The CRR model: European claim. Consider the CRR model of stock price S with T periods

August 30, 2017

Question
1. The CRR model: European claim.

Consider the CRR model of stock price S with T periods and parameters d <

1 + r < u, where r is the one-period interest rate.

ASSIGNMENT 1
MATH3075 Mathematical Finance (Normal)
Due by 4 pm on Thursday, September 17, 2015
1. [10 marks] Elementary market model.
Consider a single-period two-state market model M = (B, S) with the two dates:
0 and 1. Assume that the stock price S0 at time 0 is equal to \$27 per share, and
that the price per share will rise to either \$28 or \$31 at the end of a period, that
3
is, at time 1, with probabilities 4 and 1 respectively. Assume that the one-period
4
simple interest rate r equals 10%. We consider call and put options written on
the stock S with the strike price K = \$28.5 and the expiry date T = 1.
(a) Construct unique replicating strategies for these options as vectors (ϕ0 , ϕ1 ) ∈
R2 such that V1 (ϕ0 , ϕ1 ) = ϕ0 B1 + ϕ1 S1 . Note that V1 (ϕ0 , ϕ1 ) = V1 (x, ϕ) where
x = ϕ0 + ϕ1 S0 and ϕ = ϕ1 .
(b) Compute the arbitrage prices of call and put options through replicating
strategies.
(c) Check that the put-call parity relationship holds.
(d) Find the unique risk-neutral probability P for the market model M and
recompute the arbitrage prices of call and put options using the risk-neutral
valuation formula.
(e) How will the replicating portfolios and arbitrage prices of the call and put
options change if we assume that the interest rate r equals 5%?
2. [10 marks] Single-period market model.
Consider a single-period security market model M = (B, S) on a ﬁnite sample
space Ω = {ω1 , ω2 , ω3 }. Assume that the savings account B equals B0 = B1 = 1
and the stock price S satisﬁes S0 = 6 and S1 = (S1 (ω1 ), S1 (ω2 ), S1 (ω3 )) = (3, 6, 8).
The real-world probability P is such that P(ωi ) = pi > 0 for i = 1, 2, 3.
(a) Show that the model M is arbitrage-free by checking that no arbitrage opportunity exists.
(b) Find the class M of all risk-neutral probability measures for the model M.
Is this market model complete?
(c) Find the class A of all attainable contingent claims.
(d) Check that the contingent claim X = (X(ω1 ), X(ω2 ), X(ω3 )) = (2, 4, 7) is attainable and compute its arbitrage price π0 (X) using the replicating strategy for X.
(e) Consider again the contingent claim X = (2, 4, 7). Show that the expected
value
( )
X
EQ
B1
does not depend on the choice of a risk-neutral probability Q ∈ M.

ASSIGNMENT 1
Due by 4 pm on Thursday, September 17, 2015
1. [10 marks] Elementary market model.
Consider a single-period two-state market model M = (B, S) with parameters
S0 > 0 and 0 < d < 1 + r < u.
(a) Find the probability measure P such that
( )
B1
B0
.
EP
=
S1
S0
(b) Compute the Radon-Nikodym density of P with respect to the risk-neutral
probability P.
(c) Let X be any contingent claim. Show that the price π0 (X) satisﬁes
( )
X
.
π0 (X) = S0 EP
S1
(d) Let X = C1 = (S1 − K)+ for some K > 0. Show that the price C0 equals
C0 = S0 P(S1 > K) − K(1 + r)−1 P(S1 > K).
(e) Express the arbitrage price P0 of the put option in terms of probability measures P and P and derive the put-call parity relationship using part (d).
2. [10 marks] Single-period market model.
Consider a single-period security market model M = (B, S) on a ﬁnite sample
space Ω = {ω1 , ω2 , ω3 }. We assume that the savings account B equals B0 = 1 and
B1 = 4. The stock price S satisﬁes S0 = 2.5 and S1 = (S1 (ω1 ), S1 (ω2 ), S1 (ω3 )) =
(18, 10, 2). 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 model complete?
(b) Find the replicating strategy for the claim
X = (X(ω1 ), X(ω2 ), X(ω3 )) = (5, 1, −3)
and compute the arbitrage price π0 (X) at time 0.
(c) Compute the arbitrage price π0 (X) using the risk-neutral valuation formula
with an arbitrary martingale measure Q from M.
(d) Show that the contingent claim Y = (Y (ω1 ), Y (ω2 ), Y (ω3 )) = (10, 8, −2) is not
attainable and ﬁnd the range of arbitrage prices π0 (Y ).
(e) For the contingent claim Z = (20, 16, −4), ﬁnd the minimal initial endowment x for which there exists a portfolio (x, ϕ) with V0 (x, ϕ) = x and such
that the inequality V1 (x, ϕ)(ωi ) ≥ Z(ωi ) holds for i = 1, 2, 3.

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