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<= t ? In effect, why do we assume the portfolio value to be Markovian ? Would be grateful if you could point me to the relevant mathematical proof for this ? Otherwise, great tutorials!

Hi, thanks so much for the great video. Can I ask, did you use any reference textbook for making this video?

This is a cool video, however it feels like you have shown the construction of the so-called Stratonovich integral instead of the Itô integral. (These two integral concepts are not the same.)

Thank you! this the best way to understand the BeS formula!

Thanks for the great video! Could you please elaborate how you calculate delta at 3:13? Thank you very much in advance!

Thank you very much for this video, it was a huge help :)

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.

I can't thank you enough for this video.

Why at 3:44 do you use the equivalent expression of the probability?

I come here to understand "Thermodynamic Natural Gradient Descent".

could you or someone, elaborate how the radon-nikodym derivative can be extracted @10:16

You talk about “solutions” to equations etc without even defining what it means to be a solution for the equation 😅 Total meaningless garbage. This is not math. Nowhere did you define the Ito integral.

Total trash this “explanation”.

thanks you help, great explanation

Excellent explanation !!!

Amazing video, I will share !!

The best video I could find to have the proper understanding of Poisson Processes, thanks a lot for making such videos!!!

Hi, thank you for the great video, it truly made me understand the concept of changing probability measures way easier. I never knew it was actually that straightforward! Is it possible to share your slides? I would like to take notes on them if you don't mind :)

I wish you did your own voice over or one that is a bit better, because despite the robotic voice I sticked around as the information was so easy to understand!

Why mean of jump is divided from equstion

Thank you for this great video!! one point was not clear to me. During the moment (9:55 - 10:07), that answers the question of why Riemann-Stieltjes does not work, the video says: "... but how do we show that it converges. and you can see the infinite variation of Brownian motion which manifests itself through the zigzaggy path here, makes the Riemann-Stieltjes approach irrelevant..." I am still not convinced with the stated conclusion given the explanation. From the graph, even with the zigzaggy path, I can imagine that we would approach the area that we seek to compute as the number of partitions approaches infinity just like how we did with the function g(t) before mentioning the Brownian motion. From this visualization and following the Riemann-Stieltjes approach, I still cannot imagine why it does not converge?! What is the thing in the zigzaggy path, that g(t) does not have, which prevents the convergence? If I have to guess, the answer is the non-differentiable points that prevent that notion of convergence from existing? it would be great if a visualization was provided to help convince a viewer naive in math like me.

good work Prof 😃

this video is gold

How does E(df) can be d/dt E(f)

very well explained indeed, I just started to watch but so far this could be one of the best tutorial on the net

Hi, I am studying your videos and I have a question of the Calibration part: Why the term X_0 is not estimated? My intuition is that the actual realization shown on data is not necesarilly representative of the process beginnings, so X_0 should be estimated as the regression intercept of the model, assumming simple ordinary least squares it is given by: X_0 = exp(E[ln(X_t)] - (mu-sigma^2/2)*(#data_points)/2) where sigma^2 = Var[ln(X_{t+1}/X_t)] mu = E[ln(X_{t+1}/X_t)]+sigma^2/2 as you show in your video It is this line of thought right?

isn't at money option delta should be 0.5?

Great ! thanks!

Thanks for the amazing explanation!!

extraordinary. Have seen the previous video on (several ways) to derive Dupire PDE, excellent as well. Haven't completed this one, hope some comments on pricing behaviour for path dependent exotics (hopefully as a function of time to maturity?) Thank you so much

Great explanation though - thank you

Worth pointing out that it is the mean of the log return, not the mean of the stock price?. Seems obvious, but not always clear.

Sir can I get the solution of the equation?

Im sorry I have a maybe dumb question. I thought the moment generating function is t or θ dependent. So in 3:18 what you calculate is just a constant (or θ=1). Why do you use that one to substitute the variance of the brownian which is not constant ( var = t or θ). ? It confuses me a little bit and I would love a clarification! thanks

Thanks!

after an hour of searching on google and reading so many different definitions, i finally understand what a quasi linear equation is thanks to your video!! so well explained, thank you!

Thanks so much for this video. I am currently doing a final year college project on option pricing, and this video really helped :). Is there any way that I can formally cite this in my project? I mean, did you follow a derivation from a certain book, or do you have written notes on this? Derivations in textbooks that I've found arent as clear as this one. Thanks again, hope you can answer me!

😊 23:38

I am seeing your videos now, and I have a question about this one: Could be easier for finding \mu doing the following? \mu = E[d/dt(E[ln(S(t))]) +1/2*Var[ln(S(t))]] Could it be computationally faster?

Hi, thanks for the video and it's really insightful! I am wondering for the last step using the Borel Cantelli lemma, how to get to lim(S_nk) = 0 from P(limsup(S_nk >= epsilon)) = 0?

Can you please share 2nd and 3rd order greeks for learning

This is a fantastic video ! Really liked the points related to calendar and butterfly arbitrage check in the Call option prices before we infer the Local volatility from the Call option price surface !

I'm in love with the woman who recorded this video.❤

Be confused with the Equation at 6:28, since it implies that the particles around the position x does not participate in any movements outwards. Why is this factor not considered? If considered, next it should done like (f(x, t+\tau) - f(x, t))dx = integral of dx*f(x+\delta, t)*\phi(\delta)d(\delta).

Excellent presentation and I have benifited a lot! A more rigorous statement appears to be that both the notions f and fi by nature represent probability density functions rather than probabilities.

firstly Thanks for the awesome video, I wanna if we can use the propriety of the discounted Prices being a martingale, then concluding that the term multiplied by dt should be 0 and we can get our pde ?

At 13:45 you weigh the calls using lambda times T whereas in Joshi 2003 and several codes the call formula use lambda times m times T where m is the exponential of your mu_y. You and Joshi use different values of the uderlying asset in the summation. Joshi uses the same underlying asset value for all the calls of the summation. Would you have a look please and try to consolidate the two approaches if feasible ?

Why do you use in these examples base of 365 days/year at 9:24 (Term rate - Compounding)? Base convention for both LIBOR and SOFR is ACT/360. You could check that in BBG. Could you please comment on that?

beautiful video. the only question i have is how exactly did you get the expression for the radon-nikodym derivative exp{sigma*B_tilda - 1/2 * sigma^2 *t}?