This is the first time, I got someone who explained this Ito's lemma in layman's terms and your explanations make so much sense. Thank you, prof. I now have some understanding of what Ito's Lemma is and how it is applicable.
Thanks Very much. I thought of dropping my financial economics class because of How my Prof thought this topic. I was so lost, but I am now at home. Good work
Just one detail Mu is not the interest rate as said in the video at @0:58 but the expected rate of return or the mean if we talk about a normal distribution. Many Thanks for this video, excellent complete explanation, finally understood the mysterious origins of Black Scholes !
Thank you, this was incredibly intuitive to understand, and it feels like a fundamental step in understanding more complex derivative functions. Very helpful!!
Great video! I have an oral exam including basics of ito's lemma coming up, and this lecture really contains all the relevant information I need to be able to bring across! Thank you!
Thank you very very very much. After 1 term of stochastic calculus and a bunch of other subjects I still didn't understand why we use Ito's formula. Then I tried finding the answer on the internet and no use - Try typing in the google "why we use Ito's formula" - no reasonable answer. Then I read chapters about Ito from most referenced books and still couldn't understand it until this lecture. You explained it so simple and reasonable. I am looking forward to the rest of you videos.
What a simple and nice explanation of Ito's lemma. My prof didn't explain it this simply so it's very helpful the steps he tkaes in this video. highly recommended!!
I think the benefit is that you can define the shape of the probability density function of G over time, given just G over the stock price and a process for the stock price.
In other words if you have a a time-series of daily log returns and calculate the mean = mu, this is the expected mean log return of your stock. Not an interest rate but the specific mean rate of return of this stock calculated historically
I won't comment on the math part because I wasn't paying that much attention to it but I think you have to agree that he explained pretty well in common language why we use Ito's formula. Just an example, professor that was teaching stoch calculus course at my program didn't even know what X(t) could represent in real life. She was pure mathematician and didn't care about financial part of stochastic. This resulted un majority of people not knowing how to use it. So tell me what is better.
The formula at 36:40. Like what do I use the dz term for? Expected value is 0 so when is it usable. If I ever would need to calculate or use dz then what is it? If I want to simulate dz what is the function for doing so? Right now it feels a bit abstract. But saw there is a separate video on Brownian Motion (Wiener process) so have to check that one out as well.
+stimpen12 The easiest way to think about it is simulation 10,000 draws of a standard normal variable, i.e. a number from a uniform random distribution with mean 0 and standard deviation 1.
The reason why you can't use standard calculus is because you can't define an integral with respect to a WInner process since, as the professor pointed out, the functions is too irregular as W(i)-w(i-1) gets smaller and smaller.....Behind Ito's Lemma there's Ito integral which explains well this issue and how can it be integrated
ure sooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo much better than my lecturer! THANK YOU SIR!
Ok, now we have transformed the original equation, but we still haven't solved neither the original nor the transformed equation. Where does Ito's lemma actually help us solve the equation(s)?
The extra b^2 should be a (dt). Since we treat everything but dz as a constant we get: (Let @ denote partial) VAR[(@G/@x)*bdz] = ([(@G/@x)*b]^2)*VAR(dz) = Where: dz = epsilon*sqrt(dt) ; where epsilon~N(0,1) RV. If we let dt approach -> 0, it approaches a constant. If we treat it as such we should get: VAR(dz) = VAR(epsilon*sqrt(dt)) = ([sqrt(dt)]^2)*VAR(epsilon) = dt*VAR(epsilon) VAR of a N(0,1) RV is sigma^2 which is 1 in this case, so we get dt*1 Put it all together: ([(@G/@x)*b]^2)dt
From a Stoc.Calc perspective, this hurts a little. From an intuition perspective, it’s refreshing! A little intuition on why standard calculus doesn’t apply: We want a reasonable ’white noise’ process with continuous sample paths, to model the stochastic behavior. i.e. we want (W_t) to satisfy: i) W_t is independent wrt. W_k for all t,k not equal. ii) (W_t) is stationary. That is, the joint distributions are independent of t, for all t. iii) E[W_t] = 0 for all t Turns out the only process with continuous sample paths, satisfying i) ii) and iii) is a Brownian Motion! However, though continuous everywhere, Brownian motion is no where differentiable. Thus standard calculus doesn’t apply. (It is in fact a measurability problem. There’s no measurable processes with continuous sample paths satisfying i) and ii), but a Brownian Motion)
It's not clear to me at all why the drift rate, mu, of a stock time series is referred to as the interest rate in the video. The US stock market has an historical drift rate of 8% -> 0.08. The interest rate has been less than, say, 3% -> 0.03 for a long time.
You are confusing terms here, interest rates is a general term in finance. In this case he was just referring to the expected returns on the stock which is also an interest rate. So Interest rates are just expected payments possibly in the future that every asset is expected to pay back relative to the initial principal. If an asset has no interest, it's not worth investing in.
I went over this a couple of times. I think the answer is wrong. In the answer to the dF equation, there should be no "S" multiplied by the dt term, or the dz term.
When you calculated the variance of the ito process at around 14.15 you stated that it is ((partialG/partialx))b)^2 which I understand. But why the extra b^2???
As the good book would say, Multifractional Multistable motion or GTFO. Let me preface this by saying, I didnt watch this video. I think a fairly good intro book is Shreve: Stochastic Calc for finance II. Do you disagree, oracle?
So good. Just soo good. Watch all the way until 15:20 and at that point, you will finally get the entire thing. You will have a matrix moment at that point.
This is the first time, I got someone who explained this Ito's lemma in layman's terms and your explanations make so much sense. Thank you, prof. I now have some understanding of what Ito's Lemma is and how it is applicable.
You actually explain it like a human being, thank you
Thanks Very much. I thought of dropping my financial economics class because of How my Prof thought this topic. I was so lost, but I am now at home. Good work
this is actually the best geometric Brownian motion video after i went through +10 videos and try to understand lol.
much appreciated
My brain hurts.
Just one detail Mu is not the interest rate as said in the video at @0:58 but the expected rate of return or the mean if we talk about a normal distribution. Many Thanks for this video, excellent complete explanation, finally understood the mysterious origins of Black Scholes !
Thank you, this was incredibly intuitive to understand, and it feels like a fundamental step in understanding more complex derivative functions. Very helpful!!
Excellent step-by-step explanation that helps you to grasp the core idea behind Ito's lemma. Thank you so much!
Great video! I have an oral exam including basics of ito's lemma coming up, and this lecture really contains all the relevant information I need to be able to bring across! Thank you!
Thank you very very very much. After 1 term of stochastic calculus and a bunch of other subjects I still didn't understand why we use Ito's formula. Then I tried finding the answer on the internet and no use - Try typing in the google "why we use Ito's formula" - no reasonable answer. Then I read chapters about Ito from most referenced books and still couldn't understand it until this lecture. You explained it so simple and reasonable. I am looking forward to the rest of you videos.
Thank you. Clear explanation and extremely helpful. Keep posting these videos, they help us a lot. Great job.
What a simple and nice explanation of Ito's lemma. My prof didn't explain it this simply so it's very helpful the steps he tkaes in this video. highly recommended!!
Brilliant & simple explanation of Ito's Lemma. Thank you!
i have reached a point in maths where not seeing the proof might not be such a bad idea
The absolute best explanation of Ito's Lemma on the net. Fact!
I have to agree, passed my quant finance exams last year thanks to this..
Best Explanation of Ito's Lemma on the planet. Professor you are a saviour
This is just amazing! Great work! Thank you!
Mr Byrne, you have saved me academic life
This was so helpful! Thank you so much for explaining everything at a very accessible level.
You sir, are great! Does anyone have the presentation in a pdf or something good for printing?
Thank you for your explanation. Highly appreciated.
An Introduction to the Mathematics of Financial Derivatives, author. Neftci...very good intro book about stochastic calculus
Hi, this is a fantastic video, thanks for sharing it.
how can we apply ito formula on skew brownian motion
Thank you sooo much, this made sooo much sense!!🙏🏻☺️☺️☺️
Pretty cool work!
You're the best!!
I think the benefit is that you can define the shape of the probability density function of G over time, given just G over the stock price and a process for the stock price.
This is very helpful, thank you professor.
this is a great explanation
Cool! This really is a rather clear explanation!
Great explanation. Thank you.
In other words if you have a a time-series of daily log returns and calculate the mean = mu, this is the expected mean log return of your stock. Not an interest rate but the specific mean rate of return of this stock calculated historically
I won't comment on the math part because I wasn't paying that much attention to it but I think you have to agree that he explained pretty well in common language why we use Ito's formula.
Just an example, professor that was teaching stoch calculus course at my program didn't even know what X(t) could represent in real life. She was pure mathematician and didn't care about financial part of stochastic. This resulted un majority of people not knowing how to use it. So tell me what is better.
so what is the difference between the SDE and the ito's lemma, as we still have dz
Excellent explanation. Thank you.
great lecture. Thank you so much.
Fantastic! Thanks for sharing!
lifesaver! thanks!!!
Man , He is a Hero !!
Thank you for this nice lecture sir.
Excellent explanation. Thanks
is there an error in the variance formula at 14:54? There shouldn't be a b^2 on the outside, right?
What is the name of that book please
Great explanation! Thanks
This is good, despite you changing notation several times.
Hi! Which book do you use? Or how can I get an access for full lectures? thanks!
Best. Explanation. Ever.
Thank you for sharing!
You are my hero! Forget batman, superman and the avengers. You are my hero.
So good. Just so good.
thank u thank u thank u, i finally start understandin!!!
Awesome..... You saved my exam....
Pretty cool, nice explanation!
The formula at 36:40. Like what do I use the dz term for? Expected value is 0 so when is it usable. If I ever would need to calculate or use dz then what is it?
If I want to simulate dz what is the function for doing so? Right now it feels a bit abstract. But saw there is a separate video on Brownian Motion (Wiener process) so have to check that one out as well.
You can use dz for calculating the expected volatility. dz~N(0,t).
stimpen12
yes, E(dz) = 0
but, E(dz^2) = Var(dz) = dt
Ting Gu no… dz ~ N(0, dt)
+stimpen12 The easiest way to think about it is simulation 10,000 draws of a standard normal variable, i.e. a number from a uniform random distribution with mean 0 and standard deviation 1.
The reason why you can't use standard calculus is because you can't define an integral with respect to a WInner process since, as the professor pointed out, the functions is too irregular as W(i)-w(i-1) gets smaller and smaller.....Behind Ito's Lemma there's Ito integral which explains well this issue and how can it be integrated
quite informal, but very useful in a practical sense.
The slides are partly from Hull! (:
Thank you.
@skoules who taught you that risk free rate has to be continuous compounded.
ure sooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo much better than my lecturer! THANK YOU SIR!
Good video! Very clear actually!
What are a and b in ito's lemma? are they always mu_S and sigma_S respectively?
Lovely explanation
"we're not going to go over the proof of this"
K BYYYEEEEEEEE
Ok, now we have transformed the original equation, but we still haven't solved neither the original nor the transformed equation. Where does Ito's lemma actually help us solve the equation(s)?
Legend
The extra b^2 should be a (dt).
Since we treat everything but dz as a constant we get:
(Let @ denote partial)
VAR[(@G/@x)*bdz] =
([(@G/@x)*b]^2)*VAR(dz) =
Where:
dz = epsilon*sqrt(dt) ; where epsilon~N(0,1) RV. If we let dt approach -> 0, it approaches a constant. If we treat it as such we should get:
VAR(dz) = VAR(epsilon*sqrt(dt)) = ([sqrt(dt)]^2)*VAR(epsilon) = dt*VAR(epsilon)
VAR of a N(0,1) RV is sigma^2 which is 1 in this case, so we get dt*1
Put it all together:
([(@G/@x)*b]^2)dt
thanks for the video
From a Stoc.Calc perspective, this hurts a little.
From an intuition perspective, it’s refreshing!
A little intuition on why standard calculus doesn’t apply:
We want a reasonable ’white noise’ process with continuous sample paths, to model the stochastic behavior.
i.e. we want (W_t) to satisfy:
i) W_t is independent wrt. W_k for all t,k not equal.
ii) (W_t) is stationary. That is, the joint distributions are independent of t, for all t.
iii) E[W_t] = 0 for all t
Turns out the only process with continuous sample paths, satisfying i) ii) and iii) is a Brownian Motion!
However, though continuous everywhere, Brownian motion is no where differentiable.
Thus standard calculus doesn’t apply.
(It is in fact a measurability problem. There’s no measurable processes with continuous sample paths satisfying i) and ii), but a Brownian Motion)
It's not clear to me at all why the drift rate, mu, of a stock time series is referred to as the interest rate in the video.
The US stock market has an historical drift rate of 8% -> 0.08. The interest rate has been less than, say, 3% -> 0.03 for a long time.
You are confusing terms here, interest rates is a general term in finance. In this case he was just referring to the expected returns on the stock which is also an interest rate.
So Interest rates are just expected payments possibly in the future that every asset is expected to pay back relative to the initial principal. If an asset has no interest, it's not worth investing in.
good video!
hey thanks i understand this..but I'm going to be asked this for my dissertation and they won't accept that as a answer :(
I went over this a couple of times. I think the answer is wrong.
In the answer to the dF equation, there should be no "S" multiplied by the dt term, or the dz term.
When you calculated the variance of the ito process at around 14.15 you stated that it is ((partialG/partialx))b)^2 which I understand. But why the extra b^2???
@37:39 unless that risk minded person is a scalp trader using 1 hr to 15 minute charts:-)
The very first slide says "Ito's" not "it's." Were you joking?
Then I guess this is a perfect time to bother your advisor ;-)
32.08 - if a guy asked me that stupid question after one hour of explaining, i'd slap him in the face 'stochastically'
As the good book would say, Multifractional Multistable motion or GTFO. Let me preface this by saying, I didnt watch this video. I think a fairly good intro book is Shreve: Stochastic Calc for finance II. Do you disagree, oracle?
Shreve
So good. Just soo good. Watch all the way until 15:20 and at that point, you will finally get the entire thing. You will have a matrix moment at that point.
lol He's doing math, wrong :P
if you really wanna learn some stochastic calculus then drop the things this video tells you cuz they are totally wrong