1. The Model: Geometric Brownian Motion (GBM)

The underlying stock price $S_t$ evolves under the risk-neutral measure according to:

\text{d}S_t = (r - q)S_t\,\text{d}t + \sigma S_t\,\text{d}W_t

S_tStock price at t

rRisk-free rate

qDividend yield

σVolatility (annualised)

W_tStandard Brownian motion

Exact Closed-Form Solution

Applying Itô’s Lemma to $\ln S_t$ gives the exact terminal price:

S_T = S_0 \exp\left[\left(r - q - \tfrac{1}{2}\sigma^2\right)T + \sigma\sqrt{T}\,Z\right], \quad Z \sim N(0,1)

Why exact GBM sampling?

The SDE has a known closed-form solution for constant parameters, so Euler-Maruyama discretisation would introduce unnecessary bias. Path-wise simulation for charts uses the same exact formula stepwise.

2. Black-Scholes-Merton Analytical Benchmark & Greeks

Closed-form formulas with continuous dividend yield $q$ (Merton 1973):

d_1 = \frac{\ln(S_0/K) + (r - q + \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}

C = S_0 e^{-qT}N(d_1) - K e^{-rT}N(d_2)

P = K e^{-rT}N(-d_2) - S_0 e^{-qT}N(-d_1)

Under the risk-neutral measure, the option price is the discounted expected payoff: $V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[\text{payoff}(S_T)]$ . Monte Carlo estimates this same expectation numerically.

Analytical Greeks

Greek	Call	Put	Meaning
Delta (Δ)	$e^{-qT}N(d_1)$	$-e^{-qT}N(-d_1)$	Spot sensitivity
Gamma (Γ)	$\frac{e^{-qT}\phi(d_1)}{S_0\sigma\sqrt{T}}$	Identical to Call	Convexity
Vega (ν)	$S_0 e^{-qT}\phi(d_1)\sqrt{T}$	Identical to Call	Vol sensitivity
Theta (Θ)	$\text{Annualised} / 365$	Identical to Call	Time decay / day
Rho (ρ)	$K T e^{-rT} N(d_2)$	$-K T e^{-rT} N(-d_2)$	Rate sensitivity

3. Why Monte Carlo for a Problem Black-Scholes Already Solves?

Honest Assessment

For European vanillas under GBM, Monte Carlo is strictly worse than Black-Scholes: slower, noisier, estimates what BS computes exactly.

Why Institutions Use MC

MC generalises where no closed form exists: path-dependent payoffs, baskets, American exercise (LSM), stochastic volatility. BS is the exception, not the rule.

What PathPricer Demonstrates

Numerical machinery validated against known truth before applying it where truth is unknown. This is the intellectual foundation of the project.

4. Monte Carlo Estimator & O(N^{-1/2}) Convergence

\hat{V} = e^{-rT}\cdot\frac{1}{N}\sum_{i=1}^{N} h(S_T^{(i)}), \quad h(S_T) = \max(S_T - K,\,0)

Standard Error

\widehat{SE} = s/\sqrt{N}

Sample std dev of i.i.d. discounted payoffs

95% Confidence Interval

\hat{V} \pm 1.96\cdot\widehat{SE}

Normal approx justified by CLT for N ≥ 10⁴

Why Normal CI?

Discounted payoffs are i.i.d. with finite variance, so the CLT applies. Bootstrap targets the same asymptotics at higher cost for no benefit here.

Convergence rate: $\mathcal{O}(N^{-1/2})$ — halving error requires 4× paths. Empirically validated by regressing $\log\widehat{SE}$ on $\log N$ across a geometric grid and confirming slope ≈ −0.5.

5. Variance Reduction Techniques

5.1 Antithetic Variates

For each draw $Z_i$ , also evaluate payoff at $-Z_i$ . Monotonic payoffs guarantee negatively correlated pairs.

\hat{V}_{AV} = e^{-rT}\cdot\frac{1}{N}\sum_{i=1}^{N}\frac{h(S_T^{(i,+)}) + h(S_T^{(i,-)})}{2}

5.2 Control Variates (S_T)

Uses $S_T$ as control with known expectation $\mathbb{E}^{\mathbb{Q}}[S_T] = S_0 e^{(r-q)T}$ (Boyle 1977).

\hat{V}_{CV} = e^{-rT}\cdot\frac{1}{N}\sum_{i=1}^{N}\left[h(S_T^{(i)}) - \beta^*\left(S_T^{(i)} - \mathbb{E}^{\mathbb{Q}}[S_T]\right)\right]

Why S_T and not BS price?

BS price is the benchmark being validated. Using it as a control would be circular. S_T has a known expectation independent of the option price.

5.3 Combined Antithetic + CV

Apply antithetic pairing first, then CV correction to paired averages. The techniques target different variance components and stack without redundancy.

Method	Paths	Std Error	Rel. Efficiency
Standard MC	N	SE_{std}	$1.0 (ref)$
Antithetic	2N	SE_{AV}	$Var_{std}/Var_{AV}$
Control Variates	N	SE_{CV}	$Var_{std}/Var_{CV}$
Antithetic + CV	2N	SE_{AV+CV}	$Var_{std}/Var_{AV+CV}$
RQMC (Sobol)	N	SE_{RQMC}	$Var_{std}/Var_{RQMC}$

6. Randomized Quasi-Monte Carlo (Sobol)

Replaces pseudo-random $N(0,1)$ draws with low-discrepancy sequences — deterministic point sets covering $[0,1]^d$ more uniformly than random sampling — then randomises them via Owen scrambling for error estimation.

Z_i = \Phi^{-1}(u_i), \quad u_i \in \text{Sobol sequence}

\hat{V}_{RQMC} = \frac{1}{M}\sum_{j=1}^{M} \hat{V}_j, \quad \widehat{SE}_{RQMC} = \frac{s}{\sqrt{M}}, \quad M=20

$N$ is enforced to a power of 2 — the natural regime for Sobol optimal equidistribution. For smooth integrands, RQMC converges at $\mathcal{O}(N^{-1})$ vs standard MC’s $\mathcal{O}(N^{-1/2})$ : halving error requires only 2× paths, not 4×.

Honest Caveat: CI Interpretation

The CI uses a t-distribution on

M=20

replications. Variance between replications captures only scrambling noise, not full sampling error. The CI is a heuristic measure of uncertainty, not a strict 95% confidence statement.

7. Finite-Difference Greeks (Common Random Numbers)

Central differences on Monte Carlo prices, reusing the same seed across base and bumped scenarios:

\Delta \approx \frac{\hat{V}(S_0+h) - \hat{V}(S_0-h)}{2h}, \quad \Gamma \approx \frac{\hat{V}(S_0+h) - 2\hat{V}(S_0) + \hat{V}(S_0-h)}{h^2}

Analogous central differences for Vega ( $\sigma$ bump), Theta ( $T$ one-sided), and Rho ( $r$ bump). Default bump: 0.5–1% of the parameter value.

Why Common Random Numbers?

Without CRN, finite-difference Greeks on Monte Carlo prices are dominated by simulation noise rather than true sensitivity. Same seed across the bump pair isolates the parameter change from sampling variance.

8. Implied Volatility Solver

Given a market price, find $\sigma$ such that:

\text{BS}_{\text{price}}(S_0, K, T, r, q, \sigma, \text{type}) = P_{\text{market}}

No closed-form inverse exists (BS is transcendental in $\sigma$ ), so numerical root-finding is required.

Newton-Raphson (Primary)

\sigma_{n+1} = \sigma_n - \frac{\text{BS}_{\text{price}}(\sigma_n) - P_{\text{market}}}{\text{Vega}(\sigma_n)}

Brenner-Subrahmanyam Init

\sigma_0 \approx \sqrt{2\pi/T} \cdot P_{\text{market}} / S_0

Fallback — Brent’s method:Activates when Vega → 0 (deep ITM/OTM, near-expiry). Brent’s method requires no derivative and is guaranteed to converge on a bracketed interval.

Why This Is the Most Common Desk Task

Market prices are quoted in price space; traders think in vol space. The bid-ask spread in implied vol is informative across strikes; price is not. All volatility surface construction begins here.

9. P&L Attribution (P&L Explain)

Decomposes a scenario price change into component contributions by Greek:

\text{PnL} = \Delta\cdot\Delta S + \tfrac{1}{2}\Gamma(\Delta S)^2 + \mathcal{V}\cdot\Delta\sigma + \Theta\cdot\Delta t + \rho\cdot\Delta r + \varepsilon

Term	Greek	Driver	Interpretation
$Δ·ΔS$	Delta	Spot	Directional exposure
$\tfrac12\Gamma(\Delta S)^2$	Gamma	Spot² (convexity)	Profit from large moves; always +ve for long options
$𝒱·Δσ$	Vega	Vol	Volatility exposure
$Θ·Δt$	Theta	Time	Cost of optionality; −ve for long options
$ρ·Δr$	Rho	Rate	Interest rate exposure
$ε$	Residual	Cross-terms	Vanna, Volga, cross-Gamma, higher-order

The residual $\varepsilon$ captures everything the second-order expansion misses: Vanna ( $\partial\Delta/\partial\sigma$ ), Volga ( $\partial\mathcal{V}/\partial\sigma$ ), cross-Gamma interactions, and higher-order Taylor terms. For small scenario moves, first-order terms (especially Delta) dominate.

Operational Significance

P&L attribution distinguishes “we made money because spot moved our way” from “we made money because vol dropped.” Essential for risk management and strategy evaluation.

10. 2D Risk Grid

A $25 \times 25$ surface (625 points) computed across dual parameter axes. Every cell is evaluated in a single broadcast operation — no nested Python loops:

\text{price}_{ij} = \text{BS}_{\text{price}}(S_i, K, T, r, q, \sigma_j, \text{type}), \quad i,j = 1,\dots,25

Axis Pair	What It Reveals
Spot × Vol	Gamma as curvature along spot axis; Volga along vol axis
Strike × Expiry	Term structure of option value across strikes
Spot × Time	Option decay as expiry approaches at different moneyness

Vectorisation via NumPy broadcasting: $S_{\text{grid}}$ shape $(25,1)$ , $\sigma_{\text{grid}}$ shape $(1,25)$ → broadcast to $(25,25)$ . The heatmap renders colour intensity proportional to price, with crosshairs at the base-case parameters.

11. Heston Stochastic Volatility Model

The Heston (1993) model lets variance follow its own mean-reverting square-root process, so the volatility smile/skew is modeled rather than assumed constant:

dS_t = (r - q)S_t\,dt + \sqrt{v_t}\,S_t\,dW_t^S, \qquad dv_t = \kappa(\theta_v - v_t)\,dt + \sigma_v\sqrt{v_t}\,dW_t^v

v_tInstantaneous variance

κMean-reversion speed

θ_vLong-run variance

σ_vVol-of-vol

ρSpot/vol correlation (skew)

√v₀Initial volatility (quoted)

Pricing by Fourier Inversion

The Heston density has no closed form, but the characteristic function of $\ln S_T$ does. Prices come from integrating two risk-neutral probabilities:

C = S_0 e^{-qT} P_1 - K e^{-rT} P_2, \qquad P_j = \tfrac{1}{2} + \frac{1}{\pi}\int_0^{\infty}\operatorname{Re}\left[\frac{e^{-i\phi\ln K} f_j(\phi)}{i\phi}\right]d\phi

Evaluated by Gauss-Legendre quadrature on a fixed node grid (the nodes are parameter-independent and cached). Puts follow from put-call parity, exact under Heston. Two numerical guards matter: the branch of $d_j$ is chosen so $\operatorname{Re}(d_j) \geq 0$ , and the term $1 - g_j e^{d_j T}$ is computed in log space to avoid catastrophic cancellation in the deep-OTM wings.

Greeks & the Volatility Chain Rule

Because the Heston price is deterministic (no Monte Carlo noise), central finite differences are clean. Volga and Vanna are reported w.r.t. the volatility $\sigma_0 = \sqrt{v_0}$ , requiring a chain-rule correction:

\text{volga} = 4v_0\frac{\partial^2 V}{\partial v_0^2} + 2\frac{\partial V}{\partial v_0}, \qquad \text{vanna} = 2\sqrt{v_0}\,\frac{\partial^2 V}{\partial S\,\partial v_0}

Efficiency

The spot and vanna bumps only change

S_0

/

v_0

— one characteristic-function set serves all bumped prices, cutting 12 Fourier evaluations to 6.

12. SVI Volatility Surface

The raw SVI parameterization (Gatheral, 2004) describes the implied-vol smile at a fixed expiry as a function of log-moneyness $k = \ln(K/F)$ :

w(k) = a + b\left(\rho(k - m) + \sqrt{(k - m)^2 + \sigma^2}\right), \qquad \sigma_{imp}(k) = \sqrt{\frac{w(k)}{T}}

aTotal variance level

bWing slope (b ≥ 0)

ρSkew (−1 < ρ < 1)

mSmile minimum offset

σCurvature at minimum

Each expiry’s $(k, \sigma_{imp})$ points are fit by nonlinear least squares on total variance $w = \sigma_{imp}^2 T$ , from three restarts. The fitted $a$ is clamped so $w_{\min} = a + b\sigma\sqrt{1-\rho^2} \geq 0$ , keeping the slice arbitrage-free in strike.

A surface is a set of slices with total variance interpolated linearly in $T$ at fixed log-moneyness (sticky-strike). A calendar-arbitrage check rejects any surface where total variance decreases with time to maturity at a fixed moneyness — such a surface would admit a riskless calendar-spread arbitrage.

A second guard checks butterfly (strike) arbitrage: call prices must be convex in strike, equivalently the implied risk-neutral density must be non-negative. On a discrete strike grid this is $C(K-d) - 2C(K) + C(K+d) \geq 0$ . Each fitted slice reports an $\text{arb\_free}$ flag plus the strike of the worst violation, so the chart can badge exactly where a surface breaks.

The same fit yields the ATM volatility term structure — the at-the-money implied vol at every expiry $\sigma_{ATM}(T) = \sqrt{w(0)/T}$ — and a Greeks surface: any Greek priced across strikes × expiries where each cell uses the SVI implied vol for its own strike, so the surface reflects the smile rather than a flat vol.

13. Heston Calibration

Fits $(v_0, \kappa, \theta_v, \sigma_v, \rho)$ to observed market option prices by minimizing a blended objective via $\text{L-BFGS-B}$ :

\min_p \left[ 0.5\,\text{rel-RMSE} + 0.5\,\frac{\text{abs-RMSE}}{\overline{V^{mkt}}} \right] + \underbrace{10\,(2\kappa\theta_v - \sigma_v^2)^2}_{\text{Feller penalty, only if violated}}

A pure relative error over-penalizes deep-OTM options (tiny prices make any small misprice a huge percentage), dragging the fit into the wings. Blending in the mean-normalized absolute error keeps the ATM backbone dominant while relative error still shapes the smile.

Feller as a Soft Penalty

Hard-constraining $2\kappa\theta_v > \sigma_v^2$ forces infeasible optimizer restarts. A soft penalty lets the fit trade feasibility against quality — and reports whether Feller holds, as practitioners do.

Deterministic Multi-Start

$n$ restarts from an ATM-implied-vol seed plus log-uniformly spread seeds ( $s \leftarrow s\cdot e^{U(-0.7,0.7)}$ ) — right for positive scale parameters. Reproducible via $\text{default\_rng}(20240101+i)$ .

14. Model Validation

Scores a calibrated Heston model against the same market quotes it was fitted to, answering “how well does the model reproduce observed prices/vols, and are the market quotes internally consistent?”

Metric	Definition
Price Rel RMSE	$\sqrt{\tfrac{1}{n}\sum_i\left(\tfrac{V_i^{m} - V_i^{k}}{V_i^{k}}\right)^2}$
Price MAPE	$\tfrac{1}{n}\sum_i\left\|\tfrac{V_i^{m} - V_i^{k}}{V_i^{k}}\right\| \times 100\%$
IV RMSE	NaN-robust: over contracts with resolvable implied vols only
Market parity violation	Largest \|put-call parity RHS − market price\| across the chain

Market put-call parity

The complement option type is priced under the model and the implied parity value compared against the observed quote. A large violation flags internally inconsistent market quotes, not a bad model.

15. Assumptions & Limitations

Assumption	Reality	Treatment
Constant volatility	Smile/skew varies by strike & expiry	Single σ input for GBM; the quant workspace fits Heston & SVI models to the smile
GBM / log-normal returns	Fat tails & negative skew	GBM; Merton jump-diffusion as extension
Constant risk-free rate	Term structure, stochastic rates	Flat r; bond curve integration as fix
Continuous dividend yield	Discrete cash payments	Continuous q approx; discrete modeling as gap
European exercise only	Most US options are American	Explicit scope; Longstaff-Schwartz LSM as fix
Frictionless markets	Bid-ask spreads, market impact	Not modeled; pricing vs trading system distinction
Risk-neutral measure Q	Physical drift ≠ risk-neutral drift	Priced under Q; appropriate for hedging, not forecasting

16. Multi-Leg Strategy Engine

A strategy is a portfolio of 1–10 signed legs — a long has positive quantity, a short negative. Each option leg is priced by closed-form Black-Scholes; stock legs are valued at the forward-carried price $Se^{-qT}$ with $\Delta = 1$ .

Portfolio Greeks are quantity-weighted sums of the per-leg Greeks. The net premium is the signed sum of leg values: positive = debit (we pay to enter), negative = credit (we were paid).

\Delta_{net} = \sum_i q_i \Delta_i, \quad \text{P\&L}(S_T) = \underbrace{\text{payoff}(S_T)}_{\text{piecewise-linear in spot}} - \text{net premium}

Breakevens

The spot levels where net P&L = 0, found by linear interpolation across the net P&L's zero crossings (where payoff = net premium).

Max Profit / Loss

Exact: the payoff is piecewise-linear, so extrema live at strike kinks and the tails. Unbounded (∞) is reported when the high-tail slope is nonzero — a finite grid scan would misstate this.

Presets

Ten presets cover the classic structures — long/short straddles, strangles, bull/bear spreads, iron condor, iron butterfly, call butterfly, covered call, protective put — each with a distinct risk/reward shape visible on the payoff diagram.

17. Scenario Stress Testing

Reprices an option under named market scenarios — 2008 Crisis (−40% spot, +20 vol pts, +100 bp rates), COVID Crash, Vol Crush, Flash Crash — each defined as coordinate shifts in spot, vol, rate, and elapsed time:

S' = \max(\epsilon, S_0 + \Delta S_{abs} + \Delta S_{pct}\cdot S_0), \quad \sigma' = \max(\text{MIN\_SIGMA}, \sigma + \Delta\sigma)

Every scenario reports its repriced option value plus absolute and percentage P&L versus the base price. The engine selects the worst/best scenarios and computes an unrealized-risk metric — the largest single-scenario loss as a fraction of the base price.

Stress vs. P&L Explain

P&L attribution (§9) decomposes one observed move into Greek contributions. Stress testing imposes many hypothetical moves and reads off the outcomes — the forward-looking complement to the backward-looking attribution.

18. Put-Call Parity Data-Quality Probes

Put-call parity is the no-arbitrage identity $C - P = S_0 e^{-qT} - K e^{-rT}$ . The probes run it in reverse: given market prices, what rate or dividend is the market implicitly assuming? Consistent quotes land near consensus values; a large divergence flags stale mids, crossed markets, or mis-priced dividends.

Implied Rate

r = -\frac{1}{T}\ln\left(\frac{S_0 e^{-qT} - C + P}{K}\right)

Guarded: non-positive discounted strike ⇒ quotes inconsistent ⇒ parity_inconsistent error

Implied Dividend

q = -\frac{1}{T}\ln\left(\frac{K e^{-rT} + C - P}{S_0}\right)

Given a trusted rate, recovers the dividend the market is pricing in

Why the extracted value is never re-verified against parity

Plugging the recovered

r

or

q

back into parity is a tautology — the parameter is defined as the value that makes the identity hold, so recomputation reproduces the spread by construction and can never fail. The real checks are the positivity guard and the divergence versus a reference rate/dividend. The probes are ATM-only because ATM quotes are the most liquid and least corrupted by deep-OTM noise.

19. Delta-Hedging Comparison

Benchmarks two hedging strategies across hundreds of simulated Heston paths: BS hedging (constant implied vol) versus Heston hedging (model-informed deltas that adapt to the current variance state). The hedger is short the option — receives premium at $t=0$ , delta-hedges to expiry, and the hedging error reveals which strategy is more precise.

Hedging Error

\varepsilon = \text{cash}_T + \Delta_T \cdot S_T - \text{payoff}(S_T)

Positive = hedge over-performed (hedger profits); Negative = under-performed

BS Delta (Fixed IV)

Solve for $\sigma_{\text{IV}}$ from the Heston ATM price at $t=0$ , then use that constant vol for every rebalance:

\Delta_t^{\text{BS}} = e^{-q(T-t)} N(d_1), \quad d_1 = \frac{\ln(S_t/K) + (r - q + \tfrac{1}{2}\sigma_{\text{IV}}^2)(T-t)}{\sigma_{\text{IV}}\sqrt{T-t}}

Heston Delta (Expected Average Variance)

The delta uses the BS formula evaluated at $\sigma_h = \sqrt{\bar{v}_t}$ , where $\bar{v}_t$ is the expected time-averaged variance over the remaining life. For the CIR variance process:

\bar{v}_t = \theta_v + (v_t - \theta_v)\cdot\frac{1 - e^{-\kappa(T-t)}}{\kappa(T-t)}

Taylor expansion for small

\kappa(T-t)

:

f(x) \approx 1 - x/2 + x^2/6

avoids catastrophic cancellation

\Delta_t^{\text{Heston}} = e^{-q(T-t)} \cdot P_1 \quad (\text{calls}), \qquad \Delta_t^{\text{Heston}} = e^{-q(T-t)}(P_1 - 1) \quad (\text{puts})

P_1

is the stock-measure probability from the Heston Fourier-inversion engine (§11)

Transaction Costs

\text{TC} = \tau \cdot |\delta_{\text{trade}}| \cdot S_t, \qquad \tau = B / 10{,}000

Default: 5 bps (

B = 5

). Applied at every rebalance step.

Summary Statistics

Variance Ratio

R = \text{Var}(\varepsilon^{\text{BS}}) / \text{Var}(\varepsilon^{\text{Heston}})

>1 means Heston is more precise

RMSE

\sqrt{\frac{1}{N}\sum_i \varepsilon_i^2}

Combined bias + variance

Max Absolute Error

\max_i |\varepsilon_i|

Worst-case loss across all paths

Variance % Improvement

(1 - 1/R) \times 100\%

How much tighter the Heston distribution is

20. Design Decisions FAQ

Question	Rationale
Why MC for something BS solves?	Validation infrastructure for machinery meant to generalise to unsolvable cases
Why S_T as control, not BS price?	BS is the benchmark; using it as control would be circular
Why exact GBM sampling?	Closed-form terminal density makes Euler bias unnecessary
Why normal CI, not bootstrap?	CLT applies cleanly to i.i.d. draws; bootstrap adds cost with no benefit
Why FD Greeks need CRN?	Without CRN, bumps are swamped by MC noise, not sensitivity
Why continuous dividend yield?	Free data lacks reliable ex-div schedules; explicitly named gap
Why close-to-close vol?	Data quality across US, Indian, FX, and cryptocurrency tickers matters more than marginal efficiency
Why default_rng not RandomState?	PCG64 is superior; avoids shared global state in concurrent backend
Why Newton-Raphson + Brent?	NR fast near root; Brent handles near-zero-Vega without derivative
Why residual in P&L explain?	Taylor expansion exact only for infinitesimal moves; residual = cross-Greeks + higher-order
Why vectorise the risk grid?	625 cell-level loops dominate runtime; broadcast evaluates all at NumPy speed
Why RQMC instead of standard MC?	O(N^{-1}) vs O(N^{-1/2}) convergence for smooth integrands; CI is heuristic
Why Fourier inversion for Heston?	No closed-form density, but the characteristic function is closed form — fast, deterministic, no MC noise in Greeks
Why Volga w.r.t. √v₀, not v₀?	Traders quote volatility; chain rule 4v₀·V″ + 2·V′ converts variance bumps
Why blend relative + absolute RMSE in calibration?	Pure relative over-penalizes deep-OTM; the blend keeps ATM dominant while the smile stays shaped
Why reject calendar arbitrage at build?	Total variance decreasing in T admits a riskless spread; rejecting keeps surfaces economically sane
Why not verify an implied rate by recomputing parity?	It is a tautology — the parameter is defined as the parity-solver; it reproduces the spread by construction and never fails
Why analytic max profit/loss for strategies?	Expiration payoff is piecewise-linear; extrema are exact at kinks/tails, and unbounded (∞) is reported correctly
Why are parity probes ATM-only?	ATM quotes are the most liquid and least corrupted by deep-OTM noise — the cleanest rate/dividend signal
Why hedge with expected avg variance, not spot v_t?	Spot v_t overreacts to vol spikes and ignores mean reversion; E[v_avg] smooths the delta and produces a tighter error distribution
Why solve BS IV from the Heston price?	Traders observe market IV and hedge with it — the fixed-IV BS strategy replicates this realistic scenario, not an artificial one
Why transaction costs at 5 bps default?	A reasonable equity round-trip cost; zero TC makes both strategies look perfect, masking the practical rebalancing penalty
Why even n_paths for antithetic variates?	Antithetic pairs (+Z, −Z) require pairing; odd counts would leave one unpaired path, breaking the variance-reduction guarantee

Quantitative Methodology & Mathematical Specification