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Higher order asymptotic option valuation for non-Gaussian dependent returns
Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058 www.elsevier.com/locate/jspi
Higher order asymptotic option valuation for non-Gaussian dependent returns Kenichiro Tamaki, Masanobu Taniguchi∗ Department of Mathematical Sciences, School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan Available online 25 July 2006
Abstract This paper discusses the option pricing problems using statistical series expansion for the price process of an underlying asset. We derive the Edgeworth expansion for the stock log return via extracting dynamics structure of time series. Using this result, we investigate influences of the non-Gaussianity and the dependency of log return processes for option pricing. Numerical studies show some interesting features of them. © 2006 Elsevier B.V. All rights reserved. MSC: Primary 91B24; 62M10; secondary 62M15; 91B84 Keywords: Black and Scholes model; Edgeworth expansion; Non-Gaussian stationary process; Option pricing
1. Introduction Black and Scholes (1973) provided the foundation of modern option pricing theory. Despite its usefulness, however, the Black and Scholes theory entails some inconsistencies. It is well known that the model frequently misprices deep in-the-money and deep out-of-the-money options. This result is generally attributed to the unrealistic assumptions used to derive the model. In particular, the Black and Scholes model assumes that stock prices follow a geometric Brownian motion with a constant volatility under an equivalent martingale measure. In order to avoid this drawback, Jarrow and Rudd (1982) proposed a semiparametric option pricing model to account for non-normal skewness and kurtosis in stock returns. This approach aims to approximate the risk-neutral density by a statistical series expansion. Jarrow and Rudd (1982) approximated the density of the state price by an Edgeworth series expansion involving the log-normal density. Corrado and Su (1996a) implemented Jarrow and Rudd’s formula to price options. Corrado and Su (1996b, 1997) considered Gram-Charlier expansions for the stock log return rather than the stock price itself. Rubinstein (1998) used the Edgeworth expansion for the stock log return. Jurczenko et al. (2002) compared these different multi-moment approximate option pricing models. Also they investigated in particular the conditions that ensure the martingale restriction. As in Kariya (1993) and Kariya and Liu (2002), the time series structure of return series does not always admit a measure which makes the discounted process a martingale. Hence, we will not be able to develop an arbitrage pricing theory by forming an equivalent portfolio. In such a case, we often regard the expected value of the present value ∗ Corresponding author. Tel.: +81 3 5286 8095.
E-mail addresses: [email protected] (K. Tamaki), [email protected] (M. Taniguchi). 0378-3758/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2006.06.023
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of a contingent claim as a proxy for pricing may be with help of a risk neutrality argument. In view of this, Kariya (1993) considered pricing problems with no martingale property and approximated the density of the state price by the Gram-Charlier expansion for the stock log return. In this paper, we consider option pricing problems by using Kariya’s approach. In Section 2, we derive the Edgeworth expansion for the stock log return via extracting dynamics structure of time series. Using this result, we investigate influences of the non-Gaussianity and the dependency of log return processes for option pricing. Numerical studies illuminate some interesting features of the influences. In Section 3, we give option prices based on the risk neutrality argument. In Section 4, we discuss a consistent estimator of the quantities in our results. Section 5 concludes. The proofs of theorems are relegated to Section 6. 2. Edgeworth expansion of log return Let {St ; t 0} be the price process of an underlying security at trading time t. The j-th period log return Xj is defined as log ST0 +j − log ST0 +(j −1) = + 1/2 Xj ,
j = 1, 2, . . . , N,
(1)
where T0 is present time, N = / is the number of unit time intervals of length during a period = T − T0 and T is the maturity date. Then the terminal price ST of the underlying security is given by ⎫ ⎧ N ⎬ ⎨ 1/2 ST = ST0 exp + (2) Xj . ⎭ ⎩ N j =1
Remark 1. In the Black and Scholes option theory the price process is assumed to be a geometric Brownian motion ST = ST0 exp( + W ),
(3)
where the process {Wt ; t ∈ R} is a Wiener process with drift 0 and variance t. From (3), the log return at discreterized time point can be written as log St+j − log St+(j −1) = + 1/2 j ,
j ∼ iid N(0, 1).
(4)
The expression of (1) is motivated from (4). First, we derive an analytical expression for the density function of ST . Since from (2) the distribution of ST depends
X on that of ZN = N −1/2 N j =1 j , we consider the Edgeworth expansion of the density function of ZN . If we assume that Xj are independently and identically distributed random variables with mean zero and finite variance, it is easy to give the Edgeworth expansion for ZN (the classical Edgeworth expansion). However, a lot of financial empirical studies show that Xj ’s are not independent. Thus, we suppose that {Xj } is a dependent process which satisfies the following assumption. (A1) {Xt ; t ∈ Z} is fourth order stationary in the sense that (i) (ii) (iii) (iv)
E(Xt ) = 0, cum(Xt , Xt+u ) = cX,2 (u), cum(Xt , Xt+u1 , Xt+u2 ) = cX,3 (u1 , u2 ), cum(Xt , Xt+u1 , Xt+u2 , Xt+u3 ) = cX,4 (u1 , u2 , u3 ).
(A2) The cumulants cX,k (u1 , . . . , uk−1 ), k = 2, 3, 4, satisfy ∞
(1 + |uj |2−k/2 )|cX,k (u1 , . . . , uk−1 )| < ∞
u1 ,...,uk−1 =−∞
for j = 1, . . . , k − 1. (A3) J-th order (J 5) cumulants of ZN are all O(N −J /2+1 ).
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Under (A2), {Xt ; t ∈ Z} has the k-th order cumulant spectral density. Let fX,k be the k-th order cumulant spectral density evaluated at frequency 0 ∞
fX,k = (2)−(k−1)
cX,k (u1 , . . . , uk−1 )
u1 ,...,uk−1 =−∞
for k = 2, 3, 4. First, we state the following result. Theorem 1. Suppose that (A1)–(A3) hold. The third order Edgeworth expansion of the density function of Z = (2fX,2 )−1/2 ZN is given by (2)1/2 −1/2 fX,3 1 −1 fX,2 g(z) = (z) 1 + N N H (z) − H2 (z) 3 6 4 fX,2 (fX,2 )3/2 −1 fX,4 −1 (fX,3 )2 + H4 (z) + N H6 (z) + o(N −1 ), (5) N 12 36 (fX,2 )2 (fX,2 )3 where (·) is the standard normal density function, Hk (·) is the k-th order Hermite polynomial and fX,2 =
∞
|u|cX,2 (u).
u=−∞
Many authors have proposed to use different statistical series expansion to price options (see Jarrow and Rudd, 1982; Corrado and Su, 1996a, b,1997; Rubinstein, 1998; Kariya, 1993). Here, we give the Edgeworth expansion for the stock log return in powers of N −1/2 . A European call option can be viewed as a security which pays at time T its holder the amount XT∗ = max(ST − K, 0), where K is the exercise or strike price. As in Kariya (1993), we price XT∗ by its discounted expected value; C = exp(−r)ET0 (XT∗ ),
(6)
where r is the interest rate which is regarded as a constant for the remaining period and ET0 (·) is evaluated at T0 . Evaluate (6) based on the density in (5). Then writing d1 = (log ST0 /K + + 2fX,2 )/(2fX,2 )1/2 , d2 = d1 − (2fX,2 )1/2 , we obtain the following theorem Theorem 2. Let a1 = exp(−r) and a2 = exp( + fX,2 ). Then C = G0 + +
(2)1/2 −1/2 fX,3 1 −1 fX,2 G − G2 N N 3 6 4 fX,2 (fX,2 )3/2
−1 fX,4 −1 (fX,3 )2 N N G + G6 + o(N −1 ), 4 12 36 (fX,2 )2 (fX,2 )3
where G0 = a1 {a2 ST0 (d1 ) − K(d2 )},
(7)
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Gk = a1 a2 ST0
⎧ k−1 ⎨ ⎩
j =1
⎫ ⎬
(2fX,2 )j/2 Hk−j −1 (−d2 )(d1 ) + (2fX,2 )k/2 (d1 ) , ⎭
for k = 2, 3, 4, 6, where (·) is the standard normal distribution function. ,f From (7) it is seen that the asymptotic expansion of the option price depends on fX,2 , fX,2 X,3 and fX,4 . Hence, we can see influences of the non-Gaussianity and the dependency of the log return processes for the higher order option valuation.
Corollary 1. Write C = G0 + N −1/2 CG,2 + N −1 CG,3 + N −1 CD,3 + o(N −1 ), where CG,2 =
(2)1/2 fX,3 G3 , 6 (fX,2 )3/2
CG,3 =
fX,4 (fX,3 )2 G + G6 , 4 12 (fX,2 )2 36 (fX,2 )3
CD,3 = −
1 fX,2 G2 . 4 fX,2
If {Xt ; t ∈ Z} is independent, then CD,3 = 0. If {Xt ; t ∈ Z} is a Gaussian process, then CG,2 = CG,3 = 0. Example 1. Suppose that Xj , j = 1, . . . , N, are independently and identically distributed random variables. Let −(k−1) = 0 and f cX,k , k = 2, 3, 4. The price of a European call cX,k = cX,k (0), k = 2, 3, 4. Note that fX,2 X,k = (2) option CIID is given by cX,3 cX,4 1 1 G3 + N −1 G4 CIID = G0 + N −1/2 3/2 6 24 (cX,2 ) (cX,2 )2 +
1 −1 (cX,3 )2 G6 + o(N −1 ), N 72 (cX,2 )3
where Gk , k = 0, 3, 4, 6, are defined in Theorem 2 with fX,2 = (2)−1 cX,2 . If = r − cX,2 /2, then a1 a2 = 1 so that G0 equals the Black and Scholes formula. Example 2. In Example 1, suppose that Xj , j = 1, . . . , N, are distributed as t-distribution with degrees of freedom. Then, for > 4 Ct = Gt,0 + N −1 Gt,3 + o(N −1 ), where Gt,0 = a1 {a2 ST0 (d1 ) − K(d2 )},
, a2 = exp + 2( − 2) 1/2 d1 = log ST0 /K + + , −2 −2 1/2 d2 = d1 − , −2
K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
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option prices
20
18
16
14
12 4
5
6
7
8
9
Fig. 1. For t-distribution with degrees of freedom in Example 2, the approximation up to the first (Ct,1 , dotted line) and third order (Ct,3 , solid 30 , N = 30, r = = 0.05 and 4 < < 9. line) of the option price are plotted with ST0 = K = 100, = 365
and
⎫ ⎧ 3 ⎬ j/2 a1 a2 ST0 ⎨ 2 Gt,3 = H3−j (−d2 )(d1 ) + (d1 ) . ⎭ 4( − 4) ⎩ −2 −2 j =1
In order to show influences of higher order terms, in Fig. 1, we plotted Ct,1 = Gt,0 (dotted line) and Ct,3 = Gt,0 + 30 1 N −1 Gt,3 (solid line) of Example 2 with ST0 = K = 100, = 365 , N = 30 ( = 365 ), r = = 0.05 and 4 < < 9. From this, we observe that Ct,3 diverges as → 4. Example 3. Let {Xt : t ∈ Z} be the ARCH(1) process 1/2
Xt = ht t
and
2 ht = 0 + 1 Xt−1 ,
where 0 > 0, 1 0, { t : t ∈ Z} is a sequence of independently and identically distributed random variables with E( t ) = 0,
E( 2t ) = 1,
E( 3t ) = 0,
E( 4t ) = m,
m > 1,
and t is independent of Xt−s , s > 0. Then fX,2 = fX,4 =
1 0 , 2 1 − 1
fX,3 = 0,
fX,2 = 0,
1
20 (m − 3 + 5m 1 − 3 1 + 2m 21 − 2m 31 )
(2)3
(1 − 1 )3 (1 − m 21 )
for ma 21 < 1. Hence, CARCH(1) = GARCH(1),0 + N −1 GARCH(1),3 + o(N −1 ),
,
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14
option prices
12
10
8
6
-0.58
-0.25
0.0
0.25
0.58
1 Fig. 2. For ARCH(1) in Example 3, the approximation up to the first (CARCH(1),1 , dotted line) and third order (CARCH(1),3 , solid line) of the option √ √ 30 , N = 30, r = = 0.05, m = 3, = 0.5 and −1/ 3 < < 1/ 3. price are plotted with ST0 = K = 100, = 365 0 1
where GARCH(1),0 = a1 a2 ST0 (d1 ) − a1 K(d2 ),
0 , a2 = exp + 2(1 − 1 ) 0 1/2 0 , d1 = log ST0 /K + + 1 − 1 1 − 1 0 1/2 d2 = d1 − 1 − 1 and GARCH(1),3 =
a1 a2 ST0 m − 3 + 5m 1 − 3 1 + 2m 21 − 2m 31 24 (1 − 1 )(1 − m 21 ) ⎧ ⎫ 3 ⎨ ⎬ 0 j/2 0 2 × H3−j (−d2 )(d1 ) + (d1 ) . ⎩ ⎭ 1 − 1 1 − 1 j =1
GARCH(1),3 (solid In Fig. 2, we plotted CARCH(1),1 = GARCH(1),0 (dotted line) and CARCH(1),3 = GARCH(1),0 + N −1√ √ 30 1 line) of Example 3 with ST0 =K =100, = 365 , N =30 (= 365 ), r ==0.05, m=3, 0 =0.5 and −1/ 3 < 1 < 1/ 3. Fig. 2 illuminates influences of higher order terms under Gaussian innovations. From this, we can see that CARCH(1),3 √ diverges as 1 → ±1/ 3. In Fig. 3, we plotted CARCH(1),1 (dotted line) and CARCH(1),3 (solid line) of Example 3 with ST0 = 100, K = 95, 30 = 365 , N = 30, r = = 0.05, 0 = 0.5, 1 = 0.3 and 1 < m < 9. Fig. 3 illuminates influences of non-Gaussian innovations. From this, we observe that CARCH(1),3 decreases as m → 9. The first order term CARCH(1),1 is a constant because it is independent from m.
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14.2845
option prices
14.0
13.5
13.0
1
5
3
7
9
m Fig. 3. For ARCH(1) in Example 3, the approximation up to the first (CARCH(1),1 , dotted line) and third order (CARCH(1),3 , solid line) of the option 30 , N = 30, r = = 0.05, = 0.5, = 0.3 and 1 < m < 9. price are plotted with ST0 = 100, K = 95, = 365 0 1
Next we consider option pricing problems for a class of processes generated by uncorrelated random variables, which includes the linear process and an important class in time series analysis. Here, we are concerned with the following process: Xt =
∞
aj t−j ,
t ∈ Z,
(8)
j =0
where { t ; t ∈ Z} is a sequence of uncorrelated random variables. Instead of (A1) and (A2) we make the following assumption. (B1) { t ; t ∈ Z} is fourth order stationary in the sense that (i) (ii) (iii) (iv)
E( t ) = 0, Var( t ) = 2 , cum( t , t+u1 , t+u2 ) = c ,3 (u1 , u2 ), cum( t , t+u1 , t+u2 , t+u3 ) = c ,4 (u1 , u2 , u3 ).
(B2) The cumulants c ,k (u1 , . . . , uk−1 ), k = 3, 4, satisfy ∞
(1 + |uj |2−k/2 )|c ,k (u1 , . . . , uk−1 )| < ∞,
u1 ,...,uk−1 =−∞
for j = 1, . . . , k − 1. (B3) {aj ; j ∈ Z} satisfies ∞ j =0
(1 + |j |)|aj | < ∞.
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Under (B2), { t ; t ∈ Z} has the k-th order cumulant spectral density. Let f ,k be the k-th order cumulant spectral density evaluated at frequency 0 ∞
f ,k = (2)−(k−1)
c ,k (u1 , . . . , uk−1 )
u1 ,...,uk−1 =−∞
for k = 2, 3, 4. The response function of {aj ; j ∈ Z} is defined by A( ) =
∞
aj e−ij
j =0
for −∞ < < ∞. Under (B1)–(B3), (A1) and (A2) hold. Hence, from Theorem 1, we have Corollary 2. Suppose that (B1)–(B3) and (A3) hold. Let a1 = exp(−r) and a2 = exp + 21 2 A2 . Then C = G0 +
22 A3 −1/2 1 N f ,3 G3 − N −1 f ,2 G2 3 3 3 |A| 2A2
3 −1 24 N f ,4 G4 + 6 N −1 f 2,3 G6 + o(N −1 ), 4 3 9
+
where A = A(0), f ,2 = 2
∞
|j2 |aj1 aj1 +j2 ,
j1 ,j2 =0
Gk , k = 0, 2, 3, 4, 6, are given in Theorem 2 with fX,2 =
2 2 A . 2
Example 4. Let {Xt ; t ∈ Z} be AR(1) process Xt = Xt−1 + t ,
| | < 1.
Note that A=
1 , 1−
f ,2 =
2 (1 + )(1 − )3
.
The price of a European call option CAR(1) is given by CAR(1) = GAR(1),0 + N −1/2 GAR(1),2 + N −1 GAR(1),3 + o(N −1 ), where GAR(1),0 = a1 {a2 ST0 (d1 ) − K(d2 )}, GAR(1),2 =
22 f ,3 G3 , 33
3 24 G + f G + (f ,3 )2 G6 , 2 ,4 4 1 − 2 34 96
2 a2 = exp + , 2(1 − )2
GAR(1),3 = −
K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
1051
120
100
option prices
80
60
40
20
-1.00
-0.50
0.0
0.50
0.75
Fig. 4. For AR(1) in Example 4, the approximation up to the first (CARCH(1),1 , dotted line), second (CARCH(1),2 , dashed line) and third order 30 , N = 30, r = = 0.05, = 1, f (CARCH(1),3 , solid line) of the option price are plotted with ST0 = K = 100, = 365 X,3 = −0.1, fX,4 = 0.2 and −1 < < 0.75.
d1 = log ST0 /K + + d2 = d1 −
2 (1 − )2
1/2 , 1−
1/2 , 1−
and Gk = a1 a2 ST0
⎧ k−1 1/2 j ⎨ ⎩
j =1
1−
Hk−j −1 (−d2 )(d1 ) +
k 1/2 1−
⎫ ⎬
(d1 ) , ⎭
for k = 2, 3, 4, 6. In order to show influences of higher order terms, in Fig. 4, we plotted CAR(1),1 = GAR(1),0 (dotted line), CAR(1),2 = GAR(1),0 + N −1/2 GAR(1),2 (dashed line) and CAR(1),3 = GAR(1),0 + N −1/2 GAR(1),2 + N −1 GAR(1),3 (solid line) of 30 1 Example 4 with ST0 = K = 100, = 365 , N = 30 ( = 365 ), r = = 0.05, = 1, fX,3 = −0.1, fX,4 = 0.2 and −1 < < 0.75. From this, we observe that CAR(1),k , k = 1, 2, 3 diverges as → 1. In Examples 2 and 3, although the third order terms diverge, the first order terms do not diverge. On the other hand, in Example 4, even the first order term does not converge as → 1. This fact is attributed to finiteness of the variances. 3. Martingale restriction In the previous section, we considered pricing problems with no martingale property. Now we recall that the theoretical price of a option is based on the risk neutrality argument. In this section, to investigate influences of the martingale restriction. we derive the option price based on the risk neutrality argument (see Cox and Ross, 1976; Longstaff, 1995).
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Let d1∗ = (log ST0 /K + r + fX,2 )/(2fX,2 )1/2 , d2∗ = d1∗ − (2fX,2 )1/2 . Then we have Theorem 3. The fair price C ∗ of a European call option is given by C ∗ = G∗0 + +
(2)1/2 −1/2 fX,3 1 −1 fX,2 ∗ ∗ N N G − G 3 6 4 fX,2 2 (fX,2 )3/2
(fX,3 )2 ∗ −1 fX,4 N G∗4 + N −1 G6 + o(N −1 ), 2 12 36 (fX,2 ) (fX,2 )3
(9)
where G∗0 = ST0 (d1∗ ) − e−r K(d2∗ ), G∗k
= ST0
k−1
(2fX,2 )j/2 Hk−j −1 (−d2∗ )(d1∗ ),
j =1
for k = 2, 3, 4 and ⎡ ⎤ 2 G∗6 = ST0 ⎣ (2fX,2 )j/2 H5−j (−d2∗ ) − 2fX,2 H3−j (−d2∗ ) ⎦ (d1∗ ). j =1
Example 5. Suppose that {Xt ; t ∈ Z} is AR(1) process in Example 4. Then the fair price of a European call option ∗ CAR(1) is given by ∗ = G∗AR(1),0 + N −1/2 G∗AR(1),2 + N −1 G∗AR(1),3 + o(N −1 ), CAR(1)
where G∗AR(1),0 = ST0 (d1∗ ) − e−r K(d2∗ ), G∗AR(1),2 =
22 f ,3 G∗3 , 33
3 24 G∗2 + 4 f ,4 G4 + 6 (f ,3 )2 G∗6 , 2 1− 3 9
1/2 2 d1∗ = log ST0 /K + r + , 1− 2(1 − )2 1/2 ∗ ∗ , d2 = d1 − 1− ⎧ ⎫ k−1 1/2 j ⎨ ⎬ G∗k = ST0 Hk−j −1 (−d2∗ )(d1∗ ) , ⎩ ⎭ 1−
G∗AR(1),3 = −
j =1
for k = 2, 3, 4 and ⎡ ⎤ 2 1/2 j 2 G∗6 = ST0 ⎣ H5−j (−d2∗ ) − H (−d2∗ ) ⎦ (d1∗ ). 2 3−j 1− (1 −
) j =1
K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
1053
100
option prices
80
60
40
20
-1.0
-0.5
0.0
0.5
1.0
∗ ∗ Fig. 5. For AR(1) in Example 5, the approximation up to the first (CARCH(1),1 , dotted line), second (CARCH(1),2 , dashed line) and third order 30 , N = 30, r = 0.05, = 1, f ∗ (CARCH(1),3 , solid line) of the option price are plotted with ST0 = K = 100, = 365 X,3 = −0.1, fX,4 = 0.2 and −1 < < 1.
∗ ∗ In Fig. 5, we plotted CAR(1),1 = G∗AR(1),0 (dotted line), CAR(1),2 = G∗AR(1),0 + N −1/2 G∗AR(1),2 (dashed line) and 30 ∗ = G∗AR(1),0 + N −1/2 G∗AR(1),2 + N −1 G∗AR(1),3 (solid line) of Example 5 with ST0 = K = 100, = 365 , N = 30 CAR(1),3 1 ( = 365 ), r = 0.05, = 1, fX,3 = −0.1, fX,4 = 0.2 and −1 < < 1. Unlike Example 4, we observe that CAR(1),k , k = 1, 2, 3 converge to ST0 (=100) as → 1.
4. Estimation From (1), Xj −N0 , j = 1, . . . , N0 , are available, where N0 = T0 /. Therefore, in this section we consider to estimate ,f , fX,2 , fX,2 X,3 and fX,4 in Theorems 1 and 2 consistently based on the past observations. From (A1), is the mean of stock log returns. Hence, a natural unbiased estimator of is the sample mean ˆ =
N0 1 {log Sj − log S(j −1) }. N0
(10)
j =1
The variance of ˆ is given by Var() ˆ =
1 N0
N 0 −1 u=−(N0 −1)
1−
|u| cX,2 (u). N0
Hence, under (A2), ˆ given in (10) is a consistent estimator of . , we define the lag window function w(·) which is an Moreover, in order to construct consistent estimator of fX,2 even and piecewise continuous function satisfying the conditions, w(0) = 1, |w(x)|1
for all x,
w(x) = 0
for |x| > 1.
(11)
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Table 1 Values of consistent estimators AMOCO
ˆ fˆX,2 fˆX,3 ˆ (fX,2 )3/2 fˆX,4 (fˆX,2 )2 fˆX,2 fˆX,2
FORD
HP
IBM
MERCK
0.235103 0.002937
0.045337 0.016006
0.133815 0.016202
0.017165 0.003085
0.481340 0.004534
−0.706149
−3.078889
8.501363
0.470144
2.419969
2.278478
−0.280973
8.651378
15.0914
−5.520428
0.169291
27.18047
−22.78799
−2.249174 −37.3221
Let fˆX,2 =
N 0 −1
|u|cˆX,2 (u)w BN0 u ,
u=−(N0 −1)
where cˆX,2 (u) is the sample autocovariance function at lag u N0 −|u| 1 cˆX,2 (u) = {log S(j +|u|) − log S(j +|u|−1) − } ˆ N0 j =1
× {log Sj − log S(j −1) − }, ˆ
(12)
and BN0 → 0 as N0 → ∞, but (BN0 )3 N0 → ∞. Then we can easily see that under (A2), fˆX,2 given in (12) is a consistent estimator of fX,2 . Since fX,k , k = 2, 3, 4, are the k-th order cumulant spectral density evaluated at frequency 0, using Brillinger and Rosenblatt (1967a, b) formula, we construct consistent estimators fˆX,k of fX,k (k = 2, 3, 4). See also Brillinger (1981). Thus, we can consistently estimate all the quantities in Theorems 1 and 2 (e.g., Gj , j = 0, 2, 3, 4, 6.) by the and f and fˆ corresponding quantities replacing , fX,2 ˆ fˆX,2 X,k by , X,k (k = 2, 3, 4). For example, we discuss a consistent estimator for New York stock exchange data. The data are daily returns of AMOCO, FORD HP, IBM and MERCK companies. The individual time series are the last 1024 data points from stocks, representing the daily returns for the five companies from February 2, 1984, to December 31, 1991. We used the window functions
W (u1 , . . . , uk−1 ) =
2−(k−1)
If |u1 |, . . . , |uk−1 | 1,
0
otherwise
for fˆX,k (k = 2, 3, 4) and let w(u) = 1 for |u|1, where w(u) is defined in (11). Also we used the bandwidth in 1 1 1 frequency direction with BN0 = 50 for fˆX,2 , BN0 = 30 for fˆX,3 and BN0 = 10 for fˆX,4 and fˆX,2 (see Brillinger and Rosenblatt, 1967a, b; Brillinger, 1981). and f Table 1 show the values of consistent estimators of , fX,2 X,k (k = 2, 3, 4) for the five companies. From these results, we can see that the quantities involved in higher order terms are quite different from the Black and Scholes model. Therefore, in general the assumptions of the Gaussianity and the independency of stock log returns will not hold. Table 2 shows the values of the approximation up to the first C1 , second C2 and third order C3 of the option prices 30 , N = 30, r = 0.05. From this results, we observe that option prices are strongly affected with ST0 = K = 100, = 365 by third order terms except for AMOCO and MERCK.
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Table 2 Option prices
C1 C2 C3
AMOCO
FORD
HP
IBM
MERCK
2.776419 2.809884 2.881406
4.031663 3.979554 4.345765
4.472833 4.434833 6.392765
1.699889 1.700269 1.374588
4.689151 4.495491 4.650024
AMOCO
FORD
HP
IBM
MERCK
1.764254 1.769475 1.83751
3.827175 3.784867 4.111153
3.849221 3.954549 6.09142
1.80241 1.798532 1.481998
2.138307 2.124842 2.459177
Table 3 Fair prices
C1∗ C2∗ C3∗
Table 3 shows the values of the approximation up to the first C1∗ , second C2∗ and third order C3∗ of the fair prices with 30 ST0 = K = 100, = 365 , N = 30, r = 0.05. From these results, we observe that option prices are strongly affected by third order terms. 5. Concluding remarks The Black and Scholes model assumes the Gaussianity and the independency of stock log returns. Empirical studies, however, report that they are not Gaussian nor independent. In this paper, dropping these two assumptions, we derive a European option pricing. Then, we observed that option prices are strongly affected by the non-Gaussianity and the dependency of stock log returns. Hence, it should be noted that we use option pricing models taking into account the non-Gaussianity and the dependency of stock log returns. 6. Proofs Proof of Theorem 1. First, we evaluate the asymptotic cumulants of ZN . From (A1) and (A2), E(ZN ) = 0, N−1
cum(ZN , ZN ) = N −1
(N − |j |)cX,2 (j )
j =−(N−1) = 2fX,2 − N −1 fX,2 + o(N −1 ),
cum(ZN , ZN , ZN ) = N
−3/2
N −1
(N − Sj1 j2 )cX,3 (j1 , j2 )
j1 ,j2 =−(N−1)
= N −1/2 (2)2 fX,3 + o(N −1 ), where
Sj1 j2 =
max(|j1 |, |j2 |)
if sign(j1 ) = sign(j2 ),
min(|j1 | + |j2 |, N ) if sign(j1 ) = −sign(j2 ),
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and N−1
cum(ZN , ZN , ZN , ZN ) = N −2
(N − Sj1 j2 j3 )cX,4 (j1 , j2 , j3 )
j1 ,j2 ,j3 =−(N−1)
= N −1 (2)3 fX,4 + o(N −1 ), where
Sj1 j2 j3 =
max(|j1 |, |j2 |, |j3 |)
if sign(j1 ) = sign(j2 ) = sign(j3 ),
min{max(|j1 | + |j2 |) + |j3 |, N} if sign(j1 ) = sign(j2 ) = −sign(j3 ).
Applying the general formula for the Edgeworth expansion (e.g., Taniguchi and Kakizawa, 2000, pp. 168–170), we obtain (5). Proof of Theorem 2. From Theorem 1 and (6), ∞ C = e−r [ST0 exp{ + (2fX,2 )1/2 z} − K]g(z) dz.
(13)
−d2
Integrating by parts and using the following equality: exp{−(2fX,2 )1/2 d2 }(−d2 ) = exp(fX,2 )(d1 ), yield
∞
−d2
[ST0 exp{ + (2fX,2 )1/2 z} − K]Hk (z)(z) dz
= a2 ST0
⎧ k−1 ⎨ ⎩
⎫ ⎬
(2fX,2 )j/2 Hk−j −1 (−d2 )(d1 ) + (2fX,2 )k/2 (d1 )
⎭
j =1
for k = 2, 3, 4, 6. Inserting (14) in (13), we obtain (7).
(14)
Proof of Corollary 1. If {Xt ; t ∈ Z} is independent, then fX,2 = 0. If {Xt ; t ∈ Z} is a Gaussian process, then fX,3 = fX,4 = 0. Hence, Corollary 1 follows.
Proof of Corollary 2. From (B1)–(B3) and (8), fX,k = Ak f ,k ,
k = 2, 3, 4,
and (A2) holds. Note that ⎞ ⎛ ∞ ∞ cX,2 (u) = Var ⎝ aj1 t−j1 , aj2 t+u−j2 ⎠ j1 =0
= 2
∞
j2 =0
aj a|u|+j .
j =0 = 2 f . From above arguments Corollary 2 follows. We can see fX,2 ,2
Proof of Theorem 3. From the martingale restriction, ST0 = e−r ET0 [ST ], ∞ ST0 exp{ + (2fX,2 )1/2 z}g(z) dz. = e−r −∞
(15)
K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
Note that ∞ −∞
1057
exp{(2fX,2 )1/2 z}Hk (z)(z) dz = (2fX,2 )k/2 exp(fX,2 )
for k = 2, 3, 4, 6. Eq. (15) implies that 1 = exp(−r + + fX,2 ) 1 + 23 2 3/2 N −1/2 fX,3 − 21 N −1 fX,2 + 13 3 2 N −1 fX,4 + 29 4 3 N −1 (fX,3 )2 + o(N −1 ).
(16)
Taking the logarithm of Eq. (16) and using Taylor expansion, yield = r − fX,2 − 23 2 1/2 N −1/2 fX,3 +
1 −1 2 N fX,2
− 13 3 N −1 fX,4 + o(N −1 ).
(17)
Substituting (17) into Gk , k = 0, 2, 3, 4, 6 in Theorem 2, further expansion and collection of terms, we obtain 2 G0 = G∗0 − 2 3/2 ST0 N −1/2 fX,3 (d1∗ ) 3
1 1 2 + ST0 N −1 fX,2 − 3 2 fX,4 + 4 3 (fX,3 )2 (d1∗ ) 2 3 9 +
(fX,3 )2 (2fX,2 )5/2 (d1∗ ) + o(N −1 ), ST0 N −1 36 (fX,2 )3
(18)
G3 = G∗3 + ST0 (2fX,2 )3/2 (d1∗ ) −
(2)7/2 3 ST0 N −1/2 fX,3 (fX,2 )3/2 (d1∗ ) 6
3 (2)1/2 −1/2 fX,3 − (2fX,2 )j/2+1 H3−j (−d2∗ )(d1∗ ) ST0 N 6 (fX,2 )3/2 j =1
+ o(N −1/2 ), and Gk = ST0
⎧ k−1 ⎨ ⎩
(19) ⎫ ⎬
(2fX,2 )j/2 Hk−j −1 (−d2∗ )(d1∗ ) + (2fX,2 )k/2 (d1∗ )
⎭
j =1
for k = 2, 4, 6. From (18)–(20), Theorem 3 follows.
+ o(1)
(20)
Acknowledgments The authors are grateful to Professor S. Ralescu for kind treatment of this paper. The second author M.T. celebrates Madan’s retirement from Indiana University and his great contribution to developments in statistics. References Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. J. Polit. Economy 81, 637–654. Brillinger, D.R., 1981. Time Series: Data Analysis and Theory. Holden-Day, San Francisco. Brillinger, D.R., Rosenblatt, M., 1967a. Asymptotic theory of estimates of k-th order spectra. Spectral Analysis of Time Series. Wiley, New York, pp. 153–188.
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Brillinger, D.R., Rosenblatt, M., 1967b. Computation and interpretation of k-th order spectra. In: Spectral Analysis of Time Series. Wiley, New York, pp. 189–232. Corrado, C.J., Su, T., 1996a. S&P 500 index option tests of Jarrow and Rudd’s approximate option valuation formula. J. Futures Markets 16, 611– 629. Corrado, C.J., Su, T., 1996b. Skewness and kurtosis in S&P 500 index returns implied by option prices. J. Financial Res. 19, 175–192. Corrado, C.J., Su, T., 1997. Implied volatility skews and stock return skewness and kurtosis implied by stock option prices. European J. Finance 3, 73–85. Cox, J.C., Ross, S.A., 1976. The valuation of options for alternative stochastic processes. J. Financial Economics 3, 145–166. Jarrow, R., Rudd, A., 1982. Approximate option valuation for arbitrary stochastic processes. J. Financial Economics 10, 347–369. Jurczenko, E., Maillet, B., Negrea, B., 2002. Multi-moment approximate option pricing models: a general comparison (Part 1). University of Paris I Panthéon-Sorbonne. Kariya, T., 1993. Quantitative Methods for Portfolio Analysis. Kluwer Academic Publishers, Dordrecht. Kariya, T., Liu, R.Y., 2002. Asset Pricing: Discrete Time Approach. Kluwer Academic Publishers, Boston. Longstaff, F.A., 1995. Option pricing and the martingale restriction. Rev. Financial Stud. 8, 1091–1124. Rubinstein, M., 1998. Edgeworth binomial trees. J. Derivatives 5, 20–27. Taniguchi, M., Kakizawa, Y., 2000. Asymptotic Theory of Statistical Inference for Time Series. Springer, New York.
Higher order asymptotic option valuation for non-Gaussian dependent returns Kenichiro Tamaki, Masanobu Taniguchi∗ Department of Mathematical Sciences, School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan Available online 25 July 2006
Abstract This paper discusses the option pricing problems using statistical series expansion for the price process of an underlying asset. We derive the Edgeworth expansion for the stock log return via extracting dynamics structure of time series. Using this result, we investigate influences of the non-Gaussianity and the dependency of log return processes for option pricing. Numerical studies show some interesting features of them. © 2006 Elsevier B.V. All rights reserved. MSC: Primary 91B24; 62M10; secondary 62M15; 91B84 Keywords: Black and Scholes model; Edgeworth expansion; Non-Gaussian stationary process; Option pricing
1. Introduction Black and Scholes (1973) provided the foundation of modern option pricing theory. Despite its usefulness, however, the Black and Scholes theory entails some inconsistencies. It is well known that the model frequently misprices deep in-the-money and deep out-of-the-money options. This result is generally attributed to the unrealistic assumptions used to derive the model. In particular, the Black and Scholes model assumes that stock prices follow a geometric Brownian motion with a constant volatility under an equivalent martingale measure. In order to avoid this drawback, Jarrow and Rudd (1982) proposed a semiparametric option pricing model to account for non-normal skewness and kurtosis in stock returns. This approach aims to approximate the risk-neutral density by a statistical series expansion. Jarrow and Rudd (1982) approximated the density of the state price by an Edgeworth series expansion involving the log-normal density. Corrado and Su (1996a) implemented Jarrow and Rudd’s formula to price options. Corrado and Su (1996b, 1997) considered Gram-Charlier expansions for the stock log return rather than the stock price itself. Rubinstein (1998) used the Edgeworth expansion for the stock log return. Jurczenko et al. (2002) compared these different multi-moment approximate option pricing models. Also they investigated in particular the conditions that ensure the martingale restriction. As in Kariya (1993) and Kariya and Liu (2002), the time series structure of return series does not always admit a measure which makes the discounted process a martingale. Hence, we will not be able to develop an arbitrage pricing theory by forming an equivalent portfolio. In such a case, we often regard the expected value of the present value ∗ Corresponding author. Tel.: +81 3 5286 8095.
E-mail addresses: [email protected] (K. Tamaki), [email protected] (M. Taniguchi). 0378-3758/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2006.06.023
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of a contingent claim as a proxy for pricing may be with help of a risk neutrality argument. In view of this, Kariya (1993) considered pricing problems with no martingale property and approximated the density of the state price by the Gram-Charlier expansion for the stock log return. In this paper, we consider option pricing problems by using Kariya’s approach. In Section 2, we derive the Edgeworth expansion for the stock log return via extracting dynamics structure of time series. Using this result, we investigate influences of the non-Gaussianity and the dependency of log return processes for option pricing. Numerical studies illuminate some interesting features of the influences. In Section 3, we give option prices based on the risk neutrality argument. In Section 4, we discuss a consistent estimator of the quantities in our results. Section 5 concludes. The proofs of theorems are relegated to Section 6. 2. Edgeworth expansion of log return Let {St ; t 0} be the price process of an underlying security at trading time t. The j-th period log return Xj is defined as log ST0 +j − log ST0 +(j −1) = + 1/2 Xj ,
j = 1, 2, . . . , N,
(1)
where T0 is present time, N = / is the number of unit time intervals of length during a period = T − T0 and T is the maturity date. Then the terminal price ST of the underlying security is given by ⎫ ⎧ N ⎬ ⎨ 1/2 ST = ST0 exp + (2) Xj . ⎭ ⎩ N j =1
Remark 1. In the Black and Scholes option theory the price process is assumed to be a geometric Brownian motion ST = ST0 exp( + W ),
(3)
where the process {Wt ; t ∈ R} is a Wiener process with drift 0 and variance t. From (3), the log return at discreterized time point can be written as log St+j − log St+(j −1) = + 1/2 j ,
j ∼ iid N(0, 1).
(4)
The expression of (1) is motivated from (4). First, we derive an analytical expression for the density function of ST . Since from (2) the distribution of ST depends
X on that of ZN = N −1/2 N j =1 j , we consider the Edgeworth expansion of the density function of ZN . If we assume that Xj are independently and identically distributed random variables with mean zero and finite variance, it is easy to give the Edgeworth expansion for ZN (the classical Edgeworth expansion). However, a lot of financial empirical studies show that Xj ’s are not independent. Thus, we suppose that {Xj } is a dependent process which satisfies the following assumption. (A1) {Xt ; t ∈ Z} is fourth order stationary in the sense that (i) (ii) (iii) (iv)
E(Xt ) = 0, cum(Xt , Xt+u ) = cX,2 (u), cum(Xt , Xt+u1 , Xt+u2 ) = cX,3 (u1 , u2 ), cum(Xt , Xt+u1 , Xt+u2 , Xt+u3 ) = cX,4 (u1 , u2 , u3 ).
(A2) The cumulants cX,k (u1 , . . . , uk−1 ), k = 2, 3, 4, satisfy ∞
(1 + |uj |2−k/2 )|cX,k (u1 , . . . , uk−1 )| < ∞
u1 ,...,uk−1 =−∞
for j = 1, . . . , k − 1. (A3) J-th order (J 5) cumulants of ZN are all O(N −J /2+1 ).
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Under (A2), {Xt ; t ∈ Z} has the k-th order cumulant spectral density. Let fX,k be the k-th order cumulant spectral density evaluated at frequency 0 ∞
fX,k = (2)−(k−1)
cX,k (u1 , . . . , uk−1 )
u1 ,...,uk−1 =−∞
for k = 2, 3, 4. First, we state the following result. Theorem 1. Suppose that (A1)–(A3) hold. The third order Edgeworth expansion of the density function of Z = (2fX,2 )−1/2 ZN is given by (2)1/2 −1/2 fX,3 1 −1 fX,2 g(z) = (z) 1 + N N H (z) − H2 (z) 3 6 4 fX,2 (fX,2 )3/2 −1 fX,4 −1 (fX,3 )2 + H4 (z) + N H6 (z) + o(N −1 ), (5) N 12 36 (fX,2 )2 (fX,2 )3 where (·) is the standard normal density function, Hk (·) is the k-th order Hermite polynomial and fX,2 =
∞
|u|cX,2 (u).
u=−∞
Many authors have proposed to use different statistical series expansion to price options (see Jarrow and Rudd, 1982; Corrado and Su, 1996a, b,1997; Rubinstein, 1998; Kariya, 1993). Here, we give the Edgeworth expansion for the stock log return in powers of N −1/2 . A European call option can be viewed as a security which pays at time T its holder the amount XT∗ = max(ST − K, 0), where K is the exercise or strike price. As in Kariya (1993), we price XT∗ by its discounted expected value; C = exp(−r)ET0 (XT∗ ),
(6)
where r is the interest rate which is regarded as a constant for the remaining period and ET0 (·) is evaluated at T0 . Evaluate (6) based on the density in (5). Then writing d1 = (log ST0 /K + + 2fX,2 )/(2fX,2 )1/2 , d2 = d1 − (2fX,2 )1/2 , we obtain the following theorem Theorem 2. Let a1 = exp(−r) and a2 = exp( + fX,2 ). Then C = G0 + +
(2)1/2 −1/2 fX,3 1 −1 fX,2 G − G2 N N 3 6 4 fX,2 (fX,2 )3/2
−1 fX,4 −1 (fX,3 )2 N N G + G6 + o(N −1 ), 4 12 36 (fX,2 )2 (fX,2 )3
where G0 = a1 {a2 ST0 (d1 ) − K(d2 )},
(7)
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Gk = a1 a2 ST0
⎧ k−1 ⎨ ⎩
j =1
⎫ ⎬
(2fX,2 )j/2 Hk−j −1 (−d2 )(d1 ) + (2fX,2 )k/2 (d1 ) , ⎭
for k = 2, 3, 4, 6, where (·) is the standard normal distribution function. ,f From (7) it is seen that the asymptotic expansion of the option price depends on fX,2 , fX,2 X,3 and fX,4 . Hence, we can see influences of the non-Gaussianity and the dependency of the log return processes for the higher order option valuation.
Corollary 1. Write C = G0 + N −1/2 CG,2 + N −1 CG,3 + N −1 CD,3 + o(N −1 ), where CG,2 =
(2)1/2 fX,3 G3 , 6 (fX,2 )3/2
CG,3 =
fX,4 (fX,3 )2 G + G6 , 4 12 (fX,2 )2 36 (fX,2 )3
CD,3 = −
1 fX,2 G2 . 4 fX,2
If {Xt ; t ∈ Z} is independent, then CD,3 = 0. If {Xt ; t ∈ Z} is a Gaussian process, then CG,2 = CG,3 = 0. Example 1. Suppose that Xj , j = 1, . . . , N, are independently and identically distributed random variables. Let −(k−1) = 0 and f cX,k , k = 2, 3, 4. The price of a European call cX,k = cX,k (0), k = 2, 3, 4. Note that fX,2 X,k = (2) option CIID is given by cX,3 cX,4 1 1 G3 + N −1 G4 CIID = G0 + N −1/2 3/2 6 24 (cX,2 ) (cX,2 )2 +
1 −1 (cX,3 )2 G6 + o(N −1 ), N 72 (cX,2 )3
where Gk , k = 0, 3, 4, 6, are defined in Theorem 2 with fX,2 = (2)−1 cX,2 . If = r − cX,2 /2, then a1 a2 = 1 so that G0 equals the Black and Scholes formula. Example 2. In Example 1, suppose that Xj , j = 1, . . . , N, are distributed as t-distribution with degrees of freedom. Then, for > 4 Ct = Gt,0 + N −1 Gt,3 + o(N −1 ), where Gt,0 = a1 {a2 ST0 (d1 ) − K(d2 )},
, a2 = exp + 2( − 2) 1/2 d1 = log ST0 /K + + , −2 −2 1/2 d2 = d1 − , −2
K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
1047
option prices
20
18
16
14
12 4
5
6
7
8
9
Fig. 1. For t-distribution with degrees of freedom in Example 2, the approximation up to the first (Ct,1 , dotted line) and third order (Ct,3 , solid 30 , N = 30, r = = 0.05 and 4 < < 9. line) of the option price are plotted with ST0 = K = 100, = 365
and
⎫ ⎧ 3 ⎬ j/2 a1 a2 ST0 ⎨ 2 Gt,3 = H3−j (−d2 )(d1 ) + (d1 ) . ⎭ 4( − 4) ⎩ −2 −2 j =1
In order to show influences of higher order terms, in Fig. 1, we plotted Ct,1 = Gt,0 (dotted line) and Ct,3 = Gt,0 + 30 1 N −1 Gt,3 (solid line) of Example 2 with ST0 = K = 100, = 365 , N = 30 ( = 365 ), r = = 0.05 and 4 < < 9. From this, we observe that Ct,3 diverges as → 4. Example 3. Let {Xt : t ∈ Z} be the ARCH(1) process 1/2
Xt = ht t
and
2 ht = 0 + 1 Xt−1 ,
where 0 > 0, 1 0, { t : t ∈ Z} is a sequence of independently and identically distributed random variables with E( t ) = 0,
E( 2t ) = 1,
E( 3t ) = 0,
E( 4t ) = m,
m > 1,
and t is independent of Xt−s , s > 0. Then fX,2 = fX,4 =
1 0 , 2 1 − 1
fX,3 = 0,
fX,2 = 0,
1
20 (m − 3 + 5m 1 − 3 1 + 2m 21 − 2m 31 )
(2)3
(1 − 1 )3 (1 − m 21 )
for ma 21 < 1. Hence, CARCH(1) = GARCH(1),0 + N −1 GARCH(1),3 + o(N −1 ),
,
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K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
14
option prices
12
10
8
6
-0.58
-0.25
0.0
0.25
0.58
1 Fig. 2. For ARCH(1) in Example 3, the approximation up to the first (CARCH(1),1 , dotted line) and third order (CARCH(1),3 , solid line) of the option √ √ 30 , N = 30, r = = 0.05, m = 3, = 0.5 and −1/ 3 < < 1/ 3. price are plotted with ST0 = K = 100, = 365 0 1
where GARCH(1),0 = a1 a2 ST0 (d1 ) − a1 K(d2 ),
0 , a2 = exp + 2(1 − 1 ) 0 1/2 0 , d1 = log ST0 /K + + 1 − 1 1 − 1 0 1/2 d2 = d1 − 1 − 1 and GARCH(1),3 =
a1 a2 ST0 m − 3 + 5m 1 − 3 1 + 2m 21 − 2m 31 24 (1 − 1 )(1 − m 21 ) ⎧ ⎫ 3 ⎨ ⎬ 0 j/2 0 2 × H3−j (−d2 )(d1 ) + (d1 ) . ⎩ ⎭ 1 − 1 1 − 1 j =1
GARCH(1),3 (solid In Fig. 2, we plotted CARCH(1),1 = GARCH(1),0 (dotted line) and CARCH(1),3 = GARCH(1),0 + N −1√ √ 30 1 line) of Example 3 with ST0 =K =100, = 365 , N =30 (= 365 ), r ==0.05, m=3, 0 =0.5 and −1/ 3 < 1 < 1/ 3. Fig. 2 illuminates influences of higher order terms under Gaussian innovations. From this, we can see that CARCH(1),3 √ diverges as 1 → ±1/ 3. In Fig. 3, we plotted CARCH(1),1 (dotted line) and CARCH(1),3 (solid line) of Example 3 with ST0 = 100, K = 95, 30 = 365 , N = 30, r = = 0.05, 0 = 0.5, 1 = 0.3 and 1 < m < 9. Fig. 3 illuminates influences of non-Gaussian innovations. From this, we observe that CARCH(1),3 decreases as m → 9. The first order term CARCH(1),1 is a constant because it is independent from m.
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1049
14.2845
option prices
14.0
13.5
13.0
1
5
3
7
9
m Fig. 3. For ARCH(1) in Example 3, the approximation up to the first (CARCH(1),1 , dotted line) and third order (CARCH(1),3 , solid line) of the option 30 , N = 30, r = = 0.05, = 0.5, = 0.3 and 1 < m < 9. price are plotted with ST0 = 100, K = 95, = 365 0 1
Next we consider option pricing problems for a class of processes generated by uncorrelated random variables, which includes the linear process and an important class in time series analysis. Here, we are concerned with the following process: Xt =
∞
aj t−j ,
t ∈ Z,
(8)
j =0
where { t ; t ∈ Z} is a sequence of uncorrelated random variables. Instead of (A1) and (A2) we make the following assumption. (B1) { t ; t ∈ Z} is fourth order stationary in the sense that (i) (ii) (iii) (iv)
E( t ) = 0, Var( t ) = 2 , cum( t , t+u1 , t+u2 ) = c ,3 (u1 , u2 ), cum( t , t+u1 , t+u2 , t+u3 ) = c ,4 (u1 , u2 , u3 ).
(B2) The cumulants c ,k (u1 , . . . , uk−1 ), k = 3, 4, satisfy ∞
(1 + |uj |2−k/2 )|c ,k (u1 , . . . , uk−1 )| < ∞,
u1 ,...,uk−1 =−∞
for j = 1, . . . , k − 1. (B3) {aj ; j ∈ Z} satisfies ∞ j =0
(1 + |j |)|aj | < ∞.
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Under (B2), { t ; t ∈ Z} has the k-th order cumulant spectral density. Let f ,k be the k-th order cumulant spectral density evaluated at frequency 0 ∞
f ,k = (2)−(k−1)
c ,k (u1 , . . . , uk−1 )
u1 ,...,uk−1 =−∞
for k = 2, 3, 4. The response function of {aj ; j ∈ Z} is defined by A( ) =
∞
aj e−ij
j =0
for −∞ < < ∞. Under (B1)–(B3), (A1) and (A2) hold. Hence, from Theorem 1, we have Corollary 2. Suppose that (B1)–(B3) and (A3) hold. Let a1 = exp(−r) and a2 = exp + 21 2 A2 . Then C = G0 +
22 A3 −1/2 1 N f ,3 G3 − N −1 f ,2 G2 3 3 3 |A| 2A2
3 −1 24 N f ,4 G4 + 6 N −1 f 2,3 G6 + o(N −1 ), 4 3 9
+
where A = A(0), f ,2 = 2
∞
|j2 |aj1 aj1 +j2 ,
j1 ,j2 =0
Gk , k = 0, 2, 3, 4, 6, are given in Theorem 2 with fX,2 =
2 2 A . 2
Example 4. Let {Xt ; t ∈ Z} be AR(1) process Xt = Xt−1 + t ,
| | < 1.
Note that A=
1 , 1−
f ,2 =
2 (1 + )(1 − )3
.
The price of a European call option CAR(1) is given by CAR(1) = GAR(1),0 + N −1/2 GAR(1),2 + N −1 GAR(1),3 + o(N −1 ), where GAR(1),0 = a1 {a2 ST0 (d1 ) − K(d2 )}, GAR(1),2 =
22 f ,3 G3 , 33
3 24 G + f G + (f ,3 )2 G6 , 2 ,4 4 1 − 2 34 96
2 a2 = exp + , 2(1 − )2
GAR(1),3 = −
K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
1051
120
100
option prices
80
60
40
20
-1.00
-0.50
0.0
0.50
0.75
Fig. 4. For AR(1) in Example 4, the approximation up to the first (CARCH(1),1 , dotted line), second (CARCH(1),2 , dashed line) and third order 30 , N = 30, r = = 0.05, = 1, f (CARCH(1),3 , solid line) of the option price are plotted with ST0 = K = 100, = 365 X,3 = −0.1, fX,4 = 0.2 and −1 < < 0.75.
d1 = log ST0 /K + + d2 = d1 −
2 (1 − )2
1/2 , 1−
1/2 , 1−
and Gk = a1 a2 ST0
⎧ k−1 1/2 j ⎨ ⎩
j =1
1−
Hk−j −1 (−d2 )(d1 ) +
k 1/2 1−
⎫ ⎬
(d1 ) , ⎭
for k = 2, 3, 4, 6. In order to show influences of higher order terms, in Fig. 4, we plotted CAR(1),1 = GAR(1),0 (dotted line), CAR(1),2 = GAR(1),0 + N −1/2 GAR(1),2 (dashed line) and CAR(1),3 = GAR(1),0 + N −1/2 GAR(1),2 + N −1 GAR(1),3 (solid line) of 30 1 Example 4 with ST0 = K = 100, = 365 , N = 30 ( = 365 ), r = = 0.05, = 1, fX,3 = −0.1, fX,4 = 0.2 and −1 < < 0.75. From this, we observe that CAR(1),k , k = 1, 2, 3 diverges as → 1. In Examples 2 and 3, although the third order terms diverge, the first order terms do not diverge. On the other hand, in Example 4, even the first order term does not converge as → 1. This fact is attributed to finiteness of the variances. 3. Martingale restriction In the previous section, we considered pricing problems with no martingale property. Now we recall that the theoretical price of a option is based on the risk neutrality argument. In this section, to investigate influences of the martingale restriction. we derive the option price based on the risk neutrality argument (see Cox and Ross, 1976; Longstaff, 1995).
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Let d1∗ = (log ST0 /K + r + fX,2 )/(2fX,2 )1/2 , d2∗ = d1∗ − (2fX,2 )1/2 . Then we have Theorem 3. The fair price C ∗ of a European call option is given by C ∗ = G∗0 + +
(2)1/2 −1/2 fX,3 1 −1 fX,2 ∗ ∗ N N G − G 3 6 4 fX,2 2 (fX,2 )3/2
(fX,3 )2 ∗ −1 fX,4 N G∗4 + N −1 G6 + o(N −1 ), 2 12 36 (fX,2 ) (fX,2 )3
(9)
where G∗0 = ST0 (d1∗ ) − e−r K(d2∗ ), G∗k
= ST0
k−1
(2fX,2 )j/2 Hk−j −1 (−d2∗ )(d1∗ ),
j =1
for k = 2, 3, 4 and ⎡ ⎤ 2 G∗6 = ST0 ⎣ (2fX,2 )j/2 H5−j (−d2∗ ) − 2fX,2 H3−j (−d2∗ ) ⎦ (d1∗ ). j =1
Example 5. Suppose that {Xt ; t ∈ Z} is AR(1) process in Example 4. Then the fair price of a European call option ∗ CAR(1) is given by ∗ = G∗AR(1),0 + N −1/2 G∗AR(1),2 + N −1 G∗AR(1),3 + o(N −1 ), CAR(1)
where G∗AR(1),0 = ST0 (d1∗ ) − e−r K(d2∗ ), G∗AR(1),2 =
22 f ,3 G∗3 , 33
3 24 G∗2 + 4 f ,4 G4 + 6 (f ,3 )2 G∗6 , 2 1− 3 9
1/2 2 d1∗ = log ST0 /K + r + , 1− 2(1 − )2 1/2 ∗ ∗ , d2 = d1 − 1− ⎧ ⎫ k−1 1/2 j ⎨ ⎬ G∗k = ST0 Hk−j −1 (−d2∗ )(d1∗ ) , ⎩ ⎭ 1−
G∗AR(1),3 = −
j =1
for k = 2, 3, 4 and ⎡ ⎤ 2 1/2 j 2 G∗6 = ST0 ⎣ H5−j (−d2∗ ) − H (−d2∗ ) ⎦ (d1∗ ). 2 3−j 1− (1 −
) j =1
K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
1053
100
option prices
80
60
40
20
-1.0
-0.5
0.0
0.5
1.0
∗ ∗ Fig. 5. For AR(1) in Example 5, the approximation up to the first (CARCH(1),1 , dotted line), second (CARCH(1),2 , dashed line) and third order 30 , N = 30, r = 0.05, = 1, f ∗ (CARCH(1),3 , solid line) of the option price are plotted with ST0 = K = 100, = 365 X,3 = −0.1, fX,4 = 0.2 and −1 < < 1.
∗ ∗ In Fig. 5, we plotted CAR(1),1 = G∗AR(1),0 (dotted line), CAR(1),2 = G∗AR(1),0 + N −1/2 G∗AR(1),2 (dashed line) and 30 ∗ = G∗AR(1),0 + N −1/2 G∗AR(1),2 + N −1 G∗AR(1),3 (solid line) of Example 5 with ST0 = K = 100, = 365 , N = 30 CAR(1),3 1 ( = 365 ), r = 0.05, = 1, fX,3 = −0.1, fX,4 = 0.2 and −1 < < 1. Unlike Example 4, we observe that CAR(1),k , k = 1, 2, 3 converge to ST0 (=100) as → 1.
4. Estimation From (1), Xj −N0 , j = 1, . . . , N0 , are available, where N0 = T0 /. Therefore, in this section we consider to estimate ,f , fX,2 , fX,2 X,3 and fX,4 in Theorems 1 and 2 consistently based on the past observations. From (A1), is the mean of stock log returns. Hence, a natural unbiased estimator of is the sample mean ˆ =
N0 1 {log Sj − log S(j −1) }. N0
(10)
j =1
The variance of ˆ is given by Var() ˆ =
1 N0
N 0 −1 u=−(N0 −1)
1−
|u| cX,2 (u). N0
Hence, under (A2), ˆ given in (10) is a consistent estimator of . , we define the lag window function w(·) which is an Moreover, in order to construct consistent estimator of fX,2 even and piecewise continuous function satisfying the conditions, w(0) = 1, |w(x)|1
for all x,
w(x) = 0
for |x| > 1.
(11)
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Table 1 Values of consistent estimators AMOCO
ˆ fˆX,2 fˆX,3 ˆ (fX,2 )3/2 fˆX,4 (fˆX,2 )2 fˆX,2 fˆX,2
FORD
HP
IBM
MERCK
0.235103 0.002937
0.045337 0.016006
0.133815 0.016202
0.017165 0.003085
0.481340 0.004534
−0.706149
−3.078889
8.501363
0.470144
2.419969
2.278478
−0.280973
8.651378
15.0914
−5.520428
0.169291
27.18047
−22.78799
−2.249174 −37.3221
Let fˆX,2 =
N 0 −1
|u|cˆX,2 (u)w BN0 u ,
u=−(N0 −1)
where cˆX,2 (u) is the sample autocovariance function at lag u N0 −|u| 1 cˆX,2 (u) = {log S(j +|u|) − log S(j +|u|−1) − } ˆ N0 j =1
× {log Sj − log S(j −1) − }, ˆ
(12)
and BN0 → 0 as N0 → ∞, but (BN0 )3 N0 → ∞. Then we can easily see that under (A2), fˆX,2 given in (12) is a consistent estimator of fX,2 . Since fX,k , k = 2, 3, 4, are the k-th order cumulant spectral density evaluated at frequency 0, using Brillinger and Rosenblatt (1967a, b) formula, we construct consistent estimators fˆX,k of fX,k (k = 2, 3, 4). See also Brillinger (1981). Thus, we can consistently estimate all the quantities in Theorems 1 and 2 (e.g., Gj , j = 0, 2, 3, 4, 6.) by the and f and fˆ corresponding quantities replacing , fX,2 ˆ fˆX,2 X,k by , X,k (k = 2, 3, 4). For example, we discuss a consistent estimator for New York stock exchange data. The data are daily returns of AMOCO, FORD HP, IBM and MERCK companies. The individual time series are the last 1024 data points from stocks, representing the daily returns for the five companies from February 2, 1984, to December 31, 1991. We used the window functions
W (u1 , . . . , uk−1 ) =
2−(k−1)
If |u1 |, . . . , |uk−1 | 1,
0
otherwise
for fˆX,k (k = 2, 3, 4) and let w(u) = 1 for |u|1, where w(u) is defined in (11). Also we used the bandwidth in 1 1 1 frequency direction with BN0 = 50 for fˆX,2 , BN0 = 30 for fˆX,3 and BN0 = 10 for fˆX,4 and fˆX,2 (see Brillinger and Rosenblatt, 1967a, b; Brillinger, 1981). and f Table 1 show the values of consistent estimators of , fX,2 X,k (k = 2, 3, 4) for the five companies. From these results, we can see that the quantities involved in higher order terms are quite different from the Black and Scholes model. Therefore, in general the assumptions of the Gaussianity and the independency of stock log returns will not hold. Table 2 shows the values of the approximation up to the first C1 , second C2 and third order C3 of the option prices 30 , N = 30, r = 0.05. From this results, we observe that option prices are strongly affected with ST0 = K = 100, = 365 by third order terms except for AMOCO and MERCK.
K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
1055
Table 2 Option prices
C1 C2 C3
AMOCO
FORD
HP
IBM
MERCK
2.776419 2.809884 2.881406
4.031663 3.979554 4.345765
4.472833 4.434833 6.392765
1.699889 1.700269 1.374588
4.689151 4.495491 4.650024
AMOCO
FORD
HP
IBM
MERCK
1.764254 1.769475 1.83751
3.827175 3.784867 4.111153
3.849221 3.954549 6.09142
1.80241 1.798532 1.481998
2.138307 2.124842 2.459177
Table 3 Fair prices
C1∗ C2∗ C3∗
Table 3 shows the values of the approximation up to the first C1∗ , second C2∗ and third order C3∗ of the fair prices with 30 ST0 = K = 100, = 365 , N = 30, r = 0.05. From these results, we observe that option prices are strongly affected by third order terms. 5. Concluding remarks The Black and Scholes model assumes the Gaussianity and the independency of stock log returns. Empirical studies, however, report that they are not Gaussian nor independent. In this paper, dropping these two assumptions, we derive a European option pricing. Then, we observed that option prices are strongly affected by the non-Gaussianity and the dependency of stock log returns. Hence, it should be noted that we use option pricing models taking into account the non-Gaussianity and the dependency of stock log returns. 6. Proofs Proof of Theorem 1. First, we evaluate the asymptotic cumulants of ZN . From (A1) and (A2), E(ZN ) = 0, N−1
cum(ZN , ZN ) = N −1
(N − |j |)cX,2 (j )
j =−(N−1) = 2fX,2 − N −1 fX,2 + o(N −1 ),
cum(ZN , ZN , ZN ) = N
−3/2
N −1
(N − Sj1 j2 )cX,3 (j1 , j2 )
j1 ,j2 =−(N−1)
= N −1/2 (2)2 fX,3 + o(N −1 ), where
Sj1 j2 =
max(|j1 |, |j2 |)
if sign(j1 ) = sign(j2 ),
min(|j1 | + |j2 |, N ) if sign(j1 ) = −sign(j2 ),
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K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
and N−1
cum(ZN , ZN , ZN , ZN ) = N −2
(N − Sj1 j2 j3 )cX,4 (j1 , j2 , j3 )
j1 ,j2 ,j3 =−(N−1)
= N −1 (2)3 fX,4 + o(N −1 ), where
Sj1 j2 j3 =
max(|j1 |, |j2 |, |j3 |)
if sign(j1 ) = sign(j2 ) = sign(j3 ),
min{max(|j1 | + |j2 |) + |j3 |, N} if sign(j1 ) = sign(j2 ) = −sign(j3 ).
Applying the general formula for the Edgeworth expansion (e.g., Taniguchi and Kakizawa, 2000, pp. 168–170), we obtain (5). Proof of Theorem 2. From Theorem 1 and (6), ∞ C = e−r [ST0 exp{ + (2fX,2 )1/2 z} − K]g(z) dz.
(13)
−d2
Integrating by parts and using the following equality: exp{−(2fX,2 )1/2 d2 }(−d2 ) = exp(fX,2 )(d1 ), yield
∞
−d2
[ST0 exp{ + (2fX,2 )1/2 z} − K]Hk (z)(z) dz
= a2 ST0
⎧ k−1 ⎨ ⎩
⎫ ⎬
(2fX,2 )j/2 Hk−j −1 (−d2 )(d1 ) + (2fX,2 )k/2 (d1 )
⎭
j =1
for k = 2, 3, 4, 6. Inserting (14) in (13), we obtain (7).
(14)
Proof of Corollary 1. If {Xt ; t ∈ Z} is independent, then fX,2 = 0. If {Xt ; t ∈ Z} is a Gaussian process, then fX,3 = fX,4 = 0. Hence, Corollary 1 follows.
Proof of Corollary 2. From (B1)–(B3) and (8), fX,k = Ak f ,k ,
k = 2, 3, 4,
and (A2) holds. Note that ⎞ ⎛ ∞ ∞ cX,2 (u) = Var ⎝ aj1 t−j1 , aj2 t+u−j2 ⎠ j1 =0
= 2
∞
j2 =0
aj a|u|+j .
j =0 = 2 f . From above arguments Corollary 2 follows. We can see fX,2 ,2
Proof of Theorem 3. From the martingale restriction, ST0 = e−r ET0 [ST ], ∞ ST0 exp{ + (2fX,2 )1/2 z}g(z) dz. = e−r −∞
(15)
K. Tamaki, M. Taniguchi / Journal of Statistical Planning and Inference 137 (2007) 1043 – 1058
Note that ∞ −∞
1057
exp{(2fX,2 )1/2 z}Hk (z)(z) dz = (2fX,2 )k/2 exp(fX,2 )
for k = 2, 3, 4, 6. Eq. (15) implies that 1 = exp(−r + + fX,2 ) 1 + 23 2 3/2 N −1/2 fX,3 − 21 N −1 fX,2 + 13 3 2 N −1 fX,4 + 29 4 3 N −1 (fX,3 )2 + o(N −1 ).
(16)
Taking the logarithm of Eq. (16) and using Taylor expansion, yield = r − fX,2 − 23 2 1/2 N −1/2 fX,3 +
1 −1 2 N fX,2
− 13 3 N −1 fX,4 + o(N −1 ).
(17)
Substituting (17) into Gk , k = 0, 2, 3, 4, 6 in Theorem 2, further expansion and collection of terms, we obtain 2 G0 = G∗0 − 2 3/2 ST0 N −1/2 fX,3 (d1∗ ) 3
1 1 2 + ST0 N −1 fX,2 − 3 2 fX,4 + 4 3 (fX,3 )2 (d1∗ ) 2 3 9 +
(fX,3 )2 (2fX,2 )5/2 (d1∗ ) + o(N −1 ), ST0 N −1 36 (fX,2 )3
(18)
G3 = G∗3 + ST0 (2fX,2 )3/2 (d1∗ ) −
(2)7/2 3 ST0 N −1/2 fX,3 (fX,2 )3/2 (d1∗ ) 6
3 (2)1/2 −1/2 fX,3 − (2fX,2 )j/2+1 H3−j (−d2∗ )(d1∗ ) ST0 N 6 (fX,2 )3/2 j =1
+ o(N −1/2 ), and Gk = ST0
⎧ k−1 ⎨ ⎩
(19) ⎫ ⎬
(2fX,2 )j/2 Hk−j −1 (−d2∗ )(d1∗ ) + (2fX,2 )k/2 (d1∗ )
⎭
j =1
for k = 2, 4, 6. From (18)–(20), Theorem 3 follows.
+ o(1)
(20)
Acknowledgments The authors are grateful to Professor S. Ralescu for kind treatment of this paper. The second author M.T. celebrates Madan’s retirement from Indiana University and his great contribution to developments in statistics. References Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. J. Polit. Economy 81, 637–654. Brillinger, D.R., 1981. Time Series: Data Analysis and Theory. Holden-Day, San Francisco. Brillinger, D.R., Rosenblatt, M., 1967a. Asymptotic theory of estimates of k-th order spectra. Spectral Analysis of Time Series. Wiley, New York, pp. 153–188.
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