MATH375: Tutorial 2

Tutorial 2

1. Let r, σ, S0,K, M,T, be given positive numbers. The random variable S(T) is defined as:

where W (T) ∼ N(0,T). Calculate:

where I(·) is the indicator function.

The values inside the square brackets above are the terminal values (payoffs) of European digital (binary) options: the payoff in (i) is that of a cash-or-nothing call, whereas the payoff in (ii) is that of an asset-or-nothing call.

2. Let (Ω, F, P) be a probability space on which the random variable X is defined. Also let the σ-algebras D and G be such that D ⊂ G ⊂ F. If X is integrable, prove that:

3. Let (W (t),t ≥ 0) be a standard Brownian motion. Calculate E[X(t)] for each of the following cases: