Large losses - probability minimizing approach

0.14 MB·

by  Michał Barski

Large losses - probability minimizing approach
Micha l Barski
Faculty of Mathematics and Computer Science, University of Leipzig, Germany
Faculty of Mathematics, Cardinal Stefan Wyszyn´ski University in Warsaw, Poland
[email protected]
6 January 14, 2016
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0
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Abstract
n
a
Theprobabilityminimizingproblemoflargelossesofportfolioindiscreteandcontinuous
J
time models is studied. This gives a generalizationof quantile hedging presented in [3].
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Key words: quantile hedging, shortfall risk, transaction costs, risk measures.

]
F
AMS Subject Classification: 60G42, 91B28, 91B24, 91B30.
M
. GEL Classification Numbers: G11, G13
n
i
f
-
q 1 Introduction
[

1 Let (St) be a d-dimensional semimartingale on a filtered probability space (Ω, ,( t),P) which
F F
v represents the stock prices. We denote by the set of all martingale measures, it means that
8 Q
Q if Q P and (S ) is a martingale with respect to Q. Let H be a measurable random
8 t
∈ Q ∼ F
3 variablecalledcontingent claim. Itisknownthatonsuchmarketwehavetwoprices: thebuyer’s
3 price u = inf EQ[H] and the seller’s price u = sup EQ[H], which usually are different.
0 A natubral queQs∈tiQon arises : what price from thesso calleQd∈Qarbitrage-free interval [u ,u ] should
. b s
1
be chosen? This problem was a motivation for introducing risk measures on financial markets.
0
6 Various approaches were presented to answer this question, see for instance [1],[2],[3],[4],[6].
1 In [3] F¨ollmer, and Leukert study the quantile hedging problem. They define a random
:
v variable ϕ connected with the strategy (x,π) by:
x,π
i
X
x,π
X
ar ϕx,π = 1{XTx,π≥H}+ HT 1{XTx,π<H},
x,π
whereX is the terminalvalue of the portfolioconnected with the strategy π starting from the
T
initial endowment x. If x u then for the hedging strategy π˜ we have E[ϕ ] = 1, otherwise
s x,π˜
≥
E[ϕ ] < 1 for each π. The aim of the trader is to maximize E[ϕ ] over π from the set of
x,π x,π
all admissible strategies. Actually, the motivation of quantile hedging was a slightly different
problem, namely
x,π
P(X H) max.
T ≥ −→ π
This problem was solved by the above approach only in a particular case.
Now assume that investor has a loss function u : [0, ) [0, ), u(0) = 0, which is
∞ −→ ∞
assumed to be continuous and strictly increasing, and he accepts small losses of the portfolio. It
1
meanshehasnoobjectionstolossess.t. u((H Xx,π)+) α,whereα 0isalevelofacceptable
− T ≤ ≥
losses fixedbytheinvestor. Hewants toavoid losses which exceed α. As theoptimality criterion
we admit maximizing probability that losses are small. More precisely, the problem is
P[u((H Xx,π)+) α] max,
− T ≤ −→ π
where π is an admissible strategy. Notice that for α = 0 we obtain an original problem of
quantile hedging.
The paper is organized as follows. In section 2 we precisely formulate the problem. It turns
outthat the solution on complete markets has a clear economic interpretation. Itis presented in
section 3. Sections 4 and 5 provide examples of Black-Scholes model and the CRR model. For
B-S model explicit solution is found while for the CRR model existence is clear, but solutions
are found for some particular cases. In section 6 result for incomplete markets is proved and
presented in a one step trinomial model.
2 Problem formulation
We consider financial markets with either discrete or continuous time and with finite horizon T.
Let S be a d dimensional semimartingale describing evolution of stocks’ prices on the filtered
t
x,π
probability space (Ω, ,( ),P). X is a wealth process connected with a pair (x,π), where π
F Ft t
is a predictable process describing self-financing strategy and x is an initial endowment. Thus
x,π x,π
the wealth process is definedby: X = x, X = π S and the self-financing condition means
0 t t· t
that
π S = π S in case of discrete time model
t t t+1 t
· ·
x,π
dX = π dS , π L(S) in case of continuous time model; L(S)is the set of predictable
t t t ∈
processes integrable w.r. to S.
For simplicity assume that the interest rate is equal to zero and that the set of all martingale
measures , so that measures Q that S is a martingale with respect to Q and Q P, is not
t
Q ∼
empty. Among all self-financingstrategies wedistinguishset of alladmissiblestrategies which
x,π x,πA
satisfy two additional conditions: X 0 for all t and X is a supermartingale with respect
x,π t ≥ t
to each Q . If X 0 then the second requirement is automatically satisfied for S being a
∈ Q t ≥
continuous semimartingale, since then the wealth process is a Q-local martingale bounded from
x,π
below, so by Fatou’s lemma it is a supermartingale. In discrete time X is even a martingale,
t
see [5] Th. 2. Let H be a nonnegative , measurable random variable, called contingent
T
F
claim, which satisfies condition H L1(Q) for each Q . Its price at time 0 is given by
v = sup EQ[H]. This means tha∈t there exists a strate∈gyQπ˜ such that Xv0,π˜ H. Such
π˜0is calleQd∈aQhedging strategy. Now assume that we are given an∈inAitial capital 0 T x ≥< v . The
0 0
≤
question arises, what is an optimal strategy for such endowment? As an optimality criterion we
admit minimizing probability of a large loss. Let u : [0, ) [0, ) be a strictly increasing,
∞ −→ ∞
continuous function such that u(0) = 0. Such function will be called a loss function. Let α 0
≥
be a level of acceptable losses. We are searching for a pair (x,π) such that
P[u((H Xx,π)+) α] max,
− T ≤ −→ π
∈A
x x .
0
≤
If there exists a solution (x,π) of the problem above, then it will be called optimal.
2
3 Complete models
Let = Q , so the martingale measure is unique. Recall that in this case each nonnegative
Q { }
Q - integrable contingent claim X can be replicated. It means that there exists π˜ such that
Xv0,π˜ = X, where v = EQ[X]. In complete case the solution of our problem has a clear
T 0
economic interpretation. Let us start with the basic theorem describing the solution.
Theorem 3.1 If there exists X˜ L0 which is a solution of the problem
∈ +
P[u((H X)+) α] max
− ≤ −→
EQ[X] x
0
≤
then the replicating strategy for X˜ is optimal.
Proof : Recall thatfor (x,π),π thewealth processXx,π isasupermartingalewithrespect
to Q. Thus we have EQ[Xx,π] x∈Ax and P[u((H Xx,πt)+) α] P[u((H X˜)+) α].(cid:3)
T ≤ ≤ 0 − T ≤ ≤ − ≤
The main difficulty in this theorem is that we do not have an existence result for X˜ and any
method of constructing which could be used for practical applications. However, we show that
theproblemcan bereducedto asimpler oneby consideringanarrower class of randomvariables
than L+ and for this class in some situations the problem can be explicitly solved. This is an
0
idea of considering strategies of class which we explain below.
S
Economic motivation for introducing strategies of class
For (x,π),π consider two sets: A= ω Ω :u((H Xx,π)+S) α and its compliment Ac.
∈ A { ∈ − T ≤ }
Basing on (x,π) let us build a modified strategy (x˜,π˜) in the following way. On Ainvestor’s loss
is smaller than α. However, from our point of view it can be as large as possible, but not larger
than α. Therefore let (x˜,π˜) be such that on A holds u((H Xx˜,π˜)+) = α. On Ac investor did
− T
not manage to hedge large loss, so the portfolio value can be as well equal to 0. Such (x˜,π˜) we
will regard as a strategy of class . What is an advantage of such modification ? It turns out
S
that π˜ and the following inequalities hold:
∈ A
P[u((H Xx˜,π˜)+) α] = P[u((H Xx,π)+) α], x˜ x.
− T ≤ − T ≤ ≤
This fact is a motivation for searching the solution of the problem only among strategies of class
. Below we present this idea in a more precise way.
S
Definition 3.2 Random variable X L0 isof class ifthere existsA containing u(H)
∈ + S ∈ F { ≤
α such that
}
1. on A we have
(a) if u(H) α then X = 0
≤
(b) if u(H) > α then u(H X) = α
−
2. on Ac we have X = 0.
NoticethatonthesetAwehaveX = 0ifH u 1(α)andX = H u 1(α)ifH > u 1(α). Thus
− − −
≤ −
on A we have X = (H u 1(α))+. Since X = 0 on Ac we obtain that X = 1 (H u 1(α))+.
− A −
− −
In other words X if it is of the form X = 1 (H u 1(α))+ for some A such that
A −
∈ S − ∈ F
A u(H) α .
⊇ { ≤ }
3
Lemma 3.3 For each X L0 such that EQ[X] x there exists a random variable Z
∈ + ≤ 0 ∈ S
such that EQ[Z] EQ[X] and
≤
P[u((H X)+) α] = P[u((H Z)+) α].
− ≤ − ≤
Proof : Let us define A := ω : u(H X)+ α . Then Z := 1 (H u 1(α))+ and we
A −
{ − ≤ } − ∈ S
have
P[u((H Z)+) α] = P[u((H 1 (H u 1(α))+)+) α]
A −
− ≤ − − ≤
= P[ω A:u((H (H u 1(α))+)+) α]+P[ω Ac :u(H) α]
−
∈ − − ≤ ∈ ≤
= P[ω A u(H) α : u(H) α]+P[ω A u(H) > α :u(u 1(α))) α]
−
∈ ∩{ ≤ } ≤ ∈ ∩{ } ≤
= P(A).
On the set Ac holds Z = 0 X. On A if u(H) α then Z = 0 X and if u(H) > α then
≤ ≤ ≤
Z = H u 1(α) X. Thus we have Z X and EQ[Z] EQ[X]. (cid:3)
−
− ≤ ≤ ≤
Remark 3.4 The above calculations show that for any X = 1 (H u 1(α))+ holds
B −
− ∈ S
P(u(H X)+ α) = P(B).
− ≤
Using lemma 3.3 and remark 3.4 we can reformulate theorem 3.1 in the following form.
Theorem 3.5 If there exists set A˜ u(H) α which is a solution of the problem :
⊇ { ≤ }
P(A) max (3.5.1)
−→
EQ[1 (H u 1(α))+] x (3.5.2)
A − 0
− ≤
then the replicating strategy for 1 (H u 1(α))+ is optimal.
A˜ − −
Proof : Indeed, by lemma 3.3 the problem
P[u(H X)+ α] max, EQ[X] x , X L0
− ≤ −→ ≤ 0 ∈ +
can be replaced by
P[u(H X)+ α] max, EQ[X] x , X .
0
− ≤ −→ ≤ ∈ S
However, by remark 3.4 we know that for X = 1 (H u 1(α))+ we have
A −
− ∈ S
P[u(H X)+ α] = P(A) and the required formulation is obtained. (cid:3)
− ≤
Remark 3.6 Let us consider the optimizing problem from theorem 3.5 given by 3.5.1 and 3.5.2
but without the requirement that A˜ u(H) α . Notice that if P(u(H) α) > 0 then the
⊇ { ≤ } ≤
solution A˜ must contain u(H) α . Suppose the contrary and define A˜˜ := A˜ u(H)
{ ≤ } ∪ { ≤
α . Then EQ[1 (H u 1(α))+] = EQ[1 (H u 1(α))+] x and P(A˜˜) > P(A˜) what is a
} A˜˜ − − A˜ − − ≤ 0
contradiction. This shows that that the requirement A˜ u(H) α in the theorem 3.5 can be
⊇ { ≤ }
dropped.
4
In some particular cases the existence and construction of the set A˜ can be solved by using
Neyman-Pearsonlemma. TothisendletusintroduceameasureQ¯ whichisabsolutelycontinuous
with respect to Q by:
dQ¯ (H u 1(α))+
−
= − .
dQ EQ[(H u 1(α))+]
−
−
Then set A˜ solves the following problem
P(A) max
−→
x
Q¯(A) 0 .
≤ EQ[(H u 1(α))+]
−
−
To make the paper self-contained we present a part of the Neyman-Pearson lemma. Let P and
1
P be two probability measures such that there exists density dP1.
2 dP2
Lemma 3.7 If there exists constant β such that P dP1 β = γ then P dP1 β P (B)
2{dP2 ≥ } 1{dP2 ≥ } ≥ 1
for any set B satisfying P (B) γ.
2
≤
Proof : Let B be a set satisfying P (B) γ and denote B˜ := dP1 β . Then we have
2 ≤ {dP2 ≥ }
P (B˜) P(B)= (1 1 )dP = (1 1 )dP + (1 1 )dP
1 − ZΩ B˜ − B 1 ZddPP12≥β B˜ − B 1 ZddPP21<β B˜ − B 1
(1 1 )βdP 1 βdP
≥ ZddPP21≥β B˜ − B 2 −ZddPP12<β B 2
= β dP dP = β(γ P (B)) 0.
2 2 2
(cid:18)ZB˜ −ZB (cid:19) − ≥
(cid:3)
This lemma is useful for the Black-Scholes model since there the condition Q¯ dP β =
{dQ¯ ≥ }
x0 is satisfied. However, in case of discrete Ω this condition no longer holds. This
EQ[(H u−1(α))+]
will be−shown in the example of the CRR model.
4 Black-Scholes model
Here we follow an example presented in [3]. The stock price S is given by
t
dS = S (µdt+σdW ), S = s,
t t t 0
where µ and σ > 0 are constants and W is a standard Brownian motion. For this model
t
St = se(µ−12σ2)t+σWt.
and the unique martingale measure Q is given by
dQ = e−µσWT−12(σµ)2T.
dP
Moreover, the process W = W + µt is a Brownian motion with respect to Q. Notice that the
t∗ t σ
density of the martingale measure can be expressed in term of S , namely
T
dQ µ
= cS−σ2, where c is some constant.
dP T
5
We study a risk minimizing problem for a European call option with strike K. Recall that the
problem is reduced to constructing set A˜ being a solution of
P(A) max
−→
x
Q¯(A) 0 ,
≤ EQ[(S K u 1(α))+]
T −
− −
where measure Q¯ is as in the previous section :
dQ¯ (S K u 1(α))+
T −
= − − .
dQ EQ[(S K u 1(α))+]
T −
− −
Notice, that the superscript ”+ ” above can be dropped since for any a,b,c 0 holds ((a
≥ −
b)+ c)+ = (a b c)+. According to Neyman-Pearson lemma we are searching for the set A˜
− − −
of the form:
dP dP µ
c = c (S K u 1(α))+ = Sσ2 c c (S K u 1(α))+ ,
dQ¯ ≥ 1 dQ ≥ 2 T − − − T ≥ · 2 T − − −
(cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:27)
where c ,c are nonnegative constants such that
1 2
EQ[1 (S K u 1(α))+] = x . (4.0.3)
A˜ T − − − 0
Let us consider two cases.
1) µ σ2
≤ µ
Then the function x xσ2 is concave and has 0 in 0 and thus the solution is given by
A˜ = {ST ≤ c3} =−→{WT∗ ≤ c4}, where c3 and c4 s.t. c3 = seσc4−12σ2T are constant numbers
satisfying 3.0.3. The optimal strategy is a strategy which replicates the following contingent
claim:
1 (S K u 1(α))+ = 1 (S K u 1(α))+
A˜ T − − − {ST≤c3} T − − −
= (S K u 1(α))+ (S c )+ (c K u 1(α))1
T − − − − T − 3 − 3 − − − {ST>c3}
and the corresponding probability is equal
c µT
P(A˜)= P(W c )= Φ 4− σ .
T∗ ≤ 4 √T
(cid:18) (cid:19)
For calculating constants c and c from 4.0.3 we use formula for pricing European call option.
3 4
EQ (S K u 1(α))+ (S c )+ (c K u 1(α))1 =
T − − − − T − 3 − 3− − − {ST>c3}
h i
c +σT c
sΦ(d¯+) (K +u−1(α))Φ(d¯ ) sΦ − 4 +c3Φ 4
− − − √T − √T −
(cid:18) (cid:19) (cid:18) (cid:19)
(c K u 1(α))Q W > c =
3 − T∗ 4
− − { }
c +σT c
sΦ(d¯+) (K +u−1(α))Φ(d¯ ) sΦ − 4 +(K +u−1(α))Φ 4 = x0,
− − − √T − √T
(cid:18) (cid:19) (cid:18) (cid:19)
6
where d¯ = 1 ln K+u−1(α) +1σ√T and Φ stands for the distribution function of the
+ −σ√T s 2
N(0,1) distribution. (cid:16) (cid:17)
2) µ > σ2
µ
In this case the function x xσ2 is convex and therefore our solution is of the form
−→
A˜= S <c S > c = W < c W > c
T 5 T 6 T∗ 7 T∗ 8
{ }∪{ } { }∪{ }
µ
wherec5 < c6 are two solutions of the equation xσ2 = c¯(x K u−1(α))+, wherec¯is a constant
number s.t. 4.0.3 holds. Constants c7,c8 are given by c5−= s−eσc7−21σ2T,c6 = seσc8−12σ2T. The
optimal strategy is a strategy which replicates the following contingent claim:
1 (S K u 1(α))+ =
A˜ T − − −
(S K u 1(α))+ (S c )+ (c K u 1(α))1 +(S c )++
T − − − − T − 5 − 5− − − {ST>c5} T − 6
(c K u 1(α))1
6 − − − {ST>c6}
and the corresponding probability is equal
c µT c µT
P(A˜) = P(W < c )+P(W > c )= Φ 7− σ +Φ 8− σ .
T∗ 7 T∗ 8 √T − √T
(cid:18) (cid:19) (cid:18) (cid:19)
Now we need to determine all necessary constants. Using the same methods as in the previous
case we obtain
EQ[1 (S K u 1(α))+]= sΦ(d¯ ) (K +u 1(α))Φ(d¯ )
A˜ T − − − + − − − −
c c c c
sΦ 7 +σ√T +sΦ 8 +σ√T + K +u 1(α) Φ 7 Φ 8 = x
− 0
− √T − √T − √T − − √T
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
(cid:16) (cid:17)
(4.0.4)
Summarizing, constants are determined by 4.0.4 and by the fact that c ,c are solutions of the
5 6
µ
equation xσ2 = c¯(x K u−1(α))+, where c¯is a positive constant.
− −
5 CRR model
Let (S ) be a stock price given by
n n=0,1,2,...,N
S = S (1+ρ ), S = S,
n+1 n n 0
where(ρ )isasequenceofindependentrandomvariablessuchthatp := P(ρ = u) = 1 P(ρ =
n n n
−
d), where u > d,u > 0,d < 0. This means that at any time the price process S can increase to
n
the value S (1+u) or decrease to S (1+d). We assume that p (0,1). It is known, that the
n n
∈
unique martingale measure for this model is given by p := d .
∗ u−d
Let us study the risk minimizing problem for the call optio−n with strike K. Let us denote
(S K¯)+ := (S K u 1(α))+ and consider two measures: the objective one P
N N −
− − −
P(ω ) =pk(1 p)N k
k −
−
7
and the measure Q¯ (which is not necessarily a probability measure) given by
Q¯(ω ) := (S(1+u)k(1+d)N k K¯)+p k(1 p )N k.
k − ∗ ∗ −
− −
Here ω means an elementary event for which the number of jumps upwards is equal to k. Our
k
aim is to find set A˜ which solves:
P(A) max
−→
Q¯(A) x .
0
≤
For the CRR model existence of the required set A˜ is clear since Ω is finite. However we want
to find it explicitly. Unfortunately, the Neyman-Pearson lemma for the measures P and Q¯ can
not be applied here since Ω is discrete and the condition
dP x
Q¯ a(H u 1(α))+ = 0 for some a > 0
−
dQ ≥ − E[(H u 1(α))+]
(cid:26) (cid:27) − −
is very rarely satisfied. The first way of constructing A˜, which seems to be natural, is to find a
constant a¯ such that
dP x
a¯ = inf a :Q¯ a(H u 1(α))+ 0 .
dQ ≥ − − ≤ EQ[(H u 1(α))+]
(cid:26) n o − − (cid:27)
and then expect that
dP
A¯= a¯(H u 1(α))+
−
dQ ≥ −
(cid:26) (cid:27)
is a solution. Unfortunately, this is not a right construction as shown in the example below.
Example
Let Ω = ω ,ω ,ω and P and Q are two measures given by p = 7 ,p = 4 ,p = 4 and
{ 1 2 3} 1 15 2 15 3 15
q = 4 ,q = 3 ,q = 3 . We want to maximize P(A) subject to the condition Q(A) x = 6 .
1 10 2 10 3 10 ≤ 0 10
We have p1 = 63,p2 = 48,p3 = 48 and the above construction gives A˜ = ω . However
q1 54 q2 54 q3 54 { 1}
Q( ω ,ω ) = 6 and P( ω ,ω ) = 8 > 7 = P(ω ).
{ 2 3} 10 { 2 3} 15 15 1
Below we present a lemma which provides construction o A˜ when measures satisfy some partic-
ular condition. It turns out that this condition is satisfied by a significant number of cases in
the hedging problem of call option.
Lemma 5.1 Let Ω = ω ,ω ,...,ω and measures P and Q (not necessary probabilistic) satisfy
1 2 n
{ }
the following conditions:
p p p ... p > 0 < q q q ... q
1 2 3 n 1 2 3 n
≥ ≥ ≥ ≥ ≤ ≤ ≤ ≤
andγ beafixedconstant. LetA˜= ω ,ω ,...,ω , wherethenumberk issuchthatQ(ω ,ω ,...,ω )
1 2 k 1 2 k
{ } ≤
γ and Q(ω ,ω ,...,ω ,ω ) > γ. Then P(A˜) P(A) for any set A satisfying Q(A) γ.
1 2 k k+1
≥ ≤
Proof :
Let B Ω s.t. Q(B) γ.
⊆ ≤
1) First assume that A˜ B = . Then B k and we have P(ω˜) P(ω) for each ω˜ A˜ and
∩ ∅ | |≤ ≥ ∈
ω B. As a consequence
∈
P(A˜) = P(ω) P(ω) = P(B).
≥
ωX∈A˜ ωX∈B
8
2) If A˜ B = then by (1) A˜ A˜ B is a solution of
∩ 6 ∅ \{ ∩ }
P(A) max
−→
Q(A) γ Q( A˜ B )
≤ − { ∩ }
and so P(A˜ A˜ B ) P(B A˜ B ). As a consequence P(A˜) P(B). (cid:3)
\{ ∩ } ≥ \{ ∩ } ≥
Since P(ω ) increases with k if p > 1 and decreases if p < 1, the only point to apply the
k 2 2
lemma is to state the monotonicity of the measure Q¯. In fact we are interested in monotonicity
of Q¯ only on the set where it is strictly positive. Let us denote
a := Q¯(ω )= (S(1+u)k(1+d)N k K¯)+p k(1 p )N k,
k k − ∗ ∗ −
− −
(S(1+u)k+1(1+d)N k 1 K¯)+
− −
b := − ,
k (S(1+u)k(1+d)N k K¯)+
−
−
1+u
q := ,
1+d
where the sequence b is well defined under convention that a = for a 0. Then Q¯(ω ) is
increasing if aka+k1 ≥ 1kfor each k = 0,1,...,N −1. This conditio0n is∞equivalen≥t to that bk ≥ k1−p∗p∗
for each k = 0,1,...,N 1. But now note that the sequence b is decreasing. To see that one
k
−
can calculate that
b
k+1 1 (q 1)2 0.
b ≤ ⇐⇒ − ≥
k
The last condition is always satisfied. Thus Q¯(ω ) is increasing if
k
(S(1+u)N K¯)+ 1 p
∗
b = − − .
N−1 (S(1+u)N−1(1+d) K¯)+ ≥ p∗
−
Note that this case includes the situations when p 1.
∗ ≥ 2
By analogous arguments one can obtain condition under which Q¯(ω ) is decreasing. This is the
k
case when the bk¯ ≤ 1−p∗p∗, where k¯ is the minimal k for which bk 6= ∞. Indeed, then we have
bk ≤ 1−p∗p∗ for all k ≥ k¯ what implies that ak+1 < ak for k ≥ k¯.
Before summarizing the above consideration let us introduce the following notation
A := ω Ω s.t. the number of jumps upwards is equal to k .
k
{ ∈ }
for the set containing all elements ω . The following lemma is a consequence of lemma 5.1.
k
Lemma 5.2
1)(P increasing, Q¯ decreasing)
wLehterk¯e=themninu{mkb:erbkk6=is∞s.t}..Q¯If(Ap ≥ 21Aand bk¯..≤. A1−p∗p∗ t)henxA˜=anAdNQ¯(∪AAN−A1∪...∪..A. NA−k∪BN)−>k−x1,
N N 1 N k 0 N N 1 N k 1 0
∪ − ∪ ∪ − ≤ ∪ − ∪ ∪ − −
and the set B contains maximal number of any elements from the set A such that
N k 1 N k 1
Q¯(B ) x− − Q¯(A A ... A ). − −
N k 1 0 N N 1 N k
2)(P d−ec−reas≤ing, Q¯−increas∪ing) − ∪ ∪ −
If p ≤ 12 and (S(1(+Su(1)N+−u1)N(1−+Kd¯))+K¯)+ ≥ 1−p∗p∗ (for example when p∗ ≥ 12) then A˜ = A0 ∪A1 ∪...∪
A B , where the number−k is s.t. Q¯(A A ... A ) x and Q¯(A A ... A )>
k k+1 0 1 k 0 0 1 k+1
∪ ∪ ∪ ∪ ≤ ∪ ∪ ∪
x and the set B contains maximal number of any elements from the set A such that
0 k+1 k+1
Q¯(B ) x Q¯(A A ... A ).
k+1 0 0 1 k
≤ − ∪ ∪ ∪
9
Example
As an application of lemma 5.2 we study a risk minimizing problem for a call option with strike
K = 600 in a 3-period model with parameters : S = 1000, u = 0,1, d = 0,2, p = 1. Price
0 − 4
at time 0 of the option is u = EQ[(S 600)+] = 398 7 . Assume that we have only x = 150
0 3 − 27 0
and α = 5 is a level of acceptable losses measured by u(x) = √x. We denote by ωabc, where
a,b,c u,d elementary events with interpretation of a,b,c as a history of the price process.
∈ { }
For example ωudu means the event where the price process moved up in the first and the third
period and moved down in the second one. Since we can not hedge the original contingent claim
H = (S 600)+:
3
−
H(ωuuu)= 731, H(ωuud) = H(ωudu) = H(ωduu) = 368,
H(ωudd) = H(ωdud)= H(ωddu) = 104, H(ωddd)= 0,
we have to hedge H˜ = 1 (S 625)+. Since p = 1 and p = 2, we can apply lemma 5.2(2) for
A˜ 3− 4 ∗ 3
construction of A˜. Below we present three possible right candidates for H˜.
H˜(ωuuu) =0, H˜(ωddd)= 0, H˜(ωddu)= H˜(ωdud) = H˜(ωudd) = 79
and H˜(ωuud)= H˜(ωudu)= 343, H˜(ωddu)= 0
n o
or H˜(ωuud) = 0, H˜(ωudu) =H˜(ωddu) = 343
n o
or H˜(ωuud) = 343, H˜(ωudu)= 0, H˜(ωddu) = 343
n o
Moreover, P(A˜) = 3 3+ 1 3 2 3+ 1 2 3 2 = 15.
4 4 · 4 · 4 · 4 · 16
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
6 Incomplete markets
Now let us consider the case when the equivalent martingale measure is not unique. This means
that the market is incomplete and not every contingent claim can be replicated. We preserve
x,π
all assumptions from previous section. Recall that the wealth process X is a supermartingale
t
with respect to each martingale measure Q . In this case theorem which describes optimal
∈ Q
strategy is of the form:
Theorem 6.1 Assume that there exists set A˜ which is a solution of the problem:
P(A) max
−→
sup EQ[1 (H u 1(α))+] x .
A − 0
− ≤
Q
∈Q
Then the strategy which hedges the contingent claim 1 (H u 1(α))+ is optimal.
A˜ − −
Proof :
Let us consider an arbitrary admissible strategy (x,π), where x x . We will show that
0
P(u(H Xx,π)+ α) P(A˜). ≤
Notice,−thatTfor an≤y a,b≤,c 0wehave (a b)+ c b (a c)+ andthus u((H Xx,π)+)
α Xx,π (H u 1(α≥))+. As a cons−equen≤ce fo⇐r⇒any≥Q − we obtain − T ≤
⇐⇒ T ≥ − − ∈Q
EQ[1{u((H−XTx,π)+)≤α}(H −XTx,π)+] ≤ EQ[1{u((H−XTx,π)+)≤α}XTx,π]
EQ[Xx,π] x x ,
≤ T ≤ ≤ 0
10