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### Re: Valuation of put with reflecting barrier

Posted: **November 30th, 2020, 7:14 pm**

by **Alan**

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.

### Re: Valuation of put with reflecting barrier

Posted: **December 1st, 2020, 9:39 am**

by **complyorexplain**

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.

### Re: Valuation of put with reflecting barrier

Posted: **December 1st, 2020, 3:05 pm**

by **Alan**

reread the thread

### Re: Valuation of put with reflecting barrier

Posted: **December 1st, 2020, 4:25 pm**

by **complyorexplain**

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

### Re: Valuation of put with reflecting barrier

Posted: **December 1st, 2020, 5:14 pm**

by **Alan**

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".

### Re: Valuation of put with reflecting barrier

Posted: **December 2nd, 2020, 12:16 pm**

by **complyorexplain**

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.