代写IOE 552 Winter 2023 Financial Engineering I Final Exam代做留学生SQL语言程序
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Financial Engineering I
Final Exam
April 24, 2023
1. (5 points) Consider a market consisting of a bank account B(t), N traded assets S(t) = (S1(t), S2 (t), ··· , SN (t))T , and a contingent claim Φ(S(T)). Define the following concepts (each is worth one point):
(a) A portfolio and its value.
(b) An admissible portfolio.
(c) A self financing portfolio.
(d) A hedging/reachable portfolio.
(e) A relative portfolio.
2. (20 points) A risky asset is traded at discrete times t0 < t1 < ··· < tn with ti+1 − ti = ∆t and the price movement during each period either goes up or down by the rule:
Si+1 = { dS(uS)i(i) with probability p
with probability 1 − p
where Si is the price of the asset in period i, during time (ti, ti+1], u > 0, d > 0. There is also a bank account which will earn money at the interest rate of 100R% per period, i.e., Bi+1 = (1 + R)Bi, where Bi is the money in the bank at time ti , B0 = 1.
(a) (6 points) State conditions on u,d,R that will make the market trading the asset arbi- trage free.
(b) (6 points) State the first fundamental theorem of finance. What is the Q-martingale measure in this market. (Hint: think of qu)
(c) (8 points) Show by using the Q-measure that B(S)i(i) is a Q-Martingale. (Hint: Show that EQ ( |B(S)i(i) ) = B(S)i(i)
3. (18 points) Consider a market with two risky assets: for i = 1, 2
dSi(t) = αi(t)Si(t)dt + σi(t)Si(t)dWiP (t)
Here the Brownian motions Wi are independent and αi and σi are adapted processes.
(a) (6 points) Derive the dynamics of S1(S2)t(t) . (If you know the answer state it, if not use Ito’s lemma and derive it).
(b) (6 points) What must the dynamics of S1(S2)t(t) be in the Martingale space.
(c) (6 points) What is the dynamics of S1(t) in the Q-space.
(d) (6 points) Let the dynamics of the bank account be dB(t) = rB(t)dt. Find the dynamics of . What specific property does this dynamics have?
4. (17 points) Assume that the differential function F(t,x) solves the boundary value PDE:
Ft+ α(s)Fs+ s2 σ 2 Fss− rF = 0
F(T, s) = Φ(s) for all s.
where Φ is some deterministic function.
(a) (5 points) Write out the corresponding SDE which gives the dynamics of S(t) on which the contingent claim Φ is defined.
(b) (5 points) What is the formula for the price of the contingent claim Φ if α(s) = rs?
(c) (7 points) Describe the role of the constants λ and µ when α(s) = (µ − λσ)s in the pricing of the claim on the underlying asset. Write out the dynamics of the underlying in case α(s) is replaced by µ(t,s) for some deterministic function µ).
5. (20 points) You are given the standard Black-Scholes market, consisting of a bank account and a risky asset. The bank interest rate is r, and the risky asset follows a geometric Brownian Motion.
(a) (5 points)Write out the dynamics of these two assets.
(b) (5 points)What is an admissible portfolio in this market.
(c) (5 points) Write out value and the properties of a self financing portfolio of the bank account and the risky asset.
(d) (5 points) Consider the standard European call option
max{0, S(T) − K}
Write out the arbitrage free price of this option.
6. (20 points) Consider a European call option, with a strike price K and exercise time T, on a traded stock, in the Black-Scholes market considered in the previous question with the drift term µ, drift term σ and the risk free rate r.
(a) (6 points) Define the ∆ of this option, and give its formula.
(b) (6 points) Define the Γ of this option and give its formula.
(c) (8 points) What makes a portfolio on this market ∆ neutral, Γ neutral, and ∆ and Γ neutral. Show how each can be acheived in practice.