Brownian Motion | Part 3 Stochastic Calculus for Quantitative Finance

[@edyt4125]

Phenomenal video. I am going to use some of these visuals in lecture for my college students!

[@shreyashree.d69]

I have taken Stochastic Calculus for Finance this Fall at my uni, and man oh man I was struggling so hard to visualize what Brownian motion is like, and here you are, like a blessing, with all these videos and I can't be more grateful! Thanks a bunch! ❤

[@jesuseduardobanosgonzalez8116]

I'm so glad I found this channel right after I took my first course on financial derivatives

[@natanalbuquerque1053]

Otimo videos!

[@Deepia-ls2fo]

Hey ! Your video looks very good, I'll make sure to watch everything later. Keep it up :)

[@gensas1]

nice vids, hope your channel takes off

[@raneena5079]

BTW, thanks for making these videos!

[@mustafizurrahman5699]

Awesome

[@matveyshishov]

Thanks! When is this model NOT applicable? Do we know what situations make this impossible? Cool accent, btw!

[@maths.visualization]

Can you share video code?

[@stochastip]

I’ve got about 7000 lines of code scattered around right now. Maybe I’ll clean it up and upload it to a github when I find the time!

[@maths.visualization]

Will I have no problem if you upload it to GitHub like this? 🙂

[@raneena5079]

How did you insert \|\Pi\| into the limit in the RHS of the inequality at 9:17? And why does this only work for differentiable functions?

[@stochastip]

It's because when the function is differentiable, you can apply the Mean Value Theorem (shown in red). This gives you something of the form fʹ(t)²(tᵢ − tᵢ₋₁)². Then, you can factor out |Π| and say that (tᵢ − tᵢ₋₁)² ≤ |Π|(tᵢ − tᵢ₋₁)

[@raneena5079]

@@stochastipOhh, that makes sense. Thanks for the clarification.

[@raneena5079]

How did you get E[W_s^2] = s at 7:38?

[@raneena5079]

Does it come from var[W_s] = var[W_s - W_0] = s, so s = E[(W_s - W_0)^2] - (E[W_s - W_0])^2 = E[W_s^2]?

[@stochastip]

I used the result I proved earlier: E[Wₛ Wₜ] = min(s, t), so instead of having s and t different, I just set s = t. This simplifies to E[Wₛ²] = s. But you are right, in the case of Brownian motion, E[Wₛ²] = Var(Wₛ) = s, so you can think of it from all possible angles.

[@raneena5079]

@@stochastip Thanks for the reply, but I was asking about how you reached the last line in the proof that E[W_s W_t] = min{s, t}.

[@stochastip]

@@raneena5079 𝔼[𝑊ₜ𝑊ₛ] = 𝔼[𝑊ₜ𝑊ₛ − 𝑊ₛ𝑊ₛ + 𝑊ₛ𝑊ₛ] ⟶ (Add and subtract 𝑊ₛ𝑊ₛ) 𝔼[𝑊ₜ𝑊ₛ] = 𝔼[(𝑊ₜ − 𝑊ₛ)𝑊ₛ] + 𝔼[𝑊ₛ²] ⟶ (Distribute expectation) 𝔼[𝑊ₜ𝑊ₛ] = 𝔼[𝑊ₜ − 𝑊ₛ]𝔼[𝑊ₛ] + 𝔼[𝑊ₛ²] ⟶ (Independence of increments) 𝔼[𝑊ₜ𝑊ₛ] = 0 ⋅ 0 + 𝑠 ⟶ (Use 𝔼[𝑊ₛ] = 0 and 𝔼[𝑊ₛ²] = 𝑠) 𝔼[𝑊ₜ𝑊ₛ] = 𝑠 Hope this helps

[@raneena5079]

@@stochastip I'm sorry if I misunderstood you, but it seems like you're using a circular argument. You use E[W_s^2] = s to prove E[W_s W_t] = min{s, t} and then use E[W_s W_t] = min{s, t} to prove that E[W_t^2] = t, which I assume is the same statement as E[W_s^2] = s.