Pytorch 中用于自动调用选项的梯度流

Question

我有以下代码：

import numpy as np
import torch
from torch import autograd

# Define the parameters with requires_grad=True
r = torch.tensor(0.03, requires_grad=True)
q = torch.tensor(0.02, requires_grad=True)
v = torch.tensor(0.14, requires_grad=True)
S = torch.tensor(1001.0, requires_grad=True)

# Generate random numbers and other tensors
Z = torch.randn(10000, 5)
t = torch.tensor(np.arange(1.0, 6.0))
c = torch.tensor([0.2, 0.3, 0.4, 0.5, 0.6])

# Calculate mc_S with differentiable operations
mc_S = S * torch.exp((r - q - 0.5 * v * v) * t + Z.cumsum(axis=1))

# Calculate payoff with differentiable operations
res = []
mask = 1.0
for col, coup in zip(mc_S.T, c):
    payoff = mask * torch.where(col > S, coup, torch.tensor(0.0))
    res.append(payoff)
    mask = mask * (payoff == 0)

v = torch.stack(res).T
result = v.sum(axis=1).mean()

# Compute gradients - breaks here
grads = autograd.grad(result, [r, q, v, S], allow_unused=True, retain_graph=True)
print(grads)

我正在尝试为具有早期淘汰的可自动调用选项定价，并且需要输入变量的敏感性。

但是，优惠券的计算方式（上面代码中的 c 张量）破坏了计算图，我无法获得梯度。有没有办法让这个代码来计算导数？

谢谢

Answer 1

torch.where(col > S, coup, torch.tensor(0.0))

不是可微分运算。它没有平滑度，您要么选择

coup

要么选择

0

。该函数没有斜率。

coup * torch.sigmoid(col - S)

将为您提供与您的操作类似但可微分的结果。

在您的示例中，当

coup

时，我们从

col > S

中进行选择，否则选择

0

。在可微分版本中，当

coup

时我们得到

col >> S

，当

0

时我们得到

col << S

。在中间，我们得到了介于两者之间的东西，您可能需要调整比例和偏移量以获得应用程序所需的确切损失函数，例如

alpha * (col + beta / alpha > S)

。

import numpy as np
import torch
from torch import autograd

# Define the parameters with requires_grad=True
r = torch.tensor(0.03, requires_grad=True)
q = torch.tensor(0.02, requires_grad=True)
v = torch.tensor(0.14, requires_grad=True)
S = torch.tensor(1001.0, requires_grad=True)

# Generate random numbers and other tensors
Z = torch.randn(10000, 5)
t = torch.tensor(np.arange(1.0, 6.0))
c = torch.tensor([0.2, 0.3, 0.4, 0.5, 0.6])

# Calculate mc_S with differentiable operations
mc_S = S * torch.exp((r - q - 0.5 * v * v) * t + Z.cumsum(axis=1))

# Calculate payoff with differentiable operations
res = []
mask = 1.0
for col, coup in zip(mc_S.T, c):
    payoff = coup * torch.sigmoid(col - S)
    res.append(payoff)
    mask = mask * (payoff == 0)

v = torch.stack(res).T
result = v.sum(axis=1).mean()

# Compute gradients - breaks here
grads = autograd.grad(result, [r, q, v, S], allow_unused=True, retain_graph=True)
print(grads)