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sigma_iv(T,K) is the parameter sigma (a constant) you have to use in BS model to reproduce the value of that specific option C(T,,K). choosing this parameter the model will likely not reproduce the other option prices (they have different sigma_iv's). So there is no "one model" that reproduces all Europeans. (T,K) -> sigma_loc(T,K) is the surface you have to use in a Local Vol model to reproduce all option prices C(T,K) observed. So choosing just one point for one (T,K) is not sufficient. If the option prices observed were created with a single BS model you will find that sigma_loc(T,K) = sigma_bs (if sigma_loc is the local volatitliy in lognormal form). Hope this helps a bit. The two thinks are different quantities. Analogy: It like the velocity observed at a specific time compared to the average velocity oberserved up to a specific time. The two are related (one is an integral (average) over the other) and for constant velocity the two are the same.

Thank you for answering. I think I didn't ask the question clearly enough. I am not looking at the BS model, I am ONLY looking at the LV model dSt = r St dt + sigma(t, St) St dWt. My question relates to how we determine the sigma(t, St) given the market prices of a bunch of European call options. I can either use Dupire's formula to determine sigma_loc(T, K), or I can use BS implied volatility function to get sigma_iv(T, K). In either case I will end up with a grid of (varying) volatilities which reproduce the market prices and which I can then interpolate to construct sigma(t, St). But in both cases I want to use "one model", namely the local volatility model. I use BS only to determine the implied volatilities. So, my question is: why use Dupire if I can as well BS implied vols to determine sigma(t, St)?

@@fondueeundof3351 I am not 100% sure if I understand the question. If you interpolate sigma_iv(T,K) you have a kind of model to value all European options (just pick the corresponding sigma_iv(T,K) and plug it to BS formula, but you do not have a model for the stochastic process. The local volatility model is a model for the stochastic process S that would allow you to value - for example - an Asian option with a Monte-Carlo simulation,, where the model is consistent with the observed European options - so if you would use a Monte-Carlo simulation of S and value a European option, you will get the observed input prices (apart from numerical errors). Also: when interpolating local volatilities it is much easier to ensure absence of arbitrage. That said: if you are only interested in European option prices, a good (arbitrage-free) interpolation of BS volaitlites is an of course alternative (with possible advantages in performance or accuracy).

@@finmath6357 Ok I think I understand. I am looking at the stochastic process dSt = r St dt + sigma(t, St) St dWt, and am ONLY wondering how to estimate sigma(t, St) given market data. From your answer I gather that if I used BS impl vol function to determine the sigma(t, St), the resulting stochastic process would (probably) not reproduce the prices of the European calls that were used to calibrate the model, let alone price anything more exotic correctly. Thanks a lot for your time!

I see your point. The transpose is there to make it comparable to the "transposed" of (140). This transposed is there, because the gradient is a row-vector, not a column vector (as the Delta x). - But for m=1 that transposed is not needed - a column with 1 element is a row with one element. (It would be needed for m > 1 with f:R^n -> Rm). Thanks for the feedback. I try to improve this! Thank you.

@@finmath6357 Thank you for the anwser and thanks for your lectures, i love them! :)

Good question. One major disadvantage is, that resampling would be more complex from the implementation side. So instead: you could immediately sample a cut-of normal such that you do not require re-sampling. But then: The distribution we get from truncation and the distribution we get from reflection and the distribution we get from resampling, all have a bias. So you actually have to show if this bias improves the situation or if it makes it worse. And maybe it should be mentioned that the distribution we have to sample is known and it is possible to create an exact scheme, see Broadie and Kaya: www.columbia.edu/~mnb2/broadie/Assets/broadie_kaya_exact_sim_or_2006.pdf

@@finmath6357​⁠ Isn't sampling arguably the operation that is carried out most frequently in like an MC simulation, won't it therefore be highly efficient and shouldn't therefore an occasional resampling hardly impair performance? And also, truncation will introduce a discontinuity and reflection non-monotonicity, while resampling doesn't. That's why resampling might be preferable, but that's just my gutfeeling...

@@fondueeundof3351 I personally would not advice to do a resampling, because resampling also introduces some discontinuity and this discontinuity may be even worse: Assume you model parameters are such that V(t_i, omega_k) < 0. In that case you would trigger a resampling on this time step and path (t_i, omega_k). Now assume a change to the model parameters that will result in V(t_i, omega_k) >= 0. In the case you would not trigger a resamplling. Clearly, this is a discontinutity. The big problem is: if you use a 1-D random number generator to sample the 2-D factors, taking one more random number for the resample could result in a complete change of the random numbers for (dW1, dW2). - Of course, resampling is not completely forbidden, but one has be very careful (and there are better options).

Yes. Maybe this should be elaborates in more detail. The rK corresponds to the mu = rS in the Feynman-Kac, where the K corresponds to the S. Of course: One had to proof this correspondence. The remark here is just for illustration that one could "guess" the formula. I am sorry for being not rigorous here..Maybe I should elaborate on this. (Christian)

Thank you.

I am not 100% if I understand the request correctly. The derivation shown here is more or less taken from the book ISBN 978-0470047224. The simulation in shown there uses the code from www.finmath.net/finmath-lib/

Thank you very much

I am not (yet) covering it explicitly in this lecture, but the lecture on "The Short Period Bond" (Session 18 - ruclips.net/video/nu6LX7apbJI/видео.html ) is related. The extension is just that you "fill the gap" by simulating the short period bond (or the backward rate). If you are interested in having a model that allows to have finer (e.g. daily) tenors on the short end, you might also consider the time-homogenous forward rate model I have described here: papers.ssrn.com/sol3/papers.cfm?abstract_id=3664034 (this approach is actually a bit more general). I do not know if I will add a session on either one of the two this year, but if not, this is a good suggestion for next year! Thx.