HelloI have a question about the relation between the var fair strike Kvar, the replication portofolio (OTM calls/puts) and the vega notional fixed on the termsheet.Exemple: you are short a VS (on a stock) with a vega notional Nvol of 50,000€ and a vol fair strike of 18% (Kvar=3.24%=324).1/ First question:In which unit the vega notional is fixed? in €/(vol point)? Is it different of 50,000 vega (in terms of options)?2/ Second question:The expression of Kvar isTherefore you can compute "theoretical" weights of calls/puts of replication portofolio.But weights are very small, in the order of 10E-4 or -5.I need to link "theoretical" weights and the vega notional Nvol to have practical weights for the trader.You have the relation for the notional Nvar (in €/(vol point)²): Nvar=Nvol/(2*Kvol) with Kvar=Kvol².Which is the factor that I must multiplicate with "theoretical" weights for have "practical" weights? Is it Nvar or an other arbitrary factor? 3/ Third (and last) question:Each weight of replication portofolio option can expressed in option number or in vega.Which is the best method?Thanks for your help.

1) Vega notional is expressed in EUR/vol point. However it is given for indicative purposes: the true notional is in EUR/var point with the conventionVariance notional = Vega Notional / (2 x Strike)where Strike = Kvol x 100 = sqrt(Kvar) x 100 in your notationsIt is conceptually the same as Vega amount for options -- except that the vega of options varies with spot and maturity, whereas "notional" conveys the idea of a fixed number2) You need to discretize your integral with meaningul strikes (usually the listed ones). The theoretical weights should be multiplied by the variance notional to give you a %Spot notional (or perhaps %Forward... check this)3) I would say %Spot notional or number of options is the most meaningfulExamples of replication portfolio decomposition can be found in Derman's paper ("More than you ever wanted to know about volatility swaps") or alternatively in my introduction article published in the March issue of Wilmott Magazine.Hope this helps,SB

thanks for your answer.Just still some questions... 1/ You write that Strike = Kvol x 100 = sqrt(Kvar) x 100 in my notations. But if Kvol is quoted in %, the variance notional becomes:variance notional = vega notional x 10,000 / (2 x Kvol)Example: Nvol = vega notional = 50,000 EUR/vol point and Kvol = 18%.So you have Nvar = variance notional = 50,000 * 10,000 / (2 * 18%) = 13,888,888.88 EUR/var point.Do you know if Kvar is quoted in %: Kvar = Kvol² = a) 0.0324=18%² b) 324=18%² x 10,000 c) 3.24% ?2/ What do you mean by %spot notional ? Example: my replication portofolio is composed of call with strike 110% (of the spot). Imagine the theoretical weight is 1.10E-5, I need to buy weight x variance notional = 10E-5 x 13,888,888.88 ~ 139 calls which strikes to 110%. that's right?thx

1/ convention is to use vol/var POINTS, not %e.g. for a vega notional of 50,000 EUR per vol point (meaning: payoff is aprrox 50,000 EUR for each point of realized volatility above the strike)Nvar = variance notional = 50,000/ (2 * 18) = 1,388 EUR/var point2/ For an option, EUR payment amount = %Notional x Option Payoff in %e.g. for an ATM call and a EUR 10,000,000, if the return at maturity of the underlying is +10%, then the payoff = 10% x 10mn = EUR 1,000,000Call option payoff = Nb options x max(0, S(T) - K) = Nb options x S(0) x max(0, S(T)/S(0) - K/S(0))Hence %Spot Notional = Nb options x SpotSB

OK. and for variance strike:Do you know if Kvar is quoted in %: Kvar = Kvol² = a) 0.0324=18%² b) 324=18%² x 10,000 c) 3.24% ?With your definition, good answer is b) ie 18²=18% * 10,000=324. right?

2/ is still not clear to me, in this example the Nb options = %Spot Notional/ Spot. The %Spot Notional = weight x variance notional = 1.10E-5 x 1,388 = 0.15268. Let's assume S(T)= 100% then you would get Nb options = 0.15268/ 100% = 0.15268 call.This seems a wrong amount of calls to me, if you do this for all the strikes you never get above 1 call over all the strikes. I guess I am doing something wrong. Could phaedo or someone explain this to me?ThanksRS

Totally OK with you. I get the same pbm.In fact that depends on the value of vega notional. If you have Nvol=50,000 €/vol pt and Kvol=18% (ie Kvar=324), you compute that the var notional is Nvar=1,388 €/var pt.Therefore weights should be still below 1 after have been multiplied by Nvar...But if you put 5,000,000 €/vol pt for vega, you get Nvar=138,888 €/var pt. And here, you obtain weights>1...In conclusion: what does it pass in practical if you must hedge a VS with a small vega notional (typically 100,000 or 50,000)?

I am not sure where you get this theoretical weight of 1.10E-5If you discretize the integral with, say, dt = 1%, you get a number of 50% puts equal ton(50%) = Nvar x (1/0.5^2) x 1% = 1388 x 1/0.25 x 1% = 55.5and so on for the 51%, 52%... strikesNote that Derman proposes a different hedging procedure where the weights correspond to a piecewise approximation of [ (ST-S*)/S* - ln(ST/S*) ]. I suspect this procedure is better when only a limited number of calls and puts is available.SB