October 20th, 2006, 8:00 pm
QuoteOriginally posted by: Mainframesrho = ( v_{ab}^2 - v_{ac}^2 - v_{cb}^2 ) / (2*v_{ac}*v_{cb}) . Actually, there are three possible implied correlations:rho_ab = ( v_{ab}^2 - v_{ac}^2 - v_{cb}^2 ) / (2*v_{ac}*v_{cb}) rho_cb = ( v_{bc}^2 - v_{ab}^2 - v_{ac}^2 ) / (2*v_{ab}*v_{ac}) rho_ac = ( v_{ac}^2 - v_{ab}^2 - v_{cb}^2 ) / (2*v_{ab}*v_{cb})Taking the first implied correlation coefficient, lets look at what an "impossible" excessively-high value of correlation might mean in terms of the implied volatilities:1 < rho_ab implies1 < ( v_{ab}^2 - v_{ac}^2 - v_{cb}^2 )/(2*v_{ac}*v_{cb})(2*v_{ac}*v_{cb}) < ( v_{ab}^2 - v_{ac}^2 - v_{cb}^2 )(v_{ac}^2 + 2*v_{ac}*v_{cb}+v_{cb}^2) < v_{ab}^2( v_{ac}+v_{cb} )^2 < v_{ab}^2Given that v_{ij} > 0 by convention, we can sqrt this to yield:v_{ac}+v_{cb} < v_{ab}Thus, if the implied rho_ab > 1, then the implied v_{ac}+v_{cb} is "too low" or the v_{ab} is "too high" to hold the implied rho within the theoretical bounds. This suggests a trading strategy of shorting volatility in AB and buying volatility in AC and CB. Because one does not know which volatility is out of bounds, one would probably need a position in all three. A more sophisticated analysis would decompose rho_ab into rho_true_ab and rho_error_ab where rho_true_ab is the true value of the future correlation and thus bounded on [-1,1] and rho_error_ab represents the arbitragable error in the implied value. If rho_ab > 1, then rho_error_ab > (rho_ab - 1) assuming that the true correlation will always be less than 1. This could be algebraically run through the implied correlation equation to define how the rho_error_ab divides among the three implied volatilities. Furthermore, we have three implied correlations that might each have respective rho_error_ij estimates and thus have three quadratic inequalities for the error in the three implied volatilities. With three inequalities in three unknowns, we have some hope of isolating which implied volatility represents a mispricing.But before committing to this strategy, we would need to consider how the implied volatilities are computed and whether they contain biases that invalidate this approach. For example, interactions with the volatility smirks in the three different fx rates could easily lead one to compare an excessively large estimate of one currency pair's implied volatility (i.e., one out on the smirk) to an excessively small estimate of another currency pair's implied volatility (i.e., one at the minimum of the smirk). Because the implied volatility calculation presumes a parametric distribution (volatility being a parameter), the quality of this analysis and this arbitrage opportunity may depend on whether the true distribution for the currency fits the employed parametric model.