Option hedging is a risk management strategy designed to minimize the risks associated with transactions involving options. Among the various option hedging strategies, delta hedging is the most common and straightforward. Therefore, we will use an example of delta hedging to start this article. Delta hedging focuses on managing the risk related to price movements in the underlying asset. By adjusting the quantities of the securities held in a portfolio, delta hedging aims to maintain a neutral position, meaning the portfolio’s overall value becomes less sensitive to fluctuations in the underlying asset’s price. This is typically achieved by buying or short selling the underlying asset or its derivatives.

Traditional approaches to option hedging are often based on parametric models such as the Black–Scholes (–Merton) model (referred to as the BS model hereafter), jump-diffusions, and stochastic volatility models (Chen and Li, 2023). For example, a typical BS model-based delta hedging strategy involves buying (or short selling) the following number of shares for each option contract sold short: 100 * BS delta for call (or put) options. The BS delta for a call or put can be conveniently calculated using the standard normal cumulative distribution function as N(d1) or N(d1) – 1, respectively, where d1 is a function of the interest rate, the ratio of spot price to strike price, time to maturity, dividend yield, and volatility.

Why We Need Deep Learning and Reinforcement Learning

Deep learning and reinforcement learning are AI techniques that have demonstrated significant promise in solving complex problems. Unlike parametric models, which rely on predefined equations and assumptions, these data-driven approaches leverage large datasets and sophisticated algorithms to learn patterns and make decisions. This enables them to achieve high levels of accuracy and adaptability in domains such as image recognition, natural language processing, and autonomous systems, often surpassing traditional parametric methods in numerous applications.

In the context of option hedging, deep learning and reinforcement learning can adapt to complex and evolving market dynamics, incorporating factors such as transaction costs, liquidity constraints, and risk preferences into their decision-making processes. By leveraging these techniques, option hedging strategies can be optimized in a more flexible and responsive manner, leading to improved risk management and enhanced performance in volatile financial markets. Various studies have highlighted the advantages of deep learning and reinforcement learning compared to traditional approaches.  For example, Cao et al. (2021) and Kolm and Ritter (2019) emphasized the limitations of delta hedging in the presence of trading costs and demonstrated the superior performance of reinforcement learning under such conditions. Notably, both papers specifically incorporated trading costs into the reinforcement learning framework, further illustrating how these techniques can adapt to real-world challenges.


How Deep Learning Powers Option Hedging Strategies

In a deep learning system, such as an artificial neural network with multiple hidden layers, the focus is on learning to map inputs to outputs based on the provided training data. During training, the network adjusts its weights and biases to minimize the difference between its predicted outputs and the actual outputs from the training data. This process involves optimizing a loss function.

Chen and Li (2023) exemplify the use of deep learning in option hedging by employing a feedforward neural network (FNN) approach. An FNN is a fundamental type of artificial neural network composed of an input layer, one or more hidden layers, and an output layer. Information flows in one direction—from input to output—without forming cycles. Each neuron in a layer receives inputs from all neurons in the preceding layer, calculates a weighted sum of these inputs plus a bias, and then applies an activation function to determine its output.

The candidate features considered in Chen and Li (2023) include time to maturity, moneyness (i.e., the spot price divided by the strike price), the BS delta, and measures of market sentiment. The best-performing model identified in the paper includes the following features: time to maturity, the BS delta, and market sentiment (measured by VIX for calls and index return for puts). The output of the FNN is the hedge ratio, which represents the amount of the underlying asset to be purchased or short sold. The training process involves minimizing the loss function, defined as the mean squared hedging error, which essentially captures the variance of the hedging error. The hedging error is defined as:

The following code demonstrates a simplified example of FNN-based option hedging, using a strategy similar to that in Chen and Li (2023), implemented in Python.

import numpy as np
import pandas as pd
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Dense
from tensorflow.keras import backend as K

# Assuming 'data' is the existing DataFrame with the required features and target columns
# Example:
# data = pd.read_csv('your_data_file.csv')

# Features
features = data[['time_to_maturity', 'bs_delta', 'sentiment_score']]
change_in_option_price = data['change_in_option_price']
change_in_underlying_price = data['change_in_underlying_price']

# Split the data into training and testing sets
X_train, X_test, y_train_opt, y_test_opt, y_train_und, y_test_und = train_test_split(
    features, change_in_option_price, change_in_underlying_price, test_size=0.2, random_state=42)

# Standardize the data
scaler = StandardScaler()
X_train = scaler.fit_transform(X_train)
X_test = scaler.transform(X_test)

# Define the FNN model
model = Sequential()
model.add(Dense(64, input_dim=X_train.shape[1], activation='relu'))
model.add(Dense(32, activation='relu'))
model.add(Dense(1, activation='linear'))

# Custom loss function to minimize the variance of the hedge error
def custom_loss(y_true, y_pred):
    option_price_change = y_true[:, 0]
    underlying_price_change = y_true[:, 1]
    hedge_ratio = y_pred
    hedge_error = option_price_change - hedge_ratio * underlying_price_change
    return K.mean(K.square(hedge_error))

# Compile the model with the custom loss function
model.compile(optimizer='adam', loss=custom_loss)

# Combine option price and underlying price changes into a single array for the loss function
y_train = np.column_stack((y_train_opt, y_train_und))
y_test = np.column_stack((y_test_opt, y_test_und))

# Train the model
history = model.fit(X_train, y_train, epochs=50, batch_size=16, validation_split=0.2)

# Real-time prediction example
def predict_hedge_ratio(new_data):
    # Ensure new_data is a 2D array (e.g., [[value1, value2, value3]])
    new_data = scaler.transform(new_data)  # Apply the same scaling as training data
    predicted_hedge_ratio = model.predict(new_data)
    return predicted_hedge_ratio[0][0]

# Example of real-time prediction
new_data = np.array([[0.1, 0.6, 0.2]])  # Example new input data
hedge_ratio = predict_hedge_ratio(new_data)
print("Predicted Hedge Ratio:", hedge_ratio)

How Reinforcement Learning Works for Option Hedging

Reinforcement learning entails training an agent to learn, through trial and error, how to take actions in a dynamic environment to maximize the expected cumulative reward.

The agent comprises two key components: a policy and a learning algorithm. The policy defines the strategy or rules the agent follows to choose its actions in response to the environment. Meanwhile, the learning algorithm enables the agent to iteratively refine its policy over time by learning from the outcomes of its actions, with the goal of maximizing the cumulative reward.

There are two main types of policies in reinforcement learning:

• Deterministic Policy: A deterministic policy directly maps states to actions. For each state, the policy specifies a single action that the agent should take.
• Stochastic Policy: A stochastic policy maps states to a probability distribution over actions. Instead of selecting a single action for a given state, the agent selects actions based on probabilities assigned to each action in that state.

Cao et al. (2021) showcases the application of reinforcement learning for option hedging. In the study, the reinforcement learning system aims to maximize the expected cumulative reward, adjusted for the time value of money, with the reward for each period defined as follows:

Kolm and Ritter (2019) serves as another example of using reinforcement learning for option hedging, with a slightly different setup. The article posits that the training process involves preparing automatic hedgers to optimize the trade-off between trading costs and the variance from being unhedged. The importance of risk aversion in determining this trade-off is also discussed, showcasing the adaptability of the approach to different risk preferences. The article underscores that the proposed method’s applicability extends to various derivative securities, provided we have a clear understanding of their pricing.

In conclusion, the integration of AI techniques such as deep learning and reinforcement learning into option hedging strategies represents a significant advancement over traditional methods. By leveraging AI’s capabilities, traders and financial institutions can achieve improved risk management and enhanced performance in volatile financial markets. Future developments in this field promise to yield even more innovative applications and sophisticated techniques, further transforming option hedging practices and setting new benchmarks for financial strategy and risk management.



References:

Cao, J., J. Chen, J. Hull, and Z. Poulos (2021). Deep hedging of derivatives using reinforcement learning. Journal of Financial Data Science 3 (1): 10–27. DOI: 10.3905/jfds.2020.1.052

Chen, J., & Li, L. (2023). Data-driven hedging of stock index options via deep learning. Operations Research Letters, 51(4), 408–413. https://doi.org/10.1016/j.orl.2023.05.007

Kolm, P. N and G. Ritter (2019). Dynamic replication and hedging: a reinforcement learning approach. Journal of Financial Data Science, Winter 2019:159-171. DOI: 10.3905/jfds.2019.1.1.159

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