相依結構對多資產選擇權定價影響之實證分析

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國立臺灣大學管理學院財務金融學系碩士論文

Department of Finance College of Management National Taiwan University

Master Thesis

相依結構對多資產選擇權定價影響之實證分析 Bivariate Options Pricing with

Copula-Based GARCH Model -Empirical Analysis

許凱雯 Kai-Wen Hsu

指導教授：王耀輝博士 Advisor: Yaw-Huei Wang, Ph.D.

中華民國 98 年 6 月

June, 2009

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摘要

本論文係探討相依結構對二元選擇權定價結果影響的實證分析。所謂二元選擇權，乃指兩個相異標的資產所衍生而得的奇異選擇權，其價格與兩個資產的價格變動及相依結構具有極大關聯。在此選取四組不同的金融指數作實證分析，其組成內容包含跨國家和跨資產，以期觀察在不同的相依結構下，對三種市場上較為廣泛交易的二元選擇權，其選擇權評價結果有何異同。論文中採用 Copula-based GARCH 模型作為定價方法，並利用適合度檢定選取出一最適相依結構，最後以蒙地卡羅模擬求得選擇權價格後，比較各相依結構下的選擇權評價差異。

研究結果發現，當標的資產相關程度愈強、選擇權到期日愈長、選擇權到期時的價格在價平附近與選擇權的報酬函數為二元彩虹選擇權時，在不同相依結構下的評價結果，其價格差異顯著性會愈高。故總結而言，相依結構的設定對二元選擇權的價格確實存在影響，在評價二元選擇權時是不可被忽略的一環。

關鍵字：二元選擇權、多資產選擇權、相依結構。

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Abstract

Multivariate options have experienced significant development in the last decade,

due to their excellent abilities for hedging the risk of multiple assets. The most

important issue in the valuation of multivariate options is the dependence structure

among these underlying assets. In this paper, we use copula-based GARCH model as

pricing device to describe the dependence structures of underlying assets, rather than the

traditional linear correlation and Gaussian assumptions to price multivariate claims.

Particularly, the skewed-t GARCH model is applied to capture the marginal

distributions of underlying financial assets. To compare the impact of difference

dependence structures on option pricing, we perform Monte-Carlo simulation to

simulate the bivariate option prices, and observe the error of option prices caused from

different model dependence structures, time-to-maturities, strike prices and option

payoff functions. We use goodness-of-fit tests to choose one dependence model that fit

the empirical distributions best, and then the paired t-test is also implemented to

determine whether the pricing errors are significant enough.

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Note: u is a vector of marginal probabilities, Φ_ρxy is the standard bivariate normal distribution function with linear correlation coefficient ρ_XY, Φ is the standard normal distribution function, Ψ_ρ,ν is the cdf of a bivariate Student’s t distribution with correlation matrix ρ and degrees of freedom parameter υ> 2.

Both Gaussian and Student’s t copula belong to the class of elliptical copula. For

imperfectly correlated variables, the Gaussian copula implies tail independence. When the number of degree of freedom diverges, the Student’s t copula converges to the

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Gaussian one, but the Student’s t copula still assigns more tail events than the Gaussian copula. Moreover, the Student’s t copula exhibits tail dependence even if the correlation

coefficients equal zero.

Besides the elliptical copula family, we also use some Archimedean copula

functions, in particular to the one-parameter ones. These copula functions could be

distinguished by their own tail dependency. The Gumbel family has upper tail

dependency with λ_U = 2 − 2¹^α, the Clayton family has lower tail dependency for α > 0,

since λ_L = −2¹^α, and the Frank family has neither lower nor upper tail dependency.

3.3 IFMMETHOD

In general, people consider using maximum likelihood method to estimate the

copula parameters. Using Formula (3.3) in Sklar theorem, it is obtained that the

log-likelihood function can be represented by

𝑙 𝜃 = ^T_t=1ln 𝑐(𝐹₁(x_1t), … , 𝐹_𝑛(x_nt)) + ^T_t=1 ⁿ_j=1ln 𝑓_𝑗(x_jt) (3.5)

where 𝑐(𝐹₁(x_1t), … , 𝐹_𝑛(x_nt)) = ^∂𝐶(𝐹_∂𝐹¹^(x^1t^),…,𝐹^𝑛^(x^nt⁾⁾

1(x_1t)…∂𝐹_𝑛(x_nt) , and θ is the set of all parameters of both the marginal distributions and copula functions.

As we identify the marginal distributions and the copula, we could obtain the

previous log-likelihood function and the maximum likelihood estimators

θ _MLE = Arg max_𝜃𝑙(𝜃) (3.6)

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However, the maximum likelihood estimators could be very computationally

intensive in the case of high dimensions, because we have to estimate the parameters of

the marginal distributions and the dependence structure represented by the copula

simultaneously. We alternatively implement the inference for the margins or IFM

method proposed by Joe and Xu (1996). They suggest that these sets of parameters

should be estimated in two steps:

(i) In the first step, we estimate the parameters of the marginal distributions θ₁ by

performing the estimation of each univariate margins:

θ ₁ = Arg max_θ₁ ^T_t=1 ⁿ_j=1ln 𝑓_𝑗(x_jt; θ₁) (3.7)

(ii) In the second step, we then estimate the copula parameter θ₂ by given θ ₁: θ ₂ = Arg max_θ₂ ^T_t=1ln 𝑐(𝐹₁(x_1t), … , 𝐹_𝑛(x_nt); θ₂, θ ₁) (3.8)

The IFM estimator θ _IFM = (θ ₁, θ ₂)′ is different from the MLE θ _MLE because the

IFM estimator is the solution of:

_∂θ^∂𝑙¹

11,_∂θ^∂𝑙²

12, … ,_∂θ^∂𝑙^𝑛

1n,_∂θ^∂𝑙^𝑐

2 = 0′ (3.9) but the MLE comes from the solution of:

_∂θ^∂𝑙

11,_∂θ^∂𝑙

12, … ,_∂θ^∂𝑙

1n,_∂θ^∂𝑙

2 = 0′ (3.10)

where l is the entire log-likelihood function, lj is the log-likelihood function of the jth

marginal distribution, and lc is the log-likelihood function of the copula itself. In general,

these two estimators are therefore not equivalent.

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Also, Patton (2006) showed that the IFM estimator verifies the property of

asymptotically efficient and normality, and we have

T(θ _IFM − θ₀) → N(0, 𝒢⁻¹(θ₀)) (3.11)

where θ₀ denotes the true value of the parameter vector, and the 𝒢(θ₀) denotes the

Godambe information matrix taking the following form:

𝒢 θ₀ = D⁻¹V(D⁻¹)′ (3.12)

with

D = E ^{∂s θ}_∂θ , V = E[s(θ)s(θ)′],

and a score function

s θ = ∂𝑙₁

∂θ₁₁, ∂𝑙₂

∂θ₁₂, … , ∂𝑙_𝑛

∂θ₁_n,∂𝑙_𝑐

∂θ₂ ′

3.4 GOODNESS-OF-FIT TEST

As we obtain the marginal distributions of the underlying asset returns and

combine them into the joint distribution with copula, next we suppose to test whether

the specific distribution for a random variable accurately fits the corresponding

observations. We propose to use Kolmogorov-Smirnov (K-S) test and Anderson-Darling

(A-D) test as our measurement, which belong to the large class of goodness-of-fit tests.

They test the null hypothesis that the populations drawn from two or more

independent samples of data are identical. This test therefore can be used to decide

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which dependence structure best fit the empirical data, namely the null hypothesis H

₀

of

same population distributions cannot be rejected. Then we can compare the P-value

with the desired significance level to facilitate a decision about this null hypothesis.

Let x be a set of sample observations. According to the Glivenko–Cantelli theorem,

the empirical cumulative distribution F_E will converge to the hypothetical cumulative

distribution F_H almost surely. The test statistics of K-S test and A-D test are described

as follows:

D_KS = max

t F_E x_t − F_H(x_t) (3.13) D_AD = _F^[F^E^{x −F}^H^{x ]}²

H x [1−F_H x ]

x dF_H x (3.14) The empirical cumulative distribution F_E is defined as

F_E(x) =¹_T ^T_t=1 ^N_j=1I(x_j,t≤ x_j^u^j^∙T) (3.15)

where I(∙) is the indicator function, u is a vector of marginal probabilities, and x_j^u^j^∙T is the u_j∙ T th order statistic.

The K-S test is distribution free in the sense that the critical values do not depend

on the specific distribution being tested, and it is more sensitive to deviations in the

center of the distribution. The A-D test introduced by Anderson and Darling (1952,

1954) is a modification of the K-S test that gives more weight to deviations in the tails

than K-S test. By using these two goodness-of-fit tests simultaneously, we could have

more evidence to prove our model selection.

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15 3.5 MONTE CARLO SIMULATION

Here we propose to apply a conditional sampling method to simulate draws from a

specific bivariate copula. Given the parameters of each copula function, our task is to

generate pairs of observations (𝑢, 𝑣) of the random vector (𝑈, 𝑉), where 𝑈, 𝑉 are

both independent uniform distribution and their joint distribution function is C. The

main idea of conditional sampling is based on the conditional distribution 𝑐_𝑢 𝑣 = Pr(𝐹₂ ≤ 𝑣 𝐹₁ = 𝑢) = lim

Δ𝑢→∞

𝐶 𝑢+Δ𝑢,𝑣 −𝐶(𝑢,𝑣)

Δ𝑢 =^𝜕𝐶_𝜕𝑢 = 𝐶_𝑢 𝑣 (3.16) where 𝐶_𝑢(𝑣) is the partial derivative of the copula. The conditional distribution 𝑐_𝑢 𝑣

is a non-decreasing function and exists for all 𝑣 ∈ [0,1].

The general procedure to simulate the desired return process is stated as follows:

(i) Generate two independent random variables (𝑢, 𝑢′) ∈ Uniform(0,1), where 𝑢

is the first desired draw.

(ii) Compute the quasi-inverse function of 𝑐_𝑢 𝑣 , which depend on the parameters

of the selected copula and on 𝑢. Set 𝑣 = 𝑐_𝑢⁻¹(𝑢′) to obtain the second desired

draw. The following Table 2 reports the quasi-inverse functions of copulas.

(iii) Let 𝐹_𝑖 ∙ be the cumulative density function of the standardized marginal

innovations z_i,t~skewed − t z_i η_i, ϕ_i . We take the inverse functions 𝑆𝑇_1,𝑡, 𝑆𝑇_2,𝑡 = 𝐹⁻¹ 𝑢₁ , 𝐹⁻¹ 𝑢₂ as the observation of innovations at time 𝑡.

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Then calculate the daily asset returns 𝑟_1,𝑡, 𝑟_2,𝑡 by the following formula:

𝑟_𝑖,𝑡 = 𝑟_𝑓 −¹₂ℎ_{𝑖,𝑡−1}+ 𝑆𝑇_𝑖,𝑡, for 𝑖 ∈ 1,2 , 𝑡 ∈ 1, … , 𝑇 (3.17)

Table 2: Quasi-inverse function of copula function 𝒄_𝑢(𝒗)

Gaussian Φ 1 − 𝜌²Φ⁻¹ 𝑣 + 𝜌Φ⁻¹ 𝑢

Stud𝑒𝑛𝑡’s t

t_𝜈 𝜌 t_𝜈⁻¹ 𝑢 + 𝜈 + t_𝜈⁻¹ 𝑢 ² × 1 − 𝜌²

𝜈 − 1 t_{𝜈 +1}⁻¹ (𝑣)

Gumbel No closed-form solution

Clayton 𝑣 = u^−∝ u^′−^∝+1^∝ − 1 + 1

−1

∝

Frank 𝑣 = −1

∝ln 1 + u^′(1 − e^−∝) u^′ e^−∝u− 1 − e^−∝u

Note: The inverse function of 𝑐_𝑢(𝑣) for Gumbel copula is calculated by solving an equation by numerical way. The details are discussed in the book Copula Methods in Finance, written by Cherubini, Luciano, and Vecchiato, 2004.

(iv) After we have all daily asset returns (𝑟_1,𝑡, 𝑟_2,𝑡) for 𝑡 = {1, … , 𝑇}, we obtain the

percentage growth (𝑅_1,𝑡, 𝑅_2,𝑡) of underlying assets defined as:

𝑅_i,𝑡 = exp ^𝑡_{𝑗 =1}𝑟_𝑖,𝑗 , for 𝑖 ∈ 1,2 , 𝑡 ∈ 1, … , 𝑇 (3.18)

Finally, we can calculate the bivariate option prices under different payoff functions

with underlying asset return (𝑅_1,𝑇, 𝑅_2,𝑇) and count it in one simulation sample.

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4. D

ATASETS AND

O

PTION

P

AYOFF

S

TRUCTURES

4.1DATA AND DIAGNOSTIC ANALYSIS

We use several types of indices as proxy for the equity, commodity and foreign

exchange returns with different regions or industries: MSCI Pacific Index and MSCI Far

East Index (cross regions), MSCI World Energy Index and MSCI World Bank Index

(cross industries), Reuters/Jefferies CRB Index and MSCI Latin America Index (cross

assets), S&P 500 Index and DXY currency Index (cross assets). The linear correlations

of each pair of underlying financial assets are shown in Table 3. We would like to

compare the impact of different types of dependence structure to the bivariate options

pricing.

Table 3: The linear correlation of each pair assets

Linear Correlation Pair Composition

Pair 1 0.9828 MSCI Pacific Index and MSCI Far East Index

Pair 2 0.6330 MSCI World Energy Index and MSCI World Bank Index Pair 3 0.4160 Reuters/Jefferies CRB Index and MSCI Latin America Index Pair 4 0.0317 S&P 500 Index and DXY currency Index

All data are collected from Bloomberg over the period from July 1, 2002 to

December 31, 2008. The analysis is based on the logarithm returns of daily closing

prices which exclude non-trading days to ensure that enough observations of the tail of

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The following Table 4 reports the summary statistics on the returns in the samples.

All of them have apparently skewness and leptokurtotic, which implies the

unconditional distributions of asset returns are asymmetric and fat-tailed. Moreover, the

Jarque-Bera test also shows obviously non-Gaussian characteristic at the 5%

significance level for all underlying assets.

Table 4: Summary Statistics Asset

Statistics MSCI Pacific

MSCI Far East

MSCI

Energy MSCI Bank CRB MSCI Latin

America S&P 500 DXY Mean× 10³ 0.0273 0.0154 0.1149 -0.1114 0.0245 0.2477 -0.0270 -0.0750 Std. Dev. 0.0059 0.0062 0.0071 0.0059 0.0048 0.0086 0.0060 0.0023 Skewness -0.4306 -0.2449 -0.6977 -0.0771 -0.3822 -0.5251 -0.1874 -0.0164 Kurtosis 9.2944 8.3391 16.2382 16.1062 7.7685 14.3414 14.5425 4.7099 J-B 2854.33* 2032.39* 12294.54* 11919.17* 1577.92* 9027.30* 8980.17* 196.21*

Q(20) 33.40* 40.31* 96.83* 136.82* 60.52* 76.20* 120.07* 28.11 Q²(20) 2809.89* 1942.96* 3225.62* 2411.35* 1989.51* 3194.73* 2954.04* 587.34*

ARCH(5) 615.74* 452.75* 589.36* 303.39* 318.75* 658.89* 465.85* 104.25*

Note: The sample period for the daily logarithm returns runs from July 1, 2002 to December 31, 2008 excluding the holidays. J-B represents the Jarque-Bera test for normality; Q(20) is the Ljung-Box lack-of-fit hypothesis test for up to the 20^th order serial correlation in the returns; Q²(20) is the Ljung-Box test for the serial correlation in the squared returns; and ARCH(5) is the LM test for up to the fifth-order ARCH effects. *indicates significance at the 5% level.

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We also implement the Ljung-Box lack-of-fit hypothesis test, computing the

Q-statistic for autocorrelation lags 20 at the 5% significance level. There are significant

serial correlations in the returns and squared returns for most of samples, but the DXY

Index have no serial correlation in the returns. Moreover, the Engle's ARCH test for up

to the fifth-order shows significant evidence in support of the GARCH effects, or

namely the heteroscedasticity. These statistics support the preliminary hypothesis of

skewed-t GARCH model settings that we discussed before.

4.2OPTION PAYOFF STRUCTURES

We select three types of exotic bivariate options commonly seen in practice, the

digital options, the spread options, and the rainbow options, as our proxy to examine the impact of each dependence structure. Let X_i be the logarithm return at maturity of asset

i ∈ {1,2}, K is the pre-determined strike price, and I ∙ is the indicator function, which

equals 1 if the statement in parentheses holds and zero otherwise. The payoff structures

are shown in Table 5.

Table 5: Bivariate options payoff structures

Option Payoff Structure

Digital Options I(X₁> 𝐾 , X₂> 𝐾)

Spread Options max⁡(X₁− X₂− K , 0)

Rainbow Options max max⁡(X₁ , X₂ − K, 0]

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5. E

MPIRICAL

R

ESULTS

We implement the skewed-t GARCH model to the above dataset and derive the

empirical copula of the standardized GARCH innovations displayed in Figure 1. The

empirical copula would converge to the true copula function under the regularity

condition; for instance, Gaenssler and Stute (1987) or van der Vaart and Wallner (1996).

Figure 1: The empirical copula of the standardized GARCH innovations Pair 1: MSCI Pacific vs MSCI Far East Pair 2: MSCI Energy vs MSCI Bank

Pair 3: CRB vs MSCI Latin America Pair 4: S&P500 vs DXY

Note: The skewed-t GARCH model is described as follows:

r_i,t= μ_i+ ε_i,t, 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MSCI Far East Innovations

MSCI Pacific Innovations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MSCI Bank Innovations

MSCI Energy Innovations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MSCI Latin America Innovations

CRB Innovations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DXY Innovations

S&P500 Innovations

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21 hi,t= ωi+ βihi,t−1+ αiε_i,t−1² ,

εi,t|ℱt −1 = hi,tzi,t , zi,t~skewed − t zi η_i, ϕi

Then the standardized GARCH innovations for each pair of underlying assets are εi,t hi,t, for 𝑖 ∈ 1,2 and 𝑡 ∈ 1, … , 𝑇 .

We observe that there exists strong positive dependence between the innovations of

first two pairs of assets, particularly in the tails. On the other hand, the innovations of

last two pairs of underlying assets are quite irrelevant and scattered all over the figure.

The dependence structures are dissimilar among these four pairs of assets.

The following Table 6 reports the maximum likelihood estimation results of the

copula-based GARCH model. The estimates of parameters for marginal skewed-t

GARCH process are presented along with the P-values in parentheses in the panel A,

where 0.0000 indicates that the P-value is less than 0.00005.

The value of ω, α and β are all nonnegative that ensure the conditional variance h_i,t is positive, and the sum of α and β is close to one for all pairs of assets, which

implies the shocks between the underlying assets have high persistence in volatility and

stationary process. For the estimation of skewed-t distribution, the kurtosis parameter η

and the asymmetry parameter φ show that the conditional process of all assets is

fat-tailed and asymmetric, and this is consistence with previous studies.

The panel B of Table 6 reports the estimates of parameters for different types of

copula functions. All of these estimates are significant under 5% level, except for the

degrees of freedom ν of the last three pairs of Student’s t copulas.

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Table 6: Copula-based GARCH Model Estimations Asset

Parameters MSCI Pacific

MSCI Far East

MSCI Energy

MSCI

Bank CRB

MSCI Latin America

S&P 500 DXY

Panel A: Estimates of marginal process

ω × 10⁵ 0.0179 0.0187 0.0115 0.0029 0.0100 0.0231 0.0030 0.0025 (0.0000) (0.0000) (0.0000) (0.0001) (0.0001) (0.0000) (0.0003) (0.0000)

∝ 0.1403 0.1332 0.1107 0.1350 0.1568 0.1316 0.1113 0.1118 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) β 0.8005 0.8126 0.8533 0.8536 0.8024 0.8263 0.8779 0.8438 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) η 10.6253 10.7345 10.3980 11.9238 8.2961 9.0426 8.6330 9.7937 (0.0001) (0.0001) (0.0000) (0.0001) (0.0000) (0.0000) (0.0000) (0.0000) λ -0.0972 -0.0767 -0.1518 -0.0574 -0.0634 -0.1511 -0.1024 -0.0333 (0.0014) (0.0084) (0.0000) (0.0501) (0.0281) (0.0000) (0.0005) (0.1436) Log-likelihood 6592.83 6478.73 6394.77 6887.79 6650.99 5864.61 6597.96 7635.41

Panel B: Estimates of dependence process

Gaussian ρ 0.9842 0.5146 0.2801 0.0777

(0.0000) (0.0000) (0.0000) (0.0002)

Student's t ρ 0.9855 0.5256 0.2855 0.0772

(0.0000) (0.0000) (0.0000) (0.0000)

ν 7.2413 11.2265 38.1332 12.0364

(0.0000) (0.1077) (0.4839) (0.0976)

Gumbel 8.6708 1.4942 1.1949 1.0517

(0.0000) (0.0000) (0.0000) (0.0000)

Clayton 9.1704 0.6618 0.2560 0.0690

(0.0000) (0.0000) (0.0000) (0.0021)

Frank 35.8054 3.7587 1.8434 0.5371

(0.0000) (0.0000) (0.0000) (0.0003)

Note: The values in parentheses are P-values, where 0.0000 indicates that the value is less than 0.00005.

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We compare the model goodness-of-fit through the Kolmogorov-Smirnov (K-S)

test and Anderson-Darling (A-D) test along with the P-values in parentheses in Panel A

and B of Table 7. The log-likelihood estimator and Akaike Information Criterion (AIC)

of the above copula estimates are also reported in the Panel C and D of Table 7. The

lower values of the K-S test statistics, A-D test statistics and AIC information criterion

represent better goodness-of-fit. On the contrary, the larger of log-likelihood estimates

indicate better fit.

According to both of the K-S test and A-D test, the first pair of assets, which

components are MSCI Pacific and MSCI Far East Index, has the largest P-value that

means the best model fit under the Gumbel copula function. The Frank copula is the

most ideal model for the second pair of assets, which components are MSCI Energy and

MSCI Bank Index. Although the P-values of these five dependence structures are not

significant enough for the third and fourth pair of assets, the Frank copula is still the

model that relatively describes the dependence structure best based on the empirical

distribution under the K-S and A-D test statistics.

Nevertheless, the inferred conclusion from log-likelihood estimator and AIC

information criterion indicates that the Student’s t is the best dependence structure for

all of these four pairs of asset returns, which obviously contradicted to our above

induction by the K-S and A-D tests. We still choose to apply the K-S and A-D test as our

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basis to the following bivariate option pricing comparison.

Table 7: Goodness-of-fit test to Copula-based GARCH Models Asset

Dependence Structure

MSCI Pacific MSCI Far East

MSCI Energy MSCI Bank

CRB MSCI Latin America

S&P 500 DXY Panel A: Kolmogorov-Smirnov (K-S) test

Gaussian 0.0316 0.0597 0.0886 0.1297

(0.3558) (0.0049) (0.0000) (0.0000)

Student’s t 0.0304 0.0585 0.0886 0.1303

(0.4022) (0.0062) (0.0000) (0.0000)

Gumbel 0.0281 0.0615 0.0905 0.1291

(0.5048) (0.0034) (0.0000) (0.0000)

Clayton 0.0556 0.0901 0.1106 0.1352

(0.0097) (0.0000) (0.0000) (0.0000)

Frank 0.0298 0.0531 0.0831 0.1272

(0.4267) (0.0170) (0.0000) (0.0000)

Panel B: Anderson-Darling (A-D)Test

Gaussian 1.4626 9.0281 19.2397 43.9457

(0.1865) (0.0000) (0.0000) (0.0000)

Student’s t 1.4293 8.3982 18.8446 43.9345

(0.1951) (0.0000) (0.0000) (0.0000)

Gumbel 1.2187 10.7238 21.6726 44.4410

(0.2599) (0.0000) (0.0000) (0.0000)

Clayton 4.9203 15.9543 26.2062 46.3411

(0.0029) (0.0000) (0.0000) (0.0000)

Frank 2.3892 8.0226 17.7837 42.2574

(0.0566) (0.0000) (0.0000) (0.0000)

Panel C: Maximum Likelihood Estimation

Gaussian 2958.28 258.44 67.11 4.89

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25

Student’s t 2999.23 268.76 68.08 13.49

Gumbel 2940.42 238.37 52.00 8.17

Clayton 2322.66 187.30 52.07 5.52

Frank 2854.21 244.85 62.38 5.30

Panel D: Akaike Information Criterion (AIC)

Gaussian -5914.57 -514.89 -132.23 -7.77

Student’s t -5996.46 -535.52 -134.16 -24.98

Gumbel -5878.84 -474.74 -102.00 -14.33

Clayton -4643.33 -372.59 -102.14 -9.03

Frank -5706.42 -487.69 -122.76 -8.60

Note: The test statistics of K-S test and A-D test are described as follows:

DKS= max

t F_E x_t − F_H(xt) D_AD = _F^[F^E^{x −F}^H^{x ]}²

H x [1−F_H x ]

x dF_H x .

The copula estimates in Table 5.1 are implemented in the hypothetical cumulative distribution F_H and the empirical cumulative distribution F_E is defined as

F_E(x) =¹

T ^T_t=1 ^N_j=1I(x_j,t≤ x_j^u^j^∙T)

where I(∙) is the indicator function, u is a vector of marginal probabilities, and x_j^u^j^∙T is the u_j∙ T th order statistic. Besides, the values in parentheses are P-values, where 0.0000 indicates that the value is less than 0.00005.

After we derive all desired parameter estimates, we simulate the asset return

process that given the parameters of skewed-t GARCH model and copula functions.

Especially we apply the parameter estimates of each marginal skew-t GARCH process

with initial conditional volatility h_i,0 equals the long-term variance level ω_i 1 − α_i− β_i . The estimates of Gaussian, Student’s t, Gumbel, Clayton and Frank

copula functions are also implemented to simulate different types of dependence

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26

structure. To ensure the convergence of simulated option prices, we simulate 10,000

times and assume that the initial underlying asset return equals one for each return

process.

The following Figure 2 reports the simulation results of digital option pricing of

MSCI Pacific Index and MSCI Far East Index with different strike prices K from 0.80 to

1.20 and different time-to-maturity T from one to six months. We also fix the strike

price K to divide them into three groups of in-the-money (ITM), at-the-money (ATM)

and out-of-the-money (OTM) respectively in Figure 3.

The differences of digital option price between each type of dependence structure

tend to become wider when the maturity increases and the initial bivariate option prices

are around at-the-money. However, the tendency of price change is not very obvious,

and the same situation happens both for the spread options and rainbow options

displayed in Figure 4 to 6.

In order to confirm whether the differences among these bivariate option prices are

significant or not, we perform a paired t-test of the null hypothesis that the 10,000 times

simulated asset returns in the difference between each pairs of dependence structures are

a random sample from a normal distribution with mean zero and unknown variance,

against the alternative that the mean is not zero.

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27

Figure 1:Digital Option Pricing Result Under Different Copulas vs. Different Strike Prices - Pair 1- MSCI Pacific Index and MSCI Far East Index

A: 1 Month Maturity B: 2 Month Maturity

C: 3 Month Maturity D: 4 Month Maturity

E: 5 Month Maturity F: 6 Month Maturity

Note: The payoff function of digital option is I(X₁> 𝐾 , X₂> 𝐾), where I(∙) is the indicator function that equals to one if the statement in parentheses holds and zero otherwise. We assume that the number of trading days for a month is twenty days.

0.00 0.20 0.40 0.60 0.80 1.00 1.20

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

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28

Figure 2: Digital Option Price Result Under Different Copulas vs. Different Time to Maturities - Pair 1- MSCI Pacific Index and MSCI Far East Index

ITM (𝐾 = 0.9) ATM (𝐾 = 1) OTM (𝐾 = 1.1)

Note: ITM, ATM and OTM represent that the digital option is in-the-money, at-the-money and out-of-the-money respectively. Since both the underlying asset returns are set to start at the same value 1 in the simulation process, ATM indicates that 𝐾 equals to 1, ITM indicates that K is less than 1 and OTM indicates that K is larger than 1.

Figure 3: Spread Option Pricing Result Under Different Copulas vs. Different Strike Prices - Pair 1- MSCI Pacific Index and MSCI Far East Index

0.8 0.9 1.0

20 40 60 80 100 120

0.4 0.5 0.6

20 40 60 80 100 120 0.0 0.1 0.2

20 40 60 80 100 120

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

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29

Note: The payoff function of spread option is denoted by max⁡(X₁− X₂− K , 0). We assume that the number of trading days for a month is twenty days.

Figure 4: Spread Option Price Result Under Different Copulas vs. Different Time to Maturities - Pair 1- MSCI Pacific Index and MSCI Far East Index

ITM (𝐾 = −0.1) ATM (𝐾 = 0) OTM (𝐾 = 0.1)

Figure 5: Rainbow Option Pricing Result Under Different Copulas vs. Different Strike Prices - Pair 1- MSCI Pacific Index and MSCI Far East Index

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.09 0.10 0.11

20 40 60 80 100 120

0.00 0.01 0.02

20 40 60 80 100 120

0.00 0.01 0.02

20 40 60 80 100 120

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

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30

Note: The payoff function of rainbow option is max max⁡(X₁ , X₂ − K, 0]. We assume that the number of trading days for a month is twenty days.

Figure 6: Rainbow Option Price Result Under Different Copulas vs. Different Time to Maturities - Pair 1- MSCI Pacific Index and MSCI Far East Index

ITM (𝐾 = 0.9) ATM (𝐾 = 1) OTM (𝐾 = 1.1)

The paired t-test results among all of dependence structures with one and six

month time-to-maturity are reported in Table 8 and 9 respectively. According to our

previous goodness-of-fit tests, we take the Gumbel copula that best describes the 0.00

0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.00 0.05 0.10 0.15 0.20 0.25

0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.10 0.11 0.12

20 40 60 80 100 120

0.01 0.02 0.03

20 40 60 80 100 120

0.00 0.01 0.02

20 40 60 80 100 120

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31

dependence structure between the returns of MSCI Pacific and MSCI Far East Index as

our basis model for the comparison.

In the scenario of one month time-to-maturity in Table 8, the difference between

each pairs of dependence structures are significant under 5% level, except that the

digital and rainbow option prices between Gumbel copula and Gaussian copula are not.

However, the price differences among each pairs of dependence structure become more

significant except for the rainbow option price between Gumbel copula and Frank

copula in the scenario of six months time-to-maturity. The results of paired t-test

support our previous observation that the differences between the bivariate options

prices are more significant as the time-to-maturity increases.

Table 8: T test for the distance between the ATM (K=1) option prices of different dependence structures with 1 month time-to-maturity - Pair 1- MSCI Pacific Index and MSCI Far East Index

Panel 1: Digital Option

Student’s t Clayton Gumbel Frank

Gaussian -0.0009 0.0267 0.0014 0.0138

(0.0833) (0.0000) (0.1221) (0.0000)

Student’s t 0.0276 0.0023 0.0147

(0.0000) (0.0116) (0.0000)

Clayton -0.0253 -0.0129

(0.0000) (0.0000)

Gumbel 0.0124

(0.0000)

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32 Panel 2: Spread Option

Student t Clayton Gumbel Frank

Gaussian 0.0001 -0.0018 -0.0003 -0.0009

(0.0000) (0.0000) (0.0000) (0.0000)

Student’s t -0.0019 -0.0003 -0.0009

(0.0000) (0.0000) (0.0000)

Clayton 0.0016 0.0010

(0.0000) (0.0000)

Gumbel -0.0006

(0.0000) Panel 3: Rainbow Option

Gaussian 0.0000 -0.0014 0.0000 -0.0002

(0.0000) (0.0000) (0.4577) (0.0000)

Student’s t -0.0014 0.0000 -0.0002

(0.0000) (0.0001) (0.0000)

Clayton 0.0014 0.0012

(0.0000) (0.0000)

Gumbel -0.0002

(0.0000)

Note: The table reports the distance between the ATM (K=1) bivariate option prices of different dependence structures with 1 months time-to-maturity for the first pair of assets returns, which components are MSCI Pacific and MSCI Far East Index. The values in parentheses are p-values under the paired t tests.

Table 9: T test for the distance between the ATM (K=1) option prices of different dependence structures with 6 months time-to-maturity - Pair 1- MSCI Pacific Index and MSCI Far East Index

Panel 1: Digital Option

Student’s t Clayton Gumbel Frank

Gaussian -0.0014 0.0352 0.0048 0.0333

(0.0163) (0.0000) (0.0000) (0.0000)

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Student’s t 0.0366 0.0062 0.0347

(0.0000) (0.0000) (0.0000)

Clayton -0.0304 -0.0019

(0.0000) (0.4016)

Gumbel 0.0285

(0.0000) Panel 2: Spread Option

Gaussian 0.0002 -0.0051 -0.0007 -0.0019

(0.0000) (0.0000) (0.0000) (0.0000)

Student’s t -0.0053 -0.0009 -0.0021

(0.0000) (0.0000) (0.0000)

Clayton 0.0044 0.0031

(0.0000) (0.0000)

Gumbel -0.0012

(0.0000) Panel 3: Rainbow Option

Gaussian 0.0001 -0.0035 -0.0001 -0.0002

(0.0000) (0.0000) (0.0000) (0.0009)

Student’s t -0.0036 -0.0002 -0.0003

(0.0000) (0.0000) (0.0000)

Clayton 0.0033 0.0033

(0.0000) (0.0000)

Gumbel -0.0001

(0.2549) Note: The table reports the distance between the ATM (K=1) bivariate option prices of different dependence structures with 6 months time-to-maturity for the first pair of assets returns, which components are MSCI Pacific and MSCI Far East Index. The values in parentheses are p-values under the paired t tests.

相依結構對多資產選擇權定價影響之實證分析

國立臺灣大學管理學院財務金融學系 碩士論文

Department of Finance College of Management National Taiwan University

Master Thesis

相依結構對多資產選擇權定價影響之實證分析 Bivariate Options Pricing with

Copula-Based GARCH Model -Empirical Analysis

許凱雯 Kai-Wen Hsu

指導教授：王耀輝 博士 Advisor: Yaw-Huei Wang, Ph.D.

中華民國 98 年 6 月

June, 2009

摘要

Abstract

Contents

Tables

Figures

1. I

2. L

R

3. T

C

-

GARCH M

(A-D) test as our measurement, which belong to the large class of goodness-of-fit tests.

They test the null hypothesis that the populations drawn from two or more

independent samples of data are identical. This test therefore can be used to decide

which dependence structure best fit the empirical data, namely the null hypothesis H

of

same population distributions cannot be rejected. Then we can compare the P-value

with the desired significance level to facilitate a decision about this null hypothesis.

The K-S test is distribution free in the sense that the critical values do not depend

on the specific distribution being tested, and it is more sensitive to deviations in the

center of the distribution. The A-D test introduced by Anderson and Darling (1952,

1954) is a modification of the K-S test that gives more weight to deviations in the tails

than K-S test. By using these two goodness-of-fit tests simultaneously, we could have

more evidence to prove our model selection.

4. D

O

P

S

5. E

R

國立臺灣大學管理學院財務金融學系碩士論文

指導教授：王耀輝博士 Advisor: Yaw-Huei Wang, Ph.D.