1. A Neural Network Approach For Options Pricing By: Jing Wang Course: CS757 Computational Finance Instructor: Dr. Ruppa Thulasiram Project #: CFWin03-35 Email: jingwang@cs.umanitoba.ca jingwang@cs.umanitoba.ca Date: May 26, 2003

2. Overview Background & Motivation Problem Statement Solution Strategy Results and Comparison Conclusion Future work Reference

3. Background & Motivation Option Pricing – Determine a risk-free price for options – Approaches Lattice Models Black-Sholes Model Monte Carlo Simulation Artificial Neural Network (ANN) – Non-parametric, non-linear – Data driven – Capture the dynamics

4. Problem Statement Type of options – European call / put – American call / put Objective: – Train the ANN with a set of option pricing data with certain inputs (current stock price, strike price, maturity time, option type, risk-free rate), then use the neural network calculate risk-free option prices Assume option price upper bound: $50

5. Solution Strategy Supervised ANN Multi-Layer Perceptron (MLP) Training method: Back-Propagation One ANN for a specific type of options Benchmark: Binomial tree – Provides data to train the ANN Implementation – Sequential implementation using C – Parallel implementation using MPI

6. ANN Structure One Input Layer – 5 Inputs: current underlying asset price, strike price, maturity date, risk-free interest and sigma Hidden Layers – No threshold, but an active function – Different number of hidden layers and nodes on hidden layers are compared while implementation Output Layer – Use the above active function – Three different strategies

7. Output Layer Structure Solution 1: One node – Output value = output_node_value * 50 Sigmoid function produces a value [0,1] Above function produced a value [0,50] i.e. Output node gets value 0.249 to produce option price $12.45 Solution 2: Fifty nodes – Only node K has a value – Output Value = (K-1) + value_of_node_K i.e. For option price $12.45, only node 13 has a value of 0.45, all others produce 0

8. Output Layer Structure (cont.) Solution 3: Fifty nodes – First K nodes produce values First K-1 nodes produce 1 Node K produce the value which is option_price – (K-1) – Output Value = i.e. For option price $12.45, node 13 has a value of 0.45, node 1 to node 12 produce 1, and the rests produce 0

9. Parallel Design Use the best structure of the three strategies One processor has a portion of the ANN – all input nodes, all hidden layers, n1/p hidden nodes, and n2/p output nodes n1: # of hidden nodes on a full ANN n2: # of output nodes on a full ANN p: # of working processors.

10. Result & Comparison Learning accuracy compared between the three strategies – # of data used: European Call: Train 336, Test 2892 American Put: Train 319, Test 2600

11. Result & Comparison (cont.)

12. Conclusion Artificial Neural Network is able to pricing options An ANN produce a better result when the output value rely on more output nodes than only one An ANN with more less hidden layer and more hidden layer nodes learns quicker

13. Future Work Real data to Train and Test ANN Train ANN for different options More time to finish parallel implementation

14. Reference [1] Rashedur M. Rahman, Ruppa K. Thulasiram, and Parimala Thulasiraman. Forecasting Stock Prices using Neural Networks on a Beowulf Cluster. 2003. [2] Christian Schittenkopf. A neural network-based approach to extracting risk-neural density and to derivative pricing. [3] Rudy De Winne, Alain Francois-Heude, and Benoît Meurisse. Market Microstructure and Option Pricing: A Neural Network Approach. Book of Research, Catholic University of Leuwen. August 2001