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Cuchulainn
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Re: Valuation of put with reflecting barrier

October 30th, 2020, 7:07 pm

"What is BS formula with a 'truncated distribution'? "

My actual words were "derive the BS formula on the assumption of such a truncated distribution", which I followed up, to clarify, 

To be specific, it is trivial to derive the BS formula consistent with the assumption of such a truncated distribution, or indeed any other distribution. The distribution at expiry is neither a necessary nor sufficient condition of the derivation. 

I.e. we can derive both the PDE and the formula without making any assumptions about the terminal distribution.
What's a 'truncated distribution'? Maybe I missed that.
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Re: Valuation of put with reflecting barrier

October 30th, 2020, 7:19 pm

"What is BS formula with a 'truncated distribution'? "

My actual words were "derive the BS formula on the assumption of such a truncated distribution", which I followed up, to clarify, 

To be specific, it is trivial to derive the BS formula consistent with the assumption of such a truncated distribution, or indeed any other distribution. The distribution at expiry is neither a necessary nor sufficient condition of the derivation. 

I.e. we can derive both the PDE and the formula without making any assumptions about the terminal distribution.
What's a 'truncated distribution'? Maybe I missed that.

Refer to the paper linked in the OP, specifically equation (A.4) and passim.
 
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Cuchulainn
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Re: Valuation of put with reflecting barrier

October 30th, 2020, 9:49 pm

"What is BS formula with a 'truncated distribution'? "

My actual words were "derive the BS formula on the assumption of such a truncated distribution", which I followed up, to clarify, 





I.e. we can derive both the PDE and the formula without making any assumptions about the terminal distribution.
What's a 'truncated distribution'? Maybe I missed that.

Refer to the paper linked in the OP, specifically equation (A.4) and passim.
When I click 'pdf' to get the article I get Error 404 .. resource not found.
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Re: Valuation of put with reflecting barrier

October 30th, 2020, 10:17 pm

But you can see the html version, right?
 
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Re: Valuation of put with reflecting barrier

October 31st, 2020, 9:52 am

But you can see the html version, right?
I can see it, yes. But it's not user-friendly etc. Do you have it in pdf? Then it's possible to use "Attachments" option with posting.
Plan B: I can register with that site ... annoying.
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Re: Valuation of put with reflecting barrier

October 31st, 2020, 11:26 am

I think there may be copyright reasons why not in pdf. I could write to Guy and send you a copy, but would need your contact details.
 
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Re: Valuation of put with reflecting barrier

October 31st, 2020, 12:02 pm

I think there may be copyright reasons why not in pdf. I could write to Guy and send you a copy, but would need your contact details.
I registered on the Cambridge site and all is fine now. Thanks.
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Re: Valuation of put with reflecting barrier

November 29th, 2020, 2:37 pm

I think the problem with the reflecting barrier is located at equation (A.4). Link to paper.
 
(A.4) dZt  =  (r-q)Yt dt + v Yt dWt + Yt dLt
 
where Yt is the price of the asset without the barrier, (r-q) are carry terms (q is dividend, r is interest rate), v is vol, dWt is diffusion term, and the process Lt is a ‘reflection function’. dZ is then the change in the asset price with barrier.
 
Thomas writes “The RGBM process as specified in equation (A.4) is a semimartingale, because it can be decomposed into a local martingale (the Wiener term) and two finite variation processes, the drift and the reflection.”
 
But simulation shows that the process is negatively autocorrelated. Can a semimartingale be autocorrelated? Surely not.
 
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Re: Valuation of put with reflecting barrier

November 29th, 2020, 8:51 pm

"Most" processes you will encounter will be Semimartingales, because this is the largest class of processes with stochastic calculus, Ito's lemma, etc, etc. For example, the OU process qualifies and is a classic example of an autocorrelated process. 
 
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Re: Valuation of put with reflecting barrier

November 29th, 2020, 10:49 pm

Hmm. But the same WP article says "Fractional Brownian motion with Hurst parameter H ≠ 1/2 is not a semimartingale."
 
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Re: Valuation of put with reflecting barrier

November 29th, 2020, 11:19 pm

It feels like there should be a theorem saying that a semimartingale will have a zero co-quadratic variation with a lagged version of itself, for all non-trivial lags. Which would definitely hold for the OU example, since the autocorrelation enters via the drift term.
 
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Re: Valuation of put with reflecting barrier

November 29th, 2020, 11:46 pm

@comply,
FBM is an exception that proves the rule; that's why I said "most". 

A practical theorem might be: If the SDE of the process is built from one or more of {dt, dW, dL, dJ}, 
where L is a local time process at a boundary, and J is a jump process, it's a semimartingale.

Note that a traded price process being a semimartingale does not mean the process is arbitrage-free, with either a stock price reflecting from zero (already discussed) -- or jumping from zero --  being simple examples. 
 
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Re: Valuation of put with reflecting barrier

November 30th, 2020, 9:20 am


Note that a traded price process being a semimartingale does not mean the process is arbitrage-free, with either a stock price reflecting from zero (already discussed) -- or jumping from zero --  being simple examples. 

Thanks. So what is the condition for a process being arb-free? I.e. so that there is at most one price for our option, rather than (at least) two in the barrier case?

[EDIT] I am guessing that the full answer lies in this literature review by Schachermayer https://www.mat.univie.ac.at/~schacherm ... r0141a.pdf
 
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Re: Valuation of put with reflecting barrier

November 30th, 2020, 10:50 am

Hmm. But the same WP article says "Fractional Brownian motion with Hurst parameter H ≠ 1/2 is not a semimartingale."
Don't know if it's relevant here, but speaking of Hurst, see recent thesis on rough vol model


https://www.datasim.nl/blogs/29/msc-the ... nance-2020

//
thinking out loud..
BTW I read that Cambridge article once a few weeks ago. It seems A.4 is called the "reflected sde". They mention barrier but is it in the traditional knock {in,out} sense? Since we are talking about house prices, is the barrier not really a constraint?  
Is there a corresponding (stochastic) PDE for this? I suspect a Neumann boundary condition?
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Re: Valuation of put with reflecting barrier

November 30th, 2020, 12:24 pm

"They mention barrier but is it in the traditional knock {in,out} sense? "

No, at least not if you mean a type of barrier option (a contingent claim). It is the SDE of a barrier *process*, i.e. one which cannot possibly cross the barrier.

There are other questions about the assumption of such a process. My experience of trading FX options in the 1990s was that the central bank could try to impose such a barrier by intervening, but could fail if the market takes the bank on. E.g. September (?) 1992 when sterling collapsed out of the ERM. The whole market piled on (including me with a small put option) and the rest is history. 

The Russian Rouble was subject to a similar regime in 1998, when it collapsed after a big battle with the market. Then there was HK dollar which gave way slightly some time in the 2000s, can't remember exactly.

The option pricing tends to reflect not the barrier, but rather the probability of market forces breaching the barrier. Usually such barriers are imposed when the underlying economics is weak, hence the implied probability can be quite high.

But that's a separate issue from what we are talking about here.