Find an example of a filtered probability space (Ω, F, F t ) t∈[0,T ] , P ) and an adapted process (M t ) t∈[0,T ] with E [M t ] = E [M 0 ] for all t = 0, . . . , T such that (M t ) t∈[0,T ] is not a martingale.

Introduction to mathematical finance

Sheet 4 HS 2014

Hand-in your solutions until 15.10.2014, in Martina Dal Borgo’s mailbox on K floor (or give the sheet in person on the 15.10 in class).

Exercise 1

Find an example of a filtered probability space (Ω, F, F t ) t∈[0,T ] , P ) and an adapted process (M t ) t∈[0,T ] with E [M _t ] = E [M ₀ ] for all t = 0, . . . , T such that (M _t ) _{t∈[0,T ]} is not a martingale.

Exercise 2

Let (X _n ) n∈ N be a sequence of i.i.d. random variables. Set F _n = σ(X ₁ , . . . , X _n ) and define S _n :=

X ₁ + · · · + X _n , n ∈ N .

i) Assume that for some α ∈ [0, ¹ ₂ ],

P (X ₁ = 1) = 1

2 + α, P (X ₁ = −1) = 1 2 − α.

Show that for all k, n ∈ N , with k ≤ n,

E [S n |F k ] = S k + 2α(n − k).

ii) Assume that X ₁ ∼ N _(0,σ

₎ , with positive variance 0 < σ ² < ∞. Set Z _n ^u := exp

uS _n − nu ² σ ² 2

. Show that for every u ∈ R , the process (Z _n ^u ) _{n∈ N} is an (F _n )-martingale

Note: The relation E (XY |G) = X E (Y |G) holds for X G-measurable, as soon as both XY and Y are integrable.

Exercise 3

Let (Ω, F, (F _t ) t∈[0,T ] , P ) be a filtered probability space with F ₀ = {∅, Ω}. Assume that (X _t ) t∈[0,T] is an R ^d -valued adapted stochastic process and φ = (φ _t ) t∈[1,T ] is an R ^d -valued predictable process. We set

V ₀ = φ ₁ · X ₀ , V _t = φ _t · X _t , t = 1, . . . , T and

G ₀ = 0, G _t =

t

X

k=1

φ _k · (X _k − X k−1 ) , t = 1, . . . , T.

In the following assume that X is a martingale. Show that i) If (φ _t ) 1≤t≤T is bounded, then (G _t ) 0≤t≤T is a martingale.

ii) If (φ _t ) 1≤t≤T is bounded, then

φ _t · X _t = φ _t+1 · X _t , ∀t = 1, . . . , T − 1. (1) implies that (V _t ) 0≤t≤T is a martingale.

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Note: This exercise has the following economical interpretation: a market model consists of d risky assets whose discounted prices are denoted by X. Moreover, φ is a trading strategy with associated value process V and a gain process G, that is the sum of the variation of the portfolio’s values

G _t =

t

X

k=1

V _t − V _t−1 .

Condition (1) is the self-financing condition which means that no funds are added or removed after time t = 0. If the probability measure is risk neutral, then X will be a martingale and the above exercise shows that also the gain and value process will be martingales under the self-financing condition.

Exercise 4

Let (Y t ) 1≤t≤T be a sequence of i.i.d. normal random variables on (Ω, F , P ) with E [Y t ] = 0, Var(Y t ) = σ ² > 0.

For the sake of simplicity, interest rate is zero. We define the filtration F 0 = {∅, Ω} , F t = σ (Y 1 , . . . , Y t ) , t = 1, . . . , T and the (discounted) price process

S ₀ = 1, S _t = S ₀ +

t

X

k=1

Y _k .

We ignore the fact that S can take negative values. Furthermore we define another filtration ( G t ) 0≤t≤T by including the insider information about the final stock price S _T , i.e. G _t = σ(F _t , S _T ), t = 0, . . . , T.

i) Show that (S _t ) is a (F _t )-martingale, but not a (G _t )-martingale. Moreover prove that the process S _t ^∗ := S _t −

t−1

X

k=0

S _T − S _k

T − k , t = 0, . . . , T, is a (G _t )-martingale.

Hint: Use properties of the Gaussian to see that Y _t+1 − α (S _T − S _t ) is independent from G _t if and only if α = _T ¹ _−t .

ii) Having additional information about S _T allows one to create portfolios with positive expected profit.

Compute a trading strategy ( ˜ φ _t ) _t=1,...,T which maximizes the expected discounted profit E [G _T ] for G _T :=

T

X

k=1

φ _k (S _k − S k−1 ) ,

in the class of all (G _t )-predictable trading strategy φ (describing the holdings in the stock) satisfying

|φ _t | ≤ 1 for all t = 1, ..., T.

Hint: begin using the Exercise 3 to deduce that

E

" _T X

k=1

φ _k (S _k − S k−1 )

#

= E

" _T X

k=1

φ _k · S T − S k−1

T − (k − 1)

# .

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