the extremes of the distribution (see the chart below for an example).

We mentioned earlier that the curve that is formed by putting together the implied volatilities at all the strikes for a given expiry is typically not flat. Only in a “Black-Scholes consistent” world, where returns are normally distributed, we would see a flat smile.

In the example below, we are looking at the volatility smile/skew on two dates. We can see that from date 1 to date 2 the volatilities at lower strikes have increased further (the negative volatility skew increases). We also see that the smile becomes more convex (i.e. more curved).

 

 

The volatility skew can be monitored by looking at the risk reversal, which is the difference between the volatility at a low delta put (in this example this volatility is 29% on date 2) and a low delta call (24% on date 2). On date 1 they were 26.4% and 22%, respectively, so the risk reversal goes from 4.4% to 5%. The low deltas used are typically 25% or 10%.

A butterfly, on the other hand, is used to monitor the curvature of the smile. In the chart it is the height of the blue and grey rectangles and are calculated by subtracting the ATM volatility from the average of the low delta put and low delta call volatility.

On the right plot we can see the distribution of changes as implied by each volatility smile (here changes are expressed in terms of number of standard deviations from the mean).

Based on what we have seen so far, we can say that changes in the implied volatilities ultimately reflect the expectation of market participants that the underlying might experience a large move with a probability that is different than what a model like Black-Scholes would imply. Any change in these expectations will move the volatility smile/skew and the associated implied distribution will represent the current market expectation of potential movements in the underlying (and this distribution is often far from normal).

But how do we know that one volatility skew implies a probability that is higher than the other? To understand this, let’s look at the distributions in this example. The plot below is a zoom of the left tail in the previous graphs. By looking at the shaded area under these curves we can calculate the probability that the underlying will make a down move higher than 2 standard deviations. This probability is the area under the corresponding curves.

 

 

Clearly this area is higher under the distribution in grey, which is the implied distribution associated with the smile as of date 2. So, the market is expecting a large move like this to be more likely now and this is reflected in volatilities in the corresponding area that have lifted up compared to previous day.

A natural question to ask at this point would be: does the implied distribution have any predictive power? The answer is generally negative.

Typically, if we see for example that the implied probability that the underlying will be below 340 at expiration is 5%, this doesn’t necessarily mean that the market is expecting the underlying to be below that level with a 5% real probability.

There are 2 main reasons for that and we will discuss them below.

The impact of hedging

The first reason, the most relevant, is that implied volatilities are driven by the suppy and demand of options and they can over-react, being driven by dynamics that are not necessarily linked to expectations. To simplify the concept, we can think about the dealers to be in a position where they have sold puts at lower strikes and bought calls at higher strikes. This means that they tend to have a short skew position that can be exemplified by a risk reversal strategy, like below.

 

We can safely assume that a dealer has delta hedged the position by selling the proper amount of underlying (35 units in this case) and that, given the low gamma, this delta hedging does not need to be adjusted that often.

With the delta risk hedged, we can focus on the other major sensitivity: vega. The position has a slightly negative vega, but this significantly changes as the market moves (high vanna exposure). So, when the market starts to sell off and we start to see implied volatilities rising, we also see this short skew position becoming more and more negative vega.

To hedge this higher negative vega exposure, after the market has dropped, the dealers will need to buy vega. Since buying vega means buying options, hence volatility, this hedging activity will push volatilities even higher, particularly at lower strikes.

The skew might be expensive also for another hedging activity taking place. Traders that are usually short volatility might use long skew position, i.e. involving buying puts that are OTM. Should a shock hit the market and volatilities rise, a long skew trade might serve as a hedge for the position.

As we have seen, a skew becoming more negative will translate into an implied distribution that has a left tail that becomes fatter, i.e. the market implied probability of a larger move is increasing. But looking at the reason why volatilities have risen after the market sell off, we know that this might have been driven also by normal hedging activity which doesn’t have much to do with market expectations. For example, a professional trader might be required to hedge the vega exposure even if he expects the markets to recover and volatilities to drop.

So, this example shows that some movements in volatilities are exacerbated by hedging activity that might be triggered by a sudden market move. Using the implied distribution associated to a volatility smile/skew to make any prediction would likely prove not useful, especially after a large movement in the markets.

The risk-neutral world

The second reason why implied distributions should not be used as predictive tools is more technical, and is related to the fact that these are actually risk-neutral probabilities, so not in line with real-world probabilities.

Implied distributions are in this context also called risk-neutral distributions, to highlight the fact that they can describe the likelihood of a given event in a risk-neutral world but not in the real one.

A risk-neutral world is one where all investors are indifferent to risk and don’t require any extra risk premium for the risk they bear. In this world, all assets (irrespective of their risk) will earn the risk-free rate.
Investors’ risk appetite and true/real world probabilities of a given event both play a role in the determination of the risk-neutral probabilities. Since in the real world investors are risk averse, they are more concerned about bad outcomes (for example a market drop), so the associated risk-neutral probabilities are higher than the real ones. Similarly, the implied probabilities associated to good outcomes is lower than the real ones.

This results in an implied distribution that is typically more negatively skewed than the real one. An example could be the grey distribution below. Here we see that the possible values for the underlying at maturity are the same under both the grey (risk-neutral) and the blue (real world) distributions. What changes are the probabilities and the fact that the risk-neutral distribution corrects for the risk aversion that we have in the real world, creates a skewed shape.

 

 

But why we need to use an artificial probability distribution that does not apply to the real-world? The reason is that this choice makes the pricing of options much easier.

Every price is nothing more than the value today of the expected payoff we will receive at maturity. To get the value today, we need a discount rate while to calculate the expectation we need the possible future values of the underlying and the associated probabilities.

The difficult part is to identify the right discount rate to use because this would need to reflect the risk of the position. On the other hand, we know that in a risk-neutral world that discount rate is the risk-free rate. So if we can calculate the expectation by using the risk-neutral probabilities then we can discount that expected payoff at the risk-free rate to get the price of the option.

In practice, to obtain the implied probability density function we can follow these steps:

1. Calculate option prices P at various strikes K by using strikes with a distance ΔK extremely small. If not all strikes are available in the market, use interpolation to find the missing ones.
2. Calculate the difference between consecutive prices ΔP
3. Calculate the ratio between ΔP and ΔK (this can be seen as the first derivative of the price with respect to strike)
4. Calculate the difference between consecutive ΔP/ΔK (as calculated in previous step)
5. Use the difference from previous step and divide by ΔK

At this point, by plotting the results from step 5 against the strikes, we can see the probability distribution as implied by the prices we used.