Everything I write is for real.
I’m not an options person, so I’m trying to dig at an intuition and see how it plays out. But here are a few things that seem relevant.
The idea that a game of 1 round is equivalent to a game of 1000 rounds is predicated on the idea that the expected value of both games is 0, and therefore someone who doesn’t care about risk (i.e. a risk neutral person) wouldn’t care whether they play 1 game, 1000 games or 0 games. To such a person, there would be zero value in an option to stop, because every game has 0 value and is neither desirable nor undesirable to engage.
But a risk-averse person would note that (assuming each round cost the same price and had the same payoff probabilities), a 1000 round game could easily lose more money than the 1 round game without any increase in expected return. They would prefer not to play at all, but if forced to choose between a 1 round and a 1000 round game, they would definitely prefer the 1 round game and could almost certainly be convinced to shell out some money for that, hence the option would be valuable to them.
A lot of options pricing says that you don’t need to think about risk aversion, but that applies primarily when you have a replicating portfolio, so that you can do a long-short arbitrage between the option and the replicating portfolio that eliminates any risk aside from transaction costs. For this reason, one’s risk aversion doesn’t matter: because the arbitrage effectively eliminates risk anyway.
But I don’t see a replicating portfolio in the coin flipping case - you can’t take the heads and tails side simultaneously and mix with cash (which doesn’t pay anything here anyway), so risk aversion matters. I may be wrong about that: maybe I just can’t see the replicating portfolio and how you would construct it when you’re not allowed to be the house, only the bettor.
As for the value of the American option being meaningless for a non-dividend paying stock, the usual explanation for that is that if you want to exit the deal, you should sell your option and keep the remaining time value rather than exercise, which nets you only the intrinsic value, but this assumes that there is a liquid market where someone will buy your option whenever you feel like getting out. If there is no one to buy your option, then exercise may be your only way to exit the trade, and under those conditions, it could well be more valuable to have an American exercise option vs a European one. I never thought about it this way, but the premium for American Option rules may be at least in part a measure of market liquidity. In most cases of real world options on liquid stocks, the presence of a replicating portfolio means that a bank is almost always willing to take the other side of a trade, and so american option rules on liquid non-dividend-paying stocks are essentially value-less.
I don’t see that the coin-flipping game has either a replicating portfolio or a liquid market to buy the remainder of your 1000 games, but I do agree with you that if it did, then the premium for American Exercise would almost certainly be 0.
The american option is valueless for a Call on a non-dividend paying stock - not necessarily for a Put given rates/borrow, even though practically it’s often valueless.
Is it still valueless if you cannot sell the call? My understanding is that you never want to exercise an american option on a non-dividend paying stock because doing so will net you intrinsic value only, whereas selling the european call nets you (intrinsic value + time value). Therefore the american option is valueless since you would never use it.
The american option is valuable on a dividend paying stock when (intrinsic value + PV(dividend)) > (intrinsic value + time value), or when (PV(dividend) > time value).
If you include transaction costs for selling, then you would want to say the american option has a small value when
PV(dividend) > (time value - transaction costs).
If it’s a non-dividend paying stock, then the american option has value when
transactions costs of selling option > remaining time value.
Admittedly, that value’s probably very small, but if there is no market to sell the option, then the transactions cost is effectively very very high (infinite?) and so the american option has value in that case, and it starts being noticeable when the time value of an option is still very high.
Liquidity is not part of the coin-tossing argument, and is not part of the option argument either. There is no requirement for the American option to be listed anywhere, so that you can sell it at anytime. If you exercise early, you lose the time value, plain and simple. There doesn’t need to be a liquidity mechanism that allows you to monetize the time value by selling the option right away. Otherwise you seem to be suggesting that any unlisted option, which you cannot sell immediately to some counterparty, doesn’t have time value since you cannot cash it right away? American call option on a non-dividend paying stock is worth the same as a European call option - whether or not there is a market for the option.
For the coin tossing game, it boils down to the fact that if you are forced to play 1000 rounds, you have expected earnings of 0 and a huge variance. By being able to stop whenever you want, what you hope to achieve is minimize the variance - you won’t achieve an expected value higher than zero. So you have two zero-expectation games with different variances. What I’m saying is, they are both worth zero.
What you are saying is, you will pay a premium over zero just to reduce your variance. But this is an arbitrage opportunity - what if you must play a single round coin flip game and have to choose between betting $1 or $1000? Both have expected value of zero, but the dollar variance on the big bet is much larger - would you pay a premium over zero to bet small?
Yes, I agree with your paragraph two, and did earlier.
Also, I’m realizing that in the non-dividend paying stock department, it’s not enough to be illiquid in the options market before an american option becomes desirable, there would have to be illiquidity in the underlying market, too, otherwise you could neutralize your position by building the appropriate replicating portfolio with a mixture of shorting and cash and just wait for the position to expire. But if the underlying market were illiquid for some reason, then you might not be able to exercise the option anyway, unless the option seller had inventory on hand as collateral.
But you do need complete markets for the traditional pricing methods to work. You can’t say, “well, you build a replicating portfolio and it would be worth X, whether anyone will trade at that price or not” if it is in fact impossible to build the replicating portfolio because markets are incomplete. (and the coin-flipping market is incomplete, because you can’t be the house).
I do think if one were risk averse and for some reason forced to choose between a single flip bet on $1000 and a $1 flip, but with a $1 premium for being allowed to bet small (so the choices are win $1000 vs lose $1000, vs win $0, lose $2, there are a lot of people who would choose to part with $2 guaranteed than to risk losing $1000. Of course, it probably requires coercion just to get people to play in the first place - at least the ones who would pay $1 in order to play small.
In practice, I know that an american option on non-dividend paying stock is one of those cases where value of the american option = value of the european option. In practice, it’s easy enough to remember this.
But there are assumptions that need to hold (e.g. not paying dividends, a complete market in the underlying, volatility is known, prices are approximable by a random walk, underlying prices are visible, rebalancing the replicating portfolio is costless, etc). When those assumptions break down, american options can have some value, and I’m trying to remember/figure out which ones those are, and which ones apply to the coin flip game.
I wonder if a better analogy is options v future. When you have to play 1000 rounds, you basically hold the coin toss futures worth anywhere between -$1000 to +$1000 at expiry.
Like I posted earlier, the premium to NOT play should increase with your risk-averseness and the size of the bet. Now I wonder if it is the size of the total bet (bet per tosses * number of tosses) as well.
If forced to play all rounds, a risk-averse individual should pay the same premium for 1 round with a $1000 bet as for 1000 rounds with a $1 bet per toss. Or do you think they would pay more for the former?
If NOT forced to play all rounds, I am pretty sure a risk-averse individual should pay much more premium for 1 round with a $1000 bet as for 1000 rounds with a $1 bet per toss.