Pricing Catastrophe Equity Put Options with Conditional Jump … · 2019-06-24 · 2019...

Pricing Catastrophe Equity Put Options with Conditional Jump Intensity

使用條件跳躍強度訂價巨災權益賣權

ABSTRACT

This paper researches the pricing of catastrophe equity put (CatEPut) options under conditional jump intensity with losses generated by a compound Poisson process. We derive explicit closed-form formula for the price of CatEPut option and use the daily stock price return of RLI Corp. from 1980 to 2018 to estimate the parameters of our pricing formula. By using the American hurricane event data from 1960 to 2011 and RLI Corp. annual stock price from 1980 to 2005, we can calculate the CatEPut option price and confirm that the CatEPut option price can reflect the trend of RLI Corp. stock price. The effects of the variance of the loss process on the option’s price are illustrated through numerical experiments, and prove the importance of the jump term in the model through calculating the pricing error.

Keywords: Catastrophe equity put option, GARCH-jump model, catastrophic events, conditional jump intensity, compound Poisson process

2019 富邦人壽管理博碩士論文獎

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1 Introduction

At the beginning of 2019, the Global Risks Report is published by World Economic Forum (2019), which discussed many challenges and foremost risks presented in the future, including economic risks, environmental risks, geopolitical risks, societal risks, and technological risks. In this report, the environment-related risks stand out in the global risk landscape with every risk in the category lying in the higher-impact, higher-likelihood quadrant (Figure 1). It can be expected that the occurrence of catastrophes will also lead to enormous economic losses. According to the figure of the Evolving Risks Landscape during 2009–2019 in the Global Risks Report (Figure 2), the top 3 global risks in terms of likelihood in 2019 are extreme weather events, failure of climate-change mitigation and adaptation, and natural disasters, respectively. The three risks mentioned above are also among the top 5 of global risks in terms of impact in 2019. This report indicates that extreme weather and climate-change policy failures are seen as critical threats over a ten-year horizon.

Figure 1: Global risks landscape (The Global Risks Report 2019, WEF)

In order to hedge the enormous financial losses caused by the catastrophe, catastrophe equity put (CatEPut) options provide protection for the corporation and shareholders. Consequently, the pricing scheme of CatEPut options attracts more attention from practitioners and researchers. In 1996, RLI Corp. (NYSE: RLI), an American property and casualty insurance company, has signed catastrophe equity put option contract with Centre Reinsurance, stipulating that when RLI Corp. loses more than $200 million, it can sell a three-year, face value of $50 million convertible preferred shares to Centre Reinsurance at a fixed price. It became the first successful example in the international market that use catastrophe equity put option to resolve catastrophe risks. In general, the CatEPut option gives the owner the right to issue convertible preferred shares at a fixed price, much like a regular put option. Besides RLI Corp., other companies have also issued the CatEPut options listed in Table 1.


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Figure 2: The evolving risk landscape (The Global Risks Report 2019, WEF)

Cox, Fairchild, and Pedersen (2004) derived a closed-form solution under a jump-diffusion model with constant jump size and constant intensity rate. Jaimungal and Wang (2006) extended the result of Cox et al. (2004) to the case of random riskless interest and random jump size. Lin, Chang, and Powers (2009) further considered the valuation of CatEPut options under stochastic intensity rate of the occurrence of catastrophe. Chang, Lin, and Yu (2011) derived the pricing formula for catastrophe equity put options by assuming catastrophic events follow a Markov Modulated Poisson process whose intensity varies according to the change of the Atlantic Multidecadal Oscillation (AMO) signal. While considerable attention has been given in the literature to the pricing CatEPut options in stochastic jump-diffusion model, there are few studies on the GARCH model, which is capable of successfully depicting heteroskedasticity of financial data. Lehar, Scheicher, and Schittenkopf (2002) analyzed the fit to observed prices, and find that GARCH dominates the stochastic volatility model distinctly. Therefore, our research combines the scheme of the GARCH and conditional jump intensity model, denoted by GARCH-jump, to describe the dynamics of the underlying asset.

Heston and Nandi (2000) developed a closed-form option valuation formula for a spot asset whose variance follows a GARCH process that can be correlated with the returns of the spot asset. Christoffersen, Jacobs, and Ornthanalai (2012) extended the model of Heston and Nandi (2000), added the jump component in the return process and volatility process. Ornthanalai (2014) chose the relatively parsimonious affine GARCH model of Heston and Nandi (2000). Particularly, he estimated discrete-time models where asset returns follow a Brownian increment and a Lévy jump. Byun, Jeon, Min, and Yoon (2015) extended the research of Christoffersen et al. (2012), and considered the nonlinear pricing kernel. Babaoğlu (2016) added the jump component to the volatility process and derived the option price. Chang, Chang,


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Cheng, Peng, and Tseng (2019) considered the nonlinear pricing kernel and derived the closed-form of the option pricing model.

Table 1: The Issue of CatEPut Options

The table reports the companies which ever issued the CatEPut options from 1996 to 2002.

In this paper, with reference to Chang et al. (2019), we drive the closed-form of CatEPut options pricing formula. In particular, we use the RLI Corp. historical stock data as underlying prices in our model. Following Chang, Yang, and Yu (2018), we use the hurricane data in the United States as catastrophe loss severity in our pricing model. Figure 3 shows the annual frequency and severity of United States hurricanes between 1960 and 2011. Each hurricane event is available from the U.S. National Hurricane Center.

Through the framework of a conditional jump intensity model, we illustrate how catastrophic losses and conditional jump component, together with stock prices, affect option prices. We also demonstrated that both the variance of the loss process and conditional GARCH-jump intensity play a significant role. The results show that the Model 3, which includes conditional GARCH-jump process with jump term, generates lower pricing errors than the others and pricing errors can be reduced by 3.01%–7.32% depending on the different measurement methods. That is, we can use the formula of CatEPut option in this study to pricing the contract with reflecting stock price movement and catastrophe loss well.

The remainder of the paper is organized as follows. Section 2 describes the asset return process, the variance process with GARCH jump intensity, and the CatEPut options pricing model. In order to demonstrate the importance of jump intensity in our models, we list three cases with different terms of model to compare the results in the following sections. Section 3 describes our estimation procedure and results for the estimation. Section 4 shows the pictures of the three cases of CatEPut options prices and pricing errors. Section 5 concludes.


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Figure 3: Hurricane events in the United States 1960–2011

2 The Model 2.1 Asset returns under the physical measure

Under the physical measure (P), the underlying asset return process is given by

𝑅" ≡ ln &𝑆"𝑆"()

*

= 𝑟 + /𝜆1 − 𝜉1(1)7ℎ1," + /𝜆: − 𝜉:(1)7ℎ:," + 𝑧" − 𝛼𝐿1 (1)

where 𝑆𝑡 is the price of underlying asset, 𝑅𝑡 is a logarithmic return, 𝑟 is the risk-free rate, 𝜆𝑧and 𝜆𝐿 are risk premium, and the parameter αrepresents the percentage drop in the share value price per unit of loss, which is similar to Jaimungal and Wang (2006) and Chang et al. (2011). The normal component denotes a heteroscedastic Gaussian innovation, 𝑧𝑡 ∼ 𝑁(0, ℎ𝑧, 𝑡), and 𝜉𝑧(𝜑) = C

D𝜑2.

The jump component 𝐿𝑡 denotes a compound Poisson process:

𝐿" =F𝑙"H

IJ

HKL

∼ 𝐶𝑃𝐽/ℎ:,", 𝜇Q, 𝜎QS7 (2)

where 𝑙𝑡 ∼ 𝑁(𝜇𝐽, 𝜎𝐽) and 𝑛𝑡 ∼ 𝑃𝑜𝑖𝑠𝑠𝑜𝑛(ℎ𝐿, 𝑡) . Finally, 𝜉𝐿(𝜑) = 𝑒𝑥𝑝 &𝜇Q𝜑 +[\D]D

S* ,

𝐸"()[𝐿𝑡] = 𝜇𝐽ℎ𝐿, 𝑡,and Var"()[𝐿𝑡] = (𝜇𝐽2+ 𝜎𝐽2)ℎ𝐿, 𝑡.


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2.2 Variance process and jump intensity

Our variance process and jump intensity are similar to Chang et al. (2019). We assume the normal variance dynamics and conditional jump component are as follows:

ℎ1," = 𝑤1 + 𝑏1ℎ1,"() +𝑎11ℎ1,"()

/𝑧"() − 𝑐11ℎ1,"()7S +

𝑎1:ℎ:,"()

/𝐿"() − 𝑐1:ℎ:,"()7S

ℎ:," = 𝑤: + 𝑏:ℎ:,"() +𝑎:1ℎ1,"()

/𝑧"() − 𝑐:1ℎ1,"()7S +

𝑎::ℎ:,"()

/𝐿"() − 𝑐::ℎ:,"()7S

(3)

(4)

with initial conditions hz,0and hL,0.

In order to demonstrate the importance of conditional jump intensity in our models, we adjust terms in (4) and list three cases to compare their differences in the following sections.

Model 1 (Constant) 𝑏𝐿 = 𝑎𝐿𝑧 = 𝑐𝐿𝑧 = 𝑎𝐿𝐿 = 𝑐𝐿𝐿 = 0

Model 2 (No jump term) 𝑎𝐿𝐿 = 𝑐𝐿𝐿 = 0

Model 3 (The full model)

Our goal is to prove that the CatEPut option price of Model 3, which includes jump term, is the most accurate method to calculate the option price. That is, we can confirm that the jump term plays an important role.

2.3 The payoff formula

Following Jaimungal and Wang (2006), the CatEPut option gives the owner the right to issue convertible preferred shares at a fixed price, just like a regular put option; however, that right can only be exercised if the cumulative loss of the insurance purchaser exceeds the critical coverage limit during the lifetime of the option. Such a contract, signed at time t, is a special form of a double trigger option and has a payoff at maturity of

payoff = 𝕀{:j(:Jkℒ}(𝐾 − 𝑆o)p

= q𝐾 − 𝑆o, 𝑖𝑓𝑆o < 𝐾𝑎𝑛𝑑𝐿o − 𝐿" > ℒ0, 𝑖𝑓𝑆o ≥ 𝐾𝑎𝑛𝑑𝐿o − 𝐿" ≤ ℒ

(5)

where 𝑆𝑇 denotes the share value and 𝐿𝑇 − 𝐿𝑡 denotes the total losses of the insured over the time period [𝑡, 𝑇). The parameter 𝐿 is the trigger level of losses above which the CatEPut option becomes in-the-money, while 𝐾 represents the strike price at which the issuer is obligated to purchase unit shares if losses exceed ℒ.


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2.4 The risk-neutral dynamic

Following the affine asset pricing literature, such as Christoffersen et al. (2012) and Ornthanalai (2014), we assume that the pricing kernel 𝑀𝑡follows the affine dynamic

ln &

𝑀"

𝑀"()* = 𝑢" − 𝜌𝑅" − 𝜌Q𝐿" (6)

where Rtis the logarithm of the return on the index, and Ltis the jump component in the index return. The coefficient 𝑢" is a normalizing factor such that ln𝐸"()[𝑀" 𝑀"()⁄ ]

equals the risk

free rate r. We get

𝑢" = ln𝐸"() }exp/𝑟 + 𝜌𝑅" + 𝜌Q𝐿"7�. (7)

Substituting the dynamic of Rtfrom (1) into (6) and (7), we get the following dynamic:

𝑀"

𝑀"()=

exp/𝑟 − 𝜌𝑧" − /𝜌Q − 𝛼𝜌7𝐿"7𝐸"()}exp/−𝜌𝑧" − /𝜌Q − 𝛼𝜌7𝐿"7�

(8)

for the pricing kernel.

Motivated by the affine structure of the pricing kernel in (8), we proceed by specifying the following conditional Radon-Nikodym derivative as:

dℚ"/dℙ"

dℚ"()/dℙ"()=

exp(𝑣1𝑧" + 𝑣:𝐿")𝐸"()ℙ [exp(𝑣1𝑧" + 𝑣:𝐿")]

(9)

where 𝑣𝑧 = −𝜌 and 𝑣𝐿 = 𝛼𝜌 − 𝜌𝐽. We define𝑣𝑧 and 𝑣𝐿 as the coefficients that capture the wedge between the physical and the risk-neutral measure (ℚ). Specifically, 𝑣𝑧 and 𝑣𝐿 are linked to diffusive and jump risk premiums, respectively.

Proposition 1. Given the underlying asset return process in (1) under the physical measure, the risk-neutral probability measure defined by the Radon-Nikodym derivative of (9) is an equivalent martingale measure if and only if:

λ1 = −𝑣1

λ: = ξ:(1) − ξ:(𝑣:) + ξ:(𝑣: − α).

(10)

(11)

Given the Radon-Nikodym derivative, we are now preparing to derive the underlying asset processes under measure ℚ.


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Proposition 2. The underlying asset dynamics under the measure ℚ can be written as

𝑅" = ln &

𝑆"𝑆"()

* = 𝑟 −12ℎ1,"

∗ − ξ:∗(1)ℎ:,"∗ + 𝑧"∗ − α𝐿"∗ (12)

and the variance dynamics and jump component are as follows:

ℎ1,"∗ = 𝑤1 + 𝑏1ℎ1,"()∗ +𝑎11ℎ1,"()∗ /𝑧"()∗ − 𝑐11∗ ℎ1,"()∗ 7S +

𝑎1:∗

ℎ:,"()∗ /𝐿"()∗ − 𝑐1:∗ ℎ:,"()∗ 7S

ℎ:,"∗ = 𝑤:∗ + 𝑏:ℎ:,"()∗ +𝑎:1∗

ℎ1,"()∗ /𝑧"()∗ − 𝑐:1∗ ℎ1,"()∗ 7S +𝑎::∗

ℎ:,"()∗ /𝐿"()∗ − 𝑐::∗ ℎ:,"()∗ 7S

(13)

(14)

where 𝑧"∗ = 𝑧𝑡– 𝑣𝑧ℎ1,"∗ follows normal distribution, the mean and the variance are 0and ℎ1,"∗

respectively; Lt∗follows a compound Poisson process with jump intensity ℎ:,"∗ = 𝛱ℎ:,",

𝛱 =

𝜉:(1 + 𝑣:) − 𝜉:(𝑣:)𝜉:(1)

= exp �𝑣:𝜇Q +𝑣:S𝜎QS

2 � (15)

and the mean and the variance of the conditional jump size are 𝜇Q∗ and 𝜎Q∗S respectively.

In addition, the risk-neutral parameters can be expressed as: ξ:∗(φ) = exp &µQ∗φ +�\∗D�D

S* − 1,

𝜇Q∗ = −𝛼/𝜇Q + 𝑣:𝜎QS7, σQ∗S = αSσQS, 𝑐11∗ = 𝑐11 + λ1 , 𝑎1:∗ = 𝑎1: , 𝑐1:∗ = ��

, 𝑤:∗ = 𝑤: , 𝑎:1∗ =

𝑎:1, 𝑐:1∗ = 𝑐:1 + λ1, 𝑎::∗ = 𝑎::𝛱S, 𝑐::∗ = ��

.

After determining the parameters under the measure ℚ, we can then solve the catastrophe equity put option pricing model by first deriving the moment generating function.

Proposition 3. Consider the moment generating function as follows:

𝑔",o∗ (φ, 𝐿o∗ ) ≡ 𝐸"ℚ[exp(φ ln(𝑆o)) |𝐿o]

= exp�φ ln(S�) + A�,�(φ, L�∗ ) + B�,�(φ)h£,�p)∗ + C�,�(φ)h¥,�p)∗ ¦

As STis known at time T, the moment generating function implies the terminal condition

𝐴o,o(φ, 𝐿o∗ ) = 𝐵o,o(φ) = 𝐶o,o(φ) = 0

If 𝑡 = 𝑇 − 1, 𝐴o(),o(φ, 𝐿o∗ ) = φ(𝑟 − α𝐿o∗ ), 𝐵o(),o(φ) =�(�())

S, 𝑎𝑛𝑑𝐶o(),o(φ) =

−φξ:∗(1).

When 0≤t≤T−2,

𝐴",o(𝜑, 𝐿o∗ ) = 𝐴"p),o(𝜑, 𝐿o∗ ) + 𝜑𝑟 + 𝐵"p),o(𝜑)𝑤1 + 𝐶"p),o(𝜑)𝑤:∗

−𝑙𝑛 ©1 − 2/𝐵"p),o(𝜑)𝑎11 + 𝐶"p),o(𝜑)𝑎:1∗ 7ª

2


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−𝑙𝑛/1 − 2/𝐵"p),o(𝜑)𝑎1:∗ + 𝐶"p),o(𝜑)𝑎::∗ 7𝜎QS7

2

𝐵",o(𝜑) = −12𝜑 + 𝐵"p),o

(𝜑)(𝑏1 + 𝑎11𝑐11∗S) + 𝐶"p),o(𝜑)𝑎:1∗ 𝑐:1∗S

+/𝜑 − 2𝐵"p),o(𝜑)𝑎11𝑐11∗ − 2𝐶"p),o(𝜑)𝑎:1∗ 𝑐:1∗ 7

S

2 ©1 − 2/𝐵"p),o(𝜑)𝑎11 + 𝐶"p),o(𝜑)𝑎:1∗ 7ª

𝐶",o(𝜑) = −𝜑𝜉:∗(1) + 𝐵"p),o(𝜑)𝑎1:∗ 𝑐1:∗S + 𝐶"p),o(𝜑)/𝑏: + 𝑎::∗ 𝑐::∗D7

+𝑒𝑥𝑝 �𝒱),"p),o(𝜑)𝜇Q∗ + 𝒱),"p),oS (𝜑)𝜎QS + 2𝒱S,"p),o(𝜑)𝜇Q∗S

2/1 − 2𝒱S,"p),o(𝜑)σQS7�

where

𝒱),"p),o(𝜑) = −𝜑𝛼 − 2𝐵"p),o(𝜑)𝑎1:∗ 𝑐1:∗ − 2𝐶"p),o(𝜑)𝑎::∗ 𝑐::∗ 𝒱S,"p),o(𝜑) = 𝐵"p),o(𝜑)𝑎1:∗ + 𝐶"p),o(𝜑)𝑎::∗ .

2.5 Pricing the catastrophe equity put option

Under the risk-neutral measure, the value of CatEPut options contracts can be obtained by way of discounted expectations. Following Jaimungal and Wang (2006), let𝐶(𝑡; 𝑡0)denote the value of the option at time t, which was signed at time t0<t, and matures at time T>t, we have

𝐶(𝑡; 𝑡L) = 𝐸"ℚ}𝐷(𝑡, 𝑇)𝟙{:jk:¯pℒ}(𝐾 − 𝑆o)p�

If interest rates are deterministic, then the discount factor can be extracted from the expectation resulting in

𝐶(𝑡; 𝑡L) = 𝑃(𝑡, 𝑇)𝐸"ℚ °𝟙�:jk:J¯pℒ¦(𝐾 − 𝑆o)p± (16)

where 𝑃(𝑡, 𝑇) denotes the fixed continuously compounded discount factor e−r(T−t).

Theorem 1. Assuming the risk-neutral dynamics given above, the price of the catastrophe equity put option is

𝐶(𝑡; 𝑡L) = F𝑒(²�,j∗ /ℎ:,o∗ 7I

𝑛! ´ µ𝐾𝑃(𝑡, 𝑇) �12 −

1𝜋´ 𝑅𝑒 ·

𝐾(¸]𝑔",o∗ (𝑖𝜑, 𝐿o∗ )𝑖𝜑 ¹

º

L𝑑𝜑�

º

ℒ»

º

IKL

− �𝑆"2 −

𝑃(𝑡, 𝑇)𝜋 ´ 𝑅𝑒 ·

𝐾(¸]𝑔",o∗ (𝑖𝜑 + 1, 𝑦)𝑖𝜑 ¹

º

L𝑑𝜑�½ 𝑓:I(𝑦)𝑑𝑦

(17)

where 𝑓:I(𝑦) represents the n-fold convolution of the loss probability density function 𝑓: (𝑦), 𝑦 = ∑ 𝑙o

HIHKL ~𝑁(𝑛𝜇Q∗, 𝑛𝜎QS), 𝑃(𝑡, 𝑇) = 𝑒(À(o("), and ℒ» = 𝐿"¯ + ℒ.


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3 Model Estimation 3.1 Data description

Table 2: Descriptive Statistics of RLI Corp. Stock Price Return Panel A. RLI Stock Price Return

Observations Mean Std. Dev Skewness Kurtosis Max Min

9758 5.23 × 10−4 0.02 −0.13 20.09 0.19 −0.20

Panel B. RLI Annual Stock Price

Observations Mean Std. Dev Skewness Kurtosis Max Min

38 17.37 19.01 1.13 3.19 63.13 0.45

The table reports the summary statistics of daily RLI Corp. stock price returns and annual RLI Corp. stock price. Panel A reports the summary statistics for daily RLI Corp. stock price returns during the period from 1980 to 2018. Panels B reports the summary statistics of annual RLI Corp. stock price from 1980 to 2005.

We use March 1980 to November 2018 daily stock price return data of RLI Corp. (NYSE: RLI) to estimate parameters in our models, and use 1980 to 2005 annual stock price data of RLI to calculate CatEPut option prices of three cases. In order to save the calculation time and close to reality, because the catastrophe loss is calculated on an annual basis, we use the annual stock data instead of the daily stock data to calculate the CatEPut options price. The descriptive statistics of RLI Corp. stock price data are given in Table 2.

As seen, we have 9758 stock price returns to estimate the parameters in our model and 38 annual stock prices to calculate the price of CatEPut options. We can find that the standard deviation of RLI Corp. annual stock price in Panel B is very large, due to the sharp rise in stock prices over the past years. Therefore, we can more easily observe the relationship between the stock price and the CatEPut option price.

To calculate the coverage loss amount of RLI Corp. in catastrophe events, we multiply the market share of RLI by the loss of catastrophe event in the United States. In 2017, the market share of RLI Corp. is 0.16% in American property and casualty insurance. For the sake of convenience, we fix 0.16% as its market share and use annual stock price data to generate real CatEPut option price in Section 4.

3.2 Maximum log-likelihood estimator (MLE)

In order to determine the parameters in our model, we fit model parameters by maximizing the following likelihood function:


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𝑚𝑎𝑥Â,Â∗∈Ä

{𝐿Return} (18)

where LReturnis the log likelihood functions of the stock price returns. θand θ∗are the physical and risk-neutral parameters. Importantly, we need to filter the variance dynamics hz,tand hL,tfrom (3)–(4) and to estimate the MLE in (18). The variance dynamics in the standard GARCH framework are straightforward because the total return residual enters into variance updating. In our case, our return residual consists of two random variables: ztand Lt. One would then need to separate them, given asset returns, because of both enter into our variance updating (3) and (4).

We apply a filtering technique by Durham et al. (2015). Rewriting the underlying asset return process (1),

𝑅" = µ" + 𝑧" − α𝐿"

µ" = 𝑟 + &λ1 −12* ℎ1," +

(λ: − ξ:)ℎ:,"

(19)

(20)

Let nt to be the number of jumps at time t. Then (𝑅𝑡|ℎ𝑧, 𝑡, ℎ𝐿, 𝑡, 𝑛𝑡) ∼ 𝑁(𝜇𝑡 − 𝑛𝑡𝛼𝜇𝐽, ℎ𝑧, 𝑡 +

𝑛𝑡𝛼2𝜎2) and (𝑛𝑡|ℎ𝐿, 𝑡) ∼ 𝑃𝑜𝑖𝑠𝑠𝑜𝑛(ℎ𝐿, 𝑡) , where µ" = 𝑟 + ©λ1 −)Sª ℎ1," + /λ: − ξ:(1)7ℎ:," .

Integrating across nt, (Rt|hz,t,hL,t)is a mixture of normal with density

𝑓/𝑅"Èℎ1,", ℎ:,"7 =F𝑓/𝑗Èℎ:,"7𝑓/𝑅"Èµ" − 𝑗αµQ, ℎ1," + 𝑗αSσQS7

º

HKL

(21)

where 𝑓(𝑗|ℎ𝐿, 𝑡) is the Poisson(ℎ𝐿, 𝑡) density and 𝑓(𝑅𝑡|𝜇𝑡 − 𝑗𝛼𝜇𝐽, ℎ𝑧, 𝑡 + 𝑗𝛼2𝜎𝐽2) is the normal density.

In practice, nt, zt, Lt, hz,t, and hL,tare not observable. To estimate the model, Durham et al. (2015) apply the following filter:

µ"Ê = 𝑟 + &λ1 −12* ℎ1,"

Ë + /λ: − ξ:(1)7ℎ:,"Ë (22)

𝑧"Ê =Fℎ1,"Ë

ℎ1,"Ë + 𝑗𝛼S𝜎QS/𝑅" − 𝜇"Ê + 𝑗𝛼𝜇Q7𝑓/𝑛" = 𝑗È𝑅", ℎ1,"Ë , ℎ:,"Ë 7

º

HKL

(23)

𝐿"Ì =1𝛼(−𝑅" + 𝜇"Ê + 𝑧"Ê) (24)

ℎ1,"p)Í = 𝑤1 + 𝑏1ℎ1,"Ë +𝑎11ℎ1,"Ë

/𝑧"Ê − 𝑐11ℎ1,"Ë 7S+𝑎1:ℎ:,"Ë

/𝐿"Ì − 𝑐1:ℎ:,"Ë 7S (25)


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ℎ:,"p)Í = 𝑤: + 𝑏:ℎ:,"Ë +𝑎:1ℎ1,"Ë

/𝑧"Ê − 𝑐:1ℎ1,"Ë 7S+𝑎::ℎ:,"Ë

/𝐿"Ì − 𝑐::ℎ:,"Ë 7S (26)

with initial conditions ℎ1,LË and ℎ:,LË . Christoffersen et al. (2012) show that (19)–(20) with hz,t= ℎ1,"Ë and hL,t= ℎ:,"Ë imply that 𝑧"Ê=E(zt|Rt,ℎ1,"Ë ,ℎ:,"Ë )and 𝐿"Ì=E(Lt|Rt,ℎ1,"Ë ,ℎ:,"Ë ). Using this filter, it is straightforward to back out implied values of hz,tand hL,tconditional on data{𝑅"}"K)"(), parameter vector θ, and initial conditions hz,0and hL,0. The parameter vector is then estimated by optimization,

θÏ = max

Ñ∈ΘFln}𝑓/𝑅"Èℎ1,"Ë , ℎ:,"Ë , 𝜃7�o

"K)

(27)

where the summands are given by (21) but with ℎ1,LË and ℎ:,LË in place of hz,0 and hL,0. Christoffersen et al. (2012) refer to this as a maximum likelihood estimator.

In fact, 𝜃Ï is the maximum likelihood estimator for the model defined by (22)–(26) in conjunction with

𝑅" = µ"Ê + ε" (28)

where 𝜀" is a mixture of normal with density

F𝑓/𝑗Èℎ:,"Ë 7𝑓©⋅ Ö𝜇"Ê − 𝑗𝛼𝜇Q×, ℎ1,"Ë + 𝑗𝛼S𝜎QSËªº

HKL

where f(j| ℎ:,"Ë )is the Poisson(ℎ:,"Ë ) density. Therefore, the log-likelihood of the return is

𝐿Return ∝ ∑ ln}𝑓/𝑅"Èℎ1,"Ë , ℎ:,"Ë , 𝜃7�o"K) . (29)

3.3 Parameters estimate

We refer Pitt and Shephard (1999) to calculate the parameter values and the log-likelihood of three models through the Markov chain Monte Carlo (MCMC) algorithm. We set the same initial value, the standard deviation, the same upper bounds and lower bounds of three models. Considering that the prior distribution is normal, we randomly select a value and calculate the log-likelihood to determine whether to replace the current log-likelihood. In order to make the calculation results more accurate, we let the parameters iterate 1,000 times and record the results of the convergence.

After 1,000 times iteration, We average all of the iterations as the best parameters, calculating the standard deviation of three cases, and use these parameter value to calculate the log-likelihood of three cases respectively. Table 3 reports the results of the estimation and the log-


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likelihood values. Comparing with Model 2, we can find that the Model 1 has a higher log- likelihood value. That is, the model which has fewer parameters is better. However, if the jump term is added, the estimated result of Model 3 will be better than the Model 1.

3.4 Model selection

If parameters are added to a model, the new model will have a log-likelihood at least as great since the former model is a special case of the new model. To test whether it is worthwhile adding new parameters, we first use the likelihood ratio test that twice the increase in the log- likelihood is chi-square with degrees of freedom equal to the number of additional parameters.

The number of degrees of freedom for the likelihood ratio test is the number of free parameters in the alternative model, the model of the alternative hypothesis, minus the number of free parameters in the base model, the model of the null hypothesis. Then the likelihood ratio test accepts the alternative model if the log-likelihood of it (lnL1) exceeds the log-likelihood of the base model (lnL0) by one-half of the appropriate chi-square percentile (1minus the significance level of the test) at the number of degrees of freedom for the test. In other words, the alternative model is accepted if 2ln(L1/L0)>c, where Pr(X>c)=αfor Xa chi-square random variable with the number of degrees of freedom for the test. With 10% significance, the critical values of chi-square at 2, 3, and 5 degrees of freedom are 4.61, 6.25, and 9.24 respectively. Dividing by 2, we get 2.30, 3.13, and 4.62, which are the thresholds required to add additional parameters. The Model 2 increases the log-likelihood by 64, thus meeting the threshold; it is preferred to the Model 1. Model 3 has log-likelihoods 65 higher than the Model 2 that meet the threshold. Thus the Model 3 is selected. Log-likelihood increases as more parameters are added to a model, but parameters shouldn’t be added to a model unless they increase log-likelihood significantly. To make log-likelihoods of models with different numbers of parameters comparable, we penalize the log-likelihood for parameters. Two penalized log-likelihood methods are commonly used:

Akaike Information Criterion (AIC) estimates the relative amount of information lost by a given model: the less information a model loses, the higher the quality of that model; the model with the lowest AIC is preferred. The penalty does not vary with the number of observations. Let kbe the number of estimated parameters in the model. Let Lbe the maximum value of the likelihood function for the model. Then

AIC = 2𝑘 − 2 ln(𝐿) (30)

Bayesian Information Criterion(BIC) considers larger penalty term than AIC; the model with the lowest BIC is preferred. Let nbe the number of observations. Then


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BIC = ln(𝑛) 𝑘 − 2 ln(𝐿) (31)

The AIC and BIC calculation results for the three models are listed at the bottom of Table 3. In our cases, kis 11, 14, 16 for Model 1, Model 2, Model 3 respectively and nis 9758. We can find that the Akaike Information Criterion and the Bayesian Information Criterion both select the Model 3, which has the lowest value of the AIC and the BIC.


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Table 3: Estimates on the Variance Dynamics and Jump Components

This table reports parameters estimates for the three models defined in Section 2.2 by the maximum likelihood estimates using RLI Corp. stock price returns. The sample period is from March 1980 to November 2018. The standard errors are reported in the parentheses.


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4 Numerical Experiments 4.1 CatEPut option price in three cases

This section evaluates CatEPut option price in three cases which listed in subsection 2.2. In order to save the calculation time and close to reality, because the catastrophe loss is calculated on an annual basis, we use the annual stock data instead of the daily stock data to calculate the CatEPut options price. We assume that the maturity of CatEPut option for this paper is five years. The risk-free rate is assumed to be 5%, and the market share of RLI Corp. is fixed at 0.16%. The parameter αis assumed to be 0.01, which represents the percentage drop in the share value price per unit of loss. In practice, we assume that ℒ» is 0.0001, the upper bound of the integral is 100, and the upper bound of the summation is 10 in the CatEPut option pricing formula (17).

Figure 4: CatEPut option price with RLI Corp. stock price return in different models

The results are shown in Figure 4. As seen, CatEPut option price shows a negative correlation with RLI Corp. stock prices all three cases, much like a regular put option. Although all of the models can reflect the stock price trend, the CatEPut option prices calculated by Model 1 is different from the others. Unlike Model 2 and Model 3, the whole price calculated by Model 1 is lower than 10. We can compare the models by calculating the pricing errors in the following subsection.


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4.2 Pricing errors

After calculating the price of CatEPut option, we can use the method following by Chang et al. (2011) to compare our models. We use the method provided by Chang, et al. (2018) to obtain the real value (PR) of CatEPut option price using the frequency and loss severity data of hurricane events available from the U.S. National Hurricane Center. The theoretical value (PT) of CatEPut option price under the Model 1, Model 2, and Model 3 are calculated by using parameter values generated from the hurricane data and other values mentioned above.

For comparison purposes, we compute three measurements of pricing errors: average percentage error (APE), average absolute error (AAE), and relative measure square error (RMSE):

APE =

1𝐸(𝑃R)

F|𝑃R − 𝑃T|

𝑁

Þ

IK)

AAE = F|𝑃R − 𝑃T|

𝑁

Þ

IK)

RMSE = àF(𝑃R − 𝑃T)S

𝑁

Þ

IK)

(32)

(33)

(34)

where E(PR) is mean of the real CatEPut option price value and N is the total number of observations. In pursuit of accuracy, we calculate six prices which discount every year from the maturity date of each CatEPut option from 1980 to 2005. According to our pricing formula (17), the term t0is written date, Tis 5, and t are 0, 1, 2, ..., 5. Then we could calculate 156 theoretical CatEPut option prices and 156 real CatEPut option prices respectively to measure the pricing error.

Table 4 gives a conspectus of these three measurements of pricing errors. It indicates that the pricing errors under the Model 3 are all smaller than those under the Model 1 and Model 2 in all three measurements. Taking RMSE as an example, the improvement rate of pricing errors using the Model 2 over the Model 1 is only 5.89%. The improvement rate rises to 7.32% if we use the Model 3. Therefore, when hurricane activities are the underlying catastrophe events, our results show that the Model 3 can reduce the pricing errors by 3.01%–7.32% depending on the different measurement methods.


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Table 4: Pricing Errors and Improvement Rate

This table use RLI Corp. annual stock price and catastrophe losses data from 1980 to 2005. The parameter Nfrom (32)–(34) is 156. The parameter values for base valuation are K = 100, T = 5, α = 0.01, ℒ» = 0.0001, r = 0.05, market share is 0.16%. Panel A lists three cases calculated by three measurements of pricing errors (32)–(34). Panel B lists improvement rate of pricing errors from Model 1. For example, the improvement rate of Model 2 from Model 1 calculated by APE is (0.9503 − 0.9284)/0.9503 = 2.3045%.

5 Conclusion Because of the increasing frequency of the catastrophe event, there are more and more scholars exploring the hedging method likes catastrophe equity put contract. Many insurance companies, such as RLI Corp., Horace Mann Educators Corp., LaSalle Re Ltd., Intrepid Re Ltd., and Trenwick Group Ltd., have also issued CatEPut contracts since 1996.

We have extended the analysis of literature to pricing the contract of CatEPut by introducing the variance dynamics of the normal and conditional GARCH-jump component. Through the framework of a conditional jump intensity model, we illustrate how catastrophic losses and conditional jump component, together with stock prices, affect option prices. Consequently, we successfully obtained a closed-form formula for the price of the CatEPut option.

Through numerical experiments and simulations, we demonstrated that both the variance of the loss process and conditional GARCH-jump intensity play a significant role. The results show that the Model 3, which includes conditional GARCH-jump process with jump term, generates lower pricing errors than the others and pricing errors can be reduced by 3.01%–7.32% depending on the different measurement methods. That is, we can use the formula of CatEPut option in this study to pricing the contract with reflecting stock price movement and catastrophe loss well. Finally, the model developed here can be applied to other structured risk management products, double-trigger stop-loss products for example.


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