due on Thursday, April 20 in class
1. Price a Bermudan put option (K −ST)+ by using an implicit finite difference scheme for the Black–Scholes PDE
for S0 = 100, σ = 30%, r = 2%, T = 1, K = 115 and quarterly possibility of early exercise (i.e., possible exercise times are t˜1 = 0.25, t˜2 = 0.5, t˜3 = 0.75, and t˜4 = 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 − ST)+ by using an implicit finite difference scheme for the Black–Scholes PDE
for S0 = 100, σ = 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
(a) the Bermudan approximation for American options,
(b) 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 Black–Scholes framework
with S0 = 100, σ = 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 vs and vss in the row smax that does not depends on v at smax+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
(a) Calculate the generator of the SDE given.
(b) Derive the Cauchy problem for the price of a European put option in this model.
(c) Derive a system of ODEs approximating the solution of the PDE. Calculate the
(smax × ymax) × (smax × ymax) matrix A such that for the (smax × ymax)-dimensional vector v it holds that
6 points per problems
