• 沒有找到結果。

A GENERALIZATION OF THE BARONE-ADESI AND WHALEY APPROACH FOR THE ANALYTIC APPROXIMATION OF AMERICAN OPTIONS

N/A
N/A
Protected

Academic year: 2021

Share "A GENERALIZATION OF THE BARONE-ADESI AND WHALEY APPROACH FOR THE ANALYTIC APPROXIMATION OF AMERICAN OPTIONS"

Copied!
16
0
0

加載中.... (立即查看全文)

全文

(1)

The authors thank an anonymous referee for useful comments.

*Correspondence author, College of Management, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei, Taiwan. Tel: 886-2-3366-4988, Fax: 886-2-2369-0833, e-mail: hung@management.ntu.edu.tw

Accepted 12 May 2008

Jia-Hau Guo is from the College of Management, National Chiao Tung University, Hsinchu, Taiwan.

Mao-Wei Hung is from the College of Management, National Taiwan University, Taipei, Taiwan.

Leh-Chyan So is from the College of Technology Management, National Tsing Hua University, Hsinchu, Taiwan.

The Journal of Futures Markets, Vol. 29, No. 5, 478–493 (2009) © 2009 Wiley Periodicals, Inc.

Published online in Wiley InterScience (www.interscience.wiley.com).

BARONE-ADESI AND

W

HALEY

A

PPROACH FOR THE

ANALYTIC

APPROXIMATION OF

A

MERICAN

O

PTIONS

JIA-HAU GUO MAO-WEI HUNG* LEH-CHYAN SO

This article introduces a general quadratic approximation scheme for pricing American options based on stochastic volatility and double jump processes. This quadratic approximation scheme is a generalization of the Barone-Adesi and Whaley approach and nests several option models. Numerical results show that this quadratic approximation scheme is efficient and useful in pricing American options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:478–493, 2009

(2)

INTRODUCTION

The goal of this research is to provide an efficient analytic approximation for pricing American options in a general model that allows for stochastic volatility, return jumps, and volatility jumps. The fact that asset returns exhibit both sto-chastic volatility and jumps has been widely documented. Although some closed-form solutions for European options based on these diffusion processes have been derived in recent years, no known analytic solution for American options exists. Consequently, simulative and numerical approaches are used to calculate American option values.

Barone-Adesi and Whaley (1987) originally applied the quadratic approxi-mation method to price American options using the decomposition technique. Bates (1991) first extended this method by introducing jumps into the process of the underlying asset return. Ju and Zhong (1999) improved the accuracy of the original method for options with intermediate maturities by adding a cor-rection term onto the approximation. Chang, Kang, Kim, and Kim (2007) introduced an additional parameter into the original method and succeeded in extending its application from basic options to barrier options and lookback options. Nevertheless, these methods do not continuously evolve in conjunc-tion with the rapid growth of new opconjunc-tion pricing models in the stochastic volatility framework. Additional empirical studies provide evidence that an option model allowing for stochastic volatility and double jumps dramatically reduces option-pricing errors (see Bakshi, Cao, & Chen, 1997; Broadie, Chernov, & Johannes, 2007; and others). These studies illustrate the impor-tance of extending the quadratic approximation method for American options based on these processes.

Stochastic volatility models with double jumps proposed by Duffie, Pan, and Singleton (2000) were used as examples to highlight the generalization of the authors’ quadratic approximation scheme, followed by a comparison with the least-squares simulation approach proposed by Longstaff and Schwartz (2001). The results of the authors’ comparison show that the quadratic approximation scheme is useful and efficient in pricing American options based on these dif-fusion processes. To illustrate its generality, this approximation scheme is also applied to other existing models.

The remainder of this article is organized as follows: the second section briefly describes the stochastic volatility model with correlated double jumps and the authors’ quadratic approximation scheme. The third section gives ana-lytic approximation formulae for other jump models. The fourth section com-pares quadratic approximations with the least-squares simulation approach. Conclusions are presented in the fifth section.

(3)

ANALYTIC AMERICAN OPTION APPROXIMATIONS FOR THE STOCHASTIC VOLATILITY MODEL WITH DOUBLE JUMPS

The primary expression of the stochastic volatility model with double jumps, adopted from Duffie et al. (2000) was reproduced to generate a self-contained treatment. Under the risk-neutral measure, the underlying asset price, S(t), is posited to follow a geometric jump diffusion with the instantaneous conditional variance, Y(t), following a mean-reverting square root jump process:

(1)

(2) where r is the risk-free interest rate and d is the dividend yield. x(t) represents a percentage jump in the stock price and follows a normal distribution, , where y(t) is a level jump in the volatility and follows an expo-nential distribution, Expoexpo-nential(uy) . qS(t) and qY(t) are two correlated Poisson

counters with intensity lx,y.r denotes the instantaneous correlation coefficient

between the stock price return process and its conditional variance process. To

retain the Martingale property, the compensator, ⫽

, is subtracted from the stock price process, such that the drift of the stock return rate is equal to r⫺ d.

The partial integro-differential equation for a contingent claim price, P, on the underlying asset is given by

. (3)

Compared with European option values, American option values can be exercised at any time before maturity. The flexibility of the right-to-exercise options determines the “early-exercise premium” markup of American prices over European option prices. MacMillan (1987), Barone-Adesi and Whaley (1987), and Bates (1991) decomposed the value of an American option into its European counterpart and an early-exercise premium to obtain an approxima-tion formula. The price of a basic American call opapproxima-tion, CA(S, Y, T; K), with a

strike price, K, and a maturity date, T, can be represented as

⫹ lx,y

0

⫺⬁

[P(Sex, Y⫹ y) ⫺ P(S, Y)]£(x, y) dxdy

⫹ PS((r⫺ d) ⫺ lx,yEQ[ex⫺ 1])S ⫹ PY(Y⫺ kYY) ⫺ PT⫺ rP 0⫽1 2PSSS 2Y 1 2PYYs 2 YY⫹ PSYrsYSY

lx,y(exp[m0⫹ 0.5s2x,y]兾(1 ⫺ uymx,y)⫺1)dt

EQ[(exp[x(t)]⫺1)dqS(t)]

N(m0⫹ mx,yy, s2x,y)

dY(t)⫽ (Y ⫺ kYY(t) )dt⫹ sY2Y(t)dWY(t) ⫹ y(t)dqY(t), tⱖ 0

dS(t)

S(t) ⫽ (r ⫺ d)dt ⫹ 2Y(t)dWS(t) ⫹ (e

x(t) ⫺ 1)dq

(4)

(4) where CE(S, Y, T; K) is the price of the corresponding European call option and

␵(S, Y, T; K) is the value of the corresponding early-exercise premium. Given the linearity of Equation (4), the early-exercise premium must satisfy Equation (3) because American option values, as well as European option values, satisfy the aforementioned partial differential equation in the nonstopping region under the risk-neutral measure.

Given an analytic European option solution1 (see Appendix A), the only

unsolved problem in deriving the pricing formula of an American option is a good approximation for the early-exercise premium. Because options are homo-geneous in S and K, the premium is also homohomo-geneous in S and K:

␵(S, Y, T; K) ⫽ K␵(S/K, Y, T; 1). The Barone-Adesi and Whaley method (1987) was used to define the premium as

(5) where z ⬅ S/K and H(T) is an arbitrary function of time-to-maturity, T. The partial derivatives of ␵ are ␵S⫽ HFz, ␵SS⫽ HFzz/K, ␵Y⫽ KHFY, ␵YY⫽ KHFYY, ␵SYHFzY, and ␵T⫽ KFHT⫽ KHFHHT. Equation (6) is generated by substituting

Equation (5) into Equation (3)

. (6)

Barone-Adesi and Whaley (1987) chose H(T) as 1 ⫺ exp[⫺rT] for simplic-ity. Chang et al. (2007) further adjusted H(T) to equal 1 ⫺ exp[⫺arT] for con-trolling a to reduce barrier option pricing errors of the quadratic approximation. After substituting H(T) ⫽ 1 ⫺ exp[⫺arT] into Equation (6), Equation (7) resulted: . (7) ⫺ rKHF ⫹ lx,yKH

⬁ 0

⬁ ⫺⬁ [F(zex,Y⫹ y) ⫺ F(z,y)]£(x,y)dxdy ⫺ ar(1 ⫺ H)KF ⫺ ar(1 ⫺ H)KHFH ⫹ KHFzYrsYzY⫹ KHFY(Y⫺ kYY) 0⫽ 1 2 KHFzzz 2Y⫹ KHF z((r⫺ d) ⫺ lx,yE[ex⫺ 1])z ⫹ 1 2 KHFYYs 2 YY ⫺ rKHF ⫹ lx,yKH

⬁ 0

⬁ ⫺⬁

[F(zex, Y⫹ y) ⫺ F(z, Y)]£(x, y)dxdy

⫹ KHFzYrsYzY⫹ KHFY(Y⫺ kYY) ⫺ HT(KF⫹ KHFH) 0⫽ 1 2KHFzzz 2Y⫹ KHF z((r⫺ d) ⫺ lx,yE[ex⫺ 1])z ⫹ 1 2KHFYYs 2 YY ␵(S, Y, T; K) ⫽ KH(T)F(S兾K, Y, H) ⫽ KH(T)F(z, Y, H) CA(S, Y, T; K) ⫽ CE(S, Y, T; K) ⫹ ␵(S, Y, T; K)

1Guo and Hung (2007) suggested a simple way to avoid the branch cut difficulties arising from the choice of

(5)

As described in Barone-Adesi and Whaley (1987), Bates (1991), and Chang et al. (2007), ar[1 ⫺ H]KHFH is negligible. Substituting F(z, Y)into Equation (7) and separating variables A1 and A2, yields

(8) After further separating Equation (8) into two equations for Y-terms and non-Y-terms, respectively, Equations (9) and (10) are generated.

(9) and

. (10)

For given values of the parameters r, d, , kY, sY, r, lx,y, m0, mx,y, sx,y, uy,

and a, accurate values of f1, f2, B1, and B2can be rapidly determined from

Equations (9) and (10) using Newton’s method. Initial values are obtained from Equation (10) given by replacing exp[(⫺(B ⫹ fmx,y)uy)] ⬵ 1 ⫺ Buy

fmx,yuy and expanding in a first-order

Taylor expansion, ignoring powers of f and B higher than 2. The result of the approximation of Equation (10) is

(11)

where h0 ⫽ ar (1 ⫺ H)/H ⫹ r and

. Equation (11) indicates that for ⫺ lx,y(exp[m

0⫹ s2x,y兾2]兾(1 ⫺ uymx,y)⫺ 1)

h1⫽ (r ⫺ d) ⫹ lx,y(m0⫹ mx,yuy) B⫽ (h0⫺ h1f)兾(Y ⫹ lx,yuy)

exp[(m0⫹ mx,yuy)f ⫹ s2x,yf2兾2 ⫹ uyB]

Y ⫹ lx,yaexp[ 1 2f2s2x,y⫹ m0f] 1⫺ Buy⫺ fmx,yuy ⫺ 1b 0⫽ BY ⫺ aara1 H ⫺ 1b ⫹ rb ⫹ a(r ⫺ d) ⫺ l x,yaexp[m0⫹ 1 2s2 x,y] 1⫺ uymx,y ⫺ 1bbf 0⫽ 1 2 f 2⫹ aBrs Y⫺ 1 2bf ⫹ 1 2 B 2s2 Y⫺ BkY ⫹ lx,yaexp[ 1 2f2s2 x,y⫹ m0f] 1⫺ Buy⫺ fmx,yuy ⫺ 1b. ⫹1 2B 2s2 YY⫹ fBrsYY ⫹ B(Y ⫺ kYY) ⫺ ara 1 H ⫺ 1b ⫺ r 0⫽ 1 2 f(f ⫺ 1)Y ⫹ fa(r ⫺ d) ⫺ l x,yaexp[m0⫹ 1 2s2 x,y] 1⫺ uymx,y ⫺ 1bb A1exp[B1Y]zf1⫹ A2exp[B2Y]zf2

(6)

jumps with plausible amplitudes (兩m0兩, 兩mx,y兩, sx,y, and uysubstantially less than 1),

a one-to-one relationship exists between f and B. Substituting Equation (11) into Equation (9) generates

. (12)

Parameters that satisfy the relationship

guaran-tee that one root (f1) is negative for puts, whereas the other (f2) is positive for

calls. Because the relationships f1 ⬍ 0 and A1⫽ 0 imply that the limSS0CA

(S, Y; T, K) ⫽ q, therefore, it follows that A1⫽ 0. Once values for f2 and B2

are obtained, A2and (the critical early-exercise price for calls) can be solved from the value-match condition and the high contact condition:

(13)

. (14)

Equations (13) and (14) imply that is the implicit solution to

(15) and A2can be determined by

(16)

The resulting formula for a basic American call option is

. (17)

Although the American puts must satisfy the same partial differential equation, the boundary conditions are somewhat different compared with those for calls. The boundary conditions for puts are

(18) and (19) PSA(S, Y, T; K) ⫽ ⫺1 PA(S, Y, T; K)⫽ K ⫺ S for S ⬍ S or S ⫺ K for S ⱖ S CA(S, Y, T; K) ⫽ CE(S, Y, T; K) ⫹ (S ⫺ K ⫺ CE(S, Y, T; K) )(S兾S)f2 A2⫽ S⫺ K ⫺ CE(S, Y, T; K) K(1⫺ exp[⫺arT])exp[B2Y]a S Kb f2 S⫽ f2(S⫺ K ⫺ C E(S, Y, T; K) ) 1⫺ CEz(S, Y, T; K) SS(S, Y, T; K) ⫽ 1 ⫺ CES(S, Y, T; K) ␵(S, Y, T; K) ⫽ S ⫺ K ⫺ CE(S, Y, T; K) and S h0 ⬍ 2kY(Y⫹ lx,yuy)兾s2Y ⫹ (s2 Yh20⫺ 2kY(Y⫹ lx,yuy)h0)

⫹ (2rsY(Y⫹ lx,yuy)h0⫹ 2kY(Y⫹ lx,yuy)h1⫺ (Y ⫹ lx,yuy)2⫺ 2s2Yh0h1)f

0⫽ ((Y ⫹ lx,yu

(7)

where S is the critical early-exercise price. The positive root (f2) is precluded

for the puts because it implies that the limSSqPA(S, Y; T, K) ⫽ q. Consequently,

A2⫽ 0 and therefore,

(20) where

. (21)

Note that, after turning off the specification of stochastic volatility and volatility jumps ( , r ⫽ 0, sY⫽ 0, mx,y⫽ 0, and uy⫽ 0), the authors’

gen-eral solution reduces to that of Bates (1991). Table I demonstrates that their results are consistent with the results in this case. Data in Table I for Bates and for the finite difference method are taken from Table II in Bates (1991).

kYY⫽ Y S⫽ f1(K⫺ S ⫺ P E(S, Y, T; K)) ⫺1 ⫺ PE S(S, Y, T; K) for S ⬎ S or K ⫺ S for S ⱕ (S) PA(S, Y, T; K) ⫽ PE(S, Y, T; K) ⫹ (K ⫺ S ⫺ PE(S,Y, T; K) )(S兾S)f1 TABLE I

American Option Values Under Stochastic Volatility When sYS 0

K sY⫽ 0.5 sY⫽ 0.25 sY⫽ 0.1 sY⫽ 0.01 Bates FD P 220 0.223 0.209 0.196 0.19 0.19 0.19 (0.2196)(0.2058)(0.1923)(0.1866)(0.19) (0.19) 235 1.612 1.621 1.628 1.631 1.63 1.63 (1.6007)(1.6091)(1.6159)(1.6186)(1.62) (1.62) 250 6.773 6.807 6.839 6.852 6.85 6.82 (6.7330)(6.7664)(6.798) (6.8109)(6.81)(6.81) 265 16.918 16.923 16.925 16.926 16.91 16.90 (16.7916)(16.7942)(16.7949)(16.7949)(16.79)(16.79) 280 30.223 30.217 30.210 30.208 30.19 30.21 (29.8443)(29.8325)(29.821) (29.8162)(29.82)(29.82) C 220 30.014 30.011 30.009 30.008 30.01 30.00 (29.4789)(29.4651)(29.4516)(29.4459)(29.45)(29.45) 235 16.398 16.410 16.420 16.425 16.42 16.41 (16.2304)(16.2387)(16.2456)(16.2483)(16.25)(16.25) 250 6.789 6.824 6.857 6.870 6.86 6.84 (6.733)(6.7664)(6.798)(6.8109)(6.81)(6.81) 265 2.182 2.185 2.186 2.187 2.18 2.18 (2.162) (2.1645)(2.1653)(2.1653)(2.17)(2.17) 280 0.593 0.581 0.570 0.565 0.56 0.56 (0.585) (0.5732)(0.5617)(0.5569)(0.56)(0.56) Note. S⫽ 250, T ⫽ 0.25, r ⫽ 0.1, d ⫽ 0.1, Y ⫽ 0.01, lx,y⫽ 10, m 0⫽ 0.01005, sx,y⫽ 0.03, r ⫽ 0, mx,y⫽ 0 uy⫽ 0, kYY⫽ ¯¯

Y⫽ 0.24sY, and a ⫽ 1. Puts (P) and Calls (C), the Bates’ approximation at sY⫽ 0 (Bates), and the finite difference

(8)

ANALYTIC APPROXIMATION FORMULAE FOR OTHER JUMP MODELS

The quadratic approximation scheme can be applied to the stochastic volatility model with other jump types, such as independent return jumps and volatil-ity jumps. The partial integro-differential equation for a contingent claim price,

P, on the underlying asset is given by

(22) where x is a return-jump amplitude, lxis the arrival rate of return jumps, y is a

volatility-jump amplitude, and lyis the arrival rate of volatility jumps.

It is assumed that x follows a normal distribution, ,

and y follows an exponential distribution, Exponential(uy). Equation (22)

induces the following differential equation:

N(log(1⫹ mx) ⫺12s2x,s2x) ⫹ lx

⬁ ⫺⬁ [P(Sex)⫺ P(S)]£(x)dx ⫹ ly

⬁ 0 [P(Y⫹ y) ⫺ P(Y)]£(y)dy ⫹ PS((r⫺ d) ⫺ lEQ[ex⫺ 1])S ⫹ PY(Y⫺ kYY) ⫺ PT⫺ rP 0⫽1 2 PSSS 2Y 1 2 PYYs 2 YY⫹ PSYrsYSY TABLE II

Comparisons of American Options: Correlated Double Jumps

S European Option Simulated American (s.e.) Approx Diff

P 60 39.501 40.006 0.026 40.000 0.005 70 29.936 30.173 0.038 30.138 0.035 80 20.948 21.067 0.034 21.033 0.034 90 13.302 13.365 0.044 13.341 0.024 100 7.786 7.843 0.048 7.805 0.038 110 4.417 4.463 0.049 4.428 0.036 120 2.546 2.575 0.047 2.552 0.023 C 60 0.097 0.106 0.011 0.097 0.009 70 0.382 0.423 0.028 0.383 0.040 80 1.246 1.287 0.033 1.247 0.040 90 3.451 3.517 0.053 3.454 0.062 100 7.786 7.858 0.079 7.794 0.064 110 14.268 14.351 0.069 14.286 0.064 120 22.249 22.362 0.051 22.288 0.073 Note. K⫽ 100, T ⫽ 0.25, r ⫽ 0.06, d ⫽ 0.06, Y¯¯ ⫽ 0.49, Y ⫽ 0.0968, r ⫽ ⫺0.1, lx,y⫽ 1.64, m

0⫽ ⫺0.03, mx,y⫽ ⫺7.87,

(9)

(23) Further separating Equation (23) into two equations for Y-terms and

non-Y-terms, respectively, yields

(24) and

(25) Similarly, accurate values of f1, f2, B1, and B2can be rapidly determined

from Equations (24) and (25) using Newton’s method. Initial values are obtained from Equation (25) given by replacing 1/(1 ⫺ uyB) ⫺ 1 ⬵ uyB and

expanding in a first-order Taylor

expan-sion, ignoring powers of f higher than 2. The approximation of Equation (25) results in Equation (26)

(26)

where h0⫽ ar(1 ⫺ H)/H ⫹ r and .

Substituting Equation (26) into Equation (24) yields

. (27)

Parameters that satisfy the relationship guarantee

that one root (f1) is negative for puts, whereas the other (f2) is positive for

calls. The successive derivation is similar to those described for Equations (13)–(21). The characteristic function of this model is also provided in Appendix A.

h0 ⬍ 2kY(Y⫹ lyuy)兾s2Y ⫹ (s2 Yh20⫺ 2kY(Y⫹ lyuy)h0) ⫹ (2rsY(Y⫹ lyuy)h0⫹ 2kY(Y⫹ lyuy)h*1⫺ (Y ⫹ lyuy)2⫺ 2s2Yh0h*1)f 0⫽ ((Y ⫹ lyuy)2⫹ s2Y(h*1)2⫺ 2rsY(Y⫹ lyuy)h*1)f2 h*1⫽ (r ⫺ d) ⫺ lxmx⫹ lx(log[1⫹ mx]⫺ 1 2s2x B⫽ (h0⫺ h*1f)兾(Y ⫹ lyuy) exp[flog[1 ⫹ mx]⫹ s2xf(f ⫺ 1)兾2] ⫹ ly a 1 1⫺ uyB ⫺ 1b. ⫹ lx aexp c flog[1 ⫹ m x]⫹ 1 2 s 2 xf(f ⫺ 1) d ⫺ 1b 0⫽ BY ⫺ aar a1 H ⫺ 1b ⫹ rb ⫹ ((r ⫺ d) ⫺ l xm x)f 0⫽ 1 2 f(f ⫺ 1) ⫹ 1 2 B 2s2 Y⫹ fBrsY⫺ BkY ⫹ ly a 1 1⫺ uyB ⫺ 1b. ⫹1 2s 2 xf(f ⫺ 1) d ⫺ 1b ⫺ ar a1 H ⫺ 1b ⫺ r ⫹ l x aexp c flog[1 ⫹ m x] 0⫽ 1 2f(f ⫺ 1)Y ⫹ f((r ⫺ d) ⫺ l xm x) ⫹ 1 2B 2s2 YY⫹ fBrsYY⫹ B(Y ⫺ kYY)

(10)

NUMERICAL RESULTS AND COMPARISONS

Longstaff and Schwartz (2001) proposed a simple least-squares method (LSM) to value American options by simulation. Table II gives a comparison of the LSM and the quadratic approximation proposed in this research for the sto-chastic volatility model with correlated double jumps. The parameters reported in Bakshi and Cao (2003) are used for this computation. The simulation is based on 20,000 paths for the stock-price process and the option is exercisable 20 times before maturity. Puts (P) and Calls (C) are abbreviated as accordingly. The LSM proposed by Longstaff and Schwartz (2001), the quadratic approxi-mation, and their difference, are represented by Simulated American, Approx, and Diff, respectively. The standard errors of the simulation estimates (s.e.) are given in parentheses. As shown, the differences between the LSM and the quadratic approximation (Approx) are typically small.

Table III gives a comparison of the LSM and the quadratic approximation for the stochastic volatility model with independent double jumps using param-eters presented in Bakshi and Cao (2003). The differences between the LSM and the quadratic approximation presented in this research are also typically small. By turning off the specification of volatility jumps (ly⫽ 0), the

approxi-mation formula reduces to the solution of the stochastic volatility model with return jumps empirically examined in Bakshi et al. (1997) and Bates (1996). As shown in Table IV, the differences between the LSM and the quadratic approx-imation are small. Table V exhibits that the differences between the LSM and

TABLE III

Comparisons of American Options: Independent Double Jumps

S European Option Simulated American (s.e.) Approx Diff

P 60 39.423 39.999 0.013 40.000 ⫺0.001 70 29.749 30.065 0.028 30.090 ⫺0.025 80 20.747 20.899 0.040 20.950 ⫺0.050 90 13.261 13.328 0.046 13.389 ⫺0.060 100 7.843 7.908 0.070 7.927 ⫺0.019 110 4.381 4.426 0.047 4.439 ⫺0.013 120 2.361 2.396 0.035 2.403 ⫺0.007 C 60 0.019 0.020 0.001 0.019 0.001 70 0.195 0.198 0.005 0.197 0.001 80 1.045 1.049 0.009 1.049 0.000 90 3.410 3.425 0.018 3.419 0.006 100 7.843 7.877 0.019 7.863 0.014 110 14.232 14.287 0.039 14.274 0.013 120 22.064 22.171 0.027 22.145 0.026 Note. K⫽ 100, T ⫽ 0.25, r ⫽ 0.06, d ⫽ 0.06, Y¯¯ ⫽ 0.49, Y ⫽ 0.1623, r ⫽ ⫺0.31, lx⫽ 0.87, m x⫽ ⫺0.014, sx⫽ 0.04, ly⫽ 2.43, u y⫽ 0.0036, sY⫽ 0.54, kY⫽ 3.02, and a ⫽ 1.

(11)

TABLE IV

Comparisons of American Options: Stochastic Volatility With Jump in Return

S European Option Simulated American (s.e.) Approx Diff

P 60 39.556 39.998 0.030 40.000 ⫺0.002 70 30.156 30.400 0.046 30.309 0.091 80 21.564 21.636 0.084 21.623 0.013 90 14.436 14.505 0.044 14.461 0.043 100 9.181 9.247 0.061 9.193 0.054 110 5.686 5.722 0.048 5.692 0.030 120 3.505 3.519 0.046 3.508 0.011 C 60 0.151 0.166 0.009 0.152 0.014 70 0.603 0.628 0.027 0.605 0.024 80 1.862 1.893 0.038 1.866 0.027 90 4.585 4.660 0.082 4.594 0.066 100 9.181 9.206 0.109 9.200 0.006 110 15.537 15.611 0.063 15.572 0.039 120 23.207 23.350 0.079 23.271 0.079 Note. K⫽ 100, T ⫽ 0.25, r ⫽ 0.06, d ⫽ 0.06, Y¯¯ ⫽ 0.49, Y ⫽ 0.125, r ⫽ ⫺0.16, lx⫽ 3.05, m x⫽ ⫺0.03, sx⫽ 0.19, ly⫽ 0, uy⫽ 0, sY⫽ 0.41, kY⫽ 3.92, and a ⫽ 1. 0.05 0.05 0.05 0.05 0.05 0 0.01 0.50 0.70 0.90 2.00 4.00 10.00 30.00 50.00 70.00 90.00 Pricing Error S60 S80 S100 S120 FIGURE 1

The choice of a in reducing pricing errors for calls. Note: K ⫽ 100, T ⫽ 0.25, r ⫽ 0.06,

d⫽ 0.06, , Y⫽ 0.1623 , r ⫽ ⫺0.31, lx⫽ 0.87, m

x⫽ ⫺0.014, sx⫽ 0.04, ly⫽ 2.43,

uy⫽ 0.0036, sY⫽ 0.54, and kY⫽ 3.02. S60, S80, S100, and S120 denote cases for which

S⫽ 60, S ⫽ 80, S ⫽ 100, and S ⫽ 120.

(12)

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 Pricing Error 0.01 0.50 0.70 0.90 2.00 4.00 10.00 30.00 50.00 70.00 90.00 S60 S80 S100 S120 FIGURE 2

The choice of a in reducing pricing errors for puts. Note: K ⫽ 100, T ⫽ 0.25, r ⫽ 0.06,

d⫽ 0.06, , Y⫽ 0.1623, r ⫽ ⫺0.31, lx⫽ 0.87, m

x⫽ ⫺0.014, sx⫽ 0.04, ly⫽ 2.43,

uy⫽ 0.0036, sY⫽ 0.54, and kY⫽ 3.02. S60, S80, S100, and S120 denote cases for which

S⫽ 60, S ⫽ 80, S ⫽ 100, and S ⫽ 120.

Y⫽ 0.49

TABLE V

Comparisons of American Options: Stochastic Volatility With Jump in Volatility

S European Option Simulated American (s.e.) Approx Diff

P 60 39.445 40.012 0.032 40.000 0.012 70 29.865 30.188 0.054 30.194 ⫺0.006 80 21.066 21.245 0.053 21.273 ⫺0.028 90 13.783 13.860 0.068 13.920 ⫺0.061 100 8.439 8.488 0.063 8.534 ⫺0.046 110 4.915 4.964 0.040 4.983 ⫺0.019 120 2.770 2.778 0.044 2.821 ⫺0.043 C 60 0.040 0.046 0.006 0.041 0.005 70 0.311 0.325 0.021 0.314 0.011 80 1.363 1.402 0.031 1.369 0.033 90 3.932 3.983 0.053 3.945 0.038 100 8.439 8.498 0.056 8.466 0.032 110 14.766 14.882 0.067 14.817 0.064 120 22.472 22.652 0.104 22.564 0.088 Note. K⫽ 100, T ⫽ 0.25, r ⫽ 0.06, d ⫽ 0.06, Y¯¯ ⫽ 0.49, Y ⫽ 0.189922, r ⫽ ⫺0.26, ly⫽ 0, m x⫽ 0, sx⫽ 0, l y⫽ 1.36, uy⫽ 0.0016, sY⫽ 0.53, kY⫽ 2.58, and a ⫽ 1.

(13)

the authors’ approximation for the stochastic volatility model with volatility jumps (lx⫽ 0) are small. These computations are performed with the

parame-ters referenced from Bakshi and Cao (2003).

Figures 1 and 2 illustrate that the choice of a affects the pricing errors of the quadratic approximation for the stochastic volatility model with independent double jumps. Although the optimal choice of a, between 50 and 60, minimizes pricing errors for calls, a corresponding choice for puts in this case cannot be determined. Therefore, the approach of Barone-Adesi and Whaley (1987) is followed and a value of a ⫽ 1 is selected for simplicity in these cases because the pricing errors are small and often well within the market bid–ask spread.

CONCLUSIONS

In this article, the application of a quadratic approximation method is described to obtain efficient analytic formulae for American options on processes permitting stochastic volatility and double jumps to illustrate its gen-erality. To the best of one’s knowledge, the literature suggests no approximation formulae for American option values based on these processes. This quadratic approximation scheme is a generalization of the Barone-Adesi and Whaley approach and nests several option models.

The constant volatility model is the first special case. The solution, with parameters specified as sY⫽ 0, , ly⫽ 0, and lx⫽ 0, reduces to that of

Barone and Whaley in 1987. Bates’ constant volatility model with return jumps presented in 1991 is the second special case (sY⫽ 0, , ly⫽ 0, and lx

⫽ 0). The third special case is Heston’s (1993) stochastic volatility model with-out jumps (lx⫽ 0 and ly⫽ 0) proposed in 1993. The stochastic volatility model,

with return jumps, empirically examined in Bakshi et al. (1997) and Bates (1996) (ly⫽ 0), is the fourth special case. The fifth special case is the stochastic

volatil-ity model with volatilvolatil-ity jumps (lx ⫽ 0) proposed by Duffie et al.(2000).

Moreover, the stochastic volatility model with correlated double jumps is also included in this scheme.

Comparisons with the least-squares approach show that the quadratic approximation scheme is very useful and efficient in pricing American options with stochastic volatility and double jumps. It is anticipated that the scheme can be further applied to American exotic options in the future.

kYY⫽ Y

(14)

APPENDIX A

Stochastic Volatility Model With Correlated Double Jumps

The present value of a basic European call option can be formulated as

CE(S, Y, T; K) ⫽ EQ[e⫺rTmax{S(T) ⫺ K, 0}] and is given by

(A1) where J(T; f) is the characteristic function of the state density. The character-istic function is given by

(A2) where A(T; f) and B(T; f) are

(A3) (A4) (A5) e ⬅ 2(ifsYr ⫺ kY)2⫺ if(if ⫺ 1)s2Y B(T; f) ⬅ if(if ⫺ 1)(1 ⫺ exp[⫺eT]) 2e ⫺ (e ⫹ ifsYr ⫺ kY)(1 ⫺ exp[eT]) ⫹ 2lx,yu

yif(if ⫺ 1)exp aifm0⫺

1 2f 2s2 x,yb pq loga p ⫹ qe⫺ eT p ⫹ q b ⫹

lx,y(2e ⫺ b)exp aifm 0⫺ 1 2 f 2s2 x,ybT p ⫺ iflx,y° expam0⫹ 1 2 s 2 x,yb 1⫺ uymx,y ⫺ 1 ¢ T ⫺ l x,y TY s2 Y

c(e ⫹ ifsYr ⫺ kY)T⫹ 2log c 1 ⫺

(e ⫹ ifsYr ⫺ kY)(1⫺ exp[⫺eT]) 2e d d A(T; f) ⬅ (if(r ⫺ d) ⫺ r)T J(T; f) ⫽ exp[A(T; f) ⫹ B(T; f)Y]Sif ⫺ K a1 2 J(T; 0) ⫺ 1 p

⬁ 0 Im[J(T; ⫺ n)exp[inlog[K]]] n dn CE(S, Y, T; K) ⫽1 2 J(T; ⫺ i) ⫺ 1 p

⬁ 0 Im[J(T; ⫺ i ⫺ n)exp[inlog[K]]] n dn

(15)

(A6) (A7)

. (A8)

The proof is published in Duffie et al. (2000). Given the solution for European calls, the formula for puts, PE(S, Y, T; K), can be obtained by the put-to-call conversion equation (Grabbe, 1983):

. (A9)

Stochastic Volatility Model With Independent Double Jumps

The characteristic function of the state density has the same functional form as in Equation (A2). However, the component function A(T; f) is somewhat dif-ferent than Equation (A3).

(A10)

. (A11)

BIBLIOGRAPHY

Bakshi, G., & Cao, C. (2003). Risk-neutral kurtosis, jumps, and option pricing: Evidence from 100 most actively traded firms on the CBOE. EFA 2003 Annual Conference Paper No. 953.

Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing models. Journal of Finance, 53, 499–547.

Barone-Adesi, G., & Whaley, R. (1987). Efficient analytic approximation of American option values. Journal of Finance, 42, 301–320.

Bates, D. (1991). The crash of ‘87: Was it expected? The evidence from options mar-kets. Journal of Finance, 46, 69–107.

Bates, D. (1996). Jumps and stochastic volatility: exchange rate processes implicit in PHLX deutsche mark options. Review of Financial Studies, 9, 69–107.

q⬅ b ⫹ if(if ⫺ 1)uy ⫺ lyT l y(2e ⫺ b) 2e ⫺ q T⫹ 2ly(q⫺ b) (2e ⫺ q)q lnc 2e ⫺ q(1 ⫺ exp[⫺eT]) 2e d ⫺ Y s2 Y

c(e ⫹ ifsYr ⫺ kY)T⫹ 2log c 1 ⫺

(e ⫹ ifsYr ⫺ kY)(1⫺ exp[⫺eT]) 2e d d ⫹ lxc (1 ⫹ m x)ifexpc 1 2 if(if ⫺ 1)s 2 xd ⫺1d T

A(T; f) ⬅ (if(r ⫺ d) ⫺ r)T ⫺ iflxm xT

PE(S, Y, T; K) ⫽ CE(S, Y, T; K)⫺ Sexp[⫺dT] ⫹ Kexp[⫺rT]

q⬅ if(if ⫺ 1)uy⫹ b(1 ⫺ uyifmx,y) p⬅ 2e(1 ⫺ uyifmx,y)⫺ q

(16)

Broadie, M., Chernov, M., & Johannes, M. (2007). Model specification and risk premi-ums: The evidence from the futures options. Journal of Finance, 62, 1453–1490. Chang, G., Kang, J., Kim, H., & Kim, I. (2007). An efficient approximation method for

American exotic options. Journal of Futures Markets, 27, 1, 2007.

Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-duffusions. Econometrica, 68, 6, 1343–1376.

Grabbe, J. (1983). The pricing of call and put options on foreign exchange. Journal of International Money and Finance, 2, 239–253.

Guo, J., & Hung, M. (2007). A note on the discontinuity problem in Heston’s stochas-tic volatility model. Applied Mathemastochas-tical Finance, 14, 339–345.

Heston, S. (1993). A closed form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6, 327–343.

Ju, N., & Zhong, R. (1999). An approximate formula for pricing American options. Journal of Derivatives, 7, 31–40.

Longstaff, F., & Schwartz, E. (2001). Valuing American options by simulation: A simple least-squares approach. Review of Financial Studies, 14, 113–147.

MacMillan, L. (1987). Analytic approximation for the American put option. Futures and Options Research, 1A, 119–139.

數據

Table III gives a comparison of the LSM and the quadratic approximation for the stochastic volatility model with independent double jumps using  param-eters presented in Bakshi and Cao (2003)
TABLE IV

參考文獻

相關文件

The bivariate binomial values with 270 time steps are compared with the values generated by Hilliard and Schwartz [1996], the Hull-White stochastic volatility model [1987], and

Simonato, 1999, “An Analytical Approximation for the GARCH Option Pricing Model,” Journal of Computational Finance 2, 75- 116.

Students are asked to collect information (including materials from books, pamphlet from Environmental Protection Department...etc.) of the possible effects of pollution on our

Wang, Solving pseudomonotone variational inequalities and pseudocon- vex optimization problems using the projection neural network, IEEE Transactions on Neural Networks 17

Using this formalism we derive an exact differential equation for the partition function of two-dimensional gravity as a function of the string coupling constant that governs the

Define instead the imaginary.. potential, magnetic field, lattice…) Dirac-BdG Hamiltonian:. with small, and matrix

We propose a primal-dual continuation approach for the capacitated multi- facility Weber problem (CMFWP) based on its nonlinear second-order cone program (SOCP) reformulation.. The

Recommended Approach for Setting Regulatory Risk-Based Capital Requirements for Variable Annuities and Similar Products with Guarantees (Excluding Index Guarantees), American Academy