Volatility and Correlation

Abstract

Implied volatility is central to both the valuation and trading of options as it is the only uncertain variable in the BSM model. Over the last decade there has also been growing interest in equity market correlation, and this chapter covers some of the key characteristics relating to both of these concepts.

6.1 Introduction

Implied volatility is central to both the valuation and trading of options as it is the only uncertain variable in the BSM model. Over the last decade there has also been growing interest in equity market correlation, and this chapter covers some of the key characteristics relating to both of these concepts.

The first part of the chapter consists of a short statistics refresher covering variance, volatility, covariance and correlation. The main part of the chapter concentrates on the characteristics of both volatility and correlation highlighting some important empirical relationships. The final part considers some techniques that could be used to determine if volatility or correlation is trading at fair value.

6.2 Definitions

6.2.1 Variance and Volatility

Definitions

For a sample of data, it is often important to know how representative the mean is of all of the observations. This can be assessed by measuring the dispersion of data around the mean.

Variance is a statistical measure of how a sample of data is distributed around its mean. The variance of a sample of numbers is derived by the following ways:

(a) Calculating the mean of the series

(b) Subtracting the mean from each value within the series and then squaring the result

(c) Calculating the sum of all of the squared results

(d) Dividing the result of (c) by n−1, where ‘n’ is the sample size

However, the result of the variance calculation is expressed as ‘squared units’ of the underlying. For example, the variance of a sample of ages would be ‘squared years’ while for investment returns it would be ‘squared returns’. It is perhaps as a result of this unintuitive unit of measurement that it is more common to convert the variance into a standard deviation. This is calculated as the square root of the variance and the result is expressed in the same units of measurement as the underlying data. So in our previous examples this would be just ‘years’ or ‘returns’. Within finance, standard deviations are often used to describe the spread of asset prices or returns around a mean value which inevitably means the analysis will involve some discussion of ‘normal distributions’ such as that shown in Fig. 6.1.

Fig. 6.1

A stylized normal distribution (Source: author)

Statistical theory tells us that a normal distribution is symmetrical with 68.3 % of the observations lying within +/−1 standard deviation of the mean. Of the data 95.4 % lies within +/−2 standard deviations of the mean while +/−3 standard deviations from the mean captures 99.7 % of the observations.

Although it is tempting to think that prices can be described by a normal distribution, this would not be technically correct. Galitz (2013) argues: ‘It turns out that, while prices are not normally distributed, returns mostly are’. For example, a normally distributed variable needs to be able to take a value from positive infinity to negative infinity, which would immediately create a problem for share prices as they cannot take negative values. To avoid this issue option valuation principles hypothesize that future spot rates are lognormally distributed around the forward rate¹. A lognormal distribution is characterized by a left tail that is bounded by zero but with a right tail that extends to infinity. Although the spread of these future spot prices around the forward can be described in standard deviation terms, financial markets use the term volatility ² instead. It is important to note that the degree of uncertainty about future spot price movements does not necessarily depend on where the spot price is trading today.

Interpreting Volatility—An Intuitive Approach

To convey the intuition behind these concepts, suppose there is an index option whose current underlying forward price is 1000 points and is trading with an implied volatility of 30 %. By convention this volatility quote represents one standard deviation and so it would be possible to say that in 68.3 % of all possible outcomes the index is expected to trade within a range of +/−300 index points of the forward price over the next 12 months. It is worth noting that informally many participants will often describe this range of expected price movements relative to the current spot rate rather than the forward.

Volatilities are quoted as a percentage per annum but often traders like to know the equivalent value for another period. Market practice is to take the annualized figure and divide it by the number of applicable trading periods in a year.

• Annual to daily volatility: Divide by the square root of 252,³ that is, 15.87 (or 16 as an approximation). For an implied volatility quote of 30%, this would suggest a daily range of price movements of +/−1.89 %.

• Annual to weekly volatility: Divide by the square root of 52, that is, 7.2. So for an annualized volatility of 30 % this would equate to a weekly range of expected price movements of +/−4.16 %.

• Annual to monthly volatility: Divide by the square root of 12, that is, 3.46, which would return a monthly range of expected price movements of +/−8.67 % for an annual level of 30 %.

Another aspect of interpreting realized volatility quotes relates to the sampling period and the interval between observations. Suppose an investor sees a research piece which states that ‘3-month annualized daily realized volatility was 16 %’. This means that the sample used in the calculation comprises of daily closing prices taken from the last 3 months of trading. The result was then annualized by multiplying by the square root of 252.

6.2.1.1 Different Types of Volatility

Natenberg (1994) and Tompkins (1994) make reference to a number of different types of volatility:

• Historical volatility: the volatility over some period in the past. Sometimes referred to as ‘realized’ volatility.

• Future realized volatility: the volatility that best describes the actual future distribution of prices for an underlying contract. This parameter is unknowable in advance.

• Forecast volatility: A model-based estimate of the future volatility of a contract.

• Forward volatility: A measure of volatility whose value is known today but applies to a future time period. Analogous to a forward interest rate, that is, the 6-month rate in 6 months’ time. This is not a forecast but is a ‘no arbitrage’ value. An example calculation of forward volatility is given in the appendix to this chapter.

• Seasonal volatility: Volatility that relates to seasonal factors such as those experienced by commodities.

• Implied volatility: This is often defined as the volatility implied by an observed option price. But defining implied volatility in this manner introduces an element of circularity into the definition—from where did the other market participants obtain their volatility input? Kani et al. (1996) define implied volatility ‘as the market’s estimate of the average future volatility during the life of the option’. Natenberg (1994) defines it as ‘a consensus volatility among all market participants with respect to the expected amount of the underlying price fluctuations over the remaining life of the option’. Tompkins (1994) defines it as ‘the risk perceived by the market today for the period up until the expiration of a particular option series’. These definitions bring out the idea that volatility is a perception and is therefore subjective—or to put it crudely, a guess.

A further useful distinction to make is the difference between risk and uncertainty, which is usually attributed to the American Economist, Frank Knight. Risk relates to situations where a future outcome is uncertain but where the odds or probabilities can be objectively measured. Examples would include things like the odds of rolling a six from the throw of a die or the probability of winning a lottery. Uncertainty occurs where it is impossible to assign values to uncertain future outcomes; examples of this include where a share price or equity index will be in the future. In this sense, implied volatility could perhaps be viewed as the degree of uncertainty involving the magnitude of future price movements.

6.2.2 Covariance and Correlation

Definitions

Covariance measures how two random variables behave relative to each other, measuring the degree of linear association between the two variables. If the price of asset A generally rises (falls) at the same time that the price of asset B rises (falls), the covariance will be positive. If generally the price of asset A is associated with a fall in the price of asset B, the covariance will be negative. One of the problems relating to the interpretation of covariance is that the magnitude of the result is a function of the value of the assets under analysis—it is an unbounded number. So a higher value for covariance could be explained either by the fact that the two variables may deviate significantly from their mean values or that they display a high degree of association.

The degree of association between two variables can also be measured by converting the covariance into a correlation coefficient. This is a ‘standardized covariance’ measure whose values range from 0 to +1 or 0 to −1 and indicates the strength and direction of a linear relationship between two variables. One possible interpretation of the correlation coefficient is:

0.0 to 0.2	Negligible
0.2 to 0.4	Low
0.4 to 0.7	Moderate
0.7 to 0.9	High
0.9 to 1.0	Very high

These interpretations would apply irrespective of whether the coefficient is positive or negative.

It is worth noting:

• Correlation can also be expressed as a percentage number, for example, 40 %.

• If two assets are deemed to be correlated it does not imply any form of causality.

• A correlation measure does not give any indication of the magnitude of movement experienced by the two variables. So if asset A and asset B display a correlation of, say, +0.4 (or 40 %) it does not mean that a 1 unit move in the price of A will be associated with a 0.4 unit move in the price of asset B.

By way of illustration, the correlation between two equity markets (Hang Seng and the S&P 500) is shown in Fig. 6.2.

Fig. 6.2

Upper panel: Movement of Hang Seng (left hand side) and S&P 500 index (right hand side) from March 2013 to March 2016. Lower panel: 30-day rolling correlation coefficient over same period (Source: Barclays Live)

Similar to volatility the equity market distinguishes between realized and implied correlation. Realized correlation is the actual correlation experienced over some historic period. Implied correlation represents the market’s current expectation of future realized correlation.

Correlation and Portfolio Theory

Modern portfolio theory argues that the variability of a portfolio’s returns (i.e. the risk or volatility) is calculated as the weighted sum of the individual volatilities adjusted by the degree to which they are correlated. Provided that the assets in the portfolio are not perfectly positively correlated the volatility of the portfolio will be less than the weighted sum of the volatilities of the constituent assets that form that portfolio.

Equation 6.1 shows the composite volatility for a two-asset portfolio.

$$ {\sigma}_{\mathrm{basket}}=\sqrt{\left({w}_{x_1}^2{\sigma}_{x_1}^2\right)+\left({w}_{x_2}^2{\sigma}_{x_2}^2\right)+2\times \left({w}_{x_1}{w}_{x_2}{\rho}_{x_1{x}_2}{\sigma}_{x_1}{\sigma}_{x_2}\right)} $$

(6.1)

• where:

• σ ² _x1 = Variance of asset 1

• σ ² _x2 = Variance of asset 2

• ρ _x1x2 = Correlation between asset 1 and asset 2

• σ _x1 = Volatility of asset 1

• σ _x2 = Volatility of asset 2

• w _x1 = Proportion of asset 1

• w _x2 = Proportion of asset 1

6.3 Overview of Volatility and Correlation Trading Strategies

6.3.1 Market ‘Flows’ of Volatility

Figure 6.3 illustrates the ‘market flows’ of the demand for and the supply of volatility. The diagram shows that term structures exist for both single-stock and index volatility. The term structure of single-stock volatility is a weighted average measure derived from the individual shares within the overall index.

Fig. 6.3

The term structure of single-stock and index volatility indicating the different sources of participant demand and supply (Source: author)

In Fig. 6.3 the main participants and their respective motives are:

• Fund managers and insurance companies—they will tend to use options as a way of protecting their portfolios against downside price movements. This implies they are buyers of OTM index puts, perhaps struck 10 % away from the current price (1). From a single-stock perspective, they may also decide to sell OTM calls in order to enhance the yield on an individual position (2).

• Corporates—one source of single-stock volatility will involve some form of equity-linked financing such as the issuance of convertible bonds. This leaves the issuer with a short equity call position (3).

• Structured product vendors—these entities would typically include investment banks who issue products that are either capital protected or yield enhanced. For a capital protected structure the investor is long a call on an index while a yield-enhanced product typically results in the investor being short a put on an individual share. These are shown as positions (4) and (5) respectively, in Fig. 6.3. The options embedded within these structures may present the structuring banks with exposures that they will then seek to hedge⁴.

• Hedge funds and investment banks—these entities will be looking to trade volatility as an individual asset class. They may be buyers or sellers of index and/or single-stock volatility and may simply be looking to exploit opportunities as they arise. In some cases, they may take opposing positions in index and single-stock volatility which results in a correlation exposure.

Figure 6.3 illustrates that in general terms index volatility is in demand while single-stock volatility is in supply. At first glance these statements would appear to be inconsistent with the diagram which illustrates that average single-stock volatility will tend to trade above index volatility. At some point it would seem to be logical that the selling pressure from single-stock activity combined with the buying pressure of index volatility would force the two curves to switch position. However, this would violate the basic principle of portfolio theory shown in Eq. 6.1; that is the volatility of a portfolio (i.e. the index) is lower than the volatility of its individual components. This also gives an insight into the trading of correlation which, in essence, is based on taking views on the convergence or divergence of average single-stock volatility relative to index volatility.

6.3.2 Trading Volatility—An Overview

Within the context of option valuation, the only uncertain variable is implied volatility, which consequently offers market participants a number of trading opportunities.

At a wholesale level a market maker seeking to trade volatility may structure a quote as follows:

Bid	Offer
25.00 %	25.10 %
Motivation
Buy volatility	Sell volatility
Example trades
Buy options and delta hedge	Sell options and delta hedge
OR	OR
Buy straddles	Sell straddles

A buyer of volatility will profit if implied volatility rises, while a seller of volatility will profit if implied volatility falls. The strategy highlighted in Sect. 5.3 illustrates that there are two ways of profiting from trading volatility:

• Changes in the mark to market of the position—the trader profits from entering the trade at a given level of implied volatility and closes out the position at another level sometime in the future (‘vega trading’).

• Delta hedging—here the initial implied volatility used to value the option initially differs from the volatility actually realized over the life of the option (‘gamma trading’).

6.3.3 Trading Correlation—An Overview

Equation 6.1 demonstrated that correlation provided the link between the volatility of an index and that of the constituent stocks. A popular shorthand way of expressing this relationship is the ‘correlation proxy’ (Granger and Allen 2005):

$$ \mathrm{Index}\;\mathrm{volatility}\approx \sqrt{\mathrm{correlation}}\times \mathrm{average}\;\mathrm{single}\;\mathrm{stock}\;\mathrm{volatility} $$

(6.2)

The correlation measure in Eq. 6.2 represents the average of the correlations between all possible pairs of constituent shares within an index. Granger and Allen (2005) argue that there are two ways of measuring this value. The first approach calculates the value as the equally weighted average pairwise correlations of the constituent stocks. This is given by Eq. 6.3:

$$ {\rho}_{\mathrm{average}}=\frac{2}{N\left( N-1\right)}{\displaystyle {\sum}_{i< j}{\rho}_{i j}} $$

(6.3)

• Where:

• N = number of stocks

• ρ _ij = pairwise correlation of the i ^th and j ^th stocks

The second technique is to calculate the correlation implied by the observed implied volatilities of the index and the constituent stocks. Recall Eq. 6.1, which calculated the volatility of a portfolio based on the volatilities of the two constituent assets and their correlation. For a portfolio consisting of multiple stocks the fundamental principles of the equation would still apply except that the calculation would need to include a greater number of correlations. If the implied volatilities of the index and the constituent assets are traded in the market it should be possible to rearrange the formula to derive a single implied correlation measure that is consistent with these observations. This is illustrated in Eq. 6.4 (Granger and Allen 2005).

$$ {\rho}_H=\frac{\sigma_I^2-{\displaystyle \sum}_i{w}_i^2{\sigma}_i^2}{{\left({\displaystyle \sum}_i{w}_i{\sigma}_i\right)}^2-{\displaystyle \sum}_i{w}_i^2{\sigma}_i^2} $$

(6.4)

• Where:

• ρ _H = implied index correlation

• σ _i = volatility of constituent stock

• σ _I = volatility of index

• w _i = weight of constituent stock within the index

Generally speaking, banks tend to be short correlation by virtue of their structured product portfolios. Equation 6.2 gives an insight into how this correlation exposure could be mitigated. Implied correlation can be isolated by taking opposing positions in instruments that reference index volatility and single-stock volatility. For example, a long position in index volatility combined with a short position in single-stock volatility results in a long correlation exposure. This is referred to as dispersion trading and will be considered in greater detail in Chap. 15.

6.4 Characteristics of Volatility and Correlation

In order to profit from anticipated movements of volatility and correlation it is important to understand how these market factors evolve over time. A common technique is to chart the evolution of the variable relative to another market factor.

6.4.1 Characteristics of Volatility

6.4.1.1 Implied Volatility Versus the Level of the Market

Figure 6.4 indicates that there is an inverse relationship between implied volatility and the absolute level of the cash equity market. So as equity prices fall then implied volatility rises. Note also that implied volatility has a tendency to ‘spike’ upwards and then gradually ‘grind down’⁵.

Fig. 6.4

Level of the S&P 500 and 3-month implied volatility for a 50 delta option. March 2006–March 2016 (Source: Barclays Live)

6.4.1.2 Implied versus Realized Volatility

Figure 6.5 illustrates the relationship between index implied volatility and index historical (‘realized’) volatility.

Fig. 6.5

Implied volatility for 3-month 50 delta index option versus 3-month historical index volatility (upper panel). Implied volatility minus realized volatility (lower panel). March 2006–March 2016 (Source: Barclays Live)

Figure 6.6 shows the relationship between average single-stock implied volatility and average single-stock realized volatility.

Fig. 6.6

Average single-stock implied volatility versus average single-stock realized volatility (upper panel). Implied volatility minus realized volatility (lower panel). March 2006–March 2016 (Source: Barclays Live)

Taken together, Figs. 6.5 and 6.6 show that implied volatility has a tendency to trade at higher levels than realized volatility. That is, market participants value their option positions using a level of implied volatility that on average will be in excess of the volatility actually realized by the market. One concept used to explain why implied volatility trades higher than realized is the so-called risk premium argument. This is based on the idea that since option sellers are faced with potentially open-ended losses it is not unreasonable that they should earn some form of supplemental return.

The figures also illustrate that when implied volatility increases significantly, then realized volatility tends to increase by a greater amount.

6.4.1.3 Index Volatility Versus Single Stock Volatility

Recall that Eq. 6.1 illustrated the relationship between the volatility of a portfolio and the volatility of the constituent stocks. The equation highlighted that the volatility of a portfolio is less than the volatility of the constituents and this principle is confirmed by the data in Fig. 6.7.

Fig. 6.7

Implied volatility of 3-month 50 delta S&P index option versus average implied volatility of 50 largest constituent stocks. March 2006–March 2016 (Source: Barclays Live)

Figure 6.8 illustrates the relationship between realized volatility for the index against the average realized volatility of the constituent stocks. This shows that the realized volatility of the index is lower than that of the constituent assets by virtue of realized correlation.

Fig. 6.8

Realized volatility of 3-month 50 delta S&P 500 index option versus average realized volatility of 50 largest constituent stocks. March 2006–March 2016 (Source: Barclays Live)

6.4.1.4 Volatility Skew

In the original BSM option valuation framework, implied volatility was assumed to be constant. Empirically, this condition does not hold as volatility is seen to vary by both strike and maturity (Table 6.1).

Table 6.1 Volatility surface for S&P 500. Data as of 26th July 2014. Strikes are shown as a percentage of the spot price

Statistically, skewness measures the degree of positive or negative bias displayed by a sample of data with equity markets being characterized as ‘skewed to the downside’. Figure 6.9 shows a stylized example of a distribution that displays a negative skew.

Fig. 6.9

Example of distribution exhibiting negative skew. The columns represent the skewed distribution while a normal distribution is shown by a dotted line (Source: author)

A negatively skewed distribution has a longer fatter left tail as well as an element of ‘bunching’ on the right hand side. So from an equity perspective, although this suggests that markets generally tend to display positive returns, there is a greater probability of large negative returns, that is, stock market crashes will tend to occur more often than a normal distribution would predict.

This negative skew translates into a situation where low strike options will tend to trade with higher implied volatilities than options with higher strikes (Fig. 6.10). An option whose strike is 90 % of the current market⁶ would have to be either an ITM call or OTM put. Conversely, an option struck at 110 % would be an OTM call or ITM put option. However, the mathematics of option valuation mean that even if the market is negatively skewed, ATM options will still trade with a higher premium than OTM options.

Fig. 6.10

Volatility skew for 3-month ATM options written on the S&P 500 equity index. The X axis is the strike of the option as a percentage of the current spot price (Source: Barclays Live)

Although the type of ‘downside skew’ illustrated in Fig. 6.10 is very common in equity markets there are exceptions. For example, consider Fig. 6.11 which charts the relationship between implied volatility and a variety of different call option deltas for Blackberry. Notice that this relationship is roughly symmetrical in nature; this is referred to as a volatility smile.

Fig. 6.11

Volatility smile for Blackberry. Implied volatility (Y axis) measured relative to the delta of a 3-month call option (Source: Barclays Live)

Figure 6.11 indicates that the implied volatility for high strike, low delta OTM call options (right hand side of the diagram) is virtually the same as it is for low, high delta ITM calls (left hand side of the diagram).

In order to link this diagram to the implied volatilities for put options recall the principles of put-call parity; the combination of a call and a put with the same strike and maturity will be equal to a long forward position in the underlying asset. So, if prior to maturity an investor was short an ITM call strike with a delta of −75 then an OTM put option struck at the same price must have a delta of approximately −25, that is, the combination of the two options is equivalent to a short forward position. This means that the implied volatility of the −75 delta ITM call and the −25 delta OTM put must be equivalent as a forward position has no exposure to implied volatility. Therefore, on the left hand side of Fig. 6.11 OTM puts will trade with the same level of implied volatility as the equivalent ITM call option.

At the time Blackberry was in the process of a major restructuring having lost market share to other smartphone producers such as Apple and Samsung. Although the company had avoided bankruptcy, the smile perhaps reflects the fact that the market was still split as to their future prospects. The higher volatilities for lower strikes would be driven by those participants who had bought downside put protection, while the higher volatilities for the higher strikes was probably a result of participants buying cheap OTM call options in case the stock experienced a significant recovery.

It is also possible for an asset to display a positive skew, such as that for the Indian company Reliance Industries (Fig. 6.12). Here the volatility for OTM calls is greater than that for OTM puts.

Fig. 6.12

Volatility skew for Reliance industries (Source: Barclays Live)

Why Does the Volatility Skew Exist?

A number of reasons have been suggested to explain the existence of volatility skew. They include:

• Fat tails—this is the belief that the incidence of extreme market movements is greater than the lognormal distribution assumed by the BSM valuation framework. Since a trader knows they will occur with greater frequency than the statistics suggest, OTM puts which offer equity investors downside protection will be priced with higher implied volatilities.

• Pre-emptive pricing—De Weert (2006) argues that traders who sell OTM puts will delta hedge their directional exposures by shorting the underlying stock. As the market falls, the gamma exposure on the position will increase as the option now tends towards the ATM level. As was shown in Sect. 5.3 this rebalancing could result in losses if the underlying price movements are significant. So where a trader is delta hedging a short put position a falling spot price would require them to sell more of the asset, which would result in losses. In anticipation of these possible hedging losses the trader will charge relatively more by increasing their implied volatility quote.

• Simple demand and supply—the vast majority of participants in the cash equity market hold long positions. As a result, participants are more likely to buy downside protection in the form of an OTM put pushing up the cost in volatility terms relative to the ATM volatility. This OTM put could be financed by the sale of an OTM call with the strike set in a region that is not expected to trade. This selling pressure pushes down the volatility of higher strike options. It is also possible for implied volatility to increase for higher strike options (see Fig. 6.12). This would suggest that there is some demand for OTM call options perhaps as a result of some expectation of a sharp increase in price.

• Behavioural finance—writers such as Daniel Kahnemann have suggested that the pain suffered by investors from losing money is twice as great as the pleasure they derive from making money. This is perhaps reflected in Fig. 6.4 which illustrates the relationship between implied volatility and the underlying market. The diagram suggests that as markets fall sharply, investors will tend to panic and overreact.

How Is the Volatility Skew Measured?

There are a number of ways in which volatility skew can be measured (Deb and Brask 2009):

The cost of a collar struck at 90 % and 110 % of the market—a collar is an option combination, which from an equity investor’s perspective consists of a long OTM put option and a short ITM call option. Briefly, an investor is able to obtain downside protection on an equity position by virtue of the purchased put option, the cost of which is subsidized by the sale of an OTM call option. Although the cost of protection is reduced, the impact of the short call prevents the investor from enjoying any upside beyond the 110 % strike call.

Suppose the underlying market is trading at 2000 index points. Using the values in Table 6.1, a 3-month long OTM put struck at 90 % of this value (i.e. 1800 index points) would return a premium of approximately 10 index points. A short OTM call option struck at 110 % of spot (i.e. 2200 index points) with the same maturity would generate a premium of about 0.5 of an index point. So in this instance the cost of the structure would be approximately 9.5 index points.

Using this method of calculating the skew it would suggest that the greater the net premium the more pronounced is the skew to the downside.

Volatility spread between 90 % and 110 % options—whereas the cost of the 90–110 % collar measured the skew in terms of premium, this particular metric measures the differences in volatility terms. To illustrate how this would be calculated consider again the 3-month collar position used in the previous example:

• 90 % volatility = 17.3 %

• 110 % volatility = 9.0 %

• Volatility spread = 8.3 %

This suggests that a greater volatility spread is associated with a greater degree of skew to the downside—sometimes referred to as a ‘steep’ skew. A lower spread would be therefore indicative of a ‘flatter’ skew.

Percentage skew—empirically it has been observed that in a high volatility environment skew has a tendency to flatten. This can be illustrated by expressing the skew as a percentage. This is calculated by taking the 90–110 % volatility spread and dividing it by the ATM volatility.

Figure 6.13 shows that as implied volatility rises the degree of skewness declines and vice versa. This could be explained by mean reversion (Deb and Brask 2009): ‘when volatility is already elevated, there is less risk of a large increase in volatility should the underlying market move lower still. This calls for a flatter skew…. Equally, when volatility is low (typically when markets are rising) the primary risk to volatility is a sharp increase should markets fall. This translates into a steep skew….’

Fig. 6.13

S&P 500 index volatility (left hand side) plotted against the skew measured in per cent (right hand side). March 2006–March 2016 (Source: Barclays Live)

Delta skew—suppose an analyst is looking at a 3-month OTM 110 % S&P 500 call. With implied volatility at 20 % the delta of the option is about 17 %. However, the same option with the same strike will have a delta of about 27 % if implied volatility were to increase to 30 %, all other things being equal. So instead of holding the degree of moneyness constant (and allowing delta to change as implied volatility changes) this measure holds the delta constant and allows the degree of moneyness to vary. Delta skew can be measured either as a ratio (i.e. volatility of a 25 delta put divided by the implied volatility of a 25 delta call) or as a difference (i.e. volatility of a 25 delta put minus the implied volatility of a 25 delta call).

Normalized delta skew—this is another popular measure that is used and can be calculated in different ways (Eqs. 6.5 and 6.6):

$$ \frac{\left(\mathrm{Volatility}\;\mathrm{of}\kern0.28em 25\kern0.28em \mathrm{delta}\kern0.28em \mathrm{put}-\mathrm{volatility}\;\mathrm{of}\kern0.28em 25\kern0.28em \mathrm{delta}\;\mathrm{call}\right)}{\mathrm{Volatility}\;\mathrm{of}\kern0.28em \mathrm{a}\kern0.28em 50\kern0.28em \mathrm{delta}\;\mathrm{option}} $$

(6.5)

$$ \begin{array}{c}\frac{\left(\mathrm{Volatility}\;\mathrm{of}\kern0.28em 25\kern0.28em \mathrm{delta}\kern0.28em \mathrm{put}-\mathrm{volatility}\;\mathrm{of}\kern0.28em 50\kern0.28em \mathrm{delta}\kern0.28em \mathrm{put}\right)}{\mathrm{Volatility}\;\mathrm{of}\kern0.28em 50\kern0.28em \mathrm{delta}\kern0.28em \mathrm{put}}\\ {}\mathrm{minus}\\ {}\frac{\left(\mathrm{Volatility}\;\mathrm{of}\kern0.28em 25\kern0.28em \mathrm{delta}\;\mathrm{call}-\mathrm{volatility}\;\mathrm{of}\kern0.28em 50\kern0.28em \mathrm{delta}\;\mathrm{call}\right)}{\mathrm{Volatility}\;\mathrm{of}\kern0.28em 50\kern0.28em \mathrm{delta}\;\mathrm{call}}\end{array} $$

(6.6)

Variance skew—this is defined as the ratio of the variance swap strike expressed in volatility terms to the ATM forward volatility. Variance swaps are considered in greater detail in Chap. 12 but Deb and Brask (2009) argue ‘since a variance swap strike is essentially a weighted average of volatilities across option strikes, comparing it to ATM volatility is a measure of skew’. This is shown in Fig. 6.14.

Fig. 6.14

Variance swap strike and ATM forward implied forward volatility for S&P 500 (upper panel). Variance swap divided by ATM forward volatility for S&P 500 (lower panel). March 2011–March 2016 (Source: Barclays Live)

Characteristics of the Volatility Skew

The volatility skew itself will not remain stable but will evolve over time. Figure 6.15 is a time series that illustrates the degree of skewness exhibited by a 3-month S&P 500 option over time. The degree of skewness is measured as the difference between the implied volatility of an option struck at 90 % of spot minus the implied volatility of an option struck at 110 %.

Fig. 6.15

Evolution of the volatility skew over time. Skewness measured as the difference between the implied volatilities of an option struck at 90 % of the market less that of an option struck at 110 %. The higher the value of the number the more the market is skewed to the downside (i.e. skewed towards lower strike options) (Source: Barclays Live)

The movement of skew over time does not in itself reveal much information unless it is put into a wider context. Figure 6.16 shows how the skew moves in relation to the level of implied volatility. The figure shows that as implied volatility rises (and so the cash market is likely falling) the degree of skewness towards lower strike options increases. This is perhaps intuitive—a falling cash market should lead to an increase in the demand for OTM puts. This increased demand will translate into an increase in implied volatility which in turn will lead to an increase in cost, all other things being equal.

Fig. 6.16

Implied volatility of 3-month ATM S&P 500 option plotted against the 3-month volatility skew (Source: Barclays Live)

Does Volatility Skew Matter?

How significant is the skew? Consider Table 6.2 which revisits the net premium on a 3-month 90–110 % collar using the values in Table 6.1, the net cost of the structure being 9.64 index points. Table 6.2 shows what would happen if the absolute level of volatility increases by 1 % (column 3) against the impact of a 1 % increase in the 90–110 % volatility spread (column 4). Although perhaps somewhat simplistic the table indicates that movements in the absolute level of volatility will have a greater impact on the premium than the same change in the shape of the skew.

Table 6.2 Premium on a 90–110 % collar under different volatility assumptions

6.4.1.5 Term Structure of Volatility

Figures 6.17, 6.18 and 6.19 illustrate the existence of a term structure of volatility. The different diagrams illustrate the term structure for options struck at different levels: 80 % of spot, Fig. 6.17; ATM strike options, 100 % of spot, Fig. 6.18 and high strike options, 120 % of spot, Fig. 6.19.

Fig. 6.17

Term structure of volatility for an S&P 500 option struck at 80 % of spot (Source: Barclays Live)

Fig. 6.18

Term structure of volatility for an S&P 500 option struck at 100 % of spot (Source: Barclays Live)

Fig. 6.19

Term structure of volatility for an S&P 500 option struck at 120 % of spot (Source: Barclays Live)

The most common representation of the term structure is that of the ATM option shown in Fig. 6.18. This shows that ‘normally’ volatility is upward sloping with respect to maturity. In these cases, it would suggest that uncertainty increases with respect to maturity. For low and high strike options the charts show that shorter-dated options trade with higher volatilities which then dip down before rising. This would suggest the opposite—there is greater uncertainty with shorter-dated options struck at these levels. However, it is important to realize that although shorter-dated implied volatility may be higher than longer-dated volatility the premium on a longer-dated option will be greater in cash terms, all other things being equal.

Similar to the skew, the term structure of volatility will also evolve over time. Figure 6.20 shows how the slope of the term structure evolves with respect to time.

Fig. 6.20

Slope of term structure of S&P 500 implied volatility. Term structure is measured as 12-month implied volatility minus 3-month volatility. An increase in the value of the Y axis indicates a steepening of the slope(Source: Barclays Live)

However, similar to skew the evolution of the term structure slope needs to be put into some context. Figure 6.21 shows how the slope evolves relative to the absolute level of implied volatility.

Fig. 6.21

Implied volatility of 3-month 50 delta S&P 500 index option (Left hand axis) versus the slope of the index term structure (right hand axis). March 2006–March 2016

Figure 6.21 indicates that when the absolute level of implied volatility increases the slope of the term structure flattens and may invert (as shown by a negative value on the right hand axis).

A number of reasons have been suggested to explain the existence of the term structure (Tompkins, 1994):

• Non-stationarity in the underlying price series—if analysts anticipate that some fundamental change will occur to the underlying asset in the future then implied volatilities of options will differ for expiration periods prior to and after the economic event.

• Non-uniformity of volatility—volatility is expected to be different on different days depending on the occurrence (or non-occurrence) of ‘economic events’.

• Reversion to the mean—volatility does not remain at extreme levels but will tend towards a long-term average.

6.4.1.6 The Volatility Surface

Table 6.1 and Fig. 6.22 illustrate the concept of the volatility surface. A volatility surface is a way of representing a volatility with respect to both strike and maturity.

Fig. 6.22

Volatility skew for S&P options with different maturities. Data based on values shown in Table 6.1 (Source: Barclays Live)

Figure 6.22 illustrates two main concepts:

• The equity options market is skewed towards lower strike options.

• As the maturity of an option shortens the skew becomes more pronounced.

6.4.1.7 How Volatile Is Volatility?

Figure 6.23 illustrates that shorter-dated implied volatility is more volatile than longer-dated volatility, which has important risk management implications.

Fig. 6.23

Time series of 3-month index implied volatility plotted against 36-month index implied volatility. March 2006–March 2016 (Source: Barclays Live)

To reflect these different degrees of volatility, some traders make a distinction between headline and weighted vega. Headline vega is defined as the model-determined vega exposure for any given maturity. Since the term structure of volatility does not move in a parallel fashion (see Fig. 6.20) many traders do not add together their volatility exposures across maturities but rather use the concept of weighted vega. This is defined as the maturity-weighted vega exposure, expressed relative to a chosen maturity. Suppose a trader’s S&P 500 option portfolio has a $100,000 model-determined vega exposure in both the 3-month and 36-month maturities. This implies that a 1 % change in implied volatility will result in the option portfolio gaining or losing $100,000 in each maturity. However, from a risk management perspective the two would not be considered equivalent due to the non-parallel movement of the term structure. The trader decides to calculate the weighted vega for the 36-month exposure relative to the 3-month maturity. One of the ways to calculate the weight is to use the following formula:

$$ \sqrt{\frac{\mathrm{Base}\;\mathrm{maturity}}{\mathrm{Comparative}\;\mathrm{maturity}}} $$

(6.7)

The 36-month maturity (1080 days) is taken to be the comparative maturity while the 3-month maturity (90 days) is the base maturity. The weight is

$$ \sqrt{\frac{90}{1080}}=0.2887 $$

(6.8)

So if 3-month volatility moves by 1 % there would be a gain or loss of $100,000 (headline vega) in that maturity and this would likely be associated with a smaller move in the 36-month volatility that would result in a gain or loss of $28,700 (the weighted vega).

An alternative approach would be to use regression analysis, which is a popular technique used in similar circumstances in the fixed income market⁷. This technique describes the relationship between a ‘dependent’ and ‘independent’ variable. In this instance the dependent variable would be 36-month vega whose value is predicted based on movements in the independent variable which is 3-month vega. Figure 6.24 shows a time series of the daily changes in implied volatilities for both maturities and a casual ‘eyeballing’ of the data does confirm the assertion that shorter-dated volatility changes by more than longer-dated volatility.

Fig. 6.24

Chart shows the change in 3-month ATM spot implied volatility vs. change in 36-month ATM spot volatility for the S&P 500 index. March 2006–March 2016 (Source: Barclays Live)

The next step is to plot a scattergraph of the data and to determine a ‘line of best fit’ (Fig. 6.25) whose mathematical form would be:

$$ \varDelta {y}_{36\mathrm{m}}=\alpha +\beta \varDelta {x}_{3\mathrm{m}}+{\varepsilon}_t $$

(6.9)

• Where:

• Δy _36m = the change in 36-month vega (the dependent variable)

• α = a constant; the value of Y even if X has zero value

• β = the regression coefficient. This represents the slope of the line of best fit.

• Δx _3m = the change in 3-month vega (the independent variable)

• ε = the error term. This reflects the fact that other factors may influence the value of Y and are not captured by the specification of the equation.

Fig. 6.25

A ‘line of best fit’ for a scattergraph of changes in 36-month implied volatility (Y axis) against changes in 3-month implied volatility (x axis). S&P 500 index, March 2006–March 2016 (Source: Barclays Live)

The regression equation derived from the data is:

$$ Y=0.001244+0.338310\mathrm{x} $$

(6.10)

If, for ease of illustration, the value of the constant is ignored then Eq. 6.10 shows that a one-unit change in X (i.e. 3-month implied volatility) is associated with a 0.338310-unit change in Y (i.e. 36-month implied volatility). The regression equation also returns an R ² value of 0.84. This means that 84 % of the variance of the 36-month vega can be explained by the model. Equation 6.10 confirms that short-dated volatility is more volatile than longer-dated volatility and is also close to the short cut method outlined in Eq. 6.8. However, the value of the regression coefficient will also be a function of the sample period.

6.4.1.8 Volatility Regimes

This section covers how volatility trading has an impact on observed spot prices. All the examples are based on the hypothetical values for the S&P 500 (Table 6.3).

Table 6.3 Implied volatilities for different market levels and strikes over a 3-day period

Sticky Strike, Strike Pinning and Whippy Spot

Consider first the different levels of volatility indicated on day 1 in Table 6.3. If the underlying market were to subsequently stay unchanged but implied volatility were to rise or fall for all strikes, then this would indicate a shift in the volatility surface.

Now consider the movement in the market from day 1 to day 2. On day 1 the volatility associated with an ATM spot strike of 2000 index points was 22.0 %. The following day the market moves down by 5 % to reach a new level of 1900. The volatility associated with a specific strike price of 1900 (now the ATM strike) is the same as it was on day 1—25.0 %. This is the concept of ‘sticky strike’—a situation where the volatility associated with a particular strike price remains constant, although its relative ‘moneyness’ may have changed. Note also that the 22.1% volatility for the higher strike options on day 2 (the 105 % strike which corresponds to a level of 1995 index points) is consistent with the day 1 volatility for options struck at 2000. Citigroup (2008) argue ‘we tend to observe sticky strike moves over the short term as the market tends to anchor a certain underlying level with a certain implied vol. We would move away from this sticky strike regime once there has been a genuine rally or sell-off (or change of risk perception)’.

Another manifestation of ‘sticky strike’ is termed ‘strike pinning’. This relates to a market activity, most often observed in the single-stock markets, where the share price will trade around a popular strike price. Section 5.3 analysed a transaction where the intention was to exploit an anticipated change in the level of implied volatility. However, movements in the underlying price led to an element of directional exposure and so to neutralize this the trader had to buy and sell the underlying asset. The trade as presented involved a delta-hedged short option position and is sometimes referred to as a ‘short gamma’ exposure. A long gamma trade would require a trader to buy options (either calls or puts) and hedge the directional exposure by trading the underlying share. Share price rises would require the trader to sell shares while share prices fall would require the trader to buy shares. This long gamma hedging activity could lead to the share price being ‘pinned’ to a particular strike.

The delta hedging activities of the different volatility strategies are summarized in Table 6.4.

Table 6.4 Associated delta hedging activities when trading volatility using the four option basic ‘building blocks’

Bennett and Gil (2012) make the following observations in relation to strike pinning:

• There must be a significant amount of option trading relative to the normal traded volumes of the asset.

• It is more likely to happen in shares that are relatively illiquid where there is no strong trend to drive the price away from the strike.

• It is difficult to pin an index given the relatively high trading volumes.

‘Whippy’ spot relates to short gamma positions (i.e. either a short call or a short put) and would be characterized by significant spot price volatility around a particular strike.

Sticky Moneyness and Sticky Delta

Returning to Table 6.3, if on day 3 the market falls by another 5 % then the ATM strike is now 1805. In this example, the implied volatility for an option struck at 100 % of the market is unchanged from day 2 at 25.0 %, although the level of the underlying market has changed. This is an example of ‘sticky moneyness’⁸ and represents a situation where the implied volatility for a particular percentage of the strike (i.e. the moneyness or the delta of the option) remains the same. Citigroup (2008) argue ‘we tend to see sticky moneyness in a longer time horizon, where the perception of implied vols per given moneyness will not change even if the underlying does.’

6.4.2 Characteristics of Correlation

6.4.2.1 Correlation Versus Market Level Versus Implied Volatility

Figure 6.26 illustrates the relationship between index volatility and implied correlation, while Fig. 6.27 looks at the absolute level of the cash market and implied correlation. Taken together these charts indicate that when volatility is elevated and the market is falling, stocks within the index tend to become increasingly correlated. Similar to volatility, correlation also exhibits a tendency to mean revert.

Fig. 6.26

Implied volatility of 3-month 50 delta S&P 500 index against 3-month implied correlation. March 2006–March 2016 (Source: Barclays Live)

Fig. 6.27

Level of S&P 500 cash index against 3-month index implied correlation. March 2006–March 2016 (Source: Barclays Live)

In times of crisis both volatility and correlation have a tendency to spike up. This would suggest that a short correlation position may offset a long volatility position. If the markets are calm, then both correlation and volatility grind down. Again a short correlation position may offset a long volatility position.

One interesting exception to the concept of portfolio theory and the impact of correlation is shown in Fig. 6.28 which charts the evolution of index correlation for the Hong Kong Hang Seng Index. Notice that on several occasions short-dated index correlation exceeded the theoretical value of 1.

Fig. 6.28

One-month implied correlation of Hang Seng Index. March 2006–March 2016 (Source: Barclays Live)

6.4.2.2 Implied Versus Realized Correlation

Figure 6.29 shows that similar to implied volatility, implied correlation will tend to trade above realized volatility apart from periods of high volatility.

Fig. 6.29

Three-month Implied correlation minus realized correlation. S&P 500 equity index (Source: Barclays Live)

6.4.2.3 Correlation Skew

The concept of volatility skew was considered in Sect. 6.4.1.4. Since single stocks will also display a volatility skew then it follows from Eq. 6.2 that correlation will also display a skew.

6.4.2.4 Term Structure of Correlation

Figure 6.30 shows that similar to implied volatility, implied correlation will tend to display an upward sloping term structure.

Fig. 6.30

Term structure of S&P 500 implied correlation. Twelve-month minus 3-month implied correlation. March 2006–March 2016 (Source: Barclays Live)

However, the slope is not always positive. Figure 6.31 shows that again similar to implied volatility the slope of the term structure of correlation will flatten and possibly invert when the absolute level of implied volatility is high.

Fig. 6.31

Implied volatility of 3-month 50 delta S&P 500 index option (left hand axis) plotted against slope of correlation term structure (right hand axis). The correlation term structure is calculated as 12-month minus 3-month implied correlation. A negative value indicates an inverted term structure (Source: Barclays Live)

6.5 Identifying Value in Volatility and Correlation

The author recalls a discussion with a senior bank sales person who became frustrated with the approach of some of his junior colleagues when constructing client hedging strategies. They would often market zero premium strategies such as collars (e.g. buy an OTM call and finance with the sale of an OTM put) on the basis that the lack of premium represented an attractive proposition. Although this may be true at a simple level, the senior sales person argued that they only represented good value if the implied volatility of the purchased option was trading at a relatively low level and the implied volatility of the sold option was trading at a relatively high level. Moral of the story? Price and value are not the same thing!

6.5.1 Volatility

One of the trickiest tasks faced by an option trader is to formulate the most appropriate strategy for the prevailing volatility climate (Natenberg 1994). This usually involves some judgement as to whether implied volatility is currently trading at ‘fair value’ or some technique that forecasts how the volatility of the underlying asset is likely to evolve⁹.

The concept of fair value may suggest to some participants that there is a single ‘true’ price for every asset, which has been calculated by a top secret supercomputer, which never reveals its results to market participants. Since this is not an accurate representation it is perhaps why looking at the appropriateness of a strategy relative to the current market environment is a better approach.

A casual review of research literature produced by the various investment banks suggests that each institution has their own ‘house’ technique for identifying value relating to different strategies. Generally speaking, there are a number of common themes:

Mean reversion—over a long period volatility (realized or implied) will tend to revert to some long-term average. Consider Figs. 6.32 and 6.33. These diagrams illustrate the evolution of implied and realized volatility of the S&P 500 from 1996 to 2016. The long-term average of implied volatility is 19.13 % while the same measure for realized volatility is 17.45 %.

Fig. 6.32

Implied volatility of S&P 500 index options from 1996 to 2016 (Source: Barclays Live)

Fig. 6.33

Realized volatility of S&P 500 from 1996 to 2016 (Source: Barclays Live)

Knowing these long-term values would allow the analyst to determine if the current level of implied volatility is trading significantly away from the mean and whether it is likely to revert. However, there is no single agreed timeframe over which the mean is calculated and it is impossible to say with certainty the speed with which it will revert. This would make it difficult to rely on from a trading perspective.

Implied volatility is trading away from recent levels—if a stock has been trading at an implied volatility level of 25 % over some period but is now trading at 20 %, the investor may believe that it warrants further investigation to determine if it is trading cheap to fair value.

Predictive ability of implied volatility—this is based on the notion that observed option implied volatilities reflect market views on how the actual volatility of the underlying asset will evolve over the remaining life of the option. As a result, the different techniques analyse current implied volatility relative to realized volatility. As a simple example consider Fig. 6.34. Both implied and realized for the S&P 500 are trading away their long-term averages for the market (19.13 % and 17.45 %, respectively) but does this mean that they are both likely to mean revert in the short term? Perhaps there are a number of trade possibilities; for example:

• At the moment implied volatility has stabilized since falling from a mid-February peak. If the trader believed that implied volatility was going to increase they may decide to implement a long vega trade.

• If the trader believed that realized volatility would fall below implied, he may decide to implement a short gamma trade.

Fig. 6.34

Three-month implied and realized volatility for S&P 500 (Source: Barclays Live)

Realized volatility is expected to evolve at a different rate than implied volatility—Section 5.3 analysed a vega/gamma trade where a trader had sold options in anticipation of lower implied volatility. However, realized volatility was greater than the initial levels of implied and so the trader lost money through their delta hedge. From this it follows that:

• If the option trader sells options, the challenge is to ensure that the premium income is greater than the losses incurred from delta hedging.

• If the option trader buys options, the challenge is to ensure that the premium expense incurred is less than the profits generated from delta hedging.

• The trader will not profit if realized volatility evolves exactly in line with the initial levels of implied volatility, ignoring all transaction costs.

Volatility cones—another popular technique used to identify value is the concept of the volatility cone. There are a number of different ways in which these can be constructed and the following represents just one approach. Consider Fig. 6.35, which was constructed in the following manner:

• A number of maturities were selected for the x axis, for example 1-month, 2-month and so on.

• A sample period for the underlying asset was chosen, for example 2 years.

• The realized volatility for each maturity was measured on a rolling basis. So 1-month realized volatility is calculated using daily observations for preceding 4 weeks of data, the result of which is then annualized. The calculation is then repeated for each day of the sample period each time moving the measurement period forward by one day.

• From the data produced the minimum, maximum and average values for each maturity were collected and plotted on the graph.

• The current term structure of implied volatilities was then superimposed on the chart.

Fig. 6.35

Volatility cone for S&P 500 index options. Data as of 28th July 2014 (Source: Barclays Live)

What is the rationale for this approach? Tompkins (1994) argues:

….when evaluating the volatility input into an options pricing model, it would be fairly safe to say that this input would probably not exceed the highest actual volatility that had occurred over a comparable time period in the past or be below the lowest actual volatility that had ever occurred.

Figure 6.35 shows that as the maturity of the option increases the difference between the highest and lowest realized volatilities narrows. This would confirm the idea suggested earlier that shorter-dated volatility is more volatile higher than longer-dated volatility. The average realized volatility for each maturity over this sample period is about 11 % and perhaps the fact that this average is quite flat across all maturities is further confirmation of mean reversion.

Notice that the current term structure of implied volatility sits outside the maximum boundary for realized volatility. This could perhaps be an indication that longer-dated implied volatility may be trading above some notion of fair value and that it may revert to a lower level, even though it is still below the long-term value of 19.46 %.

6.5.2 Correlation

Figures 6.36 and 6.37 show the path of implied and realized correlation since 1996, respectively. The long-term average values are 0.47 (implied) and 0.37 (realized), indicating that similar to volatility, implied correlation trades higher than realized.

Fig. 6.36

S&P 500 index implied correlation. March 1996–March 2016 (Source: Barclays Live)

Fig. 6.37

S&P 500 index realized correlation. March 1996–March 2016 (Source: Barclays Live)

Figure 6.38 adapts the concept of the volatility cone and applies it to correlation. The figure charts realized correlation for a variety of maturities charting the maximum, minimum and average values. Overlaid on the diagram is the current term structure of implied correlation, which similar to implied volatility suggests that perhaps longer-dated implied correlation may revert to lower levels.

Fig. 6.38

Correlation cone for S&P 500. Data as of 28 July 2014 (Source: Barclays Live)

6.6 Conclusion

Two popular market factors traded in the equity derivatives market are implied volatility and correlation. An understanding of how these variables move provides a solid foundation for the construction of hedging and trading strategies.

The chapter started off with a short statistics refresher which focused on how these measures are defined and interpreted. The two measures were then linked by means of modern portfolio theory as well as the demand and supply characteristics of single-stock and index implied volatility. This illustrated that due to the effects of correlation, index volatility should trade lower than average single-stock volatility.

Section 6.4 considered some of the characteristics of volatility and correlation. Although the original option valuation framework assumed that volatility was constant it was highlighted that there is no empirical support for this assumption. For example, implied volatility displays not only a term structure but will also vary according to strike. These two characteristics taken together give rise to the concept of the volatility surface. A similar analysis was performed with respect to correlation looking at how this the metric moved in relation to such factors as the absolute level of the cash market and implied volatility.

The final section of the chapter considered a number of general principles that are used by practitioners to identify if volatility is fairly valued.

Appendices

Appendix 1

6.1.1 Calculating Variance, Standard Deviations (‘Volatility’), Covariance and Correlation

This appendix provides a short review of how measures of dispersion and association are calculated.

6.1.1.1 Calculating Variance and Standard Deviation (‘Volatility’)

Variance is often used in finance to describe risk and uncertainty and within the context of this book its most obvious use is variance swaps. Standard deviation/volatility is commonly used within an option valuation framework. Table 6.5 shows how variance and the standard deviation are calculated.

Table 6.5 Calculating variance and standard deviation

The first column shows 20 hypothetical closing prices for the FTSE 100 index. The second column calculates 19 daily returns. Rather than using the ‘traditional’ method of calculating returns ((P _t+1/P _t )−1) the calculation is performed using logarithms. The returns are therefore calculated as ln (P _t+1 /P _t ). In the same column these returns are summed and the mean is calculated.

Column 3 considers how clustered these returns are around the mean. If they are widely dispersed around the mean the difference between any individual return and the mean would be large. If they are tightly clustered around the mean the difference between the individual return and the mean would be small. Since the sum of the deviations will always be zero, the individual values in column 3 are squared and shown in column 4. These values are then summed and then divided by n−1, which in this case is 18, to return the variance which is expressed as squared units of the underlying making this result a ‘squared percentage’, which is not immediately intuitive.

Taking the square root of the variance returns the standard deviation, which is shown at the bottom of the fourth column. By convention in financial markets, volatility is measured as one standard deviation. This figure could then be converted into an annualized equivalent using the principles outlined in Sect. 6.2.1.

6.1.1.2 Calculating Covariance and Correlation

Covariance indicates how two random variables behave in relation to each other. This is often converted into a correlation coefficient which is a unit-free measure of the strength and direction of a linear relationship between two variables.

An example of how these metrics could be calculated is shown in Table 6.6, which uses hypothetical values from the FTSE 100 and S&P 500. Columns 1 and 2 are identical to those shown in Table 6.5. The third and fourth columns show the index values for the S&P 500 and their returns, respectively. The fifth and sixth columns measure the extent to which these values are clustered around the mean. Column 7 is the product of columns 5 and 6. Column 7 is then summed and divided by n−1 to give a covariance of −0.0080 %.

Table 6.6 Calculating covariance and correlation

The correlation coefficient, ρ, is calculated by dividing the covariance by the product of the standard deviations of the two underlying assets. The standard deviation for the FTSE 100 was 2.7821 % and was 4.1267 % for the S&P 500¹⁰. This returns a value of −0.0697 or −6.97 %.

Appendix 2

6.1.1 Calculating Forward Volatility

Forward volatility is an implied volatility quote, whose value is known today but applies to a future time period. The formula is:

$$ \mathrm{Forwardvol}=\sqrt{\frac{{\mathrm{vol}}_2^2\times {T}_2-{\mathrm{vol}}_1^2\times {T}_1}{\left({T}_2-{T}_1\right)}} $$

• T ₁ (in years) = Shorter option maturity

• T ₂ (in years) = Longer option maturity

• Vol₁ = Implied volatility for an option that matures at T ₁

• Vol₂ = Implied volatility for an option that matures at T ₂

What is the 1-year forward, 3-month forward implied volatility given the following parameters?

• T ₁ = 1 year

• T ₂ = 1.25 years

• Vol₁ = 13.50 %

• Vol₂ = 13.70 %

$$ 14.47\%=\sqrt{\frac{13.7\%\times 1.25-13.5\%\times 1}{\left(1.25-1.00\right)}} $$