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Marsden
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### "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

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.

Collector
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

congratulations on publication!

just glimpsed shortly at it so far, is it mostly based on technë or ëpistemë ?

Alan
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

@Marsden. I've read chunks of the paper -- thanks for the copy!

Just a couple comments about your contention above that "Black-Scholes prices are not arbitrage-free when there is any significant risk premium on equity prices.".

Technically, an arbitrage opportunity in finance requires some care to demonstrate. You have to pick a finite time horizon T and propose a trading strategy that offers the possibility of a profit with zero probability of a loss. I didn't see such a demonstration in your article and would be surprised if such a demonstration, under some plausible market assumptions, exists.

Your intuition about such an arbitrage seems to be the unease that, under the Black-Scholes model, the risk-neutral (compound) growth rate for the underlying security can be negative while, simultaneously, the real-world growth rate is positive. I have puzzled over this too, sometimes, as likely many have. This possibility leads to an "apparent" paradox that, ultimately, as T grows to infinity,  the risk-neutral stock price ends up at S=0 with probability 1, while the real-world price ends up at $S > B$ (B any positive bound) with probability 1. But, the restriction of arbitrage demonstrations to finite T prevents this from truly offering an arbitrage opportunity. Indeed, this forum has had many discussions of such $T = \infty$ "pseudo-paradoxes".

That issue aside, both Black-Scholes and your proposed modification, if I read it correctly, lead to flat smiles. Either way, offering options with a flat smile would offer some "good deal" pricing for counter-parties to take advantage of for most real-world underlying securities. So, likely both the putative Black-Scholes market-maker and you would likely go out of business at some point. But, showing some strategy is a good deal is not an arbitrage demo.

My two cents.

Marsden
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Thank you for responses.

Espen, I think mostly ëpistemë.

Alan, as noted, arguing the ability to arbitrage Black-Scholes prices is the stickiest thing about the entire enterprise. The argument made in the paper is NOT what made it "obvious" (in quotes until we agree that it's actually true ...) that Black-Scholes prices permitted arbitrage, so maybe just as an intellectual exercise I'll lay out my original reasoning here, and then I'll try to address the demonstration you find lacking.

So, a seductive quality of financial prices, and one that manifests itself strongly in risk-neutral pricing, is that any asset will have an expected rate of return, and the asset's future return distribution will have an expected discount rate ... and the two rates have to exactly cancel each other out.

$1 worth of stock has a value equal to that of$1 worth of bonds, regardless that they have completely different expected rates of return and completely different expected discount rates that apply to those returns: the round trip has to take you to \$1 in both cases.

Similarly, I and the world champion high jumper (Mutaz Essa Barshim; just looked it up. 2.37 m) share a constant when we attempt to jump over anything: we start at ground level, and we end at ground level.

So in this respect, if we want to know what would happen if Mutaz Essa Barshim tried to jump over something, I could just try to jump over it, and because we both start and end our jumps at the same level, no difference, right?

But of course that's nonsense: I might be able to jump over a stacked line of match boxes on a good day, and Barshim could probably clear 2 m just warming up.

The equity risk premium is completely invisible in Black-Scholes prices: could be 5% annually; could be 500%; it just doesn't matter!

But realistically, if we're pricing a call option with a strike price representing a 10% return over riskless, then if the risk premium is 5% ... yeah, it could make it to the strike price.

But if the risk premium is 500%, it will almost definitely make it. So sort of like if Barshim and I both tried to jump over a 1 m fence (me after a few weeks of training, of course).

Now, you use a strict definition of arbitrage: "possibility of a profit with zero probability of a loss," and over a finite time horizon T. No, I can't demonstrate that.

But I would submit that most of the world operates without that. Black-Scholes prices DO allow an expectation of profit with a profound ability to limit the possibility of loss.

While the history of investing according to models promising "losses only once in a billion years" is, as far as I know, consistent in always, always failing, it's almost always because the model isn't right.

If we confine ourselves to the fishbowl of Black-Scholes assumptions, however, then we are assured that the model IS right.

And I submit that we can create strategies using Black-Scholes prices that really do have losses only once in a billion years, at least within the fishbowl.

Do you accept that, and do you accept that, that being the case, the model is not really producing prices that we should expect to be in equilibrium?

Alan
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Sure, I'll accept that you can cook up parameters and strategies (likely, as you said, with extremely large real-world equity growth rates) where there are "losses only once in a billion years". However, this doesn't bother me like it bothers you. I would attribute the "problem" in that case, not to the Black-Scholes model, but to the unrealistic/unphysical growth rate assumption. What is a plausible growth rate becomes the issue. BTW, I have a book coming out on related questions (about the equity risk premium)

Alan
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

p.s. -- since you mentioned a 500% ERP (equity risk premium), see Fig. 10 (Mar 12, 2020!) here

Collector
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

there are risk-free arbitrage, in academia when someone talk about arbitrage they normally (my experience at least, based on limited sample points) talk about risk-free arbitrage. In some firms I work in when they talked about arbitrage they talked about very close to risk free, same as in academia, but in real life never 100% risk free, but strategies very close to.

Then I worked in other firms that when they talk about arbitrage it was just some strategy to get expected slightly better risk over returns than most standard comparable strategies. And to call it arbitrage also seemed some firm used in marketing to investors. But even if just good risk reward you mean by arbitrage, you could try to demonstrate with "practical" example.

What some firms call arbitrage strategies are unfortunately some carry strategies with  embedded short deep otm options, or short gamma of some sort. They get great Sharpe ratios for some time and therefore marketing it as arbitrage, gives impression they are smarter than the market (often think so themselves also after a few good years), until they blow up, it had nothing to do with arbitrage. But yes close to risk free arbitrages happens, some times sizeable.

Marsden
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Alan, the "once in a billion years" loss frequency doesn't require an outrageous growth assumption; 5% should work just fine.

A lower equity risk premium just requires you to go farther into the tails to execute the strategy.

I'll look at your paper once I get to a bigger screen.

Marsden
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Espen, to my knowledge there is no broadly accepted definition of "statistical arbitrage," although the term is often used. I offer a definition in an end note of my paper, but it's not terribly solid.

And maybe that ultimately means the term should just be retired.

Michael Dothan used the term "equilibrium pricing measure" (EPM, although that acronym seems familiar for reasons I can't put my finger on) in his book -- and I think got lambasted for giving a new name to risk-neutral pricing (RNP) -- but I find EPM a more elegant description: this is a pricing measure -- essentially the modified probability distribution of RNP -- where there is generally no advantage to be gained by selling outcome X and buying outcome Y.

My father once recounted that he and his classmates liked to use the justification "ADFCST" -- "any damn fool can see that" -- in math proofs, and it may be that in pricing models at some point that's as good as you can do: there is no accounting for taste ... or for risk preference.

Ultimately, using a Wiener Process as the underlying model, one way or another, for price movements, I think financial markets are all about the exchange (and relative prices) of theta and sigma squared (absent currency effects, delta should have a fixed price of zero, as ADFCS).

If Ito's Lemma applies, then the price of theta must be the same as the price of sigma squared, but Ito's Lemma does not apply to financial markets.

Absent Ito's Lemma, theta -- having riskless return -- should be preferred to risky sigma squared.

Marsden
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Marsden
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Alan, after looking at your Fig. 10 and a little bit at your ERP model description, I think an issue that can turn out 500% ERP estimates is that there essentially of necessity are more moving pieces to things that what we acknowledge.

In particular, it's almost always assumed (including in my paper) that interest rate yield curves and equity risk premiums are fixed and unchanging ... about which the yield curve demonstrably is not and of the ERP there is no real reason to believe that it is. For purposes of the prices of fundamental assets at time zero and with an infinite time horizon, this is probably fine; for purposes of the prices of derivatives that pay out at a set point in time in the future, probably not.

So (a) it's a bad assumption that stock prices follow Brownian motion because they are subject to changing appetites in the market in addition to changing information; and (b) derivative prices can probably be expected to reflect some notion about the nature of the market's change in appetite in addition to everything else going on.

As a very simple example, if we have an inverted yield curve, we pretty much don't expect that that will be sustained. The price of a bond at time zero should reflect the inverted yield curve, but a derivative based on the price of the bond at a future time should probably reflect that the yield curve is likely to revert to something more normal.

And so if we measure implied volatility of equity index options, they might include a notion that the ERP could change by an awful lot.

I wrote a (short! no equations!) follow up to the paper of this thread that lightly considers changing market appetites. I can send it to you, but you have to ask: I view sharing any formal-ish writing I do with anyone as being like a vampire getting into your house: the vampire cannot enter -- and thus he poses no danger to you -- unless he is invited in.
Last edited by Marsden on June 29th, 2022, 3:06 pm, edited 1 time in total.

Marsden
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

I think EPM for Equilibrium Pricing Measure reminded me of EMH for Efficient Market Hypothesis.

DavidJN
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

Similar issues expressed here were identified examined in appreciable detail decades ago. Get thee a copy of the 1979 JET article "Martingales and arbitrage in multiperiod securities markets" by Harrison and Kreps Fun reading. Pliska wrote pretty dense stuff about this as well. A useful recent summary is https://www.fields.utoronto.ca/programs ... liska2.pdf

Marsden
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

I'd tracked down a copy of that article some time ago, David ... which doesn't mean that I read it closely. And looking just now, my God it's loaded with terminology and symbols, so I probably wouldn't have done more than glance at it.

Also, I have a reflexive disdain for financial mathematics prior to 1987, figuring that if there wasn't a chorus of people telling Leland-O'Brien-Rubenstein that their option replication strategy wasn't theoretically sound, the state of the science at the time wasn't very good.

But I'll take a look now.

bearish
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### Re: "Option Pricing, Risk Premium, and Arbitrage: An Argument for Volatility-Modified Risk-Neutral Prices."

I’m a little late joining this party, but one perspective that I find to be kind of useful is the notion of the natural numeraire, aka the stochastic discount factor that makes discounted prices martingales under the P measure. No need for anything or anybody to be risk neutral, and the pricing results are the same. Mysteriously (at least to me), it is the cumulative return on the growth optimal portfolio, i.e. the optimal dynamic portfolio for an investor with log preferences (in a very simplified world).