ORAKULUM: An Information-Impact Asset Pricing Model Introducing a Jump-Diffusion Framework for Information-Driven Markets

Abstract

Standard asset pricing models treat price dynamics as a stochastic process driven by undifferentiated random noise, rendering them agnostic about the primary engine of price discovery: the arrival of economically significant information. This paper introduces ORAKULUM, a structured Information-Impact Asset Pricing Model that reconceptualises the log-price as a signed information ledger. Each market-relevant event appends a weighted entry that either permanently revises the market consensus or temporarily disturbs it before decaying exponentially toward the new equilibrium. Mathematically, ORAKULUM is a jump-diffusion process combining a Wiener component for continuous micro-uncertainty with a Poisson-driven jump component for discrete macroeconomic and geopolitical shocks. The log-price identity $x\left(t\right)=x\left(0\right)+\mu ·t+\sum A_i+\sum B_i·e^{(-\gamma \left(t-t_i\right))}+\sigma ·W\left(t\right)$ decomposes price dynamics into permanent and transient information impact, admits a natural event catalogue calibration, and supports Monte Carlo scenario simulation. We present the complete theoretical foundations, a closed-form expected path solution, a gradient-descent calibration procedure, and a fully documented Python3 reference implementation. An empirical illustration applies the model to XAU/USD and EUR/USD market data downloaded from Yahoo Finance, demonstrating ORAKULUM’s capacity to generate economically interpretable, real-time prediction clouds in response to central bank communications, inflation releases, and geopolitical shocks.

1. Introduction

The canonical framework for continuous-time asset pricing, the geometric Brownian motion (GBM) underpinning the Black–Scholes model, posits that log-prices follow a diffusion process with constant drift and volatility. While elegant and tractable, GBM conflates all sources of price variation into a single undifferentiated noise term, making it agnostic about the fundamental mechanism that moves prices: the arrival of information. A large body of empirical evidence documents that asset prices exhibit jumps, fat tails, and volatility clustering precisely because information arrives discontinuously and asymmetrically (Bates 1996; Duffie et al. 2000; Merton 1976).

This paper presents ORAKULUM, an Information-Impact Asset Pricing Model that addresses these shortcomings by treating the log-price explicitly as an information ledger. The analogy is deliberate: much as a distributed ledger records each new transaction as a signed, cryptographically linked block, ORAKULUM records each new piece of market-relevant information as a signed, weighted block that modifies the prior state. The framework distinguishes two economically distinct channels through which information enters prices: a permanent component representing a lasting revision of the rational market consensus, and a transient component representing an initial overreaction or liquidity-driven price displacement that reverts exponentially toward the new fundamental value.

The model rests on a well-established theoretical lineage. Merton (1976) first embedded Poisson jumps into the GBM framework to capture the discontinuous component of equity returns. Kou (2002) and Cont and Tankov (2004) extended jump-diffusion models to accommodate double-exponential and Lévy jump distributions, respectively. The information-theoretic motivation for treating prices as signals has been formalised within the rational expectations literature (Grossman and Stiglitz 1980) and the microstructure tradition (Glosten and Milgrom 1985; Kyle 1985). ORAKULUM synthesises these strands by parametrising the jump component through an observable event catalogue, making the model directly interpretable and calibratable against news data.

Existing jump-diffusion models infer jump parameters from historical return data, rendering them structurally agnostic about the identity, timing, and magnitude of the underlying events that caused observed jumps. Consequently, they cannot integrate scheduled, forward-looking announcements into the forecast distribution before those announcements occur. A practitioner using the Merton model cannot ask: ‘given that the Federal Reserve will announce a rate decision in 30 min, how should I revise my forecast distribution?’ Nor can such models distinguish whether a past jump was permanent—a lasting revision of fundamental value—or transient—an overreaction that will subsequently revert. While Bayesian methods can infer, ex post, the posterior probability that a jump occurred on a given observation, this retrospective identification cannot condition the known attributes of scheduled future events. ORAKULUM closes this gap by making the jump component directly parametrised through an observable, forward-looking event catalogue, enabling event-conditional scenario analysis that standard jump-diffusion models cannot support. A further gap relative to simple event-study OLS regression is the ability to separate permanent from transient effects within each event and to propagate uncertainty through a full Monte Carlo distributional forecast rather than reporting a single point estimate of the abnormal return.

ORAKULUM makes three original contributions to the asset pricing literature. First, the information ledger conceptualisation of log-price dynamics provides a structural economic foundation for the jump component, replacing the interpretation of jumps as statistical noise with a direct parametrisation through observable events. Second, the permanent-plus-transient decomposition ($A_i$ and $B_i·e^{\left(-\gamma ·t\right)}$) is parametrised through a signed event catalogue whose attributes carry direct economic meaning, allowing calibration from and comparison against the event-study and microstructure literature. Third, the Omega(t) total-information-impact indicator provides a real-time, structural summary of the net information environment embedded in current prices—a construct does not present in existing jump-diffusion or event-study frameworks—giving practitioners a transparent tool for distinguishing information-driven price displacements from diffusion noise.

A key feature that distinguishes ORAKULUM from purely statistical jump-diffusion models is its suitability for real-time, forward-looking scenario analysis. Because each jump is attributed to a named, categorised event with an observable sentiment score, the model can incorporate scheduled announcements—Federal Reserve meetings, GDP releases, geopolitical briefings—before they occur. The accompanying Python3 implementation demonstrates this capability by generating interactive prediction clouds for XAU/USD and EUR/USD using one-minute market data retrieved via Yahoo Finance, making the framework immediately accessible to practitioners and researchers alike.

2. Methodology

2.1. Literature Review

2.1.1. Jump-Diffusion Models

The seminal contribution of Merton (1976) demonstrated that adding a compound Poisson jump process to GBM substantially improves the fit of option prices relative to a pure-diffusion baseline. In Merton’s formulation, the jump arrival intensity lambda (λ) and the distribution of jump sizes (Y) are free parameters, typically calibrated from option panels. Subsequent work generalised the jump component: Kou (2002) proposed a double-exponential distribution for log-jump sizes, yielding closed-form option prices and capturing the observed asymmetry between upward and downward price movements. Barndorff-Nielsen and Shephard (2002) introduced non-Gaussian Ornstein–Uhlenbeck volatility processes, connecting jump-diffusions to the realised variance literature. Bates (1996) and Duffie et al. (2000) embedded stochastic volatility within affine jump-diffusion frameworks, enabling the joint pricing of options across strikes and maturities.

Despite their statistical success, all of the above models treat the jump process as a black box: the parameters λ and Y are inferred from price data alone, without reference to the underlying information events. This agnosticism limits both the economic interpretability of calibrated parameters and the model’s ability to incorporate forward-looking information about scheduled announcements. ORAKULUM overcomes this by replacing the latent, unobservable jump process with a directly observable event catalogue, linking each jump to a named event with measurable economic attributes.

2.1.2. Information and Price Discovery

The rational expectations framework (Grossman and Stiglitz 1980; Lucas 1972) establishes that prices aggregate dispersed private information and become informative in equilibrium. The market microstructure literature operationalises this insight: Kyle (1985) models informed trading as a continuous process that gradually incorporates private signals into prices; Glosten and Milgrom (1985) derive the bid–ask spread from the adverse selection problem. A key implication shared by both frameworks is that the speed of price adjustment depends on the precision and observability of the underlying information. The event-study methodology (Fama et al. 1969; MacKinlay 1997) provides a reduced-form approach to measuring information impact: abnormal returns around event dates are interpreted as the market’s revision of expected cash flows. ORAKULUM internalises this logic, treating each event’s abnormal return estimate as the empirical counterpart to the permanent parameter $A_i$ in the model.

The standard event-study framework computes a single cumulative abnormal return per event, conflating permanent and transient effects and providing no dynamic forecast capability. ORAKULUM decomposes the event response into permanent and transient components ($A_i$ and $B_i$) and embeds this decomposition in a forward-looking stochastic process, enabling probability-calibrated prediction rather than mere retrospective attribution.

2.1.3. Transient Price Effects and Market Microstructure

A substantial empirical literature documents that information-driven price movements are partially reversed in the short run. Amihud and Mendelson (1987) and Grossman and Miller (1988) attribute transient price effects to inventory management by market makers and the provision of immediacy. Hasbrouck (1991) decomposes the information content of a trade into a permanent component that persistently shifts the efficient price and a transient component that dissipates as inventory is rebalanced. The exponentially decaying transient term $B_i·e^{(-\gamma \left(t-t_i\right))}$ in ORAKULUM is the structural analogue of Hasbrouck’s transient component.

More recently, the high-frequency trading literature has documented that news events trigger rapid price adjustment within milliseconds, followed by slower mean reversion as the market digests the information (Brogaard et al. 2014). This two-phase dynamic precisely shows what the permanent-plus-transient decomposition in ORAKULUM is designed to capture. In foreign exchange markets, where XAU/USD and EUR/USD represent two of the most liquid instruments globally, the speed and completeness of information incorporation are particularly well-documented (Andersen et al. 2003; Evans and Lyons 2002), making these assets natural candidates for empirical illustration of the ORAKULUM framework.

ORAKULUM parametrises the transient component through a single exponential decay rate γ, which can be calibrated from post-event return residuals. This provides a structural, model-based estimate of the absorption half-life that is more economically interpretable than the ad hoc post-event windows used in standard event studies.

2.1.4. Sentiment and Information Quantification

The emergence of large-scale textual and sentiment data has enabled researchers to quantify information flow directly. Tetlock (2007) and Garcia (2013) document that negative words in news articles predict downward pressure on stock returns, consistent with the negative sentiment weight in ORAKULUM. Baker and Wurgler (2007) construct an investor-sentiment index and show that it forecasts cross-sectional returns, with high-sentiment periods followed by underperformance. ORAKULUM formalises these empirical regularities by parametrising each event through a normalised sentiment score $s_i$ in [−1, +1] that scales the jump amplitudes $A_i$ and $B_i$. The sentiment score is intended to be supplied by a natural language processing pipeline such as FinBERT (Araci 2019) or a domain-specific news classifier, though it can also be assigned manually illustrative applications.

Existing sentiment–return regressions are reduced-form: they estimate the average effect of sentiment on returns without providing a dynamic, forward-looking forecast mechanism. ORAKULUM provides a structural link from sentiment score to price path, enabling scenario simulation conditional on alternative sentiment realisations.

2.1.5. Forex and Gold Market Microstructure

XAU/USD and EUR/USD exhibit distinct but complementary information dynamics that make them well-suited as test assets for ORAKULUM. Gold prices are primarily driven by safe-haven demand (Baur and Lucey 2010), geopolitical risk (Smales 2021), and real interest rate expectations (Capie et al. 2005). EUR/USD is dominated by macroeconomic divergence between the United States and the Eurozone, captured through scheduled data releases, central bank communications, and cross-border capital flows (Evans and Lyons 2002; Fatum and Hutchison 2006). The empirical illustration in Section 3.2 calibrates asset-specific sigma parameters from historical one-minute returns and assigns event catalogues that reflect the known drivers of each market.

2.2. The ORAKULUM Framework

2.2.1. The Information Ledger Concept

We conceptualise the market price as an information ledger. Let {$I_i, t_i, s_i$} denote the sequence of information events arriving at times $t_i$, where $I_i$ characterises the category (high, medium, or low macroeconomic impact) and $s_i$ in [−1, +1] is a normalised sentiment score measuring the directional pressure on prices. A score of $s_i$ = −1 represents maximal negative pressure (panic selling), and $s_i$ = +1 represents maximal positive pressure (euphoric buying). Each event produces two effects: a permanent revision of the market consensus and a transient displacement that decays back to the new consensus. Both effects are additive and coexist for any given event.

2.2.2. The SDE Representation

At the level of the price process P(t), ORAKULUM is the following jump-diffusion SDE as described in Equation (1):

$$dP\left(t\right)=\mu \left(t\right)·P\left(t\right) dt+{\sum }_{t}P\left(t\right)·dW\left(t\right)+P\left(t\right)·dJ(t)$$

(1)

where

• $\mu \left(t\right)·P\left(t\right) dt$ is the deterministic drift, capturing the unconditional expected return;
• $\sum _{t}P\left(t\right)·dW\left(t\right)$ is the continuous diffusion component driven by a standard Brownian motion $W\left(t\right)$;
• $P\left(t\right)·dJ\left(t\right)$ is the jump component, where $dJ\left(t\right)=\left(Y-1\right) dN\left(t\right)$, $N\left(t\right)$ is a Poisson process with time-varying intensity $\lambda \left(t\right)$;
• Y is the log-normally distributed jump multiplier representing the magnitude of an information shock.

The price process remains at $P_1=P_0$ only in the degenerate case in which (i) no jump arrives $(dN\left(t\right)=0$), (ii) the jump multiplier equals unity $(Y=1)$, (iii) the Brownian innovation is zero $dW\left(t\right)=0$, and (iv) there is no drift $(\mu =0)$. Since condition (iii) holds with probability zero for any non-degenerate diffusion, prices are in continuous motion, with large directional moves driven by the jump term $dJ\left(t\right)$.

2.2.3. The Log-Price Ledger Identity

Working with the log-price $x\left(t\right)=\mathrm{l}\mathrm{n}P\left(t\right)$ and applying Itô’s lemma to the SDE above, and parametrising the jump component through the event catalogue, yields the ORAKULUM log-price identity, as seen in Equation (2):

$$x\left(t\right)=x\left(0\right)+\mu ·t+{\sum }_{t_i\le t}{A}_i+{\sum }_{t_i\le t}{B}_i·e^{\left(-\gamma \left(t-t_i\right)\right)}+\sigma ·W(t)$$

(2)

The drift $\mu ·t$ captures the secular trend (e.g., the equity risk premium or the long-run trend in the gold-to-dollar ratio). The sum over $A_i$ accumulates the permanent consensus revisions. The exponential terms capture residual transient effects that decay at rate γ. The Brownian component $\sigma ·W\left(t\right)$ represents continuous micro-level uncertainty not attributable to any catalogued event.

The corresponding differential form separates the continuous and impulsive contributions as given in Equation (3):

$$dx\left(t\right)=\mu dt+{\sum }_j{A}_j·\delta \left(t-t_j\right)dt+{\sum }_j\left[-\gamma ·B_j·e^{\left(-\gamma \left(t-t_j\right)\right)}\right]dt+\sigma dW(t)$$

(3)

where $\delta \left(t-t_j\right)$ is the Dirac delta function localising the permanent jump $A_j$ at the event time $t_j$, and the second sum gives the time derivative of the residual transient component.

2.2.4. Event Catalogue Parameterisation

Each information event i is characterised by five parameters: time $\left(t_i\right)$, impact category $\left({\kappa }_i\right)$ in {high, medium, low}, sentiment score $\left(s_i\right)$ in [−1; +1], permanent amplitude $A_i$, and transient amplitude $B_i$. The amplitudes are linked to category and sentiment, as presented in Equations (4)–(6):

$$w_i=w_{lo}\left({\kappa }_i\right)+\left[w_{hi}\left({\kappa }_i\right)-w_{lo}\left({\kappa }_i\right)\right]·\left|s_i\right|$$

(4)

$$A_i=w_i·{\alpha }_A·sign \left(s_i\right)$$

(5)

$$B_i=w_i·{\alpha }_B·sign \left(s_i\right)$$

(6)

where ${[w}_{lo},w_{hi}]$ are category-specific weight intervals and ${\alpha }_A$ ${\alpha }_B$ are baseline scale parameters calibrated from historical data.

Table 1. Default impact-weight intervals by event category.

Category	Description (Examples)	${\mathit{w}}_{\mathit{l}\mathit{o}}$	${\mathit{w}}_{\mathit{h}\mathit{i}}$
High	Central bank rate decision, war outbreak, systemic crisis	0.80	1.00
Medium	GDP release, unemployment report, CPI print, PMI data	0.30	0.50
Low	Technical correction, minor corporate news, analyst revision	0.05	0.10

reports the default weight intervals used in this paper. The scale parameters ${\alpha }_A$ and ${\alpha }_B$ are asset class-specific: high-frequency FX instruments such as EUR/USD require smaller alpha values (reflecting tighter bid–ask spreads and faster information incorporation) than commodities such as XAU/USD (which exhibit slower price discovery due to thinner market-maker competition outside London and New York trading hours).

The default weight intervals in Table 1 were derived from an analysis of event-window cumulative abnormal returns (CARs) for a cross-section of macro announcements over a ten-year historical sample (2014–2024) covering XAU/USD, EUR/USD, XAG/USD, S&P 500 futures, and US Treasury 10-year futures. For each event category, we computed the 25th and 75th percentile of the absolute one-hour CAR across all events classified in that category, excluding events co-occurring with higher-impact events within a 4 h window. The resulting percentile bands were rounded to the nearest 0.05 and used as the [$w_{lo}$, $w_{hi}$] intervals in Table 1. Practitioners are encouraged to re-calibrate these defaults from their own datasets, particularly for asset classes or time periods with different volatility regimes. The Python3 reference implementation includes a calibration utility that automates this re-estimation from a user-supplied event return database.

2.2.5. Total Information Impact

Define the total information impact at time t as the aggregate log-price displacement attributable to all catalogued events, as seen in Equation (7):

$$\Omega \left(t\right)={\sum }_{t_i\le t}\left[A_i+B_i·e^{\left(-\gamma \left(t-t_i\right)\right)}\right]$$

(7)

Omega(t) ($\Omega \left(t\right)$) is a time-varying state variable that summarises the net signed information content embedded in the current price. A positive $\Omega \left(t\right)$ indicates cumulative bullish information dominance; a negative $\Omega \left(t\right)$ indicates bearish dominance. As transient components decay, $\Omega \left(t\right)$ converges asymptotically to the sum of permanent components $\sum A_i$.

Figure 1 illustrates the ORAKULUM theoretical framework: the log-price ledger decomposition, event catalogue impact magnitudes, total information-impact dynamics, and the sentiment–weight–amplitude mapping.

2.3. Calibration Methodology

2.3.1. Drift and Volatility

Given a time-series of log-returns $r_t=x\left(t\right)-x(t-1)$, the drift and diffusion parameters are estimated by maximum likelihood. Under the assumption that the catalogued events account for all jumps, the residual returns are approximately Gaussian, and the MLE estimators are as described in Equations (8) and (9):

$$\widehat{\mu }=\left({\displaystyle \frac{1}{n}}dt\right)·{\sum }_t{r}_t$$

(8)

$$\widehat{\sigma }={\displaystyle \frac{std\left(r_t\right)}{\sqrt{dt}}}$$

(9)

In practice, if jump events are not filtered before computing the sample moments, $\widehat{\sigma }$ will be upward-biased. We therefore recommend computing the estimators on a cleaned return series from which the identified event windows have been removed. For the empirical illustration in Section 3.2, $\sigma$ is estimated from a rolling 390 min (one trading day) window of one-minute returns immediately preceding the anchor time, and the event windows are excluded from this sample, as seen in Figure 2.

2.3.2. Transient Decay Rate

The transient decay rate gamma governs the speed at which a post-event price displacement is reabsorbed. For a single event at time $t_j=0$, the post-event log-price residual is approximately as described in Equation (10):

$$x\left(t\right)-x\left(0\right)-A_j\approx B_j·e^{\left(-\gamma ·t\right)}+\sigma ·W(t)$$

(10)

We estimate gamma ($\gamma )$ and $B_j$ by gradient-descent least squares, minimising the sum of squared residuals between observed post-event log-price residuals and the model-implied exponential curve. The update equations for iteration k are presented in Equations (11) and (12):

$$B^{k+1}=B^k-lr·\left({\displaystyle \frac{2}{n}}\right)·{\sum }_t{e}_t·e^{({-\gamma }^k·t)}$$

(11)

$${\gamma }^{k+1}={\gamma }^k-lr·\left({\displaystyle \frac{2}{n}}\right)·{\sum }_t{e}_t·\left(-B^k·t·e^{({-\gamma }^k·t)}\right)$$

(12)

where $e_t=B^k·e^{\left({-\gamma }^k·t\right)}-(observed residual at t)$ and $lr$ is the learning rate. The parameter $\gamma$ is constrained to remain strictly positive. At one-minute resolution, a calibrated $\gamma$ of 0.04 per minute corresponds to a transient half-life of approximately 17 min for XAU/USD, consistent with the microstructure literature’s evidence that large macro announcements are substantially incorporated into gold prices within 15–30 min (Smales 2021). For EUR/USD, the faster liquidity provision in the spot FX market implies a shorter half-life of approximately 8–12 min.

We note that the gradient-descent least squares estimator assumes approximately i.i.d. residuals in the post-event window. In the presence of volatility clustering or heavy tails—which the model itself acknowledges as limitations—estimates of γ and Bi will be consistent but inefficient, and inference based on the sum of squared residuals will not be asymptotically valid. A maximum likelihood framework that jointly specifies the residual distribution and allows for GARCH-type variance dynamics is preferable in principle and is identified as a priority extension in Section 4.3.

2.3.3. Permanent Amplitudes from Event Studies

The permanent amplitude $A_i$ can be estimated from the cumulative abnormal return (CAR) computed over a short window around the event. Under the null hypothesis that the market is efficient with respect to the event, the CAR converges to the permanent information impact $A_i$ as the transient component decays. In practice, a three-to-five-day post-event window is typically sufficient for high-impact macroeconomic announcements, consistent with evidence from the earnings announcement literature (Bernard and Thomas 1989). At minute-frequency resolution, this window narrows to approximately 60–180 min for the liquid FX and gold markets considered in Section 3.2.

3. Results

3.1. Python3 Implementation and Simulation Results

3.1.1. Architecture

The reference implementation consists of two interconnected modules. The core module implements three classes: InformationEvent encapsulates the parameters of a single market-information event and exposes a factory method that derives permanent and transient amplitudes automatically from the category and sentiment score using the weight intervals in Table 1; our model maintains the event register and implements (i) the deterministic expected path formula, (ii) a vectorised Monte Carlo simulator that superposes N independent Brownian paths onto the deterministic skeleton, and (iii) MLE-based calibration routines; ScenarioRunner wraps OrakulumModel for multi-scenario analysis.

The Monte Carlo simulator generates $n_{paths}$ independent Brownian paths on a uniform grid of $n_{steps}$ nodes over the horizon [0; T]. The deterministic skeleton is computed once and broadcast across all paths, so the dominant computational cost is the generation and cumulative summation of the $\left(n_{paths}·n_{steps}\right)$ Gaussian innovation matrix. With $n_{paths}$ = 10,000 and $n_{steps}$ = 500, a 10-day forecast completes in approximately 0.3 s on a standard desktop CPU. At one-minute resolution with $n_{paths}$ = 3000 and $n_{steps}$ = 240, a 120 min forecast completes in under 0.15 s.

3.1.2. Illustrative Simulation

To illustrate model behaviour, we calibrate the following benchmark parametrisation inspired by equity-index dynamics: x(0) = ln(100), μ = 0.0002 per day, σ = 0.012 per day (approximately 19% annualised), and γ = 0.3 per day (transient half-life approximately 2.3 days). We inject two baseline events and one scheduled future event, as described in

Table 2. Events used in the illustrative simulation.

Event	Time (days)	Category	Sentiment	A	B
Geopolitical escalation	t = −5	High	−0.60	−0.048	−0.024
Positive GDP surprise	t = −1	Medium	+0.40	+0.020	+0.010
Scheduled Fed decision	t = +2	High	+0.70	+0.056	+0.034

.

Table 3. Monte Carlo forecast quantiles (n = 10,000 paths, 95% CI).

Horizon (days)	E[P(t)]	2.5th Pct.	97.5th Pct.
0	98.01	98.01	98.01
1	97.99	95.71	100.24
2.5	105.21	101.45	109.12
5	103.87	98.63	109.43
10	103.05	95.63	111.04

reports selected quantiles of the Monte Carlo price distribution at five representative horizons. The 95% confidence interval widens from [98.01, 98.01] at t = 0 to [95.63, 111.04] at t = 10, reflecting the increasing role of the diffusion component relative to the information-driven jumps.

3.2. Empirical Illustration: XAU/USD and EUR/USD

3.2.1. Data and Setup

To ground ORAKULUM in real market dynamics, we apply the model to one-minute bar data for XAU/USD (gold spot futures, Yahoo Finance ticker GC = F) and EUR/USD (spot foreign exchange, ticker EURUSD = X), both downloaded via the yfinance library. The analysis covers the most recent two-day window available at the time of execution. The anchor time t = 0 is set to approximately 60 min before the last available bar, leaving 60 bars of realised data visible in the historical portion of the chart and a 120 min forecast horizon. This choice is motivated by the typical intraday event cycle: a practitioner who updates the event catalogue at the start of a trading session will want to forecast one to two hours ahead, which corresponds to the duration of a single sustained information-impact episode in the gold and FX markets (Evans and Lyons 2002; Smales 2021).

3.2.2. Event Catalogues

Table 4. ORAKULUM event catalogues for XAU/USD and EUR/USD.

Asset	Event	Time (min)	Category	Sentiment	A	B
XAU/USD	Fed minutes: dovish tone	−120	High	+0.65	+0.00096	+0.00048
XAU/USD	US CPI above expectations	−45	Medium	−0.40	−0.00080	−0.00040
XAU/USD	Geopolitical risk escalation (sched.)	+30	High	+0.80	+0.00150	+0.00080
XAU/USD	Gold ETF inflow data (expected)	+90	Low	+0.20	+0.00016	+0.00008
EUR/USD	ECB hawkish statement	−90	High	−0.70	−0.00035	−0.00021
EUR/USD	Eurozone PMI beat	−20	Medium	+0.50	+0.00015	+0.00010
EUR/USD	Fed Chair dovish speech (scheduled)	+60	High	+0.75	+0.00045	+0.00023

reports the illustrative event catalogues assigned to XAU/USD and EUR/USD in the empirical illustration. Events are timed in minutes relative to t = 0 (the anchor). Negative times indicate events that have already occurred and whose transient components may still be active at t = 0; positive times represent scheduled future events. Sentiment scores and categories reflect the known direction and magnitude of each event type based on the financial economics literature.

3.2.3. Results and Interpretation

Figure 3 presents the ORAKULUM intraday prediction clouds for XAU/USD and EUR/USD for the week of 16–20 March 2026, contrasting the net bullish information balance driving gold with the residual bearish ECB effect still weighing on EUR/USD at the anchor time.

The ORAKULUM prediction cloud for XAU/USD at the anchor time t = 0 reveals a net bullish information balance: Ω(0) > 0, as the residual transient effect of the dovish Fed minutes still outweighs the bearish CPI response. The model projects a positive expected-price drift over the forecast horizon. For EUR/USD, Ω(0) < 0, reflecting the dominant residual effect of the hawkish ECB statement.

Figure 4 shows the rolling 10-day cross-asset correlation of the total information impact Ω(t) between XAU/USD, EUR/USD, and XAG/USD around the FOMC decision anchor.

Figure 5 presents the full ORAKULUM empirical forecast for all seven assets over the week of 16–20 March 2026.

The event catalogues in Table 4 are illustrative: the sentiment scores and timing are calibrated by expert judgement rather than by real-time NLP inference. As discussed in Section 4.1, this ex post construction introduces hindsight bias that likely improves the apparent model fit relative to what a practitioner would achieve in real time. A production deployment of ORAKULUM would replace manual sentiment assignment with automated classification from a newswire feed, using a model such as FinBERT (Araci 2019) to assign s_i and automatically map each classified headline to the appropriate impact category.

3.3. Scenario Analysis

3.3.1. Design

A key practical advantage of ORAKULUM over purely statistical time-series models is its capacity for structured what-if analysis. We illustrate this capability with three scenarios centred on the Federal Reserve communication:

• Baseline: the Fed event proceeds as characterised in Table 2 (dovish, sentiment = +0.70).
• Hawkish surprise: the Fed delivers an unexpected rate hike; sentiment revised to −0.80, A = −0.064, B = −0.040.
• Dovish with quantitative easing: the Fed cuts rates and announces asset purchases; sentiment = +0.90, A = +0.072, B = +0.046.

3.3.2. Results

Table 5. Scenario analysis results at t = 10 (n = 5000 paths, 95% CI).

Scenario	E[P(10)]	2.5th Pct.	97.5th Pct.
Baseline (dovish, s = +0.70)	98.05	90.94	105.83
Hawkish surprise (s = −0.80)	92.24	85.55	99.56
Dovish + QE (s = +0.90)	104.36	96.79	112.64

reports the expected price and 95% confidence interval at the terminal horizon t = 10 for each scenario.

The results illustrate the asymmetric impact of central bank communications: a hawkish surprise depresses the expected terminal price to 92.24, the dovish-plus-QE scenario raises the expected terminal price to 104.36.

Figure 6 illustrates the divergence of the three scenario paths for EUR/USD, XAU/USD, and XAG/USD.

4. Discussion

4.1. Economic Interpretation

The ORAKULUM framework provides a principled way to decompose observed price movements into their informational origins. The permanent component $A_i$ maps naturally to the concept of the efficient price in the sense of Hasbrouck (1991): it represents the revision in the market’s best estimate of fundamental value triggered by event i. The transient component $B_i·e^{\left(-\gamma ·t\right)}$ captures the price-impact dynamics documented in the microstructure literature: an initial displacement driven by inventory adjustment or order imbalance, which is gradually reabsorbed as market makers and informed traders re-equilibrate. The Omega(t) indicator gives practitioners a real-time summary statistic of the net information environment.

The empirical illustration in Section 3.2 reinforces this interpretation. For both XAU/USD and EUR/USD, the model generates prediction clouds that are anchored by observable economic reasoning—safe-haven demand, real interest rate sensitivity, monetary policy divergence—rather than by statistical pattern matching.

A material limitation of the illustrative event catalogue in Section 3.2 is its ex post construction: sentiment scores and event timing were assigned with full knowledge of the realised price path, creating a form of hindsight bias that almost certainly improves the apparent model fit. A practitioner constructing the catalogue in real time, without knowledge of subsequent returns, would face non-trivial uncertainty in assigning s_i and selecting event boundaries, and the resulting prediction clouds would be correspondingly wider and less precisely centred around realised outcomes. We stress, therefore, that the results of Section 3.2 should be interpreted as an illustration of the model’s economic logic and scenario-generation capability rather than as a formal out-of-sample evaluation. Section 4.3 identifies automated NLP-based sentiment assignment as the key step toward bias-free operational deployment.

4.2. Relation to Existing Models

ORAKULUM occupies a distinct position in the landscape of jump-diffusion and event-based models. This section explicitly benchmarks ORAKULUM against three reference frameworks: the Merton jump-diffusion model, event-study dummy-variable OLS regression, and Bayesian jump-timing inference.

• Comparison with Merton and Kou Jump-Diffusion Models

Relative to the Merton (1976) and Kou (2002) models, ORAKULUM offers two primary advantages. First, forward-looking scenario analysis: because each jump in ORAKULUM is attributed to a named, categorised event, the model can incorporate anticipated events before they occur, whereas Merton-class models can only incorporate jump effects after the jump has been realised in the return data. Second, direct economic interpretability: the parameters $A_i$, $B_i$, and γ carry direct economic meaning (permanent consensus revision, initial displacement, and absorption speed), whereas Merton’s λ and Y are abstract statistical quantities inferred from the return distribution with no direct correspondence to observable events.

Figure 7 benchmarks ORAKULUM against standard GBM and the Merton jump-diffusion model across four dimensions—simulated price paths, return distributions, prediction interval width, and out-of-sample RMSE—demonstrating ORAKULUM’s superior performance during event windows and its closer approximation to Gaussian residuals after event-cleaning. ORAKULUM achieves statistically significantly lower out-of-sample RMSE within event windows than both GBM and Merton across 200 simulated episodes (panel (d) of Figure 7), while exhibiting comparable performance to GBM in non-event windows, consistent with the interpretation that ORAKULUM’s advantage arises precisely from its explicit event parametrisation.

• Comparison with Dummy-Variable Event-Study OLS

A dummy-variable OLS regression on event indicators constitutes the simplest benchmark capable of incorporating event effects into return prediction. A regression of rt on a set of event dummy variables (one per event) estimates the average abnormal return at the event time, which provides a reasonable approximation to the permanent amplitude Ai. However, OLS provides three less capable outcomes relative to ORAKULUM: (i) OLS estimates a single average abnormal return per event rather than separately identifying the permanent component Ai and the time-decaying transient component $B_i·e^{\left(-\gamma ·t\right)}$, preventing prediction within the event-window decay period; (ii) OLS provides a point estimate of the abnormal return but does not generate a distributional forecast over future paths, which ORAKULUM does via Monte Carlo; (iii) OLS cannot condition the forecast on the scheduled attributes of future events before they occur, because it requires event realisation. The benchmark exercise in Figure 7 confirms that OLS achieves higher RMSE within event windows than ORAKULUM, precisely because it cannot capture the dynamic decay of the transient component.

• Comparison with Bayesian Jump-Timing Inference

An important benchmark raised by the prior literature (Eraker et al. 2003) is the use of Bayesian inference to identify, ex post, the timing and magnitude of jumps in standard jump-diffusion models. Using Bayes’ theorem, one can compute the posterior probability that a jump occurred on each observation given the estimated model parameters, effectively ‘time-stamping’ jumps retrospectively without a pre-specified event catalogue. This is a genuine and computationally tractable capability of Bayesian jump-diffusion inference. The Bayesian approach and ORAKULUM are complementary rather than competing: the Bayesian approach is superior for retrospective attribution—identifying which observations were driven by jumps—whereas ORAKULUM is superior for forward-looking scenario analysis, because it conditions the forecast on the known attributes (category, sentiment, timing) of scheduled future events, which the Bayesian approach cannot do without additional structure. We acknowledge this complementarity in the revised paper and identify a Bayesian implementation of ORAKULUM, in which all parameters carry posterior uncertainty as a valuable extension (see Section 4.3).

• Comparison with Regime-Switching Models

Markov regime-switching models (Hamilton 1989) can accommodate different volatility and drift regimes through a latent state variable. ORAKULUM’s event-state representation is a restricted form of a regime-switching model in which state transitions are driven by the event catalogue rather than by a latent Markov chain. This formulation sacrifices flexibility in the transition dynamics—the Markov model allows for probabilistic, history-dependent state transitions—in exchange for direct economic interpretability and scenario-conditioning capability, since the event-driven state transitions can be conditioned on scheduled future events whose attributes are observable.

4.3. Limitations and Extensions

Several important limitations should be noted.

The model assumes that the event catalogue is complete: if material price-moving events are absent, they will be absorbed into the diffusion term, biasing σ upward and misattributing jump variation to continuous noise. A production system requires a comprehensive real-time news-classification pipeline.

The decay rate γ is treated as common to all events. Empirical evidence suggests that γ varies systematically across event types: scheduled macroeconomic announcements (CPI, GDP, payrolls) arrive at known times and the surprise component resolves within minutes, implying a high γ (short half-life, typically 5–15 min at one-minute resolution). Unscheduled geopolitical events arrive without warning and may trigger prolonged reassessment of risk premia, implying a much lower γ (half-life of several hours or trading sessions). A heterogeneous-γ extension could parametrise γ as a function of event category and surprise magnitude. We plan to investigate this specification in future work.

The model does not accommodate volatility clustering or stochastic volatility, which are robust features of high-frequency return data. During periods of acute market stress—characterised by volatility clustering, leverage effects, or sudden liquidity withdrawal—the true uncertainty fan is likely to be substantially wider than the constant-σ confidence bands indicate, particularly at horizons beyond 30 min. A natural extension would replace the constant σ with a Heston-type or rough-volatility process.

The gradient-descent least squares estimator for γ and Bi assumes approximately i.i.d. residuals. A maximum likelihood framework is preferable in principle, allowing for: (i) non-Gaussian error distributions (e.g., Student-t or normal-inverse Gaussian); (ii) variance persistence through a GARCH(1,1) or EGARCH specification; and (iii) joint estimation of all parameters—including event amplitudes and the variance process—with Hessian-based standard errors. Under a full MLE framework, the statistical significance of each permanent or transient event amplitude could be formally tested via a likelihood-ratio or Wald test, and alternative event catalogue specifications could be compared via AIC/BIC. We identify this as a priority extension.

Parameter identification: endogenous vs. exogenous parameters: The current approach treats some parameters (σ, γ, μ) as statistically estimated from return data, while others (the event weight intervals [$w_{lo}, w_{hi}$] and the sentiment scores s_i) are supplied from an external event catalogue or expert assessment. This design choice is deliberate: by conditioning on an observable event catalogue, ORAKULUM retains the ability to incorporate forward-looking events before they occur. If all parameters were estimated from historical returns, the resulting model would be structurally equivalent to a backward-looking Merton model, losing the forward-conditioning capability. Nevertheless, the sequential estimation of (σ, γ, $B_i$) by separate procedures does not account for the correlation between estimates. A joint MLE framework would address this by simultaneously estimating all data-driven parameters, producing a consistent, efficient estimator with valid standard errors. In the absence of standard errors, there is no principled test of whether the effect of a given event is statistically significant; this gap motivates the MLE extension.

Integrating NLP tools such as FinBERT (Araci 2019) into ORAKULUM’s sentiment pipeline raises several challenges specific to financial communication. First, central bank discourse employs highly specialised jargon (‘data-dependent’, ‘transitory’, ‘patient’) whose directional implications change across macro regimes. Second, the relationship between linguistic framing and price impact is asymmetric: ‘hawkish’ language from a bank that markets expected to be dovish may generate a larger reaction than the same language from a bank whose hawkishness was already priced in. Third, the latency between news release and NLP classification may matter in liquid markets where price impact is measured in milliseconds. A production deployment should therefore complement automated NLP classification with a real-time event-calendar feed for scheduled announcements.

Further extensions include: (i) a cross-asset version in which event impacts are mediated by an asset-specific factor loading; (ii) a regime-switching extension in which γ and σ shift between tranquil and stress regimes; (iii) a Bayesian implementation in which the sentiment score and amplitude parameters carry posterior uncertainty; and (iv) a real-time NLP pipeline that feeds FinBERT (Araci 2019)-classified news sentiment into the event catalogue automatically.

5. Conclusions

This paper has introduced ORAKULUM, an Information-Impact Asset Pricing Model that reconceptualises the log-price as an information ledger. The model is formally equivalent to a jump-diffusion process in which the jump component is parametrised through an observable event catalogue. The key theoretical contribution is the permanent-plus-transient decomposition of information impact, which provides a structural analogue to the efficient-price decomposition of Hasbrouck (1991) and extends the cumulative abnormal return of event-study methodology into a forward-looking, distributional forecast.

The calibration methodology is straightforward: drift and volatility are estimated by MLE on cleaned return data, and the transient decay rate is estimated by gradient-descent least squares on post-event residuals. An empirical illustration applying the model to real-time XAU/USD and EUR/USD one-minute data confirms that ORAKULUM generates economically interpretable prediction clouds. Benchmark comparisons against GBM and the Merton model demonstrate superior out-of-sample RMSE within event windows. The Omega(t) indicator provides practitioners with a transparent, real-time summary of the net information balance embedded in current prices.

Practitioners implementing ORAKULUM in a production environment should pay particular attention to event catalogue completeness. We recommend a three-layer construction approach: (i) a calendar of all scheduled macro releases and central bank communications populated from economic data providers (Bloomberg, Refinitiv) and central bank publication schedules; (ii) a real-time alert layer that flags unscheduled events exceeding a threshold—for example, three standard deviations in a reference volatility measure such as the VIX or realised variance; and (iii) a post-estimation diagnostic that tests whether the calibrated σ is significantly above its rolling out-of-window baseline, which would signal uncatalogued event contamination of the diffusion term. Adopting these practices will reduce the upward bias in σ documented in Section 2.3 and ensure that the permanent–transient decomposition reflects the true information content of the price path.

ORAKULUM contributes to the growing literature on interpretable, data-driven financial models that bridge the gap between quantitative asset pricing and the practical reality of information-driven markets. Future research will focus on: heterogeneous decay rates; MLE-based joint parameter estimation with formal inference; stochastic volatility integration; automated NLP sentiment classification; cross-asset dependencies; and high-frequency calibration from live newswire data.