Create a function ValueDelta(char inName[], char outName[]) that reads a text file with option specifications and writes the option values as well as Delta. The inName[] file contains the input file name. The outName file contains the output file name.

As before, each line of the input file will have the following formats:
 
, Strike, Spot, Sigma, T, r 
 
Each line of the output file should have the following format:
 
, Strike, Spot, Sigma, T, r, Price, Delta 
 
Run the function for the input file ValueDeltaInput.csv which has been included with the assignment.

What are the maximum Deltas you see for a call and for a put?  What are the minimums?  *Why do the Deltas not change sign for a given option?

Challenge question*: Gamma, the second derivative of the option value with respect to stock price can be similarly approximated using the three values of the second level of the binomial tree.  As an optional challenge (not required), try approximating Gamma as well.  There are online sources that will point you in the right direction if you get stuck. 

e) Add an ExerciseBoundary() method to your binomial tree class that returns the early exercise "boundary" for an American option.  For the purposes of the homework, the exercise boundary is the maximum (or possibly minimum) value of the underlying stock for which early exercise is warranted at each time t from 0 to time of maturity.  The boundary will typically take on different values for different t, i.e., in different parts of the tree.

Tip: It is easy to keep track of the exercise boundary while calculating the option price in the binomial tree. You may therefore want to extend your generic price calculating function to take care of the exercise boundary at the same time.

Calculate the exercise boundary, for an American put with spot $90, strike $100, sigma 0.25, risk-free rate 5%, and T = 1 year using a binomial tree with 100 time steps.

Calculate the exercise boundary for an American put with spot $90, strike $100, sigma 0.45, risk-free rate 5%, and T = 1 year using a binomial tree with 100 time steps.

The only difference between the two puts above is the underlying stock's volatility. For which put is the exercise boundary higher?