Joy Omondi
Joy Omondi

LSTM models for options pricing

LSTM models for options pricing

Long Short-Term Memory Networks For Options pricing

Introduction

A while back I did a blog on derivatives specifically focusing options. In case you are new to the concept of options and options pricing you can check out my blog on that.

Traditional options pricing models like black-Scholes often assumes that there is constant volatility,Log normal returns, minimal market shocks and absence of path dependence. Deep learning models like the LSTMs come in to account for the short comings.

In this blog post, we’ll explore what LSTMs are, how they work, and why they’re so powerful for sequential data processing tasks.

What Are LSTMs?

LSTMs networks are a special kind of recurrent neural network (RNN) designed to capture long-range dependencies in sequence data.

Standard RNNs struggle with vanishing or exploding gradients, which makes it hard to learn from long sequences. LSTMs solve this with a special architecture involving gates to control what information to keep, forget, or output.

  • Vanishing gradient? - The gradients become very small, almost zero
  • Exploding gradients? - When weights or derivatives are > 1

The Architecture of an LSTM

An LSTM unit consists of different gates that form its structure and enable it to work more effectively compared to the other RNNs:

  • Cell State (Ct): The “memory” of the unit that carries information throughout the sequence processing

  • Hidden State (ht): The output state of the LSTM unit

There are three Gates namely:

  • Forget Gate: Decides what information to throw away from the cell state

  • Input Gate: Decides what new information to store in the cell state

  • Output Gate: Decides what to output based on the cell state

*Note: the forget is a binary function which either outputs 0 which means all information is forgotten or 1 which means all information is kept. These components work together to selectively remember or forget information over long sequences.

Why Are LSTMs Effective?

LSTMs address several short comings of the traditional options pricing models as well as other deep learning models:

  • Vanishing Gradient Problem: The gating mechanism allows gradients to flow unchanged, enabling learning across many time steps

  • Long-Term Dependencies: Can maintain information over hundreds of time steps

  • Selective Memory: Learns what to remember and what to forget

  • Sequence Processing: Excellent for tasks where order and context matter

Other Applications of LSTMs

LSTMs have been successfully applied to numerous domains:

  • Natural Language Processing (NLP):

  • Machine translation

  • Text generation

  • Sentiment analysis

  • Speech recognition

  • Time Series Prediction:

  • Stock market forecasting

*Implementing an LSTM

Here’s a simple example using Keras to implement an LSTM for sentiment analysis:

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from keras.models import Sequential
from keras.layers import LSTM, Dense, Embedding

model = Sequential()
model.add(Embedding(input_dim=vocab_size, output_dim=100, input_length=max_len))
model.add(LSTM(128))
model.add(Dense(1, activation='sigmoid'))

model.compile(loss='binary_crossentropy', optimizer='adam', metrics=['accuracy'])
model.fit(X_train, y_train, epochs=10, batch_size=32, validation_data=(X_val, y_val))

Challenges and Limitations

While LSTMs are powerful, they do have some limitations:

Computational Complexity: they require more parameters than simple RNNs

Training Time: they Can be slow to train, especially on long sequences

Conclusion

LSTMs represent a significant advancement in neural network architectures for sequential data. Their ability to learn long-term dependencies while avoiding the vanishing gradient problem has made them useful for many machine learning applications. While newer architectures continue to emerge, LSTMs remain a fundamental tool in the deep learning practitioner’s toolkit.