hi friends, i have been a constant beneficiary of the forum yet havent found a chance to open a topic. till now. first days of 2016, and that is my first topic. i apologise if that is already opened and discussed but not been able to locate anything on the matter. I am interested in finding a methodology to approximate low delta FX options beyond 10 delta points, which are usually contributed to financial data service companies like Reuters and Bloomberg for volatility surface construction or seldom traded in the brokers hence people have knowledge of. I have managed to find in the forum the Murex and Derivatech extrapolation methodologies which didnt shed much light so that the formula in Murex extrapolation for V(0.1c) = V(25c) + 2.3* abs[V(25c)-V(10c)] where V ( x c ) is said to be the volatility for X delta ( in % ) call derivatech formula is also similar where it uses maximum value of 10delta volatility for call or the ATM + 2.3*[10D BFLY + 0.5*10D RR ] for Bloomberg, most of the data contributed on the OVDV page are up to 10D and beyond that, it is extrapolation. it is linearly extrapolation beyond 10D. I have also managed to find Castagna and Mercurio's methodology with the usage of 3 different strikes and volatilities, but not quite sure if i got that right as what K1, K2 and K3 refers to use in the formula. For example, in the market, I had to use coarse approximation to determine 5D RR and 5D BFLY with multipliers such that 25D RR and BFLY is liquid- i know its levels and 10D RR and BFLY trade at a certain multiplier. Hence for 5D RR and BFLY, those multipliers should be that much and 5D Call= ATM + (5D RR)/2 + 5D BFLY the latter of two levels are arbitrary multipliers based on my best guesstimation. such that 25D RR x 1.9 = 10D RR hence to get 5D RR level, multiplier should be 2.5 kind of thing. Same goes with BFLY multiplier. Would really appreciate if any paper or market insight. Thanks.

Implied volatility extrapolation can be quite arbitrary. If forced to do it, what I would do is fit the existent IV points at a fixed maturity with J. Gatheral's SVI method,which is easily googled.

Thanks for the Gatheral's method Alan. I looked at the paper and presentations as well as some thesis that focused on the methodology. However since i am not a quant unfortunately after certain point I am lost. When I finished reading the paper, I understand that raw SVI parameterization equates total implied variance w = a + b*[p*(k-m)+sqrt{(k-m)^2+sigma^2}]with 5 different parameters: a,b,p,m, sigmaand since raw SVI results in arbitrage opportunities, SVI-JW parameterization is suggested with different set of parameters which are ATM variance, ATM skew, slope of put wing, slope of call wing and minimum implied variance. those new parameters, which are observable in the market, relate to the raw SVI parameters and then you can use them to calculate your low delta option's volatility. As you nicely put, I am after a fixed maturity's options whose delta is low and want to easily calculate- rather than doing calibration and iteration as per the paper. To make it practical, based on my previous experience, from time to time, in USDTRY vanilla market, ask for price for 5D Call either outright or as a spread. since in the market, available levels are ATM and 25D RR, plus 25D BFLY whose levels tend to more stable, we are okay till 25D level. Assuming a multiplier between 25D and 10D also works pretty much to get 10D RR and BFLY levels. So that is okay, too. However beyond 10D is a bit unchartered area and it's nice to have a quick way to calculate those options without passing them or making them wider. Therefore, let's say for a 3-month USDTRY 5D Call, may I apply Gatheral's method ? to my understanding, I can but would appreciate the feedback if I have got it right.for SVI-JW parameterization set values, I have 3M ATM vol hence variance. ATM skew- should I take this at 3M 25D RR ?slope of put wing: (10D Put Vol- 25D Put Vol)/(0.10-0.25) slope of call wing: similar to above minimum implied variance: here is where I am a bit confused. should i solve for the unknown parameters with what I have in hand? Eventually, if i can get all this, then i believe I can apply the equation 3.1 , w = a + b*[p*(k-m)+sqrt{(k-m)^2+sigma^2}] with strike level k, that will correspond to the 3M USDTRY 5D call level? am i on the right track or lost completely and talking nonsense? appreciate the feedback

How many reliable mid-point implied vol. quotes do you have at the maturity you are interested in? (Quotes at specific strikes)

I have ATM and 25D USD Call always in my surface. so that is 2. depending on the day, broker market also has strike specific contracts- 1 or 2 depending on the day. also i have access to other market-maker's surface through their platforms as well as through Bloomberg's OVDV page, mid-market level for variety of strikes. therefore, i can easily have at least 5 and many more if needed be

I see. Well, 5 might be enough but say 9 would be better, atm and 4 on each side.I would start by trying to fit the raw SVI expression; it will probably suffice. This takes some effort to get working;you need a decent non-linear (least squares) optimizer that accepts constraints. The trick is to give theoptimizer some helpful constraints. For example, you might be confident that theminimum implied vol might always be greater than 0.06 in your market (i.e., 6% on an annual basis).Then, work out the minimum implied vol analytically for the SVI model and apply that relation as a constraint. Once you've found an optimum parameter set, graph the IV vs strike and make sure it looks reasonable,passing very close to your market IV's.Also, work out the market implied risk-neutral density from the Black-Scholes formula combined withthe substitution of IV(k) from the SVI formula and the Breeden-Litzenberger relation. This isan exercise in differentiation with the chain rule. If you get stuck (it's tedious), it can be googled somewhere,or maybe it's in Gatheral's book -- I don't recall.Plot that density vs. K (using your fitted parameters) from K=0 to K large -- if it's everywhere non-negative, thereare no arbitrage violations (neg. butterfly spreads), and you're done! Now you have a fitted IV(k) for any log-strike k, and they are all arbitrage-free.

thanks for the insights Alan. I will take a look at Gatheral's book. seems a bit out of my scope that task but good to know the methodology and appreciate your guidance. if ever i manage to have this done, the other problem i will face is to how to have those low delta marked to market at the end of the day. why? because i currently use different system which isn't capable of calculating the deltas beyond 10D as those are my furthest contribution points for the volatility surface construction. oh well, that was a nice conversation anyhow have a nice week ahead