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IOE 552 Financial Engineering
创建日期：2024-08-19 15:29:21
所属分类：金融Finance
标签： IOE 552
Financial Engineering

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IOE 552


Financial Engineering I

Final Exam

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.

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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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