Question

Let X be derivative on [0, T] which is a positive FT measurable random variable.

Defifine Vt(X) be the value of X at time t, measured in discounted dollars.

A portfolio has bounded risk if there exists N such that for all t,

X

k

Vt(Xk) N

. Here Vk(Xk) is the negative part of V(Xk) and it represents the amount that

the portfolio is short on Xk. This basically tells us there is a limit to what we can

borrowed.

There is no arbitrage with bounded risk if no self-fifinancing portfolio with bounded

risk is an arbitrage opportunity.

(a) Let X and Y be derivatives on [0, T] . If X Y a.s., then show that

V0(X) V0(Y ). Further show that if X = Y a.s., then V0(X) = V0(Y ).

(b) Let

1. a 0,

2. 0 t T and

3. X and Y be derivatives on [0, T]

.

Show that these are all true.

(i) V0(0) = 0, V0(1) = 1, and V0(X) 0

(ii) V0(X) = 0 X = 0 a.e.

(iii) V0 is linear : V0(X + Y ) = V0(X) + V0(Y ) and V0(aX) = aV0(X)

(iv) If t 0, then V0(X) = V0(Vt(X)).

(v) If A 0 is bounded and Ft measurable, then Vt(AX) = AVt(X)