Black Scholes PDE Derivation using Delta Hedging

Explains the various approaches to derive the Black Scholes PDE using delta hedging and Ito's lemma

[sempercrescere]

Great video, the best I've seen so far. Thank you!

[vmdaiict]

@quantpie can u pls help answer below?
Q1. Why did we omit d Alpha * d B term from d( Pie ) ?
Q2. From the explanation of self financing - we see that d ( Delta ) * S + d( alpha ) * B =0. But then it means d( delta ) * d S should also be 0 to satisfy the d( Pie ) equation . But neither d ( Delta ) nor dS is ever 0 right ?
So could you pls help me understand about the higher order terms of d(Pie) becoming 0.

Btw love the video!!

[MarinaPanarina]

Thanks so much for this video. I am currently doing a final year college project on option pricing, and this video really helped :). Is there any way that I can formally cite this in my project? I mean, did you follow a derivation from a certain book, or do you have written notes on this? Derivations in textbooks that I've found arent as clear as this one. Thanks again, hope you can answer me!

[leonisvandenberg]

Thank you for your videos. I have a question in 11:18. Does the change in the portfolio value, i.e., dPi = dS - (1/Delta)dV, follows from Ito's summe rule?
To be more precise: let us first denote Pi_(V_t) as the partial derivate w.r.t. V_t
Then: Pi_(V_t) = -(1/Delta) and similarly for Pi_(S_t) = 1
And the second partial derivative is trivially zero, i.e., Pi_(V_t V_t) = 0, similarly for Pi_(S_t S_t) = 0 and also Pi_(V_t S_t) = 0.
Ito's lemma for two variable is: df(X, Y) = f_X dX + f_Y dY + 0.5 f_(XX) dX^2 + 0.5 f_(YY) dY^2 + f_(XY) dXdY,
Substituting now, we get: d(S_t - (1/Delta) dV_t ) = 1* dS_t - (1/Delta) *dV_t
Did I understand it correctly?

[dark_knight]

firstly Thanks for the awesome video, I wanna if we can use the propriety of the discounted Prices being a martingale, then concluding that the term multiplied by dt should be 0 and we can get our pde ?

[djsocialanxiety]

Man your videos are absolute gold thanks so much ❤️

[sounakmojumder]

thank you, if you can please make a discussion of modified black scholes equation for stochastic volatility,

[davidharun]

I have two questions:
1)Why do we even want to eliminate the stochastic part? I mean, isnt that there for us to understand how the options price changes over time?
2)Why do we have a bank account there? What is the point in having dB? What does that have to do with black scholes pde?

[lamja]

thanks for this, however do you have any sources/references where did you get the derivation from? Thanks!

[gaborpalovics]

Around 4:20 why do we assume delta to be constant when differentiating the portfolio price?
Why is it not the following: d_Pi = delta * dS + S * d_delta + alpha * dB
Also if delta is constant why are we setting it to -dV/dS?

Great videos
btw!

[arkabose]

why dS^2 = sigma squared*S^2*dt? Any videos on this?

[kalernikhilsingh]

I'm in love with the woman who recorded this video.❤

[michielfenaux]

Do mu and sigma need to be constant to derive the PDE?

[abhraboral_educator]

can you please show a black schole pde whre there is a time delay

[erthx]

@6:34..."and boom goes the dynamite."

[parthbhanushali]

I really thr videos of Quantpi but can be a bit slow on the part of mathematics ? 😊

[maxbob]

isn't that only true after change of measure? apriori shouldn't V be dependent on the drift of the asset as well?