Bermuda and Knock-out European Barrier Option pricing with Crank-Nicolson finite difference approximation
All options satisfy the Black-Scholes equation. What differs among these options are the boundary conditions. Different options come with different sets of boundary conditions.
Using Crank-Nicolson method, price and calculate Delta, Gamma, and Theta for the following options:
- Bermuda Call and Put options of the following parameters: (𝑆0,𝐾, 𝜎, 𝑟, 𝑑) = (100, 100, 40%, 2.5%, 1.75%)
Options are exercisable monthly with the final Maturity = 1 year (you want to have these
exercise times coincide with time grid points)
- Knock-out European Barrier Options, 𝐻 is the level of the barrier (You want to set up your grids so the barrier(s) coincide with a grid point(s)). a. Up-and-out call: (𝑆0,𝐾, 𝐻, 𝑇, 𝜎, 𝑟, 𝑑) = (100, 110, 120, 1.0, 50%, 2.5%, 1.75%) b. Up-and-out put: (𝑆0,𝐾, 𝐻, 𝑇, 𝜎, 𝑟, 𝑑) = (100, 90, 120, 1.0, 50%, 2.5%, 1.75%) c. Double knock-out put: (𝑆0,𝐾, 𝐻𝑢𝑝, 𝐻𝑑𝑜𝑤𝑛, 𝑇, 𝜎, 𝑟, 𝑑) = (100, 90, 120, 80, 1.0, 50%, 2.5%, 1.75%) Plot Gamma vs. stock price on the evaluation date for this double knock-out put.