- Email: [email protected]
General solution of the Black–Scholes boundary-value problem
Physica A 509 (2018) 546–550
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
General solution of the Black–Scholes boundary-value problem ByoungSeon Choi a , M.Y. Choi b , a b
∗
Department of Economics and SIRFE, Seoul National University, Seoul 08826, Republic of Korea Department of Physics and Center for Theoretical Physics, Seoul National University, Seoul 08826, Republic of Korea
highlights • • • •
We present infinitely many solutions of the Black–Scholes boundary problem. The Black–Scholes option valuation formula is included as a special solution. The solutions consist of many independent functions, involving Hermite polynomials. The Black–Scholes boundary-value problem violates the law of one price.
article
info
Article history: Received 19 March 2018 Received in revised form 23 May 2018 Available online xxxx Keywords: Black–Scholes formula European option Black–Scholes partial differential equation Hermite polynomials
a b s t r a c t The Black–Scholes formula for a European option price, which resulted in the 1997 Nobel Prize in Economic Sciences, is known to be the unique solution of the boundary-value problem consisting of the Black–Scholes partial differential equation and the terminal condition defined by the European call option. This has been one of the most popular tools of finance in theory as well as in practice. Here we present infinitely many solutions of the boundary value problem, involving Hermite polynomials. This indicates that the Black–Scholes boundary-value problem violates the law of one price, which is one of the fundamental concepts in economics. © 2018 Elsevier B.V. All rights reserved.
1. Introduction The Black–Scholes formula for a European option price is known to be a unique solution to the Black–Scholes partial differential equation with the terminal condition corresponding to the European option [1]. This resulted in the 1997 Nobel Prize in Economic Sciences, and has served as a paradigmatic tool of finance in theory as well as in practice [2]. To solve the Black–Scholes partial differential equation, one may conveniently consider the inverted time, in terms of which the Black–Scholes equation takes the form of the heat equation. Here we remark that the terminal condition is not differentiable and the terminal time is excluded. Related to this, we also point out that the inverted problem of the Black– Scholes equation is not exactly the same as the standard initial-value problem. Making use of this, we show that there exist infinitely many solutions to the Black–Scholes partial differential equation with the terminal condition. Such solutions include the Black–Scholes option valuation formula as a special one. Among additional solutions, in particular, there also exist solutions displaying discontinuity, which reflects the singularity in the terminal condition. Existence of such many solutions implies that the Black–Scholes partial differential equation for the European option violates the well-known law of one price, which is one of the fundamental concepts in economics [3,4]. ∗ Corresponding author. E-mail addresses: [email protected] (B. Choi), [email protected] (M.Y. Choi). https://doi.org/10.1016/j.physa.2018.06.095 0378-4371/© 2018 Elsevier B.V. All rights reserved.
B. Choi, M.Y. Choi / Physica A 509 (2018) 546–550
547
2. Black–Scholes formula Under the Black–Scholes environment, Black and Scholes [1] considered a European call option that pays [xT − K ]+ ≡ max {xT −K , 0} at expiration date T , where xt is the stock price at time t and the striking price K is a positive constant. They showed that its fair value w (xt , t) satisfies the partial differential equation (PDE): 1 (1) 2 for 0 ≤ t < T and x > 0, where r is the risk-free interest rate, v is the volatility of the stock, and ∂t w (x, t) ≡ ∂w (x, t)/∂ t, etc. with the subscript t on x suppressed for simplicity. This is nothing but Eq. (7) of Ref. [1] and is nowadays called the Black–Scholes PDE. By the definition of the European call option, its price satisfies the terminal condition (in the limit t → T from below):
∂t w (x, t) = r w (x, t) − rx∂x w(x, t) − v 2 x2 ∂x2 w(x, t)
lim w (x, t) = [x − K ]+
(2)
t ↑T
for x ̸ = K . It is well known that the Black–Scholes PDE reduces, via an appropriate transformation, to the heat equation, as sketched below: Defining the inverted time τ ≡ T − t and letting u ≡ ln(x/K ), we write the price function depending on u and τ in the form w (x, t) ≡ K w ˜ (u, τ ). We further define f (u, τ ) ≡ w ˜ (u, τ )e−αu−βτ , where α ≡ − v12 It is then straightforward to show that Eq. (1) leads to
(
r−
v2 2
)
and β ≡ − 2v12
(
r+
∂ f (u, τ ) v 2 ∂ 2 f (u, τ ) = , ∂τ 2 ∂ u2
v2 2
)2
.
(3)
which is nothing but the heat equation. Accordingly, the solution of the Black–Scholes PDE can be expressed in terms of the solution of the heat equation. In general, given the initial condition f (u, τ = 0) = g(u), the solution of Eq. (3) reads f (u, τ ) =
u
∫
du′ G(u, u′ ; τ )g(u′ ),
(4)
0
where G(u, u′ ; τ ) is the propagator (or Green’s function), i.e., a solution with the initial condition given by a delta function, G(u, u′ ; τ =0) = δ (u − u′ ). Specifically, the propagator takes the Gaussian form G(u, u′ ; τ ) = √
1 2πv 2 τ
′ 2
e
− (u−u2 ) 2v τ
,
(5)
which is the ‘‘fundamental solution’’ of the heat equation, and from the solution f (u, τ ), one obtains the solution w (x, t) of the Black–Scholes PDE, via w (x, t) = K 1−α xα f (ln(x/K ), T −t)eβ (T −t) . Black and Scholes then discussed the uniqueness of the solution of the boundary-value problem consisting of the Black– Scholes PDE (1) and the terminal condition (2) as follows: There is only one formula w (x, t) that satisfies the differential equation (1) subject to the boundary condition (2). This formula must be the option valuation formula. Specifically, the solution is given by
wBS (x, t) = xN(d1 ) − Ke−r τ N(d2 ), (6) ∫ 2 d where N(d) ≡ (2π )−1/2 −∞ dz e−z /2 is the cumulative distribution function of the standard normal random variable and ( ) ] [ x v2 1 d1 ≡ √ ln + r + τ K 2 v τ [ ( ) ] 1 x v2 d2 ≡ √ ln + r − τ . (7) K 2 v τ Eq. (6) is the Black–Scholes formula for the European call option price, which is also known as the Black–Scholes–Merton formula to acknowledge the valuable contributions by Merton [5]. For details, the readers are referred to Ref. [6]. 3. Generalized solution As commented in Section 2, the time-inversion of the Black–Scholes PDE is closely related to the heat equation. Accordingly, the solution of the Black–Scholes PDE is obtained from the heat equation, the general solution of which is given by Eq. (4). Note, however, that the inverted problem of the Black–Scholes PDE is not exactly the same as the standard initial-value problem: In particular, the time interval 0 ≤ t < T , along with the constraint x ̸ = K for the validity of the Black–Scholes PDE, transforms into the interval 0 < τ ≤ T . Namely, the initial time τ = 0 is excluded and instead the
548
B. Choi, M.Y. Choi / Physica A 509 (2018) 546–550
initial condition is given in the limit τ → 0. Here we remark that the propagator in Eq. (4) may not be unique. Specifically, it has been shown that the Gaussian propagator in Eq. (5) multiplied by appropriate polynomials not only satisfies the heat equation but also approaches the delta function in the initial limit and therefore provides another propagator [7]. Extending the idea of Ref. [7], we define ck (x, t) ≡ v −k−1 τ −
k+1 2
( Hk
1
)
x
√ ln K v τ
n(d2 )e−r τ
(8)
for nonnegative integer k (= 0, 1, 2, . . .), where n(d) ≡ (2π )−1/2 e−d /2 is the probability density function of the standard normal random variable and Hk (z) is the probabilists’ Hermite polynomial given by 2
dk
k z 2 /2
Hk (z) ≡ (−1) e
dz k
e
(
−z 2 /2
)k
d
= z−
dz
· 1.
(9)
We will now show that ck (x, t) satisfies the Black–Scholes PDE (1). Differentiating Eq. (7) leads to
∂ d2 1 ∂ d2 =− = √ ∂t ∂τ 2vτ τ ∂ d2 1 = √ . ∂x vx τ
[ ln
x K
( − r−
v2 2
) ] τ (10)
It is well-known (see, e.g., Ref. [8]) that d Hk (z) = kHk−1 (z) dz Hk+1 (z) − zHk (z) + kHk−1 (z) = 0.
(11)
Using Eqs. (10) and (11), one can easily show that
∂t ck (x, t) = −∂τ ck (x, t) = ct1 + ct2 + ct3 + ct4 ,
(12)
where k+1
k+3
v −k−1 τ − 2 Hk (z)n(d2 )e−r τ 2 ( x ) k+4 k ct2 ≡ v −k−2 ln τ − 2 Hk−1 (z)n(d2 )e−r τ
ct1 ≡
2
K
ct3 ≡ r v
1 −k−1 − k+ 2
1
ct4 ≡ − v 2 with z ≡ (1/v
√
Hk (z)n(d2 )e−r τ
τ
4 −k−2 − k+ 2
τ
[ ln
x K
( − r−
v2 2
) ] τ Hk (z)d2 n(d2 )e−r τ
τ ) ln(x/K ) hereafter. It is also straightforward to compute
∂x ck (x, t) = cx1 + cx2 ,
(13)
where cx1 ≡ kv −k−2 x−1 τ − cx2 ≡ −v −k−2 x−1 τ
k+2 2
Hk−1 (z)n(d2 )e−r τ
2 − k+ 2
Hk (z)d2 n(d2 )e−r τ
and similarly,
∂x2 ck (x, t) = cxx1 + cxx2 + cxx3 + cxx4 + cxx5 + cxx6 with cxx1 ≡ k(k − 1)v −k−3 x−2 τ − cxx2 ≡ −kv −k−2 x−2 τ −
k+2 2
k+3 2
Hk−2 (z)n(d2 )e−r τ
Hk−1 (z)n(d2 )e−r τ
cxx3 = cxx4 ≡ −kv −k−3 x−2 τ −
k+3 2
Hk−1 (z)d2 n(d2 )e−r τ
cxx5 ≡ v −k−3 x−2 τ −
k+3 2
Hk (z)(d22 − 1)n(d2 )e−r τ
cxx6 ≡ v −k−2 x−2 τ −
k+2 2
Hk (z)d2 n(d2 )e−r τ .
Using Eqs. (12), (13), and (14), one can show 1
∂t ck (x, t) − rck (x, t) + rx∂x ck (x, t) + v 2 x2 ∂x2 ck (x, t) 2
(14)
B. Choi, M.Y. Choi / Physica A 509 (2018) 546–550
=
k
2τ = 0,
549
[Hk (z) − zHk−1 (z) + (k − 1)Hk−2 (z)] (15)
where Eq. (11) has been used. Note that Eq. (15) is valid even for k = 1 and 0. Indeed for k = 1, we consider explicitly c1 (x, t) = v −2 τ −1 zn(d2 )e−r τ and differentiate it, to obtain 1
1
2
2τ
∂t c1 (x, t) − rc1 (x, t) + rx∂x c1 (x, t) + v 2 x2 ∂x2 c1 (x, t) =
[z − z ] = 0 ,
which is just Eq. (15) with k = 1. Similarly, considering explicitly c0 (x, t) = v −1 τ −1/2 n(d2 )e−r τ , we have 1
∂t c0 (x, t) − rc0 (x, t) + rx∂x c0 (x, t) + v 2 x2 ∂x2 c0 (x, t) = 0,
2 which is again Eq. (15) with k = 0. Accordingly, it is concluded that ck (x, t) for nonnegative integer k (= 0, 1, 2, . . .) satisfies the Black–Scholes PDE. Furthermore, taking the limit t → T (from below) or τ → 0 (from above) for x ̸ = K , we have lim ck (x, t) = lim v −k−1 τ −
k+1 2
τ ↓0
t ↑T
= lim v
1 −k−1 − k+ k 2
τ ↓0
= √
Hk (z)n(d2 )
τ
1 2πv 2k+1
1
z √
(
ln
x )k K
2π
( exp − 1
z2
)
2
[
lim τ −k− 2 exp − τ ↓0
1 2v 2 τ
(
ln
x )2
]
K
= 0,
(16)
where the last equality can be proved through the use of L’Hospital’s rule. Eqs. (15) and (16) thus manifest that ck (x, t) for nonnegative integer k (= 0, 1, 2 . . .) satisfies the Black–Scholes PDE (1) with the terminal condition lim ck (x, t) = 0
(17)
t ↑T
for x ̸ = K . Considering the linearity of the Black–Scholes PDE, we finally obtain the following theorem. Theorem 1. For any ζ = (ζ0 , ζ1 , . . . , ζN ) ∈ RN +1 , the expression wζ (x, t) ≡ w BS (x, t) + Scholes PDE given by Eq. (1) together with the terminal condition in Eq. (2).
∑N
k=0
ζk ck (x, t) satisfies the Black–
4. Discussion We have demonstrated that there exist infinitely many solutions to the Black–Scholes PDE subject to the terminal condition for a European option price. Among those, there is the Black–Scholes option valuation formula as a special solution for the case ζk = 0 for all k. To probe the properties of additional solutions, we first consider the limiting behavior of ck (x, t) defined in Eq. (8). Except for k = 0, Eq. (17) remains ∑ valid even at x = K . Accordingly, taking ζ0 = 0 in Theorem 1, we have the full solution wζ (x, t) = wBS (x, t) + Nk=1 ζk ck (x, t) continuous like the Black–Scholes option valuation formula. On the other hand, for k = 0, Eq. (17) does not hold for x = K , where c0 (x, t) becomes arbitrarily large as t grows toward T (from below). In consequence, c0 (x, t) has discontinuity at x = K ; this reflects the singularity in the terminal condition given by Eq. (2). Namely, the term [x − K ]+ in Eq. (2) is not differentiable at x = K . Remarkably, the existence of such many solutions implies that the well-known law of one price, which is one of the key concepts in economics, is violated. This raises a fundamental issue as to option pricing, the detailed discussion of which is left for future study. Acknowledgments B.S.C. was supported by the Overseas Training Expenses for Humanities and Social Sciences, Republic of Korea through Seoul National University in 2013. M.Y.C. acknowledges support from the National Research Foundation of Korea through the Basic Science Research Program (Grant No. 2016R1D1A1A09917318). References [1] F. Black, M. Scholes, J. Polit. Econ. 81 (1973) 637. http://dx.doi.org/10.1086/260062. [2] J.C. Hull, Options, Futures, and Other Derivatives, tenth ed., Pearson, New Jersey, 2017, pp. 321–347. [3] P.A. Samuelson, W.D. Nordhaus, Economics, nineteenth ed., McGraw-Hill, Boston, 2009, p. 718.
550 [4] [5] [6] [7] [8]
B. Choi, M.Y. Choi / Physica A 509 (2018) 546–550 N.G. Mankiw, Principles of Macroeconomics, eighth ed., Cengage Learning, Boston, 2017, p. 390. R.C. Merton, Bell J. Econ. Manage. Sci. 4 (1973) 141. http://dx.doi.org/10.2307/3003143. S. Shreve, Stochastic Calculus for Finance II: Continuous-time Models, Springer, New York, 2004, Sec. 4.5. B.S. Choi, H. Kang, M.Y. Choi, Physica A 470 (2017) 88. http://dx.doi.org/10.1016/j.physa.2016.11.095. G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999, p. 280.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
General solution of the Black–Scholes boundary-value problem ByoungSeon Choi a , M.Y. Choi b , a b
∗
Department of Economics and SIRFE, Seoul National University, Seoul 08826, Republic of Korea Department of Physics and Center for Theoretical Physics, Seoul National University, Seoul 08826, Republic of Korea
highlights • • • •
We present infinitely many solutions of the Black–Scholes boundary problem. The Black–Scholes option valuation formula is included as a special solution. The solutions consist of many independent functions, involving Hermite polynomials. The Black–Scholes boundary-value problem violates the law of one price.
article
info
Article history: Received 19 March 2018 Received in revised form 23 May 2018 Available online xxxx Keywords: Black–Scholes formula European option Black–Scholes partial differential equation Hermite polynomials
a b s t r a c t The Black–Scholes formula for a European option price, which resulted in the 1997 Nobel Prize in Economic Sciences, is known to be the unique solution of the boundary-value problem consisting of the Black–Scholes partial differential equation and the terminal condition defined by the European call option. This has been one of the most popular tools of finance in theory as well as in practice. Here we present infinitely many solutions of the boundary value problem, involving Hermite polynomials. This indicates that the Black–Scholes boundary-value problem violates the law of one price, which is one of the fundamental concepts in economics. © 2018 Elsevier B.V. All rights reserved.
1. Introduction The Black–Scholes formula for a European option price is known to be a unique solution to the Black–Scholes partial differential equation with the terminal condition corresponding to the European option [1]. This resulted in the 1997 Nobel Prize in Economic Sciences, and has served as a paradigmatic tool of finance in theory as well as in practice [2]. To solve the Black–Scholes partial differential equation, one may conveniently consider the inverted time, in terms of which the Black–Scholes equation takes the form of the heat equation. Here we remark that the terminal condition is not differentiable and the terminal time is excluded. Related to this, we also point out that the inverted problem of the Black– Scholes equation is not exactly the same as the standard initial-value problem. Making use of this, we show that there exist infinitely many solutions to the Black–Scholes partial differential equation with the terminal condition. Such solutions include the Black–Scholes option valuation formula as a special one. Among additional solutions, in particular, there also exist solutions displaying discontinuity, which reflects the singularity in the terminal condition. Existence of such many solutions implies that the Black–Scholes partial differential equation for the European option violates the well-known law of one price, which is one of the fundamental concepts in economics [3,4]. ∗ Corresponding author. E-mail addresses: [email protected] (B. Choi), [email protected] (M.Y. Choi). https://doi.org/10.1016/j.physa.2018.06.095 0378-4371/© 2018 Elsevier B.V. All rights reserved.
B. Choi, M.Y. Choi / Physica A 509 (2018) 546–550
547
2. Black–Scholes formula Under the Black–Scholes environment, Black and Scholes [1] considered a European call option that pays [xT − K ]+ ≡ max {xT −K , 0} at expiration date T , where xt is the stock price at time t and the striking price K is a positive constant. They showed that its fair value w (xt , t) satisfies the partial differential equation (PDE): 1 (1) 2 for 0 ≤ t < T and x > 0, where r is the risk-free interest rate, v is the volatility of the stock, and ∂t w (x, t) ≡ ∂w (x, t)/∂ t, etc. with the subscript t on x suppressed for simplicity. This is nothing but Eq. (7) of Ref. [1] and is nowadays called the Black–Scholes PDE. By the definition of the European call option, its price satisfies the terminal condition (in the limit t → T from below):
∂t w (x, t) = r w (x, t) − rx∂x w(x, t) − v 2 x2 ∂x2 w(x, t)
lim w (x, t) = [x − K ]+
(2)
t ↑T
for x ̸ = K . It is well known that the Black–Scholes PDE reduces, via an appropriate transformation, to the heat equation, as sketched below: Defining the inverted time τ ≡ T − t and letting u ≡ ln(x/K ), we write the price function depending on u and τ in the form w (x, t) ≡ K w ˜ (u, τ ). We further define f (u, τ ) ≡ w ˜ (u, τ )e−αu−βτ , where α ≡ − v12 It is then straightforward to show that Eq. (1) leads to
(
r−
v2 2
)
and β ≡ − 2v12
(
r+
∂ f (u, τ ) v 2 ∂ 2 f (u, τ ) = , ∂τ 2 ∂ u2
v2 2
)2
.
(3)
which is nothing but the heat equation. Accordingly, the solution of the Black–Scholes PDE can be expressed in terms of the solution of the heat equation. In general, given the initial condition f (u, τ = 0) = g(u), the solution of Eq. (3) reads f (u, τ ) =
u
∫
du′ G(u, u′ ; τ )g(u′ ),
(4)
0
where G(u, u′ ; τ ) is the propagator (or Green’s function), i.e., a solution with the initial condition given by a delta function, G(u, u′ ; τ =0) = δ (u − u′ ). Specifically, the propagator takes the Gaussian form G(u, u′ ; τ ) = √
1 2πv 2 τ
′ 2
e
− (u−u2 ) 2v τ
,
(5)
which is the ‘‘fundamental solution’’ of the heat equation, and from the solution f (u, τ ), one obtains the solution w (x, t) of the Black–Scholes PDE, via w (x, t) = K 1−α xα f (ln(x/K ), T −t)eβ (T −t) . Black and Scholes then discussed the uniqueness of the solution of the boundary-value problem consisting of the Black– Scholes PDE (1) and the terminal condition (2) as follows: There is only one formula w (x, t) that satisfies the differential equation (1) subject to the boundary condition (2). This formula must be the option valuation formula. Specifically, the solution is given by
wBS (x, t) = xN(d1 ) − Ke−r τ N(d2 ), (6) ∫ 2 d where N(d) ≡ (2π )−1/2 −∞ dz e−z /2 is the cumulative distribution function of the standard normal random variable and ( ) ] [ x v2 1 d1 ≡ √ ln + r + τ K 2 v τ [ ( ) ] 1 x v2 d2 ≡ √ ln + r − τ . (7) K 2 v τ Eq. (6) is the Black–Scholes formula for the European call option price, which is also known as the Black–Scholes–Merton formula to acknowledge the valuable contributions by Merton [5]. For details, the readers are referred to Ref. [6]. 3. Generalized solution As commented in Section 2, the time-inversion of the Black–Scholes PDE is closely related to the heat equation. Accordingly, the solution of the Black–Scholes PDE is obtained from the heat equation, the general solution of which is given by Eq. (4). Note, however, that the inverted problem of the Black–Scholes PDE is not exactly the same as the standard initial-value problem: In particular, the time interval 0 ≤ t < T , along with the constraint x ̸ = K for the validity of the Black–Scholes PDE, transforms into the interval 0 < τ ≤ T . Namely, the initial time τ = 0 is excluded and instead the
548
B. Choi, M.Y. Choi / Physica A 509 (2018) 546–550
initial condition is given in the limit τ → 0. Here we remark that the propagator in Eq. (4) may not be unique. Specifically, it has been shown that the Gaussian propagator in Eq. (5) multiplied by appropriate polynomials not only satisfies the heat equation but also approaches the delta function in the initial limit and therefore provides another propagator [7]. Extending the idea of Ref. [7], we define ck (x, t) ≡ v −k−1 τ −
k+1 2
( Hk
1
)
x
√ ln K v τ
n(d2 )e−r τ
(8)
for nonnegative integer k (= 0, 1, 2, . . .), where n(d) ≡ (2π )−1/2 e−d /2 is the probability density function of the standard normal random variable and Hk (z) is the probabilists’ Hermite polynomial given by 2
dk
k z 2 /2
Hk (z) ≡ (−1) e
dz k
e
(
−z 2 /2
)k
d
= z−
dz
· 1.
(9)
We will now show that ck (x, t) satisfies the Black–Scholes PDE (1). Differentiating Eq. (7) leads to
∂ d2 1 ∂ d2 =− = √ ∂t ∂τ 2vτ τ ∂ d2 1 = √ . ∂x vx τ
[ ln
x K
( − r−
v2 2
) ] τ (10)
It is well-known (see, e.g., Ref. [8]) that d Hk (z) = kHk−1 (z) dz Hk+1 (z) − zHk (z) + kHk−1 (z) = 0.
(11)
Using Eqs. (10) and (11), one can easily show that
∂t ck (x, t) = −∂τ ck (x, t) = ct1 + ct2 + ct3 + ct4 ,
(12)
where k+1
k+3
v −k−1 τ − 2 Hk (z)n(d2 )e−r τ 2 ( x ) k+4 k ct2 ≡ v −k−2 ln τ − 2 Hk−1 (z)n(d2 )e−r τ
ct1 ≡
2
K
ct3 ≡ r v
1 −k−1 − k+ 2
1
ct4 ≡ − v 2 with z ≡ (1/v
√
Hk (z)n(d2 )e−r τ
τ
4 −k−2 − k+ 2
τ
[ ln
x K
( − r−
v2 2
) ] τ Hk (z)d2 n(d2 )e−r τ
τ ) ln(x/K ) hereafter. It is also straightforward to compute
∂x ck (x, t) = cx1 + cx2 ,
(13)
where cx1 ≡ kv −k−2 x−1 τ − cx2 ≡ −v −k−2 x−1 τ
k+2 2
Hk−1 (z)n(d2 )e−r τ
2 − k+ 2
Hk (z)d2 n(d2 )e−r τ
and similarly,
∂x2 ck (x, t) = cxx1 + cxx2 + cxx3 + cxx4 + cxx5 + cxx6 with cxx1 ≡ k(k − 1)v −k−3 x−2 τ − cxx2 ≡ −kv −k−2 x−2 τ −
k+2 2
k+3 2
Hk−2 (z)n(d2 )e−r τ
Hk−1 (z)n(d2 )e−r τ
cxx3 = cxx4 ≡ −kv −k−3 x−2 τ −
k+3 2
Hk−1 (z)d2 n(d2 )e−r τ
cxx5 ≡ v −k−3 x−2 τ −
k+3 2
Hk (z)(d22 − 1)n(d2 )e−r τ
cxx6 ≡ v −k−2 x−2 τ −
k+2 2
Hk (z)d2 n(d2 )e−r τ .
Using Eqs. (12), (13), and (14), one can show 1
∂t ck (x, t) − rck (x, t) + rx∂x ck (x, t) + v 2 x2 ∂x2 ck (x, t) 2
(14)
B. Choi, M.Y. Choi / Physica A 509 (2018) 546–550
=
k
2τ = 0,
549
[Hk (z) − zHk−1 (z) + (k − 1)Hk−2 (z)] (15)
where Eq. (11) has been used. Note that Eq. (15) is valid even for k = 1 and 0. Indeed for k = 1, we consider explicitly c1 (x, t) = v −2 τ −1 zn(d2 )e−r τ and differentiate it, to obtain 1
1
2
2τ
∂t c1 (x, t) − rc1 (x, t) + rx∂x c1 (x, t) + v 2 x2 ∂x2 c1 (x, t) =
[z − z ] = 0 ,
which is just Eq. (15) with k = 1. Similarly, considering explicitly c0 (x, t) = v −1 τ −1/2 n(d2 )e−r τ , we have 1
∂t c0 (x, t) − rc0 (x, t) + rx∂x c0 (x, t) + v 2 x2 ∂x2 c0 (x, t) = 0,
2 which is again Eq. (15) with k = 0. Accordingly, it is concluded that ck (x, t) for nonnegative integer k (= 0, 1, 2, . . .) satisfies the Black–Scholes PDE. Furthermore, taking the limit t → T (from below) or τ → 0 (from above) for x ̸ = K , we have lim ck (x, t) = lim v −k−1 τ −
k+1 2
τ ↓0
t ↑T
= lim v
1 −k−1 − k+ k 2
τ ↓0
= √
Hk (z)n(d2 )
τ
1 2πv 2k+1
1
z √
(
ln
x )k K
2π
( exp − 1
z2
)
2
[
lim τ −k− 2 exp − τ ↓0
1 2v 2 τ
(
ln
x )2
]
K
= 0,
(16)
where the last equality can be proved through the use of L’Hospital’s rule. Eqs. (15) and (16) thus manifest that ck (x, t) for nonnegative integer k (= 0, 1, 2 . . .) satisfies the Black–Scholes PDE (1) with the terminal condition lim ck (x, t) = 0
(17)
t ↑T
for x ̸ = K . Considering the linearity of the Black–Scholes PDE, we finally obtain the following theorem. Theorem 1. For any ζ = (ζ0 , ζ1 , . . . , ζN ) ∈ RN +1 , the expression wζ (x, t) ≡ w BS (x, t) + Scholes PDE given by Eq. (1) together with the terminal condition in Eq. (2).
∑N
k=0
ζk ck (x, t) satisfies the Black–
4. Discussion We have demonstrated that there exist infinitely many solutions to the Black–Scholes PDE subject to the terminal condition for a European option price. Among those, there is the Black–Scholes option valuation formula as a special solution for the case ζk = 0 for all k. To probe the properties of additional solutions, we first consider the limiting behavior of ck (x, t) defined in Eq. (8). Except for k = 0, Eq. (17) remains ∑ valid even at x = K . Accordingly, taking ζ0 = 0 in Theorem 1, we have the full solution wζ (x, t) = wBS (x, t) + Nk=1 ζk ck (x, t) continuous like the Black–Scholes option valuation formula. On the other hand, for k = 0, Eq. (17) does not hold for x = K , where c0 (x, t) becomes arbitrarily large as t grows toward T (from below). In consequence, c0 (x, t) has discontinuity at x = K ; this reflects the singularity in the terminal condition given by Eq. (2). Namely, the term [x − K ]+ in Eq. (2) is not differentiable at x = K . Remarkably, the existence of such many solutions implies that the well-known law of one price, which is one of the key concepts in economics, is violated. This raises a fundamental issue as to option pricing, the detailed discussion of which is left for future study. Acknowledgments B.S.C. was supported by the Overseas Training Expenses for Humanities and Social Sciences, Republic of Korea through Seoul National University in 2013. M.Y.C. acknowledges support from the National Research Foundation of Korea through the Basic Science Research Program (Grant No. 2016R1D1A1A09917318). References [1] F. Black, M. Scholes, J. Polit. Econ. 81 (1973) 637. http://dx.doi.org/10.1086/260062. [2] J.C. Hull, Options, Futures, and Other Derivatives, tenth ed., Pearson, New Jersey, 2017, pp. 321–347. [3] P.A. Samuelson, W.D. Nordhaus, Economics, nineteenth ed., McGraw-Hill, Boston, 2009, p. 718.
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