Which schemes are you using, weak or strong schemes? I agree with Fingerspitzengefuehl and am impressed, that you have spelled the word without any errors.

QuoteOriginally posted by: DimathematicianWhich schemes are you using, weak or strong schemes? I agree with Fingerspitzengefuehl and am impressed, that you have spelled the word without any errors.I am using the PC method in Kloeden page 198, so it is the weak 1.0 PC (with alpha = 1 and eta = 0 factors). Is that what you mean?A scheme is a scheme and has weak and strong convergence, right? I also testedDerivative freeFRKISemi-implicitFitMilsteinOK, but they all behave alike (more or less) and applying to n-factor is not always obvious.

QuoteA scheme is a scheme and has weak and strong convergence, right? There are two types of schemes: weak and strong schemes. Of course, both of the schemes have weak and strong convergence, but they are constructed with a special goal. If you look at the Contents page of Kloedens comp book, you'll see, that he distinguishes between strong and weak approximations. Strong schemes use normally distributed random variables for dW, simplified weak schemes use for example a 2-point distributed random variable for dW, such as the one on page 195 on kloedens book. This is, why W has a hat on page 198. The purpose of weak schemes is to approximate moments only, not the paths, so dW is replaced with a simpler random variable, which can be computed faster than a normal ones.However, last schemes are called simplified weak schemes. There are normal weak schemes, which still use normally distributed rv's for dW, but they are different to the strong schemes. I think, Euler weak and Euler strong are the same. But, Taylor 1,0 weak and Taylor 1,0 strong (and schemes of higher order) are definitely different. They use different hierarchical sets in the Ito-Taylor expansion, which is derived in the more theoretical, non comp book. So, the schemes are really different, its not just a statement about the convergence.Which ones do you use?

QuoteStrong schemes use normally distributed random variables for dW, simplified weak schemes use for example a 2-point distributed random variable for dWI seeI use the former because we generate correlated/uncorr W. For FDM we need point-wise/pathwise error estimates. So I have strong schemes and even for page 198 I use dW!!!! This might be heresy!Where are weak schemes useful? BTW I have not used the Taylor schemes.

You're using dW on page 198 correctly, you can use both for PC, I think. For FDM we don't always need pathwise estimates I think. For plain vanilla calls, you're calculating the first moment of the random variable (X_T-K)+. This is exactly what weak schemes are good for, calculate the random variable, where you simulate X_T with weak schemes, since you're interested in the moment only. It works. Taylor schemes are actually the general name for most schemes imo. So: Taylor strong 0,5 = Euler strong, Taylor strong 1,0=Milstein,.... Taylor weak 1,0 = Euler weak... But I agree with you, for most applications you might need strong schemes.

QuoteOriginally posted by: DimathematicianYou're using dW on page 198 correctly, you can use both for PC, I think. For FDM we don't always need pathwise estimates I think. For plain vanilla calls, you're calculating the first moment of the random variable (X_T-K)+. This is exactly what weak schemes are good for, calculate the random variable, where you simulate X_T with weak schemes, since you're interested in the moment only. It works. Taylor schemes are actually the general name for most schemes imo. So: Taylor strong 0,5 = Euler strong, Taylor strong 1,0=Milstein,.... Taylor weak 1,0 = Euler weak... But I agree with you, for most applications you might need strong schemes.In PDE/FEM/FDM there is always a discussion on whether to use L infinity ('max norm') or L2 norms to measure convergence. I prefer the former because we need pointwise values for use in MC method. You need the underlying at each time point.The KL expansion does not make any distinction between weak and strong. why? KL uses an array of Gaussain rvs.

DimaI have tested the Call, K = S = 100, mu = 0.04 (small), vol = .2, T = 1, r = .1NT = 100, NPATH =100,000Get PC1 9.38059fit 9.42163Oiler 9.26754derivfree 9.30656FKito 9.26471Semi-Oiler 9.36352

Did you try Milstein and Euler for dt=1 and different seeds? What are your results, I mean not just one runand two numbers, but many runs with different seeds, which one is better for the parameters you've mentioned?

QuoteOriginally posted by: DimathematicianDid you try Milstein and Euler for dt=1 and different seeds? What are your results, I mean not just one runand two numbers, but many runs with different seeds, which one is better for the parameters you've mentioned?Ok, I have kind of stopped looking at Milstein. I just have 1 seed; why many seeds?

QuoteOriginally posted by: DimathematicianDid you try Milstein and Euler for dt=1 and different seeds? What are your results, I mean not just one runand two numbers, but many runs with different seeds, which one is better for the parameters you've mentioned?Ok, I have kind of stopped looking at Milstein. I just have 1 seed; why many seeds?What the end-game here? There is an interesting article by Dosty here and he has some interesting graphs. e.g. QMC convergence is monotone, while MC is oscillating around the exact value, like lattice models.

My experiments show the following:Its hard to decide, which scheme is better. If you simulate a Call with 2 schemes and say 100.000 simulations, and euler would show a better result (closer to true price), would it be ok to say euler is better? Changing the seed might show different results. I mean: Both schemes can show statistically good results, e.g. mean is in the confidence intervall, confidence intervall length is the same. Stating something about the quality, given the statistical properties are the same, is something I'm currently thinking about.