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?