Marsden, Joseph D. 2022. Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices. Wilmott, Volume 2022, Issue 120 July.
So ... did anyone read it? Understand it well enough to declare that it's crap, or (worse) not interesting?
The main points, for those whose reading habits directed them elsewhere, are:
1. Black-Scholes prices are not arbitrage free when there is any significant risk premium on equity prices.
I've thought for the last 25 years that this was pretty obvious, but I've never been able to find any published mention of it. If you were aware of it but kept it quiet to profit off of miss-pricing ... sorry, not sorry: you shouldn't take advantage of other people's ignorance like that.
And it may be that it was only obvious to me because I come from the liability side (I'm an actuary), where we use facts to determine prices; on the asset side I guess the tendency is to regard prices as providing a complete set of all relevant facts.
In any event, making the case that Black-Scholes prices allow arbitrage has been the main thing keeping me from writing this paper for 25 years. I hope I've finally done a reasonable job of it; tossing out some fancy insurance terms to fix the reader's attention for a moment was my main tool in this latest effort.
Let me know how it worked.
2. Itô's Lemma doesn't apply to financial markets because trading is not continuous.
(Note that I've changed my mind about which accent to use in "Itô" since writing the paper; I think I had seen long ago that his daughter uses "Itõ," and went by that, but apparently the professor himself used "Itô." I suspect few people would recognize "伊藤.")
This is completely obvious, but if you've been thinking that "dZ²=dt" is Itô's Lemma, then I guess it's news to you. Maybe bad news.
It's disappointing that Itô's Lemma doesn't apply, because it's very elegant. But it just doesn't.
3. Absent the constraint of Itô's Lemma, Black-Scholes prices can be modified fairly simply to eliminate the most direct target for arbitrage.
(Modified how? Read the damn paper!)
I note that I am not an academic, and I'm not even really a practicioner; I'm basically a hobbiest.
So I can't walk over to the next office and say, "Hey Steve! Who is doing the most interesting work on such-and-such these days?" (Or, more importantly, "Steve, can you tell me if this is bullshit?")
I'm generally limited to The Google and occasional trips to academic libraries for literature search, and I apologize to anyone whose work I've stepped on; it hasn't been intentional.
I haven't been secretive about any of this; see viewtopic.php?f=3&t=61129 and viewtopic.php?f=3&t=84755. But I didn't see that anyone found it interesting.
Anyway, the modification is really the only thing from the paper that I think required very much cleverness on my part; everything else was fairly obvious. After the fact, though, the modification seems obvious, too.
I have wondered if anyone else came up with the same modification, but I have been unable to find it anywhere; it does run against some fairly well set assumptions about market prices. In some respects, it comes down to choosing which to believe when logic and established practice disagree with each other.
I grew up among mathematicians, so probably I never really had a choice.
Let me know your thoughts; as I say, there is no "Steve" in the next office to set me straight, so I'm counting on all of you.