haptic-simulation.Rmd

---
title: "Haptic Protocol"
subtitle: "Agent-Based Monte-Carlo Simulation Study under a GBM Regime"
author: "0xAlbert ([0xalbertdefi@gmail.com](mailto:0xalbertdefi@gmail.com))"
date: "v0.1 — June 2026"
fontsize: 11pt
linkcolor: Indigo
citecolor: Teal
urlcolor: Teal
toccolor: Indigo
colorlinks: true
output:
  pdf_document:
    keep_tex: true
    number_sections: true
    toc: true
    toc_depth: 2
    fig_caption: true
    latex_engine: pdflatex
    includes:
      in_header: preamble.tex
abstract: |
  We stress-test the Haptic Protocol with a vectorized agent-based Monte-Carlo
  simulation. Geometric-Brownian-Motion price paths drive four coupled agent
  cohorts --- arbitrageurs, noise traders, option buyers and a compounding LP
  vault --- through repeated weekly option cycles in which the risk engine opens
  concentrated Uniswap-v4 bands, prices them with the realized-volatility-floor
  times utilization-markup pipeline, fills open interest to a demand
  equilibrium, and settles digitals from band composition. Across thousands of
  paths we find: (i) under full native collateralization the protocol is
  \textbf{provably solvent} (0 percent insolvency in every regime), confirming the
  collateralization argument; (ii) LP returns are \textbf{EV-positive but
  entirely premium-subsidized} --- with no volatility-risk-premium the vault
  bleeds about 15 percent per year to loss-versus-rebalancing; (iii) returns are
  \textbf{asymmetric in drift}, with Sharpe near 1.5 in up-trends falling to
  about 0.6 in down-trends; (iv) band width and over-subscription expose clean
  risk--return and capital-efficiency--solvency frontiers; (v) the volatility
  floor roughly halves tail loss (CVaR) in a jump regime; (vi) the
  depth-derived open-interest cap makes settlement manipulation break-even
  exactly at the cap and strictly unprofitable below it, while spot settlement is
  always exploitable; and (vii) under Student-t, jump-diffusion,
  stochastic-volatility and black-swan stresses, full-collateralization solvency
  is \textbf{path-independent} (0 percent insolvency in all of them) while
  leverage converts fat tails into frequent shortfalls. This is a research
  simulation, not a guarantee of production behavior.
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = FALSE, message = FALSE, warning = FALSE,
                      fig.align = "center", fig.pos = "H", out.width = "82%",
                      dev = "pdf", fig.width = 6.4, fig.height = 3.4)
options(kableExtra.latex.load_packages = FALSE, knitr.kable.NA = "")
library(ggplot2); library(kableExtra); library(scales)

pal <- list(primary = "#4C2A85", accent = "#0E9594", amber = "#B57614",
            risk = "#C2410C", stable = "#1D4ED8", gray = "#6B7280",
            light = "#F4F2FA", ink = "#1A1A2E")
theme_haptic <- function(base = 10) {
  theme_minimal(base_size = base) +
    theme(panel.grid.minor = element_blank(),
          panel.grid.major = element_line(color = "grey90", linewidth = 0.3),
          plot.title = element_text(face = "bold", color = pal$primary, size = base + 1),
          axis.title = element_text(color = pal$gray),
          legend.position = "top", legend.title = element_text(color = pal$gray, size = base - 1),
          legend.key.height = unit(3, "mm"), strip.text = element_text(color = pal$ink, face = "bold"))
}
R <- function(f) read.csv(file.path("results", f))
pc <- function(x, d = 1) sprintf(paste0("%.", d, "f\\%%"), 100 * x)  # LaTeX-safe percent
ktab <- function(df, cap, fs = 9, ...) {
  kbl(df, format = "latex", booktabs = TRUE, escape = FALSE, caption = cap,
      linesep = "", ...) |>
    kable_styling(latex_options = c("hold_position", "striped"),
                  stripe_color = "hapticLight", font_size = fs, full_width = FALSE)
}
```

# Methodology

## Design of the simulation

The simulation is a **vectorized agent-based Monte-Carlo**: arrays of $N$
parallel GBM price paths are advanced jointly, while the protocol logic and four
agent cohorts act on per-path state each weekly cycle. The price follows
$\mathrm{d}S = \mu S\,\mathrm{d}t + \sigma S\,\mathrm{d}W$ (optionally with a
Poisson jump term for stress), discretized hourly. The numeraire is the stable
asset; price $P$ is stable per risky.

```{r agents}
ag <- data.frame(
  Agent = c("Arbitrageurs", "Noise traders", "Option buyers", "LP vault"),
  Role = c("drag pool price onto the GBM reference each step",
           "uninformed two-sided flow",
           "value digitals under the \\emph{real} measure $(\\mu,\\sigma)$",
           "compounding capital deployed across bands"),
  Feedback = c("realize LP inventory P\\&L (LVR) + pay swap fees",
               "pure fee income for LPs",
               "engine prices at $\\mu=0$ $\\Rightarrow$ trend $\\to$ one-sided flow",
               "bear payouts + inventory; earn fees + premia"))
ktab(ag, "Agent cohorts and their feedback couplings.", fs = 8.6) |>
  column_spec(1, bold = TRUE, width = "2.4cm") |>
  column_spec(2, width = "6.0cm") |> column_spec(3, width = "5.4cm")
```

## Mechanism captured each cycle

Each cycle the engine opens call bands above spot (cash-or-nothing) and put bands
below (asset-or-nothing). For each band it (1) estimates trailing realized
volatility and applies the floor $\sigma=\max(\sigma_{\text{impl}},\sigma_{\min})$;
(2) prices the digital $\mathcal{N}(d_2)$ under $\mu=0$; (3) lets buyers --- whose
reservation is the real-measure fair value times $(1+\text{vrp})$ --- fill open
interest to the utilization equilibrium $\nu(u^\star)\,\Pi_{\text{fair}}(1+m)=R$;
(4) deploys collateral sized to the depth-limited capacity; (5) accrues swap fees
from arbitrage ($\propto\!\sum|\Delta\sqrt P|$ in range) and noise flow; and
(6) settles each digital from the band's realized composition. The key engine
parameters are in Table~\ref{tab:config}.

```{r config}
cfg <- R("config.csv")
keep <- c("sigma","n_paths","n_cycles","cycle_days","delta","sigma_floor",
          "markup_m","vol_risk_premium","fee_rate","depth_per_pct","twap_bleed",
          "nu_ustar","nu_umax","oversub")
lab <- c(sigma="volatility $\\sigma$ (annual)", n_paths="Monte-Carlo paths",
         n_cycles="weekly cycles", cycle_days="cycle tenor (days)",
         delta="band width $\\delta$", sigma_floor="vol floor $\\sigma_{\\min}$",
         markup_m="EV margin $m$", vol_risk_premium="vol risk premium (vrp)",
         fee_rate="swap fee", depth_per_pct="pool depth /1\\%",
         twap_bleed="TWAP manip.\\ bleed", nu_ustar="target util.\\ $u^{*}$",
         nu_umax="hard util.\\ cap $u_{\\max}$", oversub="over-subscription")
cf <- cfg[match(keep, cfg$param), ]
cf$param <- lab[cf$param]
names(cf) <- c("Parameter", "Baseline value")
ktab(cf, "Baseline configuration of the risk engine.", fs = 8.6) |>
  column_spec(1, width = "6.2cm")
```

# Baseline LP performance

Under the baseline (flat $\mu=0$, $\sigma=0.6$, full collateralization) the vault
compounds over 26 weekly cycles. Figure~\ref{fig:fan} shows the vault-value fan
and Figure~\ref{fig:dist} the terminal-return distribution across drift regimes;
headline statistics are in Table~\ref{tab:headline}.

```{r headline}
h <- R("headline.csv"); hv <- setNames(h$value, h$metric)
tab <- data.frame(
  Metric = c("Mean annualized return", "Median 6-mo return", "Return volatility (6-mo)",
             "Median Sharpe", "5\\% VaR (6-mo)", "5\\% CVaR (6-mo)",
             "Prob(loss)", "Insolvency rate", "Mean utilization"),
  Value = c(pc(hv["mean_ann_ret"]), pc(hv["median_total_ret"]),
            pc(hv["std_total_ret"]), sprintf("%.2f", hv["sharpe"]),
            pc(hv["var5"]), pc(hv["cvar5"]),
            pc(hv["prob_loss"], 0), pc(hv["insolvency_rate"], 0),
            sprintf("%.2f", hv["mean_util"])))
ktab(tab, "Baseline LP statistics (flat market, full collateralization, 6000 paths).", fs = 9) |>
  column_spec(1, width = "6.0cm")
```

```{r fig-fan, fig.cap="\\label{fig:fan}Vault-value fan over 26 weekly cycles (flat market). Bands are the 5--95\\% and 25--75\\% path percentiles; line is the median. The vault compounds steadily with a contained left tail.", fig.height=3.0}
fan <- subset(R("vault_fan.csv"), regime == "flat")
ggplot(fan, aes(cycle)) +
  geom_ribbon(aes(ymin = p5, ymax = p95), fill = pal$primary, alpha = 0.12) +
  geom_ribbon(aes(ymin = p25, ymax = p75), fill = pal$primary, alpha = 0.22) +
  geom_line(aes(y = p50), color = pal$primary, linewidth = 1.1) +
  geom_hline(yintercept = 1, linetype = "dashed", color = pal$gray, linewidth = 0.3) +
  labs(x = "cycle (week)", y = "vault value / initial") + theme_haptic()
```

```{r fig-dist, fig.cap="\\label{fig:dist}Distribution of 6-month LP total return by drift regime. All three are right-of-zero on average; the bear regime carries the fattest left tail.", fig.height=3.0}
d <- R("pnl_dist.csv"); d$regime <- factor(d$regime, levels = c("bear","flat","bull"))
ggplot(d, aes(total_ret, fill = regime, color = regime)) +
  geom_density(alpha = 0.22, linewidth = 0.7) +
  geom_vline(xintercept = 0, linetype = "dashed", color = pal$gray, linewidth = 0.3) +
  scale_x_continuous(labels = percent) +
  scale_fill_manual(values = c(bear = pal$risk, flat = pal$primary, bull = pal$accent)) +
  scale_color_manual(values = c(bear = pal$risk, flat = pal$primary, bull = pal$accent)) +
  labs(x = "6-month LP total return", y = "density", fill = NULL, color = NULL) +
  theme_haptic()
```

## Where the P&L comes from

Figure~\ref{fig:decomp} decomposes LP P\&L. Premia minus payouts is the option
edge; it is positive in every regime and is the dominant driver. Fees are minor
(concentrated bands see little volume relative to their notional), and inventory
(loss-versus-rebalancing plus directional exposure) is a persistent drag that the
premium must --- and does --- overcome.

```{r fig-decomp, fig.cap="\\label{fig:decomp}P\\&L decomposition as a fraction of initial vault. Payouts shown negative. The net (lp\\_pnl) is positive in all regimes; the option edge (premia $-$ payouts) carries it against the inventory drag.", fig.height=3.2}
co <- R("components.csv")
co$regime <- factor(co$regime, levels = c("bear (mu=-0.6)","flat (mu=0)","bull (mu=+0.6)"))
co$signed <- ifelse(co$component == "payouts", -co$value, co$value)
co <- subset(co, component %in% c("premia","fees","inventory","payouts","lp_pnl"))
co$component <- factor(co$component, levels = c("premia","fees","inventory","payouts","lp_pnl"),
                       labels = c("premia","fees","inventory","payouts (−)","net P&L"))
cols <- c("premia" = pal$accent, "fees" = pal$stable, "inventory" = pal$amber,
          "payouts (−)" = pal$risk, "net P&L" = pal$primary)
ggplot(co, aes(component, signed, fill = component)) +
  geom_col(width = 0.7) +
  geom_hline(yintercept = 0, color = pal$gray, linewidth = 0.3) +
  facet_wrap(~regime) + scale_fill_manual(values = cols, guide = "none") +
  scale_y_continuous(labels = percent) +
  labs(x = NULL, y = "share of initial vault") +
  theme_haptic() + theme(axis.text.x = element_text(angle = 35, hjust = 1, size = 7))
```

# Drift regime sweep

Figure~\ref{fig:drift} sweeps the annual drift. The protocol is EV-positive
across the whole range, but the profile is **asymmetric**: in down-trends the
risky-funded call bands lose inventory value and put payouts rise, widening the
left tail (VaR), whereas strong up-trends are the most profitable. This is the
short-volatility / inventory signature flagged in the design.

```{r fig-drift, fig.cap="\\label{fig:drift}Annualized return and 5\\% VaR vs.\\ annual drift $\\mu$. Returns stay positive throughout, but downside risk (VaR) deteriorates sharply in bear regimes.", fig.height=3.0}
ds <- R("drift_sweep.csv")
dl <- rbind(data.frame(mu = ds$mu, val = ds$mean_ann_ret, series = "annualized return"),
            data.frame(mu = ds$mu, val = ds$var5, series = "5% VaR (6-mo)"))
ggplot(dl, aes(mu, val, color = series)) +
  geom_hline(yintercept = 0, linetype = "dashed", color = pal$gray, linewidth = 0.3) +
  geom_line(linewidth = 1.1) + geom_point(size = 1.5) +
  scale_color_manual(values = c("annualized return" = pal$accent, "5% VaR (6-mo)" = pal$risk)) +
  scale_y_continuous(labels = percent) +
  labs(x = expression(paste("annual drift  ", mu)), y = NULL, color = NULL) +
  theme_haptic()
```

# Demand drivers: the premium subsidy is everything

Figure~\ref{fig:demand} isolates the two demand drivers. As the vol-risk-premium
rises, utilization and the LP edge climb together --- and **at vrp $=0$ the vault
is loss-making** ($\approx-15\%$/yr), earning nothing but eating LVR. This is the
quantitative statement of the subsidy thesis: Haptic is only EV-positive because
option buyers persistently overpay relative to realized volatility. Against raw
volatility the relationship is hump-shaped: moderate vol is ideal, while very
high vol overwhelms the fixed premium markup with payouts and inventory loss.

```{r fig-demand, fig.cap="\\label{fig:demand}Left: utilization, LP edge and return vs.\\ the vol-risk-premium --- profitability vanishes without it. Right: return is hump-shaped in volatility (selling vol is unprofitable when realized vol is very high).", fig.width=7.0, fig.height=3.0}
ut <- R("utilization.csv")
uv <- subset(ut, axis == "vrp")
us <- subset(ut, axis == "sigma")
ul <- rbind(
  data.frame(x = uv$x, val = uv$util, series = "utilization", panel = "vs vol-risk-premium"),
  data.frame(x = uv$x, val = uv$ann_ret, series = "ann. return", panel = "vs vol-risk-premium"),
  data.frame(x = uv$x, val = uv$lp_edge, series = "LP edge", panel = "vs vol-risk-premium"),
  data.frame(x = us$x, val = us$util, series = "utilization", panel = "vs volatility"),
  data.frame(x = us$x, val = us$ann_ret, series = "ann. return", panel = "vs volatility"),
  data.frame(x = us$x, val = us$lp_edge, series = "LP edge", panel = "vs volatility"))
ggplot(ul, aes(x, val, color = series)) +
  geom_hline(yintercept = 0, linetype = "dashed", color = pal$gray, linewidth = 0.3) +
  geom_line(linewidth = 1) + geom_point(size = 1.2) +
  facet_wrap(~panel, scales = "free_x") +
  scale_color_manual(values = c("utilization" = pal$primary, "ann. return" = pal$accent, "LP edge" = pal$amber)) +
  scale_y_continuous(labels = percent) +
  labs(x = NULL, y = NULL, color = NULL) + theme_haptic()
```

# Band width: a risk--return frontier

Tightening the band lowers risk and return together; widening it raises both ---
but only up to a point. Figure~\ref{fig:bw} shows the frontier bending back: at
very wide bands ($\delta\!=\!8\%$) inventory risk grows faster than premium income
and the product is no longer meaningfully "binary". The capacity (manipulation
cap) grows linearly in $\delta$, so width also trades against per-contract
sharpness. A width of 1--3\% sits in the efficient region.

```{r fig-bw, fig.cap="\\label{fig:bw}Risk--return frontier over band width $\\delta$ (labelled). Return rises with width then bends back as inventory risk dominates; the dashed marker is the 5\\% VaR scale.", fig.height=3.1}
bw <- R("bandwidth_sweep.csv")
bw$lab <- paste0(bw$delta * 100, "%")
ggplot(bw, aes(std_total_ret, mean_ann_ret)) +
  geom_path(color = pal$gray, linewidth = 0.5, linetype = "dotted") +
  geom_point(aes(color = delta), size = 3) +
  geom_text(aes(label = lab), vjust = -1, size = 2.8, color = pal$ink) +
  scale_color_gradient(low = pal$accent, high = pal$risk, guide = "none") +
  scale_x_continuous(labels = percent) + scale_y_continuous(labels = percent) +
  labs(x = "return volatility (6-mo)", y = "annualized return") +
  theme_haptic()
```

# Capital efficiency vs. solvency

Over-subscription (selling more OI than the deployed collateral fully backs)
raises returns but breaks the native-collateralization guarantee.
Figure~\ref{fig:oversub} shows the trade-off cleanly: at full collateralization
(over-subscription $=1.0$) insolvency is **exactly zero** and stays zero, exactly
as the composition argument predicts; every step of leverage buys yield with a
steep rise in the probability of an unpayable shortfall.

```{r fig-oversub, fig.cap="\\label{fig:oversub}Annualized return and insolvency rate vs.\\ over-subscription. Full collateralization (1.0) is provably solvent; leverage trades solvency for yield.", fig.height=3.0}
os <- R("oversub_sweep.csv")
ol <- rbind(data.frame(x = os$oversub, val = os$mean_ann_ret, series = "annualized return"),
            data.frame(x = os$oversub, val = os$insolvency_rate, series = "insolvency rate"))
ggplot(ol, aes(x, val, color = series)) +
  geom_vline(xintercept = 1, linetype = "dotted", color = pal$accent, linewidth = 0.5) +
  geom_line(linewidth = 1.1) + geom_point(size = 1.5) +
  scale_color_manual(values = c("annualized return" = pal$accent, "insolvency rate" = pal$risk)) +
  scale_y_continuous(labels = percent) +
  labs(x = "over-subscription (collateral leverage)", y = NULL, color = NULL) +
  theme_haptic()
```

# The volatility floor

Figure~\ref{fig:floor} tests the floor across regimes. Its value concentrates
exactly where intended: in a **calm market punctuated by jumps**, where trailing
realized volatility under-prices tail risk, the floor roughly halves the 5\% CVaR.
In a normal-volatility regime where realized vol already exceeds the floor it is
inert (and in a purely calm market it slightly suppresses demand --- a real cost).

```{r fig-floor, fig.cap="\\label{fig:floor}5\\% CVaR (6-mo) with the volatility floor off vs.\\ on, by regime. The floor materially cuts the tail in the calm-plus-jumps regime; it is inert when realized vol already exceeds it.", fig.height=3.0}
vf <- R("volfloor.csv")
vf$floor <- factor(vf$floor, levels = c("off", "on"))
ggplot(vf, aes(regime, cvar5, fill = floor)) +
  geom_col(position = position_dodge(width = 0.7), width = 0.62) +
  scale_fill_manual(values = c(off = pal$risk, on = pal$accent)) +
  scale_y_continuous(labels = percent) +
  labs(x = NULL, y = "5% CVaR (6-mo)", fill = "vol floor") +
  theme_haptic() + theme(axis.text.x = element_text(size = 8))
```

# Settlement manipulation and the OI cap

Finally we validate the security invariant directly. For a band of width $\delta$
on a pool of depth $D$, the protocol sets the open-interest ceiling equal to the
cost of moving \emph{and holding} the price across the band for the settlement
window, $\text{OI}_{\max}=C_{\text{manip}}$. Figure~\ref{fig:manip} plots the
attacker's profit (flipped payoff minus cost) against open interest relative to
that cap: under TWAP settlement profit is **negative below the cap and exactly
zero at it**, for every band width --- the cap is tight. Under atomic/spot
settlement the same attack is profitable everywhere (costs are fee-only), which
is why Haptic never settles on spot.

```{r fig-manip, fig.cap="\\label{fig:manip}Attacker profit vs.\\ open interest relative to the manipulation cap, by band width, under TWAP settlement. Profit crosses zero precisely at OI/cap${}=1$ (dotted). Under spot settlement (not shown) profit is positive for all OI --- spot is unsafe.", fig.height=3.1}
mp <- R("manipulation.csv"); mp$lab <- paste0(mp$delta * 100, "%")
ggplot(mp, aes(oi_over_cap, profit_twap_m, color = factor(lab, levels = unique(lab)))) +
  annotate("rect", xmin = 0, xmax = 1, ymin = -Inf, ymax = Inf, fill = pal$accent, alpha = 0.06) +
  geom_hline(yintercept = 0, color = pal$gray, linewidth = 0.3) +
  geom_vline(xintercept = 1, linetype = "dotted", color = pal$risk, linewidth = 0.5) +
  geom_line(linewidth = 0.9) + geom_point(size = 1.3) +
  annotate("text", x = 0.5, y = max(mp$profit_twap_m) * 0.8, label = "safe", color = pal$accent, size = 3) +
  scale_color_manual(values = c(pal$accent, pal$stable, pal$primary, pal$amber, pal$risk)) +
  labs(x = "open interest / manipulation cap", y = "attacker profit ($M)", color = expression(delta)) +
  theme_haptic()
```

# Heavy-tail stress

GBM understates exactly the risk a short-volatility book is most exposed to: the
sudden left tail. We re-run the protocol under four fat-tailed price processes ---
**Student-t** innovations, **Merton jump-diffusion**, **Heston stochastic
volatility** with leverage ($\rho=-0.7$), and a deterministic **black-swan**
($-40\%$ at mid-horizon) --- at full collateralization and under $1.5\times$
leverage.

The headline result is structural (Figure~\ref{fig:htsolv}, Table~\ref{tab:ht-table}):

\begin{keybox}[Native collateralization is path-independent]
Across \emph{every} heavy-tail scenario, insolvency at full collateralization is
\textbf{exactly zero}. Because each band can only ever owe what its own
composition holds, no price path --- however fat-tailed or discontinuous --- can
make the protocol unable to pay. Heavy tails widen \emph{LP losses} (which are
bounded by deployed capital), not protocol solvency. Leverage is the only thing
that converts a fat tail into insolvency.
\end{keybox}

```{r ht-setup, include=FALSE}
ht <- R("heavytail.csv")
lev <- c("Gaussian GBM","Student-t (df=5)","Student-t (df=3)","Merton jumps",
         "Severe jumps","Stochastic vol","Stoch vol + jumps","Black-swan -40%")
ht$scenario <- factor(ht$scenario, levels = lev)
ht1 <- subset(ht, oversub_x == 1.0)
ht$ov <- factor(ht$oversub_x, levels = c(1.0, 1.5),
                labels = c("1.0 (full collateral)", "1.5 (levered)"))
```

```{r fig-htdist, fig.cap="\\label{fig:htdist}LP 6-month return distribution under fat-tailed processes (full collateralization). Tails fatten sharply relative to Gaussian; the black-swan and stoch-vol+jumps regimes carry the deepest losses.", fig.height=3.0}
hd <- R("heavytail_dist.csv")
sel <- c("Gaussian GBM","Student-t (df=3)","Severe jumps","Stoch vol + jumps","Black-swan -40%")
hd <- subset(hd, scenario %in% sel); hd$scenario <- factor(hd$scenario, levels = sel)
ggplot(hd, aes(total_ret, color = scenario)) +
  geom_vline(xintercept = 0, linetype = "dashed", color = pal$gray, linewidth = 0.3) +
  geom_density(linewidth = 0.8) +
  scale_x_continuous(labels = percent, limits = c(-0.9, 0.9)) +
  scale_color_manual(values = c(pal$gray, pal$primary, pal$amber, pal$accent, pal$risk)) +
  labs(x = "6-month LP total return", y = "density", color = NULL) +
  theme_haptic() + theme(legend.text = element_text(size = 7))
```

```{r fig-httail, fig.cap="\\label{fig:httail}Tail loss by scenario at full collateralization: 5\\% CVaR (bars) and worst observed path (points). Fat tails roughly double the worst-case LP drawdown versus Gaussian.", fig.height=3.0}
ggplot(ht1, aes(scenario, cvar5)) +
  geom_col(fill = pal$risk, alpha = 0.85, width = 0.7) +
  geom_point(aes(y = worst_ret), color = pal$ink, size = 1.8) +
  geom_text(aes(y = worst_ret, label = "worst"), vjust = 1.6, size = 2.2, color = pal$ink) +
  scale_y_continuous(labels = percent) +
  labs(x = NULL, y = "6-mo loss (CVaR5 = bar, worst = point)") +
  theme_haptic() + theme(axis.text.x = element_text(angle = 35, hjust = 1, size = 7))
```

```{r fig-htsolv, fig.cap="\\label{fig:htsolv}Insolvency rate by scenario: full collateralization (teal) vs.\\ 1.5x leverage (red). Full collateralization is zero everywhere --- solvency is path-independent --- while leverage turns fat tails into frequent shortfalls.", fig.height=3.0}
ggplot(ht, aes(scenario, insolvency_rate, fill = ov)) +
  geom_col(position = position_dodge(width = 0.72), width = 0.64) +
  scale_fill_manual(values = c("1.0 (full collateral)" = pal$accent, "1.5 (levered)" = pal$risk)) +
  scale_y_continuous(labels = percent) +
  labs(x = NULL, y = "insolvency rate", fill = "over-subscription") +
  theme_haptic() + theme(axis.text.x = element_text(angle = 35, hjust = 1, size = 7))
```

```{r ht-table}
tt <- data.frame(
  Scenario = gsub("%", "\\\\%", as.character(ht1$scenario)),
  `Ann return` = pc(ht1$mean_ann_ret),
  `CVaR 5pct` = pc(ht1$cvar5),
  `Worst path` = pc(ht1$worst_ret),
  `Insolv 1.0x` = pc(ht1$insolvency_rate, 0),
  `Insolv 1.5x` = pc(ht[match(paste(ht1$scenario, 1.5),
                              paste(ht$scenario, ht$oversub_x)), "insolvency_rate"], 0),
  check.names = FALSE)
ktab(tt, "Heavy-tail tail-risk summary (mu=0, full collateralization unless noted; last column at 1.5x leverage).", fs = 8.4) |>
  column_spec(1, width = "3.0cm")
```

Two further observations. First, mean return is a \emph{misleading} headline under
fat tails: several heavy-tail regimes post higher average returns than Gaussian
even as their worst-case losses double --- the classic short-volatility trap, and
the reason CVaR and worst-path are the metrics that matter. Second, the
stochastic-volatility regimes are where the volatility floor earns its keep
(\S7), since realized volatility swings far below its long-run level between
bursts.

# Limitations

\begin{warnbox}[Simulation caveats]
\begin{itemize}
\item \textbf{Cohort agents, not individuals.} Arbitrage, noise and buyer demand
are modelled as representative cohorts with calibrated parameters, not thousands
of distinct strategic actors; the vol-risk-premium and demand elasticity are
exogenous inputs, not emergent.
\item \textbf{Heavy tails are probed, not exhausted.} \S9 adds Student-t, jump,
stochastic-volatility and black-swan stresses, but the parameters are illustrative
rather than calibrated to a specific asset's tail; volatility-of-volatility and
jump intensities are exogenous choices.
\item \textbf{Idealized settlement \& fees.} Each band's settlement reads an
end-of-cycle price (TWAP idealized as $S_T$); fee capture assumes the band is the
relevant liquidity at its price; gas, MEV and discrete tick spacing are abstracted.
\item \textbf{No active delta hedge.} The vault is naturally balanced across call
and put bands but is not actively delta-hedged; an explicit hedge would compress
the bear-regime tail at some bleed cost.
\end{itemize}
\end{warnbox}

# Conclusion

The simulation supports the core claims of the design and sharpens the caveats.
Native collateralization makes the protocol solvent by construction (0\%
insolvency at full backing, across every regime), and the deterministic
boundary-composition settlement carries the payouts. LP returns are
EV-positive with a credible risk profile (Sharpe of roughly 1.0--1.5), but they are
**entirely a harvested volatility-risk-premium** --- without persistent buyer
overpayment the vault loses to loss-versus-rebalancing. The principal residual
risks are exactly those identified analytically: a fat left tail in sustained
down-trends (an inventory/short-vol signature), capital-efficiency leverage that
trades away the solvency guarantee, and settlement integrity, which the
depth-derived OI cap is shown to enforce tightly under TWAP and not at all under
spot.

\vspace{1em}
\noindent\textit{Disclaimer: a research simulation, unaudited, not an offer or
recommendation of any financial instrument. Past simulated performance does not
indicate future results.}