Hello everyone,I want to try and price a barrier option using the Local Volatility. I have managed to construct a functional volatility surface Sigma_BS_(A(S/K),B(T-t)) <A function of Spot and Strike, B function of time to maturity --> 'T' being the maturity time and 't' being the current time>. I am wanting to simulate the following log-normal Brownian process using the local volatility :dS= mu*S*dt + Sigma_LV( S(t),t ) *S*dZ where Sigma_LV is the local volatiltiy following the Dupire approach as below :Sigma_LV( S(t), t') ^ 2 = { Sigma_BS^2 + 2*t'* Sigma_BS * dSigma_BS/dt + 2(r-q)*S*t'* dSigma_BS/dS } / { (1+S*sqrt(t') * d1 * dSigma_BS/dS ) ^2 + S*S*t'*Sigma_BS*(d2Sigma_BS/d2S -d1*sqrt(t')* (dSigma_BS/dS)^2) } I am a bit lost here with the following questions :1> The t' described in the above local volatility formula is the time to maturity (T-t) or current time (t) ?2> Everytime one is to simulate the log normal process in the 1st equation, one gets a different value of 'Stock' price and hence new levels of 'Sigma_LV' --> so we simulate the equation with the new volatility and Spot levels for every step --> Is that correct ?3> Last and most importantly --> does the value of "Sigma_BS" change at each time step ? if yes should I recalulate it according to my volatility surface parameterisation of Sigma_BS_(A(S/K),B(T-t)) <i.e. new Sigma_BS according to new levels of A(S,K) and B(T-t) > .... this does not seem intutive to me directly since we are simulating dS (log normal process) using the local volatility AT TIME t+dt, AND calculating the local vols at time 't+i*dt' from time 't+(i-1)*dt' and a different stock level FROM our parameterised BS vol surface --> where we would be using Sigma_BS AS OF TODAY <my conflict being using todays moneyness and expiry to be considered as same at a future time for the Sigma_BS surface> at a different level of Moneyness (S/K) and time of expiry (T-t).Any inputs/ feedbacks woudl be really really appreciated.ThanksA

make sure you can recover the time dependent case 1st and then everything should become clearer.ie sigma_LV(S(t,t))=alpha(t)basically on the LHS it is time, on the right it is maturity, and all the quantities are as of time zero ( eg S, etc)

Hi spv205,Thanks for your reply. I am not sure I understand you fully.In the formula Sigma_LV( S(t), t') ^ 2 = { Sigma_BS^2 + 2*t'* Sigma_BS * dSigma_BS/dt + 2(r-q)*S*t'* dSigma_BS/dS } / { (1+S*sqrt(t') * d1 * dSigma_BS/dS ) ^2 + S*S*t'*Sigma_BS*(d2Sigma_BS/d2S -d1*sqrt(t')* (dSigma_BS/dS)^2) } 1> Is every " t' " on the LHS the 'current time' and on the RHS it is the 'Time to maturity' ? If yes .. it makes me more comfortable to digest the formula !2> You mentioned that all the quantities on the RHS are as of time zero, but I am still confused with Sigma_BS_(A(S/K),B(T-t)) terms on the RHS . Say we are simulating the Brownian equation at a time 't+h' , then the value of (and hence the 1st and 2nd order derivatives of ) "Sigma_BS " being used on the RHS would be Sigma_BS_( A(S(t+h)/K) , B(T-t-h) ) OR Sigma_BS_( A(S(t+h)/K) , B(T-t) ) .Apologies for not using the Latex editor. The image doesn't seem to be being displayed for some reason.CheersAQuoteOriginally posted by: spv205make sure you can recover the time dependent case 1st and then everything should become clearer.ie sigma_LV(S(t,t))=alpha(t)basically on the LHS it is time, on the right it is maturity, and all the quantities are as of time zero ( eg S, etc)

Guys ... really stuck here .. would really really appreciate any inputs/insights.

Hi aankz,I'm not a theoretician so I give no guarantees on what I'm going to say, but I got this model running and the simulation is matching the market vanillas very well so I guess what I implemented is working. So,1> The t' described in the above local volatility formula is the time to maturity (T-t) or current time (t) ?let's make things more simple. Don't particularly care about the "maturity", your simulation will run to calculate all kinds of complicated products for which "maturity" does not necessarily have a clear meaning. Imagine you're calculating a swap with a bunch of cash flows at different dates, and your swap is path-dependent so you can't recalculate the cash flows independently. Then what would be the maturity? You'll have to calculate payoffs at many different points in time, and some paths might even be knocked out so will finish early.So let's drop the idea of maturity and just consider the local volatility as a function of two variables, current time and spot (or time and strike).Then if your spot has value St at some time t, you need the volatility at that same time t and spot value St to calculate your spot at the next time in your simulation. This local volatility at t is calculated thanks to Dupire's formula, which I guess is the big formula you wrote, at the same time t, using the implied volatility function and its differentials at that same time t, which, again, has nothing to do with any maturity. Just think of the implied and local volatilities as two functions of time without notion of maturity. Now, how you parameterize your implied volatility is up to you. In particular, it may be time dependent, i.e. depend on t. And it might also depend on other parameters, including some parameter T which will mean whatever it may mean, for example some kind of "maturity", it doesn't matter, it's just a parameter, what you differentiate on is the current time t at which you simulate, not your parameter T or whatever you call it.2> Everytime one is to simulate the log normal process in the 1st equation, one gets a different value of 'Stock' price and hence new levels of 'Sigma_LV' --> so we simulate the equation with the new volatility and Spot levels for every step --> Is that correct ?yes, the Sigma_LV that you use to evolve your SDE is different at each time step and for each spot value on each path. In pratice you might want to calculate it before running the simulation, at a pre-defined set of (t, St), and interpolate it in your simulation, at the couple (t, St^j) that you need for your simulation on path j.3> Last and most importantly --> does the value of "Sigma_BS" change at each time step ?yes, it's the one defined at the current time t and current spot St (or strike, in implied vol language). i