MFAA Chapter 16 Laboratory

Malliavin Sensitivity Engine (book §16.8)

A sensitivity layer over every engine: Greeks with error bars, estimator diagnostics, and the hedging projection. Seed 20261600.

import sys, numpy as np, matplotlib.pyplot as plt
sys.path.insert(0,'..')
from engine import ch16
from dataclasses import replace

1. Malliavin weights match Black-Scholes

Integer-by-parts estimators for delta and vega.

p = ch16.MalliavinParams()
md_ = ch16.malliavin_delta(p); mv = ch16.malliavin_vega(p)
print(f"delta: weight {md_['estimate']:.4f} ± {md_['se']:.4f} vs BS {md_['closed_form']:.4f}")
print(f"vega:  weight {mv['estimate']:.2f} ± {mv['se']:.2f} vs BS {mv['closed_form']:.2f}")

delta: weight 0.5971 ± 0.0044 vs BS 0.5987
vega:  weight 38.13 ± 0.82 vs BS 38.67

2. RMSE-versus-N laws (Proposition 16.2)

The weight estimator scales as N^{-1/2}.

rw = ch16.rmse_scaling(p, 'weight')
plt.loglog(rw['Ns'], rw['rmses'], 'o-')
plt.xlabel('paths N'); plt.ylabel('RMSE'); plt.title(f"weight estimator: slope {rw['fitted_slope']:.3f} (target -1/2)")
print(f"fitted slope {rw['fitted_slope']:.3f}")

fitted slope -0.549

3. Ex 16.12 — Clark-Ocone hedging

Floor 0.0992; residual variance rises as rebalancing coarsens (weekly < monthly < quarterly).

for rebal in ('weekly','monthly','quarterly'):
    h = ch16.hedging_projection(replace(p, M=40000), rebal)
    print(f"{rebal}: floor {h['floor']:.4f}, realized residual var {h['realized_residual_var']:.4f}")

weekly: floor 0.0994, realized residual var 0.1025
monthly: floor 0.0994, realized residual var 0.1096
quarterly: floor 0.0994, realized residual var 0.1263

4. Validation checks

v = ch16.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_delta_weight PASS
V2_vega_weight PASS
V3_rmse_weight PASS
V4_hedging PASS
V5_envelope PASS
V6_reproducible PASS
ALL: True