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Beating the Best: A Neural
Network Challenges the BlackScholes Formula
Mary Malliaris and Linda Salchenberger
Loyola University Chicago
Ninth IEEE Conference on Artificial Intelligence
for Applications
Orlando 1993
• A neural network model is developed to
estimate the market price of options
• These prices are compared to estimates
generated by the Black-Scholes model
• An option is an agreement giving the holder
the right to purchase [a call] or sell [a put]
some asset at an agreed upon future time
• The price that will be paid at this future date is
called the exercise price of the option
• The market price is the price you pay now for
the privilege of buying or selling on or before
the expiration date
Black-Scholes Model
Data obtained from Wall Street Journal
Jan 1, 1990 to June 30, 1990
Exercise price
Market price of the option
Closing price of the S&P 100 Index
Daily data
Interest rate from 3-month US treasury Bill
In the money and out of the money options
Neural Network Variables
• Previous variables plus:
• Two lagged variables: yesterday’s
closing price, and yesterday’s market
price of the option
Neural Network Approach
Single hidden layer
Fully connected
Validation set used to measure performance
Testing sets developed with two-week time
Mean absolute deviation
Mean absolute percent error
Mean squared error
Calculated for each of the 5 two-week periods
for both in-the-money and out-of-the-money
• NN: lower MAPE in 4 out of 5 periods for outof-the-money
• B-S: superior 4 out of 5 for in-the-money
Paired sample comparisons tests:
B-S consistently overprices the options
NN consistently underprices them
Standard deviation of the differences
smaller in the neural network prices
• Similarities between the individual price
estimates made by the two models
• Both have difficulty computing deep in-themoney prices
• NN better out-of-the-money
• B-S better in-the-money
• NN methodology offers a valuable alternative
to estimating option prices