API¶

This package provides functions for implementing the Almgren-Chriss model for optimal execution of portfolio transactions.

Modules¶

cost: Provides functions for calculating the expectation, variance and value-at-risk of the cost of trading.
decay_rate: Provides functions for calculating the trade decay rate.
trade: Provides functions for calculating the trading trajectory and the list of trades.

Functions¶

cost_expectation: Calculate the expected cost of trading.
cost_variance: Calculate the variance of the cost of trading.
value_at_risk: Calculate the value-at-risk of the cost of trading.
decay_rate: Calculate the trade decay rate.
trade_trajectory: Calculate the trading trajectory.
trade_list: Calculate the list of trades.

almgren_chriss.cost_expectation(lambda_: float, tau: float, sigma: float, gamma: float, eta: float, epsilon: float, X: float, T: float) → ndarray¶

Compute the expected cost of trading in the Almgren-Chriss model.

\[E(X) = \frac{1}{2}\gamma X^2+\epsilon X+\tilde{\eta}X^2\frac{\tanh(\frac{1}{2}\kappa\tau)\big(\tau\sinh(2\kappa T) + 2T\sinh(\kappa\tau) \big)}{2\tau^2\sinh^2(\kappa T)}\]

Parameters:

lambda (float) – Risk tolerance
tau (float) – Interval between trades
sigma (float) – Volatility
gamma (float) – Permanent impact slope
eta (float) – Temporary impact slope
epsilon (float) – Temporary impact intercept
X (float) – Total number of shares
T (float) – Trading duration

Returns:

The expected cost of trading

Return type:

float

almgren_chriss.cost_variance(lambda_: float, tau: float, sigma: float, gamma: float, eta: float, X: float, T: float) → ndarray¶

Compute the variance of the cost of trading in the Almgren-Chriss model.

\[V(X) = \frac{1}{2}\sigma^2X^2\frac{\tau\sinh(\kappa T) \cosh(\kappa(T-\tau))-T\sinh(\kappa\tau)}{\sinh^2(\kappa T)\sinh(\kappa\tau)}\]

Parameters:

lambda (float) – Risk tolerance
tau (float) – Interval between trades
sigma (float) – Volatility
gamma (float) – Permanent impact slope
eta (float) – Temporary impact slope
X (float) – Total number of shares
T (float) – Trading duration

Returns:

The variance of the cost of trading

Return type:

float

almgren_chriss.value_at_risk(lambda_: float, tau: float, sigma: float, gamma: float, eta: float, epsilon: float, X: float, T: float, probability: float = 0.95) → ndarray¶

Compute the value-at-risk of the cost of trading in the Almgren-Chriss model.

\[Var_p(x) = E(x) + \lambda_v \sqrt{V(x)}\]

Parameters:

lambda (float) – Risk tolerance
tau (float) – Interval between trades
sigma (float) – Volatility
gamma (float) – Permanent impact slope
eta (float) – Temporary impact slope
epsilon (float) – Temporary impact intercept
X (float) – Total number of shares
T (float) – Trading duration
probability (float) – Probability that the cost won’t exceed the value-at-risk.

Returns:

The value-at-risk of the cost of trading

Return type:

float

almgren_chriss.decay_rate(lambda_: float, tau: float, sigma: float, gamma: float, eta: float) → float¶

Compute the trade decay rate in the Almgren-Chriss model.

\[\kappa = \frac{\cosh^{-1}\left( \frac{\tau^2}{2} \tilde{\kappa}^2 + 1 \right)}{\tau}\]

Parameters:

lambda (float) – Risk tolerance
tau (float) – Interval between trades
sigma (float) – Volatility
gamma (float) – Permanent impact slope
eta (float) – Temporary impact slope

Returns:

The trade decay rate

Return type:

float

almgren_chriss.trade_trajectory(lambda_: float, tau: float, sigma: float, gamma: float, eta: float, X: float, T: float) → ndarray¶

Compute the trading trajectory in the Almgren-Chriss model.

\[x_j = \frac{\sinh(\kappa(T-t_j))}{\sinh(\kappa T)}X\]

Parameters:

lambda (float) – Risk tolerance
tau (float) – Interval between trades
sigma (float) – Volatility
gamma (float) – Permanent impact slope
eta (float) – Temporary impact slope
X (float) – Total number of shares
T (float) – Trading duration

Returns:

The trading trajectory

Return type:

np.ndarray

almgren_chriss.trade_list(lambda_: float, tau: float, sigma: float, gamma: float, eta: float, X: float, T: float) → ndarray¶

Compute the list of trades in the Almgren-Chriss model.

\[n_j = \frac{2\sinh(\frac{1}{2}\kappa\tau)}{\sinh(\kappa T)} \cosh\left(\kappa\left(T-t_{j-\frac{1}{2}} \right) \right)X\]

Parameters:

lambda (float) – Risk tolerance
tau (float) – Interval between trades
sigma (float) – Volatility
gamma (float) – Permanent impact slope
eta (float) – Temporary impact slope
X (float) – Total number of shares
T (float) – Trading duration

Returns:

The list of trades

Return type:

np.ndarray