Using Astra (checked by fable), I think I have proven and formally verified that in the Black-Schole
原帖发布于
Using Astra (checked by fable), I think I have proven and formally verified that in the Black-Scholes model, the exercise boundary of an American put option is log-convex for nonzero dividends.
It’s not really earth-shattering, but it’s a legitimate open problem in mathematical finance: it was shown in 2008 that the exercise boundary was convex for the zero-dividend (q=0) case, and in 2013 that it could be non-convex for dividends exceeding the risk-free rate r, but the very practical worlds where 0 < q <= r were unknown (barring a claimed tightening of the bound in 2017). Astra found a relatively intuitive proof that the boundary is log-convex (stronger than convexity) for the full 0 < q <= r domain. — essentially sweeping sets of candidate linear boundaries, to show that the exercise boundary would never intersect that line twice.
This result is probably not useful standalone when we already have numerical solutions for pricing American options and have empirically observed for many years that the boundary was convex (not to mention this is in pure Black-Scholes world)… but it’s a doorway to provably reliable pricing approximations and [redacted].
It’s not really earth-shattering, but it’s a legitimate open problem in mathematical finance: it was shown in 2008 that the exercise boundary was convex for the zero-dividend (q=0) case, and in 2013 that it could be non-convex for dividends exceeding the risk-free rate r, but the very practical worlds where 0 < q <= r were unknown (barring a claimed tightening of the bound in 2017). Astra found a relatively intuitive proof that the boundary is log-convex (stronger than convexity) for the full 0 < q <= r domain. — essentially sweeping sets of candidate linear boundaries, to show that the exercise boundary would never intersect that line twice.
This result is probably not useful standalone when we already have numerical solutions for pricing American options and have empirically observed for many years that the boundary was convex (not to mention this is in pure Black-Scholes world)… but it’s a doorway to provably reliable pricing approximations and [redacted].