Variational inequality for perpetual American option price and convergence to the solution of the difference equation
aa r X i v : . [ q -f i n . P R ] M a r Variational inequality for perpetual American option price andconvergence to the solution of the difference equation Hyong-chol O, Song-San Jo , Faculty of Mathematics,
Kim Il Sung
University, Pyongyang , D P R Koreae-mail: [email protected] Abstract
A variational inequality for pricing the perpetual American option and thecorresponding difference equation are considered. First, the maximum principle anduniqueness of the solution to variational inequality for pricing the perpetual Americanoption are proved. Then the maximum principle, the existence and uniqueness ofthe solution to the difference equation corresponding to the variational inequality forpricing the perpetual American option and the solution representation are providedand the fact that the solution to the difference equation converges to the viscositysolution to the variational inequality is proved. It is shown that the limits of the pricesof variational inequality and BTM models for American Option when the maturitygoes to infinity do not depend on time and they become the prices of the perpetualAmerican option.
Keywords perpetual American option; variational inequality; explicit difference equa-tion; maximum principle
Pricing financial derivatives is one of main topics in mathematical finance with clear im-plications in physics [12] and the American option is one of the widely studied financialderivatives.[9] studied discrete and differential equation models for pricing options and providedvarious pricing formulae. In particular, they provided the free boundary problem andvariational inequality models for the prices of American options and studied some propertiesof the option prices. American options are one of financial instruments that the holder mayexercise at expiry dates and any time before expiry dates.Perpetual American options are the American options without expiry dates, that is, per-petual American options can be exercised at any time in the future and the price functionsare defined on infinite time intervals. A pricing model for the perpetual American optionswas studied in [6] using expectation method. Under the diffusion model, [9] provided freeboundary problem and variational inequality models for perpetual American options andthe solution representation for the free boundary problem model. Under a general jump-diffusion risk model, [3] studied perpetual American options. In [10], [12], [13] perpetualAmerican options are studied within a market model that can be considered as a regime-switching models with only two states and where one of the transition probabilities is set tozero. In [15] perpetual American catastrophe equity put options in a jump-diffusion modelare studied and [16] studied perpetual American executive stock options (ESO), [1], [2]studied call-put dualities when the local volatility function is modified.On the other hand, financial contracts are discretely exercised, for example, every day,every month, every three months, every six months, every year and etc. So the discrete
Hyong-chol O, Song-San Jo models are considered as more realistic models for financial derivatives than continuousmodels [9].For perpetual American options, [11] provided the price representation and optimalexercise boundary by the binomial tree model which is one of most popular discrete models,and they studied the periodicity of the binomial tree price of perpetual Bermudan options.[8] found some clear relationship between the binomial tree method and a special explicitdifference schemes for the variational inequality models for American options price andusing it obtained the convergence of the binomial tree price for American option to theviscosity solution to the variational inequality model. This result is extended to the casewith time dependent coefficients in [14].In this article we are interested in the convergence of the binomial tree price for perpetualAmerican option obtained in [11]. The price functions for American options are defined onfinite time intervals whereas the price functions for perpetual American options are definedon infinite time intervals. So the theory developed in [8] or [14] could not cover this problemand the solutions to discrete models such as the binomial tree model cannot be easily foundstep by step in difference schemes as shown in [11].In this article we obtain the maximum principle and uniqueness of the solution to varia-tional inequality for pricing the perpetual American options. Then we study the maximumprinciple, the existence and uniqueness of the solution to the difference equation corre-sponding to the variational inequality for pricing the perpetual American option and thesolution representation. (From the consideration in [8], [9], the binomial tree method of [11]can be considered as a special difference equation for the variational inequality neglectinginfinitesimal and thus such study can be viewed as an extension of the results of [11].) Thenwe study the convergence of the solution to the difference equation to the viscosity solutionto the variational inequality. From this result, we have the convergence of the binomial treemethod of [11] and such study can be viewed as an extension of the results of [8] to the casewith the expiry time T = ∞ .In many literature including [9] and [11], the price of perpetual American option ismodeled under the apriori assumtion that it does not depend on time. On the other hand,it is natural to consider the price of perpetual American option as a limit of the price ofAmerican Option when the maturity goes to infinity. This approach excludes the aprioriassumtion that the price of perpetual American option does not depend on time. In thispaper we show that the limits of the prices of variational inequality and BTM models forAmerican Option when the maturity goes to infinity do not depend on time.The rest of this article is organized as follows. In Section 2 the maximum principle andthe uniqueness of the solution to variational inequality for pricing the perpetual Americanoption, the existence of the optimal exercise boundary are described. And then For thedifference equation corresponding to the variational inequality for pricing the perpetualAmerican option, the maximum principle, the existence of optimal exercise boundary, theexistence, uniqueness and representation of the solution and the convergence of the approx-imated solution are discussed. Section 3 shows that the limits of the prices of variationalinequality model, its explicit difference scheme and BTM for American Option when thematurity goes to infinity do not depend on time. ariational inequality for perpetual American option price... Let r , q and σ be the interest rate, the dividend rate and the volatility of the underlyingasset of option, respectively, then a variational inequality pricing model of American optionis provided as follows:min (cid:26) − σ S d VdS − ( r − q ) S dVdS + rV, V − ψ (cid:27) = 0 , (1)Here ψ = ( E − S ) + ( f or put ) or ψ = ( S − E ) + ( f or call ) . The Black-Scholes ordinary differential operator (BSOD operator) is defined as follows: LV = − σ S d VdS − ( r − q ) S dVdS + rV (2)If V ( S ) is the solution of (1), then we have V ( S ) = ψ, − LV > V ( S ) > ψ, − LV = 0 in continuation region.[3] Thus, the solution of (1) is alwaysnonnegative. Consider the BSOD operator (2) in the interval A = ( a, b ) , (0 ≤ a < b ≤ ∞ ) . Theorem 2.1 (Maximum principle of BSOD operator) Suppose that r > and V ( S ) ∈ C ( A ) . If − LV < ( > )0 , S ∈ A , then nonnegative maximum (nonpositive minimum) valueof V cannot be attained at the interior points of A . Moreover, if − LV ≤ ( ≥ )0 , S ∈ A , thenwe have sup x ∈ A V ( x ) = sup x ∈ ∂A V + ( x ) (cid:18) inf x ∈ A V ( x ) = inf x ∈ ∂A V − ( x ) (cid:19) . (3) Proof
Suppose that − LV < x ∈ A that V ( x ) = max x ∈ A V ( x ) = M ≤ a < x < b and thus V S ( x ) = 0 and V SS ( x ) ≤ − LV = − σ S d VdS − ( r − q ) S dVdS + rV (cid:12)(cid:12)(cid:12)(cid:12) S = x = − σ S d VdS ( x ) + rM ≥ − LV <
0. In the case of − LV ≤ u = V − ε , then − Lu = − LV − rε < x ∈ A u ( x ) = sup x ∈ ∂ p A u + ( x )Thus we havesup x ∈ A V ( x ) ≤ sup x ∈ A u ( x ) + ε = sup x ∈ ∂ p A u + ( x ) ≤ sup x ∈ ∂ p A V + ( x ) + ε. Let ε → Hyong-chol O, Song-San Jo
Corollary 1 (maximum principle for pricing function of perpetual American options) V ( S ) the solution of (1) attains non-negative maximum value at the boundary points of A = ( a, b ) ,where A is a arbitrary subdomain of ().Proof. If V ( S ) is the solution of (1),then there would hold − LV = 0 in the domain whichholds V > φ . Thus Theorem 1 leads to the result.
Lemma 2.1
Assume that V ( S ) is the price of perpetual American put options, that is,the solution to (1). If S is in the stopping region, that is, V ( S ) = ( E − S ) + , then S ≤ min { rE/ ( q ) + , E } . That is, the optimal exercise boundary could not greater thanmin { rE/ ( q ) + , E } . Here ( q ) + = ( q, q ≥ , q < Proof
First, note that if V ( S ) = ( E − S ) + , then S ≤ E [Jia]. So we can rewrite as V ( S ) = E − S On the other hand, (1) can be written asmin {− LV, V − φ } = 0Since ( V − φ )( S ) = 0, then − LV | S = rE − qS ≥
0, thus if q ≥
0, then S ≤ rE/q .(QED) Lemma 2.2
Let V ( S ) be the price of perpetual American put options.(i) If there is such a S > V ( S ) = ( E − S ) + , then we have V ( S ) = ( E − S ) + forall S ( < S ).(ii) If there is S > V ( S ) = ( E − S ) + , then V ( S ) > ( E − S ) + for all S ( > S ). Proof (i)From Lemma 1, we have S ≤ rE/q . If the conclusion were not true, that is,we assume that 0 < ∃ S < S : V ( S ) > E − S. (4)Let ( a, b ) be the largest interval where (4) holds. Then we have V ( S ) = E − S when S = a or S = b . From (4) and min {− LV, V − φ } = 0 , we have − LV = 0 and V − ( E − S ) > a, b ). Thus we have − L ( V − ( E − S )) = L ( E − S ) = qS − rE < ∵ q ≥ ⇒ S < S ≤ rE/q ) . So from Theorem 1, non-negative maximum value of V − ( E − S ) is attained at the boundarypoints. Since V ( a ) − φ ( a ) = 0 , V ( b ) − φ ( b ) = 0, we have V ( S ) − φ ( S ) ≤ a, b ). This contradicts (4).(ii) If ∃ S > S : V ( S ) = ( E − S ) + , then from the conclusion of (i) we have V ( S , t ) =( E − S ) + which contradicts the assumption.(QED)From lemma 2 and its corollary, the existence of the optimal exercise boundary is proved. Theorem 2.2 (Existence of the optimal exercise boundary) Let V ( S ) be the price functionof perpetual American put option. Then there exists an optimal exercise boundary S . ariational inequality for perpetual American option price... Proof
The interval ( E, + ∞ ) is contained to the continuation region. Let S be the infi-mum of the continuation region, then from lemma 2, ( S , + ∞ ) is the continuation regionand (0 , S ) is the stopping region. (QED) Theorem 2.3 (Uniqueness of the solution to variational inequality for the price of perpetualAmerican put option) (1) has at most one solution.
Proof
Proof. Let V and V be the two solutions of (1) and S , S be the optimal exerciseboundaries of two solutions, respectively. Without loss of generality, assume that S > S .Then we have V = V = φ for S < S and LV = LV = 0 for S > S . On the other hand,for S < S ≤ S , we have V = φ, LV = 0 , V > φ . Now consider the interval ( S , ∞ ).When S < S ≤ S , we have − L ( V − V ) = − Lφ = − Lφ = − qS + rE ≥ ∵ q ≥ ⇒ S ≤ rE/q ) . When
S > S , we have − L ( V − V ) = 0. Thus from the maximum principle (Theorem 1),the non-positive minimum value of V − V in the interval [ S , ∞ ) is attained at the bound-ary. But we have ( V − V )( S ) = 0 and ( V − V )( ∞ ) = 0, which means that ( V − V ) ≤ S , V − V )( S ) = φ ( S ) − V ( S ) <
0. So we have S = S . In (0 , S ], two solutions are equal to φ , thus, two solutions are coincided. In( S , ∞ ), we have V − V = 0 and on the boundary V − V = 0. From theorem 1, we have V = V .(QED) Remark 2.1
The solution to the free boundary problem of perpetual American optionprice considered in [jia] satisfies the variational inequality (1), so the existence of the solutionto (1) is already known. So the problem of uniqueness and existence of the solution to thevariational inequality pricing model of perpetual American put option and the existence ofthe optimal exercise boundary are completely solved.
Using transformation: u ( x ) = V ( S ) , S = e x , ϕ = ( E − e x ) + ; S = e c . the variational inequality (1) becomes the following problemmin (cid:26) − σ ∂ u∂x − ( r − q − σ ∂u∂x , u − ϕ (cid:27) = 0 , x ∈ ( −∞ , + ∞ ) (5) u ( −∞ ) = E, u (+ ∞ ) = 0For ∆ x >
0, let u j = u ( j ∆ x + c ) , ϕ j = ϕ ( j ∆ x + c ) and use difference quotients (cid:18) ∂u∂x (cid:19) j = u j +1 − u j − x , (cid:18) ∂ u∂x (cid:19) j = u j +1 − u j + u j − x . Hyong-chol O, Song-San Jo in (5), then we have the following difference equation:min (cid:26) − σ u j +1 − u j + u j − x − ( r − q − σ u j +1 − u j − x + ru j , u j − ϕ j (cid:27) = 0 , j ∈ Z (6) u j = E, j = −∞ : u j = 0 , j = + ∞ . Here, let α = ∆ t and consider the fact that min( A, B ) = 0 ⇔ min( αA, B ) = 0 ( α >
0) .Then we havemin (cid:26) − σ ∆ t x ( u j +1 − u j + u j − ) − ( r − q − σ t x ( u j +1 − u j − ) + r ∆ tu j , u j − ϕ j (cid:27) = 0 u j = E, j = −∞ : u j = 0 , j = + ∞ Consider that − σ ∆ t x ( u j +1 − u j + u j − ) − ( r − q − σ t x ( u j +1 − u j − ) + r ∆ tu j == (1 + r ∆ t ) u j − (cid:18) − σ ∆ t ∆ x (cid:19) u j − σ ∆ t ∆ x (cid:20)(cid:18)
12 + ( r − q − σ x σ (cid:19) u j − (cid:21) and denote w = σ ∆ t ∆ x , a = 12 + ( r − q − σ x σ , ρ = 1 + r ∆ t. Then we havemin { ρu j − (1 − w ) u j − w [ au j +1 + (1 − a ) u j − ] , u j − ϕ j } = 0Here, consider min ( A, B ) ⇔ min( αA, B ) = 0 , ( α >
0) again, then we havemin (cid:26) u j − ρ [(1 − w j ) u j − w [ au j +1 + (1 − a ) u j − ]] , u j − ϕ j (cid:27) = 0Then, due to min( C − A, C − B ) = 0 ⇔ C = max ( A, B ), we have a following differenceequation equal to (6). u j = max (cid:26) ρ [(1 − w ) u j + w [ au j +1 + (1 − a ) u j − ]] , ϕ (cid:27) j ∈ Z (7) u j = E, j = −∞ ; u j = 0 , j = + ∞ which is equivalent to (6)Now, we denote the discrete Black-Scholes operator as follows: I j ( U ) = ( ρ + w − U j − w [ aU j +1 + (1 − a ) U j − ] , j ∈ Z, U ∈ l ∞ ( Z )Here l ∞ ( Z ) is the Banach space of all bounded two sided sequences of real numbers. ariational inequality for perpetual American option price... Theorem 2.4 (Maximum principle for the discrete Black-Scholes operator) Assume that ∆ x is small enough. Then we have the following two facts:(i) If on some interval A of Z , we have I j ( U ) ≤ , j ∈ A , then the maximum value of U cannot be attained at interior of A . That is, there does not exist such j ∈ A that U j − < U j and U j > U j +1 hold at the same time.(ii) If on some interval A of Z , we have I j ( U ) ≥ , j ∈ A , then the minimum value of U cannot be attained at interior of A . That is, there does not exist such j ∈ A that U j − > U j and U j < U j +1 hold at the same time. Proof
If ∆ x is small enough, then 0 < a < ∃ j ∈ A, U j − < U j , U j +1 < U j . Then aU j +1 +(1 − a ) U j −
0, so we have w [ aU j +1 + (1 − a ) U j − ] < wU j < wU j + r ∆ tU j ⇒⇒ ( w + r ∆ t ) U j − w [ aU j +1 + (1 − a ) U j − ] > ⇒⇒ I j ( U ) = ( ρ + w − U j − w [ aU j +1 + (1 − a ) U j − ] > I j ( U ) < , j ∈ A . Thus, we proved the conclusion of (i). The conclusionof (ii) is derived by applying the result of (i) to − U .(QED)By using this theorem, we can prove the maximum principle of the solution to differenceequation (7) for the variational inequality for perpetual American put option price. Theorem 2.5 (Maximum principle for the difference equation for the variational inequalityfor perpetual American put option price) Let U = U j ∈ Z be the solution of (7) . Then, themaximum and minimum value of U on any subinterval A = [ j , j ] of P = { j ∈ Z | U j >ϕ j } is attained at the boundary. Thus U j is the monotone with respect to j on P . Proof
Since U is the solution of (7) and U j > ϕ j on P , we have I j ( U ) = 0. Thus, fromtheorem 4, we have the desired result. (QED) Lemma 2.3
Suppose that
E > , r > x is sufficiently small. Let U = { U j } j ∈ Z bethe solution to (7). Denote by j ∗ = min { j ∈ Z : E ≤ e j ∆ x + c } , j ∗∗ = min { j ∈ Z : rE ≤ qe j ∆ x + c } Then, we have U j = ϕ ⇒ j < inf { j ∗ , j ∗∗ } < ∞ Proof If E >
0, then −∞ < j ∗ < ∞ and U j > ϕ = 0 for j ≥ j ∗ . If U j = ϕ j , then j ≤ j ∗ − ϕ = E − e j ∆ x + c . On the other hand, if j ≤ j ∗ − U j = ϕ j , thenwe have ϕ j = U j > ρ [(1 − w ) U j + w [ aU j +1 + (1 − a ) U j − ]] ≥ ρ [(1 − w ) ϕ j + w [ aϕ j +1 + (1 − a ) ϕ j − ]] Hyong-chol O, Song-San Jo
This means that I j ( ϕ ) >
0. Since ae ∆ x + (1 − a ) e − ∆ x = 1 + ( r − q ) ∆ x σ + O (∆ x )we have I j ( ϕ ) = ( ρ + w − ϕ j − w ( aϕ j +1 + (1 − a ) ϕ j − ) == ∆ t (cid:18) ( rE − qe j ∆ x + c ) + σ ∆ x (cid:19) > x is sufficiently small, we have rE > qe j ∆ x + c and j < j ∗∗ .(QED) Remark2. If r = 0 , q ≥
0, then j ∗∗ = −∞ and the exercise region is empty, so theexercise boundary does not exist. If r = 0 , q < r > , q ≤
0, then j ∗∗ = ∞ andexercise boundary is j ∗ . Lemma 2.4
Assume that
E > , r >
0. Let U = { U j } j ∈ Z be the solution of (7) and ∆ x is sufficiently small. Then, we have( i ) ∃ j ∈ Z : U j = ϕ j ⇒ ∀ < j , U j = ϕ j ( ii ) ∃ j ∈ Z : U j > ϕ j ⇒ ∀ > j , U j > ϕ j Proof
If (i) were not true, there would exist such a set A = { j ∈ Z : −∞ ≤ j ≤ j ≤ j ≤ ∞} that j = −∞ ⇒ U j = E ; j = −∞ ⇒ U j = ϕ j ; j < j < j ⇒ U j > ϕ j ( ∗ )Without loss of generality, assume that j = j , then we have U j = ϕ j . Then, from lemma3, we have j < min { j ∗ , j ∗∗ } . U j > ϕ j ⇒ U j = 1 ρ { (1 − U j + w [ aU j +1 + (1 − a ) U j − ] } ⇒ I j ( U ) = 0We have rE > qe j ∆ x + c for j < j , and thus from (8) we have I j ( ϕ ) > I j ( ϕ ) > , ∀ j ∈ A. Thus, we have I j ( ϕ ) > , I j ( U ) = 0 , ∀ j ∈ A and thus we have I j ( U − ϕ ) < , j ∈ A ; U j = ϕ j , U j = ϕ j ariational inequality for perpetual American option price... U − ϕ in A is attained at the boundary points j = j , j = j . Therefore we have U j ≤ ϕ j j ∈ A, j = j , j This contradicts the assumption (*). If (ii) were not true, then we have ∃ j > j ⇒ U j = ϕ j and from the result of (i) we have U j = ϕ j , which contradicts the assumption.(QED) Theorem 2.6 (Existence of optimal exercise boundary of explicit difference scheme) As-sume that U j = ϕ j and U = { U j } j ∈ Z be the solution of (7) . Then there exists such j ∈ Z that j < j ⇒ U j = ϕ j and j ≥ j ⇒ U j > ϕ j . Proof
From lemma 3, if j ≥ min( j ∗ , j ∗∗ ), then U j > ϕ j . Denote j = min { j : U j > ϕ j } .Then from lemma 4 we have j ≥ j ⇒ U j > ϕ j and j < j ⇒ U j = ϕ j .In what follows, we denote optimal exercise boundary by j ∗ = j −
1. From the resultof lemma 4, the explicit difference scheme (7) can be written as the following differenceequation: u j = ρ { (1 − w ) u j + w [ u j +1 + (1 − a ) u j − ] } , j > j ∗ (9) u j ∗ = ϕ j ∗ ≥ ρ { (1 − w ) u j ∗ + w [ u j ∗ +1 + (1 − a ) u j ∗ − ] } , j = j ∗ (10) u j = 0 , j = + ∞ (11) Theorem 2.7 (Uniqueness of the solution of the explicit difference scheme of perpetualAmerican options) The solution of the equation (9) , (10) , (11) is unique if it exists. Proof
Assume that we have two different solutions ( U , j ∗ ) , ( U , j ∗ ). Without loss ofgenerality j ∗ > j ∗ . Then we have I j ( U − U ) ( = 0 , j ≥ j ∗ ,> , j ∗ ≤ j < j ∗ ,U ∞ = U ∞ = 0 , U j ∗ = U j ∗ = ϕ j ∗ Thus we have I j ( U − U ) for all ∀ j ∈ ( j ∗ , + ∞ ) and by the maximum principle, thenon-positive minimum value of U − U is attained at the boundary of ( j ∗ , + ∞ ).Since U − U | j =+ ∞ , we have U − U > ∀ j ∈ ( j ∗ , + ∞ ), that is, U > U , ∀ j ∈ ( j ∗ , + ∞ )and this contradicts the fact that U j ∗ = ϕ j ∗ < U j ∗ .Thus we have j ∗ = j ∗ . Therefore, the two solutions are equal to ϕ j in ( −∞ , j ∗ ]. In ( j ∗ , ∞ )we have I j ( U − U ) = 0 and U − U = 0 at the boundary, so from Theorem ( ?? ), we have U = U . (QED)0 Hyong-chol O, Song-San Jo
Now, we find a solution (
U, j ∗ ) to the equation (9), (10), (11). To do this, it is sufficientto find the solution satisfying the equality in (10). Substitute U j = ξ j into (9) to obtainthe characteristic equation: waξ − ( ρ + w − ξ + w (1 − a ) = 0 . This equation has two roots: ξ , = ρ + w − ± p ( ρ + w − − aw (1 − a )2 aw Obviously, ξ < < ξ , so the general solution has the form U j = c ξ j + c ξ j with constants c , c . Since ξ j → j → + ∞ , c must be zero in order to satisfy the boundarycondition (11) ( U j = 0 , j = + ∞ ).Thus, when j > j ∗ , the solution of (9) has the form U j = c ξ j . (12)If j ≤ j ∗ , then we have U j = ϕ j = E − e j ∆ x + c . In particular, when j = j ∗ , we find U j ∗ sothat U j ∗ satisfies the equality in (10). Then U j ∗ becomes the solution to (9) and thus U j ∗ have to be expressed as follows: U j ∗ = c ξ j ∗ = ϕ j ∗ = E − e j ∗ ∆ x + c . (13)From this, the constant c can be expressed as follows: c = ( E − e j ∗ ∆ x + c ) ξ − j ∗ Now, find j ∗ . From (12), (13), we have U j ∗ +1 = c ξ j ∗ +11 = c ξ j ∗ ξ = ξ ( E − e j ∗ ∆ x + c ) , U j ∗ − = E − e ( j ∗ − x + c From (10) we have U j ∗ = ϕ j ∗ = E − e j ∗ ∆ x + c = 1 ρ { (1 − w ) U j ∗ + w [ aU j ∗ +1 + (1 − a ) U j ∗ − ] } == 1 ρ n (1 − w )( E − e j ∗ ∆ x + c ) + w [ aξ ( E − e j ∗ ∆ x + c ) + (1 − a )( E − e ( j ∗ − x + c )] o ⇔ ρ ( E − e j ∗ ∆ x + c ) = (1 − w )( E − e j ∗ ∆ x + c ) + w [ aξ ( E − e j ∗ ∆ x + c ) + (1 − a )( E − e j ∗ ∆ x + c /e ∆ x )] ⇔ ρE − (1 − w ) E − waξ E − w (1 − a ) E = [ ρ − (1 − w ) − waξ − w (1 − a ) e − ∆ x ] e j ∗ ∆ x + c ⇔ e j ∗ ∆ x + c = ρ − [(1 − w ) + w ( aξ + (1 − a ))] Eρ − { (1 − w ) − w [ aξ + (1 − a ) e − ∆ x ] }⇔ j ∗ = (cid:20) x (cid:18) log { ρ − [(1 − w ) + w ( aξ + (1 − a ))] Eρ − { (1 − w ) + w [ aξ + (1 − a ) e − ∆ x ] } − c (cid:19)(cid:21) . Now, we denote f = 1∆ x (cid:18) log { ρ − [(1 − w ) + w ( aξ + (1 − a ))] Eρ − { (1 − w ) + w [ aξ + (1 − a ) e − ∆ x ] } − c (cid:19) . (14) ariational inequality for perpetual American option price... U, j ∗ ) as follows: j ∗ = [ log f ] , U j = ( ( E − e j ∗ ∆ x + c ) ξ j − j ∗ , j > j ∗ E − e j ∗ ∆ x + c j ≤ j ∗ (15)Then, ( U, j ∗ ) is the solution to (9), (10), (11). Thus we proved the existence theorem of thesolution of difference equation of perpetual American put option. Theorem 2.8
The solution of explicit difference scheme (9) , (10) , (11) of perpetual Amer-ican put option price exists and it is expressed as (15) . Corollary 2 If r = 0 in (14), then ρ = 1 , ξ = 1, thus f = 0 and optimal exercise boundarydoes not exist. Remark 3 . By Theorem 6, 7, 8, the problem of uniqueness and existence of the solutionand optimal exercise boundary of explicit difference scheme for perpetual American putoption price is solved. From the consideration in [8], [9], the binomial tree method of [11]can be considered as a special difference equation for the variational inequality neglectingan infinitesimal of the same order as ∆ x and thus such study can be viewed as an extensionof the results of [11]. Consider the concept of viscosity solutions to variational inequality of perpetual Americanoptions.If u ∈ U SC ( R ) ( LSC ( R )) satisfies the following two conditions, then u is called the viscosity subsolution (supersolution) of the variational inequality (4):(i) u ( −∞ ) ≤ ( ≥ ) E , u (+ ∞ ) ≤ ( ≥ )0.(ii) If Φ ∈ C ( R ) and u − Φ attains its local maximum (minimum) at x ∈ R , we havemin (cid:26) − σ ∂ Φ ∂x − (cid:18) r − q − σ (cid:19) ∂ Φ ∂x + ru, u − ϕ (cid:27) x ≤ ( ≥ )0 u ∈ C ( R ) is called the viscosity solution of the variational inequality (4) if it is both viscositysubsolution and viscosity supersolution of (4). For x ∈ [( j − / x + c, ( j + 1 / x ], wedefine the extension function u ∆ x ( x ) as follows: u ∆ x ( x ) := U j Here U j is the solution of (7) and given by (15).In l ∞ ( Z ), we define ( U j ) ≤ ( V j ) ⇔ U j ≤ V j , ∀ j ∈ Z . We define the operator F in l ∞ ( Z )as follows:[ F ( U )] j = max (cid:26) ρ { (1 − w ) U j + w [ aU j +1 + (1 − a ) U j − ] } , ϕ j (cid:27) , U = ( U j ) ∈ l ∞ ( Z ) . Then the solution ( U j ) of (7) is a fixed point of F . Lemma 2.5
If 0 < w ≤ (cid:12)(cid:12)(cid:12) ∆ xσ (cid:16) r − q − σ (cid:17)(cid:12)(cid:12)(cid:12) <
1, then, F is monotone in l ∞ ( Z ).That is, we have U ≤ V , U , V ∈ l ∞ ( Z ) ⇒ FU ≤ FV . Hyong-chol O, Song-San Jo
Proof
Noting that from the assumption we have 1 − α ≥
0, 0 < w <
1, the required resulteasily comes. (QED)
Lemma 2.6 If U ∈ l ∞ ( Z ) , K ≥ , K = ( ...K, K, K... ) . Then, we have F ( U + K ) ≤ FU + KProof
Since ρ >
1, we have F ( U + K ) == (cid:18) max (cid:26) ρ [(1 − w )( U j + K ) + w ( α ( U j +1 + K ) + (1 − α )( U j − + K ))] , ϕ (cid:27)(cid:19) ∞ j = −∞ ≤ (cid:18) max (cid:26) ρ [(1 − w ) U j + w ( αU j +1 + (1 − α ) U j − ))] + Kρ (cid:27)(cid:19) ∞ j = −∞ ≤ FU + K . ( QED ) Lemma 2.7
The price
U, j ∈ Z of explicit difference scheme of perpetual American putoptions is bounded. Proof
Due to (15) and 0 < ξ <
1, we have ∀ j, ≤ U j ≤ E . (QED) Theorem 2.9
Let u ( x ) be the viscosity solution of (4) . If (cid:12)(cid:12)(cid:12) r − q − σ (cid:12)(cid:12)(cid:12) ∆ xσ ≤ , then u ∆ x ( x ) converges to u ( x ) as ∆ x → . Proof
Denote u ∗ ( x ) = lim ∆ x → sup y → x u ∆ x ( y ) , u ∗ ( x ) = lim ∆ x → inf y → x u ∆ x ( y )From Lemma 7, u ∗ and u ∗ are well-defined and we have 0 ≤ u ∗ ( x ) ≤ u ∗ ( x ) ≤ E . Obviously, u ∗ ∈ U SC ( R ) and u ∗ ∈ LSC ( R ). If we prove that u ∗ is the subsolution and u ∗ thesupersolution of (4), then we have u ∗ ≤ u ∗ and thus u ∗ = u ∗ = u ( x ) becomes the viscositysolution of (4), and therefore we have the convergence of the approximate solution u ∆ x ( x ).We will prove that u ∗ is the subsolution of (4). (The fact that u ∗ is the supersolutionis similarly proved.) Suppose that φ ∈ C ( R ) and u ∗ − φ attains a local maximum at x ∈ R . We might as well assume that ( u ∗ − φ )( x ) = 0 and x is a strict local maximumon B r = { x : | x − x | ≤ r } , r >
0. Let Φ = φ − ε. ε >
0, then u ∗ − Φ attains a strict localmaximum at x and ( u ∗ − Φ)( x ) > u ∗ , there exists a sequence u ∆ x k ( y k ) such that ∆ x k → y k → x and lim k →∞ u ∆ x k ( s k , y k ) = u ∗ ( t , x ) (17) ariational inequality for perpetual American option price... u ∆ x k − Φ on B r by ˆ y k , then there exists asubsequence u ∆ x k i such that when k i → ∞ , we have∆ x k i → , ˆ y k i → x and ( u ∆ x k i − Φ) (ˆ y k i ) → ( u ∗ − Φ) ( x ) (18)Indeed, suppose ˆ y k i → ˆ y , then from (17) we have( u ∗ − Φ) ( x ) = lim k i →∞ (cid:16) u ∆ x ki − Φ (cid:17) ( y k ) ≤ lim k i →∞ (cid:16) u ∆ x ki − Φ (cid:17) (ˆ y k i ) ≤ ( u ∗ − Φ) (ˆ y ) . Therefore we have ˆ y = x , since x is a strict local maximum of ( u ∗ − Φ). Thus forsufficiently large k i , if ( ˆ S k i + ∆ t (∆ x k i )) ∈ B r , then we have (cid:16) u ∆ x ki − Φ (cid:17) ( x ) ≤ (cid:16) u ∆ x ki − Φ (cid:17) (ˆ y k i )that is, u ∆ t ki ( x ) ≤ Φ( x ) + (cid:16) u ∆ x ki − Φ (cid:17) (ˆ y k i ) (19)From (16) and (18), when k i is sufficiently large, then we have (cid:16) u ∆ x ki − Φ (cid:17) (ˆ y k i ) > . (20)For every k i , j k i = j select such that ˆ S k i ∈ [ t n , t n +1 ), ˆ y k i ∈ [( j − / x k i + c, ( j +1 / x k i + c ). Then from (19), Lemma 5 and Lemma 6 we have u ∆ x ki (ˆ y k i ) = U j = ( FU ) j = [ Fu ∆ x ki ( • )](ˆ y k i ) ≤ n F [Φ( • ) + ( u ∆ x ki − Φ)(ˆ y k i )] o (ˆ y k i ) . ≤ { F [Φ( • )] } (ˆ y k i ) + ( u ∆ x ki − Φ)(ˆ y k i ) . Thus we have Φ(ˆ y k i ) − { F [Φ( • )] } (ˆ y k i ) ≤ . Therefore using (7) we haveΦ(ˆ y k i ) − { F [Φ( • )] } (ˆ y k i ) == Φ(ˆ y k i ) − max (cid:26) ρ { (1 − w )Φ j + w [ a Φ j +1 + (1 − a )Φ j − ] } , ϕ j (cid:27) = Φ(ˆ y k i ) − max (cid:26)
11 + r ∆ t (cid:20)(cid:18) − σ ∆ t ∆ x k i (cid:19) Φ(ˆ y k i ) + σ ∆ t ∆ x k i (cid:20)(cid:18)
12 + ( r − q − σ x k i σ (cid:19) Φ(ˆ y k i + ∆ x k i )+ (cid:18) − ( r − q − σ x k i σ (cid:19) Φ(ˆ y k i − ∆ x k i ) (cid:21)(cid:21) , ϕ j (cid:27) ≤ . Hyong-chol O, Song-San Jo
This inequality is equivalent to the following.min (cid:26) ∆ t r ∆ t (cid:20) Φ(ˆ y k i ) − Φ(ˆ y k i )∆ t −− σ y k i + ∆ x k i ) − y k i ) + Φ(ˆ y k i − ∆ x k i )2∆ x k i − (cid:18) r − q − σ (cid:19) Φ(ˆ y k i + ∆ x k i ) − Φ(ˆ y k i − ∆ x k i )2∆ x k i + r Φ(ˆ y k i ) (cid:21) , Φ(ˆ y k i ) − ϕ j (cid:27) ≤ . Noting that ∆ t r ∆ t , we havemin (cid:26) − σ y k i + ∆ x k i ) − y k i ) + Φ(ˆ y k i − ∆ x k i )2∆ x k i − (cid:18) r − q − σ (cid:19) Φ(ˆ y k i + ∆ x k i ) − Φ(ˆ y k i − ∆ x k i )2∆ x k i + r Φ(ˆ y k i ) , Φ(ˆ y k i ) − ϕ j (cid:27) ≤ . Let k i → ∞ , then ∆ x k i → (cid:26) − σ ∂ Φ ∂x − (cid:18) r − q − σ (cid:19) ∂ Φ ∂x + r Φ , Φ − ϕ (cid:27) x ≤ . (Here we considered ˆ y k i → x , ϕ k i → ϕ ( x ).) Here let ε → (cid:26) − σ ∂ φ∂x − (cid:18) r − q − σ (cid:19) ∂φ∂x + rφ, φ − ϕ (cid:27) x ≤ . Since u ∗ ( x ) = φ ( x ), u ∗ is a subsolution of (18). Thus we proved theorem. (QED) Remark.
The result of this section can be viewed as an extension of the results of [JD]to the case with the expiry time T = ∞ Naturally, the price of perpetual American option can be seen as a limit of the price ofAmerican Option when the maturity goes to infinity. This approach excludes the aprioriassumtion that the price of perpetual American option does not depend on time. In thissection we show that the limits of the prices of variational inequality and BTM models forAmerican Option when the maturity goes to infinity do not depend on time.
When r > , q and σ > − LV = − ∂V∂t − σ S ∂ V∂S − ( r − q ) S ∂V∂S + rV ariational inequality for perpetual American option price... V ( S, t ; T ) be the price of American put option with the maturity T , that is, thesolution to the variational inequalitymin {− LV, V − ϕ } = 0 , < t < T, S > V ( S, T ) = ϕ ( S ) = ( E − S ) + , S > V ( S, t, T ) uniquely exists and V ( S, t, T ) is decreasing with respect to S and t and increasing on T . Furthermore, V ( S, t, T ) is bounded. That is, we have0 ≤ V ( S, t, T ) ≤ E, < S < ∞ , ≤ t ≤ T (22)Thus, the following limit exists. U ( S, t ) = lim T →∞ V ( S, t, T ) = sup T V ( S, t, T ) , < S < ∞ , ≤ t < ∞ . (23) U ( S, t ) is decreasing on S and t and it can be seen as ”the price of perpetual American putoption” but this function seems to be time dependent.On the other hand, it is natural to think that U ( S, t ) is the solution to the followingproblem. min {− LU, U − φ } = 0 , t < , S > V (0+ , t ) = E, V (+ ∞ , t ) = 0 , S > . (24)Thus (24) can be seen as ”the pricing model of perpetual American put option”. it isdifferent from the model of [9] derived under the assumption that the price of perpetualAmerican put option is independent on time. This variational inequality does not belongto the range of application of a general theory of existence and uniqueness of solution tovariational inequalities discussed in [5] or [4] because the spatial variable and time variableintervals are both infinite intervals. The price function of perpetual American option con-sructed in [9] satisfies (24). So if we can prove the uniquness of the solution to (24), thenwe will have the independence on time variable of the solution to (24). Thus we will procethe uniquness of the solution to (24).If U ( S, t ) is the solution to (24), then we have − LU > (exerciseregion or stopping region) where U ( S, t ) = ( E − S ) + and − LU = 0 on the region Σ (thecontinuation region) where U ( S, t ) > ( E − S ) + . The solution to (24) is always nonnegative.The parabolic boundary of the region A = ( a, b ) × (0 , T )(0 ≤ a < b ≤ ∞ , T >
0) isdefined as follows: ∂ p A = { a } × (0 , T ) ∪ { b } × (0 , T ) ∪ ( a, b ) × { T } . Theorem 3.1 (maximum principle of the Black-Scholes differential operator)1) For V ( S, t ) ∈ C , ( A ) , if − LV < ( > S, t ) ∈ A , the nonnegative maximum (nonpositiveminimum) value of V cannot be attained at the parabolic boundary of A . Furthermore, if − LV ≤ ( ≥ )0( S, t ) ∈ A , then we have sup x ∈ A V ( x ) = sup x ∈ ∂ p A V + ( x ) ( inf x ∈ A V ( x ) = inf x ∈ ∂ p A V − ( x )) (25)
2) Fix t > . Let A t = { ( S, t ) ∈ A } = ( a, b ) × { t } . If − LV < , ( S, t ) ∈ A t and V t ≤ ( V decreasing on t ), then we have sup V ( x ) = max { V + ( a, t ) , V + ( b, t ) } . Hyong-chol O, Song-San Jo
Proof
1) In fact, suppose that it attains the nonnegative maximum value at the parabolicinterior though − LV <
0, there exists such a point x = ( S , t ) that V ( x ) = max x ∈ A V ( x ) = M ≥ a < S < b, ≤ t < T . If a < S < b, < t < T , then we have V t ( x ) = V S ( x ) = 0 , V SS ( x ) ≤ − LV ( x ) = − V t − σ S V SS − ( r − q ) SV S + rV | x ≥ − LV <
0. If a < S < b, t = 0, then we have − V t ( x ) ≥ , V S ( x ) = 0 , V SS ( x ) ≤ − LV ( x ) ≥
0. This contradicts − LV < − LV ≤
0, let u = V − ε then have − Lu = − LV − rε <
0, thus we havesup x ∈ A u ( x ) = sup x ∈ ∂ p A u + ( x ) . Thus we have sup x ∈ A V ( x ) ≤ sup x ∈ A u ( x ) + ε = sup x ∈ ∂ p A u + ( x ) + ε ≤ sup x ∈ ∂ p A V + ( x ) + ε. and here let ε →
0, then we have (25).2) If a < S < b and V ( S , t ) = max A t V ( S, t ) = M ≥
0, then we have V S ( S , t ) =0 , V SS ( S , t ) ≤
0, and from decreasing on t , we have − V t ( S , t ) ≥
0. Thus, we have − LV ( S , t ) ≥
0. (QED)
Lemma 3.1
Assume that V ( S, t ) is the solution to (24) and V ( S, t ) = ( E − S ) + . If q ≥ S ≤ min { rE/q, E } . If q <
0, then we have S ≤ E . Proof
First, note that if V ( S, t ) = ( E − S ) + then S ≤ E [8] (That is why, in this case S is inthe stopping region and If we suppose that S > E , then ( E − S ) + = 0, that is exercise payoffis zero and thus S belongs to the continuation region. It contradicts V ( S, t ) = ( E − S ) + .)Thus if V ( S, t ) = ( E − S ) + then we can rewrite as V ( S, t ) = E − S . And (24) is written asmin {− LV, V − ( E − S ) } = 0 . Since ( V − ( E − S )) = 0 , then we have − LV = rE − qS ≥ q ≥
0, then we have S ≤ rE/q .(QED) Lemma 3.2
Let V ( S, t ) be the solution to (24) and fix t >
0. Then1) If there exists such S > V ( S , t ) = ( E − S ) + , then for ∀ S < S , we have V ( S, t ) = ( E − S ) + .2) If there exists such S > V ( S , t ) > ( E − S ) + , then for ∀ S > S , we have V ( S, t ) > ( E − S ) + . ariational inequality for perpetual American option price... Proof
1) From Lemma 1, we have S ≤ rE/q . Suppose that the conclusion were not true,that is, suppose that 0 < ∃ S < S : V ( S, t ) > E − S. (*)Let ( a, b )( b ≤ S ) be the longest interval where holds (*). Then, we have V ( S, t ) = ( E − S )at S = a and S = b . From min {− LV, V − φ } = 0, we have − LV = 0 , V − ( E − S ) >
0, onthat interval ( a, b ). Thus we have − L ( V − ( E − S )) = L ( E − S ) = qS − rE < ∵ q ≥ ⇒ S < S ≤ rE/q ) . Thus from the conclusion of 2) of Theorem 1, V − ( E − S ) attains the nonnegative maximumvalue at the boundary. But V ( a, t ) − ( E − S ) = 0 , V ( b, t ) − ( E − S ) = 0, so we have V ( S, t ) − ( E − S ) ≤ , a < ∀ S < b . This contradicts (*).2) If ∃ S > S : V ( S , t ) = ( E − S ) + , then from the conclusion of 1) we have V ( S , t ) =( E − S ) + . This contradicts the assumption (QED). Theorem 3.2 (Existence of exercise boundary) When V ( S, t ) is the solution to (24) anddecreasing on t , then for any t > , there exists such s ( t )( ≤ min { rE/q, E } ) that if < S ( E − S ) + . Proof
Fix t >
0, then ( E, + ∞ ) × { t } is included in the stopping region, so the stoppingregion is not empty. Let s ( t ) be the infimum of the stopping region, then from Lemma2, ( s ( t ) , + ∞ ) becomes the stopping region and (0 , s ( t )) becomes the continuation region(QED). Theorem 3.3 (Uniqueness of the solution) The solution to (24) that is decreasing on t isunique. Proof
Let V , V be the two solutions to (24) which are decreasing on t and be the exerciseboundaries of two solutions, respectively. Fix t >
0. Without loss of generality, assumethat s ( t ) > s ( t ). Then if S < s ( t ), then V = V = E − S , and if S > s ( t ) then LV = LV = 0. If s ( t ) < S ≤ s ( t ) then we have V = E − S, LV = 0 , V > E − S. (26)Now consider V − V in the interval ( s ( t ) , ∞ ). When s ( t ) < S ≤ s ( t ), we have − L ( V − V ) = L ( E − S ) = qS − rE < s < s ( t ) ≤ rE/q ) . and when S > s ( t ), we have − L ( V − V ) = 0, so finally we have − L ( V − V ) ≤ s ( t ) , ∞ ). From 2) of Theorem 1, the nonnegative maximum value of V − V isattained at the boundary in the interval [ s ( t ) , ∞ ). But we have ( V − V )( s ( t ) , t ) = 0 and( V − V )( ∞ , t ) = 0 in [ s ( t ) , ∞ ). Thus we have V − V ≤ s ( t ) , ∞ ). Thiscontradicts (26). So we have s ( t ) = s ( t ) , ∀ t > s ( t ) be the exercise boundary. Two solutions are equal to E − S on the interval(0 , s ( t )). In the interval ( s ( t ) , ∞ ), we have − L ( V − V ) = 0 and V − V | s ( t ) , + ∞ = 0, sowe have V = V from Theorem 1(QED).8 Hyong-chol O, Song-San Jo
Suppose that
T > N be the number of partition intervals,∆ t = T /N the length of the partition interval and t n = n ∆ t, n = 0 , , . . . , N. Especially T = t N .Let u = e σ √ ∆ t , θ = ρ/η − du − d , ρ = 1 + r ∆ t, η = 1 + q ∆ t, ud = 1 (27)Here r, q and σ are interest rate, dividend rate and volatility, respectively. In BTM welet S j = S u j , j ∈ Z . Denote by V nj the American option price at time t n with underlyingasset value S j . Then American option price by BTM is as follows [8, 9]: V Nj = ϕ, j ∈ ZV k − j = max (cid:26) ρ (cid:0) θV kj +1 + (1 − θ ) V kj − (cid:1) , φ j (cid:27) , k = N, N − , . . . , φ j = ( S j − E ) + (call) or φ j = ( E − S j ) + (put). Theorem 3.4
Let V nj ( T ) ( j ∈ Z ) be the price by BTM of American option with the matu-rity T . Then V nj ( T ) is increasing on T . That is, if T = t N < T = t M , then V nj ( T ) ≤ V nj ( T ) n = 0 , , . . . , N. (29) Proof
1) First, assume that M = N + 1. From (28), V Nj ( T ) = φ j , V N +1 j ( T ) = φ j , andwe have V Nj ( T ) = max (cid:26) ρ ( θφ j +1 + (1 − θ ) φ j − ) , φ j (cid:27) . So we have V Nj ( T ) ≥ φ j = V Nj ( T ). That is, when n = N , we have (29). Assume that wehave (29) when n = k , then we will prove (29) when n = k −
1. That is, we will prove thatif V kj ( T ) ≤ V kj ( T ) then V k − j ( T ) ≤ V k − j ( T ).In fact V k − j ( T ) = max (cid:26) ρ (cid:0) θV kj +1 ( T ) + (1 − θ ) V kj − ( T ) (cid:1) , φ j (cid:27) ≤ max (cid:26) ρ (cid:0) θV kj +1 ( T ) + (1 − θ ) V kj − ( T ) (cid:1) , φ j (cid:27) = V k − j ( T ) .
2) Now let us prove when 0 = t < t < · · · t N = T < t N +1 < · · · < t N + r = T in general.From the result of 1), when n = 0 , , . . . , N , we have V nj ( T ) = V nj ( t N ) ≤ V nj ( t N +1 ) ≤ V nj ( t N +2 ) ≤ · · · ≤ V nj ( t N + r ) = V nj ( T ) , ∀ j ∈ Z. So the required result is proved. (QED)
Remark 1.
The price by BTM of American call option φ j = ( S j − E ) + satisfies φ j ≤ V nj ( T ) ≤ S and is increasing on S , and decreasing on t ( V nj ≤ V nj +1 and V nj ≤ V n − j ).The price by BTM of American put option ( φ j = ( E − S j ) + ) satisfies φ j ≤ V nj ( T ) ≤ E and ariational inequality for perpetual American option price... S and t ( V nj +1 ≤ V nj and V nj ≤ V n − j ). And V n −∞ ( T ) = E, V n + ∞ ( T ) = 0 [8,17] So the price by BTM of American option converges when T → ∞ . We denote this limitby U nj ( j ∈ Z, n = 0 , , . . . ). That is, U nj = lim T →∞ V nj ( T ) , ∀ j ∈ Z, n = 0 , , . . . . (30) Lemma 3.3
The limit of the BTM price of call option is increasing on S ( j ), and is de-creasing on t ( n ) and the limit of the BTM price of put option is decreasing on S and t .These limit prices have the same boundedness as Remark 1. Time independence of the limit U nj of the put option BTM price. From (2), the limit U nj of the BTM price of put option satisfies the following differenceequation. U kj = max (cid:26) ρ (cid:0) θU k +1 j +1 + (1 − θ ) U k +1 j − (cid:1) , φ j (cid:27) , j ∈ Z, k = 0 , , . . .U k −∞ = E, U k ∞ = 0 . (31)Denote by l ∞ ( Z ) the Banah space formed with two-sided sequence of real numbers. For U ∈ l ∞ ( Z ), the norm k U k is defined as follows: k U k := sup j ∈ Z | U j | . Define the operator B : l ∞ ( Z ) → l ∞ ( Z ) as follows:( B U ) j = max (cid:26) ρ ( θU j +1 + (1 − θ ) U j − ) , φ j (cid:27) , j ∈ Z, U ∈ l ∞ ( Z ) . (32)Then k B U − B V k = sup j (cid:26) max (cid:20) ρ ( θU j +1 + (1 − θ ) U j − ) , φ j (cid:21) − max (cid:20) ρ ( θV j +1 + (1 − θ ) V j − ) , φ j (cid:21)(cid:27) ≤ sup j (cid:26) max (cid:20) ρ ( θ ( U j +1 − V j +1 ) + (1 − θ )( U j − − V j − )) , (cid:21)(cid:27) (33) ≤ max (cid:20) ρ ( θsup j | U j +1 − V j +1 | + (1 − θ ) sup j | U j − − V j − | ) , (cid:21) ≤ ρ k U − V k . Thus we proved the following theorem.
Lemma 3.4
Operator B : l ∞ ( Z ) → l ∞ ( Z ) is contraction mapping in Banach space andhas a unique fixed point when r > U nj defined by (30). Denote by U n = (cid:8) U nj (cid:9) j ∈ Z ∈ l ∞ ( Z ) . Hyong-chol O, Song-San Jo
From (31), U k = B U k +1 , k = 0 , , · · · and thus using ((33)) repeatedly then we have k U − U k = k B U − B U k ≤ ρ k U − U k ≤ ρ k U − U k ≤ · · · ≤ ρ n k U n − U n +1 k . So we have k U − U k ≤ E/ρ n because k U n − U n +1 k ≤ E . thus we have U − U ,and we have U k − U k +1 , k = 0 , , · · · in the same way. Thus we proved the followingtheorem. Theorem 3.5
The limit U nj of the BTM price of American put option is independent on n . Thus U j = U nj is the solution to the following problem. max (cid:26) ρ ( θU j +1 + (1 − θ ) U j − ) , φ j (cid:27) , j ∈ Z.U −∞ = E, U ∞ = 0 (34) Remark 2.
So the limit of the BTM price of American put option is the BTM price ofperpetual American option [11].
Let V ( S, t, T ) be the price of American option with expiry date T , that is, the solutionto the variational inequalitymin (cid:26) − ∂V∂t − σ S ∂ V∂S − ( r − q ) S ∂V∂S + rV, V − φ (cid:27) = 0 , < t < T, S > ,V ( T, S ) = φ ( S ) = ( E − S ) + or ( S − E ) + . (35)Then V ( S, t, T ) is increasing on T [9]. When ∆ t = T /N, ∆ x >
0, let S j = S e j ∆ x = e j ∆ x + c , j ∈ Z, t n = n ∆ t. Denote U nj = V ( S j , t n ) , w = σ ∆ t ∆ x , a = 12 + ( r − q − σ x σ , ρ = 1 + r ∆ t. Then the explicit difference scheme for (35) is privided as follows [8, 9]: U Nj = φ j , j ∈ Z (36) U nj = max (cid:26) ρ (cid:8) (1 − w ) U n +1 j + w (cid:2) aU n +1 j +1 + (1 − a ) U n +1 j − (cid:3)(cid:9) , φ j (cid:27) , n = N − , · · · , , Theorem 3.6
Let U nj ( T ) be the solution to the explicit difference scheme of AmericanOption with the maturity T . Then U nj ( T ) is increasing on T . That is, if T = t N < T = t M ,then we have U nj ( T ) ≤ U nj ( T ) . ariational inequality for perpetual American option price... Proof
1) First, let M = N + 1, that is,0 = t < t < · · · t N = T < t N +1 < T = t N +2 . Then U Nj ( T ) = φ j , U N +1 j ( T ) = φ j and we have U Nj ( T ) = max (cid:26) ρ (cid:8) (1 − w ) U N +1 j + w (cid:2) aU N +1 j +1 + (1 − a ) U N +1 j − (cid:3)(cid:9) , ϕ j (cid:27) ≥ ϕ = U Nj ( T ) . So when n = N , we have the result of the theorem. Assume that U nj ( T ) ≤ U nj ( T ) when n = k , and we will prove when n = k − U k − j ( T ) = max (cid:26) ρ (cid:8) (1 − w ) U kj ( T ) + w (cid:2) aU kj +1 ( T ) + (1 − a ) U kj − ( T ) (cid:3)(cid:9) , ϕ j (cid:27) ≤ max (cid:26) ρ (cid:8) (1 − w ) U kj ( T ) + w (cid:2) aU kj +1 ( T ) + (1 − a ) U kj − ( T ) (cid:3)(cid:9) , ϕ j (cid:27) = U kj ( T ) ∀ j ∈ Z.
2) In general, consider the case when0 = t < t < · · · < t N = T < t N +1 < · · · < t N + r = T , r ∈ N. From the result of 1), we have U nj ( T ) = U nj ( t N ) ≤ U nj ( t N +1 ) ≤ U nj ( t N +2 ) ≤ · · · ≤ U nj ( t N + r ) = U nj ( T ) , ∀ j ∈ Z. Thus, we have proves the required result.(QED)
Remark 3.
The price by explicit difference scheme of American call option ( φ j =( S j − E ) + ) satisfies φ j ≤ U nj ( T ) ≤ S j and is increasing on S and decreasing on t ( U nj ≤ U nj +1 and U nj ≤ U n − j ). The price by explicit difference scheme of American put option φ j = ( E − S ) + satisfies φ j ≤ U nj ( T ) ≤ E and is decreasing on S and t ( U nj +1 ≤ U nj , U nj ≤ U n − j ), and U n −∞ ( T ) = E, U n + ∞ ( T ) = 0 [8, 9].Thus the prices by explicit difference scheme of American options with the maturity T converge when T → ∞ . Denote the limit by W nj ( j ∈ Z, n = 0 , , . . . ). That is, W nj = lim T →∞ U nj ( T ) , ∀ j ∈ Z, n = 0 , , · · · (37)The limit of the price by explicit difference scheme of American call option is increasing on S ( j ) and is decreasing on t ( n ) and the limit of put option is decreasing on S and t . Theyhave the same boundedness as pointed in Remark 3.From (36), the limit W nj of the price by explicit difference scheme of American putoption ( φ j = ( E − S j ) + ) satisfies the following difference equation. W nj = max (cid:26) ρ (cid:8) (1 − w ) W n +1 j + w (cid:2) aW n +1 j +1 + (1 − a ) W n +1 j − (cid:3)(cid:9) , φ j (cid:27) , j ∈ Z, n = 0 , , . . . .W k −∞ = E, W k ∞ = 0 . (38)2 Hyong-chol O, Song-San Jo
This is just the difference equation of (24), the variational inequality on infinite time interval.Define an operator F : l ∞ ( Z ) → l ∞ ( Z ) as follows:( F U ) j = max (cid:26) ρ { (1 − w ) U j + w [ aU j +1 + (1 − a ) U j − ] } , φ j (cid:27) , j ∈ Z, U ∈ l ∞ Z. (39)Then we have k F U − F V k = sup j (cid:26) max (cid:26) ρ { (1 − w ) U j + w [ aU j +1 + (1 − a ) U j − ] } , φ j (cid:27) −− max (cid:26) ρ { (1 − w ) V j + w [ aV j +1 + (1 − a ) V j − ] } , φ j (cid:27)(cid:27) ≤ sup j (cid:26) max (cid:20) ρ { (1 − w )( U j − V j ) + w [ a ( U j +1 − V j +1 ) + (1 − a )( U j − − V j − )] } , (cid:21)(cid:27) (40) ≤ max (cid:20) ρ ((1 − w ) sup j | U j − V j | + w [ asup j | U j +1 − V j +1 | + (1 − a ) sup j | U j − − V j − | ]) , (cid:21) ≤ ρ k U − V k So we proved the following lemma.
Lemma 3.5
The operator F : l ∞ ( Z ) → l ∞ ( Z ) is contraction mapping and has a uniquefixed point when r > W nj defined by (38). Let W n = { W nj } j ∈ Z ∈ l ∞ ( Z ) . Then from (31), we have W n = F W n +1 , n = 0 , , . . . and thus using (40) repeatedly, wehave k W − W k = k F W − F W k ≤ ρ k W − W k ≤ ρ k W − W k ≤ · · · ≤ ρ n k W n − W n +1 k . So we have k W − W k ≤ E/ρ n → , n → ∞ because k W n − W n +1 k ≤ E . Thus wehave W = W and in the same way we have W n − W n +1 , n = 0 , , . . . . Thus we provedthe following theorem. Theorem 3.7
The limit W nj of the price by explicit difference scheme of American putoption is independent on n . That is, W j = W nj is the solution to the following problem. W j = max (cid:26) ρ { (1 − w ) W j + w [ aW j +1 + (1 − a ) W j − ] } , φ j (cid:27) , j ∈ ZW −∞ = E, W ∞ = 0 . (41) Remark 2.
So the limit of the price by explicit difference scheme of American putoption becomes the price by the difference equation (7) of perpetual American put option. ariational inequality for perpetual American option price... References [1] Aurlien Alfonsi, Benjamin Jourdain, A Call-Put Duality for Perpetual American Op-tions, arxiv:math/0612648v1[math.PR], 2006, 1–28..[2] Aurlien Alfonsi, Benjamin Jourdain, General Duality for Perpetual American Options,arXiv:math/0612649v1 [math.PR], 2006, 1–20.[3] Y. Chi, Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance, Insurance: Mathematics and Eco-nomics, 46, 2010, 385–96.[4] Crandall, M.G., Ishii, H., Lions, P.L., Users guide to Viscosity Solutions of SecondOrder Partial Differential Equations,
Bull. Amer. Math. Soc. , , 1992, 1-67.[5] Friedman, A., Variational Principles and Free Boundary Problems, John Wiley andSons, New York, 1982[6] Gerber H., Shiu E., Martingale approach to to pricing perpetual American options,Astin Bulletin, Vol. 24, No. 2, 1994, 195–220.[7] He, H., Convergence from discrete- to continuous-time contingent claims prices, Reviewof Financial Studies ,3