Marco Aita

Market estimation of implied equity correlations through
index and single stock options: a case study
There is a huge interest among practitioners in finding efficient and robust ways of
estimating equity correlations: growing volumes on correlation sensitive OTC
products, notably rainbows, are adding to existing exposure on baskets, and
increasing competition is forcing market makers to exploit all the information
available in traded instruments.
Since correlations in themselves are not market observable (in the sense that we
do dot have enough contingent claims enabling us to invert a market quote to find
out the “implied” correlations) and the market microstructure exhibits a strong
unbalance (market makers usually sell correlation embedded in various products to
customers), in most cases the correlation pricing problem simplifies into finding a
sufficient premium over historical correlations.
In a competitive environment market forces are driving this premium down, forcing
market makers to refine their “parameters discovery” process in order to avoid to
be arbitraged.
Arbitrage opportunities can in effects arise since in a typical financial market
structure we can observe options on a subset of the market index (i.e. basket
options on n liquid stocks with their weights) and options on these n stocks: the
implied basket volatility (from the basket option) and the estimated basket volatility
calculated from single stock implied volatilities and historical stocks correlations 1 do
not coincide, with the latter generally lower than the former. This is an effect
similar to the one addressed by the various volatility surface models for single stock
options, only made worse due to the great number of estimated parameters – the
correlations. According to this, any significant deviation from a correlation structure
consistent with index option quotes would imply arbitrage opportunities, e.g. given
a sufficiently diversified portfolio (i.e. a correlation exposure comparable with the
traded index/basket options), the trader would be better off selling index options
instead of complex OTC derivatives.
In order to solve this inconsistency, the market practice is to equate the implied
volatility to the estimated volatility and to invert for a single “implied correlation”
which then represents the “calibrated” global correlation.
Obviously, it would be more satisfactory to integrate in the parameter discovery
process an econometric approach, using time series on stock returns, with the
“average” market estimator embedded in index option quotes. For instance, we
would like to identify a way of “deforming” the historical correlation matrix, by
1 Some assumptions allow one to write
∑
2
σ = σ C σ , where the i
volatilities, and C is the correlation matrix (here we consider equal weights)
index
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i,
k
i
ik
k
σ are the single stock

using one or more parameters, into a new correlation matrix fitting the implied
volatilities on basket options (see for instance J.P.Morgan’s RiskMetrics Monitor 4th
Quarter 1997, "A methodology to stress correlations" by Chriss Finger 2 ).
2 Available at http://www.riskmetrics.com/pdf/journals/rmm4q97.pdf
MPS Finance Banca Mobiliare SpA - Via Nino Bixio n.2 - 53100 Siena Tel.+39-0577 209111 Fax+39-0577 209594 Aderente al Fondo Interbancario Tutela Depositi - Codice Banca 3163.3 Codice Gruppo 1030.6. Cap.Soc.130.000.000
Euro. La presente pubblicazione è stata prodotta da MPSFinance. Le informazioni qui contenute sono state ricavate da fonti ritenute da MPS FINANCE affidabili, ma non sono necessariamente complete, e l'accuratezza delle stesse
non può essere in alcun modo garantita. Per quanto non riprodotto nel presente disclaimer si fa espresso rinvio al disclaimer agreement generale posto nel sito www.mpsfinance.it ed alle relative condizioni del servizio.