Analysis of pricing American options on the maximum (minimum) of two risk assets

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1 Interfaces Free Boundaries 4, (00) 7 46 Analysis of pricing American options on the maximum (minimum) of two risk assets LISHANG JIANG Institute of Mathematics, Tongji University, People s Republic of China [Received 17 July 000 in revised form 7 June 001] We use a PDE argument to deal with the mathematical analysis of the valuation of American options on the maximum/minimum of two assets. There are several factors which affect the valuation of options, such as stock price, strike price, the time to expiry, volatilities, the correlation constant, the risk-free interest rate dividends. The first problem we are concerned with here is what happens to the prices of options if one of these factors is increasing while the others remain fixed. In the second part of this paper, the properties of the optimal exercise boundary of option as free boundary of the parabolic obstacle problem are studied such as monotonicity, convexity asymptotic behavior. Keywords: American call/put options; parabolic obstacle problem; free boundary; penalized term In finance a call (put) option is a contract which gives the holder a right to buy (sell) one of the assets, such as stock, foreign currency etc, by a certain date for a certain price. If the option can be exercised only at a certain date, then the contract is known as a European option. If the option can be exercised at any time before a certain date, then the contract is known as an American option. In the Black Scholes framework, by a simple arbitrage argument the valuation of the American option is a free boundary problem. Similar to the obstacle problem in mechanics, where the obstacle is the payoff function the free boundary is the optimal exercise boundary. In this paper we are concerned with the valuation of American options on the maximum (minimum) of two assets. These are very popular contracts traded in modern financial markets or issued by firms. Many examples can be found in [1 3]. As we know, there are several factors which affect the valuation of American options on the maximum (minimum) of two assets, such as stock prices, strike prices, the time to expiry, volatilities, the correlation constant, the risk-free interest rate dividends. The first problem considered here is what happens to the prices of the options if one of these factors is increasing while the others remain fixed. For the options of a single asset the same problem was studied in [4: Chapter 7]. For the European option on the maximum (minimum) of two assets, the problem has been solved by Stulz [] in this case there is an explicit expression of the price of the option. Unfortunately, we cannot extend his argument to the American case, because here the Black Scholes model is a variational inequality the solution cannot be expressed in a closed form. In the first part of this paper, we deal with this problem by using the PDE approach. From the view of PDE the problem is to claim the monotonicity of the solution on variables, initial data coefficients of equations respectively. In order to prove the monotonicity of the solution, we have to check the sign of related differential expressions of the solution for the variational inequality in which the obstacle initial data are only Lipschitz continuous. Some results we obtain here are of interest. For example, as is known, the monotonicity of the American option pricing on volatilities in the case of two underlying c Oxford University Press 00

2 8 LISHANG JIANG assets is not always true. It depends on their correlation constant. We find a condition prove that the monotonicity is true if only if the increments of volatilities satisfy the condition. The other contribution of this paper is to study the properties of the optimal exercise boundary which is a free boundary of a parabolic obstacle problem. For obstacle problems much is known about the existence, uniqueness regularity of the solutions. But the obstacle here is a special one in which the payoff function is also initial data of the parabolic obstacle problem. So we can get much deeper information about the properties of the optimal exercise boundary, such as evolution, monotonicity, convexity, asymptotic behavior, etc. Recently Broadie Detemple published a nice paper [1], in which they also took an interest in the similar problems by using a probability argument. We start the study of the optimal exercise boundary from a special case of the strike price E = 0, i.e. the payoff function is max(s 1, S ) (S i, the underlying asset (i = 1, )), which is the same as the one of an option to exchange one asset for another. This problem was first produced by Margrade [8] for valuing a European option. Rubinstein [9] showed that the American version of this option can be valued similarly to the case of a single asset solved using the binomial tree method. According to their idea, this problem can be reduced under some transformations to a 1D parabolic obstacle problem with two free boundaries. Some properties of free boundaries are studied, such as monotonicity, asymptotic behavior, etc. Based on all of these results, the evolutions of optimal exercise boundary is studied in the general case E > 0 using a comparison principle. The paper is organized as follows. Section 1 outlines the Black Scholes model of the American call-max (min) option. Section introduces a penalized problem corresponding to the valuation problem of the American call-max (min) option with a smooth approximate payoff function. Sections 3 6 establish the relation between the price of American call (put)-max (min) option the factors: stock price S 1, S, strike price E, time to expiration T, risk-free interest rate r, dividend q 1, q, volatility σ 1, σ correlation constant ρ respectively. Sections 7 8 focus on the properties of the optimal exercise boundary for the American call-max option. We start in Section 7 with the special case of the strike price E = 0 first. Based on the results of Section 7, the properties of the optimal exercise boundary in the general case of the strike price E > 0 are dealt with in Section Model Assume the prices of underlying assets S 1, S satisfy the stochastic differential equations ds 1 = (r q 1 )S 1 dt + σ 1 S 1 dw 1 t ds = (r q )S dt + σ S dw t where Wt 1, W t are stard Brownian motions E(dwt i i ) = 0, Var(dWt ) = dt (i = 1, ) E(dwt i dw t ) = ρ dt, ρ < 1 (1.1) where r is constant rate of interest, q i 0 is dividend rate of asset i(i = 1, ), σ i is volatility of price of asset i(i = 1, ) ρ is correlation coefficient. The price V (S 1, S, t) of an American call option on the maximum of two assets S 1, S (denoted

3 ANALYSIS OF PRICING 9 American call-max option) satisfies a boundary value problem of PDE as follows: { min V } (S LV, V G(S 1, S ) = 0, 1, S ) R + R + t 0 t < T (1.) V (S 1, S, T ) = G(S 1, S ), (S 1, S ) R + R + (1.3) V (0, S, t) = V1 0 (S, t), S R +, 0 t < T (1.4) V (S 1, 0, t) = V 0 (S 1, t), S 1 R +, 0 t < T (1.5) where E is the strike price, T is the time to expiry, ( LV = 1 σ1 V V S 1 S1 + ρσ 1 σ S 1 S + σ ) V S 1 S S S V V + (r q 1 )S 1 + (r q )S rv (1.6) S 1 S G(S 1, S ) = (max(s 1, S ) E) + (1.7) Vi 0 (s, t)(i = 1, ) is a solution of the following problem: { min V i 0 t σ i s Vi 0 s (r q i )s V i 0 s + rv 0 i, V 0 i (s E) + } = 0 (1.8) Vi 0 (s, T ) = (s E)+ (1.9) Vi 0 (0, t) = 0. (1.10) REMARK 1.1 For other American options on the maximum (minimum) of two assets, the payoff function has a different type as follows: G(s 1, s ) = (min(s 1, s ) E) + G(s 1, s ) = (E max(s 1, s )) + call-min option put-max option G(s 1, s ) = (E min(s 1, s )) + put-min option. Then the boundary conditions (1.4), (1.5) have to be changed according to different payoff function. REMARK 1. Equation (1.) is an obstacle problem of a parabolic equation. The coincidence set the separate (noncoincidence) set of the obstacle problem are called a stopping region a continuation region of the option respectively here.. Approximations Before starting our proof, we make a transformation ln s 1 = x 1, ln s = x (.1) T t = τ. (.)

4 30 LISHANG JIANG Then, the American call-max option problem is reduced to a parabolic obstacle problem with constant coefficients as follows: { } v min τ Lv, v g = 0,(x 1, x ) R, 0 <τ T (.3) v(x 1, x, 0) = g(x 1, x ) (.4) where v(x 1, x,τ)= V (e x 1, e x, T τ) g(x 1, x ) = G(e x 1, e x ) = (max(e x 1, e x ) E) + (.5) Lv = 1 [ σ v v 1 x1 + ρσ 1 σ + σ ] v x 1 x x ( + r q 1 σ 1 ) ( v + r q σ ) v rv. (.6) x 1 x As we know, (.3) is a complementary form of the variational inequality the payoff function (.5) is only a Lipschitz continuous function. First we have to consider an approximation of the problem (.3), (.4). Suppose where ε>0 g ε (x 1, x ) = Π ε (F ε (x 1, x ) E) (.7) s s ε Π ε (s) = ε s ε 0 s ε (.8) Π ε (s) C (R), 1 Π ε F ε (x 1, x ) = ex 1 + e x (s) 0, Π ε + ex 1 e x (s) 0, lim Π ε(s) = s + (.9) ε 0 ( e x 1 e x ) H (.10) ε H(s) = arctg(s). (.11) π It is easy to check that H (s) 0 0 < H (s) + sh (s) 4/π (.1) 0 1 ±[H(s) + sh (s)] lim F ε(x 1, x ) = max(e x 1, e x ) ε 0 lim g ε(x 1, x ) = lim Π ε (F ε (x 1, x ) E) = g(x 1, x ). ε 0 ε 0 (.13)

5 ANALYSIS OF PRICING 31 Consider an approximation of the complementary form of variational inequality (.3). In order to do this, we introduce a nonlinear penalized problem as follows: v τ Lv + β δ(v g) = 0, (δ > 0) (.14) where the penalized term satisfies β δ (s) C (R), β δ (s) 0,β δ (s) 0,β δ (s) 0. (.15) lim δ 0 β δ(s) = { 0 s > 0 s < 0. According to our need in the proof, we will choose the penalized function β δ (s) in the different forms. Combining the approximation (.7) (.14), we obtain an approximation of the Cauchy problem (.3), (.4) as follows: ˆv τ L ˆv + β δ( ˆv g ε (x 1, x )) = 0,(x 1, x ) Q M (.16) ˆv p Q M = g ε (x 1, x ) p Q M (.17) where Q M = { < x 1 M, < x M; 0 τ T }, M, p Q M is a parabolic boundary of domain Q M, ˆv(x 1, x,τ)= v δ,ε,m (x 1, x,τ). REMARK.1 As we know, the solution of ˆv(x 1, x,τ) of the penalized problem (.16), (.17) tends to the solution v(x 1, x,τ) of the Cauchy problem (.3), (.4) uniformly in any bounded domain D(D Q ) as δ 0 then ε 0, M, see [5, 6]. REMARK. For American call-main option the approximate payoff function is given in the form g ε (x 1, x ) = Π ε ( F ε (x 1, x ) E) (.18) where F ε (x 1, x ) = ex 1 + e x ex 1 e x ( e x 1 e x H ε ). (.19) 3. Effects from change in s 1, s in E In this section we assume that Q M = Q, β δ (s) satisfies (.15). (3.1) By the maximum principle, we have the following lemma.

6 3 LISHANG JIANG LEMMA 3.1 ˆv 0 in Q. (3.) x i As its corollary, we have the following theorem. THEOREM 3.1 If S i S i, (i = 1, ), then for the American call-max option V ( s 1, s, t) V (s 1, s, t), (3.3) where V (s 1, s, t) is a solution of the problem (1.) (1.5). THEOREM 3. If E 1 E, then for the American call-max option 0 V (s 1, s, t; E ) V (s 1, s, t; E 1 ) E 1 E. (3.4) Proof. Set ŵ = ˆv ˆv 1, where ˆv i is the solution of the penalized problem (.16), (.17) corresponding to the strike price E i (i = 1, ). Inviewof 0 ŵ τ=0 = g ε () g ε (1) E 1 E by the maximum principle, we have ŵ 0 i.e. E 1 E ˆv (x 1, x,τ) ˆv 1 (x 1, x,τ). By the same reason, we have so (3.4) is true. ˆv (x 1, x,τ) ˆv 1 (x 1, x,τ) 0 REMARK 3.1 For the American call-min option, it is easy to check that (g ε ) xi 0, therefore, the results of Theorems are still true. 4. Effects from change in r in q 1, q In this section we assume that Q M = Q β δ (s) = 1 δ ( s)+, (s R). (4.1) From the regularities of a solution of the Cauchy problem (.16), (.17) with a penalized term (4.1), we know that for any domain D Q ˆv ˆv Wp,1 p (1, ), (4.) where Wp,1 (D) is a Sobolev space Wp,1 (D) ={w w, D xw, Dx w, D tw L p (D)}.

7 ANALYSIS OF PRICING 33 LEMMA 4.1 ˆv 0 (i = 1, ). x i The proof is similar to that of Lemma 3.1 LEMMA 4. ˆv + ˆv ˆv + ε 0. (4.3) x 1 x Proof. Setting w = ˆv + ˆv ˆv + ε x 1 x we have w τ Lw + 1 δ (1 Sg(v g ε))w = 1 ( ε δ (1 Sg(v g ε)) + g ε + g ) ε g ε x 1 x w τ=0 = ε + g ε x 1 + g ε x g ε where { 1 s > 0 Sg(s) = 1 s 0. It is easy to calculate that the function ε/ + g ε / x 1 + g ε / x g ε is non-negative. Therefore by using the maximum principle the proof of the lemma is completed. REMARK 4.1 For the American call-min option, it is easy to see that if ε<eπ/, then the function ε/ + g ε / x 1 + g ε / x g ε is still non-negative, the inequality (4.3) is also true for the American call-min option. As a corollary of (4.3), we have the following theorem. THEOREM 4.1 If r 1 r, then for the American call-max option we have V (s 1, s, t; r 1 ) V (s 1, s, t; r ). (4.4) THEOREM 4. If q i q i (i = 1, ), then for the American call-max option we have V (s 1, s, t; q i ) V (s 1, s, t; q i ). (4.5) REMARK 4. Thanks to Remark 4.1 the ordering results (4.4), (4.5) are still true for the American call-min option. 5. Effects from change in expiration time T In this section we assume Q M (0 < M < ), that β δ (s) satisfies (.15): β δ (0) = 1 δ. (5.1)

8 34 LISHANG JIANG LEMMA 5.1 If δ>0 is small enough, then ˆv 0. (5.) τ Proof. Due to ˆv C (Q M ), the function ω = ˆv/ τ satisfies the following problem: ω τ Lω + β δ ( ˆv g ε)ω = 0, in Q M ω τ=0 = Lg ε β δ (0) ω x1 =M = 0 ω x =M = 0. By using the maximum principle, the estimate Lg ε β ε (0) 0 (5.3) implies (5.). According to (.9), (.1), a straightforward calculation shows that there exists a constant δ 0 (M) such that if δ δ 0 (M), the estimate (5.3) holds. As a corollary of (5.), we have the following theorem. THEOREM 5.1 If T 1 > T, then for the American call-max option V (s 1, s, t; T 1 ) V (s 1, s, t; T ), (t T ). (5.4) REMARK 5.1 What about the American call-min option? The crucial step is to claim that the inequality (5.3) is still valid. We can do this easily. 6. Effect from change in volatilities σ 1, σ correlation ρ In this section we assume Q M = Q, β δ (s) = 1 δ [( s)+ ] (s R). (6.1) LEMMA 6.1 Assume σ i = (1 + ε i )σ i, ε i 0 (i = 1, ) (ε i,ε ) D, where the domain D is enclosed by curves Γ 1 Γ { Γ 1 ={(ε 1,ε ) ε 1 = ϕ(ε 1 /ε ), 0 ε 1 /ε ω 0 } (6.) Γ ={(ε 1,ε ) ε = ϕ(ε /ε 1 ), 0 ε /ε 1 ω 0 } ω 0 = 1 ) (1 ρ 1 ρ (6.3) ϕ(ω) = (1 ρ )(1 + ω) + (1 ρ ) (1 + ω) + (1 ρ )[(1 + ω) ρ 4ω] (1 ρ ) (6.4)

9 ANALYSIS OF PRICING 35 then we have Λ ˆv = 1 ( σ 1 σ 1 ) ( ˆv x ( σ σ ) x ) ˆv ˆv + ρ( σ 1 σ σ 1 σ ) x 1 x 1 x ( ) ˆv ˆv 0 (6.5) x where ˆv is a solution of the penalized problem (.16), (.17) with a penalized term (6.1). Proof. Set w = Λ ˆv,wehave w τ Lw 1 δ Λ[(g ε ˆv) + ] = 0 (6.6) w τ=0 = Λg ε. (6.7) Computing Λ[(g ε ˆv) + ], we obtain that if ˆv >g ε, if ˆv <g ε, Λ[(g ε ˆv) + ] = 0, (6.8) Λ[(g ε ˆv) + ] = I + (( ˆv g ε ) x1,(ˆv g ε ) x ) + ( ˆv g ε )(w Λg ε ) (6.9) where I ± (λ 1,λ ) which is a quadratic form is defined as follows: Thus I ± (λ 1,λ ) = σ 1 (ε 1 + ε 1 )λ 1 + σ (ε + ε )λ ± ρσ 1σ ((ε 1 + ε ) + ε 1 ε )λ 1 λ. I ± (λ 1,λ ) 0, λ 1,λ R (6.10) iff i.e. = ρ ((ε 1 + ε ) + ε 1 ε ) + (ε 1 + ε 1 )(ε + ε ) 0 ω = ε /ε 1 (1 ρ )ε + (1 ρ )(1 + ω)ε ρ (1 + ω) + 4ω 0. Thus if (ε 1,ε ) D D is defined by (6.) (6.4), then (6.10) follows. Now we turn to check the sign of Λg ε. A straightforward calculation shows Λg ε = Π ε (F ε E)I + (F εx1, F εx ) + 1 8ε Π ε (F ε E)(H (s) + εh (s))i (σ 1 e x 1,σ e x ), ( s = ex 1 e x ). ε Therefore, according to the assumption (6.) (6.4) the inequalities (.9), (.1), we have Combining (6.8) (6.11) using the maximum principle, we have Λg ε 0. (6.11) w = Λ ˆv 0.

10 36 LISHANG JIANG THEOREM 6.1 If σ i = (1+ε i )σ i, ε i 0 (i = 1, ) (ε 1,ε ) D, where D satisfies (6.) (6.4), then for the American call-max option V (s 1, s, t; σ 1, σ ) V (s 1, s, t; σ 1,σ ). (6.1) Proof. Assume ˆv 1 (x 1, x,τ), ˆv(x 1, x,τ)be the solutions of the penalized problem with volatilities σ 1 σ σ 1, σ respectively, where the penalized term is the function (6.1), then the function ŵ =ˆv 1 ˆv satisfies the following problem: ŵ τ L σ ŵ + β (ξ)w = Λ ˆv ŵ τ=0 = 0 where L σ is the differential operator L corresponding to volatilities σ 1 σ. By using the maximum principle we have ŵ 0 according to the result of Lemma 6.1. Taking the limit as δ 0 ε 0, the inequality (6.1) holds. REMARK 6.1 What about the American call-min option? We check the proof of Lemma 6.1. Under the assumption (6.) (6.4), the inequality (6.10) still holds. But in this case, Λg ε = ΛΠ ε ( F ε E) = Π ε ( F ε E)I + ( F εx1, F εx ) 1 8ε Π ε ( F ε E)(H (s) + sh (s))i (σ 1 e x 1,σ e x ). Thus for all (x 1, x ) R the sign of Λg ε is not constant. That means the result of Theorem 6.1 is not true for the American call-min option! But for the American put option, the situation is opposite. Theorem 6.1 holds for the American put-min option, but it is false for the American put-max option. REMARK 6. It is clear that the domain D in the assumption (6.) depends on the correlation constant ρ. Denoting D by D ρ,wehave (1) D ρ1 D ρ if ρ 1 < ρ, () D 0 = R + R +, (3) D ρ D 1 ={(ε 1,ε ) ε 1 = ε, ε i R +, i = 1, } as ρ 1. Finally we study the effect of change in correlation ρ. In view of the definitions of g ε (x 1, x ) g ε x 1 x = Π Π Π ε (F ε E)(F ε ) x1 (F ε ) x + Π ε (F ε E)(F ε ) x1 x ε (E F ε)(f ε ) x1 (F ε ) x Π ε (E F ε)(f ε ) x1 x ε ( F ε E)( F ε ) x1 ( F ε ) x + Π ε ( F ε E)( F ε ) x1 x Π ε (E F ε )( F ε ) x1 ( F ε ) x Π ε (E F ε )( F ε ) x1 x call-max option put-max option call-min option put-min option.

11 ANALYSIS OF PRICING 37 TABLE 1 Max-option Min-option European American European American Call Put Call Put Call Put Call Put Stock price s 1, s Strike price E Time to expiration?? + +?? + + Volatility σ 1,σ ( ) +? +?? +? + Correlation ρ? +??? +???? Risk-free interest rate r Dividend q 1, q ( ) The increments of σ 1 σ have to satisfy conditions (6.) (6.4). (?): the conclusion of monotonicity on this variable is not true. (??): the conclusion of monotonicity on this variable is still unknown. According to (F ε ) x1 x 0, ( F ε ) x1 x 0 (.9), (.1) we have for the put-max option the call-min option g ε 0. x 1 x But for the call-max option the put-min option the sign of consequence of the above fact, we have the following theorem. g ε x 1 x is not constant. As a THEOREM 6. If ρ ρ, then for the European put-max option the European call-min option, the price of option V e (s 1, s, t) satisfies V E (s 1, s, t; ρ) V E (s 1, s, t; ρ). However, this condition is false for the European call-max option the European put-min option. Whether the result is true for the American put-max option American call-min option remains an open question. The results obtained in Sections 3 6 are summarized in Table Evolution of optimal exercise boundary in the case of E = 0 In this section the next, we confine ourselves to study of the properties of the optimal exercise boundary only for the American call-max option. We believe that for other options the proof is similar, although the results may be different. We first study an evolution of optimal exercise boundary in the case of E = 0, assume q 1 +q > 0 (wlog, q > 0). In fact, as we know, for the American call option there is not an optimal exercise boundary at all, if q 1 = q = 0. From (1.) (1.5), the price V (s 1, s, t) of the American call-max option is a solution of the

12 38 LISHANG JIANG following problem: { V min t } LV, V max(s 1, s ) = 0 (7.1) V t=t = max(s 1, s ) (7.) V s1 =0 = s (7.3) V s =0 = s 1. (7.4) Set V = s U (7.5) introduce a new variable ξ = s 1 s (7.6) define Then we have u(ξ, t) = U(s 1, s, t). { min u } t L 0u, u max(ξ, 1) = 0 (7.7) u t=t = max(ξ, 1) (7.8) u ξ=0 = 1 (7.9) where L 0 u = 1 (σ 1 ρσ 1σ + σ )ξ u ξ + (q q 1 )ξ u ξ q u. (7.10) THEOREM 7.1 A solution u(ξ, t) of problem (7.7) (7.9) satisfies the following free boundary problem: u t L 0u = 0 ξ 1 (t) <ξ<ξ (t), 0 t < T (7.11) u ξ=ξ1 (t) = 1, u ξ = 0, 0 t < T (7.1) ξ=ξ1 (t) u ξ=ξ (t) = ξ (t), u ξ = 1, 0 t < T (7.13) ξ=ξ (t) where ξ 1 (t), ξ (t) are free boundaries ξ 1 (t), ξ (t) (7.14) ξ 1 (T ) = ξ (T ) = 1. (7.15)

13 ANALYSIS OF PRICING 39 Proof. Assume C to be a continuation region of the American call-max option (E = 0), C to be the boundary of C. First, we claim T C = C {t = T }=Point(1, T ). (7.16) In C,asweknow,u(ξ, t) is the solution of the problem (7.7) (7.9). If (ξ, T ) T C ξ = 1, then from (7.7) the initial data (7.8) we have { u q 1 ξ ξ > 1 t = L 0 (max(1,ξ))= (7.17) (ξ,t ) q ξ<1. Therefore in the neighborhood of (ξ, T ), (ξ = 1), u(x, t) <max(1, ξ), (x, t) C. The contradiction shows that (7.16) is true. Thus there exist two free boundaries which start from (1, T ). The property (7.14) is a consequence of the following fact: u t The proof of (7.18) is same as for Lemma (7.18) We next study the asymptotic behavior of a solution to the parabolic obstacle problem (7.1) (7.4) as T. Consider an elliptic obstacle problem as follows. Find a triple {û 0,ξ1 0,ξ0 } such that L 0 û 0 = 0, ξ1 0 <ξ<ξ0 (7.19) u 0 (ξ1 0 ) = 1, u 0 (ξ 1 0 ) = 0 (7.0) u 0 (ξ 0 ) = ξ 0, u 0 (ξ 0 ) = 1 (7.1) where L 0 u is defined by (7.10). Problem (7.19) (7.1) is a second-order ordinary differential equation with two free boundary points. It has a solution where α ± are roots of the following quadratic equation: û = c 1 (ξ/ξ 0 )α + + c (ξ/ξ 0 1 )α (7.) 1 σ α(α 1) + (q q 1 )α q = 0 (σ = α 1 ρα 1α + α ) α < 0, α + > 1 α ± = w ± θ (7.3)

14 40 LISHANG JIANG where θ = w = σ (q 1 q ) (7.4) ω + q σ = q 1 σ + (q 1 q ) σ 4. The coefficients c 1, c free boundary points ξ1 0,ξ0 (7.1). Therefore we get the solution are defined by boundary conditions (7.0), α û 0 = β α α + ξ 0 1 = β 1 α+ α+ α ξ 0 = β α+ α+ α α+(1 α ) α+ α α 1 α+ α + β α α+ α + β β α+(1+α+) α+ α + ξ α + + α + α + α β α (α+ 1) α (1 α ) α+ α + β α+ α ξ α (7.5) (7.6) (7.7) where β ± = (α ± 1)/α ±. Thus we have proved the following theorem. THEOREM 7. The elliptic obstacle problem min{l 0 u 0, u 0 max(1,ξ)}=0 (7.8) u 0 (0) = 1, (7.9) has a solution ξ ξ ξ 0 u 0 (ξ) = û 0 (ξ) ξ1 0 ξ ξ ξ ξ1 0 (7.30) where the triple {û 0,ξ1 0,ξ0 } is a solution of the free boundary problem (7.19) (7.1), which is expressed in (7.5) (7.7). Denoting the pricing of the American option the optimal exercise boundary in the problem (7.11) (7.13) for the expiration time T by u(ξ, t; T ) ξ i (t; T ), then we have the following theorem. THEOREM 7.3 The solution u 0 (ξ) (7.30) is a limit of the solution u(ξ, t; T ) of the problem (7.7) (7.9) (also (7.11) (7.13)) as T, i.e. t < T lim T u(ξ, t; T ) = u 0(ξ), ξ R (7.31) lim ξ i(t; T ) = ξi 0, (i = 1, ). (7.3) T

15 ANALYSIS OF PRICING 41 In order to prove this, we make a transformation τ = T t in the problem (7.7) (7.9) study the asymptotic behavior of the solution as τ. The proof of the theorem is stard. We omit it here. THEOREM 7.4 Let V (s 1, s, t) be the price of the American call-max option (strike price E = 0), then the optimal exercise boundary consists of two surfaces s 1 = ξ 1 (t)s s 1 = ξ (t)s. The continuation region is {ξ 1 (t) < s 1 s <ξ (t), (s 1, s ) R + R +, 0 t T } the stopping region 0 = (1) 0 () 0, (1) 0 () 0 ={0 s s 1 /ξ (t), 0 t T,(s 1, s ) R + R + } ={0 s 1 s ξ 1 (t), 0 t T,(s 1, s ) R + R + } (7.33) where ξ i (t)(i = 1, ) are the free boundaries of problem (7.11) (7.13). As T, the price of perpetual American call-max option (strike price E = 0) is s 1 s 1 ξ 0 s α+(1 α ) α α+ α β β α+(1+α+) α+ α + s α + α V (s 1, s ) = α 1 s1 α α α (α+ 1) α (1 α ) + α+ α α+ α β+ β s α α + α 1 s1 α ξ1 0 s 1 ξ 0 s s 0 s 1 ξ1 0 s. (7.34) Theorem 7.4 is the consequence of Theorems according to the transformations (7.5), (7.6). 8. Properties of the optimal exercise boundary Let us turn to the American call-max option in the general case E > 0, study the properties of the optimal exercise boundary. Suppose (t; E) to be the stopping region of the American call-max option, i.e. (t; E) ={(s1, s ) V (s 1, s, t) = G E (s 1, s ), s 1 R +, s R + } (8.1)

16 4 LISHANG JIANG where G E (s 1, s ) = (max(s 1, s ) E) +. The optimal exercise boundary Γ (t; E) = Γ 1 (t; E) Γ (t; E), where Γ 1 (t; E) = (t; E) {s 1 s } Γ (t; E) = (t; E) {s > s 1 }. (8.) It is clear that (T ; E) ={(s1, s ) LG E (s 1, s )<0}. (8.3) In fact, if (s1 0, s0, T ) belongs to the continuation region LG E (s 0 1,s 0 ) < 0, then we have V t (s1 0,s0,T ) = LG E (s 0 1,s 0) < 0 which means that there exists t 0 < T, such that (s1 0, s0, t 0) belongs to the continuity region, V (s 0 1, s0, t 0)<V (s 0 1, s0, T ) = G E(s 0 1, s0 ). That is impossible. Therefore strightforward calculation from (8.3) shows that for the American call-max option the stopping region at expiration time t = T is { ( (T ; E) = (s 1, s ) s 1 s, s 1 max E, Er ) ( s 1 s, s max E, Er )}. (8.4) q 1 q What about the evolution of the optimal exercise boundary (stopping region) as t < T? THEOREM 8.1 (t ; E) (t 1 ; E) as t > t 1 (8.5) i.e. the free boundary is monotonic. This is a consequence of the estimate (5.) V t 0. THEOREM 8. (t; E1 ) (t; E ), (E 1 > E ). (8.6) Proof. If this is not true, there exists a point (s1 0, s0 ) time t 0 such that (s1 0, s0, t 0) (t 0 ; E 1 ) but (s1 0, s0, t 0)/ (t 0 ; E ). Then, in view of (3.4), we have G E1 (s1 0, s0 ) = V (s0 1, s0, t 0; E 1 ) V (s1 0, s0, t 0; E ) E 1 + E > G E (s1 0, s0 ) E 1 + E

17 ANALYSIS OF PRICING 43 i.e. Consequently, max(max(s 0 1, s0 ) E 1, 0) + E 1 > max(max(s 0 1, s0 ) E, 0) + E. max(s 0 1, s0 )<E 1 but in view of (8.4) (8.5), ( ( max(s1 0, s0 ) min max E 1, re ) ( 1, max E 1, re )) 1 q 1 q according to t 0 < T (s 0 1, s0, t 0) (t 0 ; E 1 ). The contradiction shows that REMARK 8.1 (s 0 1, s0, t 0) (t; E ) (t; E) (t; 0). (8.7) Due to (7.34) (8.7), we can give an evaluation figure of the optimal exercise boundary to the American call-max option. At the initial time τ = 0 (expiration time t = T ), the stopping region is (T ; E) (see (8.4)), then as τ>0(t < T ) the region is separated at the ray s 1 = s into two parts, each subregion goes to opposite side monotonical with respect to time τ. One of the subregion in {s 1 > s } is denoted by (1) (t) the other in {s 1 < s } is denoted by () (t). THEOREM 8.3 The price V (s 1, s, t) of American call-max option as a function of s 1, s is convex. The stopping subregions (1) (t) () (t) are convex sets. Proof. The convexity of the subregion (i) i = 1, is the consequence of the convexity of solution V (s 1, s, t). In fact, wlog, we assume P(s 1, s, t), P 0 (s 0 1, s0, t) (1) (t) P λ : (s λ(s 1 s 0 1 ), s0 + λ(s s 0 ), t), P λ PP 0,(0 λ 1). According to the convexity of V (s 1, s, t), wehave 0 (1 λ)g E (s1 0, s0 ) + λg E(s 1, s ) V (s1 0 + λ(s 1 s1 0 ), s0 + λ(s s 0 ), t). (8.8) In view of P, P 0 (1) (t), s 1 > s, s1 0 > s0 (8.5) ( s 1 > max E, Eq ( 1 ), s 01 r > max E, Eq ) 1. r So G E (s 1, s ) = s 1 E, G E (s 0 1, s0 ) = s0 1 E.

18 44 LISHANG JIANG Therefore, by virtue of (8.8), V (s1 0 + λ(s 1 s1 0 ), s0 + λ(s s 0 ), t) (1 λ)(s0 1 E) + λ(s 1 E) = s1 0 + λ(s s0 1 ) E = G E(s1 0 + λ(s 1 s1 0 ), s0 + λ(s s 0 )) i.e. Pλ (1),(0 λ 1). To verify the convexity of the solution of problem (1.) (1.5), we consider a penalized problem (.16), (7.17) at first claim J(ˆv; α 1,α ) = α1 ˆv ˆv x1 + α 1 α + α ˆv x 1 x x α1 ˆv α x 1 for any α 1,α 0. This is a consequence of the estimate (6.5), where σ 1 σ 1 = σ σ =,α i = 3 σ i ρ = 1. Set ˆV (s 1, s, t) =ˆv(ln s 1, ln s, t). According to (8.9), a straight calculation shows ˆv 0 (8.9) x τ ˆV (s 1 + τ s 1, s + τ s, t) = τ ˆv(ln(s 1 + τ s 1 ), ln(s + τ s ), t) ( ) s ( 1 ˆv = s 1 + τ s 1 x1 ˆv ) ( )( ) s 1 s ˆv + x 1 s 1 + τ s 1 s + τ s x 1 x ( ) s ( ˆv + s + τ s x ˆv ) ( ) s 1 s = J ˆv;, 0, x s 1 + τ s 1 s + τ s (1 λ) ˆV (s 1, s, t) + λ ˆV (s 1 + s 1, s + s, t) ˆV (s 1 + λ s 1, s + λ s, t) 1 = λ (1 τ) τ ˆV (s 1 + τ s 1, s + τ s, t) dτ λ + (1 λ) λ 0 τ τ ˆV (s 1 + τ s 1, s + τ s, t) dτ 0 for any λ [0, 1]. Taking the limits as δ 0 ε 0, the proof of Theorem 8.3 is complete. Finally, we study the asymptotic behaviour of the optimal exercise boundaries as s 1 or s. According to the convexity of the stopping regions the boundary conditions (1.4) on s 1 = 0 (1.5) on s = 0, optimal exercise boundaries Γ i (t; E), (i = 1, ) can be expressed in the form Γ 1 (t; E) s = ϕ 1 (s 1, t) Γ (t; E) s 1 = ϕ (s, t). (8.10) THEOREM 8.4 The optimal exercise boundaries Γ i (t; E)(i = 1, ) have the following properties: (A) ϕ 1 (s 1, t) ϕ (s, t) are monotonically increasing in s 1 s, (B) ϕ 1 (s 1, t) ϕ (s, t) are concave in s 1 s,

19 ANALYSIS OF PRICING 45 (C) ϕ 1 (s 1, t) lim = (ξ (t)) 1 (8.11) s 1 s 1 ϕ (s, t) lim = ξ 1 (t), (8.1) s s where ξ 1 (t), ξ (t) is defined in (7.11) (7.14), s 1 = ξ i (t)s (i = 1, ) are optimal exercise boundaries in the case of strike price E = 0. Proof. (A) Rescaling s 1 s s 1 = E s 1, s = E s, introducing a new function V = EṼ then we have Ṽ as a function of s 1 s is a solution of the problem (1.) (1.4) with the strike price E = 1, in which the payoff function is G( s 1, s ) = (max( s 1, s ) 1) +. Assume that Γ 1 (t) : s = ϕ 1 ( s 1, t) Γ (t) : s 1 = ϕ ( s, t) are optimal exercise boundaries of the American call-max problem (1.) (1.4) with E = 1. Then we have ( ) si ϕ i (s i, t) = E ϕ i E, t (i = 1, ). (8.13) Therefore we only need to prove the theorem in the case of E = 1. Due to (8.6), if E 1 > E, ( ) ( ) si si E ϕ i, t > E 1 ϕ i, t (i = 1, ) E E 1 i.e. if s i E > s i E 1, for s i > 0, ϕ i ( si E, t ) > E ( ) ( ) 1 si si ϕ i, t > ϕ i, t E E 1 E 1 (i = 1, ). (8.14) So ϕ i (s i, t) is monotonically increasing in s i. (B) is a consequence of Theorem 8.3. Now we turn to prove the property (C). We know (see (3.6)) that lim V (s 1, s, t; E) = V 0 (s 1, s, t) E 0 where V 0 is a solution of problem (7.1) (7.4). As a consequence, we have lim ϕ i(s i, t) = ϕi 0 (s i, t), (i = 1, ) (8.15) E 0 where Γ 1 (t; 0) : s = ϕ1 0(s 1, t) Γ (t; 0) : s 1 = ϕ 0(s, t). In view of (8.13) for fixed s i, s i = E s i, lim s i ϕ i ( s i, t) s i ( ) E si = lim ϕ i E 0 s i E, t ϕ i (s i, t) = lim = ϕ0 i (s { i, t) (ξ (t)) 1 i = 1 = E 0 s i s i ξ 1 (t) i = according to (7.34). The proof of Theorem 8.4 is complete.

20 46 LISHANG JIANG 9. Conclusions In accordance with the results mentioned above, even if we have not got an explicit form of the solution, we still can draw a figure of the optimal exercise boundary to the American max-call option as follows: S Γ (t) S 1 =ξ 1 (t) S Σ () (t) S 1 =ξ (t) S Γ 1 (t) Σ (1) (t) O S 1 Acknowledgements The work has been supported by the National Foundation of China. The author would like to thank the referee for pointing out the recent work [10], in which the non-emptiness the shape, especially the asymptotic behaviour near the expiry date of the stopping region, of American options on several assets are studied. REFERENCES 1. BROADIE, M.&DETEMPLE, J. The valuation of American options on multiple assets. Math. Finance 7, (1997) STULZ, R. M. Options on the minimum or the maximum of two risky assets. J. Financial Economics 10, (198) JOHNSON, H. Options on the maximum or the minimum of several assets. J. Financial Quantitative Analysis, (1987) HULL, J. C. Options, Futures their Derivatives, 3rd edn. Prentice-Hall, Englewood Cliffs, NJ (1997). 5. KINDERLEHRER, D. & STAMPACCHIA, G. An Introduction to Variational Inequalities their Applications. Academic Press, New York (1980). 6. FRIEDMAN, A. Variational Principles Free Boundary Problems. Wiley-Interscience, New York (198). 7. DEWYNNE, J. N., HOWISON, S. D., RUPF, I., & WILMOTT, P. Some mathematical results in the pricing of American options. Eur. J. Appl. Math. 4, (1993) MARGRALE, W. The value of an option to exchange one asset for another. J. Finance 33, (1978) RUBINSTEIN, M. One for another. Risk (1991). 10. VILLENEUVE, S. Exercise regions of American options on several assets. Finance Stochastics 3, (1999) 95 3.