FX Volatility smile in delta space vs. excercise price space

struggler · January 23rd, 2011, 4:19 pm

I am confused with the concept of implied volatility in delta space vs implied volatility in excercise price space. I was trying to apply Malz to reach risk neutral distribution. I got the vol in delta space using Malz equation and I backed out the strike. The papper then talks about mapping to the implied volatility in excercise price space. If for a certain delta there is an implied vol (in delta space) and an excercise price, how come for that same excercise price there is a different Implied vol (vol in excercise price space). Please explain to me that concept and how to convert from the delta space to the excercise price space.Many thanks

daveangel · January 23rd, 2011, 4:24 pm

you have vol versus delta - you imply the strike from the delta and the vol by inverting the relationshipdelta = N(d) where d is the BS moneyness parameter

struggler · January 23rd, 2011, 5:09 pm

I have done this part I am not sure of the next step. Malz is refering to moving from delta volayilty space to excercise price volatility space [sigma(Delta) to sigma(excercise prise)] by solving 2 simultaneous equations. I am confused with the concept of conversion from one space to the other.

struggler · January 23rd, 2011, 5:09 pm

I have done this part I am not sure of the next step. Malz is refering to moving from delta volayilty space to excercise price volatility space [sigma(Delta) to sigma(excercise prise)] by solving 2 simultaneous equations. I am confused with the concept of conversion from one space to the other.

MCarreira · January 23rd, 2011, 8:30 pm

In FX you quote the vol parameters (ATM, RR, Fly) for deltas, and then with {forward, delta, vol, time}, you find the strike.So a 10 delta USD Call BRL Put could be a 2.10 strike or a 4.00 strike, depending on the parameters (in 2008, a 1y 10 delta like that had a 5.92 strike and traded at around 3%).So you look at the delta space first and then find the corresponding strike.

Gmike2000 · January 23rd, 2011, 8:58 pm

I have a stupid question about this actually, since I am a rates guy and not fx-versed at all.So the fx guys tell me they look at RR numbers to "tell" whether the mkt expects to go up or down. Besides the fact that they could not cite empirical evidence for this being a true relationship, the problem I have with this statement is that they quote RR based on deltas of a lognormal model. In a lognormal model, the mean of the distribution will not be the same as the forward level in the market.So that means the RR of say 25 delta is not exactly "centered" in the middle of the distribution, especially for long term options. It is below the forward by 1/2*T*vol^2 or something.So then the RR is biased by construction and should not really tell you much about where the market thinks it will move.Yes? No?

daveangel · January 23rd, 2011, 9:08 pm

yes - you are right and if anyone claims that risk reversal tells you which way a market is going to move then its bs.but it does tell you where people will have pain if does move. in USDJPy you have (had) pronounced RR but USDEUR much less so. in the former you had carry trades whereas in the latter more directional and balanced speculative positioning.

MCarreira · January 23rd, 2011, 9:23 pm

In BRL the RR is always positive (so USD Calls are more expensive), even though most of the time the market was expecting the BRL to appreciate; it really could be seen more as: if the market moves that way, vols will be higher and the moves will be big and trending on the BRL devaluation (see May 2006, Aug 2007, Sep/Oct 2008).Even now the RR is around 5%.

struggler · January 23rd, 2011, 9:29 pm

I guess my problem is in understanding what Malz meant by "vol in excercise price space",is he simply saying once I get the vol (from the parbolic equation) and plot it against the strike (the one I drive using the method you guys mentioned) that would be the "vol in strike space"?. Thanks

MCarreira · January 23rd, 2011, 9:41 pm

That's my view.

struggler · January 23rd, 2011, 10:52 pm

thank you for the clarification. I wonder if you can help me with deriving the risk neutral distribution. I have tried to claculate the 2nd derivative of the call price with respect to strike using actual differences between the calculated call prices and strikes (trial 1) and I tried to use Breeden-Litzenberger by trying to find the value of the butterfly between 3 strikes (trial 2) in each of my trials I get negative numbers and numbers greater than one. Please refer to the attachment. Thanks alot for your help

MCarreira · January 24th, 2011, 6:05 pm

An alternative is:1. Build the 99 to 1 delta grid and find the strike range2. Build a strike grid (regularly spaced in strike)3. For each strike solve the 2-equations system in delta and vol4. Then, given strikes and vols, find the premia5. Then finding d2c/dk2 should be more straightforward (you should expect something similar to gamma)

struggler · January 26th, 2011, 9:34 pm

Thanks to your instructions MCarreira I now have the Smile in delta space and the smile in the strike space which look consitent with different publications discussing this topic. I however I am still facing a problem in Breeden-Litzenberger. the shape of the distribution looks ok but for the differnt strikes and call prices I get very high probabilites that exceeds one for some of them . Please advise what went wrong.Many thanks

MCarreira · January 26th, 2011, 11:10 pm

Divide by $G$17, not $G$17^2; once you've calculated dc/dK, you have a delta (non-dimensional, independent of the scale of S or K).

struggler · January 26th, 2011, 11:44 pm

using $G$17 makes the numbers more inline with probabilities, but can you pls elaborate on the rationale. iIam confused because by using $G$17 I am not actually calculating the 2nd derivative with respect to the strike and hence I am not really applying Breeden-Litzenberger.