Question

This problem asks you to numerically calculate Delta and Gamma for the Asian put option in Problem 1. All parametric settings are the same as that of Problem 1, so you will need to build on the program you write for Problem 1.

Choose step size = 0.1. Instead of fixing the initial stock price So at $10 as in Problem 1, we now consider the initial stock price So on grid points €, 2, 3€, 20 €, 20.

(a) (8 points) For each possible initial stock price So on the grid points, calculate the current price of the Asian put option P(So). Plot the option price P(So) against the initial stock price So. Make sure that you see a downward sloping line. Hint:

• As we discussed in class, each time your iteration changes to a different starting So, make sure you reset to the same random seed in Python using np.random.seed(0). Can you think of a smarter programming way to implement the same thing?

. Your codes could take up to 10 minutes to run depending on your computer and how efficient your codes are. To save time, first do the calculation with step size € = 1. Once you see the results and are sure your codes are correct, change back to e = 0.1 to produce the required figure.

(b) (5 points) The option Delta can be approximated by the following finite-difference equation,

P'(So)≈ P(So+) P(So) -

Use the option price P(So) in part (a) to calculate the Delta P'(So) of the Asian put option. Plot the option Delta P'(So) against the initial stock price So. Is this curve

Answer 1

In option pricing, varying the initial stock price shows that the put option value decreases as the stock price rises. Delta can be calculated using a finite-difference method. Additionally, solving market equilibrium can be done graphically by intersecting demand and supply curves.

Step-by-step explanation:

Option Pricing and Graphical Solutions

When working with option pricing, specifically Asian put options, by varying the initial stock price, So, and plotting the resulting option prices P(So), we expect to see a downward-sloping line. This relationship is because, generally, as the stock price rises, the value of put options tends to decrease. Calculating the Delta for an option can be done using a finite-difference approximation, which compares the price at a slightly higher stock price against the current stock price.

Solving Models with Graphs, if you are more visually inclined, translating demand and supply equations into demand and supply curves on a graph can give you the equilibrium price and quantity. This method is particularly helpful for those who find algebra challenging. By setting price P on the vertical axis and quantity Qd (demand) or Qs (supply) on the horizontal axis, the intersection of the two curves results in the equilibriumm market price and quantity.