Financial Engineering and Risk Management – Summer 2022
Financial 代写 | 商科代写 | 金融代写 | Risk Management代写写 – 这是一个unity面向对象设计的practice, 考察unity的理解, 涵盖了unity等方面
Final Examination
Prof. S uleyman Ba sak
Instructions
- This is an open-book take-home exam which will be released to you online on Can- vas and you are required to submit it through Canvas. The exam is an individual assignment. You may not discuss its content or the solutions with anyone.
- Please write your LBS student id numberlegiblyon the top of the first page of your solutions.
- There are 2 questions on this examination with a total of 100 points.
- The examination lasts for 2.5 hours, including the time to upload your solutions on Canvas.
- Although you have access to your notes and other materials, it is highly recommended that you use your crib sheet, and a calculator as necessary, to solve the questions.
- To receive full credit, youMUSTshow all the relevant algebraic steps in deriving your solutions. Partial credit is given to answers that are incorrect but that show a correct understanding of the solution method. If the question is not clear, state your assumptions and if they are reasonable, you will be given credit.
- Honour Code. Please note that by taking this exam, you comply with following honour code: I confirm that I have had no prior knowledge of the content of this exam and the answers to the exam are all my own work. I also confirm that I will not knowingly disclose any information from this exam to others. I also understand that during the exam, I am not permitted in any way to communicate with any person.
Questions
- Market Price of Risk and Arbitrage 40 points Assume the Black-Scholes economy with one source of uncertainty generated by a Brownian motionW. There is a riskless bond and a stock. Bond priceBand stock priceSfollow the dynamics:
dBt = rBtdt
dSt = Stdt+StdWt,
wherer,,are constants. Suppose there is an equity index that is traded and has a
price levelIfollowing the dynamics
dIt=IItdt+IItdWt,
whereIandIare constants, andWis the same Brownian motion as the one driving
the stock priceS.
Supposer= 2%, = 12%, = 25%,I= 17%,I= 20% p.a., and sitting at time
t, the bond, stock and index levels areBt=St= 100,It= 500. Is there an arbitrage
opport unity in this economy? If so, show how to exploit it, and explicitly state the
trading strategy for this arbitrage portfolio in terms of the stock, the index and the
bond and the resulting arbitrage profit.
- Power Contracts, Valuation and Replication 60 points
Today is time t= 0. Assume the Black-Scholes economy with a single source of
uncertainty generated by Brownian motionW. The riskless bondB and the non-
dividend paying stockSfollow the dynamics:
dBt = rBtdt
dSt = Stdt+StdWt
wherer,,are constants with >0.
Suppose a client of your financial institution wants a contract related to the value of
the stockSTat some dateT. In addition, the client requires that the contract exhibit
no time decay from initiation until the maturity dateTin the sense that the contract
value is insensitive to changes in time,all else fixed. Consider-powercontracts of
the form:
VT=f(ST) = (ST)
whereis a constant that is different than one ( 6 = 1), and can be chosen such that
the power contract exhibits many desired properties.
(a)(20 points)Using martingale valuation, derive a no-arbitrage price for the-
powercontract at timetin terms of the stock priceStand other relevant param-
eters, for general.
(b)(10 points)Confirm that the-powercontract price derived in part (a) satisfies
the Black ScholesPDEand the boundary condition.
(c) (10 points)Which value of the constant parametercomplies with the clients
wish that the-powercontract price exhibit no time decay? Intuitively, would
you expect this contracts price with no time decay to vary positively or nega-
tively with the stock price? Explain.
(d)(10 points)How much would the financial institution charge the client for the
No-Decay Powercontract derived in part (c) at initiationt= 0? Determine the
trading strategy, in terms of the bond and the stock, replicating theNo-Decay
Powercontract. Is it trend-chasing or contrarian?
(e) (10 points)The client additionally wishes not to pay for the contract at its
initiationt= 0, but would like to pay a prespecified amount at the contract
maturityT. Accordingly, the financial institution offers the following-power
forwardcontract and payoff upon maturity:
VT= (ST)F 0
whereF 0 is the delivery price of the-power forwardcontract chosen so that
the contract has zero value at timet= 0. Is it still possible to create aNo-
Decay Power Forwardcontract that has no time decay as in part (c) with an
appropriate constant, possibly different from thedetermined in part (c)?