We shall consider the game type Asian option. The pay-off function of the holder of the option is given by f(S,G) and that of the writer is given by fδ(S,G)= f(S,G)+δ(S,G) (> f(S,G)), whereδ(S,G) denotes the penalty payed by the writer in the case where he cancel the option. The price of the option is then given by
V(S,G,t) := inf
σ∈Tt,T
sup
τ∈Tt,T
JS,G,t(σ, τ), where
JS,G,t(σ, τ) :=E0[e−r(σ∧τ−t){f(Sτ,Gτ)1τ≤σ+ fδ(Sσ,Gσ)1σ<τ}|St =S,Gt =G].
It is finite for any S,G,tby (2.6). Further it holds f(S,G) ≤ V(S,G,t) ≤ fδ(S,G) for anyS,G,t.
Suppose that the pay-off functions satisfy f(S,G) = Sξ(G/S) and fδ(S,G)=Sξδ(G/S).Then we have
V(S,Sey/t,t)= inf
σ∈Tt,T sup
τ∈Tt,T
E∗[e−q(σ∧τ−t)S{ξ(eϕt,τ(y)/τ)1τ≤σ+ξδ(eϕt,σ(y)/σ)1σ<τ}].
ThereforeV(S,Sey/t,t)/Sis a function depending only on (y,t). It coincides with the followingW(y,t) a.s. for any (y,t)
W(y,t) := inf
σ∈Tt,T
sup
τ∈Tt,T
J∗y,t(σ, τ), (4.1)
where
J∗y,t(σ, τ)=E∗[e−q(σ∧τ−t){ξ(eϕt,τ(y)/τ)1τ≤σ+ξδ(eϕt,σ(y)/σ)1σ<τ}].
(4.2) It satisfies
ξ(ey/t)≤W(y,t)≤ξδ(ey/t)
for anyy,t. Now we define the writer’s cancellation region, the holder’s exercise region and the continuation region by
EA={(y,t);W(y,t)=ξδ(ey/t)}, EB={(y,t);W(y,t)=ξ(ey/t)}, (4.3)
CG={(y,t);ξ(ey/t)<W(y,t)< ξδ(ey/t)},
respectively. Letσ∗=σ∗y,tandτ∗=τ∗(y,t)be hitting times ofϕt,u(y),u∈[t,T]
to the setsEA,EB, respectively. Then it holds W(y,t)=Jy,t(σ∗, τ∗) (4.4)
for any y,t. (IfEA = φthen σ∗ = T holds valid. In this case we have W(y,t)=WA(y,t) for any (y,t))
Now shall consider the average strike put option of the game type. The pay-offfunctions are defined by
ξ(ey/t)=max{ey/t−1,0}, ξδ(ey/t)=ξ(ey/t)+δ, whereδis a positive constant.
The value function of the American average strike put optionWA(y,t) is a strictly decreasing function oft. It converges to the pay-offfunction ξ(ey/T) ast→T. ThereforeWA(0,t) decreases to 0 astincreases toT. Hence ifWA(0,0)> δ, then there exists a unique 0≤t∗δ<Tsuch thatWA(0,t)< δ fort>t∗δandWA(0,t)> δfort<t∗δ. IfWA(0,0)< δ, thenWA(0,t)< δholds for allt. In this case we sett∗δ=0.
Theorem 4.1. Consider the average strike put option of the game type. Assume 0≤q≤r.
1)EB is a nonempty closed set and the sectionEBt is equal to the upper half line [y∗δ(t),∞), where0<y∗δ(t)≤y∗a(t).
2) The writer’s cancellation regionEAis equal toE0 :={(0,t);t≤t∗δ},i.e., EAt =
{0},if t≤t∗δ, φ, if t>t∗δ. (4.5)
Thefirst assertion on the exercise regionEB can be verified similarly as Theorem 3.2. In order to prove the second assertion, we prepare two lemmas.
We consider the function
W(y,˜ t) :=sup
τ J∗y,t(σ0, τ), (4.6)
whereσ0is the hitting time to the setE0. We want to prove ˜W(y,t)=W(y,t).
Notefirst that ˜W(y,t) = supτJ∗y,t(T, τ) = WA(y,t) holds if t > δ∗a, where WA(y,t) is the value function of the American average strike put option.
We consider the case wheret<t∗a.
Lemma 4.2. The functionW(y,˜ t)is nondecreasing with respect to y. Further for any y>0it is nonincreasing with respect to t.
Proof. Sincce the pay-offfunctionsξandξδ are nondecreasing functions of y, the function J∗y,t(σ0, τ) is also nondecreasing with respect to y. Then W(y,˜ t) is also nondecreasing with respect to y. In order to prove the decreasing property with respect tot, we consider
V(S,˜ G,t) := sup
τ∈Tt,T
JS,G,t(σ0, τ).
(4.7) It holds
V(S,˜ G,t)= sup
τ∈Tt,T
E0[e−r(τ∧σ0−t){f(Sτ∧σ0,Gτ∧σ0)+δSσ01σ0<τ}|St=S,Gt=G].
Set Ht = tlogGt = t
0 logSuduas before. Then the above expectation is written by
E0[e−r(τ∧σ0−t){f(Sτ∧σ0,exp(Hτ∧σ0
τ∧σ0))+δSσ01σ0<τ}|St=S,Ht =tlogG].
Note that the law of (Su+h,Hu+h) with respect toP(ã|St+h = S,Ht+h = (t+ h) logG) is equal to the law of (Su,Hu) with respect to P(ã|St = S,Ht = tlogG). Then we have for anyh>0,
V(S,˜ G,t+h)
= sup
τ∈Tt,T−h
E0[e−r(τ∧σ0−t){f(Sτ∧σ0,exp(Hτ∧σ0
τ∧σ0))+δSσ01σ0<τ}|St=S,Ht=tlogG]
≤V(S,˜ G,t).
This shows that ˜Vis nonincreasing with respect tot.
Now we transform the function ˜Vto the function oftandy=tlogS/G.
The function ˜V(S,Sey/t,t)S−1coincides with ˜W(y,t). We can show that it is nonincreasing with respect totfor anyy>0, by using an equation similar to (3.3).
Lemma 4.3. Assume0≤q≤r. For any t, it holdsW(y,˜ t)< ξδ(ey/t)if y0.
Proof. The function ˜W satisfies ξ(ey/t) ≤ W(y,˜ t) for any y,t. Set ˜E = {(y,t); ˜W(y,t)=ξ(ey/t)}. We can show that ˜Etis a upper half line [ ˜y∗(t),∞) where ˜y∗(t) > 0, similarly as the case of American option. We want to show ˜W(y,t)< ξδ(ey/t) holds for anyy>0. Fixedt, consider the function U(y) :=˜ W(y,˜ t)−ξ(ey/t). It satisfies the boundary condition (terminal
condition) ˜U(y∗) = U˜y( ˜y∗) = 0 since at the boundary point y = y˜∗ the contact property ˜Wy(y,t)= ∂∂y(ξ(ey/t)) holds valid. Further fory < y˜∗(t) it satisfies the second order ordinary differential equation
(L∗(t)−q) ˜U(y)=K(y), (4.8)
where
K(y)=−∂W˜
∂t (y,t)− {q−rey/t}1{0<y<˜y∗}.
The aboveKis positive because∂∂W˜t (y,t)≤0 by Lemma 4.3 andq−rey/t <
−(r−q)<0. The solution of the differential equation (4.8) is represented by
U(y)˜ =C1eλ1(y−y˜∗)+C2eλ2(y−y˜∗)+ 1 λ1−λ2
y y˜∗
K(z)(eλ1(y−z)−eλ2(y−z))dz, (4.9)
whereλ1>0≥λ2andeλ1(y−y˜∗)andeλ2(y−y˜∗)are fundamental solutions of the homogeneous equation (L∗(t)−q)U=0. The boundary conditions imply C1=C2=0. Therefore we get
U(y)˜ = 1 λ1−λ2
y
˜ y∗
K(z)(eλ1(y−z)−eλ2(y−z))dz, −∞<y<y˜∗.
Then ˜U(y)>0 holds for 0<y<y˜∗. Further we have U˜y(y)= 1
λ1−λ2
y
˜
y∗ K(z)(λ1eλ1(y−z)−λ2eλ2(y−z))dz,
which is negative for any−∞<y<y˜∗. Therefore ˜Uis strictly decreasing.
Since ˜U(0) = δ, we have ˜U(y)< δ for any 0 < y < y˜∗, proving ˜W(y,t)<
ξδ(ey/t) for any 0<y<y˜∗.
Next, if y < 0, we have always ˜W(y,t) < δ = ξδ(ey/t). The proof is complete.
Proof of Theorem 4.1. Assume thatt>t∗δ. Then we have the inequalities W(y,t) ≤ WA(y,t) < ξδ(ey/t) for any t. ThereforeEA = φand we have W(y,t)=WA(y,t) for anyy. Assume next thatt≤t∗δ. The value function W(y,t) of the game option is less than or equal to ˜W(y,t). Therefore the functionW(y,t) satisfies the inequalityW(y,t)< ξδ(ey/t) for anyy0. This proves thatEAt ={0}ift< δ∗. The proof is complete.