Q U A N T I T A T I V E F I N A N C E V O L U M E 4 (A U G U S T 2004) 457–464 TAYLOR & FRANCIS LTD
RESEARCH PAPER tandf.co.uk
Option pricing with Weyl–Titchmarsh theory Yishen Li1 and Jin E Zhang2,3 1
Department of Mathematics, University of Science and Technology of China, Hefei 230026, PR China 2 School of Business, and School of Economics and Finance, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China Email:
[email protected] and
[email protected] Received 22 October 2003, in final form 23 June 2004 Published 3 December 2004 Online at www.tandf.co.uk/journals/titles/14697688.asp DOI: 10.1080/14697680400008643
Abstract In the Black–Merton–Scholes framework, the price of an underlying asset is assumed to follow a pure diffusion process. No-arbitrage theory shows that the price of an option contract written on the asset can be determined by solving a linear diffusion equation with variable coefficients. Applying the separating variable method, the problem of option pricing under state-dependent deterministic volatility can be transformed into a Schro¨dinger spectral problem, which has been well studied in quantum mechanics. With Weyl–Titchmarsh theory, we are able to determine the boundary condition and the nature of the eigenvalues and eigenfunctions. The solution can be written analytically in a Stieltjes integral. A few case studies demonstrate that a new analytical option pricing formula can be produced with our method.
1. Introduction In finance, the price of an underlying asset is often modelled with a diffusion process. With the no-arbitrage argument, the price of a derivative contract written on the asset can be determined by solving an initial boundary value problem of a linear partial differential equation (PDE), i:e: the generalized Black–Scholes equation. By using the separating variable method, the problem becomes a spectral problem of an ordinary differential equation (ODE). The eigenvalues and the completeness of the eigenfunctions of the ODEs in a finite interval are fundamental issues. For a simple singular second-order ODE (i:e: the coefficient of the equation is singular at the boundary point or the boundary point tends to infinity), the issue of how to give a boundary condition arises. 3
Author to whom any correspondence should be addressed.
1469-7688 Print/1469-7696 Online/04/040457–08
In 1910, a famous mathematician, H: Weyl, pointed out that, for a singular ODE, there are two different cases: a limit circle case and a limit point case. In the first case, we have to give a boundary condition. In the second case, it is unnecessary to give a boundary condition. This important result had not been realized by physicists when they first solved the Schro¨dinger equation in quantum mechanics. The same phenomenon appears in quantitative finance as well when financial mathematicians solve the generalized Black–Scholes equation. During World War II, a mathematician in the United Kingdom, E: C: Titchmarsh [17], performed serious research on the eigenvalue problem of the Schro¨dinger equation. In 1946, his work was published as a monograph by Oxford University. In 1950, B: M: Levitan [11] published a similar monograph in Russia. The self-adjoint differential operator theory in Hilbert space has been established by many mathematicians
ß 2004 Taylor & Francis Ltd
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(see e:g: Naimark [16] and Weidmann [18]). It was found that the properties for the regular and singular cases are different not only in the boundary condition, but also in the eigenvalues, such as the point spectrum and the continuous spectrum. We need to introduce the Stieltjes integral in order to prove the completeness of the eigenfunctions. These results established the mathematical foundation of quantum mechanics. By using the separating variable method, the generalized Black–Scholes equation, arising from option pricing, can be transformed into a Schro¨dinger equation, which has been well studied in quantum mechanics. In this paper we try to establish a rigorous mathematical foundation for the problem of option pricing based on the theory of the Schro¨dinger equation. We discuss how to transform a typical option pricing problem into a Schro¨dinger equation in section 2 and discuss the condition under which the equation can be transformed into a standard heat equation. By using the theory of the Schro¨dinger equation, we discuss in section 3 how to give a boundary condition and to give a solution through the Stieltjes integral. In section 4, we study a few cases, including the Black–Scholes formula. In the first case, it is unnecessary to give any boundary condition and it has a pure continuous spectrum. In the second case, we need to give one boundary condition; it has a continuous spectrum and one point spectrum. The third example is a perturbation of the first. In this case, we do not need to give any boundary condition, but it is a pure point spectrum. The next example is an exponentially decreasing volatility, which has not been studied before in the literature. It has a continuous spectrum. In the last example we do not need to give any boundary condition, it has both a continuous and a point spectrum. In section 5, we list a few potential functions q(x) for which analytical formulae were studied and presented in Titchmarsh [17]. Finally we conclude the paper in section 6.
European call option, c(S, t), satisfies the following generalized Black–Scholes partial differential equation @c 1 2 @2 c @c þ ðS, tÞS 2 2 þ ðr qÞS rc ¼ 0, @t 2 @S @S
ð2Þ
cðS, TÞ ¼ maxðS K, 0Þ,
ð3Þ
where both interest rate, r, and dividend yield, q, are assumed to be constant; K is the strike price of the option. In the general case, analytical formulae of the problem (equations (2) and (3)) cannot be obtained. Practitioners rely on numerical methods such as finite difference, binomial trees, or Monte Carlo simulation. However, for certain volatility functions, the generalized Black–Scholes equation can be transformed into the standard heat equation, which, in turn, can be solved analytically. Even for the case where the problem cannot be reduced to the standard heat equation, it is still possible to solve the problem analytically for some particular volatility functions. Identifying these solvable volatility functions then becomes an issue. Analytical solutions for the generalized Black– Scholes equation are of paramount importance to practitioners as they allow a better qualitative understanding of the solution behaviour. More importantly, volatility functions are typically fitted to market data and parametric volatility models that produce analytical solutions are in very high demand. The problem of reducibility and solvability of the generalized Black–Scholes equation has already been studied by Carr et al [5] and Bouchouev [3], and extended by many other authors over the last several years. For example, an application of spectral theory has been applied to Asian and other exotic options by Davydov and Linetsky [7] and Linetsky [12]. This paper pushes further along the same direction. With the forward price transformation S ¼ SeðrqÞ ,
c ðS , Þ ¼ cðS, tÞer ,
¼ T t,
ð4Þ
2. Typical option pricing problem In the Black–Merton–Scholes [1, 14] framework, the price of a stock, S, is modelled by a pure diffusion process4 dS ¼ S dt þ S dBt ,
ð1Þ
where is the drift, is the volatility of the stock and Bt is standard Brownian motion. In this paper, we only consider the case where the volatility, , is a deterministic function of the stock price, S, and time, t, i:e: ¼ ðS, tÞ. Standard no-arbitrage theory shows that the price of a 4
The later development of option pricing models includes Merton’s [15] jump diffusion model, Heston’s [9] stochastic volatility model, the variance Gamma process of Madan et al [13], the affine jump diffusion model of Duffie et al and time-changed Le´vy processes of Carr and Wu [6].
458
equations (2) and (3) become @c 1 2 2 @2 c S ¼ 0, @ 2 @S 2
ð5Þ
c ðS , 0Þ ¼ maxðS K, 0Þ,
ð6Þ
where the volatility is, in general, a function of forward stock price S and time to maturity , i:e: ¼ ðS , Þ: But it is difficult to solve the problem analytically for the general case. In this paper, we study a particular case where is a function of the forward stock price5, and leave the general case of time dependence for future research.
QUANTITATIVE FINANCE
Option pricing with Weyl–Titchmarsh theory
With the model set up, a typical option pricing problem is reduced to solving the following heat equation with a variable coefficient @c @2 c 2 ðSÞ 2 ¼ 0, @t @S
0 < S < 1,
cðS, 0Þ ¼ maxðS K, 0Þ,
ð7Þ ð8Þ
where ðSÞ > 0 is a sufficiently smooth function and K is a constant. For brevity we have dropped the superscript star in c and S and replaced by t and ð1=2Þ 2 S 2 by 2 ðSÞ. We introduce the following transformation: Z pffiffiffiffiffiffiffiffiffi 1 dS, cðS, tÞ ¼ ðSÞ gðx, tÞ, ¼ 1: xðSÞ ¼ ðSÞ ð9Þ Since ðSÞ > 0, x(S) is a monotonically increasing function if ¼ 1. It is a monotonically decreasing function if ¼ 1. By a simple calculation, equation (7) with initial condition (8) becomes a Schro¨dinger equation with an initial condition @g @2 g ¼ qðxÞg, @t @x2
a < x < b,
gðx, tÞjt¼0 ¼ g0 ðxÞ,
ð11Þ
ð12Þ
The variable S in (S) has to be changed to S ¼ SðxÞ by solving the equation x ¼ xðSÞ from (9). Under the condition that q(x) is a quadratic function of x, i:e: qxxx ¼ 0,
ð13Þ
equation (10) can be transformed into a standard heat equation. The mathematical result was first proved by Bluman [2] using Lie groups, and studied by Zhang [20] and Carr et al [4] in the context of option pricing. Converting the condition on g(x) to that on (S) gives ½ð 2 SSS ÞS S ¼ 0,
3. Analytical solution with Weyl–Titchmarsh theory We now look for the analytical solution of the initial value problem (equations (10) and (11)). With the separating variable method, assuming gðx, tÞ ¼ TðtÞf ðxÞ and substituting into (10) gives T 0 ðtÞ f 00 qðxÞf ¼ ¼ , TðtÞ f
We then have
f 00 qðxÞf ¼ f :
ð16Þ
Equation (16) is a Schro¨dinger spectral problem, which has been well studied in the applied mathematics literature with an application in quantum mechanics. In particular, Weyl [19] and Titchmarsh [17] have established a solid theoretical foundation for the solution of the problem. Based on Weyl–Titchmarsh (WT) theory, if the end point a is the limit point case, we do not need any boundary condition at a. If it is a limit circle case7, we need a boundary condition, f ðaÞ cos þ f 0 ðaÞ sin ¼ 0,
is a parameter:
ð17Þ
There exists a unique function a ðx, Þ that belongs to L2 ða, dÞ, d is a mid-point in ða, bÞ, where a ðx, Þ
¼ ðx, Þ þ ma ðÞðx, Þ,
Im > 0:
ð18Þ
and are solutions of (16) and satisfy boundary condition ðd, Þ ¼ 1,
0 ðd, Þ ¼ 0;
ðd, Þ ¼ 0,
0 ðd, Þ ¼ 1: ð19Þ
ð14Þ 6
which agrees with the condition in the more general setting provided by Carr et al [4]6. If condition (13) is
5
ð15Þ
where is a complex parameter. TðtÞ ¼ et , and
ð10Þ
where g0 ðxÞ is defined from cðS, 0Þ by using the transformation (9); the end points a and b, defined by (9), can be finite or infinity. The coefficient q(x), regarded as a potential function, reads 1 1 qðxÞ ¼ ðS Þ2 SS : 4 2
violated, equation (10) cannot be transformed into a standard heat equation. Therefore, the option does not have a closed-form solution of the Black–Scholes type, that is, the solution is not given in terms of the cumulative normal distribution function, but it may be given in terms of some other known special functions and/or their integrations. In this paper, we deal with the case where the potential, q, is a general function of x including a quadratic one.
A new well-posed algorithm has been proposed recently in [10] to recover the state-dependent local volatility from the market price of options on a particular maturity date.
The condition is differentiated from that of Tian, ðsss Þss ¼ 0;
cited in [21], which provides some closed-form option pricing formulae for a quadratic volatility function i.e., sss ¼ 0 or, equivalent, qx ¼ 0. 7 By limit circle case, we mean that for, Im > 0, equation (16) has two linearly independent solutions that belong to L2 ða, dÞ, where d is a mid-point in (a, b). Otherwise it is called a limit point case.
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Similarly, if the end point b is a limit point case, we do not need any boundary condition at b. If it is a limit circle case, we need a boundary condition, f ðbÞ cos þ f 0 ðbÞ sin ¼ 0, There is a unique function b ðx, Þ
is a parameter:
b ðx, Þ
ð20Þ
Im > 0:
!0 0
Z ðÞ ¼ lim
!0 0
ma ðu þ iÞ du, ma ðu þ iÞ mb ðu þ iÞ
ð23Þ
Im
ma ðu þ iÞmb ðu þ iÞ du: ma ðu þ iÞ mb ðu þ iÞ
ð24Þ
We let Z
b
ðy, Þet g0 ðyÞdy,
g1 ðt, Þ ¼ Z
a
ð25Þ
b
g2 ðt, Þ ¼
ðy, Þe
t
g0 ðyÞdy:
a
The solution of equation (10) can be written analytically in Stieltjes integral Z 1 1 gðx, tÞ ¼ ðx, Þg1 ðt, Þd ðÞ
1 Z 1 1 þ ðx, Þg2 ðt, Þd ðÞ p 1 Z 1 1 ðx, Þg1 ðt, Þd ðÞ þ
1 Z 1 1 þ ðx, Þg2 ðt, Þd ðÞ, ð26Þ p 1 which is mean-convergent. In practice, we often encounter the following two special cases: (i)
If the problem domain is ½a, þ 1Þ, and þ1 is the limit point case and a is a limit circle case, we need to give a boundary condition at x ¼ a as
Z
!0 0
f ðaÞ cos þ f ðaÞ sin ¼ 0:
Im ma ðu þ iÞ du,
then formula (26) reduces to a simpler one: gðx, tÞ ¼
(ii)
1 p
Z
Z
1
1
ðy, Þet g0 ðyÞ dy:
ðx, Þ dkðÞ a
1
If the problem domain is ð1, 1Þ and qðxÞ ! þ1 monotonically as x ! 1, then there exists a discrete spectrum only. Two linearly independent solutions are uniquely determined by the spectral problem (16). By constructing a square integrable function, we can determine m1 ðÞ from (18) and m1 ðÞ from (21). The poles of m1 ðÞ and m1 ðÞ are eigenvalues. We denote eigenvalues of the system as i, i ¼ 1, 2; . . . , and the corresponding normalized orthogonal eigenfunctions as i(x), Zb i ðxÞj ðxÞ dx ¼ ij : a
Then the integration formula (26) is reduced to a summation Zb 1 X gðx, tÞ ¼ en t n ðxÞ n ðyÞg0 ðyÞdy: ð30Þ a
n¼1
4. Case studies We now study a few cases, most of which are not well known in the finance literature. Our purpose here is to demonstrate the procedure of producing analytical option pricing formulae with Weyl–Titchmarsh theory. The standard benchmark case of the Black–Scholes formula is included in the appendix for the reader who is interested in the details of applying the theory.
4.1. A quadratic volatility Assuming ðSÞ ¼ S2 ,
0
ð27Þ
0 1 ¼ pffiffiffi > 0, 2 S0
taking ¼ 1, from (9), we have 8
By a meromorphic function, we mean that the function is analytical in the upper half plane, Im > 0, and the limit exists, but there might be singular points on the real axis.
460
ð28Þ
Define kðÞ ¼ lim
Im
¼ ðx, Þ þ ma ðÞðx, Þ:
ð21Þ
The functions ma ðÞ and mb ðÞ defined by (18) and (21) are meromorphic functions8 in Im > 0. There are simple poles at Im ¼ 0, and all these poles are the eigenvalues. Define Z 1 du, ð22Þ Im
ðÞ ¼ lim !0 0 ma ðu þ iÞ mb ðu þ iÞ
ðÞ ¼ lim
a ðx, Þ
belonging to L2 ðd, bÞ,
¼ ðx, Þ þ mb ðÞðx, Þ,
Z
Following the same argument as before, there exists a unique function a ðx, Þ that belongs to L2 ða, dÞ, d is a mid-point in ða, 1Þ, where
x¼
1 , S
or S ¼
1 , x
cðS, tÞ ¼ S gðx, tÞ:
QUANTITATIVE FINANCE
Option pricing with Weyl–Titchmarsh theory
When S 2 ð0, þ 1Þ, x 2 ð0, þ 1Þ (S ¼ 0, x ¼ þ1; S ¼ þ1, x ¼ 0), equation (10) and (11) become @g @2 g ¼ , @t @x2
gðx, 0Þ ¼ g0 ¼
0,
x > 1=ðKÞ,
1 Kx,
0 < x < 1=ðKÞ:
0
With some algebra, one may show the solution agrees with that offered in [21].
Using the separating variable method, we have g ¼ et f ,
00
f
¼ f ,
0 < x < 1:
ð31Þ
At x ¼ 1, it is a limit point case. It does not need any boundary condition; at x ¼ 0, it is a limit circle case, and we need one boundary condition f ð0Þ cos þ f 0 ð0Þ sin ¼ 0,
cot ¼ K:
> 0, < 0:
The problem has both a continuous and a discrete spectrum9. 0 ¼ s20 ¼ cot2 is an eigenvalue; the corresponding normalized eigenfunction is denoted by ðx, 0 Þ , kðx, 0 Þk
where sin is0 x is0 ¼ sin cosh s0 x sin sinh s0 x
ðx, 0 Þ ¼ sin cos is0 x cos ¼ sin es0 x , and 1
kðx, 0 Þk ¼ 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S 2 2 2 , ðSÞ ¼ S þ ln S0
S ¼ S0 eð=Þ sinh x ,
pffiffiffi sin i cos pffiffiffi : mðÞ ¼ cos þ i sin
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin3 2 ðx, 0 Þ dx ¼ : 2 cos
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z
Assuming
where > 0, S0 > 0, and > 0 is a small parameter. From (9), we have Z 1 1 S pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dS ¼ sinh1 ln , x¼ 2 2 2 S0 S þ ½lnðS=S0 Þ
pffiffiffi pffiffiffi sin x ðx, Þ ¼ cos cos x þ sin pffiffiffi , pffiffiffi pffiffiffi sin x ðx, Þ ¼ sin cos x cos pffiffiffi ,
0 ðxÞ ¼
4.2. A smile volatility
ð32Þ
Since tan > 0, we take
ðx, Þ ¼ ðx, Þ þ mðÞðx, Þ, ( pffiffiffi =ðcos2 þ sin2 Þ, Im mðÞ ¼ 0,
and the solution is pffiffiffi Z Z1 1 1 et ðx, Þ d g0 ðyÞðy, Þ dy gðx, tÞ ¼ p 0 cos2 þ sin2 0 Z1 þ g0 ðyÞ0 ðyÞ dy et cot 0 ðxÞ:
which maps S 2 ð0, 1Þ to x 2 ð1, 1Þ, and 1 1 3 2 2 , lðSÞ ¼ 2 cosh2 x: qðxÞ ¼ lðSÞ þ 2 4 4 4 lðSÞ
0 ðxÞ ¼ 9
pffiffiffiffiffiffiffi 2s0 es0 x ,
This case was missing on page 59 of [17].
ð34Þ
When ! 0, qðxÞ ! 2 =4; this is the case in the appendix. It has a pure continuous spectrum. When 6¼ 0, as x ! 1, qðxÞ ! 1 exponentially, according to WT theory; it has a pure point spectrum and both end points x ¼ 1 are limit point cases, and we do not need any boundary condition. This is the special case 2 in section 3. The spectral problem (16) with q(x) given by (34) uniquely determines the eigenvalues, denoted by n, n ¼ 1, 2, 3, . . . , and their corresponding normalized eigenfunctions, denoted by n(x) with Z1 n ðxÞm ðxÞ dx ¼ mn : 1
The initial condition function can be expanded by n(x) as Z1 g0 ðxÞn ðxÞ dx, cn ¼ 1
then the solution reads gðx, tÞ ¼
Therefore 0(x) can be written as
ð33Þ
1 X
cn en t n ðxÞ:
n¼1
This example demonstrates that a small disturbance in the coefficient of the equation can change the nature of the problem from a pure continuous spectrum to a pure point spectrum.
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Let g ¼ f eð
4.3. Exponentially decreasing volatility Assuming S
,
ð35Þ
@c @2 c e2S 2 ¼ 0, @t @S cðS, 0Þ ¼ maxðS 1, 0Þ: From (9), we have Z x ¼ eS dS ¼ eS ,
2 uðx, tÞ: 2
ð37Þ
Let ¼ þ A, then
ð47Þ
f ¼ fxx q ðxÞ f ,
ð48Þ
q ðxÞ ¼ qðxÞ A:
ð49Þ
where gðx, tÞ ¼ e
@g @2 g 1 ¼ þ g, @t @x2 4x2
S=2
cðS, tÞ:
1 < x < þ1,
pffiffiffi gðx, 0Þ ¼ g0 ðxÞ ¼ maxð xðln x 1Þ, 0Þ:
ð38Þ
ð39Þ ð40Þ
2
Let gðx, tÞ ¼ es t f ðx, tÞ, then 1 2 fxx þ s þ 2 f ¼ 0: 4x
ð41Þ
By using formula (4:10:3) on pages 74–75 of [17], we have the solution Z 1 pffiffiffi x½J0 ðsxÞY0 ðsÞ Y0 ðsxÞJ0 ðsÞ 2 sg1 ðsÞes t ds, gðx, tÞ ¼ 2 ðsÞ þ Y 2 ðsÞ J 0 0 0 ð43Þ pffiffiffi y½J0 ðsyÞY0 ðsÞ Y0 ðsyÞJ0 ðsÞg0 ðyÞdy: ð44Þ
1
To the best of our knowledge, this formula for an exponentially decreasing volatility function has never been presented before in the literature.
4.4. Asian option pricing In [20], the equation to price a continuously sampled arithmetic Asian option is reduced to @g 1 2 @2 g þ uðx, tÞg ¼ 0, @t 2 @x2
1 < x < 1,
2 1 1 1 uðx, tÞ ¼ 2 ex þ þ 2 þ ex : 2 2 2
Since q ðxÞ ! 0 as x ! þ1 and q ðxÞ ! 1 as x ! 1, the problem has a point spectrum at x ¼ 1, and has a continuous (may also have a point) spectrum at x ¼ þ1. In this case, both end points are limit point cases, and we do not need any boundary condition. The spectrum is defined by the function m1 ðÞ and m1 ðÞ, the solution of which can be obtained according to formula (26).
5. A few solvable q(x)
The end point x ¼ 1 is a limit circle case, and we need a boundary condition f ð1Þ ¼ 0. Equation (41) has two basic solutions in terms of Bessel functions pffiffiffi pffiffiffi f1 ¼ xJ0 ðsxÞ, f2 ¼ xY0 ðsxÞ: ð42Þ
462
qðxÞ ¼
ð36Þ
The domain S 2 ð0, þ1Þ is mapped to x 2 ð1, þ1Þ and the equation for g(x, t) is
þ1
, then
As x ! 1, qðxÞ ! 1 exponentially, 1 1 2 2 ¼ A: as x ! þ1, qðxÞ ! 2 þ 2 2
the original option pricing problem is
Z
=2Þt
f ¼ fxx qðxÞ f , ðSÞ ¼ e
g1 ðsÞ ¼
2
ð45Þ
We now list a few solvable q(x), for which the analytical solutions have been studied and presented in [17] (table 1). The solution for g(x, t) can be written in terms of a summation or integration of a special function. To recover the solution for option price c(S, t), one has to solve equation 1 1 ðS Þ2 SS ¼ qðxÞ 4 2 for (S) given the function q(x). This task seems nontrivial since the transformation between S and x, Z 1 dS, xðSÞ ¼ ðSÞ also involves (S). This problem has to be solved case by case. Nevertheless, these solvable q(x) will provide a clue for working out other tractable option pricing models. This is an interesting problem for future research. Another interesting direction for future research would be applying the spectral theory to an inverse problem and studying the uniqueness problem of volatility fitting to a given set of option prices with different strikes10.
6. Conclusion With some transformation, we convert the problem of option pricing under state-dependent volatility into an
ð46Þ 10
We thank an anonymous referee for pointing this out.
QUANTITATIVE FINANCE
Option pricing with Weyl–Titchmarsh theory
Table 1. List of potential functions q(x) for which the analytical solutions are available in [17]. qðxÞ 2
x ð1=4Þ tan2 x þ m2 sec2 x ð1=4Þ ðð2 ð1=4ÞÞ=x2 Þ x e2x x2 þ ðð2 ð1=4ÞÞ=x2 Þ ððrðr þ 1ÞÞ=x2 Þ c=x ðð2ð 1Þð2 1 Þ cosh xÞ=ð4 sinh2 xÞÞ þðð22 4 þ ð1 2Þ2 Þ=ð4 sinh2 xÞÞ ðð1 ÞÞ=ð4 cosh2 ð1=2ÞxÞ
x domain
Solution expanded in terms of a special function
Ref. [17] Section
1 < x < 1 p=2 < x < p=2 0a
Hermite Generalized Legendre th order Bessel or Hankel 1=3 order Bessel pffiffiffi J ðex Þ, ¼ i , Im > 0 Generalized Laguerre Generalized Laguerre Hypergeometric
4.2 4.6 4.8 4.12, 4.13 4.14 4.15 4.16 4.17
1 < x < 1
Hypergeometric
4.18
initial value problem of the Schro¨dinger equation with a certain potential. By using the separating variable method, the equation becomes a Schro¨dinger spectral problem. With Weyl–Titchmarsh theory, we are able to determine whether we need a boundary condition at each end of the domain. We are also able to determine the nature of the eigenvalues and eigenfunctions, and write the solution analytically in a Stieltjes integral. The study of a few cases demonstrates our new way of producing analytical option pricing formulae by using Weyl–Titchmarsh theory for certain volatility functions. The formula for the case where volatility is an exponentially decreasing function of the stock price seems to be a new result, and is therefore regarded as a contribution to the option pricing literature. In this paper, we focus on the pricing vanilla option. Extending our method to price exotic options, such as barrier or lookback options, is a topic for future research.
Acknowledgments We wish to thank two anonymous referees for helpful comments and suggestions. YSL is supported by the National Natural Science Foundation of China (1030). JEZ is supported by the Research Grants Council of Hong Kong under grant CERG HKU-1068/01H.
Appendix. The Black–Scholes case Assuming ðSÞ ¼ S,
1 ¼ pffiffiffi 0 , 2
where 0 is a positive constant. For brevity, we will drop the superscript star in in the following derivation. From (9), we have x¼
1 ln S,
or
S ¼ ex ,
cðS, tÞ ¼
pffiffiffiffi S gðx, tÞ,
ðA:1Þ pffiffiffi where a constant has been scaled out in the transformation between c(S, t) and g(x, t). When
S 2 ð0, 1Þ, x 2 ð1, 1Þ, equations (10) and (11) become @g @2 g 1 2 ¼ g, ðA:2Þ @t @x2 4 0, x < ðln KÞ=, gðx, 0Þ ¼ g0 ðxÞ ¼ ð1=2Þx ðA:3Þ e Keð1=2Þx , x > ðln KÞ=: 2
Let g ¼ etð1=4Þ t f , then f ¼ f 00 ; in this case, x ¼ 1 are limit point cases, and we do not need any boundary condition. We take d ¼ 0, pffiffiffi pffiffiffi 1 ðx,Þ¼cos x, ðx,Þ¼ pffiffiffi sin x, pffiffiffi pffiffiffi ma ðÞ¼i , mb ðÞ¼i , pffiffi pffiffi i x , b ðx,Þ¼ei x , a ðx,Þ¼e ( pffiffiffi 1=ð2 Þ, >0, 0
ðÞ¼ ðÞ¼0, 0, <0, ( pffiffiffi =2, >0, 0 ðÞ¼ 0, <0, Z
pffiffiffi tð1=4Þ2 t d Z 1 pffiffiffi cos x e 2 1 0 Z pffiffiffi pffiffiffi 1 1 2 cos y g0 ðyÞ dy þ sin x etð1=4Þ t p 0 Z pffiffiffi d 1 pffiffiffi sin y g0 ðyÞ dy 2 1 Z Z1 1 1 s2 tð1=4Þ 2 t cos sx e ds cos sy g0 ðyÞ dy ¼ p 0 1 Z1 1 2 2 þ sin sx es tð1=4Þ t ds p 0 Z1 pffiffiffi sin sy g0 ðyÞ dy, s ¼ 1 Z 2 1 ð1=4Þ 2 t 1 isðxyÞ e e þ eisðxyÞ es t ds ¼ 2p 0 Z1 g0 ðyÞ dy 1 Z1 Z 1 ð1=4Þ 2 t 1 isðxyÞ s2 t e ¼ e e ds g0 ðyÞ dy: 2p 1 1
1 gðx, tÞ ¼ p
1
463
Y Li and J E Zhang
QUANTITATIVE FINANCE
Using the evolution formula of the transformation pffiffiffi Z1 2 p isðxyÞ s2 t e e ds ¼ pffiffi eðxyÞ =4t , t 1
Fourier
we get 1 2 gðx, tÞ ¼ pffiffiffiffiffi eð1=4Þ t 2 pt ¼ eð1=4Þ
2
t
eðxyÞ
1 pffiffiffiffiffi 2 pt 2
=4t
Z
References
1
g0 ðyÞ e Z
ðxyÞ2 =4t
dy
1
1
eð1=2Þy K eð1=2Þy
ð1=Þ ln K
dy ¼ I1 þ I2 ,
where 2
2
2
I1 ¼ eð1=4Þ t e½x ðxþtÞ =4t Z1 2 1 pffiffiffiffiffi e½ðxþtÞy =4t dy 2 pt ð1=Þ ln K 2
2 2
¼ eð1=4Þ t eð2xtþ t Þ=4t Z 1 1 y21 =2 pffiffiffiffiffiffi dy1 , pffiffiffi e 2p ðxþtð1=Þ ln KÞ= 2t ðx þ tÞ y pffiffiffiffiffi 2t p ffiffi lnðex =KÞ t pffiffiffiffiffi þ pffiffiffi ¼ eð1=2Þx N 2t 2 y1 ¼
and 2 1 2 2 I2 ¼ K pffiffiffi eð1=4Þ t Ke½x ðxtÞ =4t Z1 2 1 pffiffiffiffiffi e½ðxtÞy =4t dy 2 t ð1=Þ ln K
1 2 2 2 ¼ K pffiffiffi eð1=4Þ t eð2xtþ t Þ=4t Z 1 1 y22 =2 pffiffiffi dy2 , pffiffi e 2 ðxtð1=Þ ln KÞ= 2t ðx tÞ y pffiffiffi 2t p ffiffi lnðex =KÞ t pffiffiffi pffiffiffi , ¼ Keð1=2Þx N 2t 2 y2 ¼
ðA:4Þ
and Z NðxÞ ¼
x
1 2 pffiffiffiffiffiffi ey =2 dy 2p 1
is the cumulative normal distribution function. Finally, transforming back to financial variables, we have the
464
Black–Scholes option pricing formula pffiffi pffiffi lnðS=KÞ 0 t lnðS=KÞ 0 t pffiffi þ pffiffi KN : cðS, tÞ ¼ SN 2 2 0 t 0 t
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