Editor’s note: Chakri Cherukuri is a speaker for ODSC Europe 2023 this June. Be sure to check out his talk, “Fast Option Pricing Using Deep Learning Methods,” there!
In finance, options are financial instruments that give the holder the right to buy (Call option) or sell (Put option) the underlying asset (price S) at a fixed price (Strike K) on (or before) a fixed date (Maturity T).
There are two types of options:
European options can be exercised only on the date of maturity.
American options can be exercised on any date before maturity.
The plot below shows the price of a European call option for two maturities (T=1 and T=0). On the day of the maturity (blue line) the price of the call is just the payoff function C = max(S – K, 0)
The plot below shows the price of a European put option. On the day of the maturity, the price of the put (blue line) is the same as the payoff function P = max(K – S, 0)
The price of the option at any time before the maturity depends on the following features:
| Feature | Description | Symbol |
| Spot Price | Price of the underlying asset (e.g., Stock price for equities) | S |
| Interest rate | Risk-free interest rate | r |
| Dividend Yield | Dividend Yield | q |
| Volatility | The measure of risk (standard deviation of returns) | Σ |
| Time to Maturity | Time left until the maturity | T |
| Strike Price | Strike of the option | K |
Quantitative Finance Pricing Methods
Stochastic processes are used to model the dynamics of the spot prices and volatilities. The most famous and most widely used model is the Black-Scholes model. In this model, the price of the European calls and puts can be obtained using analytical formulas.
Black-Scholes model makes simplistic assumptions that volatility is constant across various strikes and maturities which is not true in practice. Stochastic Volatility models are used for modeling both the spot and volatilities as stochastic processes. Pricing options in these models is more computationally intensive compared to the Black-Scholes model.
Pricing options using numerical methods typically take a few seconds for a single set of inputs!
Apart from computing the option prices, these numerical methods are also used for computing the greeks which are the partial derivatives of the option prices with respect to various inputs (for example, delta is the first derivative of the option price with respect to the spot price). The Greeks are vital tools in risk management of options positions.
ML Based Option Pricing
Since neural networks are universal function approximators they can be used to learn the option prices in various models. The recipe for ML-based option pricing is as follows:
- Generate a large representative training set of inputs to the pricing model
-
Generate the labels (option price) using the slow numerical pricers (Monte Carlo, PDEs etc.)
- Use the labeled training dataset to train deep neural networks
- Use the trained deep learning models as fast pricers
Steps 1, 2, and 3 can be performed offline. Once the neural network is trained, inference if extremely fast and can be done for batches of features at once! We can also easily recover the greeks, since they are the gradient vectors of the output with respect to inputs.
Training Data
The training dataset consists of features (inputs to the option pricing function) and labels (option prices). Features are synthetically generated using Latin Hypercube Sampling across various parameter ranges and labels are generated for each feature vector using the slow numerical pricers.
We can use domain knowledge to augment training datasets by generating more samples in regions where the pricing function becomes singular (where σ~0 or T~0).
Training Deep Learning Models
Since the data (features and labels) are all numerical we use deep fully connected neural networks to learn the pricing functions. Techniques like weight decay, reducing learning rates (when validation loss doesn’t improve) can further improve the training process.
Wrapping Up
In this post, I provided a brief overview of quantitative finance methods used for pricing options and how deep learning techniques can be used for developing fast option pricers. I will go into more detail in my upcoming talk at ODSC Europe 2023, titled Fast Option Pricing Using Deep Learning Methods.
About the author/ODSC Europe 2023 speaker:
Chakri Cherukuri is a senior researcher in the Quantitative Research group within the CTO office at Bloomberg LP. His research interests include quantitative portfolio management, algorithmic trading strategies, applied machine learning and numerical methods. Previously, he built analytical tools for the trading desks at Goldman Sachs and Lehman Brothers. Before that he worked in Silicon Valley for startups building enterprise software systems. He is a core contributor and steering council member of bqplot, a 2D plotting library for the Jupyter notebook. He has extensive experience in numerical computing and software development.
He holds an undergraduate degree in mechanical engineering from the Indian Institute of Technology, Madras, and an MS in computational finance from Carnegie Mellon University.
