IMF Theory Chapter 2

Question 1

Define financial market.

Answer 1

(Omega,F,P,(F_t)_t,S^0,S)

Fixed time horizon T in N.
1) Omega (all possible states of the world)
2) A sig-algebra F on Omega, representing all the ‘pertinent’-events, which means the ones we want to be able to give probability to. (If Omega is finite/countable, typically F=2^Omega)
3) Probability measure P on (Omega,F)
4) Filtration IF=(F_t)_t on (Omega,F,P) (information structure, where F_t=information available at time t) F_0={{},Omega} and F_T=F. (Omega,F,P,IF)=filtered prob. space
v) Financial assets:
- Non-risky asset: S^0={S^0_t : t int in [0,T]}, S^0_0=1, S^0t is F{t-1}-m-able i.e. (S^0_t)t predictable.
S^0{t+1}=S^0_t*R_t for t in [0,T-1], R_t F_t-m-able
- d in N\0 risky assets w. price process S^i={S^i_t:t in [0,T-1]}. S^i_t pos. and F_t-m-able (S^i IF-adapted, not predictable)

Question 2

Define portfolio and wealth.

Answer 2

A portfolio is an IF-adapted process (η, Δ) with values in RxR^d.
More precisely, for any t in [0,T], η_t is the number of shares of the non-risky asset in the port., while Δ^i_t is the number of shares of the i-th risky asset in the portfolio at time t, for i in {1,..,d}.
The wealth associated to this portfolio at time t is given by
η_tS^0_t+Δ_tS_t,
where Δ_tS_t=Sum[Δ^i_tS^i_t ; i=1,..,d]

Question 3

Define consumption process.

Answer 3

For any portfolio, we define the associated consumption process c, by:
c_t:= -η_tS^0t - Δ_t*S_t+( η{t-1}S^0t + Δ{t-1}*S_t)
=( η_{t-1} - η_t , Δ_{t-1} - Δ_t ) * (S^0_t, S_t)^T
A positive c_t means that the investor consumed some of his wealth at t, while c_t<0, means that he injected cash into his portfolio at time t.

Question 4

Define self-financing.

Answer 4

A portfolio (η, Δ) is self-financing if the associated consumption process is always 0.

Question 5

Define A(IR^n)
(Script A).

Answer 5

A(IR^n):={IR^n-valued, IF-adapted processes}

Question 6

Characterize portfolio.

Answer 6

A portfolio (η, Δ) can be characterized by the triplet (x,Δ,c) in IRxA(IR^n)xA(IR), where x is the initial capital of the portfolio.
Correspondence:
η_0 :=x - Δ_0*S_0
η_t=η_{t-1} - (Δ_t - Δ_{t-1})*S_t/S^0_t - c_t/S^0_t

or

η_t = x - Δ_0S_0 - Sum(Δ_t - Δ_{t-1})S_k/S^0_k + c_k/S^0_k ; k=1,..,t)

Question 7

Define the wealth process X^{x,Δ}

Answer 7

X^{x,Δ}_0:=x

X^{x,Δ}t:=X^{x,Δ}{t-1}+Δ_{t-1}(S_t-S_{t-1})+(X^{x,Δ}{t-1} - Δ{t-1}S_{t-1})*(S^0t-S^0{t-1})/S^0_{t-1}

Question 8

Define discounted value.

Answer 8

For any process V, we denote its discounted value by

V~_t := V_t / S^0_t = V_t * d(0,t)

Question 9

State the discounted wealth process

Answer 9

X~^{x,Δ}t:=[X^{x,Δ}{t-1}+Δ_{t-1}(S_t-S_{t-1})+(X^{x,Δ}{t-1} - Δ{t-1}S_{t-1})(S^0t-S^0{t-1})/S^0{t-1}] / S^0_t
=…=X~^{x,Δ}{t-1}+Δ_{t-1}(S~t -S~{t-1})

It follows that

X~^{x,Δ}t = x + Sum( Δ_k * (S~{k+1} - S~_k) ; t=0,..,T )
=x + Int( Δ_s dS~_s ; [0,t] )

lecture 8

Question 10

Define arbitrage opportunity.

Answer 10

An arbitrage opportunity is an investment strategy Δ in A(IR^d) s.t.: P(X^{0,Δ}_T >= 0)=1 and P(X^{0,Δ}_T > 0) > 0.

Question 11

State NA / AoA condition.

Answer 11

For all Δ in A(IR^d):

P(X^{0,Δ}_T >= 0)=1, then P(X^{0,Δ}_T = 0) =1

Question 12

Consider the setting where T=1, d=1, and assume that Omega is finite.
State the equivalence conditions of NA.

Answer 12

pg 62

Question 13

Define (IF,Q)-martingale.

Give an example under NA for a given finite prob space (Omega,F,Q).

Answer 13

(X_t)_{t=0,..,T} martingale if X is IF-adapted, and

i) E^Q[ |X_t| ] finite
ii) X_t = E^Q[ X_{t+1} | F_t ] Q-a.s.

Lect notes 8

Question 14

Define risk-neutral measure.

Answer 14

A prob. measure Q on (Omega,F) is called a risk-neutral measure (or sometimes an equivalent martingale measure EMM) if:

i) Q~P
ii) S~ is an IR^d-valued martingale under Q.

Question 15

Super/-submarts ineq.

Answer 15

Super: E[M_t|F_{t-1}] leq M_{t-1}
Sub: opp.

Question 16

Define stopping time.

Characterize stopping time.

Answer 16

An IF-stopping time τ is a random variable taking values in IN s.t.
{τ leq t} in F_t

τ IN-valued stopping time iff {τ = t} in F_t for all t

Question 17

Define τ_{x,[x,inf)}

Answer 17

Notation for τ_{x,[x,inf)} := inf{ t in IN : X_t>=x }, which is an IF-stopping time.

Question 18

Define stopped process.

Answer 18

Let Y be an IF-adapted process, and τ an IF-stopping time. The stopped process Y^τ is defined by:
Y^τ_t := Y_min(τ,t)

Question 19

Define information available at time τ.

Answer 19

F_τ := { A in F | for all t in IN : A n {τ=t} in F_t }

Question 20

Let Y be IF-adapted and τ an IF-stopping time.

What are Y^τ and F_τ (mathematically)?

Answer 20

i) Y^τ is IF-adapted, F_τ is a sub-sig-alg. of F, and Y_τ is F_τ-mable
ii) If Y is an (IF,P)-mart. (/sub/supermart.), then so is Y^τ.

Question 21

Define local (IF,P)-martingale

Answer 21

A process M is said to be an (IF,P)-local martingale if there exists a sequence (τ_n)_n of IF-stopping times s.t.

i) lim τ_n = inf (n->inf), P-a.s.
ii) M^{τ_n} is an (IF,P)-martingale for all n in IN.

(τ_n)_n is called a localizing sequence for the local mart. M.

Question 22

Construct a local mart. from a mart.

Answer 22

M a IR^m-val. (IF,P)-mart. and φ in A(IR^m) BOUNDED! then
Y_t=Sum(  φ_k * (M_{k+1}-M_k) ; t=0,..,t-1 )
is an (IF,P)-local mart.

BTW: if Q risk-neutral measure, this proves that X~^{x,Δ} is an (IF,Q)-locla mart. for any self-financing port. (0,Δ).

pf: slide 2, lecture 9

Question 23

Give necessary and sufficient condition for a local martingale to be a martingale.

Answer 23

If M is a IR-valued (IF,P)-local mart. on a fixed time interval [0,T], for some integer T.
Then M is an (IF,P)-martingale if and only if E[M^-_T] < inf, where x^- := max(0,x).

Question 24

State FTAP.

Answer 24

Fundamental theorem of asset pricing:
TFAE:
i) NA
ii) M(S) != {}, where M(S) is the set of risk-neutral measures on the market.

Question 25

d=T=1, Ω= {ω^u, ω^d}, IF={ { } }.

Find the set risk-neutral measures.

Answer 25

lecture 10

One-period binomial model:
If NA or 0

F=2^Ω }, S^0(ω) = (1, R)(ω) for all ω, S_0 > 0, S_1(ω) = uS_01{ω=ω^u}+dS_01{ω=ω^d}, 0

Question 26

Multinomial model:
d=1, Ω= {ω^u, ω^d}^T, F_0={ }, F_T=2^Ω,
F_t=σ( (ω_1,..,ω_s) : s in {1,..,t} ),
(R_t)_t, (u_t)_t, (d_t)_t all IF-adapted and s.t. R_t pos. and 0.
Describe the risky asset and state risk-neutral measure under conditions it exists.

Answer 26

lecture 10

Question 27

T=d=1, Ω= {ω^1, ω^2, ω^3}, F=2^Ω, IF= ( {{},Ω} , F), (p1,p2) in (0.1)^2 and
P({ω^1})=p1, P({ω^2}=p2, P({ω^3})=1-p1-p2.

S^0_0=1, S^0_1=R
S_1(ω)=u1S_01{ω=ω^1}+u2S_01{ω=ω^2}+u3S_01{ω=ω^3}

Is there a risk neutral measure? If so what is it?

Answer 27

lecture 11

Question 28

Define viable.

Answer 28

ξ payoff of an option with maturity T.

A price p in R for this option is said to be viable if buying or selling the option at this price does not create arbitrage opportunities in the market.
In other words, there does not exist Δ in A(IR^d) s.t
P(X^{p,Δ}_T - ξ >=0)=1 and P(X^{p,Δ}_T - ξ>0)>0
or
P(X^{p,Δ}_T + ξ >=0)=1 and P(X^{p,Δ}_T + ξ>0)>0

Question 29

Define to super-hedge, super-hedging price and sub-hedging price.

Answer 29

ξ payoff of an option with maturity T.

The price p allows the seller to super-hedge the payoff ξ if there exists Δ in A(IR^d) s.t.
P(X^{p,Δ}_T >= ξ)=1.

The super-hedging price of ξ is defined by:
p(ξ):=inf{p in IR : exists Δ in A(IR^d), P(X^{p,Δ}_T >= ξ)=1 }.

The sub-hedging price of ξ is defined as -p(-ξ) and verifies:
-p(-ξ):=sup{ p in IR ; exists Δ in A(IR^d), P(ξ >= X^{p,Δ}_T )=1 }.

Question 30

Define replicable.

Answer 30

The payoff ξ of an option with maturity T is said to be replicable if there exists a pair (x,Δ) in IRxA(IR^d) s.t.
P(X^{x,Δ}_T=ξ)=1.
Any such pair (x,Δ) is called a replicating strategy for ξ.

Question 31

Related viable prices with (super-/sub-)hedging prices.

Answer 31

p(ξ) is the largest possible viable price for the option with payoff ξ (it is not clear if it is one however).
Symetrically, -p(-ξ) is the lowest possible viable price.
In other words, the set of viable prices for ξ is contained in [-p(-ξ),p(ξ)].
Furthermore, the set of viable prices is either an interval or a singleton.

Question 32

State the properties of p(ξ)

super-hedging price

Answer 32

i) p is sublinear, ie p(ξ1)+p(ξ2)>=p(ξ1+ξ2)
ii) If P(ξ1>=ξ2)=1, then p(ξ1)>=p(ξ2)
iii) p is positively homogeneous, that for any λ>0:
p(λξ)=λp(ξ)
iv) We have p(0)=0 and p(ξ) >= -p(-ξ).

Question 33

Define IL_S(IR,F_T).

Answer 33

IL_S(IR,F_T) := { X ; X F_T-mable, IR-valued and for all Q in M(S): E^Q[|X|] < inf }

Question 34

State the result of replicability, initial capital and “explicit” formula for p(ξ).

Generalize the result.

Answer 34

Let ξ be a replicable payoff, with some replicating strategy (x^0,Δ^0) in IRxA(IR^d). Then
x^0=p(ξ)= -p(-ξ)=inf{ p in IR ; exists Δ in A(IR^d), P(X^{p,Δ}_T=ξ)=1 }.
In particular, all the replicating strategies for ξ have the same initial capital.
In addition, if NA holds and (d(0,T)ξ)^- in IL_S(IR,F_T), then
p(ξ)=E^Q[d(0,T)ξ] for all Q in M(S).

Now to argue why p_t(ξ)=E^Q[d(t,T)*ξ | F_t], where p_t(ξ) is the time value of the option with replicable payoff ξ….

Since ξ is replicable, there is a self-financing portfolio X^{x,Δ} s.t. P(X^{x,Δ}_T=ξ)=1.
By the no-dominance principle therefore:
P(X^{x^0,Δ^0}_t=p_t(ξ))=1 for all t in [0,T], i.e. X^{x^0,Δ^0}_t=p_t(ξ) P-a.s.
Whenever (d(0,T)*ξ)^- in IL_S(IR,F_T), NA holds and X~^{x^0,Δ^0} is an (IF,Q)-mart. for any Q in M(S), we have:
d(0,t)*p_t(ξ)=p~_t(ξ)=X~^{x^0,Δ^0}=E^Q[ X~^{x^0,Δ^0}_T | F_t ]
=E^Q[ d(0,T)*ξ | F_t ]
iff
p_t(ξ)=E^Q[ d(0,T)*ξ/d(0,t) | F_t ]=E^Q[ d(t,T)*ξ | F_t ]

Question 35

Value forward agreements using measure theoretic probability theory.

Answer 35

lecture 12, slide 8
S_t-K*B(t,T)
This uses proposition from end of previous lecture 11

Question 36

Show that payoffs in the one period binomial model with two outcomes are replicable.

Answer 36

P(X^{x,Δ}_1=ξ)=1
iff X^{x,Δ}_1(ω^u)=ξ(ω^u), X^{x,Δ}_1(ω^d)=ξ(ω^d)
iff ΔuS_0+(x-ΔS_0)R=ξ(ω^u), ΔdS_0+(x-ΔS_0)R=ξ(ω^d)
iff Δ=[ξ(ω^u)-ξ(ω^d)] / [(u-d)S_0] (=[ξ(ω^u)-ξ(ω^d)] / [S_1(ω^u)-S_1(ω^d)], x=1/R(qξ(ω^u)-(1-q)ξ(ω^d)), where q=(R-d)/(u-d)

Question 37

Show that the multi-period binomial model is replicable..

Answer 37

lecture 12, slide 3

Question 38

Find the replicating strategy for the lookback call option on slide 5, lecture 12.

Answer 38

.

Question 39

Find a replicating strategy of the one-period trinomial model.

Show that only one solution exists his his equation.

Answer 39

lecture 12, slide 7

Question 40

State the superhedging duality theorem.

Answer 40

If (NA) holds and ξ payoff s.t. (d(0,T)ξ)^- in IL_S(IR,F_T).
Then:
p(ξ)=sup {E^Q[d(0,T)ξ] ; Q in M(S) }
-p(-ξ)=inf {E^Q[d(0,T)ξ] ; Q in M(S) }
If in addition p(ξ) finite, then there is a Δ in A(IR^d) s.t. P(X^{p(ξ),Δ*}_T>=ξ)=1. Furthermore, the set of viable prices is the relative interior of [-p(-ξ),p(ξ)], and p(ξ)=-p(-ξ) if and only if ξ is replicable.

Remark: The previous theorem shows us that whenever M(S) is a singleton, all payoffs are replicable. In this case, we say that the financial market is complete. The converse is true and this constitues the 2nd FTAP.

Question 41

Define completeness of a financial market.

Answer 41

A financial market is said to be complete if all contingent claims ξ with (d(0,T)*ξ)^- in IL_S(IR,F_T), s.t. p(ξ) is finite, are replicable.

Incomplete=not complete

Question 42

Characterize completeness when NA holds.

Answer 42

If (NA) holds, then the financial market is complete iff

M(S) is a singleton.

Question 43

Three parts of the FTAP proof.

Answer 43

i) show enough to consider 1-period models, by introducing local arbitrage opportunities.
ii) Characterize local arbitrage opportunities in more “geometric” terms UPDATE?? What does he mean
iii) Use Hahn-Banach theorem to construct risk-neutral measure in one-period model

Question 44

Define local arbitrage.

Answer 44

Let Q be equivalent to P, and let t be in {1,..,T}.
A t-local arbitrage is a random variable ξ, IR-valued and F_{t-1}-mable, s.t. Q[ ξ(S~t-S~{t-1}) >= 0 ] =1
and Q[ ξ(S~t-S~{t-1}) > 0 ] >0.

Interpretation: A t-local arbitrage opportunity is a specific Δ in A(IR^d) s.t. Δ_k=0 for k in {0,..,T-1}{t-1}.
In this case X~^{0,Δ}=Δ_{t-1}*(S~t-S~{t-1})
Thus it is an arbitrage between t-1 and t.

Question 45

State NA_{loc}.

Answer 45

The financial market has no local arbitrage opportunities.
i.e.
for all Q~P and t in {1,..,T}:
If Q[ ξ(S~t-S~{t-1}) >= 0 ] = 1 for some ξ in L^0(IR,F_{t-1}),
then Q[ ξ(S~t-S~{t-1}) = 0 ] = 1

Question 46

Define K^0(IR,F_t,Q).

Answer 46

We can assume that S~t-S~{t-1} is bounded by defining
δ_t:=(S~t-S~{t-1})/(1+||S~t-S~{t-1}||), which is IF-adapted.
K^0(IR,F_t,Q):={ X in L^0(IR,F_t) : exists ξ in L^0(IR^d,F_t-1), Q(ξ*δ_t >= X) =1 }

Intuition: Forgetting division by 1+||S~t-S~{t-1}|| in δ_t, elements of K^0 are exactly the payoffs at time t, which can be super-replicated by trading with 0 initial capital between t-1 and t.

Question 47

Characterize NA_{loc}

Answer 47

NA and NA_{loc} are equivalent.

Also:
For all t in {1,..,T} and Q~P
NA_t iff K^0(IR,F_t,Q) n L^0_+(IR,F_t,Q)={0}.

Second characterization on slide 7 of lecture 14.

Also:
Fix t in {1,..,T} and some Q~P. Then
NA_t iff there exists a Z in L^inf(IR,F_t,Q) s.t.
Q(Z>0)=1 and E^Q[ Z*(S~t-S~{t-1}) | F_{t-1} ] =0.

Question 48

What topological property does K^0 have?

Answer 48

Let NA_t for some t hold, then for any A~P:

K^0(IR,F_t,Q) is closed in Q-probability.

Question 49

Define N(IR^d,F_{t-1},Q) and N^⊥(IR^d,F_{t-1},Q).
State relation between N^⊥(IR^d,F_{t-1},Q) and L^0(IR^d,F_{t-1})

Answer 49

For all t in {1,..,T}, all Q~P and all ξL^0(IR^d,F_{t-1}), there exists a bar(ξ) in N^⊥(IR^d,F_{t-1},Q), s.t.
Q[ ξ * δ_t = bar(ξ) * δ_t ]=1.
Lecture 13, slide 7

Question 50

Fix t in {0,..T} and Q~P. Let (X_n)_n sequence of r.v.s in L^0 s.t. Q(liminf ||X_n|| finite)=1.
Then?

Answer 50

There is a sequence of IN-valed r.v.s (τ_n)n and an r.v. X in L^0 s.t. Q( lim X{τ_n} = X )=1

(lim w.r.t. n->inf)

Question 51

If time add version of Hahn-Banach..

Answer 51

.

Question 52

State the Stricker-Yan theorem.

Answer 52

Let Q be a prob. measure on (Ω,F) and G s.s. F a σ-alg.. Let C be a closed convex cone of L^1(IR,G,Q) s.t.
L^1- (IR,G,Q) s.s. C and C n L^1+(IR,G,Q)={0}.
Then there exists a Z in L^inf_+(IR,G,Q) s.t. Q(Z>0)=1, and E^Q[Y*Z] non-pos. for all Y in C.

Question 53

Define calligraphic K^2(IR,F_T,P) and state a topological property.

Answer 53

Set of M in L^0(IR,F_T) s.t. there exists Δ in A(IR^d) s.t. X~^{0,Δ}_T is a.s. greater or equal to M

Question 54

Let NA hold, assume that S~t in L^2(IR^d,F_t,P) for all t in {0,..T}, that (d(0,T)ξ)^- in L_S(IR,F_T), and that p(ξ)=-p(-ξ). Let Q be an element of M(S) with bounded density. Then P(X_T^{p(ξ),Δ}=ξ)=1 for any Δ in A(IR^d) s.t. Δ_t in cal(S)t(V,Q), t in {0,..,T-1} where
V_T:=d(0,T)ξ, V_t:=E^Q[V{t+1}|F_t] and cal(S)t(V,Q) is the set of F_t-mable r.v.s which are IR^d-valued and satisfy:
Δ_t * E^Q[(S~{t+1}-S~t)(S~^1{t+1}-S~^i_t)|F_t]=E^Q[V{t+1}(S~^i_{t+1}-S~_t)|F_t] for all i in {1,..,d}.

Use this to find a replicating strategy in the binomial market.

Answer 54

.

Question 55

Define the quadratic error of hedging a contingent claim.

Answer 55

v_q(x,ξ):=inf {E^Q[(d(0,T)ξ-X~^{x,Δ}_T)^2] : Δ in A_2(IR)},
where A_2(IR) is the subset of A(IR) s.t. investment strategies are square integrable.

Question 56

State the non-degeneracy condition in the context of mean-variance hedging.

Give an equivalent condition.

Answer 56

There exists δ in (0,1) s.t. for all t in {1,..,T} the inequality:
(E^P[S~t-S~{t-1} | F_{t-1} ] )^2 leq δ * E^P[(S~t-S~{t-1})^2| F_{t-1}] holds.

BTW: We also assume S~ in L^2 before

alpha_t := E[S~t - S~{t+1} | F_{t-1}] / E[(S~t - S~{t+1})^2 | F_{t-1}] (if denominator is 0 then def =0) is finite for all t and small omega.

Question 57

Under which conditions is there a strategy s.t. the quadratic error of hedging a contingent claim is attained?

How can we get such a Δ(x,ξ)?

Answer 57

Let ND hold, then for ξ s.t. d(0,T)ξ in L^2(IR,F_T,P) and any x in R there exists a Δ(x,ξ) in A_2(IR) s.t.
V_q(x,ξ)=E^P[(d(0,T)ξ-X~^{x,Δ(x,ξ)}_T)^2].

By a backward induction algorithm:
define β_t := E^P[(S~t-S~{t-1
)Prod(1-β_j(S~j-S~{j-1}))|F_{t-1}] / E^P[(S~t-S~{t-1})^2Prod(1-β_j(S~j-S~{j-1}))^2|F_{t-1}]β
ρ_t(ξ):=E^P[d(0,T)ξ(S~t-S~{t-1})Prod(1-β_j(S~j-S~{j-1})|F_{t-1}]/E^P[d(0,T)ξ(S~t-S~{t-1})^2Prod(1-β_j(S~j-S~{j-1})^2|F_{t-1}]
All Prod’s for j=t+1,..,T

Then (if ND), Δ(x,ξ)=ρ_{t+1}(ξ)-β_{t+1}X~^{x,Δ(x,ξ)}_t

Question 58

Define the r.v. Z~ and state its properties (context quadratic hedging).

Answer 58

Lemma:
Z~ := Prod( (1-β_j*(S~_j - S~_{j-1} )) ; j=1,..,T )
* Z~ in L^2
* E[Z~] in [0,1]
* E[Z~]=0 iff Z~=0 a.s.

Question 59

State the proposition about the variance-optimal viable prices.

Answer 59

Let u_q(chi) = inf_{x in R} v_q(x, chi)
If ND holds:
Z~=0 a.s. then u_q(chi) = E[(d(0,T)chi - Sum[rho_j(S~j - S~{j-1}) Prod( (1-beta~_k(S~_k - S~{k-1}) : k=j+1,..T)] and inf. is attained for all x in R.
Z~ =/= 0 a.s., then
u_q(chi) = v_q(p_q(chi),chi) = - E[Z~d(0,T)chi]^2 / E[Z~] + above term,
where p_q(chi) := E[Z~/E[Z~]d(0,T)*chi].
Furthermore, in this case P~ exists and its density D~ is given by D~ = Z~ / E[Z~].