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Alan
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Re: Valuation of put with reflecting barrier

November 30th, 2020, 7:14 pm



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

The issue is always whether or not one can construct a trading strategy with a riskless profit. But, as the article shows, it becomes quite technical in continuous time, continuous state spaces. Personally, I just try to keep in mind "processes with known arbitrage opps". I am usually willing to skip the proofs. 
 
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complyorexplain
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Re: Valuation of put with reflecting barrier

December 1st, 2020, 9:39 am

The issue is always whether or not one can construct a trading strategy with a riskless profit.  

Which one can in this case. As I said above, go short a synthetic put whose delta is given by the standard BS delta, go long the same put, but with the delta given by the Thomas method. Then you have a guaranteed profit at expiry. The question is why, given that the barrier process apparently satisfies all the conditions for an arbitrage free process.
 
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Alan
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Re: Valuation of put with reflecting barrier

December 1st, 2020, 3:05 pm

reread the thread
 
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complyorexplain
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Re: Valuation of put with reflecting barrier

December 1st, 2020, 4:25 pm

reread the thread
I have just done so, but the only post of relevance I can find is #16 by you Alan. 

See also your number 4.
It sounds to me like you've proved my point. That is, you've constructed a trading strategy which is an arbitrage opportunity. That was my point: a reflecting diffusion process offers arbitrage opps. That makes them bad candidates for financial models. 

When there are arbitrages, there are securities with positive prices (your synthetic put) which only pay off  in "impossible states of nature". Shorting them provides the arbitrage. I think that's the resolution of your puzzle
 
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Alan
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Re: Valuation of put with reflecting barrier

December 1st, 2020, 5:14 pm

Plus there were credible references proving the existence of arbitrage opps with reflecting processes, and suggestions about how to construct a proof as a limit of discrete-time processes. 

The bottom line is that a reflecting process is not arbitrage-free and, at least in discrete-time (lattice) approximation, I think it's obvious why that is.  If you don't see it, draw some (approximating) lattices for GBM with reflection at some nodes and construct a profitable risk-free trading strategy when those nodes are hit. So, that answers "why".
 
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complyorexplain
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Re: Valuation of put with reflecting barrier

December 2nd, 2020, 12:16 pm

Thanks. 

I was spooked by your "Personally, I just try to keep in mind "processes with known arbitrage opps". I am usually willing to skip the proofs." comment above, which seemed ironic in tone. 
 
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complyorexplain
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Re: Valuation of put with reflecting barrier

December 8th, 2021, 1:10 pm

Hopefully we get a paper into print on this, after some of to and fro.  The model clearly violates a number of arbitrage constraints. Yes, I can see that others have pointed this out above, but there is nothing pointing it out in the literature that we can see. 
 
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rgthomas
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Re: Valuation of put with reflecting barrier

January 7th, 2023, 1:25 pm

Original author here. Thanks for all the critiques. I have a new paper on this: https://doi.org/10.1017/S1748499522000227
 
TLDR: I now think the previous paper was RAWR (right answer, wrong reason).
 
Or at least, an OK answer for a put, given practical limits to arbitrage. Things are more complicated, in interesting ways, for a call.
 
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Alan
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Re: Valuation of put with reflecting barrier

January 8th, 2023, 5:03 pm

I looked briefly and also tracked down one of your references: Liu and Longstaff (2004). 

Similarly to your paper, Liu and Longstaff introduce an asset process that admits an arbitrage under certain perfect market conditions. Then, they were able to show that with a fairly minimal addition of realism (margin collateral), the putative asset process could 'persist' in the market, at least for log-utility investors, as there would not be an infinite demand for the arb strategy.

Anyway, my comment is that it might be interesting to raise a similar question with your reflecting process. That is, is there a fairly minimal addition of realism that would allow the process to persist without infinite demand for the stock as the barrier was about to be hit? 

My thought is that perhaps this would require the interest rate to become stochastic and go to +infinity (briefly) on a barrier hit. The rationale would be that the interest rate would be paid by the govt intervener to keep the asset price above the barrier. If so, (i) would this translate to a finite or infinite dollar cost to the govt over a horizon T? and (ii) would it then also prevent an infinite demand for the stock -- as a riskless bond would become locally more attractive?  

If the barrier were an upper reflecting barrier, perhaps the borrow cost of the stock would (briefly) go to +infinity. Something like that seems to be going on with the AMC/APE (pseudo-) arbitrage, which is of current interest. This is q (rather than r) -> +infinity, as the (institutional) holder of the stock will reap the borrow cost.
 
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rgthomas
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Re: Valuation of put with reflecting barrier

January 8th, 2023, 11:52 pm

This paper suggests the condition that r – q goes to infinity at the barrier

 
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=974939
 
And this one by Peter Carr (RIP) suggests that r – q just needs to be stochastic
 
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2699250


 
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Alan
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Re: Valuation of put with reflecting barrier

January 9th, 2023, 12:27 am

Thanks -- Good finds!

BTW, there is another process that might be a simpler way to have reflection of a traded asset price: so-called "Slow reflection" (aka "Sticky reflection"). I discuss this for interest rates in my "Option Valuation under Stochastic Volatility II". It would be interesting to generalize it to a traded asset price.