1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
use crate::constants::*;
use crate::error::SolMathError;
use crate::arithmetic::{fp_mul_i, fp_div_i, fp_sqrt};
use crate::transcendental::{ln_fixed_i, exp_fixed_i};
use crate::trig::{cos_fixed, sin_fixed};
use crate::complex::{Complex, complex_mul, complex_sqrt, complex_exp};
/// NIG characteristic function φ(u). Internal — called by nig_call_price COS loop.
pub(crate) fn nig_char_func(
u: i128,
drift: i128,
delta_t: i128,
gamma: i128,
alpha_sq: i128,
beta: i128,
) -> Result<Complex, SolMathError> {
// α² − (β+iu)² = (α²−β²+u², −2βu)
let u_sq = fp_mul_i(u, u)?;
let beta_sq = fp_mul_i(beta, beta)?;
let inner = complex_sqrt(Complex::new(
alpha_sq - beta_sq + u_sq,
-2 * fp_mul_i(beta, u)?,
))?;
// Exponent: iu·drift + δT·(γ − inner)
let exponent = Complex::new(
fp_mul_i(delta_t, gamma - inner.re)?,
fp_mul_i(u, drift)? - fp_mul_i(delta_t, inner.im)?,
);
complex_exp(exponent)
}
/// Offline/high-precision NIG call price via COS method (17 terms, i128 arithmetic).
/// ~302K CU native, exceeds on-chain budget with Anchor overhead.
/// For on-chain use, see `nig_call_64`.
///
/// # Errors
/// - `DomainError` if s/k/alpha/delta == 0 or α ≤ |β| or α ≤ |β+1|.
///
/// # Precision
/// 95% within 0.5% of reference prices for α ≥ 10, prices > $1.
///
/// # CU cost
/// ~302,000 CU (native only — exceeds on-chain limits).
pub fn nig_call_price(
s: u128,
k: u128,
r: u128,
t: u128,
alpha: u128,
beta: i128,
delta: u128,
) -> Result<u128, SolMathError> {
if s > i128::MAX as u128 || k > i128::MAX as u128 || r > i128::MAX as u128
|| t > i128::MAX as u128 || alpha > i128::MAX as u128 || delta > i128::MAX as u128
{
return Err(SolMathError::Overflow);
}
if s == 0 || k == 0 || alpha == 0 || delta == 0 {
return Err(SolMathError::DomainError);
}
// Domain: alpha ≤ 10,000. Real NIG calibrations on equity markets have alpha in [1, 100].
// This guard prevents complex arithmetic overflow in nig_char_func (see pen test audit).
if alpha > 10_000 * SCALE {
return Err(SolMathError::DomainError);
}
let alpha_i = alpha as i128;
let delta_i = delta as i128;
let r_i = r as i128;
let t_i = t as i128;
// NIG parameters
let alpha_sq = fp_mul_i(alpha_i, alpha_i)?;
let beta_sq = fp_mul_i(beta, beta)?;
// Domain check: NIG requires α > |β| and α > |β+1|
if alpha_sq <= beta_sq {
return Err(SolMathError::DomainError); // invalid: |β| ≥ α
}
let gamma = fp_sqrt((alpha_sq - beta_sq) as u128)? as i128;
let gamma_cu = fp_mul_i(fp_mul_i(gamma, gamma)?, gamma)?; // γ³
// Convexity correction: ω = δ·(γ − √(α²−(β+1)²))
let bp1 = beta + SCALE_I;
let bp1_sq = fp_mul_i(bp1, bp1)?;
if alpha_sq <= bp1_sq {
return Err(SolMathError::DomainError); // invalid: |β+1| ≥ α
}
let omega = fp_mul_i(
delta_i,
gamma - fp_sqrt((alpha_sq - bp1_sq) as u128)? as i128,
)?;
// NIG mean and variance of log-price over period T
// c1 = ln(S) + (r−ω)T + δTβ/γ
let ln_s = ln_fixed_i(s)?;
let drift_rate = r_i - omega;
let c1 =
ln_s + fp_mul_i(drift_rate, t_i)? + fp_div_i(fp_mul_i(fp_mul_i(delta_i, t_i)?, beta)?, gamma)?;
// c2 = δTα²/γ³
let c2 = fp_div_i(fp_mul_i(fp_mul_i(delta_i, t_i)?, alpha_sq)?, gamma_cu)?;
let nig_std = fp_sqrt(c2 as u128)? as i128;
// Truncation range [a, b]
let log_k = ln_fixed_i(k)?;
let l_std = fp_mul_i(NIG_COS_L, nig_std)?;
let mut a = c1 - l_std;
let mut b = c1 + l_std;
// Extend to cover strike with 1-std margin
if log_k - nig_std < a {
a = log_k - nig_std;
}
if log_k + nig_std > b {
b = log_k + nig_std;
}
let ba = b - a;
let discount = exp_fixed_i(-fp_mul_i(r_i, t_i)?)?;
let exp_b = exp_fixed_i(b)?; // exp(b)
// Precompute drift for char func: ln(S) + (r−ω)T
let cf_drift = ln_s + fp_mul_i(drift_rate, t_i)?;
let delta_t = fp_mul_i(delta_i, t_i)?;
// COS expansion: Σ_{k=0}^{N-1} ' Re[φ(kπ/(b-a)) · e^{-ikπa/(b-a)}] · V_k
// where ' means k=0 term halved
let mut total: i128 = 0;
let mut i = 0;
while i < NIG_COS_N {
// Frequency: w = i·π / (b-a) — but i is the loop counter, use as integer
// w_ba_num = i (integer), w_ba_den = ba/π
// In SCALE: w = i * PI_SCALE / ba (but careful about overflow)
let w = if i == 0 {
0i128
} else {
// i * π / (b-a): compute as fp_div_i(i * PI_SCALE, ba)
fp_div_i((i as i128) * PI_SCALE, ba)?
};
// Characteristic function term
let char_term = if i == 0 {
SCALE_I // φ(0) = 1
} else {
// Re[φ(w) · exp(-i·w·a)]
let phi = nig_char_func(w, cf_drift, delta_t, gamma, alpha_sq, beta)?;
// exp(-i·w·a) = cos(w·a) − i·sin(w·a)
let wa = fp_mul_i(w, a)?;
let rot = Complex::new(cos_fixed(wa)?, -sin_fixed(wa)?);
complex_mul(phi, rot)?.re
};
// Payoff coefficients V_k for call: 2/(b-a) × (χ_k − K·ψ_k)
// where c = ln(K), d = b
// For k=0: χ = exp(b) − K, ψ = b − ln(K)
// For k>0: sin(kπ(d−a)/(b−a)) = sin(kπ) = 0, cos(kπ) = (−1)^k
// θ = w·(ln(K)−a)
// χ = ((−1)^k·exp(b) − K·(cos(θ)+w·sin(θ))) / (1+w²)
// ψ = −sin(θ)/w
let v_k = if i == 0 {
// 2/(b-a) × (exp(b) − K − K·(b − ln(K)))
let chi = exp_b as i128 - (k as i128);
let psi = b - log_k;
fp_div_i(2 * (chi - fp_mul_i(k as i128, psi)?), ba)?
} else {
let theta = fp_mul_i(w, log_k - a)?;
let cos_t = cos_fixed(theta)?;
let sin_t = sin_fixed(theta)?;
let w_sq = fp_mul_i(w, w)?;
let sign_k: i128 = if i % 2 == 0 { 1 } else { -1 };
let chi = fp_div_i(
sign_k * (exp_b as i128) - fp_mul_i(k as i128, cos_t + fp_mul_i(w, sin_t)?)?,
SCALE_I + w_sq,
)?;
let psi = -fp_div_i(sin_t, w)?;
fp_div_i(2 * (chi - fp_mul_i(k as i128, psi)?), ba)?
};
let weight: i128 = if i == 0 { SCALE_I / 2 } else { SCALE_I };
total += fp_mul_i(weight, fp_mul_i(char_term, v_k)?)?;
i += 1;
}
let call_i = fp_mul_i(discount, total)?;
Ok(if call_i > 0 {
call_i as u128
} else {
0
})
}