This assignment is a combination of finance and calculus
– Assignment 3 – MATH4091/7091: Financial calculus Assignment 3 Semester I 2018 Extended due date: Friday May 11 Weight 15% Total marks 65 marks Submission: Hardcopy to the submission box by 12:00pm Friday May 11. In addition, you will also need to submit a softcopy (i.e. scanned copy) of your assignment by the due date/time. Notation: “Lx.y” refers to [Lecture x, Slide y] Assignment questions 1. (15 marks) In this question, we will extend the risk-neutral pricing results for the Black- Scholes model, discussed in Lecture 5, to a generalized B-S model in which the proportional drift rate µ, volatility σ, and interest rate r are allowed to be adapted stochastic processes. Consider a filtered probability space (Ω,F , {Ft}t∈[0,T ],P), and let {Wt}t∈[0,T ] be a ( P, {Ft}t∈[0,T ] ) Brownian motion. Here, T > 0 is a fixed expiry. Note that {Ft}t∈[0,T ] is not necessarily gen- erated by the Brownian motion {Wt}t∈[0,T ]. Let {St}t∈[0,T ] be a stochastic process that satisfies the generalized Geometric Brownian motion dynamics dSt = µtStdt+ σtStdWt, S0 > 0, where {µt}t∈[0,T ] and {σt}t∈[0,T ] are stochastic processes adapted to the filtration {Ft}t∈[0,T ]. We assume that σt > 0 almost surely for all t ∈ [0, T ]. Further assume that we have a stochastic interest rate process, denoted by {rt}t∈[0,T ], which is adapted to {Ft}t∈[0,T ]. The specific form of rt is not needed for this question. Let {Bt}t∈[0,T ] be the bank-account process. That is, dBt = rtBtdt, B0 = 1. a. (3 marks) Show that d ( St Bt ) = σt St Bt ( µt − rt σt dt+ dWt ) . b. (4 marks) Define a change of probability measure from P to the risk-neutral probability measure Q under which the process { St Bt } t∈[0,T ] is a ( Q, {Ft}t∈[0,T ] ) -martingale. Then conclude that, under Q, d ( St Bt ) = σt St Bt dW̃t, where {W̃t}t∈[0,T ] be a ( Q, {Ft}t∈[0,T ] ) Brownian motion. Note: You are not required to show that Q is indeed a probability measure. However, you need to clearly state the Radon-Nikodým derivative process used in the measure change, and verify that it satisfies all key properties. Also state all key assumptions. MATH 4091/7091 – 1 – Duy-Minh Dang 2018 – Assignment 3 – c. (3 marks) Consider a portfolio Θt = (at, bt) of Xt = (St, Bt) with value process {Vt}t∈[0,T ], where Vt = Θt ·Xt. Show that under Q d ( Vt Bt ) = atσt St Bt dW̃t. What is the intuition here? d. (5 marks) Let CT be any (square-integrable) FT -measurable random variable. Show that it is possible to construct a replicating portfolio of (St, Bt) for CT . Then, conclude that Ct = EQ [ e− ∫ T t rsdsCT ∣∣Ft] . 2. (10 marks) Let r be the constant risk-free interest rate. Suppose that, under risk-neutral probabilities, the random variable YT is normally distributed with mean µ and variance σ 2. a. (5 marks) Use the risk-neutral pricing approach to find the time-0 price of an option which pays (YT −K)+ at time T , where T > 0 and K are constants. Suppose that under risk-neutral probabilities, {Wt}t∈[0,T ] is a Brownian motion, and {Xt}t∈[0,T ] is defined by dXt = (α− βXt)dt+ γdWt, X0 ∈ R, where α, β, γ, and X0 are all constants. b. (5 marks) Find the time-0 price of a K-strike, T -expiry European call on XT . Hint: Show that XT is normally distributed. Find explicit formulas for its mean and variance. Then use part (a). You can cite any results form tutorials. You don’t need to copy expressions that you’ve already written, but you should specify explicitly how they fit together to produce the answer 3. (25 marks) Consider a filtered probability space (Ω,F , {Ft}t∈[0,T ],P), where P is the physical probability measure. Let {Wt}t∈[0,T ] be a ( P, {Ft}t∈[0,T ] ) Brownian motion. Here, T > 0 is a fixed expiry. Recall the Black-Scholes dynamics under P dBt = rBtdt, B0 = 1, dSt = µStdt+ σStdWt, S0 > 0, where r, µ, σ are constants, r > 0 and σ > 0. Now, consider a European call option on the square of S, with strike K > 0 with payoff CT where CT = C(ω) : Ω→ R ω 7→ C(ST (ω)) = ( (ST (ω)) 2 −K )+ . Let C(St, t) be the time-t price of this call option, where 0 ≤ t < t="" .="" important:="" c(st,="" t)="" is="" the="" function="" which="" maps="" (st,="" t)="" to="" the="" option="" price;="" it="" is="" not="" the="" function="" which="" maps="" (s2t="" ,="" t)="" to="" the="" option="" price.="" math="" 4091/7091="" –="" 2="" –="" duy-minh="" dang="" 2018="" –="" assignment="" 3="" –="" a.="" (2="" marks)="" write="" down="" a="" pde="" for="" c="C(St," t),="" including="" the="" terminal="" condition.="" b.="" (6="" marks)="" solve="" the="" pde="" in="" part="" (a)="" by="" the="" fourier="" transform="" to="" find="" c(st,="" t).="" you="" can="" reuse="" relevant="" results="" from="" lectures.="" now="" we="" obtain="" c(st,="" t)="" by="" the="" martingale="" approach.="" c.="" (5="" marks)="" find="" the="" sde="" satisfied="" by="" s2t="" ,="" with="" respect="" to="" a="" brownian="" motion="" {w̃t}t∈[0,t="" ]="" under="" risk-neutral="" probability="" measure="" q.="" what="" is="" the="" distribution="" of="" log="" (="" s2t="" )="" under="" q?="" d.="" (6="" marks)="" find="" a="" formula="" for="" c(st,="" t).="" compare="" with="" results="" obtain="" from="" part="" (b).="" hint:="" use="" the="" b-s="" formula="" with="" appropriately="" chosen="" parameters.="" e.="" (6="" marks)="" let="" r="0.02," µ="0.1," σ="0.3," s0="10," k="100," t="1." in="" order="" to="" replicate="" the="" option="" using="" only="" shares="" of="" stock="" and="" units="" of="" the="" bank="" account,="" what="" quantities="" of="" each="" should="" the="" replicating="" portfolio="" hold="" at="" time="" 0?="" your="" final="" answers="" must="" be="" numbers="" 4.="" (15="" marks)="" in="" this="" question,="" we="" will="" outline="" a="" proof="" of="" the="" martingale="" representation="" theorem="" stated="" on="" l5.35.="" let="" {wt}t∈[0,t="" ]="" be="" a="" brownian="" motion="" on="" a="" probability="" space="" (ω,f="" ,p),="" and="" let="" {ft}t∈[0,t="" ]="" be="" the="" filtration="" generated="" by="" this="" brownian="" motion.="" note="" that="" t=""> 0 is a given fixed time. Recall the definition of Lp(Ω) (L0 or L3): Lp(Ω) = { X : Ω→ R ∣∣∣ EP [|X|p] = ∫ Ω |X|pdP <∞ }="" .="" we="" are="" particularly="" interested="" in="" the="" case="" p="2." a.="" (5="" marks)="" let="" {ht}t∈[0,t="" ]="" be="" an="" {ft}t∈[0,t="" ]-predictable="" process="" satisfying="" the="" property="" ep="" (∫="" t="" 0="" (ht)="" 2="" dt="" )=""><∞. define="" an="" ft="" -measurable="" random="" variable="" y="" as="" below="" y="exp" (∫="" t="" 0="" htdwt="" −="" 1="" 2="" ∫="" t="" 0="" (ht)="" 2="" dt="" )="" .="" (1)="" show="" that="" y="" ∈="" l2(ω)="" and="" y="EP[Y" ]="" +="" ∫="" t="" 0="" γ="" (y)="" t="" dwt,="" (2)="" where="" {="" γ="" (y)="" t="" }="" t∈[0,t="" ]="" is="" an="" {ft}t∈[0,t="" ]-predictable="" process,="" and="" ep="" (∫="" t="" 0="" (="" γ="" (y)="" t="" )2="" dt="" )=""><∞. is="" the="" representation="" (2)="" unique?="" justify="" your="" answer.="" hint:="" consider="" yt="exp" (∫="" t="" 0="" hudwu="" −="" 1="" 2="" ∫="" t="" 0="" (hu)="" 2="" du="" )="" and="" compute="" d(yt).="" also="" jensen’s="" inequality="" of="" conditional="" expectation="" may="" be="" useful.="" b.="" (1="" mark)="" conclude="" that="" any="" ft="" -measurable="" random="" variable="" x,="" where="" x="∑" i="" aiyi,="" ai="" ∈="" r,="" and="" yi="" has="" form="" (1),="" can="" be="" represented="" in="" form="" (2).="" math="" 4091/7091="" –="" 3="" –="" duy-minh="" dang="" 2018="" –="" assignment="" 3="" –="" in="" the="" below,="" we="" present="" two="" useful="" facts.="" denote="" by="" a(ω)="" the="" set="" of="" all="" linear="" combinations="" of="" random="" variables="" of="" form="" (1),="" i.e.="" a(ω)="{" x="" :="" ω→="" r="" ∣∣="" x="∑" i="" aiyi,="" ai="" ∈="" r,="" yi="" has="" form="" (1)="" }="" fact="" 1:="" a(ω)="" is="" dense="" in="" l2(ω).="" that="" is,="" any="" random="" variable="" x="" in="" l2(ω)="" is="" either="" in="" a(ω)="" or="" is="" a="" limiting="" point="" of="" a(ω).="" fact="" 2:="" any="" ft="" -measurable="" random="" variable="" x="" ∈="" l2(ω)="" can="" be="" represented="" in="" form="" (2),="" i.e.="" x="EP[X]" +="" ∫="" t="" 0="" γ="" (x)="" t="" dwt,="" where="" {="" γ="" (x)="" t="" }="" t∈[0,t="" ]="" is="" an="" {ft}t∈[0,t="" ]-predictable="" process,="" and="" ep="" (∫="" t="" 0="" (="" γ="" (x)="" t="" )2="" dt="" )=""><∞. c.="" (4="" marks)="" prove="" fact="" 2.="" you="" can="" use="" fact="" 1="" without="" proving="" it.="" d.="" (5="" marks)="" consider="" a="" martingale="" {mt}t∈[0,t="" ]="" with="" respect="" to="" (p,="" {ft}t∈[0,t="" ]).="" further-="" more,="" assume="" that="" mt="" ∈="" l2(ω)="" for="" each="" t="" ∈="" [0,="" t="" ].="" prove="" that="" mt="M0" +="" ∫="" t="" 0="" γudwu,="" t="" ∈="" [0,="" t="" ],="" where="" {γt}t∈[0,t="" ]="" is="" a="" {ft}t∈[0,t="" ]-predictable="" process,="" and="" ep="" (∫="" t="" 0="" (γt)="" 2="" dt="" )=""><∞. is this representation unique? justify your answer. you can use fact 1 without proving it, as well as fact 2 even if you could not prove it in part (c). bonus questions important: for bonus questions, you get either full marks or zero. 5. (5 marks) prove fact 1 in question 4. math 4091/7091 – 4 – duy-minh dang 2018 is="" this="" representation="" unique?="" justify="" your="" answer.="" you="" can="" use="" fact="" 1="" without="" proving="" it,="" as="" well="" as="" fact="" 2="" even="" if="" you="" could="" not="" prove="" it="" in="" part="" (c).="" bonus="" questions="" important:="" for="" bonus="" questions,="" you="" get="" either="" full="" marks="" or="" zero.="" 5.="" (5="" marks)="" prove="" fact="" 1="" in="" question="" 4.="" math="" 4091/7091="" –="" 4="" –="" duy-minh="" dang="">