import sys, numpy as np, matplotlib.pyplot as plt
sys.path.insert(0,'..')
from engine import ch02
from dataclasses import replaceMFAA Chapter 2 Laboratory
Stochastic Process and Cash-Flow Simulator (book §2.9)
Simulates the drivers of §2.5, assembles cash-flow processes by decomposition (2.3), and makes the two-filtration distinction visible. No valuation yet — cash flows are shown undiscounted. Seed 20260200.
1. Drivers and cash-flow paths
p = ch02.template('buyout')
dr = ch02.simulate_drivers(replace(p, M=2000))
fig,ax = plt.subplots(1,2,figsize=(11,3))
for j in range(12):
ax[0].plot(dr['tgrid'], dr['c'][:,j], lw=.6)
ax[1].plot(dr['tgrid'], dr['r'][:,j], lw=.6)
ax[0].set_title('cash-flow rate c_t'); ax[1].set_title('short rate r_t'); plt.tight_layout()
2. E1 — Scheme error (Euler vs exact; Theorem 2.8.1 slopes ≈ ½ and ≈ 1)
cv = ch02.convergence_diagnostics(ch02.SimParams())
for r in cv['rows']:
print(f"dt {r['dt']:.5f} strong {r['strong_error']:.5f} weak {r['weak_error']:.6f}")
print(f"strong slope {cv['strong_slope']:.3f}, weak slope {cv['weak_slope']:.3f}")dt 0.25000 strong 0.00506 weak 0.000169
dt 0.08333 strong 0.00219 weak 0.000027
dt 0.01923 strong 0.00090 weak 0.000034
dt 0.00397 strong 0.00039 weak 0.000003
strong slope 0.615, weak slope 0.8593. E2 — Intensity shape: the capital-call drawdown (Exercise 2.10)
Calibrate the age profile so expected cumulative calls reach 90% by year 4.
p = ch02.SimParams(theta=0.5, mark_a=2.0, mark_b=3.0)
lam = ch02.calibrate_lambda_call(p)
print(f'calibrated lam_call = {lam:.4f} (book ≈ 3.33)')
dd = ch02.simulate_drawdown(replace(p, lam_call=lam, M=20000))
i4 = int(round(4/p.dt))
print(f"called-by-year-4: sim {dd['mean_sim'][i4]:.4f} closed {dd['mean_closed'][i4]:.4f}")
plt.plot(dd['tgrid'], dd['mean_sim'], label='simulated mean')
plt.plot(dd['tgrid'], dd['mean_closed'], '--', label='closed form')
plt.axhline(.9, color='grey', lw=.7); plt.axvline(4, color='grey', lw=.7)
plt.legend(); plt.title('Capital-call drawdown schedule');calibrated lam_call = 3.3287 (book ≈ 3.33)
called-by-year-4: sim 0.8930 closed 0.9000
4. E3 — Two filtrations (Exercise 2.11)
The target table: volatility ratio {0, 0.333, 0.500, 0.655, 0.816}, lag-1 autocorrelation {·, 0.8, 0.6, 0.4, 0.2}.
import pandas as pd
pd.DataFrame(ch02.two_filtration_table())| alpha | vol_ratio_sample | vol_ratio_theory | rho1_sample | rho1_theory | |
|---|---|---|---|---|---|
| 0 | 0.0 | 0.000000 | 0.000000 | NaN | NaN |
| 1 | 0.2 | 0.338568 | 0.333333 | 0.806767 | 0.8 |
| 2 | 0.4 | 0.503584 | 0.500000 | 0.607492 | 0.6 |
| 3 | 0.6 | 0.656521 | 0.654654 | 0.405911 | 0.4 |
| 4 | 0.8 | 0.817247 | 0.816497 | 0.204372 | 0.2 |
The committee’s two volatility numbers from the opening problem are two points on the first row of this table — not two estimates of one quantity.
5. Validation checks
v = ch02.validation_checks()
for k,d in v.items():
if isinstance(d,dict): print(k, 'PASS' if d['pass_'] else 'FAIL')
print('ALL:', v['all_pass'])V1_poisson PASS
V2_ou_stationary PASS
V3_cir_stationary PASS
V4_reproducible PASS
ALL: True