Economic Modeling for Forecasting

Most people look at the Black-Scholes equation and see a way to find the "fair value" of an option. But when you strip away the stochastic calculus and look at the mechanics, you realize it’s actually a P&L decomposition. It doesn't tell you what the option should be worth; it tells you how to manufacture that value dynamically. I drew this sketch to visualize what is actually happening under the hood of the PDE. 1. The Engine (Taylor Expansion): The top section shows the reality of risk. Your P&L is driven by Time (Theta), Direction (Delta), and Convexity (Gamma). 2. The Cost of Business (The PDE): The equation everyone memorizes is really just a "No Free Lunch" constraint. It simplifies to: Theta + Gamma + Interest = 0 In plain English: The money you lose every day by holding the option (Time Decay) must be exactly offset by the money you make trading the volatility (Gamma), minus your financing costs. The Insight: If you are a market maker, you aren't betting on the price. You are managing a relationship between Time and Movement. 🔹 If the market doesn't move, Theta eats you alive. 🔹 If the market moves more than implied, Gamma pays the bills. The model isn't predicting the future. It's quantifying the "break-even" volatility you need to survive the time decay. When you look at a model, do you see a "Crystal Ball" (prediction) or a "Thermometer" (measurement)? #QuantitativeFinance #BlackScholes #Derivatives #RiskManagement #Mathematics #CapitalMarkets #OptionsTrading #FinancialEngineering

What can we learn from the US yield curve today? The chart below shows the US yield curve today (orange) and one year ago (blue). Before we jump into the conclusion we can derive from the change in curve shape, it's important to understand ''what'' yield curve are we looking at. This is the Overnight Index Swap (OIS) yield curve, not the standard yield curve derived using government bond yields - why? Government bond yields incorporate two dimensions: the risk-free rate set by the Fed (and investors' expectations about where it will be in the future), and something called ''asset swap spread'' on top. The asset swap spread (ASW) represents the compensation that investors require to warehouse US Treasury bonds on their balance sheet rather than simply expressing their long duration view via swaps. Being interest rate derivatives, swaps are a cash-light instrument requiring a small amount of margin to be executed while Treasuries either require the full cash amount (unlevered bond purchase) or balance sheet capacity (repo-funded purchase). Regulation has made balance sheet capacity quite scarse in the US, and so Treasury yields trade at a marked premium to swaps. This is why looking at the Overnight Index Swaps (OIS) curve gives us cleaner signals about investors' expectations for risk-free rates. So, where do we stand today? 1️⃣ The Fed has delivered a cutting cycle which has mildly exceeded expectations from a year ago 2️⃣ Yet, long-end rates today are 20+ bps higher than a year ago 3️⃣ This is because the market now prices in a higher ''neutral rate'' at around 4% Today's OIS curve (orange) sends a clear signal. The Fed is expected to cut rates to around 4%, and that's broadly considered to be the interest rate at which the US economy can operate delivering its potential growth - no overheating, no recession. This is why the yield curve is flat as a pancake, and the Fed is reinforcing this message via their ''long pause'' Fedspeak. Productivity seems to be slowly increasing, but on the other hand the US housing market is showing some early signs of distress - unsold houses are on the rise, and the construction sector stopped hiring. Do you think the new neutral rate in the US is 4%? P.S. If you enjoyed this post, follow me (Alfonso Peccatiello) to make sure you don't miss my daily dose of macro analysis.

Modeling something like time series goes past just throwing features in a model. In the world of time series data, each observation is associated with a specific time point, and part of our goal is to harness the power of temporal dependencies. Enter autoregression and lagging -  concepts that taps into the correlation between current and past observations to make forecasts.  At its core, autoregression involves modeling a time series as a function of its previous values. The current value relies on its historical counterparts. To dive a bit deeper, we use lagged values as features to predict the next data point. For instance, in a simple autoregressive model of order 1 (AR(1)), we predict the current value based on the previous value multiplied by a coefficient. The coefficient determines the impact of the past value on the present one only one time period previous. One popular approach that can be used in conjunction with autoregression is the ARIMA (AutoRegressive Integrated Moving Average) model. ARIMA is a powerful time series forecasting method that incorporates autoregression, differencing, and moving average components. It's particularly effective for data with trends and seasonality. ARIMA can be fine-tuned with parameters like the order of autoregression, differencing, and moving average to achieve accurate predictions. When I was building ARIMAs for econometric time series forecasting, in addition to autoregression where you're lagging the whole model, I was also taught to lag the individual economic variables. If I was building a model for energy consumption of residential homes, the number of housing permits each month would be a relevant variable. Although, if there’s a ton of housing permits given in January, you won’t see the actual effect of that until later when the houses are built and people are actually consuming energy! That variable needed to be lagged by several months. Another innovative strategy to enhance time series forecasting is the use of neural networks, particularly Recurrent Neural Networks (RNNs) or Long Short-Term Memory (LSTM) networks. RNNs and LSTMs are designed to handle sequential data like time series. They can learn complex patterns and long-term dependencies within the data, making them powerful tools for autoregressive forecasting. Neural networks are fed with past time steps as inputs to predict future values effectively. In addition to autoregression in neural networks, I also used lagging there too! When I built an hourly model to forecast electric energy consumption, I actually built 24 individual models, one for each hour, and each hour lagged on the previous one. The energy consumption and weather of the previous hour was very important in predicting what would happen in the next forecasting period. (this model was actually used for determining where they should shift electricity during peak load times). Happy forecasting!

OpenAI launched a new #AI model. I tested it for FP&A and #finance use cases. So OpenAI has released o1—a new series of AI models designed to spend more time thinking before they respond. What does this mean for Finance and FP&A? I have tested o1 for multiple finance and FP&A use cases and this is what I have found: ✅ o1 performs well on complex finance tasks like forecasting and risk modelling. ✅ It handled technical tasks such as machine learning-based revenue forecasting and Monte Carlo simulations effectively. ✅ For high-level strategic tasks, responses were often too general and needed refinement. ✅ The model is powerful for specific, technical use cases but lacks advanced tools like memory and file uploads. ✅ It is more suited for detailed, structured problem-solving than broad strategic ideation. You can download this guide below and I tested this 4 use cases: 1. Strategic Development I asked o1 to create a strategy for a SaaS company's FP&A department that could be improved using AI. 2. Multi-Factor Financial Risk Modelling for M&A I tasked it for support to develop a model for multi-factor financial risk modelling and scenario analysis for corporate M&A. 3. Forecasting Revenue with Machine Learning I asked o1 to create a new forecasting method for predicting revenue for a SaaS company using machine learning. 4. Monte Carlo Simulation for P&L Scenario Modeling It helped me to create a model using Monte Carlo simulations to run scenario modeling on different accounts of the P&L as a SaaS company. Have you used the new model? What do you think so far?

There will be considerable interest in Federal Reserve Chair Powell’s remarks today, particularly as they follow those of Governor Miran, whom some now call “the other chair” as he is viewed to represent the views of the likely successor to Powell, whoever that will be. In his commenta yesterday, Miran argued that the neutral rate is about 100 basis points below the median projection of Fed officials. As such, he argued that the Fed needs to cut rates meaningfully and quickly; otherwise, the risk is unnecessary and costly damage to both the labor market and the broader economy. Ahead of Chair Powell’s comments, St. Louis Fed President Mussalem—who last week appearto have been a reluctant rate cutter—presented a contrasting view, voicing concerns about inflation. As ever, this involves a view on whether tariff-related inflation pass-through may complicate the outlook. #economy #FederalReserve #inflation #growth #jobs #markets

The #Fed cut 50bps, but how low will they go? To us, this is the big question. Chair Powell stated that the neutral rate – the policy rate at which Fed policy is neither restraining not supporting economic activity – has likely moved higher since the pandemic. Whether that's the case, and how high a higher neutral rate would be, is hotly contested. The FOMC's median estimate of the neutral rate is 2.9%, but the range is 2.2 to 4.1%... a big difference. Our own best guess is that neutral is around 3%. This range of long-term interest rate estimates makes sense to us, because the path of #inflation from here depends heavily on what happens in November. A sweep in either direction is likely to bring higher spending, firmer inflation, and modestly higher #rates. It may also bring more interest rate volatility as investors grow wary of interest payments exceeding major U.S. government budget line items like defense. A split government allays these fears and points to modestly lower inflation and rates. 

📈 Building a Probabilistic Forecast with Linear Regression 👇🏼 There are many ways to model uncertainty in forecasting; a common method is using prediction intervals for a parametric model, such as linear regression, and conformal prediction when using both parametric and non-parametric models. While those methods are powerful, in some cases, you want to estimate the probability of hitting some value in the future. This is where probabilistic forecasting comes in 🎯. 🔍 What is a probabilistic forecast? A probabilistic forecast provides a distribution of possible future outcomes, not just a single predicted value. Instead of saying “sales next month will be 10,500,” we can say, for example, “there’s a 90% chance sales will fall between 9,800 and 11,300.” This gives decision-makers a clearer picture of risk and uncertainty 💡. 🛠️ How to build a probabilistic forecast with linear regression Even a simple linear regression can generate rich uncertainty estimates. Here’s the workflow: 1️⃣ Fit a linear regression model Start by estimating the relationship between your target variable and predictors. The model produces coefficients along with their standard errors. 2️⃣ Extract the coefficient distributions Under standard assumptions, regression coefficients follow approximately normal distributions centered at their estimated values. These distributions represent our uncertainty about the true parameter values. 3️⃣ Simulate coefficient draws Generate many random samples from each coefficient’s distribution. For each draw, compute a forecast using the simulated coefficients. This gives you an ensemble of possible forecast paths, each representing one plausible future scenario. 4️⃣ Aggregate the forecast distribution Once you have many simulated forecasts: ➡️ Take the median or mean as your point forecast ➡️ Use percentiles (e.g., 5th & 95th, or 10th & 90th) to form prediction intervals These intervals capture both parameter uncertainty and the variability inherent in the data. Probabilistic forecasting is essential for planning under uncertainty. And the beauty is—you don’t need a complex model. Even linear regression can provide meaningful, simulation-based uncertainty estimates that elevate the quality of your insights. The screenshot below shows an example of creating a probabilistic forecast for US monthly natural gas demand. #timeseries #forecasting #datascience

The minutes of the March 2026 #FOMC meeting show that “almost all” participants supported holding the fed funds rate at 3.50%–3.75%, judging policy to be broadly neutral and “well positioned” to assess the extent and timing of any further adjustments. Most viewed it as too early to evaluate the economic and inflationary impact of the #MiddleEast conflict and deemed it “prudent” to monitor developments before recalibrating policy. The tone of the minutes is distinctly hawkish. While “many” policymakers still see scope for further easing if #inflation evolves as expected, a couple pushed back the expected timing of rate cuts. Notably, “some” participants—more than the “several” in January—saw a strong case for a two-sided statement explicitly acknowledging that rate hikes could be appropriate if inflation remains above target. Read my latest here 👇 via EY-Parthenon EY

Macro Outlook for Capital Allocators to Consider in 2026 (Part 2 of 2): 1. Rates: Front-end lower, long-end anchored The front end of the curve should continue to rally as the Fed eases policy and maintains purchases of short-term Treasuries. By contrast, longer-dated yields are likely to remain sticky, with the 10-year UST fairly valued in a 4.00%–4.25% range. A meaningful break lower in long rates would likely require the onset of QE program or inflation to settle in at ~2%. A dove will the reins in May who will make it his mission to lower rates in an effort to allow the economy to heat up. Rate volatility should continue to compress: the MOVE index, which has oscillated between 60 and 150 over the past three years, is likely to trade below 50 later this year. I expect the Fed to cut rates three times in 2026, bringing the policy rate towards its neutral level of ~3.0%. The risk to this scenario is higher inflation. 2. Equities: High valuations, range-bound returns, subdued volatility The S&P 500 enters the year trading at roughly 23.5x forward earnings, an elevated multiple by historical standards. As a result, returns are likely to be more range-bound despite a constructive macro backdrop. I expect the S&P 500 to trade between 6,500 (-5%) and 7,666 (+12%) over the course of 2026. Equity volatility should remain well behaved, with the VIX largely confined to a 10–20 range, likely drifting lower in the coming months. This contrasts sharply with last year, when the VIX spent most of the time between 15–25, spiking above 50 in April. In this environment, volatility-selling strategies are likely to underperform. Earnings growth of ~10% is achievable given continued economic strength, but upside equity returns may be limited given starting valuation levels. The two major risks to this scenario are significantly slower GDP growth and/or AI bubble that bursts. Conclusion: 2026 is shaping up less as a year for directional conviction and more as a test of portfolio construction discipline. With policy easing expected to be incremental and asset prices already reflecting a benign macro regime, the opportunity set will increasingly favor credit selection over beta. Capital allocators should prepare for an environment where carry, relative value, and structural inefficiencies matter more than broad market exposure, and where resilience to policy or inflation surprises is as important as upside participation. If volatility continues to compress, the real edge may come from building portfolios that can compound steadily, while remaining positioned for regime shifts that markets may be underpricing today.

I loved this recent paper by James Morley & Benjamin Wong (2025). And it has provoked me a lot. My two favourite points are: "Therefore, from a small open economy perspective, shocks to global r* are essentially equivalent to shocks to U.S. r*" => This puts some serious intellectual machinery behind the laconic phrase 'when the US sneezes Australia gets the flu'. What happens to the US neutral rate is the main determinant of the Australian neutral rate. The paper makes the point that local policy can somewhat offset this, but its not super comforting reading frankly. Where the US neutral rate goes, the Australian neutral rate follows, for the most part. "Therefore, while local shocks can matter, the long-term decline in r-star appears to be due to global forces such as a lower US productivity growth and an increase in safe asset demand." => By demonstrating that heterogenous country effects are not the predominant drivers of changes in the neutral rate, the authors make an extremely persuasive point regarding the secular decline in the neutral rate. There is a common engine behind this broad based decline centred in trends in global capital accumulation (read US capital markets) and declining US productivity. +++ My speculation +++ The implications I draw from this paper makes for some vexing reading for any central bank. If the US is on the verge of a productivity improving capex boom, and is able to retain its centrality in capital markets, the neutral rate is only going up, and we will all be living with the consequences. At the same time to boost their domestic economy China must bring forward consumption and reduce their savings glut, while the US demands a capex boom they themselves could never finance. Conversely, if this will not work and the US is on the verge of a marked decline in productivity, if for whatever reason, it starts losing its centrality in capital markets, the neutral rate starts collapsing in the US at the exact moment r* in other geographies becomes unbound to US r* allowing it to rise. This basically ends up giving you a clear justification for major capital outflows from the US to other geographies that has nothing to do with "sovereign risk" perceptions and is just mechanical if you have capital mobility (for now...?). And when you stop to actually think about it, the "sovereign risk" argument for the US capital outflows doesn't make that much sense on the yield movements so far. Sovereign risk in the US is not a justification for capital outflows, as much as it is a justification for higher pricing of US assets to reflect the risk premium. But capital outflows driven by a new differential in r* now between the US and other geographies that wasn't there before makes more intuitive sense to me. And if this holds, if for whatever reason the Federal Reserve lowers rates now, that would only accelerate this process. Is it a multi-polar r* world we are now living in? #JustAnIdea #FederalReserve #r*