TEKNISKA HÖGSKOLAN I LINKÖPING
Matematiska institutionen
Matematisk statistik
TAMS 29, Exercise sheet 8
(1) Let X1 , X2 , . . . be independent random variables with E[Xi ] = 0 and E[Xi2 ] =
c < ∞, i ∈ N. Define the filtration An := σ(X1 , . . . , Xn ), n ∈ N. Show that
Yn :=
n
X
!2
Xi
− nc ,
n ∈ N,
i=1
is a martingale with respect to An , n ∈ N.
(2) Let X1 , . . . , XN be independent identically distributed random variables on
some probability space (Ω, F, P ) with E[|X1 |] < ∞, P (X1 > 0) > 0, and
P (X1 < 0) > 0.
Construct a probability measure Q on (Ω, F) which is equivalent to P such
that
Sn :=
n
X
Xi ,
n ∈ {1, . . . , N },
i=1
is a martingale with respect to Q.
(3) Consider an arbitrage free one period model with discount factor B. Let M
denote the set of all equivalent martingale measures and let H be the set of
all hedgeable claims. By convention let inf ∅ := ∞ and sup ∅ := −∞.
Let Q ∈ M be an arbitrary risk neutral measure and let EQ denote the
expectation with respect to Q. For a claim Y let
s+ (Y ) := inf {EQ [BC] : C ≥ Y , C ∈ H}
be the upper hedging price and
s− (Y ) := sup {EQ [BC] : C ≤ Y , C ∈ H}
be the lower hedging price. Assume
S := {EQ0 [BY ] : Q0 ∈ M , EQ0 [B|Y |] < ∞} =
6 ∅.
Show the following.
(a) The hedging prices s+ (Y ) and s− (Y ) do not depend on the above choice
of Q.
(b) The model extended by the claim Y with price p is arbitrage free if and
only if p ∈ S.
(4) In the context of problem (3) show
s− (Y ) = inf S
and s+ (Y ) = sup S .