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use ethers::types::{I256, U256};
use eyre::{eyre, Result};
use fixedpointmath::{fixed, FixedPoint};
use crate::{State, YieldSpace};
impl State {
/// Gets a target long that can be opened given a budget to achieve a
/// desired fixed rate.
///
/// If the long amount to reach the target is greater than the budget,
/// the budget is returned.
/// If the long amount to reach the target is invalid (i.e. it would produce
/// an insolvent pool), then an error is thrown, and the user is advised to
/// use [calculate_max_long](State::calculate_max_long).
pub fn calculate_targeted_long_with_budget<
F1: Into<FixedPoint<U256>>,
F2: Into<FixedPoint<U256>>,
F3: Into<FixedPoint<U256>>,
I: Into<I256>,
>(
&self,
budget: F1,
target_rate: F2,
checkpoint_exposure: I,
maybe_max_iterations: Option<usize>,
maybe_allowable_error: Option<F3>,
) -> Result<FixedPoint<U256>> {
let budget = budget.into();
match self.calculate_targeted_long(
target_rate,
checkpoint_exposure,
maybe_max_iterations,
maybe_allowable_error,
) {
Ok(long_amount) => Ok(long_amount.min(budget)),
Err(error) => Err(error),
}
}
/// Gets a target long that can be opened to achieve a desired fixed rate.
fn calculate_targeted_long<
F1: Into<FixedPoint<U256>>,
F2: Into<FixedPoint<U256>>,
I: Into<I256>,
>(
&self,
target_rate: F1,
checkpoint_exposure: I,
maybe_max_iterations: Option<usize>,
maybe_allowable_error: Option<F2>,
) -> Result<FixedPoint<U256>> {
// Check input args.
let target_rate = target_rate.into();
let checkpoint_exposure = checkpoint_exposure.into();
let allowable_error = match maybe_allowable_error {
Some(allowable_error) => allowable_error.into(),
None => fixed!(1e15),
};
let current_rate = self.calculate_spot_rate()?;
if target_rate >= current_rate {
return Err(eyre!(
"target_rate = {} argument must be less than the current_rate = {} for a targeted long.",
target_rate, current_rate,
));
}
// Estimate the long that achieves a target rate.
let (target_pool_share_reserves, target_pool_bond_reserves) =
self.reserves_given_rate_ignoring_exposure(target_rate)?;
let (mut target_user_base_delta, target_user_bond_delta) = self
.long_trade_needed_given_reserves(
target_pool_share_reserves,
target_pool_bond_reserves,
)?;
// Determine what rate was achieved.
let resulting_rate = self
.calculate_spot_rate_after_long(target_user_base_delta, Some(target_user_bond_delta))?;
// The estimated long will usually underestimate because the realized price
// should always be greater than the spot price.
//
// However, if we overshot the zero-crossing (due to errors arising from FixedPoint<U256> arithmetic),
// then either return or reduce the starting base amount and start on Newton's method.
if target_rate > resulting_rate {
let rate_error = target_rate - resulting_rate;
// If we were still close enough and solvent, return.
if self
.solvency_after_long(
target_user_base_delta,
target_user_bond_delta,
checkpoint_exposure,
)
.is_ok()
&& rate_error < allowable_error
{
return Ok(target_user_base_delta);
}
// Else, cut the initial guess down by an order of magnitude and go to Newton's method.
else {
target_user_base_delta /= fixed!(10e18);
}
}
// Else check if we are close enough to return.
else {
// If solvent & within the allowable error, stop here.
let rate_error = resulting_rate - target_rate;
if self
.solvency_after_long(
target_user_base_delta,
target_user_bond_delta,
checkpoint_exposure,
)
.is_ok()
&& rate_error < allowable_error
{
return Ok(target_user_base_delta);
}
}
// Iterate to find a solution.
// We can use the initial guess as a starting point since we know it is less than the target.
let mut possible_target_base_delta = target_user_base_delta;
// Iteratively find a solution
for _ in 0..maybe_max_iterations.unwrap_or(7) {
let possible_target_bond_delta =
self.calculate_open_long(possible_target_base_delta)?;
let resulting_rate = self.calculate_spot_rate_after_long(
possible_target_base_delta,
Some(possible_target_bond_delta),
)?;
// We assume that the loss is positive only because Newton's
// method will always underestimate.
if target_rate > resulting_rate {
return Err(eyre!(
"We overshot the zero-crossing during Newton's method.",
));
}
// We choose the difference between the rates as the loss because it
// is convex given the above check, differentiable almost everywhere,
// and has a simple derivative.
let loss = resulting_rate - target_rate;
// If solvent & within error, then return the value.
if self
.solvency_after_long(
possible_target_base_delta,
possible_target_bond_delta,
checkpoint_exposure,
)
.is_ok()
&& loss < allowable_error
{
return Ok(possible_target_base_delta);
}
// Otherwise perform another iteration.
else {
// The derivative of the loss is $l'(x) = r'(x)$.
// We return $-l'(x)$ because $r'(x)$ is negative, which
// can't be represented with FixedPoint<U256>.
let negative_loss_derivative = self.rate_after_long_derivative_negation(
possible_target_base_delta,
possible_target_bond_delta,
)?;
// Adding the negative loss derivative instead of subtracting the loss derivative
// ∆x_{n+1} = ∆x_{n} - l / l'
// = ∆x_{n} + l / (-l')
possible_target_base_delta += loss / negative_loss_derivative;
}
}
// Final solvency check.
if self
.solvency_after_long(
possible_target_base_delta,
self.calculate_open_long(possible_target_base_delta)?,
checkpoint_exposure,
)
.is_err()
{
return Err(eyre!("Guess in `calculate_targeted_long` is insolvent."));
}
// Final accuracy check.
let possible_target_bond_delta = self.calculate_open_long(possible_target_base_delta)?;
let resulting_rate = self.calculate_spot_rate_after_long(
possible_target_base_delta,
Some(possible_target_bond_delta),
)?;
if target_rate > resulting_rate {
return Err(eyre!(
"We overshot the zero-crossing after Newton's method.",
));
}
let loss = resulting_rate - target_rate;
if loss >= allowable_error {
return Err(eyre!(
"Unable to find an acceptable loss with max iterations. Final loss = {}.",
loss
));
}
Ok(possible_target_base_delta)
}
/// The derivative of the equation for calculating the rate after a long.
///
/// For some `$r = \frac{(1 - p(x))}{(p(x) \cdot t)}$`, where $p(x)$
/// is the spot price after a long of `delta_base``$= x$` was opened and
/// `$t$` is the annualized position duration, the rate derivative is:
///
/// ```math
/// r'(x) = \frac{(-p'(x) \cdot p(x) t
/// - (1 - p(x)) (p'(x) \cdot t))}{(p(x) \cdot t)^2} //
/// r'(x) = \frac{-p'(x)}{t \cdot p(x)^2}
/// ```
///
/// We return `$-r'(x)$` because negative numbers cannot be represented by FixedPoint<U256>.
fn rate_after_long_derivative_negation(
&self,
base_amount: FixedPoint<U256>,
bond_amount: FixedPoint<U256>,
) -> Result<FixedPoint<U256>> {
let price = self.calculate_spot_price_after_long(base_amount, Some(bond_amount))?;
let price_derivative = self.price_after_long_derivative(base_amount, bond_amount)?;
// The actual equation we want to represent is:
// r' = -p' / (t p^2)
// We can do a trick to return a positive-only version and
// indicate that it should be negative in the fn name.
// We use price * price instead of price.pow(fixed!(2e18)) to avoid error introduced by pow.
Ok(price_derivative / (self.annualized_position_duration() * price * price))
}
/// The derivative of the price after a long.
///
/// The price after a long that moves shares by $\Delta z$ and bonds by
/// `$\Delta y$` is equal to:
///
/// ```math
/// p(\Delta z) = \left( \frac{\mu \cdot
/// (z_{0} + \Delta z - (\zeta_{0} + \Delta \zeta))}
/// {y - \Delta y} \right)^{t_{s}}
/// ```
///
/// where `$t_s$` is the time stretch constant and `$z_{e,0}$` is the
/// initial effective share reserves, and `$\zeta$` is the zeta adjustment.
/// The zeta adjustment is constant when opening a long, i.e.
/// `$\Delta \zeta = 0$`, so we drop the subscript. Equivalently, for some
/// amount of `delta_base` `$= \Delta x$` provided to open a long,
/// we can write:
///
/// ```math
/// p(\Delta x) = \left(
/// \frac{\mu (z_{0} + \frac{1}{c}
/// \cdot \left( \Delta x - \Phi_{g,ol}(\Delta x) \right) - \zeta)}
/// {y_0 - y(\Delta x)}
/// \right)^{t_{s}}
/// ```
///
/// where `$\Phi_{g,ol}(\Delta x)$` is the
/// [open_long_governance_fee](State::open_long_governance_fee),
/// `$y(\Delta x)$` is the [bond_amount](State::calculate_open_long),
///
/// To compute the derivative, we first define some auxiliary variables:
///
/// ```math
/// a(\Delta x) &= \mu (z_{0} + \frac{\Delta x}{c} - \frac{\Phi_{g,ol}(\Delta x)}{c} - \zeta) \\
/// &= \mu \left( z_{e,0} + \frac{\Delta x}{c} - \frac{\Phi_{g,ol}(\Delta x)}{c} \right) \\
/// b(\Delta x) &= y_0 - y(\Delta x) \\
/// v(\Delta x) &= \frac{a(\Delta x)}{b(\Delta x)}
/// ```
///
/// and thus `$p(\Delta x) = v(\Delta x)^{t_{s}}$`.
/// Given these, we can write out intermediate derivatives:
///
/// ```math
/// \begin{aligned}
/// a'(\Delta x) &= \frac{\mu}{c} (1 - \Phi_{g,ol}'(\Delta x)) \\
/// b'(\Delta x) &= -y'(\Delta x) \\
/// v'(\Delta x) &= \frac{b(\Delta x) \cdot a'(\Delta x) - a(\Delta x) \cdotb'(\Delta x)}{b(\Delta x)^2}
/// \end{aligned}
/// ```
///
/// And finally, the price after long derivative is:
///
/// ```math
/// p'(\Delta x) = v'(\Delta x) \cdot t_{s} \cdot v(\Delta x)^{(t_{s} - 1)}
/// ```
///
fn price_after_long_derivative(
&self,
base_amount: FixedPoint<U256>,
bond_amount: FixedPoint<U256>,
) -> Result<FixedPoint<U256>> {
// g'(x) = \phi_g \phi_c (1 - p_0)
let gov_fee_derivative = self.governance_lp_fee()
* self.curve_fee()
* (fixed!(1e18) - self.calculate_spot_price()?);
// a(x) = mu * (z_{e,0} + 1/c (x - g(x))
let inner_numerator = self.mu()
* (self.ze()?
+ (base_amount - self.open_long_governance_fee(base_amount, None)?)
.div_down(self.vault_share_price()));
// a'(x) = (mu / c) (1 - g'(x))
let inner_numerator_derivative = self
.mu()
.mul_div_down(fixed!(1e18) - gov_fee_derivative, self.vault_share_price());
//(self.mu() / self.vault_share_price()) * (fixed!(1e18) - gov_fee_derivative);
// b(x) = y_0 - y(x)
let inner_denominator = self.bond_reserves() - bond_amount;
// b'(x) = -y'(x)
// -b'(x) = y'(x)
let long_amount_derivative = self.calculate_open_long_derivative(base_amount)?;
// v(x) = a(x) / b(x)
// v'(x) = ( b(x) * a'(x) - a(x) * b'(x) ) / b(x)^2
// = ( b(x) * a'(x) + a(x) * -b'(x) ) / b(x)^2
// Note that we are adding the negative b'(x) to avoid negative fixedpoint numbers
let inner_derivative = (inner_denominator * inner_numerator_derivative
+ inner_numerator * long_amount_derivative)
/ (inner_denominator * inner_denominator);
// p'(x) = v'(x) * t_s * v(x)^(t_s - 1)
// p'(x) = v'(x) * t_s * v(x)^(-1)^(1 - t_s)
// v(x) is flipped to (denominator / numerator) to avoid a negative exponent
Ok(inner_derivative
* self.time_stretch()
* (inner_denominator / inner_numerator).pow(fixed!(1e18) - self.time_stretch())?)
}
/// Calculate the base & bond trade amount for an open long trade that moves the
/// current state to the given desired ending reserve levels.
///
/// Given a target ending pool share reserves, `$z_1$`, and bond reserves,
/// `$y_1$`, the trade deltas to achieve that state would be:
///
/// From the pool's perspective:
/// ```math
/// z_1 = z_0 + \Delta z \\
/// \Delta z = z_1 - z_0
/// ```
///
/// From the trader's perspective, for a provided `base_amount`
/// `$= \Delta x$`, the pool share reserves update, `$\Delta z$`, would be:
/// ```math
/// \Delta z = \frac{1}{c} (\Delta x - \Phi_{g,ol}(\Delta x))
/// ```
///
/// Solving for the change in base:
/// ```math
/// \begin{aligned}
/// c \cdot \Delta z
/// &= \Delta x - \Phi_{g,ol}(\Delta x) \\
/// c \cdot \Delta z
/// &= \Delta x - (1 - p) \cdot \phi_c \cdot \phi_g \cdot \Delta x \\
/// c \cdot \Delta z
/// &= \Delta x \cdot (1 - (1 - p) \cdot \phi_c \cdot \phi_g) \\
/// &\therefore \\
/// \Delta x
/// &= \frac{c \cdot \Delta z}{(1 - (1 - p) \cdot \phi_c \cdot \phi_g)}
/// \end{aligned}
/// ```
///
/// These should be equal, therefore:
/// ```math
/// \Delta x = \frac{c \cdot \Delta z}{(1 - (1 - p) \cdot \phi_c \cdot \phi_g)} \\
/// \Delta x = \frac{c \cdot (z_1 - z_0)}{(1 - (1 - p) \cdot \phi_c \cdot \phi_g)} \\
/// ```
/// where `$c$` is the vault share price, `$p$` is the original spot price,
/// and `$\Phi_{g,ol}(\Delta x)$` is the
/// (open long governance fee)[State::open_long_governance_fee].
///
/// The change in bonds, $\Delta y$ is equal for the trader and the pool and
/// can be determined with (`calculate_open_long`)[State::calculate_open_long].
fn long_trade_needed_given_reserves(
&self,
ending_share_reserves: FixedPoint<U256>,
ending_bond_reserves: FixedPoint<U256>,
) -> Result<(FixedPoint<U256>, FixedPoint<U256>)> {
if self.bond_reserves() < ending_bond_reserves {
return Err(eyre!(
"expected bond_reserves={} >= ending_bond_reserves={}",
self.bond_reserves(),
ending_bond_reserves,
));
}
if ending_share_reserves < self.share_reserves() {
return Err(eyre!(
"expected ending_share_reserves={} >= share_reserves={}",
ending_share_reserves,
self.share_reserves(),
));
}
let share_delta = ending_share_reserves - self.share_reserves();
let fees = fixed!(1e18)
- (fixed!(1e18) - self.calculate_spot_price()?)
* self.curve_fee()
* self.governance_lp_fee();
let base_delta = self.vault_share_price().mul_div_down(share_delta, fees);
let bond_delta = self.calculate_open_long(base_delta)?;
Ok((base_delta, bond_delta))
}
}
#[cfg(test)]
mod tests {
use std::panic;
use ethers::types::U256;
use fixedpointmath::{uint256, FixedPointValue};
use hyperdrive_test_utils::{chain::TestChain, constants::FUZZ_RUNS};
use rand::{thread_rng, Rng};
use super::*;
use crate::test_utils::agent::HyperdriveMathAgent;
#[tokio::test]
async fn fuzz_long_trade_needed_given_reserves() -> Result<()> {
let base_reserve_test_tolerance = fixed!(1e10);
let bond_reserve_test_tolerance = fixed!(1e10);
let mut rng = thread_rng();
for _ in 0..*FUZZ_RUNS {
let state = rng.gen::<State>();
// Get a random long trade amount.
let checkpoint_exposure = rng
.gen_range(fixed!(0)..=FixedPoint::<I256>::MAX)
.raw()
.flip_sign_if(rng.gen());
let max_long_trade = match panic::catch_unwind(|| {
state.calculate_max_long(U256::MAX, checkpoint_exposure, None)
}) {
Ok(max_trade) => match max_trade {
Ok(max_trade) => max_trade,
Err(_) => continue, // Max threw an Err
},
Err(_) => continue, // Max threw a panic
};
let long_base_amount =
rng.gen_range(state.minimum_transaction_amount()..=max_long_trade);
// Do the long to see the bond delta (same amount for the user & pool in this case).
let long_bond_amount = state.calculate_open_long(long_base_amount)?;
// Get the reserve levels after the state was updated from the open long.
let updated_state = state
.calculate_pool_state_after_open_long(long_base_amount, Some(long_bond_amount))?;
let (final_share_reserves, final_bond_reserves) = (
FixedPoint::from(updated_state.info.share_reserves),
FixedPoint::from(updated_state.info.bond_reserves),
);
// Estimate the trade amounts from the final reserve levels.
let (estimated_base_amount, estimated_bond_amount) = state
.long_trade_needed_given_reserves(final_share_reserves, final_bond_reserves)?;
// Make sure the estimates match the realized transaction amounts.
let base_error = if estimated_base_amount > long_base_amount {
estimated_base_amount - long_base_amount
} else {
long_base_amount - estimated_base_amount
};
assert!(
base_error <= base_reserve_test_tolerance,
"expected abs(estimated_base_amount={}-long_base_amount={})={} <= test_tolerance={}",
estimated_base_amount,
long_base_amount,
base_error,
base_reserve_test_tolerance,
);
let bond_error = if estimated_bond_amount > long_bond_amount {
estimated_bond_amount - long_bond_amount
} else {
long_bond_amount - estimated_bond_amount
};
assert!(
bond_error <= bond_reserve_test_tolerance,
"expected abs(estimated_bond_amount={}-long_bond_amount={})={} <= test_tolerance={}",
estimated_bond_amount,
long_bond_amount,
bond_error,
bond_reserve_test_tolerance,
);
}
Ok(())
}
#[tokio::test]
async fn test_calculate_targeted_long_with_budget() -> Result<()> {
// Spawn a test chain and create two agents -- Alice and Bob.
// Alice is funded with a large amount of capital so that she can initialize
// the pool. Bob is funded with a random amount of capital so that we
// can test `calculate_targeted_long` when budget is the primary constraint
// and when it is not.
let allowable_solvency_error = fixed!(1e5);
let allowable_budget_error = fixed!(1e5);
let allowable_rate_error = fixed!(1e11);
let num_newton_iters = 7;
// Initialize a test chain and agents.
let chain = TestChain::new().await?;
let mut alice = chain.alice().await?;
let mut bob = chain.bob().await?;
let config = bob.get_config().clone();
// Fuzz test
let mut rng = thread_rng();
for _ in 0..*FUZZ_RUNS {
// Snapshot the chain.
let id = chain.snapshot().await?;
// Alice initializes the pool.
// Large budget for initializing the pool.
let contribution = fixed!(1_000_000e18);
alice.fund(contribution).await?;
let initial_fixed_rate = rng.gen_range(fixed!(0.01e18)..=fixed!(0.1e18));
alice
.initialize(initial_fixed_rate, contribution, None)
.await?;
// Small lower bound on Bob's budget for resource-constrained targeted longs.
let budget = rng.gen_range(fixed!(10e18)..=fixed!(500_000_000e18));
// Half the time we will open a long & let it mature.
if rng.gen_range(0..=1) == 0 {
// Open a long.
let max_long =
bob.get_state()
.await?
.calculate_max_long(U256::MAX, I256::from(0), None)?;
let long_amount =
(max_long / fixed!(100e18)).max(config.minimum_transaction_amount.into());
bob.fund(long_amount + budget).await?;
bob.open_long(long_amount, None, None).await?;
// Advance time to just after maturity.
let variable_rate = rng.gen_range(fixed!(0)..=fixed!(0.5e18));
let time_amount = FixedPoint::from(config.position_duration) * fixed!(1.05e18); // 1.05 * position_duraiton
alice.advance_time(variable_rate, time_amount).await?;
// Checkpoint to auto-close the position.
alice
.checkpoint(alice.latest_checkpoint().await?, uint256!(0), None)
.await?;
}
// Else we will just fund a random budget amount and do the targeted long.
else {
bob.fund(budget).await?;
}
// Some of the checkpoint passes and variable interest accrues.
alice
.checkpoint(alice.latest_checkpoint().await?, uint256!(0), None)
.await?;
let variable_rate = rng.gen_range(fixed!(0)..=fixed!(0.5e18));
alice
.advance_time(
variable_rate,
FixedPoint::from(config.checkpoint_duration) * fixed!(0.5e18),
)
.await?;
// Get a targeted long amount.
let target_rate = bob.get_state().await?.calculate_spot_rate()?
/ rng.gen_range(fixed!(1.0001e18)..=fixed!(10e18));
let targeted_long_result = bob
.calculate_targeted_long(
target_rate,
Some(num_newton_iters),
Some(allowable_rate_error),
)
.await;
// Bob opens a targeted long.
let current_state = bob.get_state().await?;
let max_spot_price_before_long = current_state.calculate_max_spot_price()?;
match targeted_long_result {
// If the code ran without error, open the long
Ok(targeted_long) => {
bob.open_long(targeted_long, None, None).await?;
}
// Else parse the error for a to improve error messaging.
Err(e) => {
// If the fn failed it's possible that the target rate would be insolvent.
if e.to_string()
.contains("Unable to find an acceptable loss with max iterations")
{
let max_long = bob.calculate_max_long(None).await?;
let rate_after_max_long =
current_state.calculate_spot_rate_after_long(max_long, None)?;
// If the rate after the max long is at or below the target, then we could have hit it.
if rate_after_max_long <= target_rate {
return Err(eyre!(
"ERROR {}\nA long that hits the target rate exists but was not found.",
e
));
}
// Otherwise the target would have resulted in insolvency and wasn't possible.
else {
return Err(eyre!(
"ERROR {}\nThe target rate would result in insolvency.",
e
));
}
}
// If the error is not the one we're looking for, return it, causing the test to fail.
else {
return Err(e);
}
}
}
// Three things should be true after opening the long:
//
// 1. The pool's spot price is under the max spot price prior to
// considering fees
// 2. The pool's solvency is above zero.
// 3. IF Bob's budget is not consumed; then new rate is close to the target rate
// Check that our resulting price is under the max
let current_state = alice.get_state().await?;
let spot_price_after_long = current_state.calculate_spot_price()?;
assert!(
max_spot_price_before_long > spot_price_after_long,
"Resulting price is greater than the max."
);
// Check solvency
let is_solvent = { current_state.calculate_solvency()? > allowable_solvency_error };
assert!(is_solvent, "Resulting pool state is not solvent.");
let new_rate = current_state.calculate_spot_rate()?;
// If the budget was NOT consumed, then we assume the target was hit.
if bob.base() > allowable_budget_error {
// Actual price might result in long overshooting the target.
let abs_error = if target_rate > new_rate {
target_rate - new_rate
} else {
new_rate - target_rate
};
assert!(
abs_error <= allowable_rate_error,
"target_rate was {}, realized rate is {}. abs_error={} was not <= {}.",
target_rate,
new_rate,
abs_error,
allowable_rate_error
);
}
// Else, we should have undershot,
// or by some coincidence the budget was the perfect amount
// and we hit the rate exactly.
else {
assert!(
new_rate >= target_rate,
"The new_rate={} should be >= target_rate={} when budget constrained.",
new_rate,
target_rate
);
}
// Revert to the snapshot and reset the agent's wallets.
chain.revert(id).await?;
alice.reset(Default::default()).await?;
bob.reset(Default::default()).await?;
}
Ok(())
}
}