MATH3075 Assignment 1: Solutions
1. Single-periodmarketmodel [12marks] Consider a single-period market modelM = (B,S)
on a sample space Ω = {ω1, ω2, ω3}. Assume that r = 3 and the stock price S = (S0, S1) satisfies
S0 = 5 and S1 = (36, 20, 4). The real-world probability P is such that P(ωi) = pi > 0 for i = 1, 2, 3.
(a) Find the class M of all martingale measures for the model M. Is the market model M
arbitrage-free? Is this market model complete?
Answer: [2 marks] We need to solve: q1 + q2 + q3 = 1, 0 < qi < 1 and (since 1 + r = 4)
EQ(S1) = 36q1 + 20q2 + 4q3 = (1 + r)S0 = 20
or, equivalently,
EQ(S1 − (1 + r)S0) = 16q1 − 16q3 = 0.
Let q2 = λ. Then q1 = q3 = 1−λ2 where 0 < λ < 1. Hence
M =
{
(q1, q2, q3)
∣∣ q1 = q3 = 1− λ
2
, q2 = λ, 0 < λ < 1
}
.
The market model M is arbitrage-free since M 6= ∅. Moreover, it is incomplete since the
uniqueness of a martingale measure forM fails to hold.
(b) Find the replicating strategy for the contingent claim Y = (10, 2,−6) and compute its
arbitrage price pi0(Y ) at time 0 through replication.
Answer: [2 marks] First solution. We may use a portfolio (x, ϕ) ∈ R2 and represent the
wealth as follows: V0(x, ϕ) = x and
V1(x, ϕ) = (x− ϕS0)(1 + r) + ϕS1 = x(1 + r) + ϕ
(
S1 − S0(1 + r)
)
= xB1 + ϕ
(
S1 − S0B1
)
.
Then we solve the following equations
4x+ 16ϕ = 10,
4x+ 0ϕ = 2,
4x− 16ϕ = −6.
From the second equation, we obtain pi0(Y ) = x = 0.5 and thus from the first (or last)
equation we get ϕ = 0.5.
Second solution. The wealth process of a portfolio (ϕ00, ϕ10) satisfies
V0(ϕ) = ϕ
0
0B0 + ϕ
1
0S0, V1(ϕ) = ϕ
0
0B1 + ϕ
1
0S1.
Replication of a claim Y means that V1(ϕ)(ωi) = Y (ωi) for i = 1, 2, 3. Hence to find a
replicating strategy for Y , we need to solve the following equations
4ϕ00 + 36ϕ
1
0 = 10,
4ϕ00 + 20ϕ
1
0 = 2,
4ϕ00 + 4ϕ
1
0 = −6.
We obtain (ϕ00, ϕ10) = (−2, 0.5) and thus pi0(Y ) = x = ϕ00B0 + ϕ10S0 = −2 + 0.5 × 5 = 0.5.
Hence at time 0 we need to buy 0.5 shares of stock. For this purpose, after receiving 0.5
units of cash from the buyer of the claim Y , we need to borrow two units of cash in the
money market.
1
(c) Recompute pi0(Y ) using the risk-neutral valuation formula with an arbitrary martingale
measure Q from the class M.
Answer: [2 marks] For any 0 < q2 = λ < 1 and q1 = q3 = 1−λ2 , the risk-neutral valuation
formula yields, for every 0 < λ < 1,
pi0(Y ) = EQ(Y/B1) =
1
2
(
5
1− λ
2
+ λ− 3 1− λ
2
)
= 0.5.
As expected, the price pi0(Y ) does not depend on λ, that is, on a choice of a martingale
measure.
(d) Check whether that the contingent claim X = (5, 4,−1) is attainable inM.
Answer: [2 marks] To find a replicating strategy, we need to solve the following equations
4ϕ00 + 36ϕ
1
0 = 5,
4ϕ00 + 20ϕ
1
0 = 4,
4ϕ00 + 4ϕ
1
0 = −1.
The strategy (ϕ0, ϕ1) =
(
11
16 ,
1
16
)
is a unique solution to the first two equations, but it does
not satisfy the last one. Hence no replicating strategy for X exists inM.
(e) Find the range of arbitrage prices for X using the class M of all martingale measures for
the modelM.
Answer: [2 marks] We compute the range of prices for X consistent with the no-arbitrage
principle. We have
pi0(X) = EQ(X/B1) =
1
4
(
5
1− λ
2
+ 4λ− 1 1− λ
2
)
= 0.5(1 + λ).
Since from part (c) we know that λ ∈ (0, 1), it is clear the range of prices pi0(X) consistent
with the no-arbitrage principle is the open interval (0.5, 1).
(f) Suppose that at time 0 you have sold the claim X for 2 units of cash. Show that there
exists a hedge ratio ϕ such that the wealth V1(2, ϕ) at time 1 strictly dominates the payoff
X, meaning that V1(2, ϕ)(ωi) > X(ωi) for i = 1, 2, 3.
Answer: [2 marks] It suffices to give any example of a portfolio (x, ϕ) with the initial
value x = 2 such that the inequality V1(x, ϕ)(ωi) > X(ωi) holds for i = 1, 2, 3. We may use
the representation of the wealth at time t = 1
V1(x, ϕ) = (x− ϕS0)(1 + r) + ϕS1 = x(1 + r) + ϕ
(
S1 − S0(1 + r)
)
= xB1 + ϕ
(
S1 − S0B1
)
.
Since x = 2 and B1 = 4 so that S0B1 = 20, it suffices to find a number ϕ ∈ R such that the
following inequalities are satisfied
8 + 16ϕ > 5,
8 + 0ϕ > 4,
8− 16ϕ > −1.
For instance, we may take ϕ = 0.5. Then the wealth of the portfolio (x, ϕ) = (2, 0.5) at time
t = 1 equals V1(2, 0.5) = (16, 8, 0) so it is clear that V1(2, 0.5)(ωi) > X(ωi) for i = 1, 2, 3.
2
2. Static hedging with options [8 marks] Consider a parametrised family of contingent claims
with the payoff Y (α) at time T given by the following expression
Y (α) = min
(
α, β + 2|β − ST | − ST
)
where a real number β > 0 is fixed and the parameter α is an arbitrary real number such that
α ≥ 0.
(a) For any fixed α ≥ 0, sketch the profile of the payoff Y (α) as a function of ST ≥ 0 and
find a decomposition of Y (α) in terms of the payoffs of standard call and put options with
maturity date T (do not use a constant payoff). Notice that a decomposition of Y (α) may
depend on the value of the parameter α.
Answer: [2 marks] It is easy to see that the payoff Y (α) is a piecewise linear and contin-
uous function, which is nonnegative and bounded from above by α.
We first consider the case α ≥ 3β. Let us take β = 1. Then for α ≥ 3β we obtain by taking,
for instance, α = 4
β = 1 β + α = 5
1
2
3
4
ST
Y (α) α ≥ 3β
We now consider the case α < 3β. We take again β = 1 and we obtain by taking, for
instance, α = 2
β = 1 β + α = 3
1
2
3
4
ST
Y (α) α < 3β
It is readily seen that for α ≥ 3β the payoff Y (α) can be represented as follows
Y (α) = 3PT (β) + CT (β)− CT (β + α)
whereas for 0 ≤ α < 3β we have that
Y (α) = 3PT (β)− 3PT
(
β − 13α
)
+ CT (β)− CT (β + α).
Notice that the second decomposition above gives Y (α) = 0 when α = 0 and the two decom-
positions of Y (α) coincide when α = 3β.
3
(b) Assume that call and put options with all strikes are traded at time 0 at some finite prices.
For each value of α ≥ 0, compute the arbitrage price pi0(Y (α)) at time t = 0 for the claim
Y (α) using the prices at time 0 of call and put options and a suitable decomposition ob-
tained in part (a).
Answer: [2 marks] By the additivity property of arbitrage pricing, we obtain, for every
0 ≤ α ≤ 3β,
pi0(Y (α)) = 3P0(β)− 3P0
(
β − 13α
)
+ C0(β)− C0(β + α) (1)
and, for every α ≥ 3β,
pi0(Y (α)) = 3P0(β) + C0(β)− C0(β + α). (2)
In particular, we deduce from (1) that pi0(Y (α)) = 0 when α = 0, which is obvious since
Y (α) = 0 when α = 0.
(c) For any α > 0, examine the sign of an arbitrage price of the claim Y (α) in any (not nec-
essarily complete) arbitrage-free market model M = (B,S) with a finite state space Ω.
Justify your answer.
Answer: [2 marks] Since the payoff Y (α) is strictly positive for every value of ST (except
for ST = β) the price pi0(Y (α)) should be strictly positive in any arbitrage-free market
model M = (B,S) since otherwise an arbitrage opportunity would arise in the extended
market model. You may use the argument that the range of prices for any contingent claim
X coincides with the range of values of the expectation EQ(B−1T X) when Q runs over the
class M of all martingale measures for the modelM.
(d) Consider a complete arbitrage-free market modelM = (B,S) defined on some finite sample
space Ω. Show that the arbitrage price of Y (α) at time t = 0 is a monotone function of the
variable α ≥ 0 and find the limits limα→0 pi0(Y (α)), limα→∞ pi0(Y (α)) and limα→3β pi0(Y (α)).
Answer: [2 marks] We observe that the payoff Y (α) increases when α increases. Specif-
ically, if we consider the payoffs Y (α1) and Y (α2) corresponding to α1 and α2, respectively,
where α1 < α2 then it is clear that Y (α1) ≤ Y (α2). Consequently, pi0(Y (α1)) ≤ pi0(Y (α2))
and thus the price pi0(Y (α)) is a nondecreasing function of the variable α.
Furthermore, limα→0 pi0(Y (α)) = 0 since limα→0 Y (α)(ω) = 0 for all ω ∈ Ω and thus
0 ≤ lim
α→0
sup
Q∈M
EQ
(
B−1T Y (α)
) ≤ lim
α→0
max
ω∈Ω
(
B−1T Y (α)(ω)
)
= 0.
Moreover, using (1) and (2)
lim
α→3β
pi0(Y (α)) = 3P0(β) + C0(β)− C0(4β), lim
α→∞pi0(Y (α)) = 3P0(β) + C0(β).
4