DEADLINE FOR THIS ASSIGNMENT IS 12/04/2018
– Assignment 2 – MATH4091/7091: Financial calculus Assignment 2 Semester I 2018 Due Thursday April 19 Weight 15% Total marks 65 marks Submission: Hardcopy to the submission box by 3:00pm Thursday April 19. In addition, you will also need to submit a softcopy (i.e. scanned copy) of your assignment. Notation: “Lx.y” refers to [Lecture x, Slide y] Assignment questions 1. (15 marks) Consider a filtered probability space (Ω,F , {Ft}t≥0,P), and let {Xt}t≥0 be a con- tinuous stochastic process adapted to {Ft}t≥0 with the following properties: • {Xt}t≥0 is a martingale with respect to {Ft}t≥0, • {X2t − t}t≥0 is also a martingale with respect to {Ft}t≥0. Let f : R → R be a function in class C2(R) so that f , f ′ and f ′′ exist, and are continuous bounded. Let 0 ≤ s < t.="" we="" denote="" by="" π="{tn}Nn=0" a="" partition="" of="" the="" interval="" [s,="" t],="" where="" s="t0">< t1="" .="" .="" .="">< tn="t," and="" by="" ‖π‖="max" 0≤n≤n−1="" (tn+1="" −="" tn)="" the="" length="" of="" the="" longest="" sub-interval="" in="" the="" partition.="" for="" simplicity,="" we="" use="" e[·]="" instead="" of="" ep[·].="" in="" this="" question,="" we="" will="" derive="" a="" itô-like="" formula="" for="" the="" conditional="" expectation="" e="" [f(xt)|fs].="" (a.)="" (8="" marks)="" using="" taylor’s="" formula="" and="" then="" taking="" conditional="" expectation,="" show="" that="" e="" [f(xt)|fs]="" can="" be="" expressed="" in="" the="" form="" e="" [f(xt)|fs]="f(Xs)" +="" 1="" 2="" n−1∑="" n="0" e="" [="" f="" ′′(xtn)|fs="" ]="" (tn+1="" −="" tn)="" +="" e="" [rπ|fs]="" ,="" where="" e="" [rπ|fs]="" is="" the="" remainder/residual="" term.="" give="" an="" explicit="" expression="" for="" rπ.="" (b.)="" (7="" marks)="" show="" that,="" as="" ‖π‖="" →="" 0,="" we="" have="" rπ="" −→="" l2="" 0,="" e="" [rπ|fs]="" −→l2="" 0,="" n−1∑="" n="0" e="" [="" f="" ′′(xtn)|fs="" ]="" (tn+1="" −="" tn)="" −→l2="" ∫="" t="" s="" e="" [="" f="" ′′(xu)|fs="" ]="" du.="" then,="" conclude="" that="" e="" [f(xt)|fs]="f(Xs)" +="" 1="" 2="" ∫="" t="" s="" e="" [="" f="" ′′(xu)|fs="" ]="" du.="" math="" 4091/7091="" –="" 1="" –="" duy-minh="" dang="" 2018="" –="" assignment="" 2="" –="" 2.="" (18="" marks)="" consider="" a="" filtered="" probability="" space="" (ω,f="" ,="" {ft}t≥0,p),="" and="" let="" {wt}t≥0="" be="" a="" brow-="" nian="" motion.="" for="" simplicity,="" we="" use="" e[·]="" instead="" of="" ep[·].="" consider="" the="" stochastic="" differential="" equation="" drt="(θt" −="" art)dt+="" σdwt,="" r0="r(0)," where="" θt="" is="" a="" positive="" deterministic="" function="" of="" t,="" a="" and="" σ="" are="" positive="" numbers.="" this="" is="" the="" popular="" hull-white="" interest="" rate="" model.="" (a.)="" (5="" marks)="" compute="" d="" (="" eatrt="" )="" ,="" and="" show="" that="" the="" time-t="" solution="" conditional="" fs,="" where="" s="">< t,="" is="" given="" as="" rt="e" −a(t−s)rs="" +="" ∫="" t="" s="" e−a(t−u)θudu+="" σ="" ∫="" t="" s="" e−a(t−u)dwu.="" (b.)="" (3="" marks)="" explain="" why="" rt="" conditional="" on="" fs="" is="" normally="" distributed.="" find="" e[rt|fs]="" and="" var[rt|fs].="" (c.)="" (3="" marks)="" if="" θu="θ," where="" θ="" is="" a="" positive="" constant,="" find="" the="" limiting="" distribution="" of="" rt="" as="" t→∞.="" (d.)="" (5="" marks)="" let="" rt="∫" t="" 0="" rsds.="" explain="" why="" rt="" is="" normally="" distributed="" and="" show="" that="" e[rt|fs]="r0" a="" (="" 1−="" e−at="" )="" +="" 1="" a="" ∫="" t="" 0="" (="" 1−="" e−a(t−u)="" )="" θudu,="" var[rt|fs]="σ2" a2="" (="" t−="" 2="" a="" (="" 1−="" e−at="" )="" +="" 1="" 2a="" (="" 1−="" e−2at="" ))="" (e.)="" (2="" marks)="" find="" e="" [="" e−="" ∫="" t="" 0="" rsds="" ]="" .="" hint:="" you="" are="" allowed="" to="" use="" results="" covered="" in="" tutorials.="" 3.="" (12="" marks)="" assume="" that="" there="" exists="" a="" bank="" account="" bt="e" rt="" and="" a="" non-dividend-paying="" stock="" s.="" for="" (a)="" below,="" we="" impose="" no="" model="" on="" the="" s="" dynamics.="" (a.)="" (2="" marks)="" state="" put-call="" parity="" for="" the="" time-t="" prices="" of="" a="" call="" and="" put="" on="" s,="" where="" the="" call="" and="" put="" have="" the="" same="" strike="" k="" and="" same="" expiry="" t="" .="" (in="" class="" we="" had="" t="0;" now="" we="" have="" general="" t="">< t ). now work under the “black-scholes” model as describe in class. (b.) (5 marks) use put-call parity to derive a formula for the time-t price of a put on s, with strike k and expiry t . simplify your formula until (like the call price formula) it has only two terms. (c.) (5 marks) derive a formula for the time-t delta and gamma of that put. hint: you may, of course, assume all of the call formulas that were presented in class. 4. (10 marks) as an employee of xyz, you participate in the employee stock purchase plan. denote by st the time-t price of a xyz share. the share price of xyz today is s0 = 50. one year from today, the company will deduct $3000 from your paycheck, and in exchange you will acquire shares of xyz at the below-market price of min(40, 0.8× s1) per share, i.e. a 20% discount to the price today or to the price in a year, whichever is lower. the entire $3000 will be spent on xyz shares (including fractional shares if necessary). there is no risk that xyz will default on this plan, and no risk that you will leave the company by t = 1 (year). math 4091/7091 – 2 – duy-minh dang 2018 – assignment 2 – (a.) (4 marks) the stock purchase plan (consisting of the debit of $3000 and the receipt of shares) is an asset to you. find its time-0 value. (b.) (3 marks) the value of the plan will fluctuate as xyz stock fluctuates. suppose you don’t want to be subject to this risk. what time-0 position, expressed in a number of european call/put options, should you take, in order to hedge perfectly that risk? does this position need to be rebalanced over the course of the year? for this part, assume that european calls and puts on xyz are available at any strikes and expiries that you desire. (c.) (3 marks) now suppose that options on xyz are not available. what time-0 position in xyz stock should you take, in order to hedge perfectly that risk? does this position need to be rebalanced over the course of the year? 5. (10 marks) consider a filtered probability space (ω,f , {ft}t≥0,p), and let {wt}t≥0 be a brow- nian motion. let a stock price be a geometric brownian motion dst = µstdt+ σstdwt, where µ and σ are positive constants. let denote by r the constant risk-free interest rate. we define the (constant) market price of risk to be θ = µ− r σ , and the stochastic process process {ζt}t≥0, where ζt = exp ( −θwt − ( r + θ2 2 ) t ) . (a.) (4 marks) show that dζt = −rζtdt− θζtdwt. (b.) (4 marks) denote by {vt}t≥0 the value process of an investor’s portfolio consisting of stock and bond. furthermore, denote by at the number of shares of stock held by the investor at time t. as shown in l4.16, dvt = rvtdt+ at(µ− r)stdt+ atσstdwt. show that ζtvt is a martingale. hint: show that the differential of d(ζtvt) has no dt term. (c.) (2 marks) let v0 be the initial capital, and furthermore, let xt be a ft -measurable random variable. show that if the investor wants to perfectly replicate xt , then the amount of initial capital must be v0 = ep [ζtxt ] . (as such, the process {ζt} is usually called the state price density process.) bonus questions important: for bonus questions, you get either full marks or zero. 6. (5 marks) we further assume in question 1 that x0 = 0. show that {xt}t≥0 is a brownian motion. math 4091/7091 – 3 – duy-minh dang 2018 t="" ).="" now="" work="" under="" the="" “black-scholes”="" model="" as="" describe="" in="" class.="" (b.)="" (5="" marks)="" use="" put-call="" parity="" to="" derive="" a="" formula="" for="" the="" time-t="" price="" of="" a="" put="" on="" s,="" with="" strike="" k="" and="" expiry="" t="" .="" simplify="" your="" formula="" until="" (like="" the="" call="" price="" formula)="" it="" has="" only="" two="" terms.="" (c.)="" (5="" marks)="" derive="" a="" formula="" for="" the="" time-t="" delta="" and="" gamma="" of="" that="" put.="" hint:="" you="" may,="" of="" course,="" assume="" all="" of="" the="" call="" formulas="" that="" were="" presented="" in="" class.="" 4.="" (10="" marks)="" as="" an="" employee="" of="" xyz,="" you="" participate="" in="" the="" employee="" stock="" purchase="" plan.="" denote="" by="" st="" the="" time-t="" price="" of="" a="" xyz="" share.="" the="" share="" price="" of="" xyz="" today="" is="" s0="50." one="" year="" from="" today,="" the="" company="" will="" deduct="" $3000="" from="" your="" paycheck,="" and="" in="" exchange="" you="" will="" acquire="" shares="" of="" xyz="" at="" the="" below-market="" price="" of="" min(40,="" 0.8×="" s1)="" per="" share,="" i.e.="" a="" 20%="" discount="" to="" the="" price="" today="" or="" to="" the="" price="" in="" a="" year,="" whichever="" is="" lower.="" the="" entire="" $3000="" will="" be="" spent="" on="" xyz="" shares="" (including="" fractional="" shares="" if="" necessary).="" there="" is="" no="" risk="" that="" xyz="" will="" default="" on="" this="" plan,="" and="" no="" risk="" that="" you="" will="" leave="" the="" company="" by="" t="1" (year).="" math="" 4091/7091="" –="" 2="" –="" duy-minh="" dang="" 2018="" –="" assignment="" 2="" –="" (a.)="" (4="" marks)="" the="" stock="" purchase="" plan="" (consisting="" of="" the="" debit="" of="" $3000="" and="" the="" receipt="" of="" shares)="" is="" an="" asset="" to="" you.="" find="" its="" time-0="" value.="" (b.)="" (3="" marks)="" the="" value="" of="" the="" plan="" will="" fluctuate="" as="" xyz="" stock="" fluctuates.="" suppose="" you="" don’t="" want="" to="" be="" subject="" to="" this="" risk.="" what="" time-0="" position,="" expressed="" in="" a="" number="" of="" european="" call/put="" options,="" should="" you="" take,="" in="" order="" to="" hedge="" perfectly="" that="" risk?="" does="" this="" position="" need="" to="" be="" rebalanced="" over="" the="" course="" of="" the="" year?="" for="" this="" part,="" assume="" that="" european="" calls="" and="" puts="" on="" xyz="" are="" available="" at="" any="" strikes="" and="" expiries="" that="" you="" desire.="" (c.)="" (3="" marks)="" now="" suppose="" that="" options="" on="" xyz="" are="" not="" available.="" what="" time-0="" position="" in="" xyz="" stock="" should="" you="" take,="" in="" order="" to="" hedge="" perfectly="" that="" risk?="" does="" this="" position="" need="" to="" be="" rebalanced="" over="" the="" course="" of="" the="" year?="" 5.="" (10="" marks)="" consider="" a="" filtered="" probability="" space="" (ω,f="" ,="" {ft}t≥0,p),="" and="" let="" {wt}t≥0="" be="" a="" brow-="" nian="" motion.="" let="" a="" stock="" price="" be="" a="" geometric="" brownian="" motion="" dst="µStdt+" σstdwt,="" where="" µ="" and="" σ="" are="" positive="" constants.="" let="" denote="" by="" r="" the="" constant="" risk-free="" interest="" rate.="" we="" define="" the="" (constant)="" market="" price="" of="" risk="" to="" be="" θ="µ−" r="" σ="" ,="" and="" the="" stochastic="" process="" process="" {ζt}t≥0,="" where="" ζt="exp" (="" −θwt="" −="" (="" r="" +="" θ2="" 2="" )="" t="" )="" .="" (a.)="" (4="" marks)="" show="" that="" dζt="−rζtdt−" θζtdwt.="" (b.)="" (4="" marks)="" denote="" by="" {vt}t≥0="" the="" value="" process="" of="" an="" investor’s="" portfolio="" consisting="" of="" stock="" and="" bond.="" furthermore,="" denote="" by="" at="" the="" number="" of="" shares="" of="" stock="" held="" by="" the="" investor="" at="" time="" t.="" as="" shown="" in="" l4.16,="" dvt="rVtdt+" at(µ−="" r)stdt+="" atσstdwt.="" show="" that="" ζtvt="" is="" a="" martingale.="" hint:="" show="" that="" the="" differential="" of="" d(ζtvt)="" has="" no="" dt="" term.="" (c.)="" (2="" marks)="" let="" v0="" be="" the="" initial="" capital,="" and="" furthermore,="" let="" xt="" be="" a="" ft="" -measurable="" random="" variable.="" show="" that="" if="" the="" investor="" wants="" to="" perfectly="" replicate="" xt="" ,="" then="" the="" amount="" of="" initial="" capital="" must="" be="" v0="EP" [ζtxt="" ]="" .="" (as="" such,="" the="" process="" {ζt}="" is="" usually="" called="" the="" state="" price="" density="" process.)="" bonus="" questions="" important:="" for="" bonus="" questions,="" you="" get="" either="" full="" marks="" or="" zero.="" 6.="" (5="" marks)="" we="" further="" assume="" in="" question="" 1="" that="" x0="0." show="" that="" {xt}t≥0="" is="" a="" brownian="" motion.="" math="" 4091/7091="" –="" 3="" –="" duy-minh="" dang="">