ASSIGNMENT 1: SOLUTIONS
MATH3075 Mathematical Finance (Normal)
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 $54 per share, and
that the price per share will rise to either $56 or $62 at the end of a period, that
is, at time 1, with equal probability. Assume that the one-period simple interest
rate r equals 10%. We consider call and put options written on the stock S, with
the strike price K = $59 and the expiry date T = 1.
(a) Construct unique replicating strategies for these options as vectors (П•0 , П•1 ) в€€
R2 such that V1 (П•0 , П•1 ) = П•0 B1 + П•1 S1 . Note that V1 (П•0 , П•1 ) = V1 (x, П•) where
x = П•0 + П•1 S0 and П• = П•1 .
(b) Compute 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 15%.
2. [10 marks] Single-period market model.
Consider a single-period security market model M = (B, S) on a finite space
Ω = {ω1 , ω2 , ω3 }. Assume that the savings account B equals B0 = B1 = 1 and
the stock price S satisfies S0 = 7 and S1 = (S1 (П‰1 ), S1 (П‰2 ), S1 (П‰3 )) = (4, 6, 9). The
real-world probability P is such that P(П‰i ) = pi > 0 for i = 1, 2, 3.
(a) Check directly that the model M is arbitrage-free.
(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(П‰1 ), X(П‰2 ), X(П‰3 )) = (2, 4, 7) is attainable and compute its arbitrage price ПЂ0 (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 в€€ M.
Solution to Question 1
Note that П•0 is the amount invested in the bank account. The initial wealth of our
portfolio is thus given by x = V0 (x, П•) = П•0 + П•1 S0 and the value of the portfolio at time
t = 1 is given by
V1 (x, П•) = (x в€’ П•1 S0 )(1 + r) + П•1 S1 = П•0 (1 + r) + П•1 S1 .
(a) For the call option with the payoff C1 = (S1 в€’ 59)+ , the unique replicating strategy
(П•0 , П•1 ) can be found by solving the following equations:
(
) ( 0) ( )
1.1 62
3
П•
=
=в‡’ (П•0 , П•1 ) = (в€’28/1.1, 1/2).
1
1.1 56
П•
0
For the put option with the payoff P1 = (59 в€’ S1 )+ , the unique replicating strategy
(П•0 , П•1 )is computed through the following system of linear equations:
(
) ( 0) ( )
1.1 62
П•
0
=
=в‡’ (П•0 , П•1 ) = (31/1.1, в€’1/2).
1
1.1 56
П•
3
(b) Using part (a), we can compute the price of the put and the call. We know that
the initial endowment x of the replicating portfolio is given by x = П•0 + П•1 S0 . Hence
the prices of the put and call options are equal to
C0 = в€’28/1.1 + 27 = 1.5454,
P0 = 31/1.1 в€’ 27 = 1.1818.
(c) On the one hand, we have C0 в€’ P0 = 1.5454 в€’ 1.1818 = 0.3636. On the other hand,
we obtain
S0 в€’ K/(1 + r) = 54 в€’ 59/1.1 = 0.3636.
Hence the put call-parity relationship holds.
(d) Recall that
P(П‰1 ) =
1+rв€’d
,
uв€’d
P(П‰2 ) = 1 в€’ P(П‰1 ).
Hence the unique risk-neutral probability measure is P(П‰1 ) = 0.56667 and P(П‰2 ) =
0.43333. The arbitrage prices can be computed using the risk-neutral valuation formula and, of course, the prices should also be 10/9 for both options. For instance, the
price of the call satisfies
(
)
(
)
C1
1
1
C0 = EP
=
EP (S1 в€’ K)+ =
Г— 0.5667 Г— 3 = 1.5454.
1+r
1+r
1.1
(e) The new replicating strategy (П•0 , П•1 ) for the call option is given by
(
) ( 0) ( )
3
1.05 62
П•
=
=в‡’ (П•0 , П•1 ) = (в€’28/1.05, 1/2).
1
П•
0
1.05 56
For the put option, we obtain
(
) ( 0) ( )
0
1.05 62
П•
=
=в‡’ (П•0 , П•1 ) = (31/1.05, в€’1/2).
1
П•
3
1.05 56
The new price of the call is C0 = 0.3333 and price of the put is P0 = 2.5238. Hence
C0 в€’ P0 = в€’2.1904 and S0 в€’ K/(1 + r) = в€’2.1904.
Solution to Question 2
(a) If x = 0 then the wealth at time 1 satisfies, for every П•1 в€€ R,
V1 (0, П•1 ) = П•1 (S1 в€’ (1 + r)S0 ) = П•1 ((4, 6, 9) в€’ (7, 7, 7)) = (в€’3П•1 , в€’П•1 , 2П•1 ).
It is not possible that all three numbers are non-negative, unless П•1 = 0. In that case,
V1 (0, П•1 ) = (0, 0, 0). Hence no arbitrage opportunity exists in the model M = (B, S)
and thus M is arbitrage-free.
(b) We need to solve q1 + q2 + q3 = 1, 0 < qi < 1 and (since r = 0)
EQ (S1 ) = EQ (S1 ) = 4q1 + 6q2 + 9q3 = S0 = 7.
The equation above is equivalent to
EQ (S1 в€’ S0 ) = в€’3q1 в€’ q2 + 2q3 = 0.
Let q3 = О±. Then q1 = 3О±в€’1
, q2 = 3в€’5О±
and thus the inequalities 0 < q1 < 1 and
2
2
0 < q2 < 1 hold when 1/3 < О± < 3/5. Consequently, the class M of all risk-neutral
probability measures is given by
{
}
3О± в€’ 1
3 в€’ 5О±
M = (q1 , q2 , q3 ) q1 =
, q2 =
, q3 = О±, О± в€€ (1/3, 3/5) .
2
2
Since the risk-neutral probability measure is not unique, we conclude market model
M is not complete.
(c) The class A of all attainable claims is the two-dimensional linear subspace of R3
spanned by the vectors 1 = (1, 1, 1) and S1 = (4, 6, 9). Formally, it equals
A = {X в€€ R3 | X = a1 1 + a2 S1 for some a1 , a2 в€€ R}
or, equivalently
A = {X в€€ R3 | X = c1 B1 + c2 S1 for some c1 , c2 в€€ R}.
Any other equivalent representation for the plane A is fine as well.
(d) We need to solve equations
П•0 + 4П•1 = 2,
П•0 + 6П•1 = 4,
П•0 + 9П•1 = 7.
The strategy (П•0 , П•1 ) = (в€’2, 1) replicates X and thus X is attainable. The price of X
at time t = 0 thus equals ПЂ0 (X) = V0 (П•) = в€’2B0 + S0 = 5.
(e) For every 1/3 < О± < 3/5, we obtain (recall that B1 = 1)
( )
X
3О± в€’ 1
3 в€’ 5О±
EQ
=2
+4
+ 7О± = 5,
B1
2
2
which shows that the expected value does not depend on the choice of a risk-neutral
probability measure Q from the class M.
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