Description
1. Price a Bermudan put option (K − ST )
+ by using an implicit finite difference scheme for
the Black–Scholes PDE
−vt(t, s) + 1
2
σ
2
s
2
vss(t, s) + rsvs(t, s) = rv(t, s)
v(0, s) = (s − k)
+
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
−vt(t, s) + 1
2
σ
2
s
2
vss(t, s) + rsvs(t, s) = rv(t, s)
v(0, s) = (s − k)
+
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.
2
3. An example where the linearity boundary condition will not work: consider a European
power call option
(ST − K)
+
2
in the Black–Scholes framework
−vt(t, s) + 1
2
σ
2
s
2
vss(t, s) + rsvs(t, s) = rv(t, s)
v(0, s) =
(s − k)
+
2
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
dSt = rSt dt +
√
Yt dW1
t
, S0 = s
dYt = κYt dt + ξYt dW2
t
, Y0 = y
E[W1
t W2
t
] = ρt.
(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
d
dtv = Av.
6 points per proble