XVA

Module: Shoals.Xva.

This module computes valuation-adjustment building blocks: survival and default probabilities, expected positive and negative exposure, two-deal netting, CVA and DVA on a time grid, and funding and capital adjustments. It also has a hazard-curve CVA and a sampled wrong-way-risk estimator.

Survival and default

def survival_probability_constant_hazard(hazard: f32, t: f32) -> f32
def default_probability_in_interval(hazard: f32, t_start: f32, t_end: f32) -> f32
def discount_factor_constant_rate(r: f32, t: f32) -> f32

survival_probability_constant_hazard is exp(-hazard * t), the probability of surviving to time t under a constant nonnegative hazard rate. It is one at time zero and does not increase with time. default_probability_in_interval is the probability of defaulting in [t_start, t_end], the difference of the survival probabilities at the two endpoints. discount_factor_constant_rate is exp(-r * t). For example:

s = survival_probability_constant_hazard(cast(0.02, f32), cast(1.0, f32))  // exp(-0.02)
p = default_probability_in_interval(cast(0.03, f32), cast(0.0, f32), cast(5.0, f32))
// p == 1 - survival(5.0)

Exposure and netting

def expected_positive_exposure[n](exposures: tensor[n, f32]) -> f32
def expected_negative_exposure[n](exposures: tensor[n, f32]) -> f32
def netted_exposure_2_deals[n](deal_a: tensor[n, f32], deal_b: tensor[n, f32]) -> tensor[n, f32]

expected_positive_exposure is the mean over the sample of the positive part of each exposure, and expected_negative_exposure is the mean of the negative part. netted_exposure_2_deals adds two exposure tensors pointwise, the netting of two deals under a single agreement. For example:

exposures = to_tensor([cast(-10.0, f32), cast(5.0, f32), cast(-3.0, f32), cast(20.0, f32)])
epe = expected_positive_exposure(exposures)  // (5 + 20) / 4 == 6.25
ene = expected_negative_exposure(exposures)  // (-10 - 3) / 4

CVA and DVA

def cva_constant_hazard[n](time_grid: tensor[n, f32], epe: tensor[n, f32], hazard: f32, recovery: f32, discount_rate: f32) -> f32
def dva_constant_hazard[n](time_grid: tensor[n, f32], ene: tensor[n, f32], hazard_own: f32, recovery_own: f32, discount_rate: f32) -> f32

cva_constant_hazard aggregates the credit valuation adjustment over the time grid: for each interval it multiplies the loss given default (1 - recovery), the default probability in the interval, the expected positive exposure, and the interval's discount factor, and sums the contributions. dva_constant_hazard is the symmetric debit valuation adjustment computed from the expected negative exposure and the institution's own hazard and recovery.

With finite intermediate values, CVA is zero when hazard is zero, recovery is one, or every exposure is zero. The example below uses exposures of 10, 15, and 12 at years 1, 2, and 3:

time_grid = to_tensor([cast(1.0, f32), cast(2.0, f32), cast(3.0, f32)])
epe = to_tensor([cast(10.0, f32), cast(15.0, f32), cast(12.0, f32)])
cva = cva_constant_hazard(time_grid, epe, cast(0.05, f32), cast(0.4, f32), cast(0.03, f32))

CVA is not generally increasing in hazard for a varying exposure profile. Raising hazard moves default probability toward earlier intervals; a profile with exposure concentrated later can therefore produce a smaller CVA.

A negative expected negative exposure yields a positive DVA, since the institution gains on its own default.

Funding, capital, and varying hazard

def fva[n](time_grid: tensor[n, f32], epe: tensor[n, f32], funding_spread: f32, discount_rate: f32) -> f32
def kva[n](time_grid: tensor[n, f32], ead: tensor[n, f32], cost_of_capital: f32, regulatory_capital_weight: f32, discount_rate: f32) -> f32
def xva_cva_stochastic_hazard[n, m](time_grid: tensor[m, f32], epe: tensor[m, f32], hazards: HazardCurve[n], recovery: f32, discount_rate: f32) -> f32
def xva_cva_wwr_constant_hazard[n](rng_key: key, time_grid: tensor[n, f32], epe: tensor[n, f32], hazard: f32, recovery: f32, discount_rate: f32, rho: f32, n_paths: i64) -> f32

fva multiplies the trapezoidal integral of discounted positive exposure by a supplied funding spread. kva multiplies the discounted exposure at default integral by supplied capital weight and cost. Both integrate between supplied grid points, without a contribution before the first point. xva_cva_stochastic_hazard uses a Shoals.Cds.HazardCurve to compute interval survival changes; despite its name, it does not draw random paths. xva_cva_wwr_constant_hazard samples correlated exposure and default shocks; it takes an explicit key, which can be derived from a seed.

Supply increasing times, matching tensor lengths, valid recoveries, and positive path counts where used. These functions do not model collateral or a general portfolio netting agreement.