Vol. I, Ch. 8seed 26108
DCT Laboratory — Volume I, Chapter 8
Stochastic Enterprise Dynamics
Seed 26108 · Companion to the chapter and AXIOM Module AXIOM-08
The fiction is dropped: the environment channel is random, trajectories become
fans, and the state at is a distribution. Three instruments here: the
enterprise diffusion (GBM) with its analytic moments, the flaw of averages
made visible (mean path ≠ median path; ),
and jump-diffusion — restructuring's stochastic home. The analytic core is
mirrored in DCT_V1_Ch08_Lab.xlsx; Monte Carlo fans live here.
import numpy as np
import matplotlib.pyplot as plt
plt.rcParams['figure.dpi']=110
import numpy as np
from math import log, sqrt, exp, erf
SEED = 26108
X0, MU, SIG, T = 100.0, 0.08, 0.25, 5.0
BARRIER = 80.0
LAM_J, MJ = 0.4, 0.9 # jump intensity /yr, mean jump multiplier
def Phi(z): return 0.5*(1+erf(z/sqrt(2)))
def mean_T(): return X0*exp(MU*T)
def median_T(): return X0*exp((MU-SIG**2/2)*T)
def p_below_barrier():
z = (log(BARRIER/X0)-(MU-SIG**2/2)*T)/(SIG*sqrt(T))
return Phi(z)
def mean_T_jumps(): # compound Poisson multiplicative jumps
return X0*exp((MU+LAM_J*(MJ-1))*T)
def mc_paths(n_paths=2000, steps=60):
rng = np.random.default_rng(SEED)
dt = T/steps
z = rng.standard_normal((n_paths, steps))
logx = np.cumsum((MU-SIG**2/2)*dt + SIG*sqrt(dt)*z, axis=1)
return X0*np.exp(np.hstack([np.zeros((n_paths,1)), logx]))
def mc_jump_paths(n_paths=2000, steps=60):
rng = np.random.default_rng(SEED+1)
dt = T/steps
z = rng.standard_normal((n_paths, steps))
nj = rng.poisson(LAM_J*dt, (n_paths, steps))
logx = np.cumsum((MU-SIG**2/2)*dt + SIG*sqrt(dt)*z + nj*log(MJ), axis=1)
return X0*np.exp(np.hstack([np.zeros((n_paths,1)), logx]))
def reference_values():
return {
"mean_t5": round(mean_T(), 4),
"median_t5": round(median_T(), 4),
"mean_median_gap":round(mean_T()-median_T(), 4),
"p_breach_t5": round(p_below_barrier(), 4),
"mean_t5_jumps": round(mean_T_jumps(), 4),
"jump_drag": round(mean_T()-mean_T_jumps(), 4),
"sigma_sqrtT": round(SIG*sqrt(T), 4),
}
if __name__ == "__main__":
[print(f"{k:18s} {v}") for k,v in reference_values().items()]mean_t5 149.1825 median_t5 127.6025 mean_median_gap 21.5799 p_breach_t5 0.2018 mean_t5_jumps 122.1403 jump_drag 27.0422 sigma_sqrtT 0.559
Panel 1 — The fan, and two "forecasts" inside it
2,000 GBM paths, seed 26108. The mean path (what the Expected Enterprise
Evolution Theorem tracks) and the median path diverge — at they are
21.6 apart on a base of 100. The mean is dragged up by a thin right tail; the
median is what the typical path does. Deterministic dynamics are the degenerate
case (Prop.): switch to 0 and the fan collapses to Chapter 7.
paths = mc_paths()
t = np.linspace(0, T, paths.shape[1])
fig, ax = plt.subplots(figsize=(8.4,4.6))
ax.plot(t, paths[:150].T, color="#1B6B52", alpha=.04)
ax.plot(t, X0*np.exp(MU*t), c="#C8A24B", lw=2.5, label=f"mean path → {mean_T():.1f}")
ax.plot(t, X0*np.exp((MU-SIG**2/2)*t), c="#0B3D2E", lw=2.5, ls="--", label=f"median path → {median_T():.1f}")
ax.axhline(BARRIER, c="#B0532F", lw=1.2, ls=":", label=f"barrier {BARRIER:.0f}")
ax.set(xlabel="years", ylabel="enterprise value", title="The fan: 2,000 futures, two summaries (seed 26108)")
ax.legend(frameon=False); ax.grid(alpha=.25); plt.tight_layout(); plt.show()
print(f"analytic mean {mean_T():.4f} MC mean {paths[:,-1].mean():.4f}")
print(f"analytic median {median_T():.4f} MC median {np.median(paths[:,-1]):.4f}")
analytic mean 149.1825 MC mean 150.3499 analytic median 127.6025 MC median 129.9764
Panel 2 — The flaw of averages, exactly
Two demonstrations. (a) The mean–median gap above: "the average outcome" and "the typical outcome" are different numbers. (b) for the breach indicator : evaluated at the mean, the breach "cannot happen" (); in expectation, it happens with probability 0.2018 — one path in five. Planning on the mean path deletes a 20% event.
p_analytic = p_below_barrier()
p_mc = float((paths[:,-1] < BARRIER).mean())
print(f"f(E[X_5]) = 1[{mean_T():.1f} < 80] = 0 (the mean-path plan)")
print(f"E[f(X_5)] = P(X_5 < 80) analytic = {p_analytic:.4f}")
print(f" MC = {p_mc:.4f} (2,000 paths)")f(E[X_5]) = 1[149.2 < 80] = 0 (the mean-path plan)
E[f(X_5)] = P(X_5 < 80) analytic = 0.2018
MC = 0.2165 (2,000 paths)
Panel 3 — Jumps: restructuring's stochastic home
A compound-Poisson jump component (/yr, mean multiplier 0.9) riding on the diffusion — the Jump-Diffusion Theorem's world. The jump drag on the mean is analytic: , a 27-point haircut at .
jp = mc_jump_paths()
fig, ax = plt.subplots(figsize=(8.4,4.4))
ax.plot(t, jp[:150].T, color="#1B6B52", alpha=.04)
ax.plot(t, X0*np.exp((MU+LAM_J*(MJ-1))*t), c="#C8A24B", lw=2.5, label=f"mean with jumps → {mean_T_jumps():.1f}")
ax.plot(t, X0*np.exp(MU*t), c="#8A8F8B", lw=1.8, ls="--", label=f"diffusion-only mean → {mean_T():.1f}")
ax.set(xlabel="years", ylabel="enterprise value", title="Jump-diffusion: discrete events on a continuous base (seed 26109)")
ax.legend(frameon=False); ax.grid(alpha=.25); plt.tight_layout(); plt.show()
print(f"jump drag at t=5 (analytic): {mean_T()-mean_T_jumps():.4f}")
print(f"MC mean with jumps: {jp[:,-1].mean():.4f}")
jump drag at t=5 (analytic): 27.0422 MC mean with jumps: 120.1459
Validation — agrees with DCT_V1_Ch08_Lab.xlsx (analytic core)
ref = reference_values()
expected = {"mean_t5":149.1825,"median_t5":127.6025,"mean_median_gap":21.5799,
"p_breach_t5":0.2018,"mean_t5_jumps":122.1403,"jump_drag":27.0422,"sigma_sqrtT":0.559}
for k,v in expected.items():
assert abs(ref[k]-v)<5e-4, f"MISMATCH {k}"
print(f"PASS {k:18s} {ref[k]}")
print("\nAll checkpoints agree — seed 26108.")PASS mean_t5 149.1825 PASS median_t5 127.6025 PASS mean_median_gap 21.5799 PASS p_breach_t5 0.2018 PASS mean_t5_jumps 122.1403 PASS jump_drag 27.0422 PASS sigma_sqrtT 0.559 All checkpoints agree — seed 26108.
Next: Exercises 8.9–8.12 (Part C) vary σ, λ, and the barrier; AXIOM-08's fan theater animates Panel 1 with a σ-slider whose zero position is Chapter 7. Solutions: IM Ch. 8.