5 step method

Hi,

I am getting a bit confused about what we are actually trying to do here. The notes suggest that we are proving the Black-Scholes call option pricing formula. This can be proved in 2 ways:
1. 5 step method
2. PDE approach from the earlier black-scholes chapter
However, i am getting confused with these:

1. the 5 step method actually just proves that V(t) = exp(-r(T-t))E(Q)[X|F(t)] which is the pricing formula from the binomial model chapter?

So the 5 step method simply proves the above result - not the actual Black Scholes Call option pricing formula?!

If we then want to prove the black-scholes call option formula we use this 5 step proof and then add in some steps after to get from V(t) = exp(-r(T-t))E(Q)[X|F(t)] to the Black Scholes call option pricing formula?
ie. sub in x=max(st-k,0)
to get V(t) = exp(-r(T-t))integral[(s-k)f(s)ds] and complete this to get the BS call pricing formula.

Is my understanding correct?

2.Looking through the core notes I cant see this PDE proof of the Black scholes pricing formula - Am i just not looking properly or is this not actually explicitly covered in the notes?
I understand the PDE formula and how we would use it (would simply apply the PDE formula to the Black scholes call option pricing formula) but looking at the black scholes call pricing formula i can imagine the calculations in differntiating it etc would be very heavy and time consuming.

Should we be able to do this approach - is it examinable? also is this in the notes somewhere and im just missing it or ?

Apologies for the wordy post,

Thanks,
James

 

2nd Q is very subjective, experienced can answer...

1.you know Black-scholes option pricing is continuos form of Binomial option pricing
and your V (t) should be correct as long as you set exhaustive integral limit subject to S-K always +ve.

 

Thanks Hemant, can anyone add to this please?

 

As James asked, anyone to add to the already mentioned ?

 

sorry to bump again, but can anyone help with this please i'm trying to complete these chapters and its bugging me that I have these questions.

- are we meant to be able to prove the Black Scholes formula using the PDE approach? I cant find it anywhere in the notes and can imagine the maths being pretty time consuming so I would assume its not examinable??

-Also, maybe my core reading notes are outdated (couple years old) but I have 2 versions of the 5 step proof - one for discrete time (chapter 14) and one for continuous time (chapter 15) yet they seem identical? They are both proving the binomial pricing formula....im confused by this, should we know both proofs ? are there actually any differences between them?

Thanks

 

are we meant to be able to prove the Black Scholes formula using the PDE approach? I cant find it anywhere in the notes and can imagine the maths being pretty time consuming so I would assume its not examinable??
Well! now, I read the syllabus objectives of CT8, especially Black-Scholes related parts. I didn't see any objectives to prove the Black-Scholes formula using the PDE approach, better leave it.
but you'd learn to " Derive the Black-Scholes partial differential equation both in its basic and Garman-Kohlhagen forms"~(ix)8

Also, maybe my core reading notes are outdated (couple years old) but I have 2 versions of the 5 step proof - one for discrete time (chapter 14) and one for continuous time (chapter 15) yet they seem identical? They are both proving the binomial pricing formula....im confused by this, should we know both proofs ?
In my 2016's ActEd notes, they're Chapter 15 and 16. If you have old ActEd notes update your notes from there.
both methods are not identical.
you can understand a proof then link with another. timing differ, trading measure differ.

are there actually any differences between them?
Let D: Discrete time 5-step method.
C: Continuous time 5-step method.
S_t is Geometric Random Walk in D, Geometric Brownian Motion in C.
suppose unit time is 1 year, and after 4 years and 6 months, you know past history up to time 4 years in D, and 4.5 years in C. you cannot measure value at any non-integer unit times in D, you can in C.