@@mohammadmunazzirhosany2854 Not to be funny but, leap frog to log. Instantaneous jump condition for early t, smooths out log dist for long t. Or you can just use HFT until you're banned from a platform, I'd never do that🤓.
Thank you! However, I could not fully understand how we set f(t,s,v) equal to the risk-neutral drift of dv at minute 17:52. In the framework of B&S it followed by the fact that we substituted weight*bond with its expression from the self-financing portfolio, where alpha = dV/dS. But I could not see the same meaning here.
I have one question please: before writing Ito’s formulation, why do we make the assumption that the value of the portfolio depends on t, S_t and v_t, and not on past realisations for s
I am studying Financial Mathematics and your content has been super helpful through my time in the course, so Thanks a lot! I have two questions. 1. Is vt following a Cox Ingersoll Ross process in your derivation as well? 2. Can you suggest me a paper where I can find the heston model being derived the way you've done it. It's easier to follow than a few papers I've seen and I plan to use it for my dissertation. Thanks a tonne!
Great to hear! Yes it is indeed a CIR process. Heston's original paper pretty much covers everything, we have just interpolated some steps, and injected a few opinions here and there! Good luck with the dissertation!
@3:10 the volitility at the modern market awaits, adorned in loving quants arms, the market friction and slippage guard the gate, she rests at last in starry eyes. The search for sigma
18:13 One thing that still confuses me. On the right hand side of the PDE in brackets. Why does it say "- lambda sigma sqrt(v)"? In the original Heston (1993) paper, Heston writes it as "- lambda (S, v, t)" and defines "lambda (S, v, t) = lambda v" where lambda is a constant. So why is it "- lambda sigma sqrt(v)" and not "- lambda v" like in Heston (1993)? Is it identical?
Thanks, that’s a very crucial point! This is mentioned in the next but one video in which we derive formula for European option. With all respect!, Heston pulled a fast one to linearise the term- turn the process into affine. He justified this step by appealing to a specialised model, which is just one of the possible forms, maybe the simplest which then enables nice analytical solution. This point is rarely highlighted in the literature so goes unnoticed but is an assumption he made!
Amazing video! but in a lot of textbooks they say that the risk neureal measure is B_1^Q = (B_1+ [ (\mu-r)/\sqrt(v) ]t ) and B_2^Q=(B_2+ [ \lambda / (\sigma \sqrt(v) )] t ) Why is your solution in minute 24:24 different?
Many thanks, yes indeed it is presented differently in the textbooks - they don't even follow Heston for some reasons! Do you have a particular reference in mind?
I think the form you mentioned is how we change the measure of the original correlated Brownian Motion. But what we did in the video is for changing of the decomposed independent processes.
Can you suggest me a paper where I can find the heston model being derived the way you've done it. It's easier to follow than a few papers I've seen and I plan to use it for my dissertation.
Thank you for your effort. I have a question if you could reply to me. how we can derive PDE in the case of Heston model with stochastic interest rate?
From around 10 minutes you construct the replicating portfolio, which you call, V, but V is already used as a variable for the first asset. Is it ok to use the same variable name? If you for an example, use Pi, instead you won't reach the same conclusion as Heston, since you will end up with (LV)(t,S,v) - rPi = -rSdV/dS - f(t, s, v) dV/dv. So I guess the question is, can we just define a replicating portfolio with same variable name as the asset we have already included? Thanks quantpie.
