Martingale property

Water Cooler
#21

I hesitatingly disagree. Forget drift for a minute, to simplify things.

Any game of chance - slot machines or blackjack or anything else - lets you stop whenever you want. But overall, people still lose money. If you play, odds are that you WILL lose money. So this freedom is worth nothing.

Compute it another way. When you start, your expected winnings are zero. No matter what round you stop in (round 31, 523, or 999) your expected future winnings are also zero. So your expected earnings till that round should also be zero.

#22

Yes, you should not play a game with negative expected return.

The option to stop playing at any point in the interim is a real option, but it is not the baseline case here.

If you have to play 1000 rounds, and your expected return is 0, then assuming no time value for money, and no discount for risk aversion, you should be willing to pay $0 to play this game, since your expected winnings are 0.

If there is risk aversion, then someone will need to pay you to play this game, or - if flipping equal heads or tails is defined to win $1, then you’d be willing to pay $0.90 or something to play this game (where the discount reflects risk aversion). If 1000 rounds of the game takes a material length of time, you’re likely to discount further to reflect time value of the money spent.

But let’s go back to the game where you are not risk averse and you play 1000 rounds and have the chance to stop at any point in the interim. Now in this game, the end value at 1000 rounds is still zero, but there is also a decent chance that you will have a positive balance in the interim, which you can lock in by stopping, and which you can try to neutralize by continuing.

What this means is that someone who forces you to play 1000 rounds would have to give you better odds than someone who doesn’t, which is perhaps why you see this as the baseline case.

I guess the way to value this option would be to run a binary tree, which allows you to exercise if the current value is worth more than the risk-neutral probabilities of flipping the coin again. If the risk-neutral probabilities equal the ordinary probabilities (which they might, if the RFR=0), then it’s very possible that they’d sum out to the same. I’d bet a tree that only has 10 flips or so would be able to do it.

I am not 100% sure, but it seems logical that a real option like that is worth paying for.

#23

By symmetry, there is still only a 50% chance that you will be ahead after any round X ends. What do you do then? Stop or continue? Also, there is a 50% chance that you are behind, and have to risk more money to break even. What do you do then? Stop or continue?

EDIT: not 50% but equal probabilities that you will be ahead or behind. There is also a small probability that you break even at the end of (an even-numbered) round X. Still, how do you decide to stop or continue?

Example - 3 tosses, $1 bet/toss, fair coin. You have 4 options - stop after 0th toss (don’t play), stop after 1st toss, 2nd toss or 3rd toss (end of game.) Call them C0, C1, C2, C3. Your expected and actual winnings are 0 if you exercise C0. Now let’s say you win toss 1. Do you exercise C1? What if you lose toss 1?

#24

It does seem to depend on how the payoffs work, and potentially what your utility is, even if they are symmetrical.

With marginally decreasing utility, it would seem that it makes sense to quit earlier in the binary tree, which would definitely make the option worth something. However, marginally decreasing utility would also imply that the person is not risk neutral, since risk aversion is a consequence of a marginal decreasing utility.

I’ll have to do the tree sometime when I’m not at risk for getting fired for not working.

#25

On roulette try it with the bets that pay 2-1 … A fun strategy is to wait 6-7 spins without it landing on one of the douzens and then betting into it… you don’t even have to double your bet… First spi. Lose 10, second spin lose 10, third spin bet 15… Figure it out on excel)

#26

On roulette try it with the bets that pay 2-1 … A fun strategy is to wait 6-7 spins without it landing on one of the douzens and then betting into it… you don’t even have to double your bet… First spi. Lose 10, second spin lose 10, third spin bet 15… Figure it out on excel) I started with $500 and after 6 hours on the same wheel won over 6k… It was incredibly lucky day

#27

On roulette try it with the bets that pay 2-1 … A fun strategy is to wait 6-7 spins without it landing on one of the douzens and then betting into it… you don’t even have to double your bet… First spi. Lose 10, second spin lose 10, third spin bet 15… Figure it out on excel) I started with $500 and after 6 hours on the same wheel won over 6k… It was incredibly lucky day

#28

On roulette try it with the bets that pay 2-1 … A fun strategy is to wait 6-7 spins without it landing on one of the douzens and then betting into it… you don’t even have to double your bet… First spi. Lose 10, second spin lose 10, third spin bet 15… Figure it out on excel) I started with $500 and after 6 hours on the same wheel won over 6k… It was incredibly lucky day

#29

i went 13 freaking spins playing this system before going broke

#30

How is the outcome of playing 1000 rounds with the option of stopping anytime any different from the outcome of playing 999 rounds with the option of stopping anytime, and then one more round (call it a new game) after? And clearly you wouldn’t pay anything other than zero for playing a single round since its expected value is zero, which implies that the price you would pay for playing 1000 rounds with the option of stopping anytime is the same as the price you would pay for 999 rounds with the option of stopping anytime… which is the same as the price you’ll pay for playing 998, 997,…,3,2 and ultimately 1 round, which is of course zero.

#31

It does seem that when you shouldn’t be playing the game in the first place, the option to stop at any time is not valuable.

Though, if one is risk averse and forced to play this game, it would seem logical that one would be willing to pay an amount >0 in order to stop before the end of 1000 throws.

#32

If I tossed a coin and got 10 heads in succession, I’d start to question whether it’s a fair coin. Odds are that it isn’t.

(The maximum likelihood estimate of P(H) is 1.0000; the Bayesian estimate of P(H) is 0.9167.)

Years ago I listened to an interview of Persi Diaconis on NPR. He had a machine that would flip a coin, and the coin would always land heads. 100% of the time.

During the interview, apparently the interviewer reached over to touch the machine, asking Persi, “Why is this piece of tape stuck on here?”

“Don’t touch that tape!” shouted Persi.

“What does it do?”

“I don’t know, but the machine won’t work without it!”

#33

This is why an American option is always sold at a premium to its identical Eurpoean option. Maybe there’s a way to use how these options are priced differently to value the right to walk away (exercise your option) at any time.

#34

That’s a loaded statement. How many heads would not arouse any suspicion that there is something wrong with the coin after only 10 tosses? Obviously the expected number of heads is 5, but usually for such small samples, people grossly overestimate the probability that you will actually get 5 heads, and similarly overestimate the chance that the coin is indeed biased based on the observed deviation from the expected value. It’s called the law of LARGE numbers for a reason…

#35

Unless it’s not optimal to exercise prior to maturity, as is the case with a call on a non-dividend paying stock, in which case the theoretical option prices for an American and European option are the same and the value of the early exercise feature is zero. Just because you have the right to do something doesn’t mean that this right is valuable or you should pay money for it, so bchad needs a more rigorous argument for his coin tossing game theory! :wink:

#36

It’s funny how every now and then, the quants come out to battle

#37

is this for real?

#38

I did concede on the last post of the previous page that if the game has no value to begin with, the option to stop is not necessarily worth anything (or is the case where the American Exercise version = European Exercise version = 0).

However, the option can still have value under certain circumstances. If you were forced to play at least one version of the game and are risk averse, you would not be indifferent between playing 1000 flip version that you had to stick to the end and an up-to-1000 flip version that you could stop at any time after the first flip. If you are not indifferent, then clearly there’s value in the option.

#39

BChad is onto something (as usual.) Basically, “how much would you pay or need to get paid to take any risk at all”? It’s like owning an option whose sole value lies in the time premium (where time is measured in number of rounds yet to be played.)

A risk neutral investor wouldn’t care whether it is 0 rounds, 3 rounds or 1000 rounds. A risk averse investor will pay more to play fewer rounds. A risk seeking investor will pay more to play more rounds.

The amount of bet also has to factor in the premium, although it could do so indirectly via change in the risk aversion/seeking factor. I might bet $1 per round with no hesitation, but $1 billion per round might cause me to be more risk averse (or risk seeking!)

I think the boundary conditions are that if you have to play 1000 rounds (the max allowed) then the premium is zero for an investor with any positive risk aversion. Similarly, zero premium to pay for any risk seeker to play 0 rounds.

#40

I am not sure how this qualitative argument holds against my counter example using options above? You might as well say, “If you were forced to buy a call option (on a non-dividend paying stock) and are risk averse, you would not be indifferent between a European and an American option.”

However, you should be indifferent according to finanical theory. The early exercise feature has no value. That is neither because “the game has no value to begin with” - both options have a positive price; nor because you assume a risk-neutral investor.