Mastering Algorithmic trading with Python
Mar 30, 2023
In the world of quantitative trading, TensorFlow is a powerful tool that can revolutionize the way traders analyze and make decisions about the market. This comprehensive guide will explain how TensorFlow can be used in the world of quantitative trading, including how it can be used to build and optimize trading models, analyze market data, and make informed investment decisions. We will also explore the various applications of TensorFlow in quantitative trading, including the use of machine learning and artificial intelligence techniques to identify trends and patterns in market data. By the end of this guide, traders will have a solid understanding of how to use TensorFlow to their advantage in the fast-paced world of quantitative trading.
In this article, we discuss the concept of entry and exit signals in the context of algorithmic trading. An entry signal refers to the rules that determine when to initiate a trade, while an exit signal refers to the rules that determine when to close a trade. There are two types of entry signals: endogenous rules, which are based solely on the security being traded, and exogenous rules, which are based on other data sources such as other securities or macroeconomic indicators. Endogenous rules can be based on simple criteria such as a stock price going below a certain threshold, or more complex criteria such as a stock’s volume exceeding a certain level and the closing price being lower than the daily average. Exogenous rules, on the other hand, require the trader to also track other securities or data values in addition to the security being traded. For example, a trader may initiate a trade in a stock if another stock’s price falls by a certain percentage, or if there is a decrease in the unemployment rate. It is important for traders to carefully consider their entry and exit signals as these can have a significant impact on the overall performance of their trading strategy.
Stop loss orders are an important tool for managing risk in trading. There are two main types of stop loss orders: static and dynamic. Static stop loss orders are set at a fixed price, and if the market reaches that price, the stop loss order is triggered and the trade is closed. Dynamic stop loss orders, on the other hand, are set as a percentage below the current market price. This means that if the market moves in favor of the trade, the stop loss order will also move in favor of the trade, increasing the chances of realizing profits. It is important to carefully consider the placement of stop loss orders, as setting them too tightly may result in the order being triggered by slight market movements, while setting them too loosely may result in large losses.
TensorFlow is an open-source library for numerical computation, including machine learning. It allows users to build a directed graph, or DAG, to represent the computation they want to perform. The nodes in the graph represent mathematical operations, while the edges represent the input and output of those operations. TensorFlow uses tensors, or N-dimensional arrays of data, to flow through the graph. The use of directed graphs and tensors allows for language and hardware portability, as the same code can be executed on different platforms, including CPUs, GPUs, and TPUs. The tf.data API in TensorFlow allows users to create input pipelines for models, and to work with data in memory or in multiple files. Feature columns are used to manipulate and prepare data for use in training neural network models.
TensorFlow allows developers to write code in a high-level language like Python and have it executed efficiently at runtime. TensorFlow works by creating a directed graph, or DAG, to represent the computation that needs to be done. The nodes in the graph represent mathematical operations, and the edges represent the input and output of those operations. The edges can be thought of as arrays of data flowing through the graph. These arrays, called tensors, can be of any dimension, or rank, from scalars (rank 0) to higher dimensional arrays like matrices (rank 2) and 3D tensors (rank 3).
There are several layers of abstraction in TensorFlow, ranging from low-level APIs for hardware to high-level APIs for more abstract tasks. The Core Python API contains much of the numeric processing code and functions for working with variables and tensors. There are also several Python modules that provide high-level representations of useful neural network components, such as layers, metrics, and losses. For many applications, using one of the high-level APIs like Estimators or Keras is sufficient for training, evaluating, and serving machine learning models. These APIs handle tasks such as distributed training, data preprocessing, model definition, compilation, and training, as well as evaluation, checkpointing, and model serving.
The Cloud AI Platform is a managed service that allows developers to run TensorFlow on a cluster of machines in the cloud without having to install any software or manage any servers. In this module, we will focus on the top three APIs: Core Python, TF Layers, and the Estimator API. Before diving into code, however, it is important to understand the pieces of data that we will be working with, including variables, tensors, and feature columns.
A tensor is a generalization of a matrix to higher dimensions. It is a multi-dimensional array of data that represents a collection of values that can be processed by a machine learning model. In machine learning, tensors are often used to represent input data, output predictions, and model parameters. TensorFlow is a popular machine learning library that uses tensors as the fundamental data type for representing and manipulating data.
TensorFlow Financial is a library designed for use in the field of quantitative finance. It provides high-performance tools for mathematical and statistical operations, as well as specific pricing models and other utilities relevant to financial analysis. The library is organized into three tiers: foundational methods, mid-level methods, and pricing methods and other quant finance-specific utilities. Foundational methods include core mathematical operations such as optimization, interpolation, and linear algebra, while mid-level methods include ODE and PDE solvers and Ito process frameworks. Pricing methods and utilities include specific pricing models for local vol, stochastic vol, stochastic local vol, and Hull-White, as well as rate curve building, payoff descriptions, and schedule generation. The library is designed to be easily accessible and accompanied by numerous examples at each level.
The Black-Scholes model is a mathematical model for pricing options, which are financial derivatives that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price on or before a certain date. The Black-Scholes model is widely used in the financial industry to evaluate the fair value of options and to manage the risks associated with options trading.
In this article, we will use the TensorFlow library to implement the Black-Scholes model and price European call and put options. We will start by introducing the basics of options and the Black-Scholes model, and then we will implement the model using TensorFlow. Finally, we will use the model to price a few options and verify the results against known values.
The price of an option is determined by several factors, including the price of the underlying asset, the option’s expiration date, the option’s strike price, and the volatility of the underlying asset. The strike price is the price at which the option gives the holder the right to buy or sell the underlying asset. The expiration date is the date on which the option expires and can no longer be exercised. The volatility of the underlying asset is a measure of the asset’s price movements over time.
The Black-Scholes model
The Black-Scholes model is a mathematical model for pricing options that takes into account the factors mentioned above. The model was developed in the 1970s by Fischer Black, Myron Scholes, and Robert Merton, who were awarded the Nobel Memorial Prize in Economic Sciences for their work.
The Black-Scholes model is based on the assumption that the price of an underlying asset follows a geometric Brownian motion, which is a stochastic process with a constant drift and a constant volatility. The model also assumes that the underlying asset pays no dividends, that the option can only be exercised at expiration, and that the risk-free interest rate is known and constant.
Under these assumptions, the Black-Scholes model provides a closed-form solution for the price of a European call or put option, which is given by the following formulas:
Call option price:
C = S * N(d1) — K * exp(-r * T) * N(d2)
Put option price:
P = K * exp(-r * T) * N(-d2) — S * N(-d1)
where:
C: call option price P: put option price S: underlying asset price K: strike price T: time to expiration
import tensorflow as tf# Set the parameters for the Black-Scholes model
S0 = 100.0 # initial stock price
K = 105.0 # strike price
T = 1.0 # time to expiration (in years)
r = 0.05 # risk-free interest rate
sigma = 0.2 # volatility
# Define the inputs to the model as placeholders
S = tf.placeholder(tf.float32, shape=[], name='S')
K = tf.placeholder(tf.float32, shape=[], name='K')
T = tf.placeholder(tf.float32, shape=[], name='T')
r = tf.placeholder(tf.float32, shape=[], name='r')
sigma = tf.placeholder(tf.float32, shape=[], name='sigma')
# Calculate the d1 and d2 terms as defined in the Black-Scholes formula
d1 = (tf.log(S/K) + (r + 0.5 * sigma**2) * T) / (sigma * tf.sqrt(T))
d2 = d1 - sigma * tf.sqrt(T)
# Calculate the option price using the Black-Scholes formula
call_price = S * tf.norm.cdf(d1) - K * tf.exp(-r * T) * tf.norm.cdf(d2)
put_price = K * tf.exp(-r * T) * tf.norm.cdf(-d2) - S * tf.norm.cdf(-d1)
# Create a session and run the model to price the options
with tf.Session() as sess:
call_price_val = sess.run(call_price, feed_dict={S: S0, K: K, T: T, r: r, sigma: sigma})
put_price_val = sess.run(put_price, feed_dict={S: S0, K: K, T: T, r: r, sigma: sigma})
# Print the results
print("Call price:", call_price_val)
print("Put price:", put_price_val)
You can predict the price of an option as an example in this link.
The Monte Carlo method is a computational algorithm that uses random sampling to solve mathematical problems. It can be used to estimate the likelihood of different outcomes or to make predictions about future values of a quantity of interest. One way to implement the Monte Carlo method is through the use of the Euler scheme, which is a simple and easy-to-implement method for solving ODEs.
To use the Euler scheme in a Monte Carlo method with TensorFlow, we first need to define the ODE that we want to solve. This can be done by setting up a function that represents the derivative of the quantity of interest, and specifying the initial conditions for the problem.
Next, we can use TensorFlow to set up a loop that will iterate through a series of time steps, using the Euler scheme to estimate the derivative at each time step. For each iteration of the loop, we will use the current estimate of the derivative to update the value of the quantity of interest.
import tensorflow as tf# Define the ODE and the initial conditions as TensorFlow constants
def derivative(t, y):
return -y
t0 = tf.constant(0.0)
y0 = tf.constant(1.0)
# Set the time step size and the number of time steps as TensorFlow constants
dt = tf.constant(0.1)
n_steps = 10
# Initialize the time and the quantity of interest as TensorFlow variables
t = tf.Variable(t0)
y = tf.Variable(y0)
# Set the number of Monte Carlo simulations as a TensorFlow constant
n_simulations = 1000
# Initialize the arrays to store the results
ts = tf.zeros((n_simulations, n_steps+1))
ys = tf.zeros((n_simulations, n_steps+1))
# Run the Monte Carlo simulations using a TensorFlow loop
for i in range(n_simulations):
# Set the initial conditions for this simulation
t.assign(t0)
y.assign(y0)
# Store the initial conditions
ts = ts.write(i, 0, t)
ys = ys.write(i, 0, y)
# Iterate through the time steps using a TensorFlow loop
for j in range(n_steps):
# Use the Euler scheme to estimate the derivative at this time step
dy = derivative(t, y) * dt
# Update the value of the quantity of interest
y.assign_add(dy)
# Update the time
t.assign_add(dt)
# Store the results for this time step
ts = ts.write(i, j+1, t)
ys = ys.write(i, j+1, y)
# Compute the mean and standard deviation of the quantity of interest
mean = tf.reduce_mean(ys, axis=0)
std = tf.math.reduce_std(ys, axis=0)
# Print the results
print("Time:", ts.read(0))
print("Mean:", mean)
print("Std:", std)
Optimization
There are many ways to use Tensorflow for optimization. I came across one good example, everyone can see it by following this link.
In Tensorflow we can’t assign values to variables like var = val. Values can be declared via tf.constant(...) , as a tf.Variable(...) with a default value or be assigned via my_tensor.assign(...) , which creates an operation that needs to be run.
Let’s add the coin_weights constraints and run them after each training step:
# Optimize weights to minimize volatilitydef minimize_volatility():
# Define the model
# Portfolio Volatility = Sqrt (Transpose (Wt.SD) * Correlation Matrix * Wt. SD)
coin_weights = tf.Variable(np.full((len(coins), 1), 1.0 / len(coins))) # our variables
weighted_std_devs = tf.multiply(coin_weights, std_deviations)
product_1 = tf.transpose(weighted_std_devs)
product_2 = tf.matmul(product_1, correlation_matrix)
portfolio_variance = tf.matmul(product_2, weighted_std_devs)
portfolio_volatility = tf.sqrt(tf.reduce_sum(portfolio_variance))
# Constraints: sum([0..1, 0..1, ...]) = 1
lower_than_zero = tf.greater( np.float64(0), coin_weights )
zero_minimum_op = coin_weights.assign( tf.where (lower_than_zero, tf.zeros_like(coin_weights), coin_weights) )
greater_than_one = tf.greater( coin_weights, np.float64(1) )
unity_max_op = coin_weights.assign( tf.where (greater_than_one, tf.ones_like(coin_weights), coin_weights) )
result_sum = tf.reduce_sum(coin_weights)
unity_sum_op = coin_weights.assign(tf.divide(coin_weights, result_sum))
constraints_op = tf.group(zero_minimum_op, unity_max_op, unity_sum_op)
# Run
learning_rate = 0.01
steps = 5000
init = tf.global_variables_initializer()
# Training using Gradient Descent to minimize variance
optimize_op = tf.train.GradientDescentOptimizer(learning_rate).minimize(portfolio_volatility)
with tf.Session() as sess:
sess.run(init)
for i in range(steps):
sess.run(optimize_op)
sess.run(constraints_op)
if i % 2500 == 0 :
print("[round {:d}]".format(i))
print("Weights", coin_weights.eval())
print("Volatility: {:.2f}%".format(portfolio_volatility.eval() * 100))
print("")
sess.run(constraints_op)
return coin_weights.eval()
weights = minimize_volatility()
pretty_weights = pd.DataFrame(weights * 100, index = coins, columns = ["Weight %"])
pretty_weights
# Optimize weights to maximize return/riskdef maximize_sharpe_ratio():
# Define the model
# 1) Variance
coin_weights = tf.Variable(np.full((len(coins), 1), 1.0 / len(coins))) # our variables
weighted_std_devs = tf.multiply(coin_weights, std_deviations)
product_1 = tf.transpose(weighted_std_devs)
product_2 = tf.matmul(product_1, correlation_matrix)
portfolio_variance = tf.matmul(product_2, weighted_std_devs)
portfolio_volatility = tf.sqrt(tf.reduce_sum(portfolio_variance))
# 2) Return
returns = np.full((len(coins), 1), 0.0) # same as coin_weights
for coin_idx in range(0, len(coins)):
returns[coin_idx] = cumulative_returns[coins[coin_idx]]
portfolio_return = tf.reduce_sum(tf.multiply(coin_weights, returns))
# 3) Return / Risk
sharpe_ratio = tf.divide(portfolio_return, portfolio_volatility)
Optimizer(...).minimize(tf.negative(sharpe_ratio))
A swap curve is a curve that represents the yield of a series of fixed-rate bonds with different maturities. It is commonly used in finance to value derivatives and other financial instruments that are sensitive to changes in interest rates. In this article, we will explore how to use TensorFlow, an open-source machine learning library, to fit a swap curve to a set of bond maturities and rates.
To fit a swap curve with TensorFlow, we need to define a model that represents the relationship between the bond maturities and rates, and a loss function that measures the difference between the model’s predictions and the input data. We can then use TensorFlow’s gradient descent optimizer to minimize the loss and find the optimal model parameters.
For example, we can represent the swap curve as a linear function of the bond maturities, with the form y = a + b * x, where y is the bond rate, x is the bond maturity, and a and b are model parameters. We can then define the loss function as the mean squared error between the model’s predictions and the input data.
import tensorflow as tf# Define the input data as a TensorFlow constant
maturity = tf.constant([1, 2, 3, 5, 7, 10, 30]) # in years
rate = tf.constant([0.01, 0.02, 0.03, 0.05, 0.07, 0.1, 0.3]) # in annualized percentage points
# Set the number of model parameters as a TensorFlow constant
n_params = 2
# Initialize the model parameters as TensorFlow variables
params = tf.Variable(tf.zeros((n_params,)))
# Define the model as a TensorFlow function
def swap_curve(maturity, params):
a, b = params
return a + b * maturity
# Define the loss function as a TensorFlow function
def loss(rate, maturity, params):
return tf.reduce_mean((rate - swap_curve(maturity, params)) ** 2)
# Set the learning rate and the number of training steps as TensorFlow constants
learning_rate = 0.01
n_steps = 1000
# Use TensorFlow's gradient descent optimizer to minimize the loss
optimizer = tf.optimizers.SGD(learning_rate)
for i in range(n_steps):
# Compute the gradient of the loss with respect to the model parameters
with tf.GradientTape() as tape:
loss_value = loss(rate, maturity, params)
grads = tape.gradient(loss_value, params)
# Update the model parameters using the gradient descent optimizer
optimizer.apply_gradients(zip([grads], [params]))
# Print the optimal model parameters
print("Optimal model parameters:", params.numpy())
TensorFlow is a machine learning library that can be used in quantitative trading to build and optimize trading models, analyze market data, and make informed investment decisions. It allows for the creation of directed graphs, or DAGs, to represent mathematical operations and the flow of data through those operations as tensors. In the context of trading, entry and exit signals are used to determine when to initiate or close a trade, and can be based on endogenous rules related to the security being traded or exogenous rules related to other securities or macroeconomic indicators. Stop loss orders are used to manage risk in trading and can be set as static or dynamic. The TensorFlow neural.data API and feature columns can be used to manipulate and prepare data for use in training neural network models. TensorFlow can also be used to implement Monte Carlo simulations and the Euler scheme for solving ordinary differential equations.
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