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# MAT3300 Homework 4: Mathematical Modeling

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Homework 4: Mathematical Modeling

Submit your code and a simple document no longer than 4 pages to illustrate your results. Compress your files to a single one and submit it to Blackboard.

The file SP500 Option Prices.xlsx contains the prices (as of November 25, 2022) of S&P500 index call options matured on February 17, 2023 with various strikes. The risk-free rate is taken as the 3-month treasury yield which is 4.1750% and the S&P500 index is 4,026.12 as of November 25, 2022. Note that the index itself can be understood as a call option with zero strike. We assume that the index can be traded without bid-ask spread.

1.     With the presence of bid-ask spreads, is there any arbitrage opportunity by trading these options and the risk-free asset? If so, construct one such strategy.

2.     Assuming that you can buy and sell the call options at the mid prices, is there any arbitrage opportunity by trading these options and the risk-free asset? If so, construct one such strategy.

3.     We may also be interested in the fair price of a call option with a strike which is not in the data. One possible way is to fit a parametric function in strike price for the call option prices. We can choose a quadratic function in the form,

C(K)=a+bK+cK2 .

Find the optimal parameters a, b, c by fitting the function above to mid prices in the data with the constraint that C(K) excludes arbitrage opportunities. (Hint: assuming that the options can be bought and sold at the same price.)

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