Feature Importance in Option Pricing with Captum

Deep neural networks map a set of features through layers of many neurons each to make a prediction of a target value. Many use cases of deep neural networks only care about the accuracy of the prediction, and don't care about how the features are related to the target. There are other sets of problems however, that are concerned about how each feature affect the target.

For example, you may be predicting default on auto leans and want to know whether debt-to-income levels affect the probability of default. If they do, you may change your auto loan origination methods. A logistic regression will give you a prediction of default, along with an estimate of how a given increase in debt-to-income will increase the probability of default. A deep neural network alone will just give you a prediction. However research in feature attribution, and the Captum library, want to change that.

In these notes we will use a different example of the need for feature attribution—option pricing. When using options pricing models (like Black-Scholes¹) the option value is only one of the important outputs of the model. The model also affords the Greeks, which are measures of how sensitive the option price is to the input values. For example, an options Delta tells how much the options price will change given some change in the underlying's price.

Here we'll use deep learning to value an interesting type of option—a Financial Transmission Right option. An FTR pays the difference between the congestion prices at two points on the electricity grid. An FTR option pays the difference if it is positive, or $0 otherwise. To simulate FTR option values we can simulate electricity price processes at two points on the grid. To do so we generate two correlated processes which follow the following stochastic differential equation:

\[dE_t = \kappa(\mu - E_t)dt + \sigma E_t dB_t\]

where:

• \(E_t\) is the electricity price at a given node
• \(\kappa\) is the rate of mean reversion
• \(\mu\) is the mean electricity price at the node
• \(\sigma\) is the price volatility at the node

Import data from Monte Carlo simulations:

     risk_free      vol1      vol2      corr        k1        k2   e1_start   e2_start        mu1        mu2  option_value
0     0.000246  0.022983  0.051694 -0.275666  0.547739  0.586659  26.736341  27.136899  25.858735  34.003442      6.797245
1     0.000058  0.029828  0.005129 -0.022649  0.720233  0.207708  33.555252  26.736119  29.976300  33.063499      5.440954
2     0.000321  0.053443  0.020939  0.391795  0.308738  0.046923  31.241577  27.143763  33.058117  27.189379     10.974798
3     0.000130  0.033620  0.016250 -0.003000  0.654085  0.382512  34.044292  33.778042  34.696238  25.894681      8.243242
4     0.000210  0.057207  0.021190  0.484918  0.575378  0.668296  30.335520  28.391513  25.502045  32.433721      6.613631
..         ...       ...       ...       ...       ...       ...        ...        ...        ...        ...           ...
995   0.000229  0.024420  0.032109  0.168135  0.370507  0.621441  26.101353  34.332895  28.746192  32.059997      3.265470
996   0.000084  0.006779  0.007712 -0.466865  0.243271  0.127567  30.873648  32.967921  34.333053  26.092348      2.313836
997   0.000076  0.015852  0.042681  0.262823  0.070187  0.369278  33.930473  34.468095  27.403043  33.487219      8.132854
998   0.000142  0.059682  0.062720 -0.049992  0.602649  0.508839  31.113604  30.427855  31.950044  32.877230     12.834470
999   0.000111  0.047834  0.032335 -0.123484  0.526323  0.366141  25.256368  28.618138  25.730874  25.107557      7.182105

[9000 rows x 11 columns]

model = nn.Sequential(
    nn.Linear(10, 400),
    nn.ReLU(),
    nn.Linear(400, 400),
    nn.ReLU(),
    nn.Linear(400, 400),
    nn.ReLU(),
    nn.Linear(400, 400),
    nn.ReLU(),
    nn.Linear(400, 400),
    nn.ReLU(),
    nn.Linear(400, 1)
)

model.load_state_dict(torch.load("./FTR_model_july_27_4000_inputs.pt"))

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