BSLookbackOption¶

class pfhedge.nn.BSLookbackOption(call=True, strike=1.0, derivative=None)[source]¶

Black-Scholes formula for a lookback option with a fixed strike.

Note

The formulas are for continuous monitoring while pfhedge.instruments.LookbackOption monitors spot prices discretely. To get adjustment for discrete monitoring, see, for instance, Broadie, Glasserman, and Kou (1999).

See also

pfhedge.nn.BlackScholes: Initialize Black-Scholes formula module from a derivative.
pfhedge.instruments.LookbackOption: Corresponding derivative.

References

Conze, A., 1991. Path dependent options: The case of lookback options. The Journal of Finance, 46(5), pp.1893-1907.
Broadie, M., Glasserman, P. and Kou, S.G., 1999. Connecting discrete and continuous path-dependent options. Finance and Stochastics, 3(1), pp.55-82.

Parameters

call (bool, default=True) – Specifies whether the option is call or put.
strike (float, default=1.0) – The strike price of the option.

Shape:

Input: \((N, *, 4)\) where \(*\) means any number of additional dimensions. See inputs() for the names of input features.
Output: \((N, *, 1)\). All but the last dimension are the same shape as the input.

Examples

>>> from pfhedge.nn import BSLookbackOption
>>>
>>> m = BSLookbackOption()
>>> m.inputs()
['log_moneyness', 'max_log_moneyness', 'time_to_maturity', 'volatility']
>>> input = torch.tensor([
...     [-0.01, -0.01, 0.1, 0.2],
...     [ 0.00,  0.00, 0.1, 0.2],
...     [ 0.01,  0.01, 0.1, 0.2]])
>>> m(input)
tensor([[0.9208],
        [1.0515],
        [1.0515]])

delta(log_moneyness=None, max_log_moneyness=None, time_to_maturity=None, volatility=None, create_graph=False)[source]¶

Returns delta of the derivative.

Parameters

log_moneyness (torch.Tensor, optional) – Log moneyness of the underlying asset.
max_log_moneyness (torch.Tensor, optional) – Cumulative maximum of the log moneyness.
time_to_maturity (torch.Tensor, optional) – Time to expiry of the option.
volatility (torch.Tensor, optional) – Volatility of the underlying asset.
create_graph (bool, default=False) – If True, graph of the derivative will be constructed. This option is used to compute gamma.

Shape:

log_moneyness: \((N, *)\) where \(*\) means any number of additional dimensions.
max_log_moneyness: \((N, *)\)
time_to_maturity: \((N, *)\)
volatility: \((N, *)\)
output: \((N, *)\)

Returns: torch.Tensor

Note

Parameters are not optional if the module has not accepted a derivative in its initialization.

forward(input)¶

Returns delta of the derivative.

Parameters: input (torch.Tensor) – The input tensor. Features are concatenated along the last dimension. See inputs() for the names of the input features.
Returns: torch.Tensor

classmethod from_derivative(derivative)[source]¶

Initialize a module from a derivative.

Parameters: derivative (pfhedge.instruments.LookbackOption) – The derivative to get the Black-Scholes formula.
Returns: BSLookbackOption

Examples

>>> from pfhedge.instruments import BrownianStock
>>> from pfhedge.instruments import LookbackOption
>>>
>>> derivative = LookbackOption(BrownianStock(), strike=1.1)
>>> m = BSLookbackOption.from_derivative(derivative)
>>> m
BSLookbackOption(strike=1.1000)

gamma(log_moneyness=None, max_log_moneyness=None, time_to_maturity=None, volatility=None)[source]¶

Returns gamma of the derivative.

Parameters

log_moneyness (torch.Tensor, optional) – Log moneyness of the underlying asset.
max_log_moneyness (torch.Tensor, optional) – Cumulative maximum of the log moneyness.
time_to_maturity (torch.Tensor, optional) – Time to expiry of the option.
volatility (torch.Tensor, optional) – Volatility of the underlying asset.

Shape:

log_moneyness: \((N, *)\) where \(*\) means any number of additional dimensions.
max_log_moneyness: \((N, *)\)
time_to_maturity: \((N, *)\)
volatility: \((N, *)\)
output: \((N, *)\)

Returns: torch.Tensor

Note

Arguments are not optional if it doesn’t accept derivative in this initialization.

implied_volatility(log_moneyness=None, max_log_moneyness=None, time_to_maturity=None, price=None, precision=1e-06)[source]¶

Returns implied volatility of the derivative.

Parameters

log_moneyness (torch.Tensor, optional) – Log moneyness of the underlying asset.
max_log_moneyness (torch.Tensor, optional) – Cumulative maximum of the log moneyness.
time_to_maturity (torch.Tensor, optional) – Time to expiry of the option.
price (torch.Tensor) – Price of the derivative.
precision (float, default=1e-6) – Precision of the implied volatility.

Shape:

log_moneyness: \((N, *)\) where \(*\) means any number of additional dimensions.
max_log_moneyness: \((N, *)\)
time_to_maturity: \((N, *)\)
price: \((N, *)\)
output: \((N, *)\)

Returns: torch.Tensor

Note

Arguments are not optional if it doesn’t accept derivative in this initialization.

inputs()[source]¶

Returns the names of input features.

Returns: list

price(log_moneyness=None, max_log_moneyness=None, time_to_maturity=None, volatility=None)[source]¶

Returns price of the derivative.

The price is given by:

\[\begin{split}\begin{cases} S(0) \{ N(d_1) + \sigma \sqrt{T} [N'(d_1) + d_1 N(d_1)] \} - K N(d_2) & (M \leq K) \\ S(0) \{ N(d_1') + \sigma \sqrt{T} [N'(d_1') + d_1' N(d_1')] \} - K + M [1 - N(d_2')] & (M > K) \\ \end{cases}\end{split}\]

where \(M = \max_{t < 0} S(t)\), \(d_1' = [\log(S(0) / M) + \frac12 \sigma^2 T] / \sigma \sqrt{T}\), and \(d_2' = [\log(S(0) / M) - \frac12 \sigma^2 T] / \sigma \sqrt{T}\).

Parameters

log_moneyness (torch.Tensor, optional) – Log moneyness of the underlying asset.
max_log_moneyness (torch.Tensor, optional) – Cumulative maximum of the log moneyness.
time_to_maturity (torch.Tensor, optional) – Time to expiry of the option.
volatility (torch.Tensor, optional) – Volatility of the underlying asset.

Shape:

log_moneyness: \((N, *)\) where \(*\) means any number of additional dimensions.
max_log_moneyness: \((N, *)\)
time_to_maturity: \((N, *)\)
volatility: \((N, *)\)
output: \((N, *)\)

Returns: torch.Tensor

Note

Parameters are not optional if the module has not accepted a derivative in its initialization.

theta(log_moneyness=None, max_log_moneyness=None, time_to_maturity=None, volatility=None)[source]¶

Returns theta of the derivative.

Parameters

log_moneyness (torch.Tensor, optional) – Log moneyness of the underlying asset.
max_log_moneyness (torch.Tensor, optional) – Cumulative maximum of the log moneyness.
time_to_maturity (torch.Tensor, optional) – Time to expiry of the option.
volatility (torch.Tensor, optional) – Volatility of the underlying asset.

Shape:

log_moneyness: \((N, *)\) where \(*\) means any number of additional dimensions.
max_log_moneyness: \((N, *)\)
time_to_maturity: \((N, *)\)
volatility: \((N, *)\)
output: \((N, *)\)

Returns: torch.Tensor

Note

Risk-free rate is set to zero.
Arguments are not optional if it doesn’t accept derivative in this initialization.

vega(log_moneyness=None, max_log_moneyness=None, time_to_maturity=None, volatility=None)[source]¶

Returns vega of the derivative.

Parameters

log_moneyness (torch.Tensor, optional) – Log moneyness of the underlying asset.
max_log_moneyness (torch.Tensor, optional) – Cumulative maximum of the log moneyness.
time_to_maturity (torch.Tensor, optional) – Time to expiry of the option.
volatility (torch.Tensor, optional) – Volatility of the underlying asset.

Shape:

log_moneyness: \((N, *)\) where \(*\) means any number of additional dimensions.
max_log_moneyness: \((N, *)\)
time_to_maturity: \((N, *)\)
volatility: \((N, *)\)
output: \((N, *)\)

Returns: torch.Tensor

Note

Arguments are not optional if it doesn’t accept derivative in this initialization.