[Solved] MA 573-Computational Methods of Mathematical Finance -Assignment 10 last assignment

1. Price a Bermudan put option (K S_T)⁺ by using an implicit finite difference scheme for the BlackScholes PDE

for S₀ = 100, = 30%, r = 2%, T = 1, K = 115 and quarterly possibility of early exercise (i.e., possible exercise times are t₁ = 0.25, t₂ = 0.5, t₃ = 0.75, and t₄ = 1). Use 1000 space discretization point on the interval [0,200] and implement the explicit boundary condition at 0 and a linearity boundary condition at 200. Calculate the price for 100, time discretization steps.

2. Price a American put option (K S_T)⁺ by using an implicit finite difference scheme for the BlackScholes PDE

for S₀ = 100, = 30%, r = 2%, T = 1, K = 115. Use 1000 space discretization point on the interval [0,200] and implement the explicit boundary condition at 0 and a linearity boundary condition at 200. Calculate the price for 100 time discretization steps using

• the Bermudan approximation for American options,
• the Brennan-Schwartz algorithm and compare the results.

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3. An example where the linearity boundary condition will not work: consider a European power call option in the BlackScholes framework

with S₀ = 100, = 20%, r = 3%, T = 1 and K = 115. As the payoff function is not linear but quadratic for large stock prices, the linearity assumption of the pricing function for large prices makes no sense. You will have to choose a finite difference approximation for the spatial derivatives v_s and v_ss in the row s_max that does not depends on v at s_max+1, i.e., your scheme cannot longer be central but has to use one-sided derivative approximations. Specifically, use 1000 space discretization point on the interval [0,200] and implement the explicit boundary condition at 0. Calculate the option price using a Crank-Nicolson scheme with 100 time discretization steps.

4. Consider the correlated Hull-White stochastic volatility model
   • Calculate the generator of the SDE given.
   • Derive the Cauchy problem for the price of a European put option in this model.
   • Derive a system of ODEs approximating the solution of the PDE. Calculate the

(s_max y_max) (s_max y_max) matrix A such that for the (s_max y_max)-dimensional vector v it holds that

6 points per problems