Local vol - MC - algorithm

[tyskland]

I'm trying to implement MC-simulation with local vol in Matlab and my algorithm is (assume one simulation)

S(tn+1) = S(tn) + r*deltaT*S(tn) + LV(K,tn)*S(tn)*sqrt(deltaT)*N(0,1)

where

tn = time
deltaT = time interval
K = strike

The LV-function is a matrix consisting of local volatilities for each pair of strike/maturity combination. I get that if I am at time tn I can look in my LV-matrix and find the local vol corresponding to that maturity but the problem is that for each tn I also have a lot of different local volatilities in my LV-matrix (one for each strike price). So which one do I choose

 

[koupparis]

You're running a simulation for a particular option it seems, hence it would make sense to use the strike of the option, K, to get your LV.

 

[tyskland]

You're running a simulation for a particular option it seems, hence it would make sense to use the strike of the option, K, to get your LV.

Not exactly.

This is how I have done:

1. Gather data, 100 option prices (same underlying) with 10 strikes and 10 maturities. Hence 100 implicit volatilities.
2. For each pair (K,T) I determine a corresponding local volatility. Hence I have a matrix 10x10 with local volatilities that a function of strke and time.

Now I want to simulate the Spot price process before determining the price of the option. I don't get which local volatility I am supposed to use when simulating the spot prices.