\Notes" Yuri Y. Boykov. 4 August Analytic approximation of. In this chapter we apply the method of lines to approximate values of several

Transcription

1 \Notes" Yuri Y. Boyov 4 August 1996 Part II Analytic approxiation of soe exotic options 1 Introduction In this chapter we apply the ethod of lines to approxiate values of several options of both European and Aerican styles. In particular, we apply this ethod to options on stoc with constant continuous dividend pay-out and to Asian 1 options. The ethod is analytic in its nature. We nd an exact general solution to a recursive syste of non-hoogeneous ordinary dierential equations (ODE's) that approxiates values of these options. However, we need to use a nuerical procedure to select certain coecients fro the boundary conditions. As the actual tests show, this ethod appears to converge to the unnown exact values of the options in a coputationally ecient anner. An option is a nancial security that gives to a purchaser the right to buy or, depending on the contract, the right to sell a xed nuber of shares of a particular stoc for a certain price at soe future date or dates. If this right is used on a particular date, then it is coon to say that the option is exercised. The aount of oney that a purchaser has to pay for the contract 1 Asian options are soeties referred to as integral options or options on average. 1

2 is called the price of an option. Options can be of European or Aerican styles. The distinction between European and Aerican styles of options can be ade according to the rules for the option's exercise. An exercise of a European option is allowed only at a aturity date specied by the contract. An Aerican option exercise can occur at any date until its expiration that is also referred to as a aturity date. Under certain assuptions, an option value can be described by a partial dierential equation (PDE) 2. If two options are identical except that one is European and the other is Aerican, then their values are described by the sae dierential equations. The only dierence is in the corresponding boundary conditions. To value an option, one can try to solve the corresponding PDE under appropriate boundary conditions. In any cases, however, an analytic solution can not be found unless soe approxiations are ade. Nuerical ethods such as nite dierences and Monte Carlo siulation can be applied in situations where neither analytic solutions nor reasonable analytic approxiations are available. No exact analytic solution is nown for the PDE's for options on stocs with constant continuous dividend pay-out and for Asian options based on the arithetic ean. The papers related to the subject of options on dividend paying stocs consider discrete dividends [3], [9], [13], [14] or continuous proportional dividends [4]. Merton [12] considered the valuation of options on stocs with constant continuous dividend pay-out. He gave a sucient condition for no early exercise of Aerican options and also valued perpetual options. However, he was unable to value or approxiate nite-lived options. The case of Asian options based on the geoetric ean is copletely solved in [6] and [10]. The perpetual case of arithetic ean options is solved in [11]. In soe papers analytic approxiations are developed for nitelived Asian options based on the arithetic ean but these results do not provide any indication of their precision. Eydeland and Gean [7] suggest a nuerical procedure to invert the Laplace Transforation of arithetic ean options developed in [8]. Even though the options on average asset values could be approxiated by the ethod of Monte Carlo [10], accurate analytic estiation of these options is still an open question. To value options that do not have an exact analytic solution, one can try the ethod of lines that was rst introduced to the nance literature by P. 2 We use [15] as a reference boo for PDE's in nancial applications. 2

3 Carr and D. Faguet [5]. The ain idea of this ethod is to approxiate a partial derivative with respect to tie in the corresponding PDE by a nite dierence and then to solve the resulting sequence of non-hoogeneous ODE's. The approxiation error of this ethod is deterined by the length of tie increents used for nite dierences. Richardson extrapolation can be applied to accelerate convergence of the approxiation errors to ero. In [5], P. Carr and D. Faguet obtained a closed for solution to the sequence of ODE's approxiating Aerican options on stocs that do not pay dividends. They also have a sei-analytic approxiation for Aerican options on stocs paying continuous proportional dividends. We will present a general analytic solution for a sequence of ODE's approxiating options on stoc with constant continuous dividend pay-out and Asian options based on the arithetic average. These two types of options are approxiated by siilar systes of dierential equations 3. In section 5 we nd their general solution. This solution is a generaliation of the conuent hyper-geoetric functions. To value an option in a particular case, we suggest an algorith that utilies the general solution and uses nuerical ethods to incorporate the eect of the boundary conditions in the solution. Although the obtained solution does not have a closed for, it has the nice property that in practice it can be applied to both European and Aerican styles of the options. Our general result could also be used to value soe interest rate sensitive options. 2 Preliinaries This section explains our terinology and reviews soe basic properties of a recursive syste of non-hoogeneous ODE's of second order. Such a syste arises in our approxiation schee. Suppose that L() is a second order linear dierential operator dened on functions in C 2. For exaple we can tae a second order linear operator L() dened by the equation L(Y ) := S 2 Y ss + (2S? 3)Y s? 4Y where Y is a twice continuously dierentiable function of S and Y s, Y ss are correspondingly the rst and the second derivatives of Y. The operator L() is of the second order because it contains the second derivative of Y. 3 Copare the systes of equations in Leas 3 and 5. 3

4 The operator L() can be used to dene a recursive syste of nonhoogeneous second order ODE's L(Y (n) ) = Y (n?1) ; n 1 (1) where fy (n) g 1 n=0 is a sequence of functions in C 2 and is a nown constant. Note that the non-hoogeneous part of the n th equation (1) contains the function Y (n?1), that is, the solution of this recursive syste fro the previous iteration. To coplete the description of the syste we ust specify the initial condition Y (0) = U (2) where U is a given function of S. The general solution of the syste (1), (2) can be obtained as follows. Suppose that functions A 0 and B 0 are any two linearly independent solutions of the corresponding hoogeneous equation L(Y ) = 0: Also assue that we now one particular sequence of functions fu n g 1 n=0 that solves (1) under initial condition (2). That is, U 0 = U and L(U n ) = U n?1 ; 8n 1: Then the general solution of equation (1) for n = 1 is Y (1) = 1 A B 0 + U 1 where 1 and 1 are arbitrary real nubers. To gure out the general solution of equation (1) for n = 2, we ust solve the non-hoogeneous ODE L(Y (2) ) = ( 1 A B 0 + U 1 ) : Suppose that soe functions A 1 and B 1 satisfy L(A 1 ) = A 0 and L(B 1 ) = B 0. Then the general solution of (1) for n = 2 is Y (2) = ( 2 A B 0 ) + ( 1 A B 1 ) + U 2 where 2 and 2 are arbitrary constants. The following lea extends this to an arbitrary n. 4

5 Lea 1 To specify the general solution of the recursive syste of ODE's (1) under initial condition (2) it is sucient to nd soe sequences of functions fa n g, fb n g, and fu n g for n 2 f0; 1; 2; : : :g such that (a) A 0 and B 0 are independent solutions of L(Y ) = 0 (b) U 0 = U (c) L(A n ) = A n?1, L(B n ) = B n?1, and L(U n ) = U n?1 for any n 1. If such functions are available then the general solution of (1), (2) is Y (n) = n i=1 i n?i A n?i + n i=1 i n?i B n?i + U n 8n 0 where coecients i and i are arbitrary real nubers. Proof: The induction arguent gives L(Y (n) ) = = n i=1 n?1 i=1 = Y (n?1) and the proof is done. i n?i L(A n?i ) + i n?i A n?1?i + n i=1 n?1 i=1 i n?i L(B n?i ) + L(U n ) = i n?i B n?1?i + U n?1 = Recursive systes of non-hoogeneous second order ODE's siilar to (1), (2) will be used in the following sections to approxiate PDE's arising in the option valuation probles. Lea 1 deterines the structure of the general solutions of these approxiating systes. 3 Options on stoc with constant dividends In this section we consider options on a stoc that pays a constant continuous dividend. First, we derive the PDE that corresponds to options of this type. Then we approxiate this PDE by a recursive sequence of non-hoogeneous ODE's. 5

6 Suppose that t represents continuous tie and S stands for a current underlying stoc price. If d denotes a dividend pay-out over a unit of tie then our odel for the stoc price evolution is given by the following stochastic dierential equation ds = Sdt + SdW? d dt if S > 0 ds = 0 if S = 0; (3) where W is a Brownian otion, is an average growth rate, and is a paraeter that represents volatility of the underlying stoc. Note that this odel allows for banruptcy in contrast to ost other odels of price processes for dividend paying stocs 4. To be specic, we consider a standard European call option on a stoc that follows (3). The ain property of the European call option is that it gives to its holder the right to buy one share of stoc at a xed price K when the option reaches its aturity date T. The price K and the aturity T are specied at the contract initiation. K is usually referred to as the strie price. Suppose that function C(; S) denotes the value of the call when there are = T? t units of tie left until the option's aturity and if the current stoc price is S. The value of the European call option at its aturity is 5 C(0; S) = (S? K) + = ax(0; S? K): (4) There are other conditions which the value of the European call has to satisfy: C(; 0) = 0 (5) C(; S) = O(S) as S! +1: (6) Equations (4), (5), and (6) are the boundary conditions for C(; S). We now derive a PDE for the function C(; S) 6. Fro It^o's rule and fro (3), we conclude that for S > 0 and for 0 = (T? t) T the call value 4 (3) is also a reasonable description for the value of the assets of a r which has debt outstanding. If the debt accrues interest continuously and the assets are liquidated continuously to nance the interest pay-out of d, then (3) describes the asset value until the debt is fully paid o. 5 In fact, condition (4) holds for Aerican calls too. 6 We follow Merton's paper [12]. 6

7 C(T? t; S) satises dc =?C dt + C s ds C sss 2 2 dt = =?C dt + C s Sdt + C s SdW? C s d dt S2 2 C ss dt: (7) At tie t we can for a portfolio that consists of one call and?c s (T? t; S) shares of stoc. The value of this portfolio is V = C? C s S: Since the stoc pays constant continuous dividends then at tie t dv = dc? C s (ds + d dt) = dc? C s (Sdt + SdW ): As follows fro (7) dv =?C? C s d S2 2 C ss dt: Therefore, at tie t the portfolio V is risless and the assuption of no arbitrage would iply that it should grow at the sae rate as a risless ban account would grow. That is, dv = rv dt = r (C? C s S) dt where r is an interest rate that we assue to be a nown constant paraeter. The last two equations iply that C(; S) satises the PDE obtained by Merton in [12] 1 2 S2 2 C ss + (rs? d) C s? rc = C : (8) Note that the sae PDE wors for Aerican style calls 7 and for put options 8 as long as the underlying stoc follows odel (3). We will use PDE (8) as a starting point of our search for the value of the corresponding options. Choose an arbitrary positive integer N and tae 7 The doain of this PDE for Aerican style options is the continuation region. 8 A standard put is a right to sell one share of soe stoc at a xed strie K. The value of a put upon its exercise is (K? S) +. 7

8 = T. We will concentrate on the values of C(; S) at ties = n N where n is a nonnegative integer which is less than or equal to N. Let (S) denote our approxiation of C(n; S): (S) C(n; S); 8n 2 f0; : : : ; Ng: The ethod of lines suggests the approxiation C (n; S) (S)? C (n?1) (S) : Then (8) yields the recursive syste of non-hoogeneous ODE's where S 2 ss + (S? ) s? ( + ) =? C (n?1) (9) = 2r ; = 2d 2 ; = ; Fro (4) we now that the initial condition is n 2 f1; : : : ; Ng: C (0) (S) = (S? K) + : (10) Since the total nuber of tie steps N could be arbitrarily large then we should solve (9) for all integers n 1. The structure of the general solution of the syste (9), (10) is given by Lea 1. First, we will nd one particular sequence of functions fu n g that satises equation (9) and the initial condition U 0 (S) = 8 < : S? K if S K 0 if 0 < S < K: The general solution of the syste (9), (10) could be found separately on the intervals 0 < S < K and S K. On the interval S 2 (0; K) we can obviously tae U n = 0 for all n. The next lea provides us with a sequence of functions that wors for S 2 [K; +1). Lea 2 The sequence of linear functions U (S) = S + u n where u 0 =?K; u n = u n?1? d 1 + r (n 1) satises equation (9) and the initial condition C (0) U (S) = S? K. 8

9 Proof: Substitute the forula for U into (9) and use induction. Lea 1 iplies that we also have to nd two sequences of functions f A g and f B g such that C (0) A and C (0) B are independent solutions of the hoogeneous ODE S 2 C (0) ss + (S? )C (0) s? ( + )C (0) = 0 (11) and such that each of these two sequences satises the recursive equation S 2 ss + (S? ) s? ( + ) = C (n?1) ; n 1: (12) Specic forulas for functions A and B will be obtained in the end of section 5 9. The following lea provides a substitution that transfors the syste of equations (11), (12) to a standard for. Lea 3 Suppose that = and ( ) = S e? p h (n) () where p is either p 1 or p 2 :? 1 s? 1 2 p 1;2 = + ( + ): 2 2 Then the sequence f g satises equation (12) and the initial condition (11) if and only if the sequence fh (n) g satises the recursion h (n) and the initial condition + (b? )h (n) where a = p + 2? and b = 2p + 2?.? ah (n) = h(n?1) (13) h (0) + (b? )h (0)? ah (0) = 0 (14) Proof: It is easy to chec that if = S and (S) = e? p h (n) () then s (S) =? p+1 ss (S) = p See equations (49) and (50). e? h (n) () + (p? )h (n) () h e? 2 h (n) () + 2(p + 1? )h (n) () (p? 2)(p + 1) h (n) () i : 9

10 Substituting these forulas into (12) gives 2 h (n) + h(n) (b? ) + h (n)?a + p 2 + p(1? )? ( + ) = h (n?1) : This equation copletes the proof of Lea 3 since p 1;2 are the roots of the quadratic equation p 2 + p(1? )? ( + ) = 0. 4 Asian options Consider now an Asian option based on the arithetic ean. To be specic we consider a European style put struc at the average. This option is a nancial security which gives to an owner a right to sell one share of soe stoc upon aturity T for the average price that this stoc exhibits fro the contract initiation until its expiration at T. For siplicity, we will assue that the stoc does not pay any dividends 10 and that its price satises the stochastic odel ds = Sdt + SdW for S > 0: (15) Suppose that I(t) denotes the following integral: I(t) = Z t 0 S(t 0 )dt 0 : One can show 11 that the value of the put on the average is a function of the tie to aturity = T? t, the stoc price S, and the integral I. We will denote this function by A(; S; I). The value of this option at aturity = 0 is I + A(0; S; I) = T? S : (16) Ito's rule and equation (15) iply that for 0 = T? t T da =?A dt + A I di + A s ds S2 2 A ss dt = =?A dt + A I Sdt + A s Sdt + A s SdW S2 2 A ss dt: 10 Continuous proportional dividends are easily handled. 11 See [15] for reference. 10

11 At tie t we for a portfolio that consists of one option and?a s (T? t; S; I) shares of stoc. The value Z of this portfolio is We see that at tie t dz = da? A s ds = Z = A? A s S:?A + A I S S2 2 A ss dt: Applying the sae no arbitrage condition as before, we obtain dz = rzdt = r (A? A s S) dt: The last two equations yield the following PDE for A(; S; I): SA I + rsa s S2 2 A ss? ra = A : It is convenient to ae a substitution A(; S; I) = S V (; ); (17) where = I. The function V (; ) satises the PDE S 1 2 Condition (16) iplies that 2 2 V + (1? r )V = V : (18) V (0; ) = T? 1! + : (19) Note that equation (18) will wor for Aerican style puts struc at the average 12 and for European and Aerican style calls struc at the average 13. An equation siilar to (18) is considered by Kraov and Mordeci in [11]. This paper solves the perpetual case of the integral options where the tie derivative is irrelevant. Kraov and Mordeci obtained the exact analytic solution of (18) in the case where V = Doain for the PDE is the continuation region. 13 A European call struc at the average pays (S? I T )+ at its exercise at tie = 0. 11

12 We will approxiate (18) using the ethod of lines. Tae an arbitrary positive integer N and = T. Let V (n) ( ) denote our approxiation of N V (n; ): V (n) ( ) V (n; ); 8n 2 f0; : : : ; Ng: Assue that V (n) ( )? V (n?1) ( ) V (n; ): Then (18) yields the following approxiation 2 V (n) + (? )V (n)? V (n) =?V (n?1) (20) where = 2r ; = 2 2 ; = ; Fro (19) we derive the initial condition n 2 f1; : : : ; Ng: V (0) ( ) = T? 1! + : (21) We want to nd a general solution of the recursive syste of ODE's (20), (21) for all integers n 1. As follows fro Lea 1, we need to nd one particular sequence of functions fu n g that satises (20) and the initial condition U 0 ( ) = 8 < : T? 1 if > T 0 if 0 T: We will nd a general solution of the syste (20), (21) separately on the intervals 2 [0; T ] and 2 (T; +1). On the rst interval we obviously can tae U n ( ) = 0 for all n. A sequence of functions that wors for the interval 2 (T; +1) is given in the next lea. Lea 4 The sequence of linear functions V (n) U ( ) = p n + q n where p 0 = 1 T ; p n = p n?1 1+r q 0 =?1; q n = q n?1 + p n?1 1+r (n 1) satises equation (20) and the initial condition V (0) U ( ) = T? 1. 12

13 Proof: Substitute the forula for V (n) U into (20) and use induction. Lea 1 also requires soe sequences of functions fv (n) A g and fv (n) B g such that V (0) A and V (0) B are independent solutions of the hoogeneous ODE 2 V (0) + (? )V (0)? V (0) = 0 (22) and such that each of these two sequences satises equation 2 V (n) + (? )V (n)? V (n) = V (n?1) ; n 1: (23) Specic forulas for functions V (n) and F (n) are found in the end of section The next lea converts the syste of equations (22), (23) to a standard for. Lea 5 Suppose that = and V (n) ( ) = p h (n) () where p is either p 1 or p 2 : + 1 s p 1;2 =? + : 2 2 Then the sequence fv (n) g satises equation (23) and the initial condition (22) if and only if the sequence fh (n) g satises the recursion h (n) and the initial condition + (b? )h (n)? ah (n) = h(n?1) (24) where a = p and b = 2p h (0) + (b? )h (0)? ah (0) = 0 (25) Proof: It is easy to chec that if = and V (n) ( ) = p h (n) () then V (n) ( ) =? p+1 V (n) ( ) = p+2 2 h (n) 14 See equations (51) and (52). 2 h (n) () + ph (n) () () + 2(p + 1)h(n) () + p(p + 1)h (n) () : 13

14 Substituting these forulas into (23) gives 2 h (n) + h (n) (b? ) + h (n)?a + p 2 + p(1 + )? = h (n?1) : This equation copletes the proof of Lea 5 since p 1;2 are the roots of the quadratic equation p 2 + p(1 + )? = 0. Note that the syste of equations (24), (25) is identically the sae as the syste of equations (13), (14) in Lea 3. 5 General solution of approxiating ODE's In this section we establish two sequences of functions fh (n) g and fg (n) g, n 0, that satisfy recursive equation (13) and such that H (0) and G (0) are independent solutions of the corresponding hoogeneous equation (14). As follows fro Leas 1, 3, and 5, this would coplete the search for the general solutions of the recursive systes of non-hoogeneous ODE's (9), (10) and (20), (21). The hoogeneous equation (14) h + (b? )h? ah = 0 is the well nown Kuer's equation. In Abraowit and Stegun [1], we nd that its independent solutions are H (0) () = M(a; b; ) (26) G (0) () = 1?b M(1 + a? b; 2? b; ): (27) Note that M(a; b; ) is a standard notation for Kuer's function 15 : M(a; b; ) = 1 i=0 (a) i i (b) i i! where (a) i = a(a + 1)(a + 2) : : : (a + i? 1); (a) 0 = 1. Therefore, the proble is to nd fh (n) g and fg (n) g, for n 1, that satisfy the recursive syste of ODE's (13) with the initial eleents H (0) and 15 Kuer's function belongs to the class of conuent hyper-geoetric functions. 14

15 G (0) as in (26) and (27). It is convenient to introduce notation for the linear operator on the left hand side of (13). Denote Then (13) can be rewritten as ^L () := () + (b? ) ()? a () : ^L h (n) = h (n?1) ; n 1: (28) Theore 1 Suppose that functions, 0 n, satisfy the recursive syste of equations f (0) 0 () = h (0) () = M(a; b; ) (29) ^L 0 = Fh 1 n 1 (30) n = f (n?1) n?1 n(b? 1) = f (n?1)?1 (b? 1) (n) ( + 1)f+1? (b? 1) n 1 (31) 1 n? 1 (32) where F h () :=?2() + () is a rst order linear operator. Then H (n) = n =0 solves the syste of equations (26), (28). (ln ) (33) Proof: First we want to show that equations (29)-(32) iply ( + 1)F h +1 0 n? 1 ^L 8 < = : 0 = n: (34) Property (34) is obvious for = 0 due to (30). Equations (29) and (31) iply that n is equal to Kuer's function M(a; b; ) ultiplied by soe constant. Therefore (34) holds for = n. One can prove (34) for other values 15

16 of by induction. Assue that f (n?1) satises (34) for all 0 n? 1. Then chec that for property (34) extends fro = n to all saller values of. Equation (33) agrees with (26) if n = 0. For n 1 we have ^L H (n) = (ln ) n ^L n + (ln ) n?1 ( + n?2 =0 (ln ) ( ^L n?1 + n " b? 1 n #) + 2? n n ^L " b? 1 + ( + 1) ? +1 Equations (34), (31), and (32) conclude the proof of (28). Siilar theore holds for the sequence fg (n) g. ( + 1)( + 2) + # ) +2 : Theore 2 Suppose that functions, 0 n, satisfy the recursive syste of equations f (0) 0 () = g (0) () = 1?b M(1 + a? b; 2? b; ) (35) ^L 0 = Fg 1 n 1 (36) n = f (n?1) n?1 n(1? b) = f (n?1)?1 (1? b) (n) ( + 1)f+1? (1? b) n 1 (37) 1 n? 1: (38) where F g () :=?2 1?b () + () is a rst order linear operator. Then 1?b G (n) = n =0 solves the syste of equations (27), (28). 16 (ln ) (39)

17 Proof: Repeat the steps of the proof of Theore 1. The property ( + 1)F g +1 0 n? 1 ^L 8 < = : 0 = n: wors as an analogue of property (34). We dene the following functions of the arguent : Y (a; b; ) := 1 =0 (a) (b)! () (a; b) 8 0; (40) where the coecients () (a; b) solve the recursive equations (0) (a; b) := 1 0 () (a; b) := 8 >< >: P?1 i=0 (?1) i a+i? 2 (?1) i+1 b+i 1; 1 0 = 0; 1: The functions Y (a; b; ) are generaliations of Kuer's function. Y will serve as a basic function for constructing sequences fh (n) g and fg (n) g that solve (28). Lea 6 The functions Y (a; b; ) have the properties: 8 < : 8 < : Y 0 (a; b; ) = h (0) = M(a; b; ) ^L (Y (a; b; )) = F h (Y?1 (a; b; )) 1?b Y 0 (a 0 ; b 0 ; ) = g (0) = 1?b M(a 0 ; b 0 ; ) ^L 1?b Y (a 0 ; b 0 ; ) = where a 0 = 1 + a? b and b 0 = 2? b. =0 (41) F g 1?b Y?1 (a 0 ; b 0 ; ) (42) Proof: We will prove (41). The upper equation is obvious fro the denition of Y 0. One can chec that for 1 1 (a) F h (Y?1 (a; b; )) = (?1)? 2 a + (b)! b + (?1) +1 : 17

18 Fro [2] we now that that the function +1 f() = ( + 1)( + b) 1 n=0 (1 + + a) n n ( + 2) n ( b) n solves the dierential equation ^L (f) =. Therefore, (41) will be satised if we choose Y (a; b; ) = = 1 =0 1 =1 (a) (b)! +1 ( + 1)( + b) (a) (b)! (?1)? 2 a + b + (?1) +1?1 i=0 0 1 (?1) i (a + + 1) n n = : : : = ( + 2) n (b + + 1) n (?1) i+1 a + i? 2 b + i which agrees with the denition of Y (a; b; ) in (40). The proof of (42) is siilar to the proof of (41). that solve the systes of equa- The following theore gives functions tions in Theores 1 and 2. 1 A Theore 3 Suppose that y is a function of the arguent and that is a real constant. Assue also that functions, 0 n, satisfy 0 = = 1! n =0 n? i=0 R (n) y n 0 (43) (?1) i C () i 2i f (n??i) 0 1 n (44) where are the constants dened by forula (58) in the appendix and are the coecients dened by the equations R (n) C () i R (0) 0 = 1; R (n) 0? any real nubers for n 1 R (n) = n? i=0 (?1) i C (1) i 2i R (n?1?i)?1 1 n: (45) 18

19 If y () = Y (a; b; ) and = b? 1 then functions solve the syste of equations (29)-(32) fro Theore 1. If y () = 1?b Y (1 + a? b; 2? b; ) and = 1? b then functions solve the syste of equations (35)-(38) fro Theore 2. Proof: Suppose that F := F h in case of y () = Y (a; b; ) and that F := F g in case of y () = 1?b Y (1 + a? b; 2? b; ). We have to show that conditions (43) and (44) iply the syste of equations n = f (n?1) n?1 n ^L 0 = f (n?1)?1 = F 1 (n) ( + 1)f+1? n 1 (46) 1 n? 1 (47) n 1: (48) Relations (46) and (47) are proved in the appendix. The proof of (48) is based on Lea 6 which iplies the properties ^L (y 0 ) = 0 ^L (y ) = F (y?1 ) 1: Equation (48) follows fro these properties by eans of induction. Equations (33), (43), and (44) explicitly dene the sequence of functions fh (n) g that satises (26), (28). Equations (39), (43), (44) specify the sequence of functions fg (n) g satisfying (27), (28). The next lea establishes a siple relationship between the functions H n and G n. Lea 7 Suppose that H n and G n are the functions deterined by forulas (33), (39), and by the syste of equations fro theore 3. Then G n (a; b; ) = 1?b H n (1 + a? b; 2? b; ); 8n 0: Proof: The theore is trivial for n = 0. Consider n 1. Suppose that function (a; b; ) satises (43), (44) for y = Y (a; b; ) and = b? 1. Assue also that function (a; b; ) satises (43), (44) for y = 1?b Y (1 + 19

20 a? b; 2? b; ) and = 1? b. As follows fro (33) and (39), it suces to chec that (a; b; ) = 1?b (1 + a? b; 2? b; ): This relationship easily follows fro equations (43), (44), and (45). Finally, Leas 3 and 7 iply that the functions A (S) = e? p H (n) (a; b; ) B (S) = e? p+1?b H (n) (1 + a? b; 2? b; ); where =, solve equations (11) and (12) fro section 3. Using notations S in Lea 3 we equivalently rewrite A (S) = e? p 1 H (n) (p 1 + 2? ; 2p 1 + 2? ; ) (49) B (S) = e? p 2 H (n) (p 2 + 2? ; 2p 2 + 2? ; ): (50) Siilarly, Leas 5 and 7 iply that the functions V (n) A ( ) = p 1 H (n) (p 1 ; 2p ; ) (51) V (n) B ( ) = p 2 H (n) (p 2 ; 2p ; ); (52) where = section 4. and p 1;2 are dened in Lea 5, solve equations (22), (23) fro 6 Boundary Conditions and Richardson Extrapolation The results of the preceding sections calculate the general solutions of the approxiating systes of ODE's (9), (10) in the case of options on stoc paying constant continuous dividends and (20), (21) in the case of Asian options based on the arithetic ean. The proble of solving the corresponding boundary conditions has not been addressed yet. We suggest doing this part of the valuation nuerically. To be specic, we will consider the case of a European call on a stoc that pays constant continuous dividends. As follows fro Lea 1 and the 20

21 results in sections 3 and 5, the general solution of the recursive sequence of non-hoogeneous ODE's (9), (10) is 1 = 2 = U + n i=1 n i=1 h i (?) n?i (i) 1 C (n?i) A + (i) 1 C (n?i) B h i (?) n?i (i) 2 C (n?i) A + (i) 2 C (n?i) B for S 2 [0; K] for S K where coecients (i) 1, (i) 1 and (i) 2, (i) 2 should be deterined fro the corresponding boundary conditions. We nd these coecients iteratively. Suppose that (i) 1, (i) 1 and (i) 2, (i) 2 for i 2 f1; ; n? 1g are already nown fro the preceding iterations. At the n th iteration we need to deterine the coecients (n) 1, (n) 1 and (n) 2, (n) 2. This can be done by considering the following four conditions: 1 (S) = 2 (S) for S = K (53) 1 (S) s = 2 (S) s for S = K (54) 2 (S) = O(S) as S! 1 (55) 1 (0) = 0: (56) Conditions (53) and (54) follow fro the continuity of the stoc price and fro the continuity of the stoc price derivative. The other two conditions, (55) and (56), are iplied by equations (5) and (6). Conditions (53) and (54) can be solved nuerically in a straightforward anner. Conditions (55) and (56), however, require soe analytic consideration. One can chec that! (ln S)n A = O as S! 1 S p1! (ln S)n B = O as S! 1: S p2 Without loss of generality we can assue that p 1 > 0 and p 2 < 0. Since n 0, is a linear function of S then condition (55) is satised if (n) 2 = 0 21 U,

22 for all n 1. More diculties arise in solving (56). It is ipossible to obtain nuerically values of 1 at S = 0 and at any S which is suciently sall. The trouble coes fro estiating the function H (n) for very large arguents =. (n) If S is sall then values of C S A and B overow a coputer and it looses its precision in arithetic operations between A and B. To avoid this proble we suggest the following approach. Hoogeneous equation (14) is nown to have only one solution that grows polinoially as goes to innity 16. This function is called Tricoi's function. For large values of the arguent Tricoi's function can be approxiated by 17 T (0) (a; b; ) =?a M T =0 (a) (1 + a? b) (?1)! : (57) Polinoial growth of T (0) as! 1 ensures that the function C (0) T (S) := e? p T (0) (p + 2? ; 2p + 2? ; ); = S converges to ero as S =# 0. We obtained a sequence of polinoially growing functions T (n) that solve recursive syste (13). Theore 4 The functions where M T T (n) (a; b; ) =?a =0 (a) (1 + a? b) (?1) (n)! (a; b) (0) = 1 2 f0; : : : ; M T g (n) = M T i= (n?1) i (a + i)(1 + a? b + i) solve the syste of equation (13). n 1; 2 f0; : : : ; M T g Proof: Apply the sae arguents as in the proof of Lea Other solutions grow exponentially as! See [1]. The integer constant M T is a paraeter of approxiation. 22

23 It follows that the sequence of functions T (S) := e? p T (n) (a; b; ) n 0; = S solves (11), (12) and that each of these functions converges to ero as S =# 0. Finally, condition (56) can be substituted by two equations T (S) = 1 (S) for S = s 0 (S) T s = 1 (S) s for S = s 0 where is an extra unnown paraeter and where s 0 is any positive nuber less than the strie K such that T (0) is a good approxiation of Tricoi's function at = s 0 and such that the values of A (s 0) and B (s 0) do not overow the coputer. Therefore, at the n th iteration we have ve straightforward equations for ve unnown paraeters (n) 1, (n) 1, (n) 2, (n) 2, and. The approxiation error of this sei-analytic approach is ostly deterined by. In each particular exaple, we can obtain several estiations corresponding to dierent and then apply Richardson extrapolation to the. Figure 1 shows the values obtained by our ethod for one European call on a stoc with constant continuous dividends. The horiontal axis corresponds to the nuber of tie steps N that were used to subdivide the tie to aturity T. The lower graph shows the option's value C (N ) obtained for = T. The upper graph corresponds to Richardson extrapolation. For N each N this graph indicates the extrapolation obtained fro C (1) ; : : : ; C (N ). 23

24 14.4 r=0.04, d=5.0, vlt=0.32, T= 1.5, S=100.0, K=100.0 for N={0,.,10} call value C(N) value of N Figure 1: European Option. Siilar approach wors for solving boundary conditions corresponding to the Asian options. Figure 2 provides data for one European style put struc at the average. 24

25 r=0.04, vlt=0.43, T= 1.0, S= 1.0, for N={0,., 7} put value P(N) value of N, Put= Figure 2: Asian Option. Figures 1 and 2 indicate convergence of the approxiation errors to ero. Richardson extrapolation signicantly iproves the rate of convergence. 7 Appendix This appendix contains soe technical parts of the proof of Theore 3. We dene the coecients 18 C () 0 := 1 1 C () :=?1 i=0 i+1 i?1=0 i 3 +1 : : : i 2 =0 i 2 +1 i 1 =0 These coecients have the following properties: 1 1; 1: (58) Property 1: C (1) +1 = i=0 C (1) i C (1)?i = 18 In soe sources these coecients are refered to as Catalan's nubers. 25

26 = C (1) 0 C (1) + C (1) 1 C (1)?1 + C (1) 2 C (1)?2 + : : : + C (1) C (1) 0 This property provides a siple tool for calculating C (1). One can chec that C (1) 0 = 1; C (1) 1 = 1; C (1) 2 = 2; C (1) 3 = 5; C (1) 4 = 14; C (1) 5 = 42; : : : Property 2: Proof of Property 1: C (1) +1 = C () = C (?1) + C (+1)?1 ; 1; 2 (59) C (1) = C (2)?1; 1 (60) = 0 i +1 =0 1 i=0 = C (1) + i i=0 i+1 i?1=0 1 i=1 = C (1) + C (1)?1 : : : +C (1) 1 1 : : : i 2 +1 i 1 =0 i?1+1 i?2=0 (C (1)?1 + 1 i=1 i=1 i?1=1 = C (1) + C (1)?1 : : : +C (1) 1 0 i+1 0 i=0 i+1 i=0 i?1=0 + 1 = : : : i+1 i 2 +1 i 1 =0 i?1=1 i?1+1 i?2=1 1 + C 1?2 i 3 +1 : : : i 2 =1 1 + C (1) 1 = (C (1)?2 + (: : : (C (1) 1 + i 3 +1 : : : 1 + i 2 = i+1 i=1 i?1=1 1 i+1 i=1 i?1=1 0 i+1?2 i=0 i?1=0 0 i+1 i=0 i?1=0 i 2 +1 i 1 =1 1 + : : : i 2 +1 : : : 1 + : : : 1) : : :))) = i 1 =1 1 = i 2 +1 : : : 1 = i 1 =0 = C (1) + C (1)?1C (1) 1 + C (1)?2C (1) 2 + C (1) 1 C (1)?1 + C (1) : 26

27 Proof of Property 2: For 2, 2?1 C () = i+1 i=0 i?1=0 C () 1 =?1 i 1 =0 (: : :) = For = 1, 2 we have 1 =?2?2 i 1 =0 i+1 i=0 i?1= = C (?1) 1 + C (+1) 0 : (: : :) + ((+1)?1) i?1=0 (: : :) = C (?1) which proves (59). We also have to show (60). Indeed, for = 1 C (1) 1 = 0 i 1 =0 1 = 1 = C (2) 0 + C (+1)?1 and for 2 C (1) = 0 i=0 i+1 i?1=0 (: : :) = 1 i?1=0 (: : :) = C (2)?1 : Property 3: Suppose that, 0 n, satisfy equation (44) = 1! n? i=0 (?1) i C () i 2i Then equations (46) and (47) hold. That is, f (n??i) 0 1 n: n = f (n?1) n?1 n = f (n?1)?1 (n) ( + 1)f+1? n 1 1 n? 1: Proof of Property 3: Equation (44) iplies (46) since C () 0 = 1 for all 1 and relationship (47) follows fro (44) by eans of properties (59) and (60). 27

28 References [1] Milton Abraowit and Irene Stegun. Handboo of Matheatical Functions. Dover, [2] A. W. Babister. Transcendental Functions Satisfying Nonhoogeneous Linear Dierential Equations. MacMillan, New Yor, London, (see par. 4.18, page 121). [3] G. Barone-Adesi and R. Whaley. On the valuation of aerican put options on dividend paying stocs. Advances in Futures and Options Research, 3:1{14, [4] E. Bloeyer. An analytic approxiation for the aerican put price for options on stocs with dividends. Journal of Financial and Quantitative Analysis, 21:229{233, [5] Peter Carr and Diitri Faguet. Valuing nite-lived options as perpetual. Morgan Stanley woring paper, [6] M. Curran. Valuing asian and portfolio options by conditioning on the geoetric ean price. Manageent Science, 40(12):1705{11, [7] A. Eydeland and H. Gean. Doino eect. Ris, pages 65{67, [8] H. Gean and M. Yor. Quelques relations entre processes de bessel, options asiatiques, et fonctions conuentes hypergeoetriques. C.R.A.S. Paris Serie I, 314:471{474, (Loo up in Math. Finance 92-95). [9] R. Gese. A note on an analytical forula for unprotected aerican call options on stocs with nown dividends. Journal of Financial Econoics, 7:375{80, [10] A. Kena and T. Vorst. A pricing ethod for options based on average asset values. woring paper, [11] D. O. Kraov and E. Mordeci. An integral option. Probability Theory and its Applications, (in Russian). [12] Robert C. Merton. Theory of rational option pricing. Bell J. Econ. and Manageent Sci.,

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