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

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

MATH3975 Mathematical Finance (Advanced)

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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**$ 5**

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