Hi allI have a square root function of the form: sqrt(a1*v(t)+a2*v(t+dt)), where a1 and a2 are constants, v(t) and v(t+dt) are price variances at time t and t+dt, respectively. Does anyone know what is a good series or polynomial approximation for this square root function---in terms of sqrt(v(t)) and sqrt(v(t+dt))?Many thanks

QuoteOriginally posted by: wdlHi allI have a square root function of the form: sqrt(a1*v(t)+a2*v(t+dt)), where a1 and a2 are constants, v(t) and v(t+dt) are price variances at time t and t+dt, respectively. Does anyone know what is a good series or polynomial approximation for this square root function---in terms of sqrt(v(t)) and sqrt(v(t+dt))?Many thankswell, you could try sqrt(a1*v(t)+a2*v(t+dt)) = sqrt{(v(t)(a1+a2)+a2(v(t+dt)-v(t))} = sqrt(a1+a2) sqrt(v(t)) sqrt{1 + e X} = sqrt(a1+a2) sqrt(v(t)) {1 + e/2 X -1/8 e^2 X^2 + ... },where e = a2/(a1 + a2), and X = [v(t+dt)/v(t)] - 1.You can certainly write X in terms of sqrt(v(t)) and sqrt(v(t+dt)) if you like ...

QuoteOriginally posted by: AlanYou can certainly write X in terms of sqrt(v(t)) and sqrt(v(t+dt)) if you like ... Do you mean to write X =v(t+dt)/v(t)-1=(sqrt(v(t+dt))/sqrt(v(t)) +1)*(sqrt(v(t+dt))/sqrt(v(t)) -1) ? I would like to have separate terms of sqrt(v(t)) and sqrt(v(t+dt)). If I approximately write sqrt(a1*v(t)+a2*v(t+dt))~=a1*sqrt(v(t))+a2*sqrt(v(t+dt)), should there be some convex correction terms?Many thanks for your answer.

Do you mean to write X =v(t+dt)/v(t)-1=(sqrt(v(t+dt))/sqrt(v(t)) +1)*(sqrt(v(t+dt))/sqrt(v(t)) -1) ? ... Well, that's true. I just meant X = (sqrt(v(t+dt))^2/(sqrt(v(t)))^2 - 1.I would like to have separate terms of sqrt(v(t)) and sqrt(v(t+dt)). If I approximately write sqrt(a1*v(t)+a2*v(t+dt))~=a1*sqrt(v(t))+a2*sqrt(v(t+dt)), should there be some convex correction terms?... Yes, the correction terms are the difference between your answer and mine. Seriously, you got an answer, but didn't like the form of it. Apparently, you are looking for a double powerseries in positive powers of the two sqrt roots. Any other form makes you unhappy.Such a form may not work for this function.