Best teacher for stochastic calculus. People Like you(For Stochastic Calculus) and Prof leonard(For Calculus) are out there saving people's degree.Lots of love.
At 14:03 and 15:36 :- Ito's isometry is not same as quadratic variation. Because Ito isometry is expectation of the integral and hence a non random quantity. While quadratic variation is a random/stochastic process and it depends on the path.
At 30 min ½ Sigma (0,n1) (wj+1-Wj)^2, this concept was introduced. I wanted to understand why this concept was introduced. At 39.17min, the presenter refers to the above as computation of Quadratic variation. Can anyone explain the concept introduction and its reference to QV?
At 32.16 min The presenter changed the variable and counter(increment) from j to K. The presenter later (32.45 min)segregated the counter into two parts K=n and Kbelongs to (0, (n-1)) What did we achieve by this math operation?
Δn(t) is defined on the left board, its just a regular function: Wj dW(t) is the difference between two Brownian Motion steps: Wj+1 - Wj Δn(t).dW(t) -> Wj( Wj+1 - Wj) The idea of Ito calculus is that the dW(t) or differentiation in the classical calculus sense can be applied to probabilistic functions (brownian motion in this case) where dW(t) is the difference in two subsequent outputs of the function output. We don't know the output because it will happen in the future so we can just say we have an expectation (both in the English sense and in the mathematical sense) of what the difference between those two steps are. To explain another way: Take an example of a stochastic process of a drunk man walking around. We have a reasonable expectation of how far a step he takes each time, and assuming he can only walk in left or right direction, we can then start to come up with a mathematical model to calculate how far left or right this person has walked after N steps. Δn(t) is the current step distance, dW(t) is the next expected step size. Assuming he can only take 10 steps, so if you come up with all combination of step sizes and directions, it would be an infinite number of distiances that he has gone left or right after 10 steps. Imagine he can take 40cm left and 40.000000001 cm right. See what I mean by infinite combinations? Integrating across all combinations of left right steps would give you a distribution of distance left or right after 10 steps. We can't hand calculate infinite combinations, but we can take statistics of how this drunk person perfers left vs right stepping (maybe he has a bad foot) and we can take his preference of how far he goes left when he goes left or how far he goes right, use that information into Ito integral and get an expectation. Where things go wrong: Ito best applies if the drunk man walking is memoryless, the asymmetry you can mathematically build into Ito, the memoryless-ness of it has to be assumed since it is fundamental to Ito and martingales. What can go wrong in Finance? People are greedy and fearful. People are not memoryless. People on average follow a crowed due to fear, but rush to safety as soon as they see other people freak out. You can't model this in Ito. These extremes are due to the fact that the system no symmetric and not memory less and has multiple feedback loops that make it none linear. You can't predict these events with any math and the show up as "fat tails" in the distribution. The look innocent, in fact they are very dangerous. In the drunk man model, imagine the drunk man breaking his leg after two steps. You can' model any future with that. Smart people know this, price options with this math, and remove the fat tail by buying deep out of the money options.
@@ilredeldeserto this describes the state of the option given the strike and price of the underlying. google in the money and out of the money options. Many better explanations than I can write here.
Best teacher for stochastic calculus. People Like you(For Stochastic Calculus) and Prof leonard(For Calculus) are out there saving people's degree.Lots of love.
At 14:03 and 15:36 :- Ito's isometry is not same as quadratic variation. Because Ito isometry is expectation of the integral and hence a non random quantity. While quadratic variation is a random/stochastic process and it depends on the path.
Maybe this comment is not in order, he corrects himself almost immediately
At 30 min
½ Sigma (0,n1) (wj+1-Wj)^2, this concept was introduced. I wanted to understand why this concept was introduced.
At 39.17min, the presenter refers to the above as computation of Quadratic variation.
Can anyone explain the concept introduction and its reference to QV?
best prof ever
is there any prerequisite and/or books in mathematical finance and stochastics?
At 32.16 min The presenter changed the variable and counter(increment) from j to K. The presenter later (32.45 min)segregated the counter into two parts K=n and Kbelongs to (0, (n-1))
What did we achieve by this math operation?
Wow, extremely detailed explanation, Thank you!!!
tks from Brazil
Thank you for the video
respect
Nice explanation, thanks!
This is so clear, great teacher!
Pleaz i dont inderstand the passage between lim of integral ans lim of the sum in 30 min
Δn(t) is defined on the left board, its just a regular function: Wj
dW(t) is the difference between two Brownian Motion steps: Wj+1 - Wj
Δn(t).dW(t) -> Wj( Wj+1 - Wj)
The idea of Ito calculus is that the dW(t) or differentiation in the classical calculus sense can be applied to probabilistic functions (brownian motion in this case) where dW(t) is the difference in two subsequent outputs of the function output. We don't know the output because it will happen in the future so we can just say we have an expectation (both in the English sense and in the mathematical sense) of what the difference between those two steps are.
To explain another way:
Take an example of a stochastic process of a drunk man walking around. We have a reasonable expectation of how far a step he takes each time, and assuming he can only walk in left or right direction, we can then start to come up with a mathematical model to calculate how far left or right this person has walked after N steps. Δn(t) is the current step distance, dW(t) is the next expected step size. Assuming he can only take 10 steps, so if you come up with all combination of step sizes and directions, it would be an infinite number of distiances that he has gone left or right after 10 steps. Imagine he can take 40cm left and 40.000000001 cm right. See what I mean by infinite combinations? Integrating across all combinations of left right steps would give you a distribution of distance left or right after 10 steps. We can't hand calculate infinite combinations, but we can take statistics of how this drunk person perfers left vs right stepping (maybe he has a bad foot) and we can take his preference of how far he goes left when he goes left or how far he goes right, use that information into Ito integral and get an expectation.
Where things go wrong:
Ito best applies if the drunk man walking is memoryless, the asymmetry you can mathematically build into Ito, the memoryless-ness of it has to be assumed since it is fundamental to Ito and martingales. What can go wrong in Finance? People are greedy and fearful. People are not memoryless. People on average follow a crowed due to fear, but rush to safety as soon as they see other people freak out. You can't model this in Ito. These extremes are due to the fact that the system no symmetric and not memory less and has multiple feedback loops that make it none linear. You can't predict these events with any math and the show up as "fat tails" in the distribution. The look innocent, in fact they are very dangerous. In the drunk man model, imagine the drunk man breaking his leg after two steps. You can' model any future with that. Smart people know this, price options with this math, and remove the fat tail by buying deep out of the money options.
@@mehdiAbderezai what are "money options"? Do you mean "call" and "put" options?
@@ilredeldeserto this describes the state of the option given the strike and price of the underlying. google in the money and out of the money options. Many better explanations than I can write here.
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