Chapter 8 Laboratory — Module 8: The Pricing Machine¶

Mathematical Foundations of Modern Finance · Part II · Week 8

Four panels: the PDE panel (finite differences vs Monte Carlo — Feynman–Kac split-screen), the Greeks panel, the smile panel, and the barrier panel. Reproduces Example 8.3: put(90) = 1.7228, call(110) = 4.4803, collar = −2.7574.

Seeds: 20260801–20260804.

In [1]:
import numpy as np
import matplotlib.pyplot as plt
plt.rcParams.update({"figure.figsize": (7, 4.2), "axes.grid": True,
                     "grid.alpha": 0.3, "font.size": 11})
BRASS, INK = "#B4884A", "#0F1E3D"
import mfmf_engine_ch08 as eng

E1 — The collar, three ways (supports LOS 8.1–8.2)¶

With $S_0=100$, $\sigma=17\%$, $r=3.98\%$, $T=1$: the closed-form Black–Scholes collar is $-2.7574$, and the Monte Carlo estimate of the Feynman–Kac expectation converges to the same value.

In [2]:
r1 = eng.E1_collar()
print(f"put(90)   : {r1['put90']:.4f}   (book 1.7228)")
print(f"call(110) : {r1['call110']:.4f}   (book 4.4803)")
print(f"collar    : {r1['collar']:.4f}   (book -2.7574)")
print(f"Monte Carlo: {r1['mc']:.4f} ± {r1['mc_se']:.4f}")

put(90)   : 1.7228   (book 1.7228)
call(110) : 4.4804   (book 4.4803)
collar    : -2.7576   (book -2.7574)
Monte Carlo: -2.7753 ± 0.0153

E2 — The Greeks and the jump-day P&L (supports LOS 8.3)¶

Read the dealer's collar gamma at $S=92$ and $S=100$, and the P&L a gamma position takes on a large gap day.

In [3]:
r2 = eng.E2_greeks()
for S, d in r2['by_S'].items():
    print(f"S = {S}: collar gamma = {d['collar_gamma']:+.5f}, jump P&L (-8%) = {d['jump_pnl_-8pct']:+.4f}")

S = 92.0: collar gamma = +0.00356, jump P&L (-8%) = +0.0963
S = 100.0: collar gamma = -0.00769, jump P&L (-8%) = -0.2461

E3 — The volatility smile (supports LOS 8.5)¶

Reprice the collar with the put at $18.4\%$ and the call at $16.5\%$ — a downward skew. The fair value moves, and so does the zero-cost strike.

In [4]:
r3 = eng.E3_smile()
print(f"Collar at flat 17% vol : {r3['collar_flat']:.4f}")
print(f"Collar at skewed vols  : {r3['collar_skewed']:.4f}  (put {r3['put_vol']:.1%}, call {r3['call_vol']:.1%})")
print(f"Fair-value move        : {r3['fair_value_move']:+.4f}")

Collar at flat 17% vol : -2.7576
Collar at skewed vols  : -2.1947  (put 18.4%, call 16.5%)
Fair-value move        : +0.5629

E4 — Barrier knockout and monitoring frequency (supports LOS 8.6)¶

Price the dealer's knockout put ($B=80$) and compute its discount to the vanilla 90-put. Sliding monitoring from continuous to daily changes the price.

In [5]:
r4 = eng.E4_barrier()
print(f"Vanilla 90-put       : {r4['vanilla_put']:.4f}")
print(f"Knockout (daily)     : {r4['knockout_daily']:.4f}")
print(f"Knockout (continuous): {r4['knockout_continuous']:.4f}")
print(f"Discount to vanilla  : {r4['discount_to_vanilla']:.4f}")
print(f"Monitoring gap       : {r4['monitoring_gap']:+.4f}")

Vanilla 90-put       : 1.7228
Knockout (daily)     : 0.3286
Knockout (continuous): 0.2986
Discount to vanilla  : 1.3941
Monitoring gap       : +0.0300

Validation checks¶

Every laboratory report must reproduce these; a report whose checks do not pass is returned ungraded.

In [6]:
checks = eng.validation_checks()
for k, v in checks.items():
    if k.startswith("_"): continue
    print(f"[{'PASS' if v else 'FAIL'}] {k}")
assert checks["ALL_PASS"], "Validation failed — do not submit."
print("\nAll Module validation checks pass.")

[PASS] V1_put90_1.7228
[PASS] V2_call110_4.4803
[PASS] V3_collar_-2.7574
[PASS] V4_mc_matches_closed_form
[PASS] V5_smile_moves_value
[PASS] ALL_PASS

All Module validation checks pass.