Discretize the above pde using the implicit method write

Question

1. Calculate American put option values for the following parameters:

Stock price: S0 = {80, 100, 120},                          Strike price: K = {80, 100, 120},
Risk-free rate: r = 4.75%,                                     Dividend rate: q = 1.75%,
Maturity: T = 1,                                                  Volatility: σ = {15%, 30%, 50%}.

Use the three approaches in lecture notes 8. Use the fully implicit method for discretization with Neumann boundary condition on the second derivative:

lim_S→0∂²V/∂S² = lim_S→∞∂²V/∂S² = 0.

Also compute the corresponding European options by the closed form Black-Scholes formula. Requirements on coding and outputs:

- Use Matlab object oriented programming. This means, except a single driver script, all are Matlab class files. Refer to the last section in lecture notes 4. All your Matlab files must sit in the same folder. No sub-folder structure is in your code.

- Your PDF file must contain necessary mathematical formulas that are used in your code.

- Your Matlab code should dump the tabular data like this (copy this output into your PDF):

Spot    Strike  Vol    European   Bermudan    SyntheticDiv   BrennanSchwartz

  80       80     0.15     ?              ?                   ?                  ?
...

2. For K < S_b, the forward PDE for up-and-out calls (Carr-Hirsa) is as follows:

1/2σ²(K, T)K²∂²U/∂K² - (r-q)K.∂U/∂K - qU = ∂U/∂T + 1/2σ²(S_b, T)S_b²∂³U/∂K³(S_b, T) (K - S_b)

Here U = U (K, T ) is the price function - the option price function of different strikes and expirations. Local volatility function σ(K, T) is known. Use initial condition

U (K, 0) = (S₀ - K)⁺, for K < S_b, S₀ < S_b,

and boundary conditions

∂²U/∂K²(0, T) = ∂²U/∂K²(S_b, T) = 0.

(a) Discretize the above PDE using the implicit method. Write down the recursive equations in matrix form. For the term ∂³U/∂k³(S , T), use explicit scheme, i.e., discretize at time level j when ∂K3 b you work out recursive equation from j to j + 1. Also use the result of problem 2 in homework 3. This is a math problem - No coding is required.

(b) Compute the option price using both backward PDE (BS equation) and the above forward PDE with vol surface σ(K, T) = 0.5e^-T(100/K)^0.3 and

spot: S₀ = 100,   expiration: T = 0.25,

risk-free rate: r = 4.75%,     dividend rate: q = 1.75%,

strike: K = 110,                  barrier: Sb = 130.