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Mainframes
Topic Author
Posts: 89
Joined: June 1st, 2005, 5:37 pm

### FX currency correlation

Hi,Given a currency triangle one can derive an implied correlation from knowing all three implied volatilities. My question does this correlation ever exceed 1 (or go less than -1). Can you give an example if it ever does. Also. assuming that it did, would there be a trade that one could carry out in order to profit from it?

cemil

### FX currency correlation

Correlation is between -1 to 1. So, you can t have a correlation under -1 and over 1

fuez
Posts: 16
Joined: September 9th, 2004, 1:56 pm

### FX currency correlation

QuoteOriginally posted by: MainframesHi,Given a currency triangle one can derive an implied correlation from knowing all three implied volatilities. Really? if yes: How? Doubting: Only by knowing the implied volatilities?

Mainframes
Topic Author
Posts: 89
Joined: June 1st, 2005, 5:37 pm

### FX currency correlation

Hi,Let us consider three currencies A,B,C. Let S_ab, S_{ac}, S_{cb} denote the three exchange rates. We are using the notation such that S_{ij} is the price of i in terms of j. That is to say that i=S_{ij} * j. As there are only two independent exchange rates due to the triangle constraint we have that S_{ab} = s_{ac } * S_{cb}. (***)Let v_{ab}, v_{ac}, v_{cb} be the implied volatilities of the the respective exchange rates. It has been shown that (google implied correlation to find out who by), by assuming the "S" processes are lognormal and by using constraint (***), the implied correlation obeysv_{ab}^2 = v_{ac}^2 + v_{cb}^2 + 2* rho *v_{ac}*v_{cb},where rho is the correlation. This can be rearranged to giverho = ( v_{ab}^2 - v_{ac}^2 - v_{cb}^2 ) / (2*v_{ac}*v_{cb}) . I believe that this is quite easy to prove by using stoch calculus and (***).

Mainframes
Topic Author
Posts: 89
Joined: June 1st, 2005, 5:37 pm

### FX currency correlation

Oh yeah, the equation in the previous post means that implied correlation can become greater than 1 or less than -1 if the implied volatilies are arranged in a certain way. Maybe this never happens in real markets?

cemil

### FX currency correlation

Mainframes can you explain to me what it means "correlation over 1" or "correaltion under -1"?This assertion has no sense.Don't forget the reality of things!

Mainframes
Topic Author
Posts: 89
Joined: June 1st, 2005, 5:37 pm

### FX currency correlation

Sorry, I'm not being clear enough and am not an expert in these matters either. I don't mean "actual correlation" which must be between 1 and -1, I mean "implied correlation", where "implied correlation" is given by the formula in my previous post and is not a real quantity. "Implied correlation" is dependent upon the 3 "implied volatilities" and these "implied volatilities" are numbers that are decided by traders who are buying and selling options (i.e it's their perception of how the volatility will behave). I suppose I could rephrase the question and ask, are there situations where the implied volatilies of a currency triangle (as quoted in the market) are such that the equation in my previous post is greater than 1 or less than -1?

Jonathan81
Posts: 122
Joined: April 22nd, 2005, 6:25 am

### FX currency correlation

Proof:Proof of the formula :v_{ab}^2 = v_{ac}^2 + v_{cb}^2 + 2* rho *v_{ac}*v_{cb},It is very easy ito on X*Y and egalization of variancei see on ly brownian term X = S_ab and Y = S_ac and Z = S_cbIntegration by part : dX = YDZ + ZdY + d<Y,Z>so v_{ab} dW1 = v_{ac} dW2 + v_{cb}dW3 with d<W2,W3> = rhothen v_{ab}^2 = v_{ac}^2 + v_{cb}^2 + 2* rho *v_{ac}*v_{cb}However V(A) = V(B+C) = V(B) + V(C) + 2*Cov(B,C) / sqrt(V(B)*V(C)) * sqrt(V(B)) * sqrt(V(C) )By definition of rh0 = 2*Cov(X,Y) / sqrt(V(X)*V(Y)) is between -1 to 1J
Last edited by Jonathan81 on October 19th, 2006, 10:00 pm, edited 1 time in total.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### FX currency correlation

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.

farmer
Posts: 13479
Joined: December 16th, 2002, 7:09 am

### FX currency correlation

If volatility between B and C is zero, and between A and C is 1 and A and B 2, what do you get?

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### FX currency correlation

QuoteOriginally posted by: farmerIf volatility between B and C is zero, and between A and C is 1 and A and B 2, what do you get?BOOM!rho_ab = +INF (assuming we have v_{cb} approaching zero from the positive side)rho_cb = -1.25rho_ac = -INF (assuming we have v_{cb} approaching zero from the positive side)If volatility between B and C is zero, then currencies B and C are locked together (e.g., a pegged exchange rate). Then we have A's rate of exchange with C being more stable than it's rate of exchange with B. This seems more like a classic fx price arbitrage where you pair-trade AB and AC when the rates violate the known, fixed exchange rate for BC.Farmer, are you suggesting that anomalous values for implied correlation coefficients could be the basis for understanding 3-way fx arbitrage situations between three currencies?
Last edited by Traden4Alpha on October 19th, 2006, 10:00 pm, edited 1 time in total.

Mainframes
Topic Author
Posts: 89
Joined: June 1st, 2005, 5:37 pm

### FX currency correlation

Okay are there situations in the market where v_{ab} > v_{ac} + v_{cb}where a,b,c form a currency triangle. I don't think it happens with the EUR/CHF/USD triangle, however, is it possible to occur with any of the emerging market currencies?

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### FX currency correlation

QuoteOriginally posted by: MainframesOkay are there situations in the market where v_{ab} > v_{ac} + v_{cb}where a,b,c form a currency triangle. I don't think it happens with the EUR/CHF/USD triangle, however, is it possible to occur with any of the emerging market currencies?First, I don't trade currencies, so I can't speak to whether specific triads might show anomalous values of implied correlation. However, I would look for triads that are tightly correlated. If the normal correlation between the currencies is approaching 1, then its more likely that a tradable pricing inefficiency can drive the implied correlation above 1.Note that there may be tradable errors in implied correlation where the implied correlation is within the [-1,1] interval. The key is to have a confident estimate for the true rho and observations that the implied value is different from that. It's not as pure an arbitrage situation as the one in which implied rho violates the [-1,1] bounds on correlation because one's estimate of future rho could be wrong and the implied rho could be right. One could call this correlation speculation rather than correlation arbitrage.

flairplay
Posts: 130
Joined: September 26th, 2006, 1:34 pm

### FX currency correlation

No such trivial arbitrage exists in FX markets.The question one normally asks is if current implied vols are a good consensus for future historicals, and if not how to get meat out of it. The question is trivial, the answer is fun to conceptualise as it is dependent on your judgement and imagination, and forming an implementable strategy within the remit your trading mandate is the best bit.But as you see this is pure trading rather than anything quanty. Just as you may decide that the price of Microsoft shares is too high or too low - no arb there. Just trading...

Mainframes
Topic Author
Posts: 89
Joined: June 1st, 2005, 5:37 pm

### FX currency correlation

Fairplay, are you saying that there are no situations where v_av > v_ac + v_cb? Or are you saying that if such a situation arose it wouldn't be an aribtrage?
Last edited by Mainframes on October 21st, 2006, 10:00 pm, edited 1 time in total.