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.