# 考慮有隨機跳躍頻率強度的GARCH模型之巨災選擇權定價

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(2) 謝辭 兩年的碩士生活、也是人生第一次在外地求學，想不到時間過的這麼快，即 將要邁入下個階段。感謝週遭這麼多人的照顧，才可以把碩士階段如期完成。 首先要感謝的是指導教授─黃士峰老師。老師是論文得以完成的幕後推手， 老師很忙碌，可是總是願意抽空出來和我討論問題，甚至在周末的私人時間。在 這邊要跟師母說聲抱歉，我實在是不該在週末時打擾到老師，佔用了你們相處的 時間。老師與學生們的相處，亦師亦友，沒有距離。課業上，老師不厭其煩的指 正錯誤與共同討論問題該如何解決，讓我對模型的盲點有豁然開朗之感。閒暇 於，像是所上的聚餐活動老師和學生們豪邁吃酒，排球場上帶領學生們過關斬 將。在老師身上學到的除了專業知識之外，還有老師對問題處理的嚴謹態度，相 信這對我在未來工作上有很大的幫助。謝謝您兩年來的照顧。 其次，感謝口試委員郭美惠教授和林士貴教授在口試過程中諸多的指導，使 得論文得以盡善盡美。接著，感謝所上所有老師的教導與扶持，從課堂上學到許 多專業知識，讓我知道我對統計這個學門，認識的只是皮毛，還有更多知識要去 追求。還有蘭屏姐的關心與支持，蘭屏姐可是我來高雄認識的第一個朋友，和她 聊天，總是可以讓心情更加平靜，可以說是所上的心靈輔導老師。感謝總是會來 研究室關心的佳芳學姊和三國殺的好戰友基祥學長，謝謝你們幫忙解決好多 Latex 和 Matlab 的程式問題。 感謝碩班的同窗好友建宇和其女友孟儒，認識你們，也認識了花蓮。黃昏的 七星潭好美，自強夜市好多其他夜市沒有的小吃，磯崎露營好累，可是很有趣， 謝謝你們花蓮四天三夜的招待，讓研究所生活更加豐富。感謝碩班的同窗帥凱、 思淳和珮甄，碩一時老是麻煩你們載我趴趴走。感謝碩班的同窗樵樵、正修和安 婷，你們讓碩班生活增加許多歡笑聲。感謝論文口試前一起在 320 教室奮鬥的夥 伴雅婷、維迦和澤初。 感謝小黃跟小黑，還記得第一次見面你們的咆哮嚇到我了，這也表示你們真 的有好好守護家園，不會讓陌生人隨意靠近。熟悉後你們總是護送我和雅婷到理 學院，不只讓我們安全，也介紹了其他夥伴讓我們認識，像是黑肥還有小鹿班比。 我不會忘記小黃你那楚楚可憐的臉龐，還有騎車時老是愛跳上踏墊要一起出發的 小黑，雖然不知道你們去了哪裡，相信你們現在還是快快樂樂的在奔馳。 感謝我的家人。爸，謝謝你支持我來高雄還有辛勞工作，讓我生活無後顧之 憂，也讓我生活有了不一樣的體驗。媽，謝謝你不厭其煩的提醒我，要我好好照 顧自己，我真的很不客氣的好好照顧自己，您白操心了，我從未讓自己挨餓過， 飲食也相當均衡。姊、哥，謝謝你們開先例去異地求學，這也讓我有了機會，可 以離開家園，你們很優秀，是從小我學習的榜樣。品方、麗維，你們也很優秀，.

(3) 從小我們一起長大，沒有你們，沒有現在的我。感謝好友宏德、育辰、育辰女友 衣怜，高中死黨阿超、貴龍、彥廷、盈漢、迪函、凱元和維峻，雖然我不在台北， 可是你們沒忘記我，保持聯絡，祝我們都工作順利。 最後謝謝雅婷這兩年的陪伴，妳把我照顧的很好。 曾品源 撰 中華民國一○一年七月.

(4) Pricing Catastrophe Options in GARCH-Jump with Stochastic Intensity Models. by Pin-Yuan Tseng Advisor Shih-Feng Huang. Institute of Statistics National University of Kaohsiung Kaohsiung, Taiwan 811, R.O.C. June 2012.

(5) Contents dddd. ii. Abstract. iii. 1 Introduction. 1. 2 Literature Review. 3. 2.1. Esscher transform (Gerber and Shiu, 1994; Siu, et al., 2004; Huang, et al., 2012) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. 3. CatEPut pricing (Cox, et al. 2004; Jaimungal and Wang, 2006; Lin, et al. 2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 3 Pricing CatEPut in GARCH-Jump Model. 5. 4 Sensitivity Analysis. 8. 4.1. The influence of the magnitude of intensity rate . . . . . . . . . . . .. 8. 4.2. The influence of the noise ratio . . . . . . . . . . . . . . . . . . . . . 10. 4.3. The influence of maturity. 4.4. The influence of stochastic intensity rate . . . . . . . . . . . . . . . . 12. 4.5. The influence of risk-neutral measures . . . . . . . . . . . . . . . . . . 12. . . . . . . . . . . . . . . . . . . . . . . . . 11. 5 Conclusion. 13. Appendix. 13. i.

(6) 考慮有隨機跳躍頻率強度的 GARCH 模型之 巨災選擇權定價 指導教授：黃士峰 博士 國立高雄大學應用數學系 學生：曾品源 國立高雄大學統計學研究所. 摘要 本文利用 Esscher 轉換法推導出在有隨機跳躍頻率強度的 GARCH 的風險中 立測度模型，並且用來定價巨災權益賣權(CatEPut)。CatEPut 會處理巨災所造成 的損失進而保護公司的股東權益。模擬研究指出當巨災發生頻率，跳躍項的分 佈其平均數、變異數和到期日增加時巨災權益賣權的定價上升。. 關鍵字：巨災選擇權; Esscher 轉換法; 跳躍 GARCH 模型. ii.

(7) Pricing Catastrophe Options in GARCH-Jump with Stochastic Intensity Models Advisor: Shih-Feng Huang Department of Applied Mathematics National University of Kaohsiung. Student: Pin-Yuan Tseng Institute of Statistics National University of Kaohsiung. ABSTRACT. A risk-neutral GARCH-Jump model with stochastic intensity is established by the Esscher transform and is applied to evaluate the price of a catastrophe equity put (CatEPut) option. The CatEPut contract provides protection for the shareholders of the underlying company to handle catastrophic losses. Numerical results indicate that the CatEPut prices increase as the intensity rate, the expected value of the jump size, the variance of the jump size or the time to maturity increasing. Keywords and phrases: Catastrophe options; Esscher transform; GARCH-Jump model.. iii.

(8) 1. Introduction. A catastrophe equity put (CatEPut) option was first issued by the RLI corporation in 1996 to cover the potential losses caused by catastrophe such as typhoons, hurricanes, storms, floods, waves and earthquakes. Recently, the frequency of the occurrence of catastrophe tends to be increasing and leads to enormous financial losses. For example, the Sumatra-Andaman earthquake (2004) in Indonesia, the Hurricane Katrina (2005) in the Gulf of Mexico, the Wenchuan earthquake (2008) in Sicuan, the Haiti earthquake (2010) in Haiti, the New Zealand earthquake (2011) in Christchurch and the Great East Japan earthquake (2011). The total amount of loss caused by these catastrophes is over 457 billion U.S. dollars (around 3.03% of U.S. GDP in 2011). In Taiwan, recent catastrophes like the 921 earthquake (1999) in Nantou, the Jiaxian earthquake (2010) in Kaohsiung, and several typhoons such as the Mindulle (2004), Morakot (2009), Fanapi (2010) and Nanmado (2011) also lead to more than 450 billion loss in N.T. dollars (around 32.74% of Taiwan’s GDP in 2011). To hedge these enormous catastrophic losses, CatEPut options provide protection for the corporation and shareholders. Consequently, the pricing scheme of CatEPut options attracts more attention from practitioners and researchers (Cox, et al., 2004; Jaimungal and Wang, 2006; Lin, et al., 2009). In financial derivative pricing, the GARCH models and jump-diffusion models are commonly used to describe the dynamics of the underlying processes (Merton, 1976; Engle, 1982; Bollerslev, 1986; Duan, 1995; Kou, 2002; Lin, et al., 2009; Huang and Guo, 2009; Huang, 2012). For CatEPut option pricing, Cox, et al. (2004) derived a closed-form solution under a jump-diffusion model with constant jump size and constant intensity rate. Jaimungal and Wang (2006) extended Cox, et al.’s result to the case of random riskless interest and random jump size. Lin, et al. (2009) further considered the valuation of CatEPut options under stochastic intensity rate of the occurrence of catastrophe. While considerable attention has been given in the literature to the valuation of CatEPut options in jump-diffusion model, less attention has been devoted to GARCH model, which is capable of successfully depicting. 1.

(9) heteroskedasticity of financial data. Therefore, this study combines the ideas of the GARCH and jump-diffusion model, denoted by GARCH-Jump, to depict the underlying dynamics. In particular, the intensity rate is assumed to follow a geometric Brownian motion and the jump size is constant, gamma or normally distributed. Since there is usually no closed-form solution in GARCH option pricing, practitioners can only compute the derivative prices by simulation or numerical methods (Duan, 1995; Duan and Simonato, 1998; Huang and Guo, 2009) based on a riskneutral model. To obtain a risk-neutral GARCH-Jump model, we use the Esscher transform to derive a change of measure process (Gerber and Shiu, 1994; Siu, et al., 2004; Huang, et al., 2012). In the literature, two popular economic considerations on the jump part are proposed and yield different risk-neutral measures. Merton, (1976), Cox, et al. (2004) and Jaimungal and Wang (2006) assumed that the risk caused by the jump is non-diversifiable while Amin (1993) took the opposite point of view that the jump risk is diversifiable. In stead of jumping into the debate of which measure is appropriate, this study proposes a general approach of deriving risk-neutral GARCH-Jump models to accommodate both economic assumptions. Moreover, to investigate the influence of the different risk-neutral measures, the intensity rate, the mean and variance of jump size and time to maturity on the CatEPut prices, we conduct several simulation scenarios. Numerical results indicate that the CatEPut prices increase as the intensity rate, the expected value of the jump size, the variance of the jump size or the time to maturity increasing. The remainder of the paper is organized as follows. Section 2 briefly introduces the Esscher transform and several methods of pricing CatEput options in jumpdiffusion models. Section 3 establishes the corresponding risk-neutral GARCH-Jump model by the Esscher transform. Sensitivity analyses are given in Section 4. Conclusions are in Section 5. All the tables, figures and theoretical proofs are in the Appendix.. 2.

(10) 2. Literature Review. 2.1. Esscher transform (Gerber and Shiu, 1994; Siu, et al., 2004; Huang, et al., 2012). Let Rt = log St /St−1 denote the log return process of a stock price process St at time t, for t = 1, 2, . . . , T . A papular approach to derive an equivalent probability measure of Rt for financial derivative pricing is the Esscher transform (Gerber and Shiu, 1994). We sketch the scheme of it in the following : Let ft (·) denote the conditional probability density function (pdf) of Rt |Ft−1 and MRt |Ft−1 (z) = Et−1 (ezRt ) denote the conditional moment generating function (mgf) of Rt |Ft−1 under the physical (or dynamic) measure P , where Et−1 (X) denotes the conditional expectation of X given Ft−1 under P and Ft−1 is an information set including the prices of the underlying asset and the riskless bond prior to time t − 1. To simplify the illustration, we assume that the riskless interest rate r is constant herein. 1. Define a new conditional pdf as ft (xt ; δt ) =. eδt xt MRt |Ft−1 (δt ). ft (xt ). (1). with an extra parameter δt . 2. Define an equivalent probability measure Q by dQ = ΛT dP with a RadonNikod´ ym derivative ΛT =. T Y t=1. where. δt∗ ’s. ∗. eδt Rt , MRt |Ft−1 (δt∗ ). (2). are obtained in the next step.. 3. By (2), define a change of measure process Λt by Λt = Et (ΛT ), t = 0, 1, . . . , T . Then, the conditional pdf of Rt |Ft−1 under Q is ft (xt ; δt∗ ) defined in (1) and the parameter δt∗ is determined by solving the following equation: Rt r EQ t−1 (e ) = e ,. where EQ t−1 (X) denotes the conditional expectation of X under Q. 3. (3).

(11) Siu, et al. (2004) and Huang, et al. (2012) applied the Esscher transform to GARCH option pricing with non-normal innovations.. 2.2. CatEPut pricing (Cox, et al. 2004; Jaimungal and Wang, 2006; Lin, et al. 2009). Let VT be the payoff of a CatEput option with maturity time T , VT = I{LT −Lt0 >L} (K − ST )+ ,. (4). where ST denotes the price of the underlying asset at time T , LT − Lt0 denotes the total loss-percentage rate process from time t0 to T , L is the pre-determined limit of loss on the contract, K is the strike price, and I{B} is the indicator function of an event B. Cox, et al. (2004) considered the following jump-diffusion model to depict the dynamics of the underlying processes {St , t ≥ 0}, St = S0 exp{−Lt + σWt + (µ − 0.5σ 2 )t},. (5). where S0 is the initial price of the underlying asset, Wt is a standard Brownian motion, and Lt = ANt with a constant jump size A and a pure Poisson process Nt . Note that Lt in (5) causes a drop in underlying asset prices. Cox, et al. (2004) derived the closed-form solution for CatEPut options of Model (5). To extend Cox’s result to more general settings, Jaimungal and Wang (2006) considered the case of random riskless interest and assuming Lt being a compound Poisson process with Gamma distributed jump size. The explicit closed-form formulae for the price of the option is derived and the hedging parameters such as Delta, Gamma and Rho are obtained. They found that accounting for stochastic interest rates, the Rho hedging can significantly reduce the expected conditional loss of the hedged portfolio. Moreover, Lin, et al. (2009) considered the case of compound Poisson process Lt with Nt having stochastic intensity rate, called the doubly stochastic Poisson process. They also established the pricing formulae of CatEPut options by the Merton measure. 4.

(12) 3. Pricing CatEPut in GARCH-Jump Model. Assume that the log returns Rt , t = 1, 2, . . . , T, are generated from the following GARCH-Jump model under the physical P measure: Rt = RtG − Lt , RtG = r − 1 σt2 + λσt + σt εt , εt i.i.d. ∼ N(0, 1) 2 2 σt+1 = α0 + α1 σt2 (εt − γ)2 + β1 σt2 , ∆Nt X i.i.d. Yt,n , Yt,n ∼ fYt,n (xt ), Lt =. (6). n=0. where r is the one-period risk-free interest rate, λ denotes the risk premium, γ is a nonnegative constant to describe asymmetric effects between positive and negative stock returns, α0 > 0, α1 ≥ 0, β1 ≥ 0 and α1 + β1 < 1 in the GARCH part RtG . For the jump part Lt , ∆Nt := Nt − Nt−1 denotes the total number of catastrophe during [t − 1, t). Nt is a Poisson process with stochastic intensity rate θt , which is assumed to be governed by the diffusion, dθt = µθ θt dt + σθ θt dWt ,. (7). where µθ and σθ are constants and Wt is a Brownian motion. The jump size Yt,n , n = 1, 2, . . . are assumed to be independent and identically distributed and Yt,0 is set to be zero for all t. Moreover, assume that εt , Wt , Nt and Yt,n are independent. A risk-neutral counterpart of Model (6) is derived by the Esscher transform. Before illustrate the main result, we first give the following useful lemma. Lemma 3.1. Let Mθt∗ |Ft−1 (h) denotes the conditional moment generating function Rt of θt∗ given Ft−1 , where θt∗ = t−1 θs ds with θs satisfying (7). Then, ∗. Mθt∗ |Ft−1 (h) = Et−1 (ehθt ) < ∞. Furthermore, Mθt∗ |Ft−1 (h) can be approximated by hµθ∗. Mθt∗ |Ft−1 (h) ≈ e. t. 5. (1 +. h2 2 σ ∗ ), 2 θt.

(13) where µθt∗ = θt−1 (eµθ − 1)/µθ is the conditional expectation of θt∗ given Ft−1 and σθ2t∗ =. 1 2 h 2θt−1 1 eµθ i 1 2 + − µ2θt∗ e2µθ +σθ − − µθ µθ + σθ2 2µθ + σθ2 2µθ + σθ2 µθ + σθ2. is the conditional variance of θt∗ given Ft−1 . By Lemma 3.1, a change of measure process derived from the Ecccher transform is obtained in the following proposition. Proposition 3.1. For Model (6), a change of measure process derived by the Esscher transform is dQ G ∗ L|θ∗ = ΛR × ΛθT , T × ΛT dP. (8). where G ΛR T. =. = L|θ∗ ΛT. =. T Y t=1 T Y t=1 T Y. G RG t. eδt. MRtG |Ft−1 (δtG ) exp{− L. eδt. 1 1 [(c + λσt )2 + 2(c + λσt )(RtG − r − λσt + σt2 )]}, 2 2σt 2. Lt. L. t=1. ,. ∗. eh(δt ) θt. ,. and. ∗ ΛθT. =. T Y t=1. L. (9). ∗. eh(δt ) θt L ∗ , Et−1 [eh(δt ) θt ]. in which δtG = −. c + λ σt , σt2. (10). c is a constant, h(δtL ) = MYt,n |Ft−1 (δtL ) − 1 and δtL is the solution of MYt,n |Ft−1 (δtL − 1) − MYt,n |Ft−1 (δtL ) =. c . θt∗. (11) L. ∗. Let MYt,n |Ft−1 (δtL ) denotes the conditional mgf of Yt,n given Ft−1 and Et−1 [eh(δt ) θt ] can be obtained either by numerical methods or by the approximation in Lemma 3.1. Corollary 3.1. In Proposition 3.1, if θt is a deterministic function of t, then the change of measure process becomes dQ G L = ΛR T × ΛT , dP. 6.

(14) G. where ΛR is defined the same as in Proposition 3.1 and T ΛLT =. L T Y eδt Lt , h(δtL ) θt∗ e t=1. where δtL and h(δtL ) are defined the same as in Proposition 3.1. Corollary 3.2. If the jump risk is assumed to be non-diversifiable, then the jump activity rate and distribution are not altered by the measure changed (Merton, 1976; Cox et al., 2004; Jaimungal and Wang, 2006). Therefore, the constant c in (9), (10) and (11) is set to be c = θt∗ [MYt,n |Ft−1 (−1) − 1]. (12). such that δtL = 0. By the change of measure process obtained in (8), the following proposition establishes the corresponding risk neutral counterpart of Model (6). Proposition 3.2. The risk neutral counterpart of Model (6) corresponding to the change of measure process obtained in Proposition 3.1 is: G Rt = Rt − Lt , 1 i.i.d RtG = r − c − σt2 + σt ξt , ξt ∼ N(0, 1) 2 2 σt+1 = α0 + α1 σt2 (ξt − λ − cσt−1 − γ)2 + β1 σt2 , ∆Nt X i.i.d Yt,n , Yt,n ∼ ft (xt ; δtL ), L = t. (13). n=0. −1 where ξt = εt + λ + cσt−1 , Nt is a Poisson process with stochastic intensity rate. θ˜t = MYt,n |Ft−1 (δtL )θt , which satisfies dθ˜t = µθ θ˜t dt + σθ θ˜t dWt . Moreover, ξt , Wt , Nt and Yt,n are independent under the risk-neutral measure Q. Note that if θt is a deterministic function of t, then the intensity rate under Q is θ˜t = MYt,n |Ft−1 (δtL )θt , which is still a deterministic function of t. 7.

(15) 4. Sensitivity Analysis. In this section, several simulation scenarios are conducted to investigate the sensitivity of the parameters in the GARCH-Jump model (6). Under the Merton measure, Sections 4.1, 4.2 and 4.3 investigate the influence of the constant intensity rate θ, the mean µy and the variance σy2 of the jump size, and the time to maturity T , respectively, on pricing CatEPut options. Section 4.4 presents a sensitivity study of µθ and σθ in the stochastic intensity rate model (7). Moreover, to investigate the influence of different choices of the risk-neutral measures, Section 4.5 conducts a comparison study by choosing different c such that (23) and (24) hold in the GARCH-Jump model, where the intensity rate is a constant and the jump-size is N (µy , σy2 ) distributed. Throughout this section, let L = E[LT − Lt0 ], which is the same as in Jaimungal and Wang (2006), S0 = 25, K = 80, Lt0 = 0, r = 0.05, α0 = 3.3 × 10−3 , α1 = 0.2, β1 = 0.7, λ = 0.01, γ = 0.3, initial conditional variance σ12 = α0 [1 − α1 (1 + γ 2 ) − β1 ]−1 , and 10,000 sample paths.. 4.1. The influence of the magnitude of intensity rate. In this section, the following GARCH-Jump model with constant intensity rate is considered, Rt = RtG − Lt , 1 2 i.i.d G R = r − c − σt + σt ξt , ξt ∼ N(0, 1), t 2 2 σt+1 = α0 + α1 σt2 (ξt − λ − cσt−1 − γ)2 + β1 σt2 , ∆Nt X i.i.d Yt,n , Yt,n ∼ ft (xt ), Lt =. (14). n=0. where Yt,n is assumed to be a constant A, Ga(α,β) or N (µy , σy2 ) distributed, and the 2. corresponding c is θ[e−A − 1], θ[e−µy +0.5σy − 1] or θ[(1 + β)−α − 1], respectively. For the case of constant jump size, which is considered by Cox, et al. (2004), Figure 1 plots the CatEPut prices under various initial stock price S0 and intensity rate θ with A = 0.02 and T = 1. Figure 1 has the same shape as that in Figure 2 8.

(16) of Cox, et al. (2004) because the GARCH-Jump model with T = 1 is the same as the jump-diffusion Model (5) considered in Cox, et al. (2004). To investigate the impacts of the magnitude of intensity rate and the above three distributions, constant, normal and gamma of the jump size on the CatEPut prices, we compute the value of P (LT > L), which is proportional to the CatEPut price and has a simpler analytic representation than the CatEPut price, under various settings of θ, (α, β) and (µy , σy2 ). Table 1 shows the simulation results of θ = 0.2, 0.4, . . . , 1.2. In Table 1, the CatEPut prices increase when P (L1 > L) (in parentheses) increases in each column. We also plot the CatEPut prices versus the value of P (L1 > L) in Figure 3, which shows that the CatEPut prices are positively correlated to P (L1 > L). More interesting findings are illustrated as follows. 1. If the jump size is assumed to be a constant A, then LT = NT A, where NT is a Poisson(θT ) distribution. Hence, we have P (LT > L) = 1 −. [θT ] −θT X e (θT )j. j!. j=0. ,. (15). where [x] denotes the largest integer less and equal to x. Figure 2 plots P (L1 > L) with θ = 0, 0.1, 0.2, . . . , 5 and shows that P (L1 > L) increases in θ ∈ [n, n + 1), n = 0, 1, 2, . . . , 5 but drops when θ = 1, 2, . . . , 5. Since the CatEPut option is proportional to P (LT > L) for a fixed T , thus the values of the CatEPut prices in Table 1 increases when θ ∈ [0.2, 0.8] and [1, 1.2] but drops at θ = 1. 2. If the jump size is assumed to be Ga(α,β) or N (µy , σy2 ), then P (LT > L) =. Z ∞ X e−θT (θT )k k!. k=0. ∞. fY |NT (y|k)dy,. (16). L. which is a continuous function of θ, where fY |NT (y|k) is the conditional density of the jump size Y conditional on NT = k. Figure 4 plots P (L1 > L) defined in (16) with θ = 0, 0.1, 0.2, . . . , 5, and three different settings of the distribution parameters, where the values of the Ga(α,β) case are denoted in red dots and 9.

(17) the values of the N (µy , σy2 ) case are in blue circles. Note that the means of the gamma and normally distributions considered in Figure 4 are all equal to 0.02, which is the same as the previous constant jump size, and (α,β) are set to be satisfying αβ = µy and αβ 2 = σy2 . The middle and lower panels of Figure 4 show that the values of P (L1 > L) defined in (16) increase in θ, but does not increase in θ in the upper panel. In particular, if Yt,n is gamma distributed, the values of P (L1 > L) drops dramatically when σy2 increases. However, if Yt,n is normally distributed, then the three patterns of P (L1 > L) of our settings are more stable than the gamma case. From the above discussion, we find that the CatEPut prices are proportional to the values of P (LT > L), P (LT > L) does not necessary increase as θ increasing if the jump size Yt,n is a constant or has relatively small variance, and P (LT > L) decreases more dramatically as the variance of Yt,n increasing when Yt,n is gamma distributed than Yt,n is normally distributed.. 4.2. The influence of the noise ratio. In this section, the sensitivity analysis of µy and σy2 /µy on pricing CatEPut is presented under Model (14). Figure 5 plots the CatEPut prices under various θ, µy and σy2 /µy with T = 1. The jump size is assumed to be Ga(α,β) or N (µy , σy2 ) distributed, where the ratio is set to be σy2 /µy = 1, 2, . . . , 10 with µy = 0.01, 0.02, 0.05, 0.5. Similar to Section 4.1, let α = µ2y /σy2 and β = σy2 /µy such that αβ = µy and αβ 2 = σy2 . Figure 5 indicates that 1. the CatEPut prices increase when µy increases, 2. if Yt,n is gamma distributed, the CatEPut prices decrease in σy2 /µy ; which is consistent with the results reported in Figure 5 in Jaimungal and Wang (2006); 3. if Yt,n is normally distributed, the CatEPut prices does not have significant changes in σy2 /µy ; 10.

(18) 4. the CatEPut prices increases as θ increasing.. 4.3. The influence of maturity. In this section, the sensitivity analysis of T on pricing CatEPut is presented. In addition to Model (14), we also consider the following jump-diffusion model under the Merton measure : ˜ t − ct + σ W ˜ t + (r − 0.5σ 2 )t}, St = S0 exp{−L ft = µ + c − r t + Wt , W σ. (17). where S0 is the initial stock price, Wt is a standard Brownian motion under the ft is a new Brownian motion under the Merton measure, c is physical measure, W ˜ t = PNt Yt,n , in which {Nt : t > 0} is a Poisson defined the same as in (14), and L n=0 process with intensity rate θ. Figure 6 plots the CatEPut prices under various θ and T , where the jump size is constant. In Figure 6, there is no significant difference between the GARCH-Jump model and jump-diffusion model for CatEPut prices. Interestingly, the CatEPut prices shown on the left panel of Figure 6 do not increase as T increasing. Nevertheless, recall that the CatEPut prices are proportional to P (LT > L). In Figure 6, the blue, yellow and green points are used to denote the CatEPut prices (left panel) and P (LT > L) (right panel) of T = 1, 3 and 5, respectively. In particular, note that the green points on the right panel are lower than most of the blue and yellow points, which explains why the CatEPut prices do not increase as T increasing since the corresponding values of P (LT > L) do not increasing with θT . Figure 7 plots the CatEPut prices under various θ, T and σy2 . The jump size is assumed to be Ga(α,β) or N (µy , σy2 ) distributed, where the variance of jump size is set to be σy2 = 4 × 10−3 , 4 × 10−1 with fixed µy = 0.02. The results shown in Figure 7 are consistent with those in Figures 4 and 6, except for T = 3, 5 in the normally distributed case, where the CatEPut prices increase as σy2 increasing. Moreover, comparing to the case of constant jump size shown in Figure 6, the CatEPut prices in Figure 7 increase as T increasing. 11.

(19) 4.4. The influence of stochastic intensity rate. In this section, we consider Model (14) with stochastic intensity rate and normally distributed jump sizes. The solution of the stochastic intensity rate model (7) is θt = θ0 exp[(µθ − 0.5σθ2 )t + σθ Wt ],. (18). where θ0 is the initial intensity rate. Table 2 presents the CatEPut prices under various θ0 , µθ and σθ with T = 1, 3, 5. Numerical results in Table 2 indicate that the CatEPut prices increase in θ0 , µθ − 0.5σθ2 for fixed µθ , and T . Moreover, if µθ − 0.5σθ2 is fixed, then different combinations of (µθ , σθ ) yield different CatEPut prices.. 4.5. The influence of risk-neutral measures. Recall Model (6) and let the intensity rate be a constant θ. By Proposition 3.2, if i.i.d. the jump-size is N (µy , σy2 ) distributed under the P measure, then Yt,n ∼ N (˜ µy , σy2 ) under the Qc measure, where Qc denotes the risk-neutral measure corresponding to a specific constant c satisfying (23) and (24), µ ˜y = µy + δtL σy2 , and δtL is obtained from solving (11) with a fixed c. In addition, the intensity rate of the Poisson process Nt under Qc becomes θ˜ = MYt,n |Ft−1 (δtL )θ. Since different choices of c correspond to different risk-neutral measures Qc , we investigate the influence of the choices of c on the CatEPut prices. Note that the parameters δtG and δtL defined in (10) and (11), respectively, when proceeding of the the Esscher transform in the GARCH and Jump parts, are determined by c. 1. If c = θ[exp(−µy + 0.5σy2 ) − 1], which corresponds to the Merton measure, then δtL = 0 and δtG = −{θ[exp(−µy + 0.5σy2 ) − 1] + λσt }/σt2 . 2. If c = 0, which corresponds to the risk-neutral measure in Amin (1993) when T = 1, then δtL = 0.5−µy /σt2 and δtG = −λ/σt , which is the same as the results of Siu, et al. (2004) and Huang, et al. (2012) in the GARCH framework. 3. In particular, if δtL = δtG = δt∗ , then by (10) and (11) δt∗ satisfies exp[µy (δt∗ −1)+ 0.5σy2 (δt∗ −1)2 ]−exp[µy δt∗ +0.5σy2 (δt∗ )2 ] = −(λσt +δt∗ σt2 )/θ and c = −(λσt +δt∗ σt2 ). 12.

(20) Table 3 presents the CatEPut prices under various k = c {θ[exp(−µy + 0.5σy2 ) − 1]}−1 with θ = 0.4, µy = 0.02, 0.04, 0.08, σy2 = 0.4 and T = 1, 3, 5. For the case of δtL = δtG = δt∗ , we report the values of k with c = −(λσ1 + δ1∗ σ12 ), that is, k = −0.3322, −0.3401 and −0.3608 if µy = 0.02, 0.04 and 0.08, respectively. The CatEPut prices presented in Table 3 versus k are shown in Figure 8, which indicates that the CatEPut prices decrease as k (or c, since c is proportional to k) increasing. That is, different Qc measure computes different CatEPut prices. From Table 3, we find that the CatEPut prices increase in T for all the different Qc measures, which is the same as the phenomenon observed in Section 4.3, where only Merton measure is considered. Interestingly, as µy increasing, the CatEPut prices increase for all the different Qc measures if T = 1 while the CatEPut prices decrease for all Qc if T = 5.. 5. Conclusion. This study establishes a general approach to derive a risk-neutral GARCH-Jump model with stochastic intensity rate by the Esscher transform. The CatEPut option prices are computed under the GARCH-Jump model. Sensitivity analysis of the influence of the model parameters on the CatEPut prices are conducted by simulation. Numerical results indicate that there is no significant difference between the GARCH-Jump model and jump-diffusion model for CatEPut prices. Table 4 summarizes the influence of the intensity rate of the occurrence of catastrophes, the distribution assumption of the jump size, the time to maturity, and the selection of risk-neutral measure on pricing CatEPut options in GARCH-Jump model. In general, the CatEPut prices increase as the intensity rate, the expected value of the jump size, the variance of the jump size or the time to maturity increasing.. Appendix Proof of Lemma 3.1. Let ηt = (log θt )/σθ and α = (µθ − 0.5σθ2 )/σθ . By (7), we have dηt = αdt + dWt . 13. (19).

(21) Further let Mt = max{ηs : s ∈ [t − 1, t]}.. (20). By (19), (20) and Corollary 7.2.2 in Shreve (2004), the pdf of Mt is 2 2 fMt (m) = √ e−0.5(m−α) − 2α e2α m Φ(−m − α), 2π. (21). where m ≥ 0 and Φ(·) denotes the cdf of N (0, 1). By Taylor’s expansion and (20), we have Mθt∗ |Ft−1 (h) =. ∞ X hj j=0. j!. Et−1 [(θt∗ )j ]. ≤. ∞ X hj j=0. j!. Et−1 [ejσθ Mt ].. (22). By (21) and noting that Z ∞ 2 2 2 ejσθ m−0.5(m−α) dm = O(ejασθ +0.5j σθ ) 0. and Z. ∞. ejσθ m+2αm Φ(−m − α)dm = O(j −1 ejασθ +0.5j. 2 σ2 θ. ),. 0. the right-hand-side of (22) is finite. Consequently, Mθt∗ |Ft−1 (h) exists. However, it is difficult to obtain closed-form representation of Mθt∗ |Ft−1 (h). In this lemma, we approximate it by the following second order Taylor’s expansion: hµθ∗. Mθt∗ |Ft−1 (h) ≈ e. t. (1 +. h2 2 σ ∗ ), 2 θt. where µθt∗ = Et−1 (θt∗ ) and σθ2t∗ = Vart−1 (θt∗ ) are the conditional mean and variance of θt∗ given Ft−1 , respectively. In the following, we derive the closed-form representations for µθt∗ and σθ2t∗ .. 14.

(22) Let Xn = n−1. Pn−1 i=0. θt−1+i/n . First note that Xn converges to θt∗ in L2 sicne. Et−1 |Xn − θt∗ |2 n−1 Z t−1+(i+1)/n

(23) X

(24) 2

(25)

(26) (θt−1+ i − θs )ds

(27) = Et−1

(28) n t−1+i/n. i=0 n−1 Z X. ≤. t−1+(i+1)/n. Et−1 |θt−1+ i − θs |2 ds n. t−1+i/n. i=0 n−1 X Z t−1+(i+1)/n. =. n. t−1+i/n. i=0. 2 1 θt−1. =. Et−1 Et−1+ i |θt−1+ i − θs |2 ds. n−1 X. n. i=0. n. h i 1 n µθ n 2 i (2µθ +σθ2 ) n e(2µθ +σθ ) n 1 − 2 en −1 + e − 1 µθ 2µθ + σθ2. → 0 as n → ∞, where the inequality holds by Cauchy-Schwarz inequality. By Theorem 4.5.4 in Chung (2001), Et−1 (θt∗ )r = lim Et−1 (|Xn |r ), for 0 < r ≤ 2. Therefore, the condin→∞. tional mean and conditional variance of θt∗ given Ft−1 can be computed as µθt∗ = P µθ ni = θµt−1 (eµθ − 1), and σθ2t∗ = Et−1 (θt∗ ) = lim Et−1 (Xn ) = lim n1 n−1 i=0 θt−1 e θ n→∞. n→∞. Et−1 (θt∗ )2 − µ2θt∗ , where Et−1 (θt∗ )2 =. lim Et−1 (|Xn |2 ). n→∞. n−1 n−2 n−1 h1 X 2 X X 2 (µθ +σθ2 ) i µθ j i 2 (2µθ +σθ2 ) ni ne n θ e + θt−1 e t−1 2 n→∞ n2 n i=0 i=0 j=i+1 2 h 2θt−1 2µθ +σ2 1 1 1 eµθ i θ e − + − . = µθ µθ + σθ2 2µθ + σθ2 2µθ + σθ2 µθ + σθ2. =. lim. Proof of Proposition 3.1. Let ΛT be a Radon-Nikod´ ym derivative and the corresponding risk-neutral measure Q be defined by dQ = ΛT dP . Recall the scheme of the Esscher transform introduced in Section 2.1 and the equation, Rt = RtG − Lt , in Model (6). By using similar arguments as in Shreve (2004), ΛT is decomposed by G. L ΛT = ΛR T × ΛT , G. where ΛR and ΛLT are two independent Radon-Nikod´ ym derivatives such that T G ΛR G t RtG EQ eRt = er−c (23) (e ) = E t−1 t−1 RG Λt−1 15.

(29) and −Lt ) = Et−1 EQ t−1 (e. ΛL t −L e = ec , ΛLt−1. (24). G. G. L L = Et (ΛR in which ΛR t T ) and Λt = Et (ΛT ), t = 0, 1, . . . , T .. In Model (6), the conditional distribution of RtG |Ft−1 under P is N(r − 0.5σt2 + σt , σt2 ) and the corresponding conditional mgf is 2. 2 2. MRtG |Ft−1 (z) = e(r−0.5σt +σt )z+0.5σt z .. (25). Define a new conditional pdf of RtG |Ft−1 with a parameter δt fRtG |Ft−1 (xt ; δt ) =. e xt δ t MRtG |Ft−1 (δt ). fRtG |Ft−1 (xt ).. (26). Then, choose a δt = δtG in (26) for 0 ≤ t ≤ T such that (23) holds, that is, δtG = −. c + λσt . σt2. (27) G. Consequently, the Radon-Nikod´ ym derivative ΛR T derived by the Esscher transform is G ΛR T. =. =. T Y t=1 T Y t=1. G RG t. eδt. MRtG |Ft−1 (δtG ) exp{−. ,. 1 1 [(c + λσt )2 + 2(c + λσt )(RtG − r − λσt + σt2 )]}. (28) 2 2σt 2. In Model (6), the conditional density of Lt |Ft−1 under P is fLt |Ft−1 (xt ) =. ∗ ∞ X Et−1 [e−θt θ∗k ]. t. k=0. k!. fYt,n (xt |∆Nt = k),. (29). and the conditional mgf of Lt |Ft−1 , denoted by MLt |Ft−1 (z), is MLt |Ft−1 (z) = Et−1 {exp[θt∗ (MYt,n |Ft−1 (z) − 1)]}.. (30). Define a new conditional pdf of Lt |Ft−1 with a parameter δt fLt |Ft−1 (xt ; δt ) =. e xt δ t MLt |Ft−1 (δt ). 16. fLt |Ft−1 (xt ),. (31).

(30) Then, choose a δt = δtL in (31) for 0 ≤ t ≤ T such that (24) holds where δtL is the solution of (11). Consequently, the Radon-Nikod´ ym derivative ΛLT derived by the Esscher transform is ΛLT. T Y. =. t=1. L. eδt Lt L ∗ , Et−1 [eh(δt ) θt ]. (32) ∗. L. where h(δtL ) = MYt,n |Ft−1 (δtL ) − 1 and Et−1 [eh(δt ) θt ] can be approximated either by Lemma 3.1 or Monte Carlo simulation. In addition, if ΛLT in (32) is further decomposed by L|θ∗. ΛLT = ΛT L|θ∗. where ΛT. ∗. × ΛθT ,. (33). is the Radon-Nikod´ ym derivative of Q with respect to P for LT condiL|θ∗. ∗. tional on θT∗ , and ΛθT is the Radon-Nikod´ ym derivative for θT∗ . Note that Λt. can. be obtained by similar arguments used in (29)-(32), that is, L|θ∗. ΛT. =. L T Y eδt Lt . h(δtL ) θt∗ e t=1. (34). By (32)-(34), we have ∗ ΛθT. =. T Y t=1. ∗. L. eh(δt ) θt L ∗ . Et−1 [eh(δt ) θt ]. (35). Finally, the desire result holds by (33)-(35).. . Proof of Proposition 3.2. By (28) in the proof of Proposition 3.1, the change of G. G. G. G measure processes ΛR be defined by ΛR = Et (ΛR t t T ) for t = 1, . . . , T . Choosing δt. in (10) the conditional mgf of RtG |Ft−1 under measure Q is G. MQ (z; δtG ) RtG |Ft−1. =. z RtG ) EQ t−1 (e 2. ΛR G = Et−1 ( Rt G ez Rt ) Λt−1 2 2. = e(r−c−0.5σt )z+0.5σt z ,. (36). which has the same form as the conditional mgf of RtG |Ft−1 under P given in (25). That is, RtG |Ft−1 is normally distributed with mean r − c + σt2 /2 and variance σt2 /2 under Q. 17.

(31) By (34) and (35) in the proof of Proposition 3.1, the change of measure processes L|θ∗. Λt. ∗. and Λθt are defined by L|θ∗. Λt. L|θ∗. = Et (ΛT. ∗. ∗. ) and Λθt = Et (ΛθT ),. (37). for t = 1, . . . , T . By (33), the conditional mgf of Lt |Ft−1 under measure Q is Q z Lt L ) MQ Lt |Ft−1 (z; δt ) = Et−1 (e ∗. L|θ∗. Λ Λθ ΛL = Et−1 ( Lt ez Lt ) = Et−1 [ θt∗ Et−1 (ez Lt tL|θ∗ |θt∗ )] Λt−1 Λt−1 Λt−1 L ∗ n eh(δt ) θt o L L MYt,n |Ft−1 (z + δt ) ∗ )[ M (δ = Et−1 exp{θ − 1]} Yt,n |Ft−1 t L ∗ t MYt,n |Ft−1 (δtL ) Et−1 (eh(δt ) θt ) (38) where the last equality holds by (34), (35) and (37). In addition, let ψθt∗ |Ft−1 (x) denote the conditional pdf of θt∗ |Ft−1 under P . By (35) and (37), the conditional density of θt∗ |Ft−1 under Q can be represented as L. ψ˜θt∗ |Ft−1 (x) =. eh(δt ) x ∗ L ∗ ψθ |Ft−1 (x). Et−1 (eh(δt ) θt ) t. (39). By (39) and let θ˜t∗ = θt∗ MYt,n |Ft−1 (δtL ), the conditional mgf obtained in (38) can be rewritten as L Q L ˜∗ ( MYt,n |Ft−1 (z + δt ) − 1)]}. MQ (z; δ ) = E {exp[ θ t−1 t Lt |Ft−1 MYt,n |Ft−1 (δtL ). (40) . 18.

(32) Table 1: The CatEPut prices and P (L1 > L) (in parentheses) versus intensity rates under the GARCH-Jump model with constant, gamma and normally distributed jump sizes in the Merton measure with T = 1. Case 1 θ 0.2. 0.4. 0.6. 0.8. 1. 1.2. A = 0.02. Ga(10, 0.002). Case 2 N (0.02, 4 × 10−5 ). Ga(0.1, 0.2). Case 3 N (0.02, 4 × 10−3 ). Ga(10−3 , 20). N (0.02, 4 × 10−1 ). 9.1009. 9.1013. 9.0808. 3.1014. 5.9534. 0.0663. 5.6801. (0.1813). (0.1813). (0.1803). (0.0566). (0.1098). (0.0016). (0.0926). 17.4452. 17.3270. 17.0452. 4.9773. 10.2147. 0.1987. 10.0838. (0.3297). (0.3275). (0.3219). (0.0924). (0.194). (0.0029). (0.1678). 23.1722. 21.8047. 21.6306. 6.4104. 13.9292. 0.3020. 14.3979. (0.4512). (0.4236). (0.4172). (0.1202). (0.2584). (0.0041). (0.2289). 28.3179. 23.1340. 23.4905. 7.6496. 16.1245. 0.3949. 17.6613. (0.5507). (0.4488). (0.4554). (0.1428). (0.3074). (0.0052). (0.2786). 13.5460. 22.1135. 22.6213. 8.7856. 17.9866. 0.4018. 20.8239. (0.2642). (0.4321). (0.4458). (0.162). (0.3447). (0.0063). (0.319). 17.7409. 21.9585. 21.9910. 9.5113. 19.7644. 0.4333. 22.5657. (0.3374). (0.4204). (0.4246). (0.1784). (0.3729). (0.0073). (0.3518). Table 2: The CatEPut prices under GARCH-Jump model with stochastic intensity rate and jump sizes is N (0.02, 0.4) distributed in the Merton measure. θ0 µθ −. 0.5σθ2. T. (µθ , σθ ). 0.2. 0.4. 0.6. 0.8. 1. (0.1, 0). 0.1. 6.2771. 11.3545. 15.4044. 18.9942. (0.1, 0.3). 0.055. 6.2392. 11.1265. 15.1625. 18.5459. (0.1, 0.4). 0.02. 6.0868. 10.9545. 15.0098. 18.3812. (0.2, 0.6). 0.02. 6.5768. 11.5850. 15.4185. 18.2758. (0.1, 0). 0.1. 14.7899. 22.6858. 26.8348. 29.3594. (0.1, 0.3). 0.055. 14.3457. 21.6676. 25.6937. 28.5816. (0.1, 0.4). 0.02. 13.9540. 21.0800. 25.2308. 27.6481. (0.2, 0.6). 0.02. 14.5843. 21.1960. 24.8104. 27.2773. 3. 5. (0.1, 0). 0.1. 19.1988. 25.8191. 28.1968. 29.1458. (0.1, 0.3). 0.055. 18.1266. 24.2215. 26.9488. 28.4127. (0.1, 0.4). 0.02. 17.4332. 23.4287. 26.2077. 27.5207. (0.2, 0.6). 0.02. 17.7179. 23.0994. 25.7644. 27.1952. 19.

(33) Table 3: The CatEPut prices versus k under normally distributed jump sizes, where k = c{θ[exp(−µy + 0.5σy2 ) − 1]}−1 with θ = 0.4 and σy2 = 0.4. µy = 0.02. µy = 0.04. µy = 0.08. k. -0.3322. 0. 0.25. 0.5. 0.75. 1. T =1. 14.4340. 13.3650. 12.5233. 11.6947. 11.0095. 10.3423. T =3. 28.0865. 25.8917. 24.3212. 22.7853. 21.3443. 20.1021. T =5. 31.8047. 29.4976. 28.0237. 26.1909. 24.6224. 23.2531. k. -0.3401. 0. 0.25. 0.5. 0.75. 1. T =1. 14.3354. 13.2432. 12.4245. 11.7216. 11.1213. 10.5368. T =3. 27.6199. 25.5841. 24.1402. 22.7616. 21.6054. 20.4345. T =5. 31.2082. 29.1659. 27.3588. 26.1471. 24.6395. 23.2393. k. -0.3608. 0. 0.25. 0.5. 0.75. 1. T =1. 13.9434. 13.1623. 12.4645. 11.9243. 11.5145. 10.9894. T =3. 26.2002. 24.6243. 23.6016. 22.6098. 21.6747. 20.7056. T =5. 29.5389. 27.5814. 26.5712. 25.3519. 24.2673. 23.0902. Table 4: Summary of sensitivity analysis on the CatEPut prices. Yt,n Constant. Ga(α, β). N (µy , σy2 ). ↑. ↑. Merton measure. θ. ↑↓. and constant intensity. µy ( if T = 1). ↑. ↑. ↑. σy2. –. ↓. ↑. T. ↑↓. ↑. ↑. Merton measure. θ0. –. –. ↑. and stochastic intensity. µθ − 0.5σθ2. –. –. ↑. Qc and constant intensity. c. –. –. ↓. Figure 1: The CatEPut prices under various initial stock price S0 and intensity rate θ in the GARCH-Jump model, where T = 1.. 20.

(34) 0.7. 0.6. > L). 0.5. P( L. 1. 0.4. 0.3. 0.2. 0.1. 0. 0. 0.5. 1. 1.5. 2. 2.5 θ. 3. 3.5. 4. 4.5. 5. Figure 2: The value of P (L1 > L) versus the intensity rate θ in the case of constant jump size.. 30. 25. CatEPut price. 20. 15. 10. 5. 0. 0. 0.1. 0.2. 0.3 P( L. 1. 0.4 >L). 0.5. 0.6. 0.7. Figure 3: The CatEPut prices given in Table 1versus P (L1 > L).. 21.

(35) 0.5. 0.5. 1. 1.5. 2. 2.5 θ. 3. 3.5. 4. 4.5. 5. 0.5. 1. 1.5. 2. 2.5 θ. 3. 3.5. 4. 4.5. 5. 0.5. 1. 1.5. 2. 2.5 θ. 3. 3.5. 4. 4.5. 5. 0.5. P( L. 1. >L). 0 0. 0 0. 0.5. 0 0. Figure 4: The probability of P (L1 > L) versus the intensity rate θ in the cases of normally (blue circle) and gamma (red dot) distributed jump size. The upper panel is the case of Ga(10,0.0002) and N(0.02,4 × 10−5 ). The middle panel is the case of Ga(0.1,0.2) and N(0.02,4 × 10−3 ). The lower panel is the case of Ga(10−3 ,20) and N(0.02,4 × 10−1 ).. Gamma distribution. θ = 0.4. CatEPut price. 20. Normal distribution 25. µy = 0.01. 15 10. µy = 0.02. 20. µy = 0.05. 15. µy = 0.5. 10. 5 0. 5 0. 2. 4. 6 σ2y / µy. 8. 0. 10. θ = 0.8. CatEPut price. 20. 0. 2. 4. 0. 2. 4. 6 σ2y / µy. 8. 10. 6. 8. 10. 25 20. 15. 15 10 10 5 0. 5 0. 2. 4. 6 σ2y / µy. 8. 0. 10. σ2y / µy. Figure 5: The CatEPut prices versus the ratio σy2 /µy = 1, 2, · · · , 10 under GARCHJump model with gamma and normally distributed jump sizes in the Merton measure, where µy = 0.01, 0.02, 0.05, 0.5, θ = 0.4, 0.8 and T = 1.. 22.

(36) Constant 30. 25. CatEPut price. 20. 15. 10 T=1 T = 3 (JD) T=3 T = 5 (JD) T=5. 5. 0 0.1. 0.2. 0.3. 0.4. 0.5 θ. 0.6. 0.7. 0.8. 0.9. Figure 6: The CatEPut prices versus the intensity rate θ = 0.2, 0.4, 0.6, 0.8 under GARCH-jump model and jump-diffusion (JD) model with constant jump size in the Merton measure, where T = 1, 3, 5 .. CatEPut price. σ2y = 4*10−3. Gamma distribution 20. 20. 10. 10. CatEPut price. 0. σ2y = 4*10−1. Normal distribution. 0.2. 0.4. 0.6. 0. 0.8. 0.2. 0.4. 0.2. 0.4. 0.6. 0.8. 0.6. 0.8. 30 4 20 2 0. 10. 0.2. 0.4. θ. 0.6. 0. 0.8. θ. Figure 7: The CatEPut prices versus the intensity rate θ under GARCH-Jump model and jump-diffusion (JD) model in the Merton measure, where the notations for T = 1, 3, 5 are the same as those in Figure 6.. 23.

(37) T=1 15 14. µy = 0.02. 13. µy = 0.04. 12. µy = 0.08. 11 10 −0.4. −0.2. 0. 0.2. 0.4 k. 0.6. 0.8. 1. 1.2. 0.6. 0.8. 1. 1.2. 0.6. 0.8. 1. 1.2. T=3 CatEPut price. 30 25 20 15 −0.4. −0.2. 0. 0.2. 0.4 k T=5. 35 30 25 20 −0.4. −0.2. 0. 0.2. 0.4 k. Figure 8: The CatEPut prices given in Table 4 versus k.. References [1] Amin, K. I. (1993). Jump diffusion option valuation in discrete time. Journal of Financial, 48, 1833-1863. [2] Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometics, 31, 307-327. [3] Cox, H., Fairchild, J., and Pedersen, H. (2004). Valuation of structured risk management preducts. Insurance: Mathematics and Economics, 34, 259-272. [4] Duan, J. C. (1995). The GARCH option pricing model. Mathematical Finance, 43, 13-32. [5] Duan, J. C. and Simonato, J. G. (1998). Empirical martingale simulation for asset prices. Management Science, 44, 1218-1233. [6] Engle, R. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica, 50, 987-1008.. 24.

(38) [7] Gerber, H. U. and Shiu, S. W. (1994). Option pricing by Esscher transform. Transactions of Society of Actuaries, 46, 99-191. [8] Huang, S. F. and Guo, M. H. (2009) Financial derivative valution - A dynamic semiparametric approach. Statistica, 19, 1037-1054. [9] Huang, S. F. (2012). A modified empirical martingale simulation for financial derivative pricing. Communications in Statistics - Theory and Methods. Accepted. [10] Huang, S. F., Liu, Y. C., and Wu, J. Y. (2012). An empirical study on implied GARCH models. Journal of Data Science, 10, 87-105. [11] Jaimungal, S. and Wang, T. (2006). Catastrophe options with stochastic interest rates and compound Poisson losses. Insurance: Mathematics and Economics, 38, 469-483. [12] Kou, S. G. (2002). A jump-diffusion model for option priceing. Management Science, 50, 1178-1192. [13] Lin, S. K., Chang, C. C., and Michael, R. P. (2009). The valuation of contingent capital with catastrophe risks. Insurance: Mathematics and Economics, 45, 6573. [14] Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125-144. [15] Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous Time Models, (Springer, New York). [16] Siu, T. K., Tong, H., and Yang, H. (2004). On pricing derivatives under GARCH models: A dynamic Gerber-Shiu approach. North American Actuarial Journal, 8, 17-31.. 25.

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