Transcription
[Music] welcome back today what I want to do is talk about Martin Gail betting Theory. we'll use a casino as an example. it's kind of the canonical example for any sort of betting strategy. we'll take a look at meanor version and or lback process and how that's typically used to model volatility. you may have seen an extension of the black schs model, maybe like a hon model, where volatility is treated as a stochastic process.
then what we're going to do is we're going to try to apply the Martin Gill betting Theory to volatility trading. we'll see if we can continue to take a short position in say the vix and then double down on that short position every day or week to try to cover our losses and gain some money, some p&l. and that's what it's all about at the end of the day.
so for starters, let's take a look at roulette. we're going to simplify roulette, no zeros, no greens. we're just going to talk about black and red here. and we're going to take a look at what happens every time you lose and then double down. so suppose that you have an initial bet size B. after n losses, the total amount wagered is going to be B plus 2B plus 4B plus so on until you get 2 to the NB, where n is your total number of losses. so if I bet 100 bucks and my first loss is going to yield, you know, a loss of 100 bucks, I'm going to go ahead and bet 200 the next time around. and then my total amount wagered is going to be T of 1, which is going to go ahead and be 2 to the first power times B, which is 100, and then we have our initial bet of 100, so 300 total wagered. and you can go ahead and do this recursively. you can continue to compute the total amount wagered. it's not super difficult, really. all we're doing is every time we lose, we double down, double down, double down. you know, depending on how how much capital you got, this may or may not be a good idea. regardless, at the n+1 bet, your total profit is going to be B minus T sub n plus 2 to the NB. and this strategy assumes that you have some sort of infinite capital capacity, of course, maybe unrealistic in practice, but some people do have quite a bit of money, maybe close to infinite money. so this may be even somewhat practical for them.
now, why is this an issue for casinos? well, the expected value of the Mart Gale strategy, so this particular betting strategy, is actually your initial bet size. so after one round of Martingale, the EV is going to be B, which is positive. and if you've seen any of my previous videos, I talk about, you know, making decisions that yield positive expected value to accumulate profit over time. we don't know what's going to happen in the short run, but if you can make decisions that contribute to positive EV over time, that's how you accumulate p&l, that's how you generate wealth. well, this is a problem for casinos because if I could walk in and I could do this Martin Gale strategy, continue to double down on my losses, then I have positive EV and I can continue to do that and continue to take money from the casino in the long run. again, it's not about the short run, it's about the long run. so let's take a look at the expected value here. what we do is we are going to weight each outcome by the associated probability, that is, each profit gets a probability weight by the likelihood of the outcome. and this turns into some sort of geometric series which will eventually converge to B.
okay, so what is the issue with this? well, clearly the profit has a positive expectation for the player. the casino is going to lose money over time. therefore, we really don't want that. we really don't want that as the house. we really don't want that as a casino. we don't want to give the players the.
now, some games you may consider fair. you know, others, you know, there's an edge towards the casino, and that's how they generate money in the long run. and, you know, we talk a lot about probability, statistics. we talk about casinos in the realm of of finance because that is a very canonical example of, you know, just odds in general. it's not necessarily saying that finance itself is gambling. depending on what you do, it may be gambling, but, you know, there's a very big difference between just arbitrarily gambling, investing, and, you know, we we got to try to draw a line somewhere there. but nevertheless, let's continue to look at this. so through this strategy, we get positive expected value. casinos don't want this. the house doesn't want this. we're going to throw limits on roulette, for example, so that they can't double down at some point. and, you know, those short-term losses are just going to accumulate, and and you're not going to be able to recover this. so if I lose $10,000 and I can't double down, then how can I possibly hope to have positive expected value? if I do 10,000 again, that's not going to yield the positive expected value that I'm looking for. so that's an important idea, right? just because it's capped at 10,000 doesn't mean if I keep betting 10,000, I'll eventually regain those losses. you need that doubling down. that's the process itself that yields the positive expected value.
now, we talked about getting to the long run, and that's true positive EV to get to the long run. but what does the short run look like? what is the probability of surviving the short run with some sort of Martin Gale strategy? well, if you just take a look, the total amount wagered is going to be proportional to 2 to the N, which is exponential. so that can be a problem because after 10 losses, you take your initial bet B, and you have a a coefficient of 2,047. then after 20 losses, you have a coefficient of 2 million. so your bet size blows up, right? and that's really the issue with this strategy. is if you don't survive the short run, you blow out your account. and now there's no more recovering losses. there's no more positive expected value. and that's really the game that we play, right? whether you're a market maker, a casino, a player in a casino, or a market participant, you need to figure out what it is you're trying to do, what your strategy is, what your expected value is, and how you can survive the short run to get to the long run to to kind of reap the reward of that expected value over time.
what I have here is a simulation of the Martin Gale strategy with roulette. so every time I run this, we're going to essentially try to win win money from roulette, and we are going to play until we win. so if we lose, we'll double down. if we lose again, we'll double down. and every time I run this, so we won, we won, we won. I run it again, we win here. we lost the first round, so we double down. we win the second round. as you can see, eventually you always end up at this 1,000. so my, you know, initial bet is 10, my max rounds is 10 here, just for the sake of the simulation, and then the bankroll is 1,000. so you can see as I keep running this, here's a very good example. is we keep doubling down, we keep doubling down. so 990, 970, 930, 850. very quickly we're starting to increase. you know, this is this is not going linearly, right? this is going 30, then it's 40, and then it's going down all the way to 80. so that is that 2 to the N. and here we can see eventually we do hit that win and we get back to the 1010. but this is the problem, right? eventually this is going to hit a really, really big loss streak, maybe 20 losses, and we're going to completely blow out the account. in fact, with a bankroll of a thousand, it's not going to take 20 losses. it's going to take roughly, what is that, close to 10, maybe 10 losses or something like that. so it's very important to understand that idea.
now, here's the deal. if you do survive the short run, you don't blow out your account, or you have infinite money, which, you know, you wouldn't be doing this, but you can go ahead and actually simulate the outcome of this Martin Gale strategy and roulette over time. so as you can see, you kind of have these manic highs and lows locally, but over time, you still have this cumulative wealth that is positive. and this is really what it means to have a positive expected value.
now, this is all well and good, but what does the actual bet sizing look like? here's the issue, right? take a look at this bet sizing. so based on all these game numbers, you have a thousand games. eventually, the bet size spikes up. look at this. we're betting, you know, over $1,200 here a variety of different times. so you're risking a significant amount of capital cumulatively to attempt to recoup your losses, but also regain that, you know, initial wager back. you know, that's how you accumulate the wealth. is you bet 100, or you bet 10, and then, you know, you get that that 10 back plus 10 here. you know, in this case scenario, you could see that it's not a a nice, low, constant, stable bet size, but rather, you know, over the significant number of games that we play here, it does tend to blow up. so you can see the positive expected value, but if you blow out your account in the short run due to this bet size, and you can't double down, or there's limits, then you can't use this strategy, and it's it's not productive. okay, so that's the idea of Martin Gale, at least in a nutshell.
what about applied to the financial markets? well, we don't really have just a way to 50/50 the the financial markets. of course, there's, you know, different types of contracts you might be able to engage in, and you may argue that they're 50/50. I I would argue that more likely than not, the implication is that they're 50/50, or there's some sort of implied probability that they're 50/50. but let's take a look at this, this sort of example in the context of volatility trading, mean reversions. so the on sign L back process models, uh, mean reversion. so it could be used for rates, it could be used for volatility. here, I'm I'm kind of just showing the process itself. it's a stochastic differential equation, and you can simulate this stochastic differential equation, and it's going to give you a path over time.
now, this red line represents the mean level. and what you'll notice is as I continue to generate paths, the path tends towards the mean. and if it goes too low, it starts to tend back. if it goes too high, it starts to tend back. and there are processes in the financial markets that emulate this behavior, specifically volatility. if you take a look at the VIX, right, the VIX based off of the implied volatility of the S&P 500 options. we take a look here, and this is the mean level here. typically, it's below 20. this is the mean level. you can see that as there are these spikes, it does revert back to this mean level here. so one one strategy perhaps to, you know, apply what we had just talked about in roulette to the financial markets is realizing, hey, well, if volatility tends, you know, tends towards the mean, and the mean tends to be around 20, then if we deviate from the mean, so if we go up, shouldn't we just take a short position? and if it continues to go up, double down on that short position and continue to do that until, you know, we start to ride the vault down and we just collect essentially the the cash for borrowing those shares? then we can buy back the shares, cover the shares in the market at a lower level, and, you know, we'll make out with that profit. that is one theoretical implementation of Martin Gale. and I actually have a simulation of it here.
so what I'm going to do is I'm going to go ahead and I'm going to apply that Martingale strategy to the the VIX historical data. and I'm going to enter a short position every day. if there's a loss, I'll double down. if there's not a loss, I'm going to collect that premium. and you can take a look at what this equity looks like over time. it's like, wow, you know, this is this is tremendous. we're we're certainly gaining a lot of equity value here. and what you'll have to notice is the the final capital may be, you know, roughly, what is this, 700,000 or something, but your exposure is significantly higher than your final capital. so if you took a look at the sharp ratio, this would just be absolutely exogenous for the amount of capital that you risk to acquire this equity curve. moreover, another issue is you may not have the capital to continue to short the the VIX itself. so you're on margin when you go ahead and and short the VIX here. you're eventually going to run out of capital, right? so if I continue to try to double down, then eventually I might get a margin call, and I'll have to liquidate the position at a at a loss, or I'll blow out my account, I'll be bankrupt, or I might owe money. so that's an important idea as well to remember. so even though going short, collecting that, you know, collecting the cash for going short, eventually you could buy back those shares at a lower level, theoretically, yeah, that works great if you could do that, survive the short run, get to the long run. yeah, fantastic. however, you know, with events like 2020, if you go all the way back to events like 2008, 2001, even these spikes here to like 40 and 50, they can completely blow out your account. so when you're taking a short position, let's say you start doubling down at at, you know, 25, if we ride all the way up to 40, there's no way that you're going to be above water. you're going to have to liquidate at a pretty massive loss. so important to keep that in mind.
that is just, uh, I think this is a, this was a fun video to make. I wanted to kind of talk a little bit about, you know, an application of probability to not just like, not just casinos and and trading, but I wanted to kind of go from from theory to practice, theory to practice. so, you know, the idea here was we started with the Martin Gil betting Theory. we talked about it theoretically in roulette, positive EV. we talked about it in practice, how eventually you're going to have a bet size that kind of just blows up and it's not practical. similarly, we talked about theoretically with mean reversion, how well, hey, if, you know, we have this process that reverts to a level over time, then we can take a position and maybe double down on our losses when it goes against us to apply that theoretical betting strategy to the process that is mean reverting. but of course, in practice, you're going to have a crazy amount of exposure and margin to answer to. and that is no good. that is no good for getting to the long, getting to the long run. so I hope you enjoyed this video. this was a fun one to make, and I'll see you in the next one.