XVA

Module: Shoals.Xva.

This module computes the building blocks of valuation adjustments under a constant-hazard credit model: survival and default probabilities, a constant-rate discount factor, expected positive and negative exposure, pointwise two-deal netting, and the credit and debit valuation adjustments aggregated over a discrete time grid.

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 hazard rate. It is one at time zero and decreases 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). From tests/xva.ch:

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. From tests/xva.ch:

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.

From tests/xva.ch, the CVA is zero when there is no default risk, when recovery is full (a recovery rate of one), or when the exposure is zero, and it increases with the hazard rate:

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))

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