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complyorexplain
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### Valuation of put with reflecting barrier

The paper here (“Valuation of no-negative-equity guarantees with a lower reflecting barrier”, Annals of Actuarial Science) is ostensibly about valuing no-negative-equity guarantees, but clearly applies to the case of valuing any simple European put when the underlying asset is underpinned by what the author calls a ‘reflecting barrier’.

I have reproduced the valuation model presented by the author which is essentially a bear spread (short put with long put at the barrier, plus a minor adjustment factor). This unsurprisingly gives a price of zero when the put is struck at the barrier. If the distribution of prices at expiry consists of prices that are only above the barrier, then clearly a model based on average payoff must give a result of zero.

But here’s the puzzle. If I simulate prices using the barrier model, then model the effect of being short a put at the barrier by rebalancing at the delta, i.e. continuously buying/selling the underlying to maintain a position equal to the theoretical Black 76 delta for a short put, I make a profit by being short the synthetic put. I.e. while the synthetic never pays off at maturity (since the price always expires above the barrier), I still make the time value of the put at inception.

This suggests that even if the price has an underlying barrier, we should still use the standard model (here Black 76) rather than the non-standard model presented by the author. But how can an option which has no value at expiry have a  value at inception?

Alan
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### Re: Valuation of put with reflecting barrier

Instantaneous reflection of a tradable underlying probably always leads to contradictions because of the "instantaneous arbitrage" opportunity available in continuous-time.

I see the author says:

"The instantaneous nature of the reflection means that the price does not spend any finite time at the barrier, so no arbitrage opportunities are created (we can never buy at the barrier with certainty of a price rise)."

While the "finite time" remark is correct, a continuously monitored diffusion "signals" a barrier touch. So you can buy the underlying when they are "arbitrarily close" to the barrier. You have to treat a diffusion process as allowing a purchase at any price in it's sample space, meaning the barrier price + $\epsilon$, for arbitrary $\epsilon>0$. This is probably "close enough" to an arbitrage opportunity to render standard risk-neutral arguments wrong.

There is some literature on instantaneous arbitrage". A simpler case might be: why don't we allow a stock price to reflect off of zero? The answer is probably that it leads to the same issues. To avoid arbitrage, zero has to be absorbing in the absence of a cash infusion (recapitalization).

Finally, continuous-time arbitrage opps can be subtle, their analysis very technical, and I don't pretend to be an expert on them.
Last edited by Alan on October 26th, 2020, 5:47 pm, edited 2 times in total.

complyorexplain
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### Re: Valuation of put with reflecting barrier

Here is my simulation. I don't follow the point about instantaneous reflection. I simulated prices every minute for 25 years, which is about (perhaps exactly) 9m prices.  But I rebalanced at end of day only. Very rarely does the price hit the barrier at end of day.

The simulation is of forward prices, so I rebalance a short synthetic put using the Black76 delta, struck at the barrier of 100. The synthetic always makes the time value of a short put struck at 100, priced on B76. There is also a fairly simple proof to demonstrate this.

The puzzle is why the statistical value of the put is zero (for there is no possible non-zero payoff in a world where price always >= 100, on a put struck at 100.

we can never buy at the barrier with certainty of a price rise
Sure, but we can buy or sell at any point above the barrier. As the price approaches the barrier, the delta rises so we buy. As it moves away, delta falls so we sell. Anyone looking at the price action can instantly see we make a profit at expiry.

Alan
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### Re: Valuation of put with reflecting barrier

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

complyorexplain
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### Re: Valuation of put with reflecting barrier

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

But the author says, right at the beginning "Reflection at the barrier is instantaneous, so the no-arbitrage property is preserved, and hence risk-neutral valuation of NNEG is possible." You can't both be right. (For the record, I think you are right, and obviously so).

Alan
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### Re: Valuation of put with reflecting barrier

It would be nice if there was a peer-reviewed paper in a good journal, say Mathematical Finance, titled "Arbitrage in Reflecting Diffusions". Unfortunately, the closest I could find was "Arbitrage in skew Brownian motion models", and, being behind a paywall, I haven't looked at it. You might write the author for a copy and also ask him if his work extends to proving arbitrage in models with instantaneous reflection. If you cite him that quote we both posted, it might get him interested. It seems like it should be a long-resolved topic, but maybe it's not.

complyorexplain
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### Re: Valuation of put with reflecting barrier

To be honest, I occasionally correspond with him. On this topic, he refuses to believe that the synthetic put is possible.

The equity release saga is a long one. His is the latest of of many attempts by actuaries to justify the absurd pricing models used by some insurance firms.

bearish
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### Re: Valuation of put with reflecting barrier

I used to know something about this stuff, and I’m with Alan. It’s perfectly fine for something like an interest rate or a volatility to have a reflecting barrier at zero, but not for the price of a traded asset. And once you’re a little bit pregnant, there is no going back. Well, actually there is, but that is a separate discussion.

Cuchulainn
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### Re: Valuation of put with reflecting barrier

a) Instantaneous reflection of a tradable underlying probably always leads to contradictions because of the "instantaneous arbitrage" opportunity available in continuous-time.

b) It’s perfectly fine for something like an interest rate or a volatility to have a reflecting barrier at zero, but not for the price of a traded asset.

Is there a mathematical way to explain a) and b)?
"Compatibility means deliberately repeating other people's mistakes."
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complyorexplain
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Posts: 176
Joined: November 9th, 2015, 8:59 am

### Re: Valuation of put with reflecting barrier

a) Instantaneous reflection of a tradable underlying probably always leads to contradictions because of the "instantaneous arbitrage" opportunity available in continuous-time.

b) It’s perfectly fine for something like an interest rate or a volatility to have a reflecting barrier at zero, but not for the price of a traded asset.

Is there a mathematical way to explain a) and b)?

I would have asked the same question.

bearish
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### Re: Valuation of put with reflecting barrier

One constructive approach to a proof would be to generate a sequence of binomial lattices with increasing density, constructed so as to always have a layer of nodes exactly at zero. From each such node there is only one possible path, so an adapted and finite trading strategy of always buying one stock at a price of zero and selling it after one time step will generate an “arbitrage” profit of order $N^{-1/2}$ per trade, where N is the number of time steps in the lattice. The remaining work would now be to show that this lattice model converges in a suitable norm to the continuous model (that should be standard), and that the number of these arbitrage opportunities grows at least at the rate of $N^{1/2}$. My intuition suggests that it will indeed grow at the rate of N, so that a non-trivial arbitrage opportunity is baked into the modeling setup.

complyorexplain
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### Re: Valuation of put with reflecting barrier

Barrier is at 100, not zero. Or have I misunderstood?

Alan
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### Re: Valuation of put with reflecting barrier

bearish has a good idea. Suggest you take his "zero" to mean the difference between the underlying and the barrier, and try to develop his construction. Even a sequence of Monte Carlo's on those lattices might be convincing.

bearish
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### Re: Valuation of put with reflecting barrier

I hadn’t really looked at the details of the original post and was kind of addressing Alan’s simplified question of why we don’t have stock price processes with a reflecting barrier at zero. But zero is not all that special in this case.

Cuchulainn
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### Re: Valuation of put with reflecting barrier

One constructive approach to a proof would be to generate a sequence of binomial lattices with increasing density, constructed so as to always have a layer of nodes exactly at zero. From each such node there is only one possible path, so an adapted and finite trading strategy of always buying one stock at a price of zero and selling it after one time step will generate an “arbitrage” profit of order $N^{-1/2}$ per trade, where N is the number of time steps in the lattice. The remaining work would now be to show that this lattice model converges in a suitable norm to the continuous model (that should be standard), and that the number of these arbitrage opportunities grows at least at the rate of $N^{1/2}$. My intuition suggests that it will indeed grow at the rate of N, so that a non-trivial arbitrage opportunity is baked into the modeling setup.
This answer seems to be diverging (no pun intended) from the original assertions. Some quick retorts.

1. What are sequence of binomial lattices,  suitable norm to the continuous model (that should be standard),??
2. I can't visualise what's going on
3. I can't see the maths/numerics. BTW what does the continuous model look like? SDE, PDE, analytic, whatever?

inter os atque offam
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl