Discrete Financial Mathematics Models. Fall of 2020.
1st Midterm exam.
Prepared by Martynas Manstavičius
At the top of your solutions’ page, please, write the last 4 digits of your
student ID. They will be needed later.
• Consider a one-period market model consisting of a riskless security (with rate
of return r = 1/25) and 3 risky assets (their discounted prices are provided in the
table below). Leave your answers exact, avoid rounding of fractions.
i S0i
(i)
(ii)
(iii)
(iv)
(v)
S̃1i
ω1 ω2 ω3 ω4
1 4 3 4 5 3
2 8 7 7 11 5
3 4 1 6 3 5
(3 points) Is there a dominating strategy in this market? Is this market
arbitrage-free? Does the law of one price hold? Justify your answers.
(1 point) Is the given market complete? Justify your answer.
(2 points) Which contingent claims are attainable in this market?
(2 points) Let ni , i = 1, 2, 3, 4 be the last digits of your student ID number.
Is the discounted claim X̃ = (n1 , n2 , n3 , n4 ) attainable? If your answer was
“YES”, find all replicating strategies for X and the fair value V0X . Otherwise,
find the interval of fair values [V0X ] = [V−X , V+X ].
(2 points) Can we define a state price density in this market? If your answer
was “YES”, how many different state price densities can we have? Is at
least one of them attainable? The initial (physical) probability measure is
P(ωi ) = 1/4, i = 1, . . . , 4.
Extra credit question
(1 point) For which value(s), if any, of K is the claim
YK = max{S11 , S12 , S13 } − K
+
,
attainable in the above-described market? Justify your answer. Here (a)+ denotes the
positive part of number a, i.e., (a)+ = max{a, 0}.
1