GBM is arb-free -- end of story.

?

What do you mean, Alan? That doesn't even seem to be the end of a thought: GBM isn't even a pricing model; it's an*aspect* of a lot of pricing models.

What do you mean, Alan? That doesn't even seem to be the end of a thought: GBM isn't even a pricing model; it's an

Long version. Generally, one defines (i) a market environment; then (ii) a set of admissible trading strategies; then (iii) an arbitrage portfolio. The simplest (i) is a market with a GBM stock price S(t) plus a zero-interest-rate bond price B(t)=1. Then, (ii) are arbitrary portfolio processes: V(t) = n(t)S(t) + m(t)B(t), as long as unlimited borrowing is excluded (BSM option model is a special case, by dynamic replication). Finally, (iii) is standard and already discussed in this thread. Result: GBM is arb-free.

Please, can we call it “a replicating portfolio”? As I was typing that, the word tallywhacker came to mind, for reasons that should be obvious.

Sorry, I interrupted myself. By the time our continental colleagues got to NFLVR, I lost a bit of enthusiasm, but I like this little note by Freddy and Wally: https://www.ams.org/notices/200405/what-is.pdf

Aside from the fact that they manage to repeatedly say Cox-Ingersoll-Ross when they meant Cox-Ross-Rubinstein.

Aside from the fact that they manage to repeatedly say Cox-Ingersoll-Ross when they meant Cox-Ross-Rubinstein.

So can a GBM stock price with expected return 10000000% annually and variance of return 1% annually coexist with your zero-interest bond? To my thinking, it wouldn't: demand would drive the price of the stock up -- and the rate of return down -- until ... something.Long version. Generally, one defines (i) a market environment; then (ii) a set of admissible trading strategies; then (iii) an arbitrage portfolio. The simplest (i) is a market with a GBM stock price S(t) plus a zero-interest-rate bond price B(t)=1. Then, (ii) are arbitrary portfolio processes: V(t) = n(t)S(t) + m(t)B(t), as long as unlimited borrowing is excluded (BSM option model is a special case, by dynamic replication). Finally, (iii) is standard and already discussed in this thread. Result: GBM is arb-free.

If you disagree, then why wouldn't investors be eager to exchange the bond for more of the stock?

And if you agree, what are the parameters that would have to be met for the two financial assets to coexist? Or is that something that falls into the "here there be dragons" category?

So you want to postulate a model with totally unreasonable parameters, making no sense whatsoever, and then use the existence of said model to argue that the modeling framework is wrong? Isn’t that a bit like saying that a car, driving at a speed of 99% of the speed of light, is very implausible, and thus the theory of relativity must be wrong? Now, 101% would be bad. That would cause arbitrage opportunities all over the place. One may consult Marc Reinganum’s 1986 JPM paper on the impossibility of time travel, which puts forth a no-arbitrage argument to disprove the possibility of time travel, now and in the future (to the extent that sentence makes sense in a world where time travel is at least considered - see related work on grammar by Dr Dan Streetmentioner). Alas, he assumed that interest rates would remain positive…

Is this your way of saying that you agree that the stock and bond as I described them cannot coexist, just as I suggested -- ?

Why does it seem like you want to make a pissing contest out of it?

Why does it seem like you want to make a pissing contest out of it?

I’m sorry. I don’t. I’m very mild mannered. At the same time, I do think the standard paradigm is a pretty reasonable starting point, into which one can try to insert various kinds of real world complications. Like transaction costs and other problems that prevent even simple claims from being perfectly replicable. It’s like efficient markets. They clearly aren’t, but it’s a decent null hypothesis.

No worries.

I note that in my comment that you responded to, I was responding to Alan's comment, "GBM is arb-free." I presented a GBM stock price (didn't I?) that I asserted did not work in his model, something with which apparently you agree. Whether Alan agrees, I don't know.

Alan pretty clearly was relying on a technical definition of arbitrage: a zero investment portfolio with no possibility of loss and a non-zero possibility of gain.

And that, at least to my thinking, is not what makes my GBM stock with "totally unreasonable parameters" incompatible with his zero interest bond: there is, however minuscule, a possibility that the stock will underperform the bond.

But there's something going on, I guess, that both you and I can recognize that my stock doesn't belong in the same universe as Alan's bond. "Arbitrage" is probably not the best term for it, but apparently there is something going on.

At the same time, I think all of us agree that there are*some* GBM stocks that possibly do fit in the same universe as Alan's bond.

So, you and I (I think) concur that there are some GBM stocks that reasonably coexist with Alan's bond, and others that do not.

How do we tell the difference? We might just use Potter Stewart's explanation of obscenity -- "I know it when I see it" -- but that's not very useful.

Could we come up with a model that would make a clear distinction?*A* model, I'm sure is possible; *the* model ... maybe a little more work.

But does it make any sense whatsoever to even consider such a thing?

I note that in my comment that you responded to, I was responding to Alan's comment, "GBM is arb-free." I presented a GBM stock price (didn't I?) that I asserted did not work in his model, something with which apparently you agree. Whether Alan agrees, I don't know.

Alan pretty clearly was relying on a technical definition of arbitrage: a zero investment portfolio with no possibility of loss and a non-zero possibility of gain.

And that, at least to my thinking, is not what makes my GBM stock with "totally unreasonable parameters" incompatible with his zero interest bond: there is, however minuscule, a possibility that the stock will underperform the bond.

But there's something going on, I guess, that both you and I can recognize that my stock doesn't belong in the same universe as Alan's bond. "Arbitrage" is probably not the best term for it, but apparently there is something going on.

At the same time, I think all of us agree that there are

So, you and I (I think) concur that there are some GBM stocks that reasonably coexist with Alan's bond, and others that do not.

How do we tell the difference? We might just use Potter Stewart's explanation of obscenity -- "I know it when I see it" -- but that's not very useful.

Could we come up with a model that would make a clear distinction?

But does it make any sense whatsoever to even consider such a thing?

So can a GBM stock price with expected return 10000000% annually and variance of return 1% annually coexist with your zero-interest bond? To my thinking, it wouldn't: demand would drive the price of the stock up -- and the rate of return down -- until ... something.Long version. Generally, one defines (i) a market environment; then (ii) a set of admissible trading strategies; then (iii) an arbitrage portfolio. The simplest (i) is a market with a GBM stock price S(t) plus a zero-interest-rate bond price B(t)=1. Then, (ii) are arbitrary portfolio processes: V(t) = n(t)S(t) + m(t)B(t), as long as unlimited borrowing is excluded (BSM option model is a special case, by dynamic replication). Finally, (iii) is standard and already discussed in this thread. Result: GBM is arb-free.

If you disagree, then why wouldn't investors be eager to exchange the bond for more of the stock?

And if you agree, what are the parameters that would have to be met for the two financial assets to coexist? Or is that something that falls into the "here there be dragons" category?

No, it's a perfectly fine question and has been answered since Daniel Bernoulli by "risk aversion". Risk averse investors require the expected return on the stock to be higher than the riskless rate to want to buy any stock.

How much higher? Empirically (and discounting some nonsense), the coefficient of relative risk aversion of the typical investor seems to lie in the range 2 to 4. It's this fact that prevents your particular example from living in the real world. What determines "reasonable" GBM (or other market model) coefficients is thus a separate and complementary inquiry. The theory all fits together quite nicely. I think you are tilting at windmills.

Probably a good assessment.I think you are tilting at windmills.

Still, there are three obvious zones for the GBM stock (as noted previously, I follow the more distinguished and honorable convention of basing things off of the drift μ of the mean of the logarithm of the stock price as opposed to the expected rate of return on the stock α. α = μ + ½σ², for those following at home. And r = 0 in this example.): μ ≤ -½σ² ; -½σ² < μ < 0; and 0 ≤ μ. (Restoring r makes these μ ≤ r - ½σ² ; r - ½σ² < μ < r; and r ≤ μ.)

The first range is probably easy to reject: the risky asset has a lower expected rate of return than the riskless asset, AND over time Pr{S

The second range is something of a sweet spot, at least to my thinking: the risky asset has a higher expected rate of return than the riskless asset, but over time Pr{S

The third tests one's belief in risk aversion at least a little: the risky asset has a higher expected rate of return than the riskless asset, AND over time Pr{S

With just fundamental assets, cheating into the third range a bit isn't too troubling: after a hundred years, there's very little possibility the stock will have underperformed the bond? And -- ?

But with derivative assets -- as someone noted earlier -- you can have payout probabilities approaching zero without significant time delay.

And in some respects, that's not too troubling: a derivative that pays out only in a six standard deviation event costs twice times its expected value? That's not going to break the bank; it might not even beat the bid-ask spread.

But extreme values are more a proof of concept matter: if you don't like the probability of loss or the expected loss of one derivative, no worries -- there's a derivative available where BOTH are lower!

For what WOULD affect prices, imagine the investor willing to sell something with a 48.65% chance of loss and only a 2.78% price over expected value ... which is what the house gets on a single zero roulette wheel bet on red or black. And then they need to net out the croupier's and the cocktail waitress' wages, the free drinks, the rent, etc.

Well, that's enough tilting at windmills for me for awhile.