I have not got a clue who he is but his lecture is the best way to explain a difficult subject. Not many people can stand on his feet, think and solve problem in front of the students.
Thanks for the great video! One question please may I: 27:51 how did you derive that diffusion equation of dx at the top? Do you have another lecture for the details of derivation? Many thanks!
ice cold explanation man!!!! altho at 21:13 you talked about the amount added at each increment is sqrt(t/n), i wonder where does the Normal distribution come in? I thought the amount added at each increment was based on a Normal Distribution but it looks like the amount added (or subtracted??) is sqrt(t/n), a constant. What am I missing?
There is no implication between martingale and Markovian. You should remove the note on the slide shown around the 6th minute stating that all martingales are Markovian and make sure you do not mix up both concepts as they are distinct.
Prof. Bill, It might be helpful, when explaining your random walk Markovian Martingales, where the expectation is zero (50/50 probability) to turn your coin toss slide sideways. It then becomes a histogram with a mean of approximately zero. Good lecture.
When he modifies a standard walk to a brownian motion, why is Ri equal to the square root of t/n? Maybe i don't understand what Ri really is because im thinking that Ri is the normal random variable score that occurs at increment i.
great class, even for me (I´m a lready familiar with brownian motions). By the way, the third name contributing to the Black-Scholes model is Merton, not Morton.
So considering martingale "older values" is as "gamblers fallacy"? Expecting a coin to become "fair" in the direction opposite to what it has already shown... if that makes any sense?
This is very good video and very helpful to understand basic of BM. thanks prof bill. thank you so much. one request, if you upload video on BSM, solve equation starting from basic i.e "x-w(1/delta)" to D1 and D2. thank you
I think it's interesting how he adds letters to stochastic, instead he's fantastically stoked! Stoke-tastic! Also, Wiener is pronounced with a V (Vee-ner). I realize they are just implications of the local vernacular, but it always makes me jump a little bit when he says them. Overall, a great high-level overview for non-mathematicians.
Hi Bill, it may be a bit late for me to ask this question, but why exactly should we care about the volatility of a stock when assuming Brownian Motion? The expected value is always going to be 0 if I understand correctly, so shouldn't we just focus on the non-Brownian part of the equation? The factor "b" in the differential equation surely has no influence on the stocks expected value over time, only the factor "a" would be relevant, right?
Hey so I know I'm too late but the b does matter , since when we predict the future prices of stocks we do have uncertainty regarding its future path of prices so unless and until we have 0 uncertainty and we are absolutely sure of what the future path of the stock price is gonna be (in which case the b=0) b or the volatility of the stock does matter.
Are you sure? I thought all martingales were Markovian? Markovian means that the expected value of the process at any future value depends only on the current value and not on any previous history. Martingales means that the value at any future value is expected to be the current value (and not on any previous history's value).
+Dasha Y You can think of Ri as the the standard deviation of each movement (increment). I'm not sure about this statement so don't take my word for it. In an informal way, think of the Var(Ri): Var(Ri) = E[Ri^2] - E^2[Ri]; where E^2[Ri] = 0, E[Ri^2] = t/n -> sd(Ri) = sqrt(Var(Ri)) = sqrt(t/n) On the other hand Since Ri is a martingale: E[Ri^2] = Ri^2 = t/n -> sqrt(Ri) = sqrt(t/n) Hope it helps!
+Dasha Y Me neither... plus if the increments are sqrt(t/n), then the increments are always positive, and E(Si) can never tend to zero. I believe he is actually stating that the stdev(Ri) is sqrt(t/n)... but without really showing why.
@@changantonio Maybe I'm too late, but the reason is when you consider a random walk realization Xn with an equal like likely realization of +/-1 , then E[X] is zero, but E[X^2] will equal to x1^2 +x1*x2 + x2^2+x2*x1...etc. here you can see that x1^2 and x2^2 (which correspond to the stepsize N) will equal to 1 regardless if they are +/-1 since they get squared. All the other combination terms f.e. x1*x2 imagine which combinations x1 and x2 could be. both can be 1 in that case the combination would equal to +1, both can be -1 in that case the combination would again be +1, and twice one can be positive and the other negative, where the combination would result in -1, since +1*-1 is negative. So you have 4 combination possibilities with twice +1 and twice -1, which in sum is zero. So all the combination terms equal to zero and only the squared single terms are left, which correspond to the amount of N steps taken - hence E[X^2] = N. Since its a martingale and today is the best estimator for tomorrow X^2 = N and therefore X = sqrt(N).
I feel like the definition of Markov Process should be "a sequence of possible events in which the probability of each event depends only on the state attained in the previous event." Simply put, future is independent of the past, given the present. Doesn't this contrast with your slide in 4:25?
Hi Robin, I believe your definition "[...] each event depends only on the state in the previous event." is slightly flawed. Each event does NOT depend on ANY previous event, not even the one just before it. Each event is completely random. The expected value is solely dependent on the current value, maybe you mixed those two things up. Hope I could help.
Actually, even if the weather for the last 3 days predicts weather for today, that can still be modeled as a markov process. If each day, there are n possible different states of weather, then the weather for the past 3 days gives n^3 possible states, so the past 3 days can be considered the "current" state, as long as the number of days the next day's weather depends on is finite.
But how do I model the Wiener process. Say I have a value for b and want to simulate a Wiener process. What do I do with b? Do I run a random number generator picking a number from the normal distribution and then take it times b? And then do that again with the previous result and take it times a new random number from a normal distribution? What would the characteristics of the normal distribution be that I should use for the random numbers? Expected value =0 and what about the variance, I did not really understand that part. It´s only t? But what is t? Is it years? Say I want to model a stockprice over a day should I use 1/365 then? And to simulate a Wiener process I crank up the number of observations on a day to say 1 000 000 to simulate that the n in t/n goes to infinity?
Had a question. At 21:13 we talk about the amount added at each increment is sqrt(t/n). Where does the Normal distribution come in? I was thinking that the amount added at each increment was based on a Normal Distribution but it looks like the amount added (or subtracted??) is sqrt(t/n), a constant...What am I missing??? All help appreciated :)
2 counter examples: - the Ito integral is a martingale but not Markovian - a biased coin scoring +1 if H and -1 if T. The score is Markovian but not martingale I am happy to provide more explanation if needed. Otherwise check online and the link below. wilmott.com/messageview.cfm?catid=8&threadid=11322
@@hansa9159direct observation with perhaps particles with different flourescent dyes . Different particle sizes . Different tempratures in uv light with high powered microscopes. Vary the parameters. Collect the data. Try to figure out the mathematical relationships between the various variables. there is too much theory and not enough experimenting in the subject of brownian motion
@@hansa9159I did but did not persue it. I'm not made for regular school and it is a lot of work and does not pay well anyway. If you are interested , I am convinced that Brownian Motion is the result of microeddies and microcurrents that .shift and change at very high velocity. The water molecules are not ricocheting around. They are polar for one and water is incompressible for another. There may be tiny ricochets which push and pull at the fully connected water matrix causing it to behave like a complex current carrying whatever is immersed in it , with it. That's my take and it should be verifiable
Lamentably, the presenter does not seem to have much clue about his subject and of math in general, he explains things badly if at all, and there are numerous mistakes both in slides and talk. Skip this.
If so, can you kindly produce a lesson and slides without errors and with better explanations, we as learners would really appreciate that. Meantime I am thankful to Billbyrne that he made all this effort and produced stuff that we all can read from all around the world.
CompuViz I'd love to do that, I could do it (I'm a university lecturer on stochastic modelling, among other topics), but I don't have the time. Sorry. I still hope that my comment will help viewers, as well as the lecturer himself, to more realistically assess the quality of this presentation.
I agree with L2K4D44L4R, If this is the first video someone is watching on Brownian process..I am afraid they are going to have lot of misconception about the subject..The section of Martingales esp is very badly explained
L2K4D44L4R being critical can be constructive but I can say what ever I want too without any back up to my claims. I'm a masters of stats student and currently working in a hedge fund and this vid although maybe not perfect did help clear up some concepts for me. thanks to Bill for this.
L2K4D44L4R Your comment was useless, why would it help anyone trying to learn the subject? You made no specific, concrete critique of the presentation, you just claimed it was wrong. So no, it's not helpful.
Thank you for simplifying and making me understand what was such a difficult field for me.
I have not got a clue who he is but his lecture is the best way to explain a difficult subject. Not many people can stand on his feet, think and solve problem in front of the students.
If you are still active well you should know that after 12 years this video is helping a lot. I wish you were my professor
Your lectures are heaven-sent
Thanks for the great video! One question please may I: 27:51 how did you derive that diffusion equation of dx at the top? Do you have another lecture for the details of derivation? Many thanks!
ice cold explanation man!!!! altho at 21:13 you talked about the amount added at each increment is sqrt(t/n), i wonder where does the Normal distribution come in? I thought the amount added at each increment was based on a Normal Distribution but it looks like the amount added (or subtracted??) is sqrt(t/n), a constant. What am I missing?
i think maybe there should be an 'e' before the sqrt(t/n) so it is e*sqrt(t/n), which e~N(0,1).
@@thesupersimon Could it be we assume Central Limit theorem to be applicable?
You are a very good teacher!
There is no implication between martingale and Markovian. You should remove the note on the slide shown around the 6th minute stating that all martingales are Markovian and make sure you do not mix up both concepts as they are distinct.
Prof. Bill,
It might be helpful, when explaining your random walk Markovian Martingales, where the expectation is zero (50/50 probability) to turn your coin toss slide sideways. It then becomes a histogram with a mean of approximately zero. Good lecture.
I actually understand all your videos. I highly recommend them
When he modifies a standard walk to a brownian motion, why is Ri equal to the square root of t/n? Maybe i don't understand what Ri really is because im thinking that Ri is the normal random variable score that occurs at increment i.
great class, even for me (I´m a lready familiar with brownian motions). By the way, the third name contributing to the Black-Scholes model is Merton, not Morton.
So considering martingale "older values" is as "gamblers fallacy"? Expecting a coin to become "fair" in the direction opposite to what it has already shown... if that makes any sense?
Thank you for this great explanation!
This is very good video and very helpful to understand basic of BM. thanks prof bill. thank you so much.
one request, if you upload video on BSM, solve equation starting from basic i.e "x-w(1/delta)" to D1 and D2. thank you
The lecturer is great
I think it's interesting how he adds letters to stochastic, instead he's fantastically stoked! Stoke-tastic! Also, Wiener is pronounced with a V (Vee-ner). I realize they are just implications of the local vernacular, but it always makes me jump a little bit when he says them.
Overall, a great high-level overview for non-mathematicians.
Hi Bill, it may be a bit late for me to ask this question, but why exactly should we care about the volatility of a stock when assuming Brownian Motion? The expected value is always going to be 0 if I understand correctly, so shouldn't we just focus on the non-Brownian part of the equation? The factor "b" in the differential equation surely has no influence on the stocks expected value over time, only the factor "a" would be relevant, right?
By the way I much enjoyed the video, thanks for uploading!
Hey so I know I'm too late but the b does matter , since when we predict the future prices of stocks we do have uncertainty regarding its future path of prices so unless and until we have 0 uncertainty and we are absolutely sure of what the future path of the stock price is gonna be (in which case the b=0) b or the volatility of the stock does matter.
Great lecture, loved your examples very straightforward to understand
Cool material!!
Are you sure? I thought all martingales were Markovian?
Markovian means that the expected value of the process at any future value depends only on the current value and not on any previous history.
Martingales means that the value at any future value is expected to be the current value (and not on any previous history's value).
I am really enjoying this video, thanks for sharing!
20:48 I don't really understand why Ri = square root of (t/n)? Why is it the square root?
+Dasha Y You can think of Ri as the the standard deviation of each movement (increment). I'm not sure about this statement so don't take my word for it.
In an informal way, think of the Var(Ri):
Var(Ri) = E[Ri^2] - E^2[Ri]; where E^2[Ri] = 0, E[Ri^2] = t/n
-> sd(Ri) = sqrt(Var(Ri)) = sqrt(t/n)
On the other hand
Since Ri is a martingale:
E[Ri^2] = Ri^2 = t/n
-> sqrt(Ri) = sqrt(t/n)
Hope it helps!
+Dasha Y Me neither... plus if the increments are sqrt(t/n), then the increments are always positive, and E(Si) can never tend to zero. I believe he is actually stating that the stdev(Ri) is sqrt(t/n)... but without really showing why.
@@changantonio Maybe I'm too late, but the reason is when you consider a random walk realization Xn with an equal like likely realization of +/-1 , then E[X] is zero, but E[X^2] will equal to x1^2 +x1*x2 + x2^2+x2*x1...etc. here you can see that x1^2 and x2^2 (which correspond to the stepsize N) will equal to 1 regardless if they are +/-1 since they get squared. All the other combination terms f.e. x1*x2 imagine which combinations x1 and x2 could be. both can be 1 in that case the combination would equal to +1, both can be -1 in that case the combination would again be +1, and twice one can be positive and the other negative, where the combination would result in -1, since +1*-1 is negative. So you have 4 combination possibilities with twice +1 and twice -1, which in sum is zero. So all the combination terms equal to zero and only the squared single terms are left, which correspond to the amount of N steps taken - hence E[X^2] = N. Since its a martingale and today is the best estimator for tomorrow X^2 = N and therefore X = sqrt(N).
This is really good
Nice Lecture :)
Thank you! Great explanations
Good videos but I think you should cite Hull, which your content draws heavily on.
I feel like the definition of Markov Process should be "a sequence of possible events in which the probability of each event depends only on the state attained in the previous event." Simply put, future is independent of the past, given the present. Doesn't this contrast with your slide in 4:25?
wait, I think there is a difference between a Markov process and a Markov chain.
Hi Robin, I believe your definition "[...] each event depends only on the state in the previous event." is slightly flawed. Each event does NOT depend on ANY previous event, not even the one just before it. Each event is completely random. The expected value is solely dependent on the current value, maybe you mixed those two things up. Hope I could help.
this is so good
Actually, even if the weather for the last 3 days predicts weather for today, that can still be modeled as a markov process. If each day, there are n possible different states of weather, then the weather for the past 3 days gives n^3 possible states, so the past 3 days can be considered the "current" state, as long as the number of days the next day's weather depends on is finite.
But how do I model the Wiener process. Say I have a value for b and want to simulate a Wiener process. What do I do with b?
Do I run a random number generator picking a number from the normal distribution and then take it times b? And then do that again with the previous result and take it times a new random number from a normal distribution?
What would the characteristics of the normal distribution be that I should use for the random numbers? Expected value =0 and what about the variance, I did not really understand that part. It´s only t? But what is t? Is it years? Say I want to model a stockprice over a day should I use 1/365 then?
And to simulate a Wiener process I crank up the number of observations on a day to say 1 000 000 to simulate that the n in t/n goes to infinity?
Thanks for such a nice explanation
Thank you Sir.
Absolutely ... Amazing teaching ....
very nice
thank you so much for this lecture!
Had a question. At 21:13 we talk about the amount added at each increment is sqrt(t/n). Where does the Normal distribution come in? I was thinking that the amount added at each increment was based on a Normal Distribution but it looks like the amount added (or subtracted??) is sqrt(t/n), a constant...What am I missing??? All help appreciated :)
This is awesome!
Thanks! Really good explanation!
dW(t) a derivative ? Wiener process is nowhere differentiable...
So a= constant as inteterst rate in a bank/bond and b is what? beta of the stock?
Thanks great lecture!
interesting video, thanks a lot .
2 counter examples:
- the Ito integral is a martingale but not Markovian
- a biased coin scoring +1 if H and -1 if T. The score is Markovian but not martingale
I am happy to provide more explanation if needed. Otherwise check online and the link below.
wilmott.com/messageview.cfm?catid=8&threadid=11322
great video
Great Video... Next lecture link please?
13:00 for brownian material discussed
black scholes option cost variation formulae...
nice study
I need help could u
ممكن الترجمة الى العربي وشكرا
You are never going to understand what Brownian Motion is with Math . You might learn more about math but not Brownian Motion.
What makes you say that? What tool do you suggest?
@@hansa9159direct observation with perhaps particles with different flourescent dyes . Different particle sizes . Different tempratures in uv light with high powered microscopes. Vary the parameters. Collect the data. Try to figure out the mathematical relationships between the various variables. there is too much theory and not enough experimenting in the subject of brownian motion
Thanks for the insight. Did/do you study physics?
@@hansa9159I did but did not persue it. I'm not made for regular school and it is a lot of work and does not pay well anyway. If you are interested , I am convinced that Brownian Motion is the result of microeddies and microcurrents that .shift and change at very high velocity. The water molecules are not ricocheting around. They are polar for one and water is incompressible for another. There may be tiny ricochets which push and pull at the fully connected water matrix causing it to behave like a complex current carrying whatever is immersed in it , with it. That's my take and it should be verifiable
haha wiener
WIENER!
(Vee-nur)
Not bad
Lamentably, the presenter does not seem to have much clue about his subject and of math in general, he explains things badly if at all, and there are numerous mistakes both in slides and talk. Skip this.
If so, can you kindly produce a lesson and slides without errors and with better explanations, we as learners would really appreciate that. Meantime I am thankful to Billbyrne that he made all this effort and produced stuff that we all can read from all around the world.
CompuViz I'd love to do that, I could do it (I'm a university lecturer on stochastic modelling, among other topics), but I don't have the time. Sorry. I still hope that my comment will help viewers, as well as the lecturer himself, to more realistically assess the quality of this presentation.
I agree with L2K4D44L4R, If this is the first video someone is watching on Brownian process..I am afraid they are going to have lot of misconception about the subject..The section of Martingales esp is very badly explained
L2K4D44L4R being critical can be constructive but I can say what ever I want too without any back up to my claims. I'm a masters of stats student and currently working in a hedge fund and this vid although maybe not perfect did help clear up some concepts for me. thanks to Bill for this.
L2K4D44L4R Your comment was useless, why would it help anyone trying to learn the subject? You made no specific, concrete critique of the presentation, you just claimed it was wrong. So no, it's not helpful.