I'm just starting to learn this area of math/finance/statistics and these videos have been amazing to get an intuitive understanding of what's really going on
Glad to see more uploads. Been a bit busy getting into statistical field theory and exploring more stochastics, but looking forward to getting back into finance soon :-). Cannot thank you enough once again for these uploads.
@@WeiXing25 People did not like the voiceless series, partly because they were to meant to facilitate a future video, so we removed them for now. Please send us the email address that you use to access RU-vid, and will send you a link to the normal model video!
Thank you for this video! One question: As some people have pointed out, the "t" disappears from the expression of the jump part of S_0 - can the "t" really be absorbed by S_0 as you suggest? Since it is not a constant parameter?
Would be great if we expand what we learned from the videos to encompass the merton jump model and the stochastic volatility model to arrive at the Bates model! thanks a lot for the videos anyway really helpful.
Another amazing video. I hope to see a video on the Levy-Khintchine Theorem. I understand the parts of the characteristic exponent due to the Brownian motion and due to the compound Poisson process. However, I am having a hard time understanding why it appears a: -i \theta x 1_{|x|
Great suggestion! It is on the list, it is just that the levy series does not seem to be of much interest to people, so the progress has been slow. Actually we had the LK video a while back, but never quite polished it because of interest! We shall try to get back to it soon! thanks for the patience!
Thanks for yet another instructive video! Absolutely loved it. Is the Jump feature only important for short dated OTM options? How about Barriers and other exotics with discontinuous payoffs? What would be the challenges of implementing the Jump feature with the Local Volatility model in terms of parameters observability, hedging etc.
You are welcome! Indeed people do report that models with jumps produce fit for exotics (e.g., things that involve forward vol), but others think that it makes the model too complex. Regime switching is probably a more pragmatic alternative, though this might not solve all the problems!
At 5:55, when there are more than one jumps occur, why didn't you write (\prod(Y_j -1) dN_t. Where did the dN_t term go in the case of multiple jumps? Can you explain please.
The prod will require the number of terms to be multiplied right? And that's where the dN_t went - i.e., into the upper limit of the product. Does that make sense?
Many thanks and sorry for the slow response! Assuming you have process without jump as base, and you then introduce jump, which can occur randomly, but when the jump occurs the price on average changes by some positive amount. In this simple case, you you can see that adding the jump will introduced a drift? A positive drift? Same reasoning applies to the case in video though there are a few more moving elements.
Many thanks! Could you elaborate a bit more as to what are you looking for please? We have a couple of videos in the Simplified playlist which covers the risk neutral valuation - these are the Change of probability and Girsanov theorem videos - so if you could have a look, and let us know what additional items would you like us to cover, that would be very much appreciated!
if I want to do a simulation of this model. The S0n factor of the solution: Is it added in the model at the instant the first jump occurs? o Is it added from the initial moment? This with the idea of comparing it with the numerical solution of euler-maruyama. What could be an initial condition of Yt?
Many thanks for the question! Have created some basic Python code that calculates the Merton price using the analytical formulae (two versions) and basic simulation. You can compare it to Black Scholes. You can find the code here, warning may contain typo: github.com/quantpie/Merton-Jump-Diffusion-Model-Python-code/blob/main/Merton%20Jump%20Diffusion%20model.py
Hi.I am looking for the bates model and how the parameters impact on the implied volatility curve. if you have a tutorial like the heston model it will helps me a lot.thank you in advance
Very nice video but I'm a little confused in 5:54 you write that the stochastic differential equation is given by dS_t/S_t = (µ- λκ) + σ W_t + ( \prod^{dN_t}_{j=1} Y_j - 1), but in textbooks the SDE for Merton's jump-diffusion model is given by dS_t/S_t = (µ- λκ) + σ W_t + d( \SUM^{N_t}_{j=1} Y_j - 1), see for example KOU or Steven E.Shreve Can you explain that some more?
Ah the textbook version has to be compared with the infinitesimal version. It is just that we write it in terms of the new jumps (dN_t), and they write in terms of the change in the aggregate counter. Hope this helps!
great video. I think there is a typo in the derivation. At 11:16 is a "t" missing from the first parenthesis ("first set of brackets") in the 3rd line: lnSt - ln So = (-\lambda*k +n* u_y + 0.5*n*\sigma_y^2 ) + ...?? or at least the first term inside those parenthesis, ie, \lambda*k should be multiplied by t
At 11:16, while moving from the second step to the third, where did the t factor go from -lambda*k? Shouldn't it be -lambda*k*t in the third statement?
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 ?
I think that you have mad a mistake. Around 11 minutes when putting apart terms which doesn't depend on t nor Wt .you have put lambda mulitplied by k but it was originally in factor of t not constant. Great content though
Thanks, could you clarify a bit further please. Is this when we try to reformat the equation to the Black Scholes form? You mean k depends on t? If yes then dependence on t is not the key splitting criteria here as this solution now works for any t, and for European options only the terminal distribution matters, so t is more like a constant in this context.
@@lanvu437 It is all about adapting Black Scholes to account for jumps, so American options are not in scope as Black Scholes work for European options. Including t in the S0 won't change anything - btw, \sigma_n also depends on t. Hope this helps!
