Assume that the risk-neutral dynamics of the price S of an underlying asset is given by:

dS = rSdt +V Sdz₁

dV = V dt + V dz₂

where z1 and z2 are two Wiener processes with instantaneous correlation .

Let f_i, (i = 1, 2) be the value the value of the down-and-in (put) and the up-and-out (call) barrier

options on S with barrier level H_i, strike price X_i and maturity T_i, (i = 1, 2).

1. What is the payoff of each option at their respective maturity dates.

2. By numerical simulation, give the price of each option when Ti = 0.5 (6 months), H1 = 95 and

H2 = 105. Consider the current stock price S₀ = 100 and the parameters values: V₀ = 0.15², X₁=98, X₂=103

= 0, the risk-free rate r = 3%, = 1 and = 0.2. (Give the value of B and n simulated.)

3. Obtain the prices when S follows a geometric Brownian motion with volatility = 0.15

4. Compare the prices obtained at 2. and 3.