very excellent videos. However, I have a fundamental question: In miniute 22:22 (here) you take the function y from your riccati differential equation's video (see there min 25:12) but there you take the auxiliary characteristic function (see there 22:38) for real & distinct roots. Don't we have complex roots here?
Many thanks! Great attention to details, and many thanks for taking the time to link the concepts across two very long videos! You mean complex because the coefficients of the Riccati equation here are complex numbers?
@@leonisvandenberg6157 Many thanks, much appreciated! In terms of exp, the two cases ( real and distinct, and complex) don't differ much. The main concern in the solution of constant coefficient linear 2nd order equations that we see in the ODE books is the fact that the ODE coefficients there are real, so there you see complex conjugate pair, and they manipulate the solution a bit to get the trig function. It is not really necessary here. Hope it makes sense but let me know if it is not clear!
I would love to see a deterministic volatility option pricing model (like for GARCH). Given they are usually similar to the derivation of the Heston model it would fit well with the sort of videos you do. :D
Hello, Thank you for the detailed explination, at 6:31 you write out the equation with respect to V_{tau}, However if tau is the time to maturity from a given t( so tau= T-t), should the equation at 6:31 have V_{t} instead of V_{tau}? You are discounting by time tau to arrive at time t or tau? Please help me understand. Thankyou
Hello, it is just a representation change, there are a bit more detail on reversing the time in the Black Scholes PDE transformation video, but pls do let me know if it still does not make sense!
Thank you very much for your videos. I think there is a small notation error in minute 15:17. I think you have to write P_2=E^Q[...] and not in general P_j=E^Q[...], because P_1=E^S[..] is defined under the stock measure.
Thanks for the excellent exposition! I have a minor question at 8:10 when you say because the price is a linear combination of P1 and P2, when we solve the price satisfies the PDE the two terms will also satisfy the PDE, can you elaborate a bit on this?
thanks for the comment! For simple explanation, say you have z=x+y, where x and y are positive. Now if z satisfies a linear differential equation, say dz/dt=0, then dz/dt=dx/dt+dy/dt=0, and so dx/dt=0, dy/dt=0. The key is the linearity
hello! And many thanks for the question! Assuming this relate to the equation where we write the function f as integral of exponential times P, this is just one of the definitions of a characteristic function - treating x as the single random variable.
This is another spectacular video: thank you so much! Do you have a dedicated video to SABR? I couldn't find one, but maybe I didn't look hard enough? :)
@quantpie. Thank you so much for this.. Beautifully elaborated and provided so much clarity on a topic which is quite difficult to grasp from standard textbooks alone..Have a request..Could you pls do a similar derivation of the popular SABR model including the asymptotic expansion process to derive it's implied volatility.. That'll be really appreciated..Thanks again!!!
Thanks for your clear explanation. May I have two questions? 1. How do we determine lambda in practice? 2. I'm using your python project and try to price a European call, with these parameters I got negative value. Do you know what is wrong? hc=Heston(S0=100,K=100*1.514,tau=335/360,r=0.0157,kappa=1.538,theta=0.0127,v0=0.071,lamda=0.5,sigma=0.708,rho=-0.796); Thanks again.
Thanks @Yuchen Yue! For 1): Lambda, like other parameters, will be estimated via calibration - you get the current market prices and determine all the parameters so that the model implied prices are as close to the market prices as possible. There are a quite a few alternative algorithms one can use - e.g., simple weighted error minimisation, differential evolution, simulated annealing etc. For 2): could you check whether the range and the number of intervals used in the integration cover the spectrum of the two Ps please?
@@quantpie Thanks for your reply. I guess I should ask what is lambda first. As we know, kappa is the rate of reversion, theta is the asymptotic variance, etc.
Hi and many thanks for the very helpful series of videos! Following the question of Yuchen Yue I wanted to ask, in case we want to calibrate and the approach we follow involves simulation of the parameters, what would be a reasonable interval for the parameter λ (given that we will simulate using a uniform distribution)?
This is really helpful thank you Quantpie! Just a small question in the whole video: how do we know that the f characteristic function would also satisfy the same PDE as the original probability P function?
Hello i just saw your question and i am not sure but i think it is because of the Feyman-kac theorem, where we know that the solution to a PDE of the form as in min 14:58 is the conditional expected value of a function at a future time. But i am not sure because there are many fenyman-kac theorems, maybe @quantpie will help us with a more detailed answer.
@@leonisvandenberg6157 thanks Leonis I was thinking of Fenman Kac too, probably we are on the right track but just an opinion that the video might be too quick on that point, a bit more of explanation would help!
thanks both! There are various ways to look at it with different levels of rigour! The PDE is really Kolmogorov equation, and we know conditional expectation and conditional probability satisfy the Kolmogorov equation. There is some coverage of this kinda reasoning in the SABR part 2A video as well. If one does not want to reference Feynman-Kac or Kolmogorov, then it is relatively easy to see that the two conditional expectations (price and char function) are very similar for some minor differences, so one can deduce that they must satisfy the same PDE. Hope this makes sense but happy to answer any follow up questions! many thanks!
Hello and thanks for the question! You would calibrate these, in addition to the other parameters, using the current market prices. So you determine their values so that the formula reproduces the market prices as closely as possible. Hope this helps!
Thank you so much for this video! Could you please help me, you are talking about Levy Inversion formula (hope I understood you correctly). Where can I find any information about it? I can only find information about Fourier inversion?
Hello! It is more or less Fourier inversion, but when applied to the relationship between probability and characteristic functions becomes Levy. Since Levy, this has been generalised and specialised, so there are several forms, which carry their own names - e.g., Gil-Pelaez, which, hoping you can access this doc, is available here: academic.oup.com/biomet/article-abstract/38/3-4/481/258591?redirectedFrom=PDF. Note we used the version with real part, but one can easily translate between them. For further references, please see the references in this article: citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.571.882&rep=rep1&type=pdf
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Thank you so much for making these excellent videos on stochastic volatility models! Helped me a lot!
Your explaining is most clear, intuitive, and novice-friendly. Thanks for the good video. :D
This is simply exceptional stuff! Love this channel
Glad you enjoy it! thank you!
Thank you a lot for your brilliant job! You are fantastic. Keep on doing this fabulous stuff
thanks!
This is simply fascinating!
very excellent videos. However, I have a fundamental question:
In miniute 22:22 (here) you take the function y from your riccati differential equation's video (see there min 25:12) but there you take the auxiliary characteristic function (see there 22:38) for real & distinct roots. Don't we have complex roots here?
Many thanks! Great attention to details, and many thanks for taking the time to link the concepts across two very long videos! You mean complex because the coefficients of the Riccati equation here are complex numbers?
@@quantpie yes that is exactly what I mean :)
@@leonisvandenberg6157 Many thanks, much appreciated! In terms of exp, the two cases ( real and distinct, and complex) don't differ much. The main concern in the solution of constant coefficient linear 2nd order equations that we see in the ODE books is the fact that the ODE coefficients there are real, so there you see complex conjugate pair, and they manipulate the solution a bit to get the trig function. It is not really necessary here. Hope it makes sense but let me know if it is not clear!
I would love to see a deterministic volatility option pricing model (like for GARCH). Given they are usually similar to the derivation of the Heston model it would fit well with the sort of videos you do. :D
Very excellent explanation. I read your python code and just one minor suggestion: please remove ";" at the end of each line.
yes this semi-colon happens because we have been spending more time with c++, but it is good python is forgiving unlike the VBA!
Thank you, best wishes from Sweden
thanks @Henrik Swedish, and best wishes to you too! We are working on levy processes, so coming up soon!
Hello, Thank you for the detailed explination, at 6:31 you write out the equation with respect to V_{tau}, However if tau is the time to maturity from a given t( so tau= T-t), should the equation at 6:31 have V_{t} instead of V_{tau}? You are discounting by time tau to arrive at time t or tau? Please help me understand. Thankyou
Hello, it is just a representation change, there are a bit more detail on reversing the time in the Black Scholes PDE transformation video, but pls do let me know if it still does not make sense!
Thank you for such a nice video. One query is at 30:48, how we say that M is martingale?
Thank you very much for your videos. I think there is a small notation error in minute 15:17. I think you have to write
P_2=E^Q[...] and not in general P_j=E^Q[...], because P_1=E^S[..] is defined under the stock measure.
many thanks! Yes that is right we used this as an example but should have used some generic symbol in the superscript! thanks!
Thanks for the excellent exposition! I have a minor question at 8:10 when you say because the price is a linear combination of P1 and P2, when we solve the price satisfies the PDE the two terms will also satisfy the PDE, can you elaborate a bit on this?
thanks for the comment! For simple explanation, say you have z=x+y, where x and y are positive. Now if z satisfies a linear differential equation, say dz/dt=0, then dz/dt=dx/dt+dy/dt=0, and so dx/dt=0, dy/dt=0. The key is the linearity
Very nice video, can you please link the derivation of the Levy inversion formula you used to get the probability in the end?
Thanks in advance
hi, could you please elaborate on how you got the relationship between f(j) and P(j) at 15:34?
hello! And many thanks for the question! Assuming this relate to the equation where we write the function f as integral of exponential times P, this is just one of the definitions of a characteristic function - treating x as the single random variable.
@@quantpie Thank you
@@quantpie Could you please elaborate more on why f(j) must satisfy the same pde as p(j) ?
This is another spectacular video: thank you so much! Do you have a dedicated video to SABR? I couldn't find one, but maybe I didn't look hard enough? :)
Thanks @John Van Prague! Hope you are keeping well! Not yet, but it is on the list!
@quantpie. Thank you so much for this.. Beautifully elaborated and provided so much clarity on a topic which is quite difficult to grasp from standard textbooks alone..Have a request..Could you pls do a similar derivation of the popular SABR model including the asymptotic expansion process to derive it's implied volatility.. That'll be really appreciated..Thanks again!!!
Thanks! SABR is indeed an important model, and it is on the list now 👍
@@quantpie Thanks a lot!!Really appreciated..:-D
Thanks for your clear explanation. May I have two questions?
1. How do we determine lambda in practice?
2. I'm using your python project and try to price a European call, with these parameters I got negative value. Do you know what is wrong? hc=Heston(S0=100,K=100*1.514,tau=335/360,r=0.0157,kappa=1.538,theta=0.0127,v0=0.071,lamda=0.5,sigma=0.708,rho=-0.796);
Thanks again.
Thanks @Yuchen Yue! For 1): Lambda, like other parameters, will be estimated via calibration - you get the current market prices and determine all the parameters so that the model implied prices are as close to the market prices as possible. There are a quite a few alternative algorithms one can use - e.g., simple weighted error minimisation, differential evolution, simulated annealing etc. For 2): could you check whether the range and the number of intervals used in the integration cover the spectrum of the two Ps please?
@@quantpie Thanks for your reply. I guess I should ask what is lambda first. As we know, kappa is the rate of reversion, theta is the asymptotic variance, etc.
@@yuchenyue1243 It refers to the market price of risk (volatility risk) - how much extra would you want to be rewarded for bearing volatility risk.
Hi and many thanks for the very helpful series of videos!
Following the question of Yuchen Yue I wanted to ask, in case we want to calibrate and the approach we follow involves simulation of the parameters, what would be a reasonable interval for the parameter λ (given that we will simulate using a uniform distribution)?
This is really helpful thank you Quantpie!
Just a small question in the whole video: how do we know that the f characteristic function would also satisfy the same PDE as the original probability P function?
Hello i just saw your question and i am not sure but i think it is because of the Feyman-kac theorem, where we know that the solution to a PDE of the form as in min 14:58 is the conditional expected value of a function at a future time.
But i am not sure because there are many fenyman-kac theorems, maybe @quantpie will help us with a more detailed answer.
@@leonisvandenberg6157 thanks Leonis I was thinking of Fenman Kac too, probably we are on the right track but just an opinion that the video might be too quick on that point, a bit more of explanation would help!
thanks both! There are various ways to look at it with different levels of rigour! The PDE is really Kolmogorov equation, and we know conditional expectation and conditional probability satisfy the Kolmogorov equation. There is some coverage of this kinda reasoning in the SABR part 2A video as well. If one does not want to reference Feynman-Kac or Kolmogorov, then it is relatively easy to see that the two conditional expectations (price and char function) are very similar for some minor differences, so one can deduce that they must satisfy the same PDE. Hope this makes sense but happy to answer any follow up questions! many thanks!
Great video, really going into the detail! Please keep up! Just wondering you may go through SABR model?
thanks @Can Jin, yes it is on the to-do list!
HI, thanks for the video. I have a doubt, how do we know the value of the initial condition when \tau is zero f(x,v,0)=exp(i*\phi*X)?
Hello and thanks for the question! You would calibrate these, in addition to the other parameters, using the current market prices. So you determine their values so that the formula reproduces the market prices as closely as possible. Hope this helps!
Thank you so much for this video! Could you please help me, you are talking about Levy Inversion formula (hope I understood you correctly). Where can I find any information about it? I can only find information about Fourier inversion?
Hello! It is more or less Fourier inversion, but when applied to the relationship between probability and characteristic functions becomes Levy. Since Levy, this has been generalised and specialised, so there are several forms, which carry their own names - e.g., Gil-Pelaez, which, hoping you can access this doc, is available here: academic.oup.com/biomet/article-abstract/38/3-4/481/258591?redirectedFrom=PDF. Note we used the version with real part, but one can easily translate between them. For further references, please see the references in this article: citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.571.882&rep=rep1&type=pdf
@@quantpie thank you very much! You helped me a lot!