Risk estimation model on epidemic
outbreaks for an insurer

Prepared by Naoyuki Ishimuraco-authored with Daniel Komadel and Yasukazu Yoshizawa

Presented to the ASTIN COLLOQUIUMLISBOA 201631 MAY – 03 JUNE 2016

This paper has been prepared for the ASTIN COLLOQUIUM LISBOA 2016
IAP wishes it to be understood that opinions put forward herein are not necessarily those of the IAP is not responsible,
the author is the only responsible for those opinions.

<Naoyuki Ishimura, Daniel Komadel, Yasukazu Yoshizawa>

IAP will ensure that all reproductions of the paper acknowledge the author(s)and include the above copyright statement.

IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/

Risk estimation model on epidemic outbreaks for an insurer
Naoyuki Ishimura (Chuo University, Tokyo, Japan)
Daniel Komadel (University of Economics in Bratislava, Slovakia)
Yasukazu Yoshizawa (Tokio Marine and Nichido Fire Insurance Co. Ltd., Tokyo, Japan)
Abstract. Epidemic outbreaks, which have been observed in recent years, have caused a big impact on modern
society once if they are occurred. We recall, for examples, SARS (Severe Acute Respiratory Syndrome), Ebola
virus diseases, MERS (Middle East Respiratory Syndrome), dengue and/or Zika fever and so on. Insurancecompanies have a potential to mitigate the effect of such events. Here we propose a simple stochastic model for
an insurer to estimate the risk of epidemic outbreaks.
The construction of our model is based on several apparently separate aspects. The first one is a roughestimate on ultimate number of removals once the epidemic burst takes place. This is given through thethreshold theorem, which is originally due to Kermack and McKendrick. The second point is the onset of
epidemic outbreaks, which is modelled by the doubly stochastic Poisson process. This part is motivated by thecatastrophe options as risk transfer instruments. Finally, the market risk is involved to derive a financial product.
Key words: Epidemic outbreaks, Kermack-McKendrick model, Threshold theorem, Doubly stochastic Poissonprocesses, Catastrophe options, Risk estimation model for an insurer

I. INTRODUCTION
It has been observed in recent years that epidemic outbreaks, such as SARS (Severe AcuteRespiratory Syndrome), Ebola virus diseases, MERS (Middle East Respiratory Syndrome),dengue and/or Zika fever, and so on, sometimes made a large impact on the modern society.
These phenomena are challenging issues for whole world.
Apart from the scientific research, the function of insurance companies will be important inorder to mitigate and stabilize the effect due to epidemic bursts. The overall estimate of therisk from these tragic events is then necessarily investigated.
In this article, we introduce a simple stochastic model to estimate such risk of epidemic
outbreaks for an insurer.
Our strategy is a combination of several apparently independent ingredients. The first
aspect is of course a mathematical modeling of diseases, which has been studied over 300
years and much progress has been made so far. Here we just employ the classicaldeterministic model of Kermack and McKendrick, one of whose important outcomes is theso-called threshold theorem; we can estimate a rough number of ultimate removed
individuals from the population suffering from an infectious disease.
The second point is the onset of epidemic outbreaks, which is regarded as a trigger variable
and is described by a stochastic process. From the financial viewpoint of an insurer, the
IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/phenomenon of epidemic bursts is of a catastrophic nature because it may bring enormous
financial expenditures within the short period of time. Such a feature makes it similar to thecatastrophic events such as a huge earthquake, a big typhoon, and so on. We utilize a doubly
stochastic Poisson process to model the onset.
Finally, in addition to above, the point to be involved is the market risk. Once an epidemic
outbreak takes place, a depression will follow and we should be concerned with thedepreciation of securities. We simply use a standard Black-Scholes-Merton type model toevaluate the depreciation process. Combining all together, we are then able to estimate therisk of the considered epidemics bursts by taking the expectation.
The present paper is a slight extension as well as modification with necessary corrections ofour previous work [7] together with establishments made in [8]. The organization is asfollows: In Section II, preliminary issues of our classical epidemic model as well as
stochastic process are recalled. Our model for the estimation of risk is explained in Section III.
Section IV is devoted to the proof and the computation of risk. Section V concludes withdiscussion.
II. PRELIMINARY
Here we recall several known basic concepts for the readers' convenience.
A. Structured risk management productsCompared to the traditional approach to risk management, the emerging approach has beendeveloped and applied extensively since 1980s. The former approach is based onstandardized insurance products and treated on a case-by-case basis. The latter approach, on
the other hand, combines insurance risks and financial risks and is handled on a long termrisk management program, which involves multiple long term risks and is implemented by
the so-called structured risk management products. Examples of transactions include dual
trigger instruments, some catastrophe options, and so on.
Our proposed model for estimating the risk for an insurer of epidemic outbreaks is on theline of this concept of structured risk management products; we observe common
characteristics to the catastrophe options. The writer of these products may expect higher
return if no epidemics bursts takes place. For further details and other information, we refer
for instance to a nice paper by Cox, Fairchild, and Pedersen [1].
B. Epidemic ModelIn the present article, for our epidemic modeling, we simply employ the classicaldeterministic Kermack-McKendric model, since it gives a benchmark result for later
sophisticated researches. For background issues and other details, see for instance, nicemonographs by Dalley and Gani [3], Murray [11].
IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/

Let 𝑥(𝑡) denote the number of susceptibles, 𝑦(𝑡) infectives, and 𝑧(𝑡) removals, which arecounted according to the disease status. We note that removals mean dead, isolated, or
immune individuals. The total size of the population 𝑁 = 𝑥 (𝑡) + 𝑦(𝑡) + 𝑧(𝑡) is assumed andturned out to be fixed for all 𝑡 ≥ 0.
The system of ordinary differential equations due to Kermack and Mckendrick is thenexpresses as follows:𝑑𝑥(𝑡)= −𝛽𝑥(𝑡)𝑦(𝑡)
(1)
𝑑𝑡
𝑑𝑦(𝑡)
= 𝛽𝑥 (𝑡)𝑦(𝑡) − 𝛾𝑦(𝑡)
(2)
𝑑𝑡
𝑑𝑧(𝑡)
= 𝛾𝑦(𝑡).
(3)
𝑑𝑡
The equations is analysed under the initial conditions (𝑥(0), 𝑦(0), 𝑧(0)) = (𝑥0 , 𝑦0 , 0) with
𝑥0 + 𝑦0 = 𝑁. Here, 𝛽 denotes the infection parameter representing the strength of epidemics,
and 𝛾 is the removal parameter indicating the rate of infectives becoming immune. We define
the critical parameter 𝜌 = 𝛾/𝛽 as the relative removal rate. It is easy to see that 𝑥(𝑡) ismonotone non-increasing, and 𝑧(𝑡) is non-decreasing. If 𝑥0 ≤ 𝜌 , then 𝑦(𝑡) is monotone
decreasing for all 𝑡 > 0.
Now the well-known Kermack-Mckendrick threshold theorem is summarized as follows,whose way of formulation is essentially taken from [3].
Theorem (Kermack and Mckendrick). (i) Let 𝑥∞ = lim 𝑥(𝑡) and 𝑧∞ = lim 𝑧(𝑡). Then, when𝑡→∞infection ultimately ceases spreading, it follows that

𝑡→∞

𝑧
− ∞

𝑁 − 𝑧∞ = 𝑥0 + 𝑦0 − 𝑧∞ = 𝑥0 𝑒 𝜌 ,where 𝑦0 < 𝑧∞ < 𝑥0 + 𝑦0 = 𝑁.
(ii) A major outbreak occurs if and only if 𝑥0 > 𝜌.
(iii) If 𝑥0 = 𝜌 + 𝜈 with small 𝜈 > 0 and 𝑦0 is small relative to 𝜈, then the total number ofsusceptibles left in the population are approximately 𝜌 − 𝜈, and we have𝜌𝑧∞ ≈ 2𝜈≈ 2𝜈 .
(4)
𝜌+𝜈Here we just exhibit a sketch of proof for the readers’ convenience. First we note that𝑑(𝑥(𝑡) + 𝑦(𝑡) + 𝑧(𝑡)) = 0 ,𝑑𝑡
and hence𝑥 ( 𝑡 ) + 𝑦 (𝑡 ) + 𝑧 (𝑡 ) = 𝑁
IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/

for all 𝑡 ≥ 0.
Next we divide the equation (1) by (3) to obtain𝑑𝑥
𝛽1=− 𝑥=− 𝑥,𝑑𝑧𝛾𝜌where 𝜌 = 𝛾/𝛽 denotes the relative removal rate. Integration directly gives
𝑥 (𝑡) = 𝑥0 𝑒

𝑧(𝑡)
−
𝜌 .

In a similar way we find that𝑑𝑦𝜌
= −1 + ,𝑑𝑥
𝑥and so that𝑥 (𝑡) + 𝑦(𝑡) − 𝜌 log 𝑥 (𝑡) = 𝑥0 + 𝑦0 − 𝜌 log 𝑥0 .
Within the region considered where 𝑥 , 𝑦 , 𝑧 are positive, we easily deduce that 𝑦∞ =lim 𝑦(𝑡) = 0, which implies that the identity of (i) holds. The properties (ii) and (iii) of the𝑡→∞theorem will be deduced by a direct and an approximation argument. We may safely omit thedetails.
It is known that a detailed analysis on (1)(2)(3) is possible. Indeed, Kendall [9] has
succeeded in an exact handling and has deduced that 𝑧∞ = 𝜁, where 𝜁 is a positive root of−𝑧

𝑁 − 𝑧 − (𝜌 + 𝜈 )𝑒 𝜌 = 0(5)

See also § 2.3 of [3].
We will utilize its outcomes to estimate the risk of epidemics.
C. Doubly Stochastic Poisson Process
The onset of epidemic bursts is regarded as a trigger process; as a general model for suchrare event, we here employ the so-called doubly stochastic Poisson process or a Cox process.
We refer to Lando [10] or a book of Rolski, Schmidli, Schmidt, and Teugels [12] for otherbackground issues.
The reason why we prefer a doubly-Poisson process to a usual homogeneous Poissonprocess is that the latter has deterministic intensity and hence it is rather inappropriate formodelling the resulting claims for epidemics outbreaks. On the other hand, a doubly
stochastic Poisson process or a Cox process are known to provide flexibility in modelling,
since the intensity process is allowed to be stochastic.
Now we turn our attention to our process. Let 𝛬 = {𝜆(𝑡)}𝑡≥0 be an intensity process;namely, a nonnegative, measurable, and locally integrable stochastic process. A counting
process {𝑁(𝑡; 𝛬)}𝑡≥0 is called a Cox process or a doubly stochastic Poisson process with
IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/

intensity 𝛬 if for each sequence {𝑘𝑗 }𝑗=1,2,3,⋯,𝑛 of nonnegative integers, and for 0 < 𝑡1 ≤ 𝑠1 ≤𝑡2 ≤ 𝑠2 ≤ ⋯ ≤ 𝑡𝑛−1 ≤ 𝑠𝑛−1 ≤ 𝑡𝑛 ≤ 𝑠𝑛 , there holds
𝑛

𝑛

𝑗=1

𝑗=1

𝑘𝑗

𝑠𝑗
𝑠𝑗
1
𝑃 (⋂{𝑁(𝑠𝑗 ; 𝛬) − 𝑁(𝑡𝑗 ; 𝛬) = 𝑘𝑗 }) = ∏ 𝐸 [ (∫ 𝜆(𝑢)𝑑𝑢) exp (− ∫ 𝜆(𝑢)𝑑𝑢)] .
𝑘𝑗 ! 𝑡𝑗
𝑡𝑗

One typical example, which is favorably used to measure the effect of catastrophic events,
and therefore suitable to epidemic bursts also, is the shot noise process. See for example
Dassios and Jang [4]. However, we will not treat this process in the sequel.
The next section is devoted to the introduction of a model for estimating the risk concerning
epidemic outbreaks, which is performed by the use of a doubly Poisson process or a Cox
process.
III. MODEL
Now we formulate a model to estimate the risk due to epidemic outbreaks for an insurer.
Main ingredients are as follows.
(1) The onset of epidemic outbreaks is monitored by a doubly stochastic Poisson process
𝑁(𝑡) = 𝑁(𝑡; 𝛬) with intensity 𝛬, which should be combined with the epidemic model; forexample in the Kermack-Mckendrick model, the term like 1{𝑁(𝑇)>𝜌/𝜀} is included. Here 𝜌 > 0means the threshold value in the threshold theorem, 1/𝜀 is the magnification parameter withsmall 𝜀 > 0, and 1𝐵 stands for the indicator function; that is,1𝜔 ∈𝐵
1𝐵 (𝜔) = {.
0𝜔 ∉𝐵(2) Once the epidemic burst takes place, the total number of removals is presumed by thethreshold theorem; with the original Kermack-McKndrick approximation, we have 𝑧∞ ≈ 2𝜈,and with the Kendall's analysis, 𝑧∞ = 𝜁 , where 𝜁 is a positive root of (5).
(3) The occurrence of epidemic outbreaks will result in the market disturbance. Indeed, it wasobserved that South Korea economy experienced a slight depression after the occurrence of
MERS. Here the market model we consider for the security {𝑆(𝑡)}𝑡≥0 is a standard BlackScholes-Merton type, which is expressed as follows:
𝜎2
𝑆(𝑡) = 𝑆0 exp [−𝛼𝑁(𝑡) + (𝜇 − ) 𝑡 + 𝜎𝑊(𝑡)] ,2
where 𝑆0 > 0, α > 0, μ > 0 and the volatility 𝜎 are given positive constants. The process
𝑆(𝑡) is accompanied by the bond process 𝐵(𝑡) = 𝑒 𝑟𝑡 with constant interest rate 𝑟. This is justa well-known Black-Scholes-Merton model. 𝑁 (𝑡) = 𝑁(𝑡; 𝛬) is a doubly stochastic Poissonprocess as above and 𝑊(𝑡) denotes the standard one-dimensional Brownian motion, which is
IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/independent of 𝑁(𝑡). The assumption of this independence is acceptable in view of theirrelevant occurrence of epidemic bursts.
Furthermore, under this circumstance, insurance companies keep a policy for the insured.
For example, if the stop-loss policy is held, the term max{𝑆(𝑇) − 𝐾, 0} is included, where 𝐾
denotes a retention level and 𝑇 > 0 is the time of maturity, which is interpreted as the period
of insurance. We note that although 𝑇 does not directly correspond to the ultimate time when
infection ceases spreading, we may consider that 𝑇 is related to this critical time from theviewpoint of estimating the risk.
In summary, our proposed model for the considered risk 𝑅(𝑡) (0 < 𝑡 ≤ 𝑇) is described as
the discounted expected value of the combination of above all risk factors. To be precise, we
have, in view of (4)𝑅(𝑡): = 𝐸𝑄 [𝑒 −𝑟(𝑇−𝑡) 1{𝑁(𝑇)>𝜌 } ∙ 2𝜈 max{𝑆(𝑇) − 𝐾, 0}] ,(6)
𝜀for the Kermack-McKendrick original approximation, and, with 𝜁 being a positive root of (5),𝑅(𝑡): = 𝐸𝑄 [𝑒 −𝑟(𝑇−𝑡) 1{𝑁(𝑇)>𝜌/𝜀} ∙ 𝜁max{𝑆(𝑇) − 𝐾, 0}] ,(7)
for the analysis by Kendall. We remark that the difference is just that 2𝜈 in (6) is replaced by
𝜁 in (7). In both cases, the calculation of the expectation is implemented under the riskneutral probability measure 𝑄 defined below in the next section.
Now our main analytical estimate of this article is read as follows.
Theorem. We infer that the risk 𝑅(𝑡) for an insurer can be expressed as∞

𝑅(𝑡) = ∑ 2𝜈( 𝑆(𝑡)𝑒 −𝛼𝑙+log(𝑀(𝛬(𝑇)−𝛬(𝑡))(𝑘))

𝛷(𝑑𝑙 + 𝜎√𝑇 − 𝑡) − 𝐾𝑒 −𝑟(𝑇−𝑡) 𝛷(𝑑𝑙 ))

𝑙=𝜈+1
𝑙

(𝛬(𝑇) − 𝛬(𝑡)) −𝑙(𝛬(𝑇)−𝛬(𝑡))
],
∙𝐸[
𝑒
𝑙!
for (6), and for (7), 2𝜈 in 𝑅(𝑡) above is replaced by 𝜁 with 𝜁 being a positive root of (5). Here
we have put
𝑆(𝑡)log ( 𝐾 ) + 𝑟(𝑇 − 𝑡) − 𝛼𝑙 + log(𝑀(𝛬(𝑇)−𝛬(𝑡)) (𝑘 )) 1𝑑𝑙 =− 𝜎√𝑇 − 𝑡 .
(8)2
𝜎 √𝑇 − 𝑡with 𝑘 ≔ 1 − 𝑒 −𝛼 and 𝛷 (𝑑 ) denotes the cumulative distribution function for the standardized
normal distribution𝑑𝑥2
1∫ 𝑒 − 2 𝑑𝑥 .
𝛷 (𝑑 ) =√2𝜋 −∞
IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/

In the next section, we compute 𝑅(𝑡) in the spirit of no arbitrage principle. In this respect,
we refer to Cummins and Geman [2].
IV. PROOF OF THEOREM
A. Key Lemma
In order to estimate the risk above, the next lemma is useful, which is taken from [5]. See
also [1].
Lemma. Let {𝑁(𝑡) = 𝑁(𝑡; 𝛬)}𝑡≥0 be a doubly stochastic Poisson process with intensity 𝛬 ={𝜆(𝑡)}𝑡≥0 and 𝑀𝛬(𝑡) (𝑘 ) denotes the moment generating function of the aggregated process𝑡𝛬(𝑡) ≔ ∫0 𝜆(𝑠)𝑑𝑠; namely𝑀𝛬(𝑡) (𝑘 ) = 𝐸[exp(𝑘𝛬(𝑡))] .
Then, for 𝑘 ≔ 1 − 𝑒−𝛼, the process{exp{−𝛼𝑁(𝑡) + log(𝑀𝛬(𝑡) (𝑘 ))}}

𝑡≥0

gives a martingale with respect to the minimum augmented filtration ℱ𝑡𝛬 of𝜎{𝑁(𝑠) | 0 ≤ 𝑠 ≤ 𝑡} .
Sketch of proof. We wish to show that for 0 ≤ 𝑠 ≤ 𝑡
𝐸 [exp {−𝛼𝑁(𝑡) + log (𝑀𝛬(𝑡) (𝑘 ))} | ℱ𝑠𝛬 ] = exp{−𝛼𝑁(𝑠) + log(𝑀𝛬(𝑠) (𝑘 ))} .
To ensure this, we firstly learn that∞𝐸[𝑒−𝛼𝑁(𝑡) ]= 𝐸 [∑ 𝑒∞−𝛼𝑙

𝑃 (𝑁(𝑡) = 𝑙 )] = 𝐸 [∑ 𝑒 −𝛼𝑙

𝑙=0

𝑙=0

𝛬(𝑡)𝑙 −𝛬𝑒 𝑡]𝑙!

= 𝐸 [exp(𝛬(𝑡)(𝑒 −𝛼 − 1))] .
We then compute𝐸 [exp {−𝛼𝑁(𝑡) + log (𝑀𝛬(𝑡) (𝑘 ))} | ℱ𝑠𝛬 ]= 𝐸 [exp {−𝛼(𝑁(𝑡) − 𝑁(𝑠)) + log (𝑀(𝛬(𝑡)−𝛬(𝑠)) (𝑘 ))} | ℱ𝑠𝛬 ]∙ exp {−𝛼𝑁(𝑠) + log (𝑀𝛬(𝑠) (𝑘 ))}= 𝐸[exp{−𝛼(𝑁(𝑡) − 𝑁 (𝑠))} | ℱ𝑠𝛬 ]𝑀(𝛬(𝑡)−𝛬(𝑠)) (𝑘 ) ∙ exp {−𝛼𝑁(𝑠) + log (𝑀𝛬(𝑠) (𝑘 ))}= 𝑀(𝛬(𝑡)−𝛬(𝑠)) (−𝑘 )𝑀(𝛬(𝑡)−𝛬(𝑠)) (𝑘 ) ∙ exp {−𝛼𝑁(𝑠) + log (𝑀𝛬(𝑠) (𝑘 ))}= exp{−𝛼𝑁(𝑠) + log(𝑀𝛬(𝑠) (𝑘 ))} .
This is what we wanted to know and the proof is completed.
IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/

B. Pricing Formula
Now we turn our attention to evaluate (6) and (7). The case (6) is treated. First we learn
that 𝑄-mesaure is defined by(𝜇 − 𝑟)𝑡 + log (𝑀𝛬(𝑡) (𝑘 ))𝑊 𝑄 ( 𝑡 ) ≔ 𝑊 𝑃 (𝑡 ) +,𝜎where the notation 𝑊 𝑃 (𝑡) ≔ 𝑊(𝑡) is just used for the sake of avoiding a confusion. The
Radon-Nikodym derivative process is
2𝑡𝑑𝑄
1 𝑡 𝜇 − 𝑟 + 𝛾𝑘 (𝑠)𝜇 − 𝑟 + 𝛾𝑘 (𝑠)= exp {− ∫ () 𝑑𝑠 − ∫𝑑𝑊 𝑃 (𝑠)} .
𝑑𝑃
2 0
𝜎
𝜎
0
Here 𝛾𝑘 (𝑡): = 𝐸 [𝜆(𝑡)exp(𝑘𝛬(𝑡))]/𝑀𝛬(𝑡) (𝑘 ).
Consequently. in view of the key lemma, we see that the process 𝑒 −𝑟𝑡 𝑆(𝑡) defined by1𝑒 −𝑟𝑡 𝑆(𝑡) = 𝑆0 exp[−𝛼𝑁(𝑡) + log(𝑀𝛬(𝑡) (𝑘 ))] ∙ exp [𝜎𝑊 𝑄 (𝑡) − 𝜎 2 𝑡]
2gives a 𝑄-martingale. Thus we are able to compute the discounted value of (6) under 𝑄measure. See a relevant discussion in [5].
V. DISCUSSIONSThe phenomena of epidemic outbreaks, once it takes place, brings a huge confusion to thestability of modern society. We recall an example of MERS, which occurred in South Korea
in the spring of 2015 and has resulted in a severe damage in economy, public activity, social
mind, and other aspects. The risk management on this issue, both from natural and social
scientific standpoint, is certainly indispensable. This attempt is nothing but the structured risk
management product.
In this paper, we have introduced and formulated a simple stochastic model for an insurer
to estimate the risk which originates from epidemic bursts. The onset of pandemic ismodelled by a doubly stochastic Poisson process or a Cox process. After the trigger, theongoing of epidemics follows the story described by the Kermack-Mckendrick thresholdtheorem. We then estimate the risk of an insurer subject to above mentioned risk factors. The
method is based on a similar argument as employed in the case of catastrophe options. The
closed form solution is obtained.
As is well known, one of important functions of insurance includes the mitigation of tragic
events. To conduct this role, however, quantitative estimate on the risk is necessary and forsuch purposes an effective mathematical model is surely needed. Today, so many models forthe spread of infectious diseases have been analyzed and applied; it may be possible to utilize
some of them for further studies. Concerning various epidemics models, we refer for instance
IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/to a review by Hethcote [6].
As to applications of our model to real world, we remark that various hedging parameters,
which are needed for considering hedging strategies, can be computed within our proposedmodel. Indeed, the next proposition is computed. We treat only the case of (6) for simplicity.
Proposition. Let the risk 𝑅(𝑡) for an insurer be given by (6). Then the Delta, Gamma, Rho,
and Vega of 𝑅(𝑡) are computed as the next expressions, respectively.
∞−𝛼𝑙+log(𝑀(𝛬(𝑇)−𝛬(𝑡)) (𝑘))𝜕𝑅(𝑡)∆ (𝑡 ) == ∑ 2𝜈 𝑒𝛷(𝑑𝑙 + 𝜎√𝑇 − 𝑡)
𝜕𝑆
𝑙=𝜈+1

𝑙

(𝛬(𝑇) − 𝛬(𝑡)) −𝑙(𝛬(𝑇)−𝛬(𝑡))
],
∙𝐸[
𝑒
𝑙!
𝜕 2 𝑅 (𝑡 )𝛤(𝑡) =
𝜕𝑆 2∞= ∑ 2𝜈
𝑙=𝜈+1𝑒−𝛼𝑙+log(𝑀(𝛬(𝑇)−𝛬(𝑡)) (𝑘))𝑆(𝑡)𝜎√𝑇 − 𝑡𝜑(𝑑𝑙 + 𝜎√𝑇 − 𝑡)𝑙(𝛬(𝑇) − 𝛬(𝑡)) −𝑙(𝛬(𝑇)−𝛬(𝑡))
],
∙𝐸[
𝑒
𝑙!
∞

𝑙

(𝛬(𝑇) − 𝛬(𝑡)) −𝑙(𝛬(𝑇)−𝛬(𝑡))𝜕𝑅(𝑡)],𝜌 (𝑡 ) == ∑ 2𝜈 𝐾(𝑇 − 𝑡)𝑒 −𝑟(𝑇−𝑡) 𝛷(𝑑𝑙 ) ∙ 𝐸 [
𝑒
𝜕𝑟
𝑙!
𝑙=𝜈+1
∞

−𝛼𝑙+log(𝑀(𝛬(𝑇)−𝛬(𝑡))(𝑘))𝜕𝑅(𝑡)𝜈 (𝑡 ) == ∑ 2𝜈 𝑆(𝑡)√𝑇 − 𝑡𝑒𝜑(𝑑𝑙 + 𝜎√𝑇 − 𝑡)
𝜕𝜎
𝑙=𝜈+1

𝑙

(𝛬(𝑇) − 𝛬(𝑡)) −𝑙(𝛬(𝑇)−𝛬(𝑡))
],
∙𝐸[
𝑒
𝑙!
where 𝑑𝑙 is defined by (8) and 𝜑 = 𝛷′.
Despite its importance, the quantitative research on an insurer against epidemic outbreaks,
to the authors' knowledge, is not so common in the literature. We believe that mathematicalstudies on this subject should be pursued much further. We do hope that our study will put
something forward on this line. It is to be noted that some empirical investigations should be
IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/

undertaken, which will be our future project of researches.
Many questions are still open to further investigations. One of important directions is thatthe character of the pattern of outbreaks should be taken into account. Indeed, the number ofremovals may be different from each epidemics; one is modeled by double peak graphs andthe other by one peak graph with the tail of dumped oscillations. We will conduct a better
management if such differences are involved in the models. The other is to discuss certain
practical applications combined with real data and numerical computation; the evaluation of
our model may be performed. These will be our another theme for continuing researches.
ACKNOWLEDGMENT
We are grateful to Professor Takahiko Fujita for helpful discussions. The first author ispartially supported by Grant-in-Aid for Scientific Research (C) (Kakenhi) No. 15K04992
from Japan Society for the Promotion of Science (JSPS).
REFERENCES[1] S.H. Cox, J.R. Fairchild, and H.W. Padersen, “Valuation of structured risk management products,”
Insurance: Mathematics and Economics, vol. 34, pp. 46-57, 2004.
[2] J.D. Cummins and H. Geman, “Pricing catastrophe insurance futures and call spreads: An arbitrage
approach,” Journal of Fixed Income, vol. 4, pp. 46-57, 1995.
[3] D.J. Dalley and J. Gani, Epidemic Modelling, Cambridge University Press, 1999, ch. 2.
[4] A. Dassios and J.-W. Jang, “Pricing of catastrophe reinsurance and derivatives using the Cox process withshot noise intensity,” Finance and Stochastics, vol. 7, pp. 73-95, 2003.
[5] T. Fujita, N. Ishimura, and D. Tanaka, “An arbitrage approach to the pricing of catastrophe options
involving the Cox process,” Hitotsubashi Journal of Economics, vol. 49, pp. 67-74, 2008.
[6] H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2001), 599{653.
[7] C. Ishii, N. Ishimura and T. Fujita, Estimate on the risk of epidemic outbreaks for an insurer: A simple
stochastic model, International J. Model. Optim., 5 (2015), 161-165.
[8] C. Ishii, On the risk of an insurer due to epidemic outbreaks (Densenbyo ga hassei sitasai no hoken-gaisyano shiharai risk ni tuite), Thesis for the Master Course Degree, Graduate School of Economics, Hitotsubashi
University, March, 2015 (in Japanese).
[9] D.G. Kendall, “Deterministic and stochastic epidemics in closed populations,” in “Proceedings of the Third
Berkeley Symposium on Mathematical Statistics and Probability,” vol. 4, Ed. J. Neyman, University ofCalifornia Press, Berkeley, pp. 149-165, 1956.
[10] D. Lando, “On Cox processes and credit risky securities,” Review of Derivatives Research, vol. 2, pp. 99120, 1998.
[11] J.D. Murray, Mathematical Biology, I. Introduction, 3rd Edition, Springer, 2002.
[12] T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels, Stochastic Processes for Insurance and Finance,
Springer-Verlag, New York, 1998.

IAP: Instituto dos Atuários Portugueses
Campo Grande 28, 8 C 1700-093 Lisboa
Portugal
TEL: + 351 21 846 38 82
Email: astincolloquium2016@gmail.com
http://www.actuaries.org/lisbon2016/