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.

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 %

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