Assignment Details:
Recall that given two probability measures P and Q, the Radon-Nikodym derivative dP/dQ links the expectations of random variables F under P and under Q via the relation
The following 10 questions are interdependent and should be treated in sequence.
1. Neyman-Pearson Lemma. Given P and Q two probability measures, consider the event
Show that for A any event, Q(A) ≤ Q(Aα) implies P(A) ≤ P(Aα).
Hint: Start by proving that we always have
2. Let C ≥ 0 denote a non-negative claim payoff on a financial market with risk-neutral measure P∗. Show that the Radon-Nikodym density
defines a probability measure Q∗.
Hint: Check first that dQ∗/dP∗ ≥ 0, and then that Q∗(Ω) = 1. In the following questions, we consider a non-negative contingent claim C ≥ 0 with maturity T >0, priced e^−rT IEP∗ [C] at time 0 under the risk-neutral measure P∗.
Budget constraint. In the sequel, we will assume that no more than a certain fraction β ∈(0, 1] of the claim price e^−rT IEP∗ [C] is available to construct the initial hedging portfolio V0 at time 0.
Since a self-financing portfolio process (Vt)t∈R+ started at V0 = βe^−rT IEP∗ [C] may fall short of hedging the claim C when β < 1, we will attempt to maximize the probability P(VT ≥ C) of successful hedging, or, equivalently, to minimize the shortfall probability P(VT < C).
For this, given A an event we consider the self-financing portfolio process 
 hedging the claim C1A, priced 
T.
3. Show that if α satisfies Q∗ (Aα) = β, the event
maximizes P(A) overall possible events A, under the condition
Hint: Rewrite Condition (2) using the probability measure Q∗, and apply the Neyman-Pearson Lemma of Question 1 to P and Q∗.
4. Show that P(Aα) coincides with the successful hedging probability
Hint: To prove an equality x = y we can show first that x ≤ y, and then that x ≥ y.
One inequality is obvious, and the other one follows from Question 3.
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