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Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

to Black Scholes pricing

After my Heston remark on stochastic volatility
https://www.slideshare.net/list2do/crit ... g-80339422
https://papers.ssrn.com/sol3/papers.cfm ... id=3046261

the idea to look more thoroughly the Black Scholes option derivation did lead me to a critical point to their derivation. It presented at

https://www.slideshare.net/secret/K20W7nhXBcTKJs
https://papers.ssrn.com/sol3/papers.cfm ... id=3049495

Your critical comments are welcome.

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: to Black Scholes pricing

Here is the essence of my comment. The change in the value of the Black Scholes hedged portfolio at $t \in [ 0 , T )$
$d \Pi ( t ) = d C ( t , S ( t ) ) - C ^{\prime}_S ( t , S ( t ) ) d S ( t )$
does not have risky term dw ( t ). This fact lead them to the statement that
$d \Pi ( t ) = r \Pi ( t ) dt$
from which Black Scholes equation is followed.
My point is to calculate rate of return of the BS portfolio. Then we note that
$\frac{d\,\Pi \,( t )} {\Pi \,( t ) } = \frac { C^{\prime}_{t} ( t , x ) - C ^{\prime \prime}_{S S} ( t , x )) \, \sigma^{2} S ^2 ( t )}{ C ( t , x )) - C _{S}^{\prime} ( t , x )\, x ) } | \, _ { x = S ( t ) } \, dt$     (1)
The BS logic of statement that if the change in the value of the portfolio does not hold 'dw' then it is equal to risk free rate r is correct for a single point of time. The formal statement that is correct on arbitrary interval is that  [ 0 , T ] is that
$\frac{d\,\Pi ( t ) } {\Pi ( t ) } = r$           (2)
[font=Times New Roman, serif]and it is wrong on the [ 0 , T ].If (2) is true then BSE is takes place on [ 0 , T ]. [/font] We can not say that rate of return of the portfolio (2) is nonrandom on [ 0 , T ] as far as it depends on random process S ( t ). It is incorrect to write that rate of return of the hedged portfolio which is a random function is equal to nonrandom constant r.
Details is in
https://www.slideshare.net/list2do/remark-on-no-arbitrage-black-scholes-options-pricing

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: to Black Scholes pricing

It's likely I was wrong in my conclusion about BS pricing

outrun
Posts: 3913
Joined: April 29th, 2016, 1:40 pm

Re: to Black Scholes pricing

Are you considering a one time static hedge at t=0, or a continuous dynamic re-heding at all points in time in the interval [0,T] ..like B&S assume?

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: to Black Scholes pricing

outrun wrote:
Are you considering a one time static hedge at t=0, or a continuous dynamic re-heding at all points in time in the interval [0,T] ..like B&S assume?

From my point of view BS pricing states that at any point during lifetime of the option the solution of the BSE provides risk free rates of return of the BS hedged portfolio. The keep hedging over the whole interval [ 0 , T ]  we need additional stochastic cash flow. Nevertheless this is another problem, which can effect BS price to make non zero deviation between theoretical BS price and observed market prices but such fact does not make BSE wrong.
I started to verify BS in integral form because stochastic calculus should be written in integral form and to write dw ( t ) dw ( t ) = dt  is rather informal and figured out more accurate way to present BSE derivation. I think I will do it soon.
The idea to use ' the rate of return' rather than 'the change in the value' that I thought leads me to rejection of BS construction can be adjusted to BS by the statement. If C is a solution of the BSE then at any moment rate of return at ( t , S ( t )) does not depend on distribution S ( t ). This statement has a different flavour than it commonly stated when we used standard derivation of the BSE but does not actually reject the value of BSE. In such interpretation we do not state that market uses BSE solution for pricing option. When we state that market uses BSE then it sounds like a sufficient condition. In my interpretation BSE solution looks like a price of no arbitrage strategy which in general does not coincides with general price which is the settlement price.

kermittfrog
Posts: 63
Joined: September 9th, 2010, 10:25 am
Location: Frankfurt

Re: to Black Scholes pricing

It seems that there is an error in your mathematical argument.

The hedge portfolio is held fixed from t to t+dt, so there is no derivative of Delta with respect to S involved.

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: to Black Scholes pricing

kermittfrog wrote:
It seems that there is an error in your mathematical argument.

The hedge portfolio is held fixed from t to t+dt, so there is no derivative of Delta with respect to S involved.

I did not take derivative of Delta.

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: to Black Scholes pricing

There is something that I could not comprehend in derivation (BSE). In one step [ t , t + dt ) everything looks quite good but this is informal style.
To be formal we should make several steps:
1) take a subinterval ( s , t ) $\in$ [ 0 , T ] make a partition $s=t_0 < t_1 < ... < t_m = t$ .
2) Then add and subtract portfolio value at each point.
3) On each $[ t_{k - 1} , t_k ]$ to make transformations similar to BS did which lead us to BSE on each small subinterval.
4) Then take a limit which leads us to integrated over [ s , t ] BSE and
5) take derivatives with respect to upper limit of integral , ie t. T
his steps bring us formal proof of BSE. Recall that differential form of stoch calculus is informal representation of the integral form of corresponding formulas.
First difficulty is if we use portfolio value as it usually written as
$\Pi ( t ) = C ( t , S ( t )) - C_{S}^{\prime} ( t , S ( t )) S ( t )$                        (1)
we can not apply BS technics as far as t as variable and as constant at Delta are formally equal. I could use my adjustment to consider portfolio as a function of 2 variables
$\Pi ( u , t ) = C ( u , S ( u )) - C_{S}^{\prime} ( t , S ( t )) S ( u )$                    (2)
where u ≥ t . It will be ok to consider discrete time approximation and take the continuous time transition from discrete approximation. The problem comes at the last 5). When we consider the change in the value of the continuous time the change in the value formula I again arrive at formula (1) .
As it was remarked earlier starting with that formula I could not developed BSE and also could not establish time discrete approximation for which it is true, It somewhat unusual though I do not see why it is wrong.
It might be as for 2 dimensional projection we could not establish its 3 dimensional image but I am not sure.

kermittfrog
Posts: 63
Joined: September 9th, 2010, 10:25 am
Location: Frankfurt

Re: to Black Scholes pricing

I do not fully grasp your argument.

Where do you do the portfolio rebalancing step?
At each time step t, we have a value before and after rebalancing / rehedging... I miss this step in your last post, but maybe I didn't get what you are saying

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: to Black Scholes pricing

kermittfrog wrote:
I do not fully grasp your argument.

Where do you do the portfolio rebalancing step?
At each time step t, we have a value before and after rebalancing / rehedging... I miss this step in your last post, but maybe I didn't get what you are saying

In order to present BS price we choose arbitrary point t, t < T and define BS portfolio for future moments u ≥ t, ie call and stock are functions of a variable u and delta argument is a fixed parameter t. Following BS we arrive at BSE at any point of time during lifetime of the option. This present option price based on no arbitrage principle. It might be good or bad approximation for the market pricing which is basically represents price as a settlement price.
If we talk about adjustments which is represented by stochastic cash flow then we actually deal with hedging of the portfolio and cash flow is related to the hedging problem [ 0 , T ] .Of course one can ask about something like what is mean average of the call option price in term of S in perfectly hedged portfolio on [ 0 , T ]. Then we should take into account values adjustments

outrun
Posts: 3913
Joined: April 29th, 2016, 1:40 pm

Re: to Black Scholes pricing

In BS the cashflows are values against a constant rate you can hedge with a bond. Just like the stock's expected return is linked to that same rate.

edit:
money in the bank, money invested in stock, borrowed money, .. al assumed having the same rate/expected yield in B&S

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: to Black Scholes pricing

outrun wrote:
In BS the cashflows are values against a constant rate you can hedge with a bond. Just like the stock's expected return is linked to that same rate.

edit:
money in the bank, money invested in stock, borrowed money, .. al assumed having the same rate/expected yield in B&S

BS hedged portfolio values are related to risk free rate and therefore option which is priced synthetically based on BS portfolio also depends on r. The stock's expected return is linked to mu and if risk free does not appeared in stock equation to tell that stock return formally relates to risk free rate somewhat incorrect.

outrun
Posts: 3913
Joined: April 29th, 2016, 1:40 pm

Re: to Black Scholes pricing

Your mu is set to r for stock options!

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: to Black Scholes pricing

outrun wrote:
Your mu is set to r for stock options!

I interpret BS formula in other way.
1) the real stock eq ( mu, sigma ) does not depend on the fact whether options on this stock are traded or not, ie it holds its original sense itself.
2) the option underlying is a heuristic asset, which actually does not exist on the market.
3) this non existed asset makes sense only for non arbitrage pricing strategy, which might be close or might be not close to the price of the option observed on the market.
4) the BS price is a price of a strategy. It does not comprise settlement between buyer and seller.

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

Re: to Black Scholes pricing

5) In the hedged portfolio we also use delta shares of the real stocks ( mu, sigma) not ( r, sigma).
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