17. (Requires Writing Code) Using Octave, write a short recursive program to implement the Cox-Ross-Rubinstein model. Run this out six periods. Use the following parameters: initial stock price of $100, strike price of $101, risk-free rate of 5%, volatility is 25% per annum, and maturity of half-year.
(a) Make sure that the program provides reasonably accurate prices by checking your results against the Black-Scholes formula (the prices will be within the ballpark of the correct prices even though there are very few periods in the model). Use both puts and calls in your validation. Report your results.
(b) Extend the program to allow for an extra negative jump in stock returns of 20% per jump. Thisjumpoccurswith probability of 5% (risk-neutral). (Now you have three branches emanating from each node.) For the calls and puts reported in the previous question, also report the prices from the jump-enhanced model. What can you say from your comparisons about the effect of jumps?
(c) Now extend the basic program in (a) to incorporate switching volatility. This is a simple volatility process where the volatility can take just one of two values, i.e., 10% or 40%, with equal risk-neutral probability. Volatility is not correlated with the stock price movement. With this addition, there will now be four branches emanating from each node. What can you say from your comparisons about the effect of stochastic volatility? Start with the initial volatility of 25% and then let it switch between 10% and 40%.
(d) Run your program from the previous question with volatility at levels 20% and 30%. Start with initial volatility of 25%. How do prices change in comparison? Explain why. Note: Make sure that in each question, you set up the risk-neutral probabilities correctly. You will need to calculate it differently for each of the subparts of this problem.
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