Merton Jump Diffusion Model

Рет қаралды 14,325

quantpie

3 жыл бұрын

Derives formula for the price of a European call option under the Merton's Jump Diffusion model.

Пікірлер: 75

@sharatvyas 3 жыл бұрын

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

@quantpie 3 жыл бұрын

Great to hear! many thanks!

@rileyheiman1161 Жыл бұрын

very few people on earth know stochastic calculus this well!

@hyunpang8267 2 жыл бұрын

OMG this deserve more views and likes, making such a complicated model so simple to understand with your notations and highlighting.

@quantpie 2 жыл бұрын

thank you! glad you found it useful!! many thanks

@Ignoramus.et.Ignorabimus 3 жыл бұрын

Many thanks Quantipe for the lovely teaching and patience and illustration put into this splendid work. God bless!

@quantpie 3 жыл бұрын

Glad you enjoyed it! thank you!

@TheChalarock 3 жыл бұрын

It is impossible for me not to thank you for the time you invest in teaching us

@quantpie 3 жыл бұрын

so nice of you! you are welcome, and thank you!

@williamqiu1917 Жыл бұрын

In awe of this amazing presentation!!

@JaGWiREE 3 жыл бұрын

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.

@quantpie 3 жыл бұрын

Good to hear you are having fun! Keep it up, and thanks for the kind words as always!!

@yiranzi8678 2 жыл бұрын

Thank you. I spent hours for homework about this before watching your video. You saved my life.

@quantpie 2 жыл бұрын

You're welcome! thanks! What you did is the best approach, give it a go and then watch, one learns a lot more using this approach!

@JBasics 3 жыл бұрын

what happens to the t at 11:16? On the second line -λk is multiplied by t and on the third line it isn't anymore

@quantpie 3 жыл бұрын

many thanks for highlighting this! Last time we saw the t(ea) had been drinking itself! Sorry this is a typo, the t should not have disappeared.

@madaragrothendieckottchiwa8648 3 жыл бұрын

Thank you !!!! Good session

@quantpie 3 жыл бұрын

Thank you for the kind words as always!

@amrabou-senna9838 3 жыл бұрын

Thanks alot very much for this great videos

@quantpie 3 жыл бұрын

Glad you like them! Very nice of you! thank you!

@carolinify11 Жыл бұрын

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?

@WeiXing25 3 жыл бұрын

Thank you!! I have been checking you site everyday for the latest video!

@quantpie 3 жыл бұрын

You are so welcome! Thank you!

@WeiXing25 3 жыл бұрын

Hey buddy what happened to the normal option pricing video you had up a few weeks ago?

@quantpie 3 жыл бұрын

@@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 KZfaq, and will send you a link to the normal model video!

@WeiXing25 3 жыл бұрын

@@quantpie Thanks!

@WeiXing25 3 жыл бұрын

Can I ask what future videos you plan to release?

@caetanocardeliquio7174 3 жыл бұрын

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|

@quantpie 3 жыл бұрын

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!

@qiguosun129 6 ай бұрын

Thanks!

@Sobriquet80 3 жыл бұрын

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.

@quantpie 3 жыл бұрын

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!

@yizhangli2434 3 жыл бұрын

great explanation!

@quantpie 3 жыл бұрын

Glad it was helpful!

@mohamadghrayeb6923 2 жыл бұрын

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.

@quantpie 2 жыл бұрын

thank you!! Yes we do intend to cover Bates at some point, just never got around it! but it is on the list. thank you very much!

@surendrabarsode8959 3 жыл бұрын

Nice video and well explained. Can you have another follow up video wit some specific examples to understand actual calculations. Thanks

@quantpie 3 жыл бұрын

thanks yes calibration and implemenation will follow!

@HungDuong-dt3lg 2 жыл бұрын

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.

@quantpie 2 жыл бұрын

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?

@shogoh8684 Жыл бұрын

At 9:00, can we take the expectation of the random sum instead of assuming the number of jumps is equal to some fixed number n?

@irfayasseyf1603 2 жыл бұрын

Brillant derivation.

@quantpie 2 жыл бұрын

thank you @Irfayas Seyf!

@Vaggos16 2 жыл бұрын

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

@quantpie 2 жыл бұрын

noted! thank you, yes Bates is on the to-do list! many thanks for the suggestion! much appreciated!

@steviezecevic123 Жыл бұрын

Could anyone explain further how dS^2/s@ was derived? The point in the video I am referring to is 7:53 . TY.

@ItachiUchiha-ge6em 2 жыл бұрын

Hi thanks for your explanation, very helpful! Could you maybe explain (at min 4:40) why the jump introduces a drift?

@quantpie 2 жыл бұрын

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.

@yinyl 3 жыл бұрын

great video ! would you mind explainning the risk neutral world?

@quantpie 3 жыл бұрын

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!

@TheChalarock 3 жыл бұрын

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?

@quantpie 3 жыл бұрын

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

@khansari4532 3 жыл бұрын

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?

@quantpie 3 жыл бұрын

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!

@glongoria8004 2 жыл бұрын

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

@quantpie 2 жыл бұрын

Thank you! Yes you are right!

@kingshukdutta2064 3 жыл бұрын

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?

@quantpie 3 жыл бұрын

hello this is a typo! thanks for highlighting!

@Gilloup 8 ай бұрын

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 ?

@sounakmojumder5689 3 ай бұрын

Why mean of jump is divided from equstion

@WeiXing25 3 жыл бұрын

Hey buddy when is your first SABR video? Can't wait to see it!!

@quantpie 3 жыл бұрын

sorry to keep you waiting - will be out in a week (wish!)!

@athinansutchada3649 2 жыл бұрын

Do you have some references?

@quantpie 2 жыл бұрын

Hello! Merton’s original paper is quite readable, and probably better than most textbooks! Many thanks!

@nicolastorres147 Жыл бұрын

2:28 Oil futures would like to introduce themselves

@alexgold4965 3 жыл бұрын

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

@quantpie 3 жыл бұрын

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.

@alexgold4965 3 жыл бұрын

at 11:20 : (-lambda * k + n*mu_y +(n*sigma_y**2)/2) in the part So(n) but in fact you should have ( -k*lambda*t +...)

@quantpie 3 жыл бұрын

ah thank you! good spot! Indeed should be \lambda k t!

@lanvu437 3 жыл бұрын

Well then it should not be embedded in S0(n) don't you think? For american option, t is not a constant for example

@quantpie 3 жыл бұрын

@@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!