Model
Definitions and closed forms
Klima Protocol: Fee-Based vs. Principal Model
A carbon-retirement intermediary buys kVCM tokens and burns them on behalf of clients. The intermediary can charge clients in one of two ways. A fee is a markup on the pass-through token cost; it carries no inventory and no price exposure. A principal price is a fixed USD quote per tonne, set at inception, and the intermediary absorbs the spot risk.
This note has three parts. It derives closed-form moments for both books. It splits the principal book into a zero-capital operating leg and a balance-sheet treasury. It then studies three ways to reshape the principal loss tail: pre-purchased inventory \((k, C_{\mathrm{basis}})\), syndication at fraction \(\beta\), and switching to fee mode above a threshold \(h\). The companion code checks every identity below.
Setup and notation
The intermediary retires \(\lambda\) tonnes per unit time over \([0, T]\). Each tonne needs \(P\) kVCM, sourced at spot \(S_t\). Demand is deterministic, so the only random driver is the spot price. We model \(S_t\) as geometric Brownian motion,
\[ dS_t = \mu S_t \, dt + \sigma S_t \, dW_t, \qquad S_0 = \pi_0 / P, \]
and write \(\pi_t := P S_t\) for the carbon price per tonne. Throughout, \(\Pi\) denotes a book’s terminal P&L in USD over \([0, T]\). The expectations \(\mathbb{E}\), \(\mathbb{P}\), \(\mathrm{Var}\), \(\mathrm{SD}\), and \(\mathrm{Cov}\) are taken under the GBM law above. We write \(\mathbf{1}\{A\}\) for the indicator of event \(A\) and \(a \wedge b := \min(a, b)\).
The table below lists the price-process symbols.
| Symbol | Meaning |
|---|---|
| \(S_t\) | kVCM spot (USD / kVCM) |
| \(S_0\) | spot at \(t = 0\); \(S_0 = \pi_0 / P\) |
| \(\pi_0\) | initial carbon price per tonne (USD) |
| \(\pi_t = P S_t\) | carbon price per tonne |
| \(\mu, \sigma\) | GBM drift and volatility (annualised) |
| \(W_t\) | driving Brownian motion |
| \(\mathcal{F}_t\) | natural filtration |
The next table lists the contract and demand symbols.
| Symbol | Meaning |
|---|---|
| \(P\) | kVCM per tonne (protocol constant) |
| \(\lambda\) | retirement flow, tonnes / unit time |
| \(T\) | horizon |
| \(N = \lambda T\) | total tonnes retired |
| \(f\) | fee rate |
| \(Q\) | fixed USD quote per tonne (principal) |
The treasury symbols describe a pre-purchased inventory of tokens.
| Symbol | Meaning |
|---|---|
| \(k\) | treasury token notional (kVCM) |
| \(C_{\mathrm{basis}}\) | sunk USD basis |
| \(\alpha = \min(1, k / (N P))\) | coverage fraction |
| \(\tau_{\mathrm{cov}} = \min(T, k / (P \lambda))\) | inventory-exhaustion time |
The syndication symbols describe a quota-share cession of the principal book to a counterparty.
| Symbol | Meaning |
|---|---|
| \(\beta \in [0, 1]\) | ceded fraction |
| \(\theta \ge 0\) | counterparty risk load |
| \(\pi_{\mathrm{syn}}\) | up-front premium (USD) |
The switching symbols describe the rule that flips the book between fee mode and principal mode.
| Symbol | Meaning |
|---|---|
| \(h \ge 1\) | threshold multiple |
| \(H := h S_0\) | absolute threshold |
| \(\mathbf{1}^{\mathrm{fee}}_t := \mathbf{1}\{S_t \ge H\}\) | fee-mode indicator |
| \(\tau := \inf\{t : S_t \ge H\} \wedge T\) | first-passage time |
| \(f_{\mathrm{post}}\) | fee rate in fee mode |
The last table lists the compensated Merton jump parameters, introduced in §Adding jumps.
| Symbol | Meaning |
|---|---|
| \(\lambda_J\) | Poisson intensity (/yr) |
| \(\mu_J\) | mean log-jump |
| \(\sigma_J\) | log-jump SD |
| \(\kappa = e^{\mu_J + \sigma_J^2 / 2} - 1\) | Merton compensation |
P&L figures are totals over \([0, T]\) in USD. Divide by \(N\) for a per-tonne reading.
We report risk as value-at-risk and expected shortfall at 95 % and 99 %,
\[ \mathrm{VaR}_p[\Pi] := -\inf\{ x : \mathbb{P}[\Pi \le x] \ge 1 - p \}, \qquad \mathrm{CVaR}_p[\Pi] := \mathbb{E}\!\left[ -\Pi \mid \Pi \le -\mathrm{VaR}_p[\Pi] \right], \]
together with a horizon-absolute Sharpe ratio \(\mathrm{Sharpe}[\Pi] := \mathbb{E}[\Pi] / \mathrm{SD}[\Pi]\). As a Monte Carlo cross-check, we report the z-score of the MC mean against the closed-form mean,
\[ z := (\mathbb{E}_{\mathrm{mc}}[\Pi] - \mathbb{E}_{\mathrm{cf}}[\Pi]) / \mathrm{stderr}. \]
We treat \(|z| \le 2\) as sampling noise and \(|z| > 3\) as suspect. Monte Carlo means carry \(\pm \mathrm{CI}_{95} = 1.96 \cdot \mathrm{SD} / \sqrt{n}\). Fee revenue is non-negative, so the fee book’s VaR and CVaR read as low-end revenue rather than loss, and its Sharpe does not depend on \(f\).
Every operating book below reduces, up to sign and scale, to the spot integral
\[ I_T := \int_0^T S_t \, dt, \]
for which Dufresne (2001) gives the moments
\[ \mathbb{E}[I_T] = S_0 \cdot \frac{e^{\mu T} - 1}{\mu} \quad (\to S_0 T \text{ as } \mu \to 0), \]
\[ \mathbb{E}[I_T^2] = \frac{2 S_0^2}{\mu + \sigma^2} \left[ \frac{e^{(2\mu + \sigma^2) T} - 1}{2\mu + \sigma^2} - \frac{e^{\mu T} - 1}{\mu} \right], \qquad \mathrm{Var}[I_T] = \mathbb{E}[I_T^2] - \mathbb{E}[I_T]^2. \]
\(I_T\) is not log-normal, so tail quantiles need Monte Carlo.
The fee book
The intermediary quotes each tonne at \((1 + f) \pi_t\), remits \(\pi_t\) to the spot market, and keeps \(f \pi_t\). Total revenue over the horizon is therefore the fee rate times the spot integral,
\[ R_{\mathrm{fee}} = \int_0^T f \pi_t \lambda \, dt = f P \lambda \cdot I_T, \]
and the first two moments follow from those of \(I_T\),
\[ \mathbb{E}[R_{\mathrm{fee}}] = f P \lambda \cdot \mathbb{E}[I_T], \qquad \mathrm{Var}[R_{\mathrm{fee}}] = (f P \lambda)^2 \cdot \mathrm{Var}[I_T]. \]
Revenue is non-negative almost surely. Variance is driven by \(\sigma\) alone. The Sharpe ratio is invariant in \(f\), since \(f\) scales the mean and the SD by the same factor.
The back-to-back book
The intermediary fixes the quote \(Q\) at inception and sources each tonne at spot. Terminal P&L is the fixed payout minus the stochastic sourcing cost,
\[ \Pi_{\mathrm{b2b}} = \int_0^T (Q - P S_t) \lambda \, dt = Q N - P \lambda \cdot I_T, \]
so the moments are
\[ \mathbb{E}[\Pi_{\mathrm{b2b}}] = Q N - P \lambda \cdot \mathbb{E}[I_T], \qquad \mathrm{Var}[\Pi_{\mathrm{b2b}}] = (P \lambda)^2 \cdot \mathrm{Var}[I_T]. \]
The two books share the same random driver, so their variances satisfy \(\mathrm{Var}[\Pi_{\mathrm{b2b}}] / \mathrm{Var}[R_{\mathrm{fee}}] = 1 / f^2\) exactly, and one Monte Carlo pass prices both. Upside is capped at \(Q N\), and downside is unbounded. The position is economically equivalent to shorting a continuous strip of forwards on kVCM struck at \(Q / P\).
The book’s \(S_t\)-delta follows from Itô,
\[ \frac{\partial \, \mathbb{E}[\Pi_{\mathrm{b2b}} - \Pi_{\mathrm{b2b}}(t) \mid \mathcal{F}_t]}{\partial S_t} = -P \lambda \cdot \frac{e^{\mu(T - t)} - 1}{\mu} \approx -P \lambda (T - t). \]
The fee book satisfies the same identity with the sign reversed and magnitude \(f P \lambda (T - t)\). The natural static hedge for the b2b book at time \(t\) is therefore \(P \lambda (T - t)\) tokens of spot kVCM, which is exactly the treasury schedule at \(k = N P\).
The active treasury
The treasury opens at \(t = 0\) with \(k\) tokens at basis \(C_{\mathrm{basis}}\), feeds retirement at spot, and marks any over-hedge leftover at the terminal spot \(S_T\). Two derived quantities organise the P&L. The inventory-exhaustion time \(\tau_{\mathrm{cov}} := \min(T, k / (P \lambda))\) is the date the inventory runs out. The leftover stack \(k_{\mathrm{left}} := \max(0, k - N P)\) is the tokens still in the treasury at \(T\) when the initial inventory exceeds demand. Terminal P&L is
\[ \Pi_{\mathrm{trea}} = P \lambda \int_0^{\tau_{\mathrm{cov}}} S_t \, dt + k_{\mathrm{left}} \cdot S_T - C_{\mathrm{basis}}, \qquad \Pi_{\mathrm{trea}}(0) = k \cdot S_0 - C_{\mathrm{basis}}. \]
When the treasury is under- or exactly hedged (\(k \le N P\)),
\[ \mathbb{E}[\Pi_{\mathrm{trea}}] = P \lambda \cdot S_0 \cdot \frac{e^{\mu \tau_{\mathrm{cov}}} - 1}{\mu} - C_{\mathrm{basis}}, \qquad \mathrm{Var}[\Pi_{\mathrm{trea}}] = (P \lambda)^2 \cdot \mathrm{Var}\!\left[\int_0^{\tau_{\mathrm{cov}}} S_t \, dt\right]. \]
When it is over-hedged (\(k > N P\)),
\[ \mathbb{E}[\Pi_{\mathrm{trea}}] = P \lambda \cdot \mathbb{E}[I_T] + k_{\mathrm{left}} \cdot S_0 \, e^{\mu T} - C_{\mathrm{basis}}, \]
\[ \mathrm{Var}[\Pi_{\mathrm{trea}}] = (P \lambda)^2 \mathrm{Var}[I_T] + k_{\mathrm{left}}^2 \, \mathrm{Var}[S_T] + 2 P \lambda \, k_{\mathrm{left}} \, \mathrm{Cov}[I_T, S_T], \]
with \(\mathrm{Var}[S_T] = S_0^2 e^{2\mu T} (e^{\sigma^2 T} - 1)\) and
\[ \mathrm{Cov}[I_T, S_T] = S_0^2 e^{\mu T} T \left( \frac{e^{(\mu + \sigma^2) T} - 1}{(\mu + \sigma^2) T} - \frac{e^{\mu T} - 1}{\mu T} \right), \]
derived from \(\mathbb{E}[S_t S_T] = S_0^2 e^{\mu(t + T) + \sigma^2 t}\) for \(t \le T\), integrated over \([0, T]\). The opening MTM \(k S_0 - C_{\mathrm{basis}}\) translates the distribution without reshaping it.
Strategies as operating plus treasury
Every desk in this note is an operating book plus a treasury,
\[ \Pi_{\mathrm{desk}} = \Pi_{\mathrm{op}} + \Pi_{\mathrm{trea}}. \]
The table below names the desks and the treasury parameters that recover each one. Each row pairs the operating book with the \((k, C_{\mathrm{basis}})\) that, summed with it, gives the named desk.
| Strategy | Operating | Treasury \((k, C_{\mathrm{basis}})\) |
|---|---|---|
| Fee-only | fee | \((0, 0)\) |
| B2b | b2b | \((0, 0)\) |
| Matched | b2b | \((N P, N P S_0)\) |
| Partial (\(\alpha\)) | b2b | \((\alpha N P, \alpha N P S_0)\) |
| Custom | b2b | \((k, C_{\mathrm{basis}})\) |
| Syndicated | retained | \((0, 0)\) |
| Syndicated-matched | retained | \((N P, N P S_0)\) |
| Switching | switching | \((0, 0)\) |
| Switching-partial (\(\alpha\)) | switching | \((\alpha N P, \alpha N P S_0)\) |
| Switching-matched | switching | \((N P, N P S_0)\) |
At \((k, C_{\mathrm{basis}}) = (N P, N P S_0)\) the \(I_T\) kernel cancels path by path between the b2b operating leg and the treasury consumption,
\[ \Pi_{\mathrm{matched}} = (Q N - P \lambda I_T) + (P \lambda I_T - N P S_0) = N (Q - P S_0). \]
The matched desk is therefore deterministic, and the identity holds to machine precision in test/models.test.ts.
Partial coverage sits between the naked and matched cases. Let \(J_\alpha := \int_{\alpha T}^T S_t \, dt\). Then
\[ \Pi_{\mathrm{partial}} = \Pi_{\mathrm{b2b}} + \Pi_{\mathrm{trea}} \bigl|_{(\alpha N P, \alpha N P S_0)} = Q N - \alpha N P S_0 - P \lambda \, J_\alpha. \]
By the strong Markov property at \(\alpha T\), \(J_\alpha = S_{\alpha T} \cdot Y\) with \(Y := \int_0^{(1-\alpha) T} S'_s \, ds\) an independent unit-start GBM integral, so
\[ \mathbb{E}[J_\alpha] = S_0 e^{\mu \alpha T} \cdot (1-\alpha) T \cdot \frac{e^{\mu (1-\alpha) T} - 1}{\mu (1-\alpha) T}, \]
\[ \mathrm{Var}[J_\alpha] = S_0^2 e^{(2 \mu + \sigma^2) \alpha T} \cdot \mathbb{E}[Y^2] - \bigl(S_0 e^{\mu \alpha T} \cdot \mathbb{E}[Y]\bigr)^2, \]
with \(\mathbb{E}[Y^2]\) given by the Dufresne identity at \(S_0 = 1\) and horizon \((1-\alpha) T\). Hence
\[ \mathbb{E}[\Pi_{\mathrm{partial}}] = Q N - \alpha N P S_0 - P \lambda \, \mathbb{E}[J_\alpha], \qquad \mathrm{Var}[\Pi_{\mathrm{partial}}] = (P \lambda)^2 \, \mathrm{Var}[J_\alpha]. \]
At \(\alpha = 0\), \(J_0 = I_T\) and the desk is the naked b2b book; at \(\alpha = 1\), \(J_1 = 0\) and the matched identity recovers. Between them, variance decays with the length of the uncovered window, not with \((1-\alpha)^2\).
Switching to fee above a threshold
The treasury and syndication rescale the loss tail. Switching cuts it. Whenever the spot sits above the threshold \(H = h S_0\), the book quotes at the fee-mode rate \(f_{\mathrm{post}}\) and accrues non-negative revenue on that sub-interval. The mode indicator tracks the spot symmetrically,
\[ M_t := \begin{cases} \text{fee} & S_t \ge H, \\ \text{b2b} & S_t < H, \end{cases} \qquad \mathbf{1}^{\mathrm{fee}}_t := \mathbf{1}\{S_t \ge H\}. \]
We split the \(I_T\) kernel and the occupation times by mode,
\[ I_{\mathrm{b2b}} := \int_0^T (1 - \mathbf{1}^{\mathrm{fee}}_t) S_t \, dt, \qquad I_{\mathrm{fee}} := \int_0^T \mathbf{1}^{\mathrm{fee}}_t S_t \, dt, \qquad I_{\mathrm{b2b}} + I_{\mathrm{fee}} = I_T, \]
\[ T_{\mathrm{b2b}} := \int_0^T (1 - \mathbf{1}^{\mathrm{fee}}_t) \, dt, \qquad T_{\mathrm{fee}} := T - T_{\mathrm{b2b}}. \]
The switching book’s P&L is
\[ \Pi_{\mathrm{sw}} = Q \lambda T_{\mathrm{b2b}} - P \lambda I_{\mathrm{b2b}} + f_{\mathrm{post}} P \lambda I_{\mathrm{fee}}, \qquad \Pi_{\mathrm{sw}}(0) = 0. \]
At the boundary, \(h \to \infty\) recovers \(\Pi_{\mathrm{b2b}}\), and \(h \le 1\) starts the book in fee mode.
With \(\nu := \mu - \tfrac12 \sigma^2\) and \(\log S_t \sim \mathcal{N}(\log S_0 + \nu t, \sigma^2 t)\),
\[ \mathbb{P}[S_t \ge H] = \Phi\!\left( \frac{\nu t - \log h}{\sigma \sqrt{t}} \right). \]
The lognormal partial-expectation identity at \((m, v^2) = (\nu t, \sigma^2 t)\) gives
\[ \mathbb{E}[S_t \mathbf{1}\{S_t \ge H\}] = S_0 e^{\mu t} \cdot \Phi\!\left( \frac{\mu t + \tfrac12 \sigma^2 t - \log h}{\sigma \sqrt{t}} \right), \]
so by Fubini
\[ \mathbb{E}[T_{\mathrm{fee}}] = \int_0^T \Phi\!\left( \frac{\nu t - \log h}{\sigma \sqrt{t}} \right) dt, \]
\[ \mathbb{E}[I_{\mathrm{fee}}] = S_0 \int_0^T e^{\mu t} \Phi\!\left( \frac{\mu t + \tfrac12 \sigma^2 t - \log h}{\sigma \sqrt{t}} \right) dt. \]
Both integrals are Simpson-tractable and are implemented as expectedTimeAboveBarrier and expectedIntegralAboveBarrier in src/core/moments.ts. The first-passage time \(\tau\) is a path property. Under pure GBM it follows the Harrison / Borodin-Salminen law,
\[ \mathbb{P}[\tau \le T] = \Phi\!\left(\frac{-\log h + \nu T}{\sigma \sqrt{T}}\right) + h^{2\nu / \sigma^2} \Phi\!\left(\frac{-\log h - \nu T}{\sigma \sqrt{T}}\right), \]
with \(\mathbb{E}[\tau \wedge T] = \int_0^T (1 - \mathbb{P}[\tau \le t]) \, dt\). In general \(T_{\mathrm{fee}} \ne T - \tau\), because the spot can re-enter after its first crossing. The two quantities \(\mathbb{E}[\tau \wedge T]\) and \(\mathbb{E}[T_{\mathrm{fee}}]\) are therefore independent. The switching P&L density has no closed form, so tail quantiles need Monte Carlo.
We can nonetheless bound the tail. Partitioning by whether the spot ever crosses the threshold, we split the paths into the two events \(\{\tau = T\}\) (no crossing) and \(\{\tau < T\}\) (at least one crossing),
\[ \mathrm{CVaR}_{95}[\Pi_{\mathrm{sw}}] \le \mathbb{P}[\tau = T] \cdot \mathrm{CVaR}_{95}^{\{\tau = T\}}[\Pi_{\mathrm{b2b}}] + \mathbb{P}[\tau < T] \cdot \mathrm{CVaR}_{95}^{\{\tau < T\}}[\Pi_{\mathrm{sw}}]. \]
On \(\{\tau = T\}\) the switching book coincides with the b2b book. On \(\{\tau < T\}\) every fee-mode sub-interval contributes non-negative revenue. Lowering \(h\) shrinks the first term and grows the second. Re-entries temper the cut, because paths that cross back below \(H\) resume b2b exposure. The simulator reports CVaR95|no-switch and CVaR95|switched separately. Sweeping \(h\) at fixed \((\mu, \sigma, f,
f_{\mathrm{post}})\) moves \(\mathbb{E}[\Pi_{\mathrm{sw}}]\) and \(\mathrm{CVaR}_{95}[\Pi_{\mathrm{sw}}]\) monotonically in opposite directions. Choosing \(h\) to minimise a given risk measure is a control problem and is not addressed here.
Side-by-side comparison
The table below collects the pure-GBM results for each operating book and the treasury, one book per column. Each column’s \(\Pi\) is that book’s terminal P&L, matching the earlier \(\Pi_{\mathrm{b2b}}\), \(\Pi_{\mathrm{ret}}\), \(\Pi_{\mathrm{sw}}\), \(\Pi_{\mathrm{trea}}\) (\(R_{\mathrm{fee}}\) for the fee column). The rows report, in order, the mean, the variance, the sign of the kVCM exposure, the downside shape, the capital required, and the counterparty exposure.
| Fee | B2b | Retained | Switching | Treasury \((k, C_{\mathrm{basis}})\) | |
|---|---|---|---|---|---|
| \(\mathbb{E}[\Pi]\) | \(f P \lambda \mathbb{E}[I_T]\) | \(Q N - P \lambda \mathbb{E}[I_T]\) | \((1 - \beta) \mathbb{E}[\Pi_{\mathrm{b2b}}] + \pi_{\mathrm{syn}}\) | MC; anchors on \(\mathbb{E}[T_{\mathrm{fee}}]\), \(\mathbb{E}[I_{\mathrm{fee}}]\) | \(P \lambda \mathbb{E}[I_{\tau_{\mathrm{cov}}}] + k_{\mathrm{left}} S_0 e^{\mu T} - C_{\mathrm{basis}}\) |
| \(\mathrm{Var}[\Pi]\) | \((f P \lambda)^2 \mathrm{Var}[I_T]\) | \((P \lambda)^2 \mathrm{Var}[I_T]\) | \((1 - \beta)^2 \mathrm{Var}[\Pi_{\mathrm{b2b}}]\) | MC | branch on \(k_{\mathrm{left}}\) |
| kVCM exposure | long | short | short, \(\times (1 - \beta)\) | short on \(\{S_t < H\}\), long on \(\{S_t \ge H\}\) | long on \([0, \tau_{\mathrm{cov}}]\), plus \(S_T\) on leftover |
| Downside | \(\ge 0\) | unbounded | \((1 - \beta) \times\) b2b | truncated on fee-mode intervals | \(-C_{\mathrm{basis}}\) if \(S \equiv 0\) |
| Capital | 0 | 0 | 0 | 0 | \(C_{\mathrm{basis}}\) |
| Counterparty | none | none | \(\beta \Pi_{\mathrm{b2b}}\) upside | as retained on the fee leg | none |
Reading one column gives the full profile of that book. A desk total is the row-wise sum of an operating column and the treasury column, as laid out in §Strategies as operating plus treasury.
Setting \(\mathbb{E}[R_{\mathrm{fee}}] = \mathbb{E}[\Pi_{\mathrm{b2b}}]\) solves for the break-even quote
\[ Q^* = (1 + f) P S_0 \cdot \frac{e^{\mu T} - 1}{\mu T}, \]
with \(Q^* \to (1 + f) P S_0\) as \(\mu \to 0\). A positive drift pushes \(Q^*\) above that level, and a negative drift pushes it below. At \(Q = Q^*\) the two books share \(I_T\) and therefore share variance, but they enter it with opposite signs. The fee book is bounded below. The b2b book carries a left-skewed loss tail. This asymmetry survives the moment equalisation at \(Q = Q^*\) and is precisely what the three dials of the principal book work on.
Adding jumps
Replace pure GBM with
\[ \frac{dS_t}{S_{t-}} = (\mu - \lambda_J \kappa) \, dt + \sigma \, dW_t + (J - 1) \, dN_t, \]
where \(N_t\) is Poisson(\(\lambda_J\)) independent of \(W_t\), \(J = e^Y\) and \(Y \sim N(\mu_J, \sigma_J^2)\) i.i.d. The compensation \(\kappa := \mathbb{E}[J - 1] = e^{\mu_J + \sigma_J^2/2} - 1\) removes the jump mean from the drift.
Means survive the overlay. The compound-Poisson identity gives \(\mathbb{E}[S_t] = S_0 e^{\mu t}\) for every \((\lambda_J, \mu_J, \sigma_J)\), so the mean of \(I_T\) is still the GBM expression,
\[ \mathbb{E}[I_T] = S_0 \cdot \frac{e^{\mu T} - 1}{\mu}. \]
Every mean-level identity above carries over unchanged, including \(\mathbb{E}[R_{\mathrm{fee}}]\), \(\mathbb{E}[\Pi_{\mathrm{b2b}}]\), \(\mathbb{E}[\Pi_{\mathrm{trea}}]\), \(\mathbb{E}[\Pi_{\mathrm{ret}}]\), the break-even quote \(Q^*\), and the matched-desk identity. Variances do not. Jumps inflate \(\mathrm{Var}[S_t]\), and with it \(\mathrm{Var}[I_T]\) and every downstream tail metric. The simulator reports the pure-GBM variance as a GBM anchor and leaves the jump-aware tails to Monte Carlo. The test test/jump-gbm.test.ts verifies both predictions.
Baselines this note assumes
The table below lists the simplifying assumptions this note makes and points to where the simulator relaxes each one.
| Baseline | Lifted in |
|---|---|
| Deterministic demand | Simulator: compound-Poisson order flow |
| GBM dynamics | Simulator: compensated Merton (above); regime switching pending |
| No calibration | Simulator: kVCM proxy |
| No discounting / gas / slippage | Simulator: parameterised |
| Passive treasury | Simulator: active consumption schedule; dynamic delta hedge pending |
| Static syndication / switching | Simulator: quota-share cession; symmetric threshold; optimal-\(h\) pending |
| No credit layer | Out of scope |
References
- Dufresne, D. (2001). The integral of geometric Brownian motion. Adv. Appl. Probab. 33(1), 223-241.
- Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering, §3.4.
- Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems.