Probability Maths Flowchart

This document is a maths-first companion to the Probability page. It describes the intended statistical object as one terminal price distribution over a grid of terminal prices K, not one distribution per listed strike.

The model starts with the vanilla market-implied terminal distribution from the IV surface. It then optionally tilts that distribution with historical forward returns from similar gex_hourly gamma-regime states. Touch probabilities and candidate ranking inherit the same density-quality and stationarity gates.

Integrated Calculation Flow

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Formula Notes

1. Vanilla terminal distribution

S_T ~ f_Q(K)
E_Q[S_T] = integral K f_Q(K)dK
F = S0 exp(rT)

2. IV surface to strike/IV points

K_i = strike_i
sigma_i = IV_i

If strike is missing, invert Black-Scholes delta:

d1 = N^-1(delta_call)
K = S0 exp((r + 0.5 sigma^2)T - d1 sigma sqrt(T))

For puts:

d1 = N^-1(1 - abs(delta_put))

3. Synthetic Black-Scholes calls

d1 = [ln(S0 / K) + (r + 0.5 sigma^2)T] / [sigma sqrt(T)]
d2 = d1 - sigma sqrt(T)
C(K,T) = S0 N(d1) - K exp(-rT) N(d2)

4. Breeden-Litzenberger density

f_Q(K) = exp(rT) d2C(K,T)/dK2
raw_density(K) = max(0, exp(rT) C''(K))
f_vanilla(K) = raw_density(K) / integral raw_density(K)dK

5. GEX-regime reweighting

R_K = (K - S0) / S0
R_t,h = (S_t+h - S_t) / S_t
L(R_K) = NormalPDF(R_K; mean(R_matches), std(R_matches))
raw f_cond(K) = f_vanilla(K) L(R_K)
f_cond(K) = raw f_cond(K) / integral raw f_cond(K)dK

6. GEX regime ranking and matching

percentile = count(gex_history <= current_gex) / N * 100

percentile < 25       => short_gamma
25 <= percentile <=75 => neutral
percentile > 75       => long_gamma

low_pct = max(0, percentile - 15)
high_pct = min(100, percentile + 15)
matching observations = rows with gex percentile in [low_pct, high_pct]

7. Stationarity check and blend

baseline = first half of R_matches
recent = second half of R_matches

KS_stat, p_value = KS_2sample(baseline, recent)
mean_shift_bps = (mean(recent) - mean(baseline)) * 10000

if len(R_matches) < 24:
  status = insufficient
  w = 0.70
elif p_value < 0.05 or abs(mean_shift_bps) > 15:
  status = unstable
  w = 0.45
else:
  status = stable
  w = 1.00

f_final(K) = w f_cond(K) + (1 - w) f_vanilla(K)

8. Touch probability

upside finish_probability = P(S_T >= target)
downside finish_probability = P(S_T <= target)
touch_probability ~= min(1, 2 * finish_probability)

This is a reflection-principle approximation, not a full barrier-pricing model.

9. Distribution moments and conditional shift

mean = integral K f(K)dK
variance = integral (K - mean)^2 f(K)dK
std = sqrt(variance)
conditional_shift = conditional_mean - vanilla_mean
conditional_shift_pct = conditional_shift / S0

10. Candidate ranking

payoff_call(K) = max(K - option_strike, 0)
payoff_put(K) = max(option_strike - K, 0)
expected_payoff = integral payoff(K) f_final(K)dK
edge_long = expected_payoff - premium
edge_short = premium - expected_payoff

The candidate screen then applies DTE, budget, open-interest, premium, and objective filters before ranking.

11. Quality and no-data flow

bad or sparse IV surface      => withhold vanilla density and downstream outputs
too few matching GEX returns  => return vanilla or partially blend toward vanilla
unstable stationarity         => materially discount conditional overlay
missing option chain          => withhold candidate ranking

Unavailable or discounted model states should be shown explicitly. They should not be silently converted into neutral probabilities or zero-valued metrics.

Worked Miniature Example

S0 = 80000
r = 4.5%
T = 7 / 365

historical_gex = [-120, -80, -30, 10, 25, 40, 55, 70, 95, 140]
current_gex = 100

count(gex <= 100) = 9
N = 10
percentile = 9 / 10 * 100 = 90
90 > 75 => long_gamma

R_matches = [-1.0%, -0.4%, 0.2%, 0.5%, 0.8%]
mean(R_matches) = 0.02%
std(R_matches) ~= 0.70%

K = 81600
R_K = (81600 - 80000) / 80000 = 2.0%
L(2.0%) = NormalPDF(0.020; mean=0.0002, std=0.0070)

f_vanilla(81600) = 0.00050
f_cond(81600) = 0.00080
w = 0.45

f_final(81600) = 0.45 * 0.00080 + 0.55 * 0.00050
               = 0.000635

Interpretation: the regime evidence raises the density at that terminal price, but the stationarity/control layer only allows part of the conditional adjustment through.