1. Consider a Cox-Ross-Rubinstein model with two periods (N = 2) and with interest rate r = 0.03. Assume the risky asset S has initial value S₀ = 100 and at every step it can move up by a factor 1 + u with u = 0.05 or move down by a factor 1 + d with d = -0.05.

(a) Compute the unique risk neutral probability measure?

(b) Compute the price of an European put with strike price K = 100.

(c) Compute the price of an European call with strike price K = 100.

(d) Compute the price and the hedging strategy of an American put with strike price K = 100.

(e) Compute the price of an American call with strike price K = 100.

2. Under the Black-Scholes model compute the price and the hedging strategy of an European derivative with payoff G = (K - S_T)².

3. Consider the stochastic differential equation

dX_t = -bX_tdt + σdB_t, t > 0, X₀ = 0,

where b and σ are positive parameters and B is a standard Browian motion over the stochastic basis (?, F, {F_t}_t≥0, Q) with its natural completed filtration. Assume Q is a risk neutral measure.

Assume the short term interest rate is given by

r_t = r + X_t

where r > 0 is a given constant. Recall that the Markov property allows us to write the price of a zero coupon bond

P(t, T) = E[exp{- _t∫^Tr_sds}|F_t]

as a function F(t, X_t).

a) Derive the partial differential equation satisfied by F(t, x).

b) Solve the stochastic differential equation satisfied by X.

c) Show that

₀∫^tX_sds = - σ/b ₀∫^t (e^-b(t-s) - 1)dB_s, t > 0.

d) Show that

E[_t∫^TX_sds|F_t] = X_t/b(1 - e^{-b(T -t)}).

e) Show that

V[_t∫^TX_sds|F_t] = σ²/b² _t∫^T(e^{-b(T -s)} - 1)²ds

f) What is the distribution _t∫^T X_sds given F_t?

g) Show that

P(t, T) = e^{A(t,T)-r(T -t)+X_tC(t,T)}

where

A(t, T) = σ²/2b²_t∫^T(e^{-b(T -s)} - 1)²ds

and

C(t, T) = 1/b(e^{-b(T - t)} - 1)

h) Check that function P(t, T) = F(t, X_t) solves explicitly the partial differential equation obtained in question (a).

4. Consider the exponential Vasicek short rate model

dr_t = r_t(η - a log r_t)d_t + σr_tdB_t,                      (1)

where η, a, σ are positive parameters.

a) Find the solution of the stochastic differential equation

dY_t = (θ - aY_t)d_t + σdB_t

as a function of the initial condition y₀.

b) Let X_t = e^Yt. Determine the stochastic differential equation satisfied by X.

c) Find the solution r of the equation 1 in terms of the initial condition r₀.

d) Compute E[r_t|F_s] for 0 ≤ s ≤ t.

e) Compute V[r_t|F_s] for 0 ≤ s ≤ t.

f) Compute the asymptotic mean and variance, that is

lim_t↑∞E(r_t)

and

lim_t↑∞V(r_t).