of Business,
Washington University
Stochastic Foundation of Finance (FIN 538)
Problem Set 1
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(Please submit to Canvas on Wed, Sep 18 before class. TA: ,
Problem 1 (An extension of Example 1 of Lecture 1) Suppose the the economy in Exam- ple 1, Lecture 1 lasts for three quarters. Similar to Example 4 of Lecture 1, consider a security that pays dt = $1 if the economy state in quarter t is G and dt = $0 if the economy state in quarter t is B. Let X = d1 + d2 + d3 (assuming d0 = 0).
1. What is the sample space?
Answer: Ω={GGG,GGB,GBG,GBB,BGG,BGB,BBG,BBB}.
2. What is the filtration that corresponds to the σ-algebras Ft at t = 0, 1, 2, 3?
Answer: F0 = {∅, Ω}, F1 = {∅, Ω, {GGG, GGB, GBG, GBB}, {BGG, BGB, BBG, BBB}},F2 =
{∅, Ω, {GGG, GGB}, {GBG, GBB}, {BGG, BGB}, {BBG, BBB}, all possible unions of these sets}, F3 = 2Ω, where 2Ω is the set of all subsets of Ω.
3. Calculate the conditional expectations Zt = E[X|Ft] (note that Zt’s are random variables themselves).
Answer: Z0 = E[X|F0] = E[X] = 3∗q3 +2∗3∗q2(1−q)+1∗3∗q(1−q)2 +0∗(1−q)3 = 3q Z1(GGG) = Z1(GGB) = Z1(GBG) = Z1(GBB) = E[X|{GGG,GGB,GBG,GBB}] = 1+2∗q2 +1∗2q(1−q)=1+2q
Z1(BGG) = Z1(BGB) = Z1(BBG) = Z1(BBB) = E[X|{BGG,BGB,BBG,BBB}] = 2q2(1−q)+2q(1−q)2 = 2q
Z2(GGG) = Z2(GGB) = E[X|{GGG, GGB}] = 2 + q Z2(GBG) = Z2(GBB) = E[X|{GBG, GBB}] = 1 + q Z2(BGG) = Z2(BGB) = E[X|{BGG, BGB}] = 1 + q Z2(BBG) = Z2(BBB) = E[X|{BBG, BBB}] = q
and Z3 = E[X|F3] = E[X|2Ω] = X.
4. Find the σ-algebras generated by Zt, t = 0, 1, 2, 3.
Answer: σ(Z0) = F0, σ(Z1) = F1, σ(Z2) =
{∅, Ω, {GGG, GGB}, {GBG, GBB, BGG, BGB}, {BBG, BBB}, all possible unions of these sets}, note that σ(Z2) ̸= F2. σ(Z3) is generated by the following set
{{GGG}, {GGB, GBG, BGG}, {GBB, BGB, BBG}, {BBB}} 1
Problem 2 Let (Ω,F,P) be a probability space where Ω = {a,b,c,d,e,f} and F = 2Ω, P is uniform. Consider the following random variables
X(a) = X(b) = 1,X(c) = X(d) = 3,X(e) = X(f) = 5 Y (a) = Y (b) = Y (c) = 2, Y (d) = Y (e) = 4, Y (f ) = 6
Calculate Z1 = E[X|Y ] and Z2 = E[Y |X]. Answer.
Z1(a) = E[X|Y (a)] = E[X|Y = 2] = E[X|{a, b, c}]
= P (a|{a, b, c}) ∗ X (a) + P (b|{a, b, c}) ∗ X (b) + P (c|{a, b, c}) ∗ X (c)
Similarly, Z1(b) = Z1(c) = 53. Z1(d) = Z1(e) = 4, Z1(f) = 5.
Z2(a) = Z2(b) = 2,Z1(c) = Z1(d) = 3,Z1(e) = Z1(f) = 5. Problem 3 Consider the geometric Brownian motion Xt = eμt+σBt
Let Ft be the natural filtration associated with Bt. Compute E [Xt], V ar [Xt], E [Xt|Fs], V ar [Xt|Fs] for s < t. Show your calculations.
x2−2(μ+σ2)tx+μ2t2
e− 2σ2t dx
Answer. Xt = eN(μt,σ2t).
∞ (x−μt)2 ∞ x2−2(μ+σ2)tx+μ2t2 ∞
E[Xt] = exe− 2σ2t dx = e− 2σ2t dx = 000
∞ (x−(μ+σ2)t)2+μ2t2−(μ+σ2)2t2 σ2 = e− 2σ2t dx = e(μ+ 2 )t
V ar(Xt) = E[Xt2] − E[Xt]2 = E[e2μt+2σBt ] − e(2μ+σ2)t = e(2μ+2σ2)t − e(2μ+σ2)t
E[X |F ] = E[eμt+σ(Bt−Bs)+σBs |F ] = eμs+σBs E[eμ(t−s)+σ(Bt−Bs)] = X e(μ+ σ2 )(t−s) tsss2
V ar[Xt|Fs] = E[Xt2|Fs] − E[Xt|Fs]2 = Xs2{e(2μ+2σ2)(t−s) − e(2μ+σ2)(t−s)}
Problem 4 Which of the following three processes is adapted to the natural filtration of the
Brownian motion: Xt = Bt2 + t + 1, Yt = Bt + Bt+ 12 , Zt = mins≤t Bs?
Answer. Xt, Zt.
Problem 5 LetX bearandomvariableonaprobabilityspace(Ω,F,P)andletFt beafiltration. Define a continuous-time stochastic process Yt by Yt = E[X|Ft]. Show that Yt is a martingale with respect to Ft.
Answer. Assuming E[|X|] < ∞, then E[|Yt|] = E[|E[X|Ft]|] ≤ E[E[|X||Ft]] = E[|X|] < ∞ (law of iterated expectation). Yt is adapted to Ft by definition and by law of iterated expectation,
E[Yt|Fs] = E[Xt|Fs] = Ys 2
Problem 6 1. Show that if X, Y are independent random variables then E[XY ] = E[X]E[Y ]. 2. Calculate E[BtBs] for t ≥ s where Bt is the standard Brownian motion.
3. Calculate V ar[Bt + Bs] for t ≥ s where Bt is the standard Brownian motion.
1. We showed this in class.
2. E[BtBs]=E[(Bt −Bs)Bs +Bs2]=E[(Bt −Bs)Bs]+E[Bs2]=0+s=s.
3. Var(Bt+Bs)=Var(Bt)+Var(Bs)+2Cov(Bt,Bs)=t+s+2(E[BtBs]−E[Bt]E[Bs])=t+3s.
Problem 7 1. Calculate E[BtBs|Fs] for t ≥ s where Bt is the standard Brownian motion..
2. Calculate E[Bt2|Fs] for t ≥ s where Bt is the standard Brownian motion..
3. Suppose that Bt2 − ct is a martingale with respect to the natural filtration of the Brownian motion Bt. Find c.
1. E[BtBs|Fs] = E[(Bt−Bs)Bs+Bs2|Fs] = E[(Bt−Bs)Bs|Fs]+Bs2 = BsE[(Bt−Bs|Fs]+Bs2 = Bs2.
2. E[Bt2|Fs] = E[(Bt − Bs)2 + 2BtBs − Bs2|Fs] = E[(Bt − Bs)2|Fs] + 2E[BtBs|Fs] − Bs2 = E[(Bt −Bs)2]+Bs2 =Var(Bt −Bs)+Bs2 =t−s+Bs2.
3. E[Bt2 −ct] is a martingale if E[Bt2 −ct|F∫ ] = Bs2 −cs. Consider E[Bt2 −ct|F∫ ] = Bs2 +t−s−ct, whichisequaltoBs2 −csifc=1.
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