Markov Chains (and HMM) for Quantitative Finance Modeling

Welcome to the Deep Dive. Today, we're really jumping into something core to quant finance. We're looking at these powerful probability tools, mark-off chains, and the even more advanced hidden mark-off models or HMMs. Yeah, and not just the theory, right? We're focusing on how they're actually used, you know, how they manage serious capital in the real world. Exactly. Because there's often this huge gap isn't there between the clean models you learn in a course. Oh, definitely. The textbook stuff, the sort of mechanical construction. And what happens when you hit a trading desk or a risk team? Suddenly those simple models, they just don't hold up. They really don't. They can be fragile, fundamentally unable to cope with the wild swings, the extreme events you actually see in markets. So that's our mission today, essentially. To bridge that gap, we want to unpack why the basic static models fail so badly. Show how practitioners had to adapt, how they started embracing dynamics, and really get into why modeling uncertainty, especially in finance, means you have to move past assuming things are independent. You need models that get the world's volatility, it's constant shifting. That's the heart of it. When your job is to quantify risk, maybe it's figuring out the chance of a huge loss or even a massive gain, you have to impose some kind of mathematical structure on all that uncertainty. But the structure can't be rigid? Absolutely not. Finance is, well, it's a game where the rules feel like they're changing all the time. So the structure you use has to be dynamic and needs to adapt its view of what's likely based on what's happening right now in the market. Okay, so let's start with the basics. Where does it all begin? It's the random variable, right? The RV. Yep. Standard stuff. In any intro, finance or scat's course, you model something like stock returns, it's called the return S as an RV. Yeah. And usually, usually it's assumed to follow a normal distribution. Yeah. The bell curve. And the appeal is its elegance, its simplicity. You're trying to put structure on likelihoods, which outcomes are likely, which aren't. And the normal distribution gives a clear picture. Things cluster around the average. The mean which for daily returns is often near zero. Right. And the further you get from that center, the less likely the outcome, those big gains, those big losses out in the tails. Very improbable, according to the model. We all learn that rule, the empirical rule, 68% within one standard deviation, 95% within two over 99.7% within three. It feels neat, comfortable, like you've got to handle on things predictable, but it suggests an orderly world and finance. Well, that's often not very orderly. So this is where the math hits the wall of reality. Pretty much. You take that fixed, static, normal distribution. You try to fit it to real data, say daily returns for a stock like Nvidia, something known for volatility. It breaks down dramatically. Yeah. It fails in two really key ways. Okay. First is the peakness, the shape around the middle. Exactly. It's called leptocrytosis. Real returns often bunch up much more sharply around the mean than the normal curve predicts. Too many small boring moves. But the bigger problem, the catastrophic one is out in the tails. That's the killer. The model just massively, systematically underestimates how often the really big moves happen. The extreme events. Yeah. And I think this point, it really needs to sink in for anyone working in quant rules. The scale of this underestimation is just shocking. The analysis we looked at had this incredible number. If that fixed normal distribution really was the true process for Nvidia returns, just observing one single return beyond four standard deviations. Statistically, you'd expect that to happen on average only once every 125.3 years. Once every 125 years. Think about that. Practitioners, traders, risk managers, they see four sigma moves, five sigma, even six sigma moves way more often than that. Sometimes multiple times a year in a crisis. So the model says once a century, reality says maybe next quarter. And the research pointed out something even starker, just to explain the extreme returns already seen in that data set. The fixed normal model implies you needed to be collecting data for over a thousand years. A thousand years of data. Just to explain the volatility you already saw maybe over a few weeks or months. Exactly. That's not just a small error. That's a complete model failure. It's a fundamental breakdown. And think about the consequences. If you're setting value at risk, your mar or your capital buffers based on a model that says a disaster only happens every 125 years. But it actually happens every few years or even more frequently. You're drastically under capitalize. You're exposed. That's how firms go bust. That's how you get systemic crises like 2008. Lots of supposedly impossible tail events hitting everyone at once because the models were wrong in the same way. So what's the diagnosis then? Why does it fail so badly? It's simple. Really. The underlying assumption is wrong. The true data generating distribution, as we call it, isn't fixed. Is parameters aren't constant. Especially the variance, the standard deviation. That's the key. The shape of the real distribution changes. It breathes, you know. So when things are calm, the distribution is tight, peaked, small moves are likely. But when fear spikes, when volatility goes up, the distribution flattens out, the tails get much fatter. Suddenly, those extreme outcomes become dramatically more probable. Okay. So we need to ditch the static picture. We need a model that can change its shape, change its view of probability, depending on the environment. Precisely. We need the probability of return S to be conditional on the market regime at that specific time. And that regime is always shifting. Okay. So if the distribution isn't static, if it's changing, what's making it change? What's the engine under the hood? This brings us to latent variables, things we can't directly see. Exactly. Layton processes, unobservable factors. These are the hidden drivers influencing the returns we actually observe. And the classic example in finances volatility. Volatility is absolutely central. Realize volatility or maybe the current volatility regime you're in. That's probably the main latent variable dictating how likely different returns are. But you said latent unobservable. So how do we handle it? You can't just look it up on a screen like a stock price. That's the tricky part. Stock prices, yeah, they're observable. Terminal variables. Volatility, though, it's a process. It's this sort of instantaneous measure of market intensity, market choppiness. You have to model it, release fine, a good proxy for it. Ah, so if we can't see the real thing, we use proxies, standins. Right. And that leads straight into what the research called the proxy challenge. Okay, what's the challenge? Well, the simplest way to proxy volatility is just to calculate it from reason history. Hmm. You take say the standard deviation of the last 20 days of returns, or maybe the last 60 days. That historical volatility becomes your proxy for current volatility. Yep. But here's the problem. Which window do you choose? 20 days, 60 days, 10 days. And the choice matters. Usually a short window like 20 days will react fast to sudden shocks, which sounds good. But it might also be really noisy jumping around based on short-term blips. Okay, in a longer window. A 60 day window will be smoother and more stable, but it might be too slow. It could completely miss a sudden spike in risk until it's too late, because it's averaging over too long a period. So the model's output suddenly depends heavily on this arbitrary choice. The analyst made the window size. Exactly. It's not driven purely by the data anymore. Yeah. It introduces this element of subjectivity, potential instability. It's a source of model error. So how do quants get around that? Do they just pick one and hope for the best? Well, more sophisticated approaches try to mitigate that arbitrariness. They move beyond just picking one single window. Ah, okay. This is where those dimensionality reduction techniques come in. Like PCA. Profisely. Principal component analysis is a good example. Instead of relying on just one historical vol measure. You calculate a whole bunch. Yeah. Maybe you compute realized dollars over 10, 20, 40, 60 days. Maybe you also pull in implied volatility from the options market. What the market itself is pricing in. Maybe you even look at bid-ask spreads as a proxy for liquidity stress. So you have this whole basket of indicators related to volatility and risk. Right. And then PCA does its magic. It takes that whole basket of features and mathematically compresses them. It extracts the most important underlying signal, creating a single combined volatility component that captures the most shared information, the maximum variance from all those inputs. That sounds much more robust. You're letting the data itself wait the different factors rather than just picking a single window size. It's definitely more data driven, less reliant on that one arbitrary choice. But fundamentally, it's still in admission, right? We're dealing with something crucial volatility that we can't perfectly observe. We're always trying to capture this moving hidden target. And volatility isn't the only hidden factor, is it? Oh, no, not at all. You can think about liquidity regimes times when it's easy to trade, versus times when it's hard. Trend or momentum states is the market generally going up, down, or sideways. Even things like broader macroeconomic sentiment or confidence levels. So the true picture is even more complex. The distribution of returns is really a function of multiple latent variables all interacting. That's it. And we need models that can handle that dynamic interplay somehow efficiently. Because if we don't capture those hidden dynamics, we're back to square one, massively underestimating risk, stuck in that thousand year time warp calculation. Exactly. Our predictions about the extremes, the tails will just stay fundamentally flawed. Okay, so the journey so far, static models fail because reality is dynamic. That dynamism is driven by hidden latent factors. So we need models that capture dynamics that capture dependence. Yes, dependence. We have to explicitly move away from that naive assumption of independence. Where, you know, what happened yesterday has absolutely no bearing on what might happen today. Which just doesn't reflect how markets or many financial processes actually work. To make this really concrete, maybe let's quickly touch on why pure independence fails conceptually. Sure. A simple stochastic process, technically, is just a sequence of random outcomes over time. Think about rolling a fair die every day to decide your portfolio return. Each roll is totally independent of the last one. Completely. You rolled a six yesterday, tells you nothing about the odds of rolling a six today. But that's just not realistic for most things we care about in finance. And the loan portfolio example you mentioned earlier that really drives this home. It's a perfect illustration. Imagine you're a bank, modeling your loan book. You categorize loans based on how late the payments are. Let's say you have states like the CUNLY. 30, 59 days past due, 60, 89 days past due, and 90 plus days past due, which is usually considered default. Okay. Four states. Now, if you try to model transitions between these states using a simple independent model, what would happen? The model would allow a loan that is perfectly current this month to suddenly jump straight to 90 plus days late next month in one 30 day step. Which is impossible in reality. Physically impossible. You have to pass through the intermediate stages. You have to be 30 days late before you can be 60 days late. So the model needs constraints. It needs to know where it is before deciding where it can go next. Precisely. And that constraint, that idea is the absolute core innovation of the Markov chain. We impose what's called local conditional dependence. Often called the Markov property. Right. And what it means is that the probability of the system's future states say the loans to the Inquancy Status next month depends only on its immediate current state. Not the entire history of how it got there. Exactly. It's often called the memoryless property. Knowing a loan was current six months ago, then went 30 days late, then back to current, then 30 days late again. That history doesn't matter for predicting next month's state if you know its status right now. Only the current state matters for the next step. Now that memoryless part that often trips people up doesn't. It sounds counterintuitive. Surely the market or a borrower's history does have memory. Why ignore it? That's a really important question. And it's true the market does remember past crises. A borrower's long term history does matter for ultimate default risk. The Markov property isn't claiming literal amnesia. Okay. So what is it claiming? It's more of a simplifying assumption that buys this incredible mathematical power and tractability. The justification is that for many short term forecasting tasks like predicting the probability distribution one step ahead, say 30 days, the immediate current state genuinely captures the vast majority of the relevant information needed to realistically constrain the possibilities for that next step. So it's less about ignoring history entirely and more about saying the current state summarizes the relevant information for the immediate future. Exactly. It prunes the tree of possibilities effectively for the next transition. While the long term path depends on the accumulation of these steps, the probability of the next single step is dominated by where you are right now. And we can visualize these allowed transitions. Yep. So when you transition diagram, you draw the state's current, 30, 59, etc. And draw arrows only between states that can transition in one step. So an arrow from current to current and arrow from current to 30, 59 days late. But crucially, no arrow directly from current to 90 plus days late. That diagram then gets turned into math into the heart of the model. Transition matrix usually called T. Okay. What's in T? Square matrix. If you have N states, it's N by N. So for our four lone states, it's a four by four matrix. Each row represents the state you're starting in, the from state, and each column represents the state you're transitioning to. And the entries are probabilities. Exactly. The entry and row I column J is the probability of moving from state I to state J in one time step. And critically, those impossible transitions we talked about, like current, straight to 90 plus bell. The probability in that cell of the matrix is just zero. Right. So matrix encodes the rules of movement. Once you have T, what can you do with it? This is where the power comes in. You can forecast transitions over multiple steps, multiple months, very efficiently using something called the Chapman-Komagara equation, which sounds complicated. But the result is surprisingly simple. If T is your one step transition matrix, then the matrix of transition probabilities after N steps is just T raised to the power of N. So I was on. So matrix multiplication does the heavy lifting. Instead of simulating month by month, thousands of times, you just calculate T12 using matrix algebra and boon. You instantly have the probability distribution after 12 months. So back to the lone example. If we calculate T12 dollars, we can look up the entry for starting in current and ending in 90 plus days late. Precisely. And that single number in the resulting matrix tells you the probability of that specific path unfolding over the entire year. The research example give a figure like maybe 18.26% probability for a long going from current to default within 12 months based on the one month T matrix. That's incredibly useful for things like lone loss provisioning, setting capital reserves, absolutely indispensable for financial institutions. And there's another key property many of these Markov chains have, a steady state. Right. The idea that eventually the system settles down kind of. It means that if you run the process for a very long time, if you calculate $10 for a really large and like 10,000 steps, the probability stabilize the distribution of states stabilizes. The vector that tells you the proportion of loans in each category current 3059, etc. Converges to fixed long run equilibrium values. And this happens regardless of where the portfolio started. For many common types of Markov chains, yes, whether you started with mostly current loans or mostly delinquent loans after enough time steps, the long run proportion of the 90 plus day category might always settle at say 3.8%. It represents the long term average state of the system, which is also vital for long term strategic planning and capital allocation. Definitely. Okay, but this all hinges on getting those probabilities in the T matrix right in the first place. How do we estimate them? And what are the pitfalls? Good question. They're typically estimated from historical data using maximum likelihood estimation or MLE. MLE sounds technical. It can be, but for basic Markov chains, the MLE estimator is wonderfully intuitive. It's just the observed frequency. You mean like counting? Pretty much. If you look back at your data and see that 1,000 loans were in the current state last month. And of those 1,000, you see that 990 state current this month and maybe 10 moved to 3059 days late. Then the probability of current to current is 990 divided by 1,000.99. Exactly. And current to 3059 is 10 divided by 1,000 or 0.01. It's just the simple proportion of transitions observed out of each starting state. Okay, that seems straightforward enough, simple counting. But there must be assumptions baked in there, right? Especially when we think about chaotic markets. Oh, absolutely. Two huge assumptions underpin this simple MLE approach. And they both tend to break down precisely when you need the model most during stress or crises. What are they? First is time homogeneity. This assumes the transition probabilities in your T matrix are constant over time. That the 0.01 probability of going from current to 3059 days late is the same today as it was last year and as it will be next year. Which is unlikely in a recession, surely that probability jumps up. Dramatically. So homogeneity gets violated. The second assumption is independence between the entities being modeled. Meaning each loan's transition is independent of all the other loans. Yes. The model assumes loan aid defaulting doesn't change the probability of loan be defaulting. Also unlikely in a crisis defaults to cluster people lose jobs in the same industries house prices fall affecting many homeowners. Precisely. You get correlation, contagion, transitions are not independent. So both core assumptions, constant probabilities and independence often get violated in the real world, especially during volatile periods. That's a major reason why models calibrated on calm data can fail so badly when things get rough. Okay. So we've got the Markov chain framework. It imposes local dependence, tracks transitions between discrete states using the T matrix. Let's us forecast. Now, let's connect this back to where we started that failing static normal distribution for returns. Right. How can we use this Markov structure to fix that original problem, the problem of dynamic volatility? We need to model volatility itself as a Markov process. That's the key insight. This is where theory meets practice in quant finance. Instead of modeling loan delinquency states, we define our Markov states to be different volatility regimes like low, vol, medium, vol, high, vol. Exactly. You define a small number of discrete states that represent the latent volatility process. Say, three states, low, mid, and high volatility. How do you draw the lines between these states? If volatility is continuous, how do you make it discrete? Good point. It's usually done using historical data. You might calculate some proxy for realized volatility every day for the past few years. Then you find the percentiles. Ah, okay. So, maybe the lowest 33% of historical volatility readings define the low, vol state? Precisely. Then maybe days between the 33rd and 66%ile are mid-vol, and anything above the 66%ile is defined as being in the high-vol regime. So it's data-driven, transparent. You can explain why you chose those boundaries. Yes. And the real power comes when you link these regimes back to the returns. Absolutely. The core idea that each volatility regime state low, mid, high, gets assigned its own conditional distribution for stock returns. Each state has its own specific mean return. and crucially its own variance. - So when the Markov chain model determines where currently in the high-volt state. - The model automatically switches to using the return distribution associated with high volatility, a distribution that will have a much larger variance, a higher standard deviation. - Making those extreme returns much more likely, fitting what we see in reality. - Exactly. It builds that dynamic shape shifting directly into the model's structure. - Can you give a feel for how different those parameters might be like the standard deviations in each regime? - Sure. Based on typical market data, you might find that in the low-volume, the estimated daily standard deviation is quite small, maybe around 0.5%. - Okay. - In the mid regime, it might jump up significantly, maybe to 1.2%. - Right. - But then in the high-voltage, it could usually be 2.5% or even higher during really stressed periods. - Wow, so the standard deviation in the higher regime could be like five times larger than in the low regime? - Easily. And think what that does. - A return that would have been a shocking four-sigma event if you were stuck assuming the low-vol distribution was always true. - Suddenly becomes maybe less than a one-sigma event, just a routine move. Once you correctly identify, you're in the high-volt regime. - That's it. The model adapts its definition of normal, based on the current volatility context. - This feels so much more realistic. You're conditioning the expected outcomes on the market's current mood or state. - And it directly addresses that major failure of the static model. The inability to capture the fat tails, the excess kurtosis. - Does it work? When you simulate data from this kind of regime switching Markov chain model, does it look more like real financial data? - Much more so. The research we reviewed showed exactly this. Remember, a perfect normal distribution has a kurtosis of three. - Right. - When they simulated returns from the three-state volatility MC model, the resulting data showed an excess kurtosis of 0.687. That means a total kurtosis of three plus 0.687 equals 3.687. - Which is significantly fatter tailed than the normal three. It's moving in the right direction. - It's a huge step closer to empirical reality. It proves the model is successfully generating more extreme events than the basic normal curve allows. - That's a big win statistically. But the source is also really emphasized another advantage of this specific Markov chain approach, interpretability. - Yes, and this is often a massive deal in practice, especially in banking or insurance. Where models face regulatory scrutiny. - Why is it so interpretable? - Because the states are explicitly defined by the practitioner based on something tangible volatility percentiles. So when the model spits out a risk number, say a higher var estimate. - You can explain why. - Exactly, you can say, look, the model estimates there's a 95% chance we are currently in the high volatility regime. And our definition of that regime inherently uses a standard deviation of 2.5%. That's why the risk number is elevated. It's transparent, you can trace the logic. - That clarity is gold when talking to management or regulators. - Absolutely, it builds trust and understanding. - It's still a simplification, isn't it? What are the limitations here? - For sure, it's not perfect. First, you're taking something continuous volatility and forcing it into a small number of discrete boxes. Three, in our example. You inevitably lose some information, some nuance in that discretization. - Okay, the boundaries are a bit artificial. What else? - Another common simplification is that, while you've made the variance conditional on the regime, you often still assume that the returns within each regime follow a normal distribution. - Ah, so you might assume low-val returns are normal with a low sigma and high-val returns are normal with a high sigma. - Exactly, which is way better than assuming one fixed normal distribution for everything, but it might still be wrong. Maybe returns in the high-val regime aren't just higher variance normal. Maybe they follow a different shape of distribution altogether, perhaps skewed. - So we've pushed the problem down a level maybe, made it smaller, but not eliminated entirely. - That's a good way to put it. It's a very pragmatic compromise. You gain enormous ground in capturing dynamics and maintaining interpretability, but you accept that there are still some underlying statistical assumptions that might not perfectly hold. - It's a powerful, practical tool, but maybe not the final word if you need ultimate statistical fidelity. - Right, and it work well as long as you can reasonably define the key latent states explicitly. But what if there are multiple important latent factors driving returns? - Yeah, that leads us right to the next level, that explicit Markov chain model focusing just on volatility regimes. It's effective, definitely. But we already has discussed the volatility probably isn't the only latent factor that matters. - Right, you mentioned trends, liquidity, maybe other things too. - Exactly, so what happens if a quant says, "Okay, I need to capture volatility and trend explicitly in my Markov model to get better performance?" - They try to add more states. - They do, but they immediately slam into a major practical problem. - Right. - The state space explosion. - Sounds dramatic, how does it explode? - Let's walk through it. We start with our three volatility states, low, mid-high. Now let's say we also define three simple trend states, bull market, bear market, sideways market. - Okay, three volustates, three trend states. - To capture both explicitly, every possible combination has to become its own unique state in the model. - Ah, so you need a low-vulable state, a low-vulbear state, low-vul sideways. - A mid-vulable, mid-vulbear, mid-vul sideways. - High-vulable, high-vulbear, high-vul sideways. Okay, that's $3 times three equals 99 states already. - Nine states, manageable, maybe. But now, what if you decide you also need to capture liquidity? Let's keep it simple, just two states. Tight liquidity or loose liquidity. - Okay, so now each of those nine states has to be combined with the two liquidity states. - Yep, you now need low-vul-tight, low-vulable, loose, and so on, for all nine previous combinations. - That's $9 times two, it was 18-ite states. - 18 distinct states. And think about the transition matrix. For 18 states, your team matrix is $18 times 18-ite. - That's 324 transition probabilities you need to define an estimate. - Exactly. Plus, you need to define the specific conditional return distribution, mean and variance. For each of those 18 unique regime combinations, it gets computationally nightmarish very quickly. You need vast amounts of data to estimate all those parameters reliably, and just defining and justifying each state becomes incredibly complex. - It just doesn't scale well in practice. - It really doesn't. And this practical bottleneck, this state space explosion, is the primary motivation behind using hidden Markov model or HMMs. - So HMMs offer a way out of this explosion, how? - They offer a solution through compression. - Compression, like data compression. - Sort of analogous, yeah. Instead of the practitioner trying to explicitly define every relevant latent factor and every possible combination. - The HMM does something different. - Yes, the HMM essentially takes all the underlying complex, interacting latent processes, volatility, trend, liquidity, momentum, whatever else might be driving the returns, and it compresses their combined effect into a much smaller, pre-specified number of hidden states. - Hidden states, okay, so maybe just three or four states total. - Typically yes, maybe three or four or five states, depending on how complex the data seems to be. The key is that this number is chosen by the practitioner, not dictated by trying to enumerate every possible explicit combination. - And the crucial difference is, we don't actually know what these hidden states represent beforehand. - That's the absolute core distinction. With the explicit Markov chain, we define state one as low-vul, state two as mid-vul, et cetera. With an HMM, we just say, okay, model, find the best three hidden states. We label them state one, state two, state three. We don't give them names like low-vul, bullish. - So how does the model know what state one or state three is? How are they defined? - They are learned directly from the data. The HMM algorithm looks at the entire time series of observed returns, and it adjusts the parameters of the hidden states, their means, their variances, and the transition probabilities between these hidden states. - Until it finds the configuration that makes the observed data sequence most likely. - Exactly. It optimizes the parameters to maximize the likelihood of the entire return series having been generated by transitions between these few hidden states. It essentially lets the data itself define what combination of underlying factors each hidden state is implicitly capturing without us needing to specify it. - So it avoids us having to argue about whether high-volt starts at the 66th or 70th percentile, or how exactly to define bull trend? - Precisely. It shifts that burden of definition from the human analyst to the estimation algorithm. - What kind of algorithms do this learn? It sounds computationally intensive. - It is. The main work courses are algorithms like the forward-backward algorithm, and particularly for training the Bound Welch algorithm. - Bound Welch, can you give us the basic idea, not the full math, but the intuition? - Sure. Bound Welch is actually a specific type of a more general algorithm called expectation maximization or EM. Think of it as an iterative process. It starts with a random guess for all the parameters, the initial state probabilities, the transition probabilities between hidden states, and the parameters like mean and variance of the return distribution within each hidden state. Just a guess. - Step one. - Guess. Step two is the expectation or e-step. Using the current guesses for the parameters and the observed data, it calculates the expected probability of being in each hidden state at every single point in time in your data series. It uses the forward-backward algorithm for this part. - So it figures out, given the current model guess, how likely it was that the market was in hidden state one, state two or state three on each particular day. - Exactly. Then comes step three, the maximization or M step. Now that it has those probabilities of being in each state at each time, it uses them to re-estimate the model parameters. It calculates new, better estimates for the transition probabilities and for the means and variances within each. state, choosing the parameters that maximize the expected likelihood based on the state probabilities from the E step. So E step estimates state probabilities given parameters, M step estimates parameters given state probabilities. You got it. And then it just repeats. It takes the new parameters from the M step, goes back to the E step to recalculate the state probabilities, then back to the M step to refine the parameters again. And it keeps iterating like this. Thousands of times, potentially, until the parameters stop changing much, until the overall likelihood of observing the actual data given the model stops improving significantly. At that point, it's converged. It's found the best fit definition for those hidden states and their dynamics. That's pretty clever. It's like the model is bootstrapping its own understanding of the market regimes directly from the return data itself. It is. It's a powerful way to let the data reveal the underlying structure without imposing too many preconceived notions. Oh, there's always a bite, isn't there? We gained this power, this compression, this data-driven definition. What did we lose? We lost what we just praised about the explicit Markov chain interpretability. Because the states are hidden. We don't know exactly what they mean. That's the fundamental trade-off. It's the classic quant dilemma, really. Bias versus variance, complexity versus interpretability. Since hidden state three, for example, is just a mathematical construct that represents some optimal compression of potentially multiple underlying factors. Volatility trend, maybe others. You can't just point to it and say, "Ahh, state three, that's our high volatility regime." Nope. You can't. It might mostly capture high volatility, but it might also be mixed with, say, a specific trend behavior that was prevalent during those high-ball periods in historical data. It's like looking at a principal component in PCA. Capture variance, but explaining precisely what it represents can be tricky. So how do you interpret these hidden states, then? You have to infer their meaning. You infer it by examining the parameters the algorithm settled on for that state. You look at the estimated conditional distribution associated with hidden state three. So you look at its mean and standard deviation. Exactly. If hidden state three ended up with a very low mean, negative drift, and a very high standard deviation, say 6.46 percent, like in one of the research examples. Then, you can infer, okay, this state clearly captures periods of high volatility and negative market performance. You infer its character from its statistical output, but it's still an inference, non-explicit predefined label, like high volatility regime. Okay. This brings us right back to that practical modeling dilemma. The HMM might actually fit the data better, maybe give slightly more accurate risk forecasts, because it's capturing these complex compressed interactions we didn't explicitly think to model. Often. Yes. That's the potential performance benefit. The explicit mark off chain where we did define the states as low mid-high-vol is much easier to explain, to stakeholders, to management, crucially, to regulators. And that can be the deciding factor. When you're building models for things like regulatory capital or explaining portfolio risk to a board. The regulators, like the Fed or the ECB, they need transparency. They need models they can understand, audit, backtest, justify. Absolutely. Trying to explain a model failure by saying, well, the model suddenly jumped into hidden state seven, which seems to be some complex mix of volatility and liquidity stress, but we can't pin down the exact waiting. That's a much tougher conversation. Compared to saying, the model shifted to the high volatility regime, which we explicitly defined based on the 66 percentile, and that regime has a higher standard deviation. The simpler explicit mark off chain often wins in those situations, purely because of its interpretability, even if the HMM showed a slightly better fit to the historical data and testing. The constant tension, predictive power versus explainability, performance versus governance. It's a tension at the heart of modern quantitative finance. The choice of where you draw that line depends heavily on the specific application, the audience, and the regulatory environment. Wow. Okay, that was quite a journey, really, into the probabilistic core of finance. We started way back with that simple, static, random variable model. The normal distribution. Which, frankly, sounded almost comical in how badly it underestimated risk. Including a thousand years of data to explain what you already saw. A clear sign, something was deeply wrong with the static assumption. Then we moved on. Ignolish reality is dynamic driven by these hidden latent factors with volatility being key. Which led us naturally to the explicit mark off chain, using that concept of local conditional dependence, defining clear states like low, mid, high volatility regimes. Which did a much better job capturing the fat tails, the excess kurtosis we see in real returns, and critically remained highly interpretable. You could explain why the risk changed. Big advantages. And then, pushing further, we got to the hidden mark off model, the HMM powerful data-driven. Using compression to handle the explosion of states when you consider multiple interacting latent factors. But at the cost of that clear interpretability, the hidden states are learned, not defined, making them harder to explain. The ultimate trade off. So if there's one core takeaway for anyone listening who deals with financial data, financial outcomes, what is it? I think it's that you simply must accept that probability isn't fixed. Distributions change. The world is dynamic. You have to move beyond assuming independence where it doesn't exist. You need models that embrace dependence that can handle regime shifts, whether through explicit states like in an MC or compressed hidden states like in an HMM. You need tools that can cope with the reality of tail risk, which means understanding the latent drivers. Which leaves us with that final, really thought-provoking question, doesn't it? The one every quant faces on the job. The Balancing Act. Yeah. When you have to deploy a model in the real world, how do you make that call? How do you balance the potential, maybe marginal performance gain from a super complex opaque model like an HMM? It's the absolute necessity, often, for transparency, for explainability, for satisfying stakeholders and regulators who need to understand why the model is giving the answer it is. Where do you draw that line between mathematical sophistication and practical governance? It's not just a technical question, is it? It requires real judgment, commercial awareness, regulatory understanding. The math might give you an edge, but can you explain it? Can you defend it? That's the challenge. Absolutely. It's crucial, but communicating it effectively is just as vital. A perfect place to wrap up this deep dive. Thanks for joining us.