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20

An alternative method for estimating /dS is presented in §6.3. There we show how Dermans sticky models are limited to only parallel shifts in the equity skew, corresponding to a volatility surface that is linear in the space dimension. Section 6.3 presents a different model that has a quadratic parameterization of the volatility surface and includes non-parallel as well as parallel shifts in the skew. Using principal component analysis, the model will be used to provide an estimate of the /OS term that is so important for dynamic delta hedging.

2.4 Implied Correlation

In some circumstances an implied correlation may be derived from associated implied volatilities - for example, in currency markets, when there are options available on two foreign exchange rates and the cross rate, or whenever there are traded spread options. Rearranging the formula for the variance of a difference,

CT.v-y - ol + cr2 - 2crvcrvp,

to solve for the correlation p gives

<J2x + a? - cr?-,. 2crxc>v

If there are traded options on X, Y and X - Y, putting these implied volatilities in the above formula gives the associated implied correlation. For example, X and Y could be two US dollar foreign exchange rates (in logarithms) so X - Y is the cross rate, and the implied correlation between the two foreign exchange rates is calculated from the implied volatilities of the two US dollar rates < and crv. and the implied volatility of the cross rate crw.

There is a convenient graphic representation of this implied correlation in the unique triangle with the length of each side equal to the implied volatility. The implied correlation is the cosine of the opposite angle, as shown in Figure 2.11. This idea can be extended to higher-dimensional simplices.

Similar calculations can be used for equity implied correlations, but they are based on some rather severe assumptions. If the correlation between all pairs of equities in an index is constant then the implied value of this constant correlation is approximated from implied volatilities of stocks in the index, cr, and the implied volatility of the index, crj as

p = (cr2 - 2 2)/(2 - .- (2.12)

where , denotes the weight of stock i in the index. Of course, the constant correlation assumption is very restrictive, and not all equities will be option-able, so they have to be omitted from the formula. The approximation (2.12) is therefore very crude and can lead to implied correlation estimates that are



Side length is a

Cosine of angle is implied correlation between x and

Side length is a

Side length is ax

Figure 2.11 Graphical representation of implied correlation.

Figure 2.12 Implied correlation in FTSE 100 equities.

greater than 1. Figure 2.12 shows that this was in fact the case at the end of 1992, for a constant implied correlation of the form (2.12) calculated using all the optionable stocks in the FTSE 100.

A quanto correlation is the correlation between an equity and an exchange rate. The issuer of a quanto option offers purchasers a protection against currency risk by determining the pay-off at a fixed exchange rate. The Black-Scholes type formula for pricing a quanto option on an underlying foreign equity with foreign currency price S and strike is:

= 5 ( ) ( ) - ~ ( - / )), (2.13)

where is the domestic discount rate, X is the predetermined domestic/foreign exchange rate, x = ln(S/Ke~Rl)/o/x + and R is determined by the

foreign discount rate rf and the covariance between the equity and the exchange



rate, as R = rr - , where p is the quanto correlation and and are the equity and the exchange rate implied volatilities.

Quanto implied correlations can be obtained by inverting this formula to back out the correlation between the equity and the exchange rate: The market price of a quanto option, the domestic and foreign interest rates, the equity and exchange rate implied volatilities (backed out from vanilla options) and values for all the quantities that appear in (2.13) except p must be obtained. Then they can be substituted in (2.13) to back out an implied correlation between the equity and the exchange rate.

In summary, there are some ways in which implied correlations can be backed out from option prices, but they can be very unstable. Instability arises because the true underlying asset returns have a non-linear relationship, and/or they are not jointly stationary, and/or because there are a number of rather questionable assumptions that are used for calculating implied correlations. In short, implied correlations should be used with more caution than implied volatilities.



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