Position & Greeks
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Pricei
18.006
long 10× · $18,006
Delta Δi
0.5720
position +572 sh
Gamma Γi
0.01047
position +10.47 / $1
Theta / dayi
-0.2074
position −$207/d
Vega νi
106.624
position +$1,066 / vol pt
P&Li
+0
entry 18.01 / sh
Net debit$18,006
Max profitUnlimited
Max loss−$18,006
Break-even773.01
$ delta+$432,690
Gross premium$18,006
At expiry47d
Strategy Payoff
P&L Surface
spot × vol · USD
| Spot shockΔS \ Δσ | −50% | −25% | 0% | +25% | +50% |
|---|---|---|---|---|---|
| −10% | -18006 | -17970 | -17680 | -16916 | -15652 |
| −5% | -17717 | -16381 | -14203 | -11535 | -8580 |
| 0% | -7269 | -3664 | +0 | +3687 | +7386 |
| +5% | +25742 | +26469 | +28088 | +30359 | +33057 |
| +10% | +63492 | +63516 | +63760 | +64469 | +65711 |
lossgainFull revaluation · vol shocks are relative to σ 13.8% · outlined cell = current
One portfolio, three pricing models. Black-Scholes-Merton is the closed-form reference; the CRR lattice and Monte Carlo must converge to it. Monte Carlo error is reported as standard error, a 95% interval and |z| against the reference, so an unlucky draw is visible rather than hidden.
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Model formulas and assumptions
Black-Scholes-Merton
d₁ = [ln(S/K) + (r − q + σ²/2)T] / (σ√T) d₂ = d₁ − σ√T C = S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂) P = K·e^(−rT)·N(−d₂) − S·e^(−qT)·N(−d₁)
European exercise, constant σ and r, continuous dividend yield q. Greeks are analytic.
Cox-Ross-Rubinstein lattice
Δt = T/N, u = e^(σ√Δt), d = 1/u p = (e^((r−q)Δt) − d) / (u − d) V = e^(−rΔt)·[p·V_up + (1 − p)·V_down]
Backward induction from terminal payoffs; European, so no early-exercise check.
Monte Carlo
S_t = S₀·exp((r − q − σ²/2)t + σW_t) V̂ = mean(Σ legs e^(−rT)·qty·100·payoff) SE = s/√N, 95% CI = V̂ ± 1.96·SE |z| = |V̂ − V_BS| / SE
Seeded mulberry32 + Box-Muller. One Brownian path is observed at every distinct expiry, so mixed-maturity legs stay correctly correlated. Antithetic mode averages each draw with its mirror (−Z).