In this paper we study the existence of an optimal hedging strategy for the shortfall risk measure in the game options setup. We consider the continuous time Black–Scholes (BS) model. Our first result says that in the case where the game contingent claim (GCC) can be exercised only on a finite set of times, there exists an optimal strategy. Our second and main result is an example which demonstrates that for the case where the GCC can be stopped on the whole time interval, optimal portfolio strategies need not always exist.
Complete marketgame optionsshortfall riskstochastic optimal control91G1091E20Supported in part by the GIF Grant 1489-304.6/2019 and the ISF grant 160/17.Introduction
A game contingent claim (GCC) or game option, which was introduced in [8], is defined as a contract between the seller and the buyer of the option such that both have the right to exercise it at any time up to a maturity date (horizon) T. If the buyer exercises the contract at time t then he receives the payment Yt, but if the seller exercises (cancels) the contract before the buyer then the latter receives Xt. The difference Δt=Xt−Yt is the penalty which the seller pays to the buyer for the contract cancellation. In short, if the seller will exercise at a stopping time σ≤T and the buyer at a stopping time τ≤T then the former pays to the latter the amount H(σ,τ) where
H(σ,τ):=XσIσ<τ+YτIτ≤σ
and we set IQ=1 if an event Q occurs and IQ=0 if not.
A hedge (for the seller) against a GCC is defined here as a pair (π,σ) which consists of a self–financing strategy π and a stopping time σ which is the cancellation time for the seller. A hedge is called perfect if no matter what exercise time the buyer chooses, the seller can cover his liability to the buyer (with probability 1). The option price is defined as the minimal initial capital which is required for a perfect hedge. Recall (see [8]) that pricing a GCC in a complete market leads to the value of a zero sum optimal stopping (Dynkin’s) game under the unique martingale measure. For additional information about pricing of game options see, for instance [5–7, 10–12].
In real market conditions an investor (seller) may not be willing for various reasons to tie in a hedging portfolio the full initial capital required for a perfect hedge. In this case the seller is ready to accept a risk that his portfolio value at an exercise time may be less than his obligation to pay and he will need additional funds to fulfill the contract.
We consider the shortfall risk measure which is given by (see [2])
R(π,σ):=supτEPH(σ,τ)−Vσ∧τπ+
where {Vtπ}t=0T is the wealth process of the portfolio strategy π and EP denotes the expectation with respect to the market measure. The supremum is taken over all exercise times of the buyer and corresponds to the case where the investor has no information on the buyer exercise strategy. The only assumption is that the buyer exercise strategy is a stopping time with respect to a given filtration.
A natural question to ask, is whether for a given initial capital there exists a hedging strategy which minimizes the shortfall risk (an optimal hedge). For American options the existence of an optimal hedging strategy is proved by applying the Komlós lemma and relies heavily on the fact that the shortfall risk measure is a convex functional of the wealth process (see [14, 16]). For the game options setup, the shortfall risk measure, as a functional of the wealth process is given by
R(π):=infσsupτEPH(σ,τ)−Vσ∧τπ+.
This functional is not necessarily convex (because of the inf) and so the Komlós lemma can not be applied here.
In this paper we treat the simplest complete, continuous time model, namely the Black–Scholes (BS) model. Our first result (Theorem 1) which is proved in the next section says that for the case where the option can be exercised only on a finite set of times, there exists an optimal hedging strategy. The proof is based on the dynamical programming approach and the randomization technique developed in [17, 18]. Up to date there are several existence results for risk minimization in the game options setup (see [2, 3] and Section 5.2 in [9]). The above papers treat essentially discrete time trading and due to admissability conditions the trading strategies are compact. In the current setup trading is done continuously, and so it requires a new method of proof.
In Section 3 we provide the second result of the paper (Theorem 2). This is an example which demonstrates that for the case where the GCC can be stopped on the whole time interval, optimal portfolio strategies need not always exist. We combine the machinery developed in [13] with additional ideas which allow us to treat the shortfall risk measure for game options. Formally, we show that the inf in (2) which ruins the convexity leads to non existence of optimal hedging strategies.
Existence result
Consider a complete probability space (Ω,F,P) together with a standard one–dimensional Brownian motion {Wt}t=0∞, and the filtration Ft=σ{Ws|s≤t} completed by the null sets. We consider a simple BS financial market with time horizon T<∞, which consists of a riskless savings account bearing zero interest (for simplicity) and of a risky asset S, whose value at time t is given by
St=S0expκWt+(ϑ−κ2/2)t,t∈[0,T]
where S0,κ>00$]]> and ϑ∈R are constants.
Define the exponential martingale
Zt:=exp−ϑκWt−ϑ22κ2t,t∈[0,T].
From the Girsanov theorem it follows that the probability measure Q which is given by
dQdP|Ft:=Zt,t∈[0,T]
is the unique martingale measure for the risky asset S.
Next, let T:={0=T0<T1<⋯<Tn=T} be a finite set of a deterministic times. Consider a game option that can be exercised on the set T. Denote by TT the set of all stopping times with values in T. For any k=0,1,…,n the payoffs at time Tk are path–independent and given by YTk=fk(STk) and XTk=gk(STk) where fk,gk:(0,∞)→R are measurable functions and 0≤fk≤gk. The payoff function H is given by (1). We will assume the following integrability condition
EP[XTk]<∞,k=0,1,…,n.
A portfolio strategy with an initial capital x≥0 is a pair π=(x,γ) such that γ={γt}t=0T is a predictable S–integrable process and the corresponding wealth process
Vtπ:=x+∫0tγudSu,t∈[0,T]
satisfies the admissibility condition Vtπ≥0 a.s. for all t.
Let us recall some elementary properties that will be used in the sequel (for details see Chapters IV-V in [19]). The continuity of S implies that the wealth process {Vtπ}t=0T is continuous as well. Moreover, since {St}t=0T is a Q–martingale then the wealth process {Vtπ}t=0T is a Q–local martingale, and so from the admissibility condition we get that {Vtπ}t=0T is a Q–super martingale. On the other hand, due to the martingale representation theorem, for any nonnegative Q–martingale {Mt}t=0T there exists a portfolio strategy π such that Vtπ=Mt for all t a.s.
For any x≥0 denote by A(x) the set of all portfolio strategies with an initial capital x. A hedging strategy with an initial capital x is a pair (π,σ)∈A(x)×TT.
The shortfall risk measure is given by
RT(π,σ):=supτ∈TTEPH(σ,τ)−Vσ∧τπ+,(π,σ)∈A(x)×TT,RT(x):=inf(π,σ)∈A(x)×TTRT(π,σ).
Now, we are ready to formulate our first result.
For anyx≥0there exists a hedging strategy(πˆ,σˆ)∈A(x)×TTsuch thatRT(πˆ,σˆ)=RT(x).
We emphasis that in contrast to previous work on game options (see [2, 3] and Section 5.2 in [9]) the trading in our setup is done continuously. Namely, the investor trades the risky asset continuously, but the GCC can be exercised only on a finite set of deterministic times. This can be viewed as a game version of the Bermudan options.
Proof of Theorem <xref rid="j_vmsta164_stat_001">1</xref>
We start with some preparations. Let U:[0,∞)×(0,∞)→R be a measurable function such that for any y>00$]]>, U(·,y) is a bounded, nondecreasing and continuous function. Let Uc:[0,∞)×(0,∞)→R be the concave envelop of U with respect to the first variable. Namely, for any y>00$]]> the function Uc(·,y) is the minimal concave function which satisfies Uc(·,y)≥U(·,y). Clearly, Uc is continuous in the first variable. Thus, for any y>00$]]> the set {x:U(x,y)<Uc(x,y)} is open and so can be written as a countable union of disjoint intervals
{x:U(x,y)<Uc(x,y)}=⋃n∈N(an(y),bn(y)).
From Lemma 2.8 in [17] it follows that Uc(·,y) is affine on each of the intervals (an(y),bn(y)). Since U, Uc are continuous in the first variable, then the functions an,bn:(0,∞)→R+, n∈N are determined by the countable collection of functions U(q,·),Uc(q,·):(0,∞)→R for nonnegative rational q, and so they are measurable.
For any 0≤t1<t2≤T and a Ft1 measurable random variable Θ1≥0 denote by Ht1,t2(Θ1) the set of all random variables Θ2≥0 which are Ft2 measurable and satisfy EQ(Θ2|Ft1)≤Θ1.
The following auxiliary result is an extension of Theorem 5.1 in [17].
Let0≤t1<t2≤Tand letΘ1≥0be aFt1measurable random variable. For a function U as above, assume that there exists a functionG:R→Rsuch that|U(x,y)|≤G(y)for all x, y andEP[G(St2)]<∞. Then there exists a random variableΘ∈Ht1,t2(Θ1)such thatEPU(Θ,St2)|Ft1=esssupΘ2∈Ht1,t2(Θ1)EPU(Θ2,St2)|Ft1=esssupΘ2∈Ht1,t2(Θ1)EPUc(Θ2,St2)|Ft1.
Since Uc≥U, it is sufficient to show that there exists Θ∈Ht1,t2(Θ1) such that
EPU(Θ,St2)|Ft1=esssupΘ2∈Ht1,t2(Θ1)EPUc(Θ2,St2)|Ft1.
Choose a sequence Θ(n)∈Ht1,t2(Θ1), n∈N such that
limn→∞EPUc(Θ(n),St2)|Ft1=esssupΘ2∈Ht1,t2(Θ1)EPUc(Θ2,St2)|Ft1.
From Lemma A1.1 in [4] we obtain a sequence Λ(m)∈conv(Θ(m),Θ(m+1),…), m∈N converging P a.s. to a random variable Λ. The Fatou lemma implies that Λ∈Ht1,t2(Θ1).
By applying the dominated convergence theorem, the inequality |Uc(·,St2)|≤G(St2) and the fact that Uc is concave and continuous in the first variable we obtain
EPUc(Λ,St2)|Ft1=limn→∞EPUc(Λ(n),St2)|Ft1≥limn→∞EPUc(Λ(n),St2)|Ft1.
This together with (7) gives
EPUc(Λ,St2)|Ft1=esssupΘ2∈Ht1,t2(Θ1)EPUc(Θ2,St2)|Ft1.
Next, introduce the normal random variable
Γ:=(Wt1+t22−Wt1)−12(Wt2−Wt1).
Observe that EP[ΓWt2]=0 and so we conclude that Γ is independent of the σ–algebra generated by Wt,t∈[0,t1]∪{t2}. From Theorem 1 in [20] it follows that there exists a measurable function Φ:C[0,t1]×R2→R such that we have the following equality of the joint laws
W[0,t1],Wt2,Λ;P=W[0,t1],Wt2,ΦW[0,t1],Wt2,Γ;P.
In particular ΦW[0,t1],Wt2,Γ∈Ht1,t2(Θ1) and
EPUcΛ,St2|Ft1=EPUcΦW[0,t1],Wt2,Γ,St2|Ft1.
Thus, without loss of generality we assume that Λ=ΦW[0,t1],Wt2,Γ.
We arrive to the final step of the proof. Introduce the normal random variable
Γˆ:=(Wt1+2t23−Wt1)−23(Wt2−Wt1)−23Γ.
Observe that EP[ΓˆWt2]=EP[ΓˆΓ]=0. Thus, Γˆ is independent of the σ–algebra generated by Wt,t∈[0,t1]∪{t2} and Γ. Let F−1 be the inverse function of the cumulative distribution function F(·):=P(Γˆ≤·). Recall (6) and define the random variable
Θ:=ΛIΛ∉⋃n∈N(an(St2),bn(St2))+∑n∈Nbn(St2)IΛ∈(an(St2),bn(St2))IΓˆ<F−1Λ−an(St2)bn(St2)−an(St2)+∑n∈Nan(St2)IΛ∈(an(St2),bn(St2))IΓˆ>F−1Λ−an(St2)bn(St2)−an(St2).{F^{-1}}\left(\frac{\Lambda -{a_{n}}({S_{{t_{2}}}})}{{b_{n}}({S_{{t_{2}}}})-{a_{n}}({S_{{t_{2}}}})}\right)}}.\end{array}\]]]>
Let G be the σ–algebra generated by Wt,t∈[0,t1]∪{t2} and Γ. From the Bayes theorem, the tower property for conditional expectation and (4) we get
EQΘ|Ft1=EPΘZt2Zt1|Ft1=EPEPΘZt2Zt1|G|Ft1=EPΛZt2Zt1|Ft1=EQΛ|Ft1.
Thus Θ∈Ht1,t2(Θ1). Finally, let us notice that U(Θ,St2)=Uc(Θ,St2), and so, from the tower property of conditional expectation and the fact that Uc(·,y) is affine on each of the intervals (an(y),bn(y)) we obtain
EPU(Θ,St2)|Ft1=EPEPU(Θ,St2)|G|Ft1=EPEPUc(Θ,St2)|G|Ft1=EPUc(Λ,St2)|Ft1.
This together with (8) completes the proof. □
We arrive at the following Corollary.
LetB:[0,∞)×(0,∞)→Rbe a measurable function such that for anyy>00$]]>,B(·,y)is a bounded, nonincreasing and continuous function. LetBc:[0,∞)×(0,∞)→Rbe the convex envelop of B with respect to the first variable.
(i). Let0≤t1<t2≤Tand letΘ1≥0be aFt1measurable random variable. Assume that there exists a functionG:R→Rsuch that|B(x,y)|≤G(y)for all x, y andEP[G(St2)]<∞. Then there exists a random variableΘ∈Ht1,t2(Θ1)such thatEPB(Θ,St2)|Ft1=essinfΘ2∈Ht1,t2(Θ1)EPB(Θ2,St2)|Ft1=essinfΘ2∈Ht1,t2(Θ1)EPBc(Θ2,St2)|Ft1.
(ii). Lett1=0. The functionb:[0,∞)→Rwhich is defined byb(x):=infΘ∈H0,t2(x)EPB(Θ,St2)=infΘ∈H0,t2(x)EPBc(Θ,St2),x≥0is convex and continuous.
(i). The result follows immediately by applying Lemma 1 for U:=−B.
(ii). The convexity of b follows from the convexity of Bc in the first variable and the fact that for any x1,x2≥0 and λ∈(0,1),
λA(x1)+(1−λ)A(x2)⊂A(λx1+(1−λ)x2).
In particular b is continuous in (0,∞). It remans to prove continuity at x=0. Since B is nonincreasing in the first variable, then b is nonincreasing as well. Thus, it is sufficient to show that b(0)≤limn→∞b(1/n). To that end, choose Θ(n)∈H0,t2(1/n), n∈N such that
limn→∞EPBc(Θ(n),St2)=limn→∞b(1/n).
From Lemma A1.1 in [4] we obtain a sequence Λ(m)∈conv(Θ(m),Θ(m+1),…), m∈N converging P a.s. to a random variable Λ. The Fatou lemma implies that Λ=0, and so by applying the dominated convergence theorem together with convexity and continuity of Bc in the first variable we get,
b(0)=limn→∞EPBc(Λ(n),St2)≤limn→∞EPBc(Θ(n),St2)=limn→∞b(1/n)
and continuity follows. □
Now we are ready to prove Theorem 1.
Let x≥0. For any π∈A(x) we define RT(π) as in (2) where the infimum and the supremum are taken over the set TT.
Moreover, define the random variables Ψkπ, k=0,1,…,n by
Ψnπ:=YT−VTπ+
and for k=0,1,…,n−1 by the recursive relations
Ψkπ:=minXTk−VTkπ+,maxYTk−VTkπ+,EP(Ψk+1π|FTk).
In view of (5) the random variables Ψkπ, k=0,1,…,n are well defined. From the standard theory of zero–sum Dynkin games (see [15]) it follows that
Ψ0π=RT(π).
Moreover, for the stopping time
σ:=T∧mint∈T:Ψtπ=Xt−Vtπ+
we have RT(π)=RT(π,σ).
Thus, in order to conclude the proof we need to show that there exists πˆ∈A(x) such that
Ψ0πˆ=infπ∈A(x)Ψ0π.
We apply dynamical programming. Introduce the functions Bk:[0,∞)×(0,∞)→R, k=0,1,…,n by
Bn(z,y):=(fn(y)−z)+,
and for k=0,1,…,n−1 by the recursive relations
Bk(z,y)=mingk(y)−z+,maxfk(y)−z+,infΘk+1∈H0,Tk+1−Tk(z)EPBk+1(Θk+1,ySTk+1−Tk).
Let us argue by backward induction that for any k, Bk(z,y) is measurable, and for any y the function Bk(·,y) is continuous and nonincreasing. For k=n this is clear. Assume that the statement holds for k+1, let us prove it for k. From Corollary 1(ii) it follows that for any y the function Bk(·,y) is continuous and nonincreasing. For any z>00$]]> the measurability of the function Bk(z,·) follows from the fact that the set H0,Tk+1−Tk(z) is separable (with respect to convergence in probability). Since Bk is continuous in the first variable we conclude joint measurability and complete the argument.
Next, from Corollary 1(i) it follows that we can construct a sequence of random variables D0, D1, ..., Dn such that D0=x and for any k=1,…,nDk∈HTk−1,Tk(Dk−1) satisfies
EPBk(Dk,STk)|FTk−1=essinfΘk∈HTk−1,Tk(Dk−1)EPBk(Θk,STk)|FTk−1.
Since Bk, k=0,1,…,n are nonincreasing in the first variable then without loss of generality we assume that EQ[Dk|FTk−1]=Dk−1 for all k.
Finally, the completeness of the BS model implies that there exists πˆ∈A(x) such that VTkπˆ=Dk for all k=0,1,…,n. Observe that STkSTk−1 is independent of FTk−1 and has the same distribution as STk−Tk−1. Thus, from (9) and (11) we obtain (by backward induction)
Bk(VTkπˆ,STk)=Ψkπˆa.s.∀k=0,1,…,n.
On the other hand, for an arbitrary π∈A(x) we have VTkπ∈HTk−1,Tk(VTk−1π), k=1,…,n. Hence, similar arguments as before (12) yield
Bk(VTkπ,STk)≤Ψkπa.s.∀k=0,1,…,n.
By combining (12)–(13) for k=0 gives that for any π∈A(x)Ψ0πˆ=B0(x,S0)≤Ψ0π
and (10) follows. □
We observe that the proof of Theorem 1 and Lemma 1 can be adjusted to the case where the volatility and the drift are deterministic functions of time. However, in order to make the presentation more friendly we assume constant parameters.
Example where no optimal strategy exists
In this section we consider a game option which can be exercised at any time in the interval [0,1]. The payoffs are given by
Xt=(1+sin(πt))max(Zt,1/2),t∈[0,1]Y1=X1,Yt=0,fort<1
where Zt was defined in (3). Notice that EP[sup0≤t≤1Xt]<∞.
Denote by T the set of all stopping times with values in the interval [0,1]. Obviously, the equalities Y[0,1)≡0 and Y1=X1 imply that (the buyer of the game option will not stop before t=1) the shortfall risk measure is given by
R(π,σ)=EPXσ−Vσπ+.
As in (2), for a portfolio strategy π we have
R(π):=infσ∈TR(π,σ)=infσ∈TEPXσ−Vσπ+.
Similarly to Section 2, for an initial capital x we define
R(x):=inf(π,σ)∈A(x)×TR(π,σ)=inf(π,σ)∈A(x)×TEPXσ−Vσπ+.
For any π the process {(Xt−Vtπ)+}t=01 is continuous, and so from the general theory of optimal stopping (see Section 6 in [9]) it follows that there exists σ=σ(π) such that R(π,σ)=R(π). Namely, the existence of an optimal hedging strategy is equivalent to the existence of an optimal portfolio strategy. We say that π∈A(x) is an optimal portfolio strategy if R(π)=R(x).
We arrive at the main result.
Assume that the drift termϑ≠0and letν:=12EPZ1IZ1<1/2,observe thatϑ≠0implies thatν>00$]]>. Then for any initial capitalx∈(0,ν)there is no optimal strategy.
If ϑ=0 then P=Q. In this specific case (see Theorem 7.1 in [2]) there exists an optimal hedging strategy.
Let us notice that for a given stopping time σ∈T the functional
π→R(π,σ):=supτ∈TEPH(σ,τ)−Vσ∧τπ+
is convex. Thus, by following the same arguments as in [14] (which are based on the Komlós lemma) one can prove that (for a given σ) the infimum in the expression infπ∈A(x)R(π,σ) is attained. Hence, Theorem 2 implies that for any x∈(0,ν) we have the following
infπ∈A(x)R(π,σ)=minπ∈A(x)R(π,σ)>R(x)∀σ∈TT.R(x)\hspace{2.5pt}\hspace{2.5pt}\forall \sigma \in {\mathcal{T}_{T}}.\]]]>
Before we prove Theorem 2 we will need some auxiliary results. We start with the following lemma.
The functionR:[0,∞)→[0,∞)is convex and continuous. Namely, the shortfall risk measure R is convex and continuous as a function of the initial capital.
The proof will be done by approximating R(·). For any n∈N let Tn be the set of all stopping times with values in the set {1/n,2/n,…,1} (0 is not included). Set,
Rn(π):=infσ∈TnR(π,σ)Rn(x):=infσ∈Tninfπ∈A(x)R(π,σ).
We argue that Rn converge uniformly to R. First, we have the obvious observation Rn(·)≥R(·). Next, let x≥0 and (π,σ)∈A(x)×T. Define σn∈Tn by
σn:=1nmin{k∈N:k/n≥σ}.
Clearly, σn≥σ. Thus, there exists a portfolio πn∈A(x) such that Vσnπn=Vσπ. From the inequality σn−σ≤1/n we obtain
R(πn,σn)−R(π,σ)≤EP|Xσ−Xσn|≤EPsup|t−s|≤1/n|Xt−Xs|.
Since (π,σ)∈A(x)×T was arbitrary we conclude that
0≤Rn(x)−R(x)≤EPsup|t−s|≤1/n|Xt−Xs|.
From the dominated convergence theorem
limn→∞EPsup|t−s|≤1/n|Xt−Xs|=0
and uniform convergence follows.
It remains to argue that for any n the function Rn:[0,∞)→[0,∞) is convex and continuous. Fix n∈N. For any k=1,…,n let gk:(0,∞)→(0,∞) be such that Xk/n=gk(Sk/n). Introduce the functions Bˆk:[0,∞)×(0,∞)→R, k=0,1,…,n by
Bˆn(z,y):=(gn(y)−z)+,
for k=1,…,n−1 by the recursive relations
Bˆk(z,y)=mingk(y)−z+,infΘk+1∈H0,1/n(z)EPBˆk+1(Θk+1,yS1/n),
and for k=0Bˆ0(z,y)=infΘ1∈H0,1/n(z)EPBˆ1(Θ1,yS1/n).
Observe that Rn(·) is “almost” as RT(·) defined in Section 2 for the set T:={0,1/n,2/n,…,1}, the only difference is that for Rn(x) stopping at zero is not allowed. This is why in (15) we do not take minimum with (g0(y)−z)+. Using similar arguments as in the proof of Theorem 1 we obtain that Rn(x)=Bˆ0(x,S0). Finally, from Corollary 1(ii) we get that for any y, Bˆ0(·,y) is convex and continuous. This completes the proof. □
Next, we observe that for any stopping time σ∈T and λ>00$]]>infΥ≥0(Xσ−Υ)++λZσΥ=Xσmin(1,λZσ).
This brings us to introducing the function
F(λ)=infσ∈TEPXσmin(1,λZσ),λ>0.0.\]]]>
Obviously F:(0,∞)→[0,∞) is concave and nondecreasing. Inspired by Corollary 8.3 in [13] we prove the following.
(i). For anyx≥0andλ>00$]]>,R(x)≥F(λ)−λx.
(ii). Letλ>00$]]>be such that F is differentiable at λ. Then forx=F′(λ)we have the equalityR(x)=F(λ)−λx.
(i). Let x≥0 and λ>00$]]>. Choose arbitrary (π,σ)∈A(x)∈T. Then, from the super–martingale property of an admissible portfolio we have
x=V0π≥EQ[Vσπ]=EP[ZσVσπ].
This together with (16) gives
R(π,σ)+λx≥EP(Xσ−Vσπ)++λZσVσπ≥F(λ).
Since (π,σ)∈A(x)∈T was arbitrary we complete the proof.
(ii). In view of (i), it is sufficient to show that R(x)≤F(λ)−λx. Let σλ∈T be an optimal stopping time in (17), i.e.
F(λ)=EPXσλmin(1,λZσλ).
Such stopping time exists because the process {Xtmin(1,λZt)}t=01 is continuous. Set Υλ=XσλIZσλ<1/λ. From (16) it follows that for any λ˜>00$]]>F(λ˜)≤EP(Xσλ−Υλ)++λ˜ZσλΥλ.
On the other hand from (19)
F(λ)=EP(Xσλ−Υλ)++λZσλΥλ.
Thus,
F(λ˜)−F(λ)λ˜−λ≤EPZσλΥλ,forλ˜>λandF(λ˜)−F(λ)λ˜−λ≥EPZσλΥλforλ˜<λ.\lambda \\ {} & \displaystyle \text{and}\hspace{2.5pt}\hspace{2.5pt}\frac{F(\tilde{\lambda })-F(\lambda )}{\tilde{\lambda }-\lambda }\ge {\mathbb{E}_{\mathbb{P}}}\left[{Z_{{\sigma _{\lambda }}}}{\Upsilon _{\lambda }}\right]\hspace{2.5pt}\hspace{2.5pt}\text{for}\hspace{2.5pt}\hspace{2.5pt}\tilde{\lambda }<\lambda .\end{array}\]]]>
From the fact that F′(λ)=x we conclude that
x=EPZσλΥλ=EQΥλ.
The completeness of the BS model implies that there exists π∈A(x) such that Vσλπ=Υλ. From (19) we get
R(x)+λx≤R(π,σλ)+λx=EP(Xσλ−Υλ)++λZσλΥλ=EPXσλmin(1,λZσλ)=F(λ)
as required. □
While Lemmas 2–3 are quite general, the following lemma uses the explicit structure of the payoff process {Xt}t=01.
(i). For anyλ≥2,F(λ)=1.
(ii). The derivative of F from the left (exists because F is concave) satisfiesF−′(2)≥νwhere ν is given by (14).
(i). Let λ≥2. Obviously, EP[Zσ]=1 for all σ∈T. Hence, from the simple formula max(z,1/2)min(1,2z)≡z we obtain
F(λ)≥F(2)=infσ∈TEPZσ(1+sin(πσ))≥1.
On the other hand, taking σ≡0 in (17), we get F(λ)≤1 and so F≡1 on the interval [2,∞).
(ii). Choose λ<2. Clearly, (we take σ≡1 in (17))
F(λ)≤EPmax(Z1,1/2)min(1,λZ1)≤EPZ1IZ1>1/2+λ2Z1IZ1<1/2=1−2−λ2EPZ1IZ1<1/2.1/2}}+\frac{\lambda }{2}{Z_{1}}{\mathbb{I}_{{Z_{1}}<1/2}}\right]=1-\frac{2-\lambda }{2}{\mathbb{E}_{\mathbb{P}}}\left[{Z_{1}}{\mathbb{I}_{{Z_{1}}<1/2}}\right].\end{array}\]]]>
This together with the equality F(2)=1 gives F−′(2)≥ν. □
Now, we have all the ingredients for the proof of Theorem 2.
From Lemma 3(i) and Lemma 4(i) it follows that for any xR(x)≥F(2)−2x=1−2x.
Let us prove that
R(x)=1−2x,∀x≤F−′(2).
Since R is convex (Lemma 2) then it is sufficient to show that R(0)≤1 and R(F−′(2))≤1−2F−′(2).
The first inequality is trivial, R(0)≤X0=1. Let us show the second inequality. The concavity of F implies that there exists a sequence λn↑2 such that for any n the derivative F′(λn) exists. Hence, from the continuity of R (Lemma 2), the concavity of F and Lemma 3(ii) we obtain
R(F−′(2))=limn→∞R(F′(λn))=limn→∞[F(λn)−λnF′(λn)]=1−2F−′(2)
and (20) follows.
Next, let x∈(0,ν). Assume by contradiction that there exists a hedging strategy (π,σ)∈A(x)×T such that R(π,σ)=R(x). From Lemma 4(ii) and (20) we obtain
R(π,σ)=1−2x.
Observe that if σ takes on values (with positive probability) in the interval (0,1) then
EPXσmin(1,2Zσ)=EPZσ(1+sin(πσ))>EP[Zσ]=1.{\mathbb{E}_{\mathbb{P}}}[{Z_{\sigma }}]=1.\]]]>
Thus, from (16) and (18)
R(π,σ)+2x≥EP(Xσ−Vσπ)++2ZσVσπ>11\]]]>
which is a contradiction to (21). On the other hand if σ≡0 then
R(π,σ)=X0−x=1−x,
also a contradiction to (21).
We conclude that the only remaining possibility is σ≡1. Let us show that there is a contradiction in this case as well. Introduce the event
A:={max(Z1,V1π)<1/2}.
Observe that on the event A we have
(X1−V1π)++2Z1V1π=(1/2−Z1)(1−2V1π)+Z1>Z1=X1min(1,2Z1).{Z_{1}}={X_{1}}\min (1,2{Z_{1}}).\]]]>
From (18) and the fact that x<ν it follows that
EP[Z1V1π]<ν=12EPZ1IZ1<1/2.
This together with the inequality V1π≥0 gives P(A)>00$]]>. Thus, by combining (16), (18) and (22) we obtain
R(π,σ)+2x=EP(X1−V1π)++2Z1V1π>EPX1min(1,2Z1)=EP[Z1]=1{\mathbb{E}_{\mathbb{P}}}\left[{X_{1}}\min (1,2{Z_{1}})\right]={\mathbb{E}_{\mathbb{P}}}[{Z_{1}}]=1\end{array}\]]]>
which is a contradiction to (21). □
We end this section with the following two remarks.
The message of Theorem 2 is that the inf in (2) which ruins the convexity of the shortfall risk functional R(π) can lead to non existence of an optimal strategy. Observe that in the above constructed example, the payoff process X is continuous and the payoff process Y has a positive jump in the maturity date.
One can ask, what if we require that both of the payoff processes X and Y will be continuous, is there a counter example in this case as well?
The answer is yes. Let us apply Theorem 2 in order to construct a counter example with continuous payoffs.
Consider a simple BS financial market with time horizon T=2 which consists of a riskless savings account bearing zero interest and of a risky asset S, whose value at time t is given by
St=S0expκWt+(ϑ−κ2/2)t,t∈[0,1]St=S1expκ(Wt−W1)−κ2(t−1)/2,t∈(1,2]
where, as before, S0,κ>00$]]> and ϑ≠0 are constants. Namely, this is a BS model which has a drift jump in t=1. Obviously this market is complete and the unique martingale measure is given by dQdP|Ft:=Zt∧1 where Zt is given by (3). Consider a game option with the continuous payoffs
Xˆt=(1+sin(πt))max(Zt,1/2),t∈[0,1]Xˆt=Xˆ1,t∈(1,2],Yˆt=0,t∈[0,1]Yˆt=(t−1)Xˆ1,t∈(1,2].
Denote by Rˆ the corresponding shortfall risk. We argue that for an initial capital 0<x<ν:=12EPZ1IZ1<1/2 there is no optimal hedging strategy.
Indeed, let π be an admissible portfolio strategy and σ be a stopping time with values in the interval [0,2]. From the super–martingale property of the portfolio value and the fact that Z is a constant random variable after t=1 we obtain
Vσ∧1π≥EP[Vσπ|F1].
This together with the Jensen inequality and the fact that Xˆ is a constant random variable after t=1 gives
EP(Xˆσ∧1−Vσ∧1π)+≤EPEP(Xˆσ−Vσπ)+|F1=EP(Xˆσ−Vσπ)+.
From (23), and the relations Yˆ[0,1]≡0, Yˆ2=Xˆ2 we obtain
Rˆ(π,σ∧1)=EP(Xˆσ∧1−Vσ∧1π)+≤EP(Xˆσ−Vσπ)+≤Rˆ(π,σ).
Namely, we can restrict the investor to stopping times in the interval [0,1], but this is exactly the setup that was studied in Theorem 2. From Theorem 2 we conclude that there is no optimal hedging strategy for x∈(0,ν).
From Theorem 1 it follows that for any n there exists a hedging strategy (πn,σn)∈A(x)×Tn such that
Rn(x)=EPXσn−Vσnπn+
where Rn was defined in the beginning of the proof of Lemma 2.
Theorem 2 implies that we should not expect that these optimal hedging strategies (πn,σn)∈A(x)×Tn, n∈N will converge in a strong sense when n goes to infinity. Indeed, if these hedging strategies would converge in a strong sense, then we can argue that the limit is an optimal hedging strategy for the continuous time problem. This is a contradiction to Theorem 2 (at least for x∈(0,ν)).
By applying the weak convergence theory we can show that (πn,σn)∈A(x)×Tn, n∈N has a cluster point (with respect to convergence in law). However, in view of Theorem 2 we conclude that the representation of the cluster point will require an enlargement of the probability space, i.e. an additional randomization.
An interesting question which is left for the future, is whether by allowing the investor to randomize from the start (in the spirit of [1]) will provide an existence of an optimal hedging strategy.
Acknowledgments
I would like to thank Yuri Kifer for introducing me to the problems which are treated in this paper and also for related fruitful discussions. I also would like to thank Walter Schachermayer for sharing some ideas a while ago which turned out to be helpful for proving Theorem 1.
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