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MH4514 Financial Mathematics - Assignment 1 February 9, 2020 - 12:57 Due Wednesday April 1st, 2020 Questions 1-5 can be treated immediately. Question 6 requires knowledge from Chapters 5-6, and Questions 7-10 require knowledge from Chapters 5-8. 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 IEQ[F ] = ∫ Ω F (ω)dQ(ω) = ∫ Ω F (ω)dQ(ω) dP(ω) dP(ω) = IEP [ F dQ dP ] . The following 10 questions are interdependent and should be treated in sequence. 1. (10 marks) Neyman-Pearson Lemma. Given P and Q two probability measures, consider the event Aα := { dP dQ > α } , α ≥ 0. Show that for A any event, Q(A) ≤ Q(Aα) implies P(A) ≤ P(Aα). Hint: Start by proving that we always have( dP dQ − α ) (21Aα − 1) ≥ ( dP dQ − α ) (21A − 1). (1) 2. (10 marks) Let C ≥ 0 denote a nonnegative claim payoff on a financial market with risk-neutral measure P∗. Show that the Radon-Nikodym density dQ∗ dP∗ := CIEP∗ [C] defines a probability measure Q∗. Hint: Check first that dQ∗/dP∗ ≥ 0, and then that Q∗(Ω) = 1. In the following ques- tions we consider a nonnegative 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 porftolio 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="" (v="" at="" )t∈[0,t="" ]="" hedging="" the="" claim="" c1a,="" priced="" v="" a0="erT" iep∗="" [c1a]="" at="" time="" 0,="" and="" such="" that="" v="" at="C1A" at="" maturity="" t="" .="" https://en.wikipedia.org/wiki/radon%e2%80%93nikodym_theorem#radon%e2%80%93nikodym_derivative="" https://en.wikipedia.org/wiki/neyman%e2%80%93pearson_lemma="" 3.="" (10="" marks)="" show="" that="" if="" α="" satisfies="" q∗(aα)="β," the="" event="" aα="{" dp="" dq∗=""> α } = { dP dP∗ > α dQ∗ dP∗ } = { dP dP∗ > αC IEP∗ [C] } maximizes P(A) over all possible events A, under the condition e−rT IEP∗ [ V AT ] = e−rT IEP∗ [C1A] ≤ βe−rT IEP∗ [C]. (2) Hint: Rewrite Condition (2) using the probability measure Q∗, and apply the Neyman- Pearson Lemma of Question 1 to P and Q∗. 4. (10 marks) Show that P(Aα) coincides with the successful hedging probability P ( V AαT ≥ C ) = P(C1Aα ≥ C), i.e. show that P(Aα) = P ( V AαT ≥ C ) = P(C1Aα ≥ C). 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. 5. (10 marks) Check that the self-financing portfolio process ( V Aαt ) t∈[0,T ] hedging the claim with payoff C1Aα uses only the initial budget βe−rT IEP∗ [C], and that P ( V AαT ≥ C ) maximizes the successful hedging probability. In the next Questions 6-10 we assume that C = (ST − K)+ is the payoff of a European option in the Black-Scholes model dSt = rStdt+ σStdBt, (3) with P = P∗, dP/dP∗ = 1, and S0 := 1 and r = σ2 2 := 1 2 . (4) 6. (10 marks) Solve the stochastic differential equation (3) with the parameters (4). 7. (10 marks) Compute the successful hedging probability P ( V AαT ≥ C ) = P(C1Aα ≥ C) = P(Aα) in terms of the parameter α > 0. 8. (10 marks) From the result of Question 7, express the parameter α in terms of the successful hedging probability P ( V AαT ≥ C ) . 9. (10 marks) Compute the minimal initial budget e−rT IEP∗ [C1Aα ] required to hedge the claim C = (ST −K) in terms of α > 0. 10. (10 marks) Taking K := 1 and assuming a successful hedging probability of 0.9, compute numerically: a) The European call price e−rT IEP∗ [(ST −K)+] from the Black-Scholes formula. b) The value of α > 0 obtained from Question 8. c) The minimal initial budget needed to successfully hedge the European claim C = (ST −K)+ with probability 90% from Question 9. 2