You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


Q:

Why is it so rare for finance theory to depart from the normal distribution?

I understand almost all of the theory that has been built upon in quantitative finance is based on the normal distribution, and obviously you wouldn't want to throw all of it out the window on a whim but since stock returns are clearly not as normal as we like to believe why is it so uncommon to treat them as not being normal. For example, could you not as easily simulate a price process based on, say, a student T random distribution as well as a normal one? Looking at stock returns we often see that a student T distribution fits it better than a normal one so how come something like this is never done (at least I have never seen it done)?
Would is the biggest limitation that prevents us from going further with these ideas?

A:

The Geometric Random Walk: The Starting Point
Let me begin by being a little more specific. The simplest, yet relatively sound model of asset prices that we have is this one:
\begin{equation}
   ln S(t+1) = \mu - \Psi_{t+1}(-1) + ln S(t) + \epsilon(t+1), \; \epsilon(t+1) | F_t \sim N(0,\sigma^2).
\end{equation}
where $\Psi_{t+1}(u) := ln E_t \left( \exp( -u \epsilon_t) \right)$ is the log conditional MFG of the normal distribution and $F_t$ is the natural filtration of the process. You normally do not see the term $\Psi_{t+1}(-1)$ in a geometric random walk, but I appended this term because it ensures:
\begin{equation}
   E_t\left( \frac{S(t+1)}{S(t)} \right) := E_t(R(t+1)) = \mu.
\end{equation}
In essence, the log MFG is a convexity correction. Unsurprisingly, when $\epsilon(t+1) \sim N(0,\sigma^2)$, we have $\Psi_{t+1}(-1) = \sigma^2/2$. In continuous time, we have the equivalent to a geometric Brownian motion:
\begin{equation}
   \frac{dS(t)}{S(t^-)} = \mu dt + \sigma dW(t)
\end{equation}
where $(W(t))_{t \geq 0}$ is a standard Brownian motion under the physical measure. If you apply Ito's Lemma to move toward $dlnS(t)$ you will see that convexity correction term appear as well. This way, everything is treated the same way.
The nice properties:

Over any future interval of time, expected returns are just compounds of $\mu$. One thing we do know about financial markets is that first conditional moments are hard to estimate, so it's not stupid to just rule it out;
This ensures stock prices never become negative;
Prices, conditional on today, are log-normally distributed, so you have a bit of an heavy tail thing going on. However, it does say that returns are conditionally normally distributed.
The first two moments summarizes the normal distribution, so you can more or less equate risk with variance and, since it is subadditive, you immediately have an obvious advice: diversify.

That's probably why the first tight framework for the valuation of European options was constructed under a simple geometric Brownian motion (Black and Scholes, 1973). But people were quick to work on departures.

A Few Common Examples of Departures from Conditionally Normal Arithmic Returns
Heston (1993) proposed to model the dynamic of stock prices using a stochastic volatility model where the volatility followed an Ornstein-Uhlenbeck process where both Brownian motions were correlated. This model takes into account the fact that volatility estimates seem to exhibit temporal dependance and that they tend to be negatively correlated to returns (at least for stock market indexes). Note that, now, returns no longer are conditionally normal because they are build from a mixture of two normal distributions. Going forward, there is a window where the model will build sknewness and kurtosis.
Cool addition: As for the Black-Scholes-Merton world of geometric Brownian motion, Heston's model allows for a quasi-closed form European option pricing formula. In fact, all models which allows for an exponentially linear conditional MFG of log prices allow this as well.
Duan (1995) proposed a GARCH model for option prices. The model uses normal innovations and since for a GARCH model the conditional variance is known one step ahead, returns are conditionally normal only for one period ahead. Like the Heston (1993) model, this builds up conditional non-normality over time as past shocks enter the variance dynamic in a non-linear manner.
Bakshi, Cao and Chen (1997) have a nice (and famous) study where they compare the pricing and hedging performance of many option pricing models. They have time-varying conditional volatility like Heston, but they also look into adding jumps. When you add jumps in continuous time (or heavy tailed innovations as in the IG-GARCH model of Christoffersen, Heston and Jacobs (2006)), you have an immediate departure from conditional normality that can be quite severe, even in the very short run unlike in GARCH and SV models where it sorts of builds up over time.
As a sidenote with regards to option pricing, conditional nonnormality has consequences on risk-neutralization. Christoffersen, Elkamhi, Fenou and Jacobs (2010) showed that it is one way to force a variance risk premium at all horizons in a GARCH model (otherwise, the Q and P conditional expectations of variances only diverge over time) and a negative VRP is a pervasive empirical feature (it's one way to resolve a variance forecasting puzzle).

Conclusion
I barely scratched the surface for one very narrow part of financial literature and we have many examples of people departing from models that impose the conditional nonnormality of arithmetic returns. It's not absurdly complicated to talk about these things. If you know a bit of econometrics and a bare minimum of stochastic calculus, you can learn these things just by reading the relevant papers.