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- Volatility Derivatives 1
- The world of Structured Products 4
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- Table of Contents
- Vanilla Options
- Volatility, Skew and Term Stru
- Option Sensitivies: Greeks
- Option Strategies
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- Volatility, Skew and Term Structure

# Volatility, Skew and Term Structure

__4. Volatility, Skew and Term Structure__

**4.1. Volatility**

**4.1.1. Realized Volatility **

Realized volatility (= historical volatility) is a measurement of how much the price of the asset has changed during a period of time. Often, volatility is taken to be the **standard deviation** of the movements in the price.

We can specify the period over which we want to look at the realized volatility, and the frequency of price observations, although often volatility is computed using the daily closing prices of the asset over a year, the **annualized standard deviation**.

Both volatility and variance are useful measurements with each having its own advantages. **Volatility** as a standard deviation is a good measurement of the price variability of an asset because it is expressed in the same units as the price data, thus making it easier to interpret.

A higher volatility means more uncertainty about the size of an asset’s fluctuations and, as such, it can be considered a **measurement of uncertainty**.

Volatility is **dynamic** and changes a great deal over time. It experiences high and low regimes, but that it also has a **long-term mean** to which it reverts. Also, as a stock market witnesses a large decline, volatility shoots up: we therefore generally see a **negative correlation between such assets and their volatilities**.

**4.1.2. Implied Volatility **

The IV of an asset is a representative of **what the market is implying in terms of volatility**.

If one looks at the prices of liquidly traded instruments, one can extract an IV. In fact vanilla options are quoted in terms of their IVs since this, or a given price, amounts to the same information. The IVs are in fact the **market’s consensus on the forward-looking volatility of the asset**. This IV incorporates the forward views on all market participants on the asset’s volatility.

**Implied volatility and realized volatility do not necessarily coincide**, and although they may be close, they are typically not equal.

Using the correct IV of an asset allows one to price other derivatives on the asset, in particular those that are not liquidly traded. Where the IV of an asset cannot be implied from traded instruments, one may resort to using the realized volatility as a proxy for IV to get an idea of what volatility would be correct to use. In contrast, the realized volatility of an asset can be used as a sanity check to ensure that the IVs being used make sense. The two are different, with **IV generally being higher than realized volatility**, but too far a spread could imply a mistake, or if correct, an arbitrage opportunity.

**4.2. The (Implied) Volatility Surface**

The **IV surface** is the **three-dimensional surface** obtained when one plots the market IVs of European options with different strikes and different maturities.

By fixing a maturity and looking at the IVs of European options on the same underlying but different strikes, we obtain the **IV skew / smile**.

Fixing a strike of options on the same underlying and looking at their IVs, we see what is known as the **term structure of volatilities**.

**4.2.1. The Implied Volatility Skew**

**Although the volatility skew is dynamic, in equity markets it is almost always a decreasing function of strike**. Other asset classes such as __FX and commodities have differently shaped skews__. To say there is a skew means that European options with low strikes have higher IVs than those with higher strikes.

There are **several reasons** for the existence of this IV skew. Put options pay on the downside and are thus good hedging instruments against market crashes. If an asset drops in price, this is generally accompanied by an increase in its volatility. In this case **fear** manifests itself because of the increased uncertainty and risk involved in such a drop. This is reflected in the IVs of the OTM puts being higher than the OTM calls because puts pay on the downside. The market also tends to consider a **large downward move in an asset to be more probable than a large upward move**. A **downward jump also increases the possibility of another such move**, again reflected by higher volatilities. Additionally, one can discuss the **leverage effect**: a leverage increase given by a decline in the firm’s stock price, with debt levels unchanged, generally results in higher levels of equity volatility.

Note that the fact that if markets go down, they tend to become more volatile does not explain the skew as this realized vol is the same regardless of any strike price.

The existence of skew is actually saying that this increase of vol has a bigger impact on lower strike options than on higher strike options.

The reason behind skew becomes apparent when thinking in terms of **realized gamma losses** as a result of rebalancing the delta of the option in order to be delta hedged.

In downward spiraling market, the gamma on lower strike increases, which combined with a higher realized vol causes the option seller to rebalance his delta more frequently, resulting in higher losses for the option seller.

Therefore option sellers of low strike options want to get compensated for this and charge the option buyers by assigning a higher IV for these options.

*Measuring and Trading the Implied Skew *

The first thing in measuring the skew is to note **its level**, which is given by the **ATM IV**. The word skew is also used to refer to the slope of the IV skew. Equity markets have a negative skew since his slope is negative.

Assuming we had the set of IVs as a function of strike σ_{Implied(K)}, then the slope is given by the first derivative, at a specific point, possibly the ATM point.

In reality, we only have IVs for a discrete set of strikes. One can use some form of interpolation to obtain the function σ_{Implied(K)} in order to have a parametric form, but in practice, and to have a standard method of measuring skew, we take the **difference between the IVs of the 90% and 100% (or 110%) strike vanillas**.

If we compare the IV skews of an index and that of a stock we find that **index volatilities are more skewed than those of a single stock**. The reason for this is that if stocks are all dropping during a market decline, the realized correlation between them rises, and an equity index is a weighted average of different stocks.

This is a useful property as one can use the skew of an index as a proxy for pricing skew-dependent payoffs on stocks whose implied skews are not as liquid as those of the index. Knowing that the index’s IVs are more skewed than those of the single stock, it is possible to take a percentage of the index skew and use this in the pricing. What percentage to use is primarily a function of whether the structure in question sets the seller short skew or long skew, and from there it is a function of how aggressive/conservative the trader wants to be on the skew position.

*Measuring and Trading the Implied Skew’s Convexity/Curvature*

To quantify the skew convexity, one can consider the **sum of the 90% and the 110% implied volatilities minus twice the 100% strike volatility and dividing by the difference in strikes squared**^{[1]}. In fact the combination of vanillas with the above strikes is known as a **butterfly spread**. If we go long a butterfly spread, we are long a 90% and a 110% strike call option, meaning that we are long the IVs at these two strikes. If the implied skew becomes more convex, it means that these two IVs have increased, making the butterfly spread more valuable.

**The IVs of a single stock generally have more curvature than those of an index**. The reason for this is that downward jumps have a larger impact on single stocks than they do on an index, and the risk of a single stock crashing completely is greater than that of a whole index doing so. So, although a stock may have less negatively skewed IVs than an index, the former’s IVs are more convex in strike than those of the index.

**4.2.2. Term Structure of Volatilities **

For a given strike, IVs vary depending on the maturity of the option. **In most cases, the term structure is an increasing function of maturity**. It is generally the case in calm periods where short-term volatilities are relatively low. This curve could be decreasing if the market is volatile and short-term volatility is exceptionally high. This term structure can also reflect the market’s expectations of an anticipated near term event in terms of the volatility that such an event would imply. **The term structure also reflects the mean-reversion characteristic of volatility**.

One can take a view on the term structure’s shape. A simple trade to provide this is the **calendar spread**, which is the difference of two call options of the same characteristics but different maturities.

*Skew through Maturities*

If we look at the IV skew for various maturities we notice that the **short-term skew is much steeper than the long-term skew**, and generally flattens out as maturities increase. **A long maturity could have a skew at a level higher than the short-term maturity, but the short-term skew will be more pronounced.** This has to do with **supply and demand, since people may be less keen to sell OTM puts in the short-term**. A jump in the underlying’s price in the immediate future would have a large impact on the price of the put; for the short term this is more severe as the market may not have time to recover. There is also an increase in the OTM call implied volatility compared to the longer maturities, again for similar reasons.

**4.3. Volatility Models**

Our goal is to explain what to do with the many models that already exist. Ultimately, a model is still just a model, and our goal is to find – given a specific derivative – the model that is best suited. Models can be useful only if properly understood, in both their limits and their strengths. Here we discuss these with respect to which forms of volatility each of the different models captures.

**4.3.1. Model Choice and Model Risk **

Essentially, the choice of which model to use depends on the different risks involved in the option.

Option’s value is a convex function of the underlying price. This second-order effect (non-linearity) in the underlying’s price is what gives the option value, and we must assume that the underlying’s price has some randomness in order to see this effect in whatever model we choose.

When specifying any model, we are faced with model risk: the risk that a derivative is modelled incorrectly. This is divided into three forms:

- The model being used has been incorrectly implemented
- The correct inputs are used in such models
- The correct model is chosen for the correct derivative

Since many of the more exotic structures to come are illiquid, it is imperative that we reduce these two last forms of model risk as much as possible.

To give an example regarding correct model inputs, one must take account of the liquidity of the underlying and one’s ability to trade it in order to know the hedging costs accurately.

An example of the right choice of model can be related to the volatility skew: if the option has skew dependence, that is, its price is sensitive not just to one implied volatility but to more than one, then we need to use models that capture this effect.

We discuss these issues from the point of volatility to understand how to capture the different characteristics of volatility when necessary. Other risks will become apparent as we move along (e.g. correlation risk in multi-asset options) and at each stage we will explain the models that can capture these risks.

**4.3.2. Black- Scholes or Flat Volatility**

The BS model is the market standard in the sense that the prices of vanilla options are quoted in terms of IVs rather than in dollar values. There is a **one-to-one relationship** between the price of a vanilla option and its IV in the BS formula.

We refer to this as the case of **flat volatility**, since the BS model assumes that **volatility is constant across strikes**. The **model therefore does not know about the implied volatility skew, nor does it know about the term structure of IVs**. However, the IVs that one obtains for vanilla options in the market across strikes and maturities are those that should be fed into the BS formula to obtain the correct values.

Asian options can be priced using the flat volatility Black–Scholes model. As long as we use the correct IV we can apply the model to obtain the price of the Asian option. The reason why we can apply such a model in this case is that the Asian option does not have skew sensitivity or any hidden convexities other than that to the underlying’s price.

Even though they involve skew, call spread can be priced using the BS model as they are a combination of a long call option and a short call option of a strike greater than the first call with all else remaining the same. The reason is that **although the call spread is sensitive to skew, it is only sensitive to two specific points on the skew: the volatilities at the correct strikes.**

Even in the case of some skew dependence it may still be possible to apply the BS model as long as we are cautious about the various effects, while keeping an eye on how the skew affects the price.

It is possible to extend the case of a constant volatility across maturities to having a time-dependent but deterministic volatility, thus allowing for a term structure of volatilities.

**4.3.3. Local Volatility**

LV models offer a way of **capturing the implied skew without introducing additional sources of randomness**; the only source of which is the underlying asset’s price that is modelled as a random variable. In the BS model, the asset’s price is modelled as a log-normal random variable, which means that the asset’s log-returns are normally distributed. However, the fact that we have a **skew is the market telling us that the asset’s log-returns have an implied distribution that is not Normal**.

LV is still a **one-factor model** and it also **allows for risk-neutral dynamics**, which means that, like BS, the model is preference free from the financial point. The LV model is the **simplest one to account for skew and offers a consistent structure for pricing options**.

How does local volatility work?

We can agree that the market is telling us that log-returns are not normally distributed. In fact, **the market is implying some distribution**. If we are given a set of prices of vanilla options for a fixed maturity across strikes, can we find a distribution that corresponds to these prices? This is to say, **can we find a distribution for the asset price so that if we used this distribution to price vanilla options on this asset, it would give the same prices as the vanillas on this asset seen in the market?** Yes, theoretically, there is a way to find the distribution (LV model) which corresponds exactly to all vanilla prices taken from the skew. In fact, **LV extends beyond skew and can also capture term structure**. It can therefore **theoretically supply us with a model that gives the exact same prices for vanillas taken from a whole implied volatility surface**.

*Local Volatility Models and Calibration *

Finding this function σ(S(t),t) is a process known as **calibration**. The inputs for these models are not only the current level of the asset, the curve of riskless interest rates, and the size and timing of known dividends to come, but also the implied volatility skew (possibly a whole surface). Given the set of IVs of vanilla options, calibration is the process where we search for these volatilities σ(S(t),t) so that the model matches these prices.

There are computational difficulties in finding this function that will exactly fit all market prices, which is why Dupire’s formula, though theoretically correct, has some practical drawbacks. In particular, fitting all points may lead to unrealistic model dynamics. **In practice, there may be more than one LV model that fits a set of vanillas, so one must lay down a set of criteria to follow when choosing the model to use.** The surface is two-dimensional, one in time and one in strike, and the focus on one or both must be determined in order to correctly capture the effect of the volatility surface on certain payoffs.

When IV data is not available, one must resort to interpolation or extrapolation of the surface. Doing this in an arbitrage-free manner will be discussed later.

**4.3.4. Stochastic Volatility**

In SV models, the asset price and its volatility are both assumed to be random processes. Recall, in Black–Scholes, that volatility is assumed to be constant, and in LV models the volatility is a deterministic function of the asset’s level.

In allowing the volatility to be random, SV models give rise to implied volatility skews and term structures. **SV models can explain in a self-consistent manner the actual features we see in the empirical data from the market**. Once such a model is specified, the skews generated by the model are a function of its parameters, and finding the parameters that fit a certain skew (or a surface) is again the act of calibration.

Do we need such models? Do we need to always model volatility as a random variable, or can we still use constant or local volatility?

The answer is that it depends on the derivative. Options that can be broken down into vanillas can be priced using BS as long as one uses the right IV for each option. For options that have skew dependency yet cannot be broken down into vanillas, we can use LV assuming it is correctly calibrated to the skew (or surface) in a manner consistent with the skew sensitivity of the option. **SV, on the other hand, goes beyond just skew and term structure allowing for Vega convexity and forward skew.**

A derivative exhibits **Vega convexity** when its **sensitivity to volatility is non-linear**: there is a non-zero second-order price sensitivity to a change in volatility. Vanilla options are convex in the underlying’s price, but are they also convex in volatility? **ATM vanillas are not, but OTM vanillas do have Vega convexity**. However, these options are liquidly traded and their prices are obtained by using their IVs in Black–Scholes. **These IVs give the market’s consensus of the right price**; therefore the cost of Vega convexity of OTM vanillas is already included in the skew.

In more complex payoffs, almost all the payoffs will exhibit some form of Vega convexity, although in many cases this is captured in the skew and can be correctly priced by getting the skew right (for example, with a LV model). Other payoffs exhibit such convexities that are not captured in the skew and we must in these cases use stochastic volatility. **Since volatility is taken to be random, it must also have its own volatility, and this is known as the volatility of volatility, or vol-of-vol**. This parameter corresponds to the **Vega convexity term**. The second-order sensitivity to volatility is known as **Volga**.

The second feature of SV models is that they can generate **forward skews**. If a derivative has exposure to forward skew, one must use a model that knows about forward skews in order to get a correct price.

Although LV models can capture the market’s consensus on the prices of vanilla options by matching the volatility surface, **the evolution of future volatility implied by these models is not realistic**. In the case of forward skews we are faced with the problem that **the forward skews generated by LV models flatten out as we go forward in time**, even though, in reality, forward skews have no reason to do so. The LV model, therefore, does not provide the correct dynamics for products with sensitivities such as these.

This issue is related to the question of smile dynamics. **By smile dynamics we refer to the phenomena of how the skew moves as the underlying moves**: if the underlying moves in one direction, how should the skew move? The answer is that LV models can provide inaccurate smile dynamics, while the dynamics of SV models are, in fact, more consistent with the dynamics observed in the market. If an option is sensitive to smile dynamics, then getting the smile dynamics wrong will have a large impact on both the price and the computation of the subsequent hedge ratios. We will see this concept in the context of option sensitivities as we move along.

**On the calibration side, SV models have difficulty fitting both ends of the surface**, that is, fitting the skew for both short and long maturities at the same time. One remedy for this is to add jumps to a SV model. **Jumps are able to explain the short-term skew** quite well, and we recall that the reason for the existence of the steep short-term skew has to do with jumps. Adding jumps to such a model does not generally affect the long-term skews which remain relatively flatter; the long-term implied skew is not driven by jumps in the underlying.

[1] An approximation of the second derivative of a function *f *at the point *x *is given by

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