MATH375: Tutorial 4

1. Consider the following nonlinear stochastic differential equation (called the Cox-Ingersoll-Ross (CIR) equation):

where a, b, σ, are positive constants. Show that its solution (X(t),t ≥ 0) satisfies the mean-reverting property.

[Hint: Integrate both sides of the equation from 0 to t, take the expected value of both sides, and solve the resulting equation for E[X(t)].]

2. Consider the following market of two assets:

Let us introduce a new asset in this market with price Y (t) satisfying:

dY (t) = uB (t)dB(t) + uS (t)dS(t),   t ≥ 0,

for some F(t)-adapted processes ((uB (t), uS (t)),t ≥ 0) such that this equation has a unique solution. Show that this enlarged market of three assets does not admit arbitrage opportunities.

3. Consider the following market of two assets:

where r, µ1, σ1, are given positive constants. Let us introduce a new asset in this market with price process Y (t) satisfying

for some positive constants µ2, σ2. Show that in this market of three assets, if

there is an arbitrage opportunity.