## Overview

Because the outcome of the proposed strategy is dependent on SSV’s price trajectory, which is largely uncontrollable, a numerical analysis was performed to help evaluate its resulting risk and efficiency.

A Monte Carlo simulation of the strategy’s execution was evaluated over a large number of random outcomes (skewed to be slightly pessimistic for caution), and statistics were computed for the total quantity of SSV minted and the low points of the DAO's treasury balance.

Note that this simulation is simply a “tool” used to guide the design, not to predict any precise outcome. Rather, the purpose was to exhaustively evaluate the outcomes over a large number of widely-varying possible conditions, with the goal of minimizing the danger of pessimistic cases, while balancing the secondary goal of maximizing efficiency over all outcomes collectively.

## Treasury Simulation

The treasury balance was simulated using a given schedule of expenditures (defined by the proposed budget) and the execution steps of the proposed strategy. Since the strategy depends on the SSV price at any given time, the price is also calculated using a simulation, which is nested within the treasury simulation and described below.

Note that some liberties were taken to simulate the flexible elements of the proposed strategy, primarily around when the DAO will choose to spend USDC from its reserve instead of SSV for expenses that may be paid in either. In reality, this decision would be considered carefully by the DAO based on many variables and perspectives. But to perform the analysis here, an extremely simplified policy is implemented to efficiently utilize the treasury, where the USDC reserve is used to pay expenses during times of very poor market conditions.

**This is only used as a placeholder for the purpose of analysis. It is not binding, will not be implemented in practice, and is not a strategy that is being recommended.** With that said, the following simple rules were used for leveraging the USDC reserve in the simulation:
* If SSV price < $15 and USDC balance > $5M, then pay 100% in USDC. Otherwise pay 100% SSV.
* If price is between $15-25 and USDC balance > $10M, then pay 50%/50% SSV/USDC. Otherwise pay 100% SSV.
* If price > $25, pay 100% SSV.

## SSV Price Simulation

To model SSV prices within the simulation, Geometric Brownian motion (GBM) can be used. GBM is used to model asset prices in the Black–Scholes model and can be thought of as the standard choice for such applications.

It is governed by a stochastic differential equation (SDE):
\begin{equation}
\tag{1}
dS_t = \mu S_t dt + \sigma S_t dW_t
\end{equation}

where,
\begin{aligned}
& \bullet S_t: \text{ SSV price at time } t \\
& \bullet \mu: \text{ drift coefficient (mean rate of return)} \\
& \bullet \sigma: \text{ volatility (i.e. standard deviation)} \\
& \bullet dW_t: \text{ Wiener process (i.e. Brownian motion)}
\end{aligned}

If used in this form, the result will be unnatural— samples will drift to ludicrously high prices ($10k+), have unrealistic movement, etc. The basic GBM model was modified to increase realism here, where **four additional features were added:**

### 1) Mean Reversion

The first is *mean reversion*, which causes the price to slightly drift toward a *mean function*. The mean function should be set as an overall long-term prediction of SSV's price movement. This could be as simple as a linear gain over some period of time, or something complex and stochastic.

This change to the GBM model has been named an [Ornstein-Uhlenbeck Process](https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) and alters the governing equations to:

\begin{equation}
\tag{2}
dS_t = \theta (\mu_t - S_t) dt + \sigma dW_t
\end{equation}

where,
\begin{aligned}
& \bullet \mu_t: \text{Mean reversion value at time } t \\
& \bullet \sigma: \text{volatility (i.e. standard deviation)} \\
& \bullet \theta: \text{Speed of mean reversion for the SSV price to mean value} \\
& \bullet dW_t^1: \text{ Wiener process (i.e. Brownian motion) for SSV price}
\end{aligned}

For additional accuracy and realism, a Ornstein-Uhlenbeck Process is nested inside itself to represent the mean function, which gives randomness to underlying price trends. Therefore, the mean function is governed by the SDE:

\begin{equation}
\tag{3}
d\mu_t = \alpha (m - \mu_t) dt + \xi dW_t^3
\end{equation}

\begin{aligned}
& \bullet \alpha: \text{Speed of mean reversion for the mean value} \\
& \bullet m: \text{Ultimate mean reversion value that the mean drifts towards} \\
& \bullet \xi: \text{Volatility (i.e. standard deviation) of the mean reversion value} \\
& \bullet dW_t^3: \text{ Wiener process (i.e. Brownian motion) for mean reversion}
\end{aligned}

### 2) Stochastic Volatility

The second feature is stochastic volatility, which can be thought of as "adding randomness to the randomness" to better approximate real markets. The common choice for doing so is the [Heston Model](https://en.wikipedia.org/wiki/Heston_model) and changes the volatility parameter, $\sigma$, to be dynamic and vary around a user-selected mean variance value, $\theta_v$, using a similar mean reversion method to what was used previously. This introduces the following governing equation for $\sigma$:

\begin{equation}
\tag{4}
d\sigma_t^2 = \kappa (\theta_v - \sigma_t^2) dt + \eta \sqrt{\sigma_t^2} dW_t^2
\end{equation}

where,

\begin{aligned}
& \bullet \sigma_t: \text{Volatility (i.e. standard deviation) of SSV price at time $t$} \\
& \bullet \kappa: \text{Rate of mean reversion for volatility} \\
& \bullet \theta_v: \text{Long-term mean volatility (variance) value for mean reversion} \\
& \bullet \eta: \text{Volatility of volatility} \\
& \bullet dW_t^2: \text{ Wiener process (i.e. Brownian motion) for volatility}
\end{aligned}

### 3) Jump Diffusion

The third feature is [Jump Diffusion](https://en.wikipedia.org/wiki/Jump_diffusion), which adds the ability for the model to make sudden jumps, as they occur in real markets. The *Merton Model* is used, which adds the following term to the governing equations:

\begin{equation}
\tag{5}
dS_t = S_t J_t dN_t
\end{equation}

where,

\begin{aligned}
& \bullet J_t: \text{Jump size at time $t$, sampled from log-normal distribution} \\
& \bullet N_t: \text{Poisson process governing jump occurrence} \\
\end{aligned}

such that,

\begin{equation}
\tag{6}
J_t \sim \exp(\mu_{\text{jump}} + \sigma_{\text{jump}} Z_t) - 1
\end{equation}

\begin{equation}
\tag{7}
N_t \sim \text{Poisson}(\lambda_{\text{jump}} dt)
\end{equation}

and,

\begin{aligned}
& \bullet \mu_{\text{jump}}: \text{Mean of jump size} \\
& \bullet \sigma_{\text{jump}}: \text{Standard deviation of jump size} \\
& \bullet Z_t: \text{Standard normal random variable} \\
\end{aligned}

### 4) Absorbing Boundary at Zero

The fourth feature is an *absorbing boundary condition* at zero, which implements the assumption that the price will never rebound if it reaches zero.

\begin{equation}
\tag{8}
S_t = 0 \quad \text{if} \quad S_t = 0 \quad \text{at any prior time}
\end{equation}

### Combined Model

Combining the mean reversion, stochastic volatility, jump diffusion, and absorbing boundary features into a single modified GBM model of stochastic differential equations gives:

\begin{equation}
\tag{9}
dS_t = \theta (\mu_t - S_t) dt + \sigma_t S_t dW_t^1 + J_t S_t dN_t
\end{equation}

\begin{equation}
\tag{10}
d\sigma_t^2 = \kappa (\theta_v - \sigma_t^2) dt + \eta \sqrt{\sigma_t^2} dW_t^2
\end{equation}

\begin{equation}
\tag{11}
d\mu_t = \alpha (m - \mu_t) dt + \xi dW_t^3
\end{equation}

\begin{equation}
\tag{12}
J_t \sim \exp(\mu_{\text{jump}} + \sigma_{\text{jump}} Z_t) - 1
\end{equation}

\begin{equation}
\tag{13}
N_t \sim \text{Poisson}(\lambda_{\text{jump}} dt)
\end{equation}

\begin{equation}
\tag{14}
S_t = 0 \quad \text{if} \quad S_t = 0 \quad \text{at any prior time}
\end{equation}

### SSV Price Simulation Results

Note that this simulation aims to be an estimation of not just what is predicted, but also of the "unknown unknowns". Therefore, this simulation is attempting to capture "long-tail" outcomes like macro-level worldwide catastrophes and spectacular failures and successes of SSV.

The following is a simulation of SSV price over 4 years for 10,000 sampled outcomes. The results are represented as "probability distributions", which represent the relative probabilities of each possible outcome.

These are illustrated as histograms of the SSV price at time = 90 days, 1 year, 2 years, and 4 years. Each bar of the plots show the number of simulated samples that have the corresponding SSV prices at the given times. The vertical dotted lines denote the 5th, 50th (median), and 95th percentiles, which are meant to represent the "extremely optimistic", "baseline", and "extremely pessimistic" outcomes. Also remember that all of this analysis is skewed to be slightly pessimistic, for the sake of caution.


Additionally, to illustrate how individual price simulations look, here are 3 randomly-selected example price trajectories from the SSV price simulation:

## Treasury Simulation Results

### Quantifying the Total Amount of SSV Minted
To evaluate the total amount of SSV minted over this time period, the proposed strategy was simulated over a large number of random samples, recording the amount of SSV minted for each. The result is a "probability distribution" of values, which represents the relative probabilities of each possible outcome.

This is illustrated as a histogram in Figure 1 below, where each bar of the plot shows the number of simulated samples that have minted the corresponding amount of SSV. The vertical lines represent the 5th, 50th, and 95th percentiles, which can be thought of as “extremely optimistic”, “base”, and “extremely pessimistic” outcomes. These evaluate to 1.51 million, 1.85 million, and 2.85 million SSV minted, respectively, or in terms of average token inflation rate, 2.6%, 3.2%, and 4.9% inflation per year (not considering the supply reduction from burning Network Fees).

Although this result might seem too uncertain or imprecise, understand that this is simply the mathematical consequence of being dependent on an extremely volatile and unpredictable market. Because this uncertainty cannot be escaped, this analysis is important to estimate realistic bounds of outcomes and ensure preparedness for a possible future of unfavorable market conditions.

<div style="text-align: center;">
<img src="https://i.imgur.com/XGHlYeD.png" alt="Plot Image" width="600">
<p><em>Figure 1: Probability distribution of total SSV minted from the proposed plan, simulated using 10,000 sampled market trajectories.</em></p>
</div>

Note that there is a small “long tail” of the distribution (i.e., the far right side of the histogram) that represents very unlikely cases of bad outcomes, where large amounts of SSV would be minted. But given that this simulation includes extremely pessimistic market trajectories (e.g., sustained <$10 SSV), the likelihood of these worst-case outcomes was determined to be sufficiently unlikely. In the event of a true catastrophe that would create such an unfortunate scenario, there are emergency actions that can be taken, such as reducing expenditures.

One of the main uses for this simulation and analysis was to modify the plan's design to attempt to minimize the right side of this distribution (the negative outcomes from the pessimistic cases), thereby minimizing the risk of unforeseen catastrophic outcomes.

### Ensuring Adequate Cash Flow

It's also important to confirm that the liquidity of the DAO's treasury will always be sufficient to pay expenses and cover liabilities. In other words, it must be ensured that the treasury will never be depleted to dangerously low levels, especially in the event of sudden market downturns.

This can be confirmed by examining the trajectories of the DAO treasury balances and recording the lowest point of the treasury's value for each sample of the simulation. These low points are plotted as a histogram in Figure 2 below.

Note that every sample has the same value (the initial treasury value), which indicates that the treasury value only increases over time and never falls below the initial value. This is the desired outcome, as it shows that the DAO will always have sufficient funding in its treasury to cover its liabilities.

<div style="text-align: center;">
<img src="https://i.imgur.com/zSASkns.png" alt="Plot Image" width="600">
<p><em>Figure 2: Probability distribution of the minimum treasury value at any time when executing the proposed plan, simulated using 10,000 sampled market trajectories. Note that samples where SSV prices reaches 0 are omitted, which is why the number of samples plotted is fewer than 10,000.</em></p>
</div>