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Weather Derivatives and their Applications in Hong Kong

YAO Li

A Thesis Submitted in Partial FulfiUments of the Requirements for the Degree of

Master of Philosophy in

Finance

© Chinese University of Hong Kong June, 2004

The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School.

(卜(统

2 ffl^iPjl)

Abstract

In this thesis we attempt to design a weather derivative risk management tool for the recreation industry in Hong Kong. After an analysis of the weather risks in Hong Kong, we form and develop a Markov model with transitive density for predicting rainfall amounts. This model was first proposed by Grunwald and Jones (2000) and is generally used for meteorological data with a mass cluster at zero. With some modifications the model can meet the requirements of different geographic locations and climates. We further elaborate the model and add another typhoon signal duration factor into the model to fit the subtropical climate in Hong Kong, The estimated parameter results for the rainfall amount model are used in further simulation process for a weather cap contract. The Monte-Carlo simulation gives an evaluation of the option (cap) contract we proposed. Further risk management analysis is proposed to analyze the relationship between the park attendance and the rainfall event. Due to the lack of historical visitor flow data, we discuss the methodology rather than the actual outcome. The derivative risk management tool introduced in this thesis can be used together with the analysis of the revenue stream and other risk-control mechanisms to hedge the weather risk in Hong Kong.

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摘要

在這篇論文中我們嘗試用一種衍生工具來對沖對香港的旅遊業造成不利影響的

天氣風險.經過對香港天氣風險的分析，我們設計了一種預測雨量的隨機變量

模型.Gmnwald和Jones在2000年首次將此類模型運用於雨量預測•這類模型

通常用於估値天氣數據’特別適用於大量樣本爲零的數據•稍作修改後，這個

模型也可適用於預測不同地理位置以及不同氣候的天氣數據•爲符合香港的亞

熱帶氣候及地域特色’我們在此模型中加入了台風信號的因素•模型的最大擬

然估計値可用於下一部分的天氣期權估値•使用蒙特卡羅模擬爲具體的雨量期

權估値’而進一步的風險管理分析可用於硏究主題公園的訪客量及每日雨量的

關係•然而，由於歷史訪客量數據不足，我們只能針對分析方法進行討論，使

用少量數據的實際分析結果可能並不理想•本文中提出的風險管理工具（雨量

期權）也可配合收益流分析及其他風險控制機制，從而更好的對沖香港的天氣

風險•

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Table of Contents

Page

Chapter 1 Weather Derivatives: A Review 1 1.1 Introduction 1 1.2 Types of weather risk 1 1.3 Key weather derivative elements 3 1.4 Methods for pricing weather derivatives 5 1.5 Current Situation in Hong Kong: the Recreation

Industry 8 Tables and Figures 10

Chapter 2 Markov Models with Application to Hong Kong's Rainfall 13

2.1 The Model 14 2.2 Maximum Likelihood Estimation 17

2.2.1 Estimates for Occurrence Model 18 2.2.2 Estimates for Intensity Model 23

2.3 Model for Amount 28 Tables and Figures 29

Chapter 3 Contract Specifications and Option Evaluation 42 3.1 The Contract 42 3.2 The Monte-Carlo Simulation 44

3.2.1 The Rainfall Event 45 3.2.2 The Aggregate Payoff 47 3.2.3 Some Simulation Results 48

3.3 Further Applications 49 Tables and Figures 55

Chapter 4 Concluding Remarks and Discussions 64

References 66

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Chapter 1 Weather Derivatives: A Review

1.1 Introduction

Contracts, where payments are determined by weather conditions, are known as

weather derivatives. Since the beginning of the 1970s, investors have realized that

some of the weather phenomena had significant effects on the risks and rewards in

the energy sector. Weather derivatives have started trading on the electronic platform

of the Chicago Mercantile Exchange (CME) on September 29’ 1999. Nowadays

these derivative instruments protect not only utility and energy sector, but

agriculture, construction, and the recreation industry, as well as insurance companies

and financial institutions. Although these financial products cannot undo adverse

weather conditions, they offer companies a good shelter from any economic loss

arising from the unfavorable weather conditions.

Ill this thesis a model for a proper underlying weather variable in Hong Kong is

established and estimated with historical weather data. A weather product with such

underlying weather event is designed and priced based on simulation. Further

applications of the weather product and hedging strategies are discussed in the end.

Weather derivatives are a relatively new risk management tool in Hong Kong. The

strategic use of weather products can have significant impact on the recreation

industry and help to stabilize the revenue stream of theme parks in Hong Kong.

1.2 Types of Weather Risk

Weather phenomena have significant effects on the value generation prospects of

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any economic activity. Different aspects of weather phenomena range from

temperature levels, humidity levels, precipitation levels to hurricanes and tornadoes.

Weather risk is the uncertainty of cash flow caused by such weather events. The

energy sector, e.g. heat or gas provider, and the recreation industry, e.g. theme parks

and recreational product makers, whose profits depend heavily on weather

conditions, are directly exposed to weather risks. As such, weather derivatives offer

these companies the chance to lessen the weather risk and ease the economic

consequences.

There are basically two types of weather risk, insurable weather risk and uninsurable

weather risk. Different approaches should be taken to mitigate different weather risk

exposures. Note that not all the business risks arising from adverse weather

conditions can be fully or even partially hedged or insured against.

The first type of weather risk, the insurable weather risk, includes mostly extreme

weather events, such as tornadoes, floods and hurricanes. Business losses arising

from these extreme weather events - such as a tornado shutting down power in a

certain district - cannot be hedged against using a weather derivative. But some

form of business interruption insurance and catastrophic insurance can be helpful in

these unpredictable situations. Although these extreme weather events are rare,

many companies have long purchased insurance policies to protect themselves

against large losses resulting from these meteorological events. In this situation a

company identifies the catastrophic weather events that have an impact on its

revenue stream and arranges catastrophic insurance coverage. The insurance

companies carefully evaluate the risk probability and set an appropriate premium.

The other type of weather risk is non-catastrophic, but it still has an impact on the

2

revenue generating prospect of a company. These weather risks talks about adverse

weather events such as severe and continuous precipitation, rainstorms or typhoons.

This type of risk mitigation seeks to provide protection against fluctuation in the

revenue streams deviating fi-om the norm and would employ a weather derivative as

hedge. It has only been within the past decade that derivatives have allowed

companies to hedge against weather that is not necessarily catastrophic, but which

could still devastate regular earnings. In this case it is important to note the effect of

weather events on the value generation prospects of any economic activity. The

impact of the weather event and the related economic activity would affect the

construction of a hedging portfolio involving weather derivatives.

In this paper, we focus on the application of weather derivatives on non-catastrophic

weather risks in Hong Kong, mostly affecting the recreation industry. These

derivative products can be highly customized to meet specific needs and the design

of such weather contracts will be further elaborated in Chapter 3.

1.3 Key Weather Derivative Elements

Weather derivatives are becoming increasingly common in industries whose profits

are adversely affected by weather and are one of the most rapidly growing sectors of

risk management. As one kind of derivative instrument, they share some attributes

with ordinary derivatives. As a weather product, they are also unique in design and

application. A typical weather option traded on Chicago Mercantile Exchange can

be used as an example to explain these elements. (See Table 1.1 and Table 1.2 at the

end of this chapter)

The construction of a weather contract requires the following elements.

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Weather variable the underlying asset for weather derivatives

Location an official weather station from which the meteorological record is observed

Contract period time span of the contract Contract type call or put Strike price predetermined price or measuring level of the underlying

variable to trigger exercise of an option Premium price paid for the option

The unique characters of weather variable determine the uniqueness of weather

derivatives. To be a properly defined underlying weather variable:

• The underlying product should be standardized and uniform.

Consistency of measurement is required. E.g. once Centigrade is used to

measure the temperature as a weather variable, Fahrenheit cannot be used

instead anytime in the future, unless the product is carefully re-designed for the

change.

• The underlying prices or measures should be widely and frequently

disseminated.

The authorized observatory should disseminate the weather information to the

public at a predetermined rate, daily, weekly or monthly.

Common examples of weather variables include rainfall amounts, sunshine hours,

snow depth, air temperature, wave height, wind speed, or a combination of these, if

appropriate.

The primary observables on which temperature derivative contracts are based and

traded in the market are cooling degree days (CDD) and heating degree days (HDD).

They are also the most actively traded weather derivatives in the market. In Table

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1.1 and Table 1.2，the heating degree days are used as the weather variable.

• Cooling degree day (CDD) is a summer measure of how hot it is on any given

day at a specific location.

CDD for a given day = Max [Average temperature for the day - 65 °F, 0]

• Heating degree day (HDD) is a winter measure of how cold it is on any given

day at a specific location.

HDD for a given day = Max [65 "F - Average temperature for the day, 0]

By calculating HDD and CDD we can also get a measure of the cumulative HDD

and CDD over different intervals of time. HDD and CDD serve as an important

measure of the revenue generating prospects of the US energy sector, e.g., a put

option (or a 'floor') based on HDD (or cumulative HDD) can be used to hedge low

revenue due to low energy consumption of heating in an exceptionally warm winter.

According to Bank (2002), further application of weather derivatives can be

achieved by a combination of the two basic contract types or by much more

complicated derivative products. As we mentioned above, weather contracts are

highly customizable. Pricing and designation of these derivatives can be very

complicated. In this thesis we focus on the design and evaluation of weather

contracts to meet the needs of recreational entertainment companies in Hong Kong.

1.4 Methods for pricing weather derivatives

Weather derivatives are classic examples of incomplete markets. As the underlying

weather variables are usually very illiquid and even not replicable, the standard

'risk-neutral' point of view is not applicable to evaluate the derivatives based on

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weather variables. Therefore, a direct application of the standard derivative pricing

theory, based on the no-arbitrage and market completeness assumptions, is

inadequate.

In addition, although weather derivatives share features with options and futures, the

structures are not identical. The statistical processes followed by temperatures or

rainfall amounts are quite different from those governing price movements. There

have been many previous works about the pricing of weather products. Figlewski

and Levich (2002) and German (1999) have proposed several pricing and simulation

methods for catastrophic bonds and weather instruments. Pricing of a weather

derivative for non-catastrophic weather risk is generally carried out following one of

the following procedures:

1. Utility optimization method

Pauline and Nicole (2002) determine the optimal structure of derivatives written on

an illiquid asset, such as a catastrophic or a weather event. The modeling for the

optimal design of such derivatives involves the definition of a choice criterion for

the different agents. For simplicity, the agents, the bank and the investor, are

assumed to be risk averse and to have an exponential utility criterion. The bank

wants to hedge its position at maturity for exposure to a non-financial risk. The bank

sells a contract to the investor by choosing the optimal structure of this contract

according to its utility. On the other hand, the investor finds the transaction

interesting only when its expected utility is the same whether the investor buys the

contract or not. The optimal structure can be determined by maximizing the bank's

expected utility under the constraint that the investor's expected utility is unharmed.

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2. Expected discounted value approach

Since there is no liquid market in these contracts, Black-Scholes style pricing is not

entirely satisfactory. Mark Davis (2001) and Brody, Syroka and Zervos (2002)

suggested that valuation of weather derivatives is generally conducted on an

'expected discounted value' basis, discounting at the risk-free rate but under the

physical measure of the weather variable. The exposure or loss for each outcome is

estimated and a corresponding probability of occurrence is obtained from a sample

of historical events. When the pricing method is quite straightforward, the

empirical/statistical distribution of the underlying weather variable is essential to

make the discounting process accurate. Therefore model building and estimation of

the weather variable is very important in this method.

3. Option pricing theory

It is assumed that a valuation technique similar to that employed for pricing options

and other claims on marketable assets, such as stocks and bonds, can be used (e.g.,

Black-Scholes pricing formula). The critical distinction between pricing an ordinary

stock derivative and a weather derivative is that the underlying is not tradable in our

problem, which makes it impossible to construct a replicating portfolio. Cao and

Wei (2000) suggested that although the assumptions under this valuation method do

not hold, a proxy market asset can be used for replication if possible. The idea

behind this method is that if we can find a suitable proxy asset, we can mimic the

value dynamics of the weather variable and evaluate it. The problem is whether such

a proxy is feasible and reasonable.

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1.5 Current Situation in Hong Kong: the Recreation Industry

Hong Kong's economy relies heavily on the tourism industry and the tourism

industry, one part of the recreation industry, is often affected by weather volatility.

Bad weather can deter people from going outdoors, thus the revenues for public

transportation, retailers, theme parks and other tourism-related business, will

decrease. For theme parks in Hong Kong, such as the Ocean Park or the up-coming

Disneyland, weather conditions greatly affect park attendance and therefore

influence the revenue stream. Traditionally the weather risks were considered

non-diversifiable and beyond human control. Some kinds of weather insurance may

have been possible but they were too costly and the insured weather events were not

highly correlated with the revenue stream. Therefore it makes good sense to develop

a comprehensive risk management strategy to enable theme parks in Hong Kong to

diversify their weather risk. The forming of such strategy requires a pricing

methodology of weather derivatives with proper underlying weather events, together

with alternative weather risk control tools and certain regulatory concerns.

While designing the weather option, we notice that the weather variable should be

carefully selected to show its influence on the revenue stream. To satisfy the needs

of theme parks in Hong Kong, this weather product should be customized; therefore

the climate of Hong Kong should be studied to analyze theme park's needs.

According to Hong Kong Observatory's resources 1, we can identify the characters

of the climate in Hong Kong. Hong Kong's climate is sub-tropical, tending towards

temperate for nearly half the year. Severe weather phenomena that can affect Hong

Kong include tropical cyclones, strong winter monsoon winds, and thunderstorms

‘Climate of Hong Kong: Hong Kong Observatory

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with associated squalls that are most frequent from June to October. During the time

period, September is the month during which Hong Kong is most likely to be

affected by tropical cyclones, although gales are quite common at any time between

May and November. Moreover, June to October is the time period recorded as the

highest park attendance period for Ocean Park2, and most probably for other

up-coming theme parks in Hong Kong as well, such as the Disneyland. The

matching of time period between severe weather phenomena and park attendance

fluctuation indicates the need of hedging with a weather product.

We concluded that in Hong Kong, excessive rainfall, thunderstorms and typhoons

can be more suitable underlying weather events than temperature fluctuations.

Unlike in U.S., where the temperature (HDD/CDD) contracts are actively traded in

the CME to meet the needs for the energy sector, the temperature fluctuations in

Hong Kong in the different seasons do not influence the recreation industry so much

as rainfall amount and typhoons. In Chapter 2, we propose a Markov model with

mixed transition density, incorporating the two weather factors, rainfall amounts and

typhoon signal durations, to estimate the underlying weather variable for the weather

option.

2 Source: Ocean Park daily visitor flow from June, 2001 to July 2003

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Table 1.1: CME Weather Degree Day Index Futures: U.S. Contracts^

MONTHLY CONTRACTS SEASONAL CONTRACTS Ticker symbols Clearing codes Ticker symbols Clearing codes HDD CDD HDD CDD HDD CDD HDD CDD

Boston HW KW HW K.W Atlanta HSl KSl AH AK

Houston HR K.R HR KJR Chicago HS2 KS2 HH ICH

Kansas City HX IOC HX KX Cincinnati HS3 KS3 HT KT

Minneapolis HQ KQ HQ KQ New York HS4 K:S4 HY KY

Sacramento HS KS HS K.S Dallas HS5 KS5 TH TK：

Philadelphia HS6 KS6 FH FlC

Portland HS7 K.S7 RH RK

Tucson HS8 KS8 VH VK

Des Moines HS9 1CS9 JH JK

Las Vegas HSO KSO WH WK

3 Source: Chicago Mercantile Exchange http://www.cme.com/weather

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Table 1.1 (continued): CME Weather Degree Day Index Futures: U.S. Contracts

U. S. CONTRACT SPECIFICATIONS FUTURES OPTIONS ON FUTURES

Contract Size: $100 tiroes the Degree Day Index Contract Size: 1 CME weather futures contract

Minimum Price Increment: 1 Degree Day Index Point Minimum Price Increment: 1 Degree Day Index Point

Degree Day Index: HDD(winter) CDD(suramer) (cabinet- .5 degree day index)

Degree Day Metric: Temperature measured in Fahrenheit

Tick Value: $100.00 Tick Value: 1=$ 100.00

Seasonal Contracts Traded: Seasonal Products Traded:

Heating Season - Nov through Mar Heating Season - Nov through Mar

Cooling Season - May through Sept Cooling Season - May through Sept

Monthly Contracts Traded: Monthly Products Traded:

Heating Degree Days (HDD) Heating Degree Days (HDD)

Oct, Nov, Dec, Jan, Feb, Mar, Apr Oct, Nov, Dec, Jan, Feb, Mar, Apr

Cooling Degree Days (CDD) Cooling Degree Days (CDD)

Apr, May, Jun, Jul, Aug, Sep, Oct Apr, May, Jun, Jul, Aug, Sep

Trading Hours: GLOBEX"' Mon. - Thurs. 5P.M. to 4P.M. Termination of Trading: Same date and time as

the following day (Sun. and holiday trading starts at underlying fiitures.

5:30P.M., LTD closing is 9:00A.M.). Strike Price Interval:

Currency: Contracts settled in US dollars. Monthly Contracts

Termination of Trading: The first exchange business days Month 1 = 10 Index Points

that is at least 2 calendar days after the last calendar day of e.g., 700, 710, 720 (±100 points)

the contract month/season. Month 1-2 = 50 Index Points

Settlement: Based on the relevant Degree Day Index on e.g., 700, 750’ 800(±100 points)

the first exchange business day at least 2 calendar days Seasonal Contracts

after the contract raontli/season. 50 Index Points, e.g., 700’ 750’ 800(±250 points)

Exercise: European Style

(Exercised on last trading day)

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Table 1.2: New York HDD options

New York HDD Options: settlement prices as of 02/27/04 07:00 pm (est)

MTH/ --- DAILY ---- PT ---- PRIOR DAY ----

STRIKE OPEN HIGH LOW LAST SETT CHGE EST.VOL SETT VOL INT

14FEB04 NEW YORK HDD OPTIONS CALL

850 ---- ---- ---- —— 23.0 -6 29.0 40

14FEB04 NEW YORK HDD OPTIONS PUT

850 ---- --— —— —— 5.0 -6 11.0 40

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Chapter 2 Markov Models with Application to Hong

Kong's Rainfall

As valuation of the exotic option is conducted on an 'expected discounted value'

basis, the empirical distribution of the underlying weather variable is essential to

make the discounting process accurate. There have been many research works in

meteorology and quantitative analysis for precipitation models, Aitchison (1995)

introduced the methodology of modeling positive random variables having a discrete

probability mass at the origin. Gabriel and Neumann (1962), Jones and Brelsford

(1967), Katz (1977)，Stem and Coe (1984), Smith (1987), Gregory, Wigley and

Jones (1993), Hyndman and Gmnwald (2000) all proposed different Markov models

with various applications to daily precipitation.

When modeling a time series of amounts of a quantity when the amount can at times

be zero (as is the case with rainfall amount), Gmnwald and Jones (2000) proposed a

stochastic Markov model with mixed transition density. In this chapter, the Hong

Kong daily rainfall data amount (in millimeters/day) and typhoon durations (in

minutes/day) of the past 57 years (from January 1，1947 to February 12, 2004) ^ are

used to fit this model. These data comprise 20862 daily observations, out of which

11878 daily observations contain positive rainfall data. The model parameters are

estimated by maximum likelihood using standard Generalized Linear Model

methods proposed by McCullagh and Nelder (1989) and the most accurate model is

selected based on Akaike Information Criteria by Akaike (1974). The results of

4 Data source: Hong Kong Observatory. To simplify the analysis of seasonality, the leap days are omitted from the time-series.

13

Chapter 2, i.e., the selected model for the weather variable in Hong Kong, are

further elaborated in Chapter 3 in order to design and evaluate an exotic option.

2.1 The Model

Meteorological or environmental data share the common feature that the amount of

quantity may at times be zero. In our example, during autumn or winter, when

precipitation is rare, there may be no recordable amount of rain or typhoons. As a

substantial amount of zeros exist in Hong Kong daily rainfall and typhoon

observations, a Markov model of random variables with mixed distribution is

proposed. We can further include the factor of typhoon durations into the original

one factor model for estimating the rainfall amounts. Hyndman and Grunwald (2000)

suggested that this model is generally applicable to a time-series data with a mixed

density composed of a discrete component at zero and a continuous component on

the positive real line. Other suggested models to forecast rainfall data are ARIMA

models. The seasonal effects can be included in the previous observations in the

ARIMA model. For both models, proper model selection criteria should be applied

to determine the number of autoregressive or retracing terms should be used.

In our study we apply the Markov model with transition density. The notations used

are as follows. Ft is a random variable denoting the rainfall amount at time t, t = 0,

1，...’ n. The the stochastic process {Ft} is refer to as the amount process. Take

Hong Kong daily rainfall amount at time t (t = 0’ 1, , " ’ n) as a time-series process

[yt}, yt is the observed value of the amount process Ft. Let pt(yt| 少t-i’ 0) denote the

transition density for Ft. Given the previous day rainfall amount observation _yt-i, the

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transition density for _yt, varies with time t, ^ is a vector of parameters including

sinusoids and shape parameters, which will be described later in both occurrence

and intensity process. As the rainfall records frequently reach zero, the time-series

process is a mixed density composed of a discrete component at zero and a

continuous component on the positive real line (as is the case with most

meteorological data). Thus the transition density p � i s composed of a discrete

component at 少t 二 0 and a continuous component foryt > 0 (positive tail). It is useful

to model occurrence and intensity process separately, and model the transition

density of Ft based on the occurrence process and the intensity process.

1. Occurrence Process

fO if r = 0 H (2.1)

The occurrence process is 7t = 1 when it rains (Ft > 0) and Jt = 0 otherwise. It is an

indicator process of whether Y is positive.

Thus, the occurrence process / t has conditional Bernoulli distribution with

probabilities

n r , . I V ZD if it = 0 , 0 0 � P r / , = 人 二兄’没0.= ( . X .. . , (2.2)

Here the Bernoulli random variable has a mean n between 0 and 1. Similar to 0�0q is

a vector of parameters describing the variation in time (seasonal patterns for rainfall

15

amounts and typhoon durations) and any other covariates or interventions in the

model.

2. Intensity Process

The intensity process is defined when there is recordable amount of rainfall, i.e.,

when Ft > 0. Define Xt= [Ft | / t = 1] and Xt follows conditional density p^(Xi| yt-i’ 权i)

for Xt > 0 and 0 otherwise. 0\ contains parameters describing variations in time

(seasonal patterns for rainfall amounts and typhoon durations) and other shape or

scale parameters of the distribution. As the conditional mean for the density is

always positive, general assumptions for the form ofPtO^tl 少t-i,权i) are Gamma (Stem

and Coe, 1984) or log-nomial (Katz and Parlange, 1995), Conditional gamma

density with log link is selected in this paper for the intensity process.

3. Amount Process

Given models for occurrence and intensity, the transition density of Ft can now be

written as

Pt (y, I y.-i ’没)=t -冗,(兄-1，^ ) k � G O + 双 , ， ) p , (x, 1 , o,) (2,3)

The first part of the transition density model represents the point mass clustering at

zero, where OaO) represents a point mass at a and 0 = {0o', 0/)'. To simplify further

likelihood analysis,仇 is assumed to have no parameters in common with 0、. The

distribution of positive amounts contributes to the second part of the amount process,

with probability TitCvt-i’ 外).

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2.2 Maximum Likelihood Estimation

As the model for amount process is defined, we need to estimate the parameters

through maximum likelihood estimation. Through some model selection criteria, we

may select an appropriate model.

The likelihood function for {少2，yi---yn ) conditional on Y[ = yi for the amount

process model is

= (2.4) 1=2

The mixed transition density is not of a standard form, so standard methods and

software are not directly applicable. However, Grunwald and Jones (2000) show that

if there are no common parameters in the occurrence and intensity model, the

likelihood function can be factorized into separate parts for occurrence and intensity

models, where standard GLM may apply.

To simplify the maximization process, through some mathematical manipulation, we

can rewrite in this form

= n [ 1 -冗,u - i A ) ] r i A � j v p � '=2,.v,=0 t=2,y,>Q i=2,y,>0 (2.5)

=11[1一冗'(兄-1’氏)]'力'疋'0^-1’氏)'n厂I少M，没 1) t=2 t=2,y,>0

17

Assuming Oo has no parameters in common with 0�the likelihood function for the

amount process is factorized into two products. The two products are the likelihood

of the occurrence process Jt and the likelihood of the intensity process Xt. It is much

easier to estimate the maximum likelihood estimates by maximizing Lj{0q) over 0q

and LxiPi) over 0\ separately. We can use the observed occurrence process {/\} to

estimate the parameters for Lj{0q) and the observed intensity process {xt} to estimate

the parameters for Lx{P\). Therefore, we need to estimate parameters for the

following two likelihood functions, Equation 2.6 and 2.7.

L, (^o)=n[i-�，)]'"'兀 t Ov,，y' (2.6) 1=2

rUOU^y,—丨’⑴ (2.7) /=2,y,;-0

In the following section, the maximum likelihood estimation is calculated for the

occurrence process and the intensity process separately.

2.2.1 Estimates for Occurrence Model

As shown in Equation 2.2, the occurrence process has a conditional Bernoulli

distribution

D f , . I V ^ J l - ^ r U - P ^ o ) if J, = 0 My,-I A ) =1

18

We can use the logit link function in the binary response model for occurrence:

冗 , ’ "(J - — — p w — ^ (2.8)

The logit link function ensures that the estimates of Tit lie between 0 and 1. An

identity link function is also acceptable, but we must impose constraints to

manipulate 兀tto lie in the required range [0，1]. The constraints will increase the

computational time while finding the maximum likelihood estimates.

Since the binomial distribution is a member of the exponential family for

distributions, the model is a generalized linear model (McCullagh and Nelder, 1989)

if nf(yt-i, 0o) is a linear combination of all the parameters to be estimated in 0q. But

the function m^Ot-i,权o) may take different forms, which may not be strictly linear

for all parameters in Oq. This point will be further discussed in the section with

maximum likelihood estimation.

Generally, a time series model with parametric sinusoidal seasonal effects is used,

走 = 1

+ + W + W]iog(y,-, +c) (2.9) k=� A=1

where c > 0，SiiJc) = sin(27r汝/365) and Ct{k) = cos(27rt/:/365) for 众=1，2, m, and n-,

denotes the number of sinusoids for model term i. We consider the simplest model

19

with fewest terms, m冗(yt-i, Oq) = y i + Pi h log(yt-i 十 c)十 giT^t-i + g2T3,t.i +

g3r8’t-i as a non-seasonal model. The sinusoidal seasonal effects are included by

adding the additional 3 terms involving y, S and rj as in Equation 2.9. The terms

involving y describe seasonal changes in rainfall probability following a dry day.

The terms involving S describe the difference in patterns over the year between days

following a wet day and days following a dry day. The terms involving rj describe

the difference among previous day's intensity throughout the season. Taking larger

m, n2 and ns in Equation 2.9, we are taking more sinusoids into account and

retracing more time effect for the model. log(yt-i + c) is used instead ofyt-i because

the likelihood analysis shows a much improved fit in the occurrence data.

Ti,t-i’ T3J.1 and Ts.t-i denote the typhoon signal duration in minutes at time t-1, for

typhoon signal No.l, No.3 and N0.8 or above accordingly/

Models with increasing values of «’s are fitted successively until no improvement in

fit is gained by including additional terms. Maximum likelihood estimation is used,

so likelihood ratio tests, or certain information criteria, may apply to assess the

increase in goodness of fit. Here we use Akaike Information Criteria (Akaike, 1974)

as model selection criteria to find the best fitting model.

Akaike (1974) pointed out that a tradeoff takes place when the number of parameters

increases with sinusoid terms and when the improvement of the likelihood function

may not be satisfactory enough. To determine the number of sinusoidal terms giving

good models, we use AIC = -2 log(l) — 1 却 df�where L is the likelihood and df

(degree of freedom) is the number of model parameters.

5 For the meaning o f typhoon warning signals, please refer to Table 2.1 at the end o f Chapter 2.

20

We also note that log(yt-i + c) is the only term that makes the predictor non-linear. If

we can fix the value for c, we can turn iif(yt-i’ 权o) into a linear predictor and use

standard GLM software to obtain the maximum likelihood estimates.

The maximum likelihood estimate of 0q can be found by fitting the model with

rainfall occurrence observations using traditional optimal searching functions in

Matlab. As the function uses a direct simplex search method which applies to

unconstrained local minimum only, we define the negative of the log-likelihood

function and search for the local maximum with different initial estimates.

Table 2.2 shows the detailed parameter estimates using direct search method. Table

2.3 shows a summary of the estimated results in Table 2.2. In Table 2.3, a 'Y' or 'N'

is placed in the column if the term is included or not included in the model.

Numbers under column m’ 叱 and n) denotes the number of sinusoidal terms, y, 5

and T|, in Equation 2,9 accordingly in the model. After estimating several of the

seasonal models, we can observe that c gives a value of around 0.10,

The maximum likelihood estimate of Oq can also be obtained by fitting the same

model using standard GLM (Generalized Linear Model) software with binary

response, logit link and covariates 人i, log(yt-i + 0.10)，T terms, and the sinusoids.

Compared with the direct search method for likelihood maximization, GLM takes

less computational time and produces more stable results. We use GLM to process

other seasonal models, adding more sinusoids and trying to find an appropriate

model by AIC.

21

Table 2.4 and 2.5 shows the parameter estimates using GLM. Table 2.6 summarizes

the estimation results for occurrence models. All tables list the AIC values in

descending order. According to Duong (1984), a decrease of about 2 in the Akaike

Information Criteria represents a significant improvement. With indistinctive

difference in AIC values, the model with fewer degrees of freedom, i.e. the number

of parameters, is preferred. AIC chose model 1 in Table 2.4 and Table 2.6 with n\ =

2, = 4, «3 = 1. Higher autoregressive terms associated with «i, and 均 make

little improvement in the AIC value. The chosen model gives AIC = 22688,

including seasonality in occurrence probability = 2)，difference in this

seasonality following wet and dry days {ni = 4) and seasonality in the effect of

log(yt-i 十 0.10) («3 = 1). In Table 2,6, we find that strong dependence on previous

day's rainfall occurrence (/Vi) and intensity (log(yt-i + 0.10)) is indicated. Models

with the two terms generally have lower AIC values than models without the two

terms. Moreover, comparing the models 1 to 15 with models 16 to 30 in Table 2.6

we find that models including typhoon signal effects have significantly lower AIC

values. We will find quite a different observation when analyzing intensity models in

Section 2.4.

Obtaining the exact parameter estimates, we use the Model 1 in Table 2.4 for all

subsequent analyses.

l + e x p K U-P^o) .

Applying the estimated parameters, we have

22

= (-0.1003) + 1.167M+ 0.2279 log(y,., + 0.10)

+ 0.00057/+ 0.0032rj+ 0.00447：

+ [0.1665 sm(27it/365) + -0.7565 cos(27rr/365)

+ 0.1221 sm(4;r//365) + -0.0936 cos(47r"365)]

+ [0.162 sin(2;rr/365) + 0.4202 cos(27r?/365)

+ (-0.093) sin(47r"365) + 0.1187 cos(4;rr/365)

+ 0.0382 sin(67rr/365) + -0.1575 cos(6对/365)

+ -0.1076 sin(87rr/365) + 0.0229 cos(67rr/365)]yt-i

+ [(-0.0172) sin(27rr/365) + (-0.0437) cos(27r"365)] logOvi + 0.10)

(2.10)

2.2.2 Estimates for Intensity Model

We now consider modeling rainfall intensity X,, as X, = F, if Y, > 0. Units of rainfall

intensity are mm per wet day. We assume that intensity given previous rainfall data

follows a Gamma distribution G (jli, r). G (//, r) denotes a Gamma distribution with

mean ^ and shape parameter r. The intensity data follows a conditional Gamma

distribution with constant shape parameter r not depending on season or 少t.i. The

seasonal effects and other covariates are included through mean /u. We can

parameterize Equation 2.7 as:

with

p{x I exp ( - rx 丨 I / r ( r ) • for A： > 0, r > 0 and // > 0

23

Furthermore

•_ , ’6> ,* )=exp(<(y ,_"0 / ) ) (2.12)

"I "2 1 + i k A W + W ] + Z [么,A W + K c , � ( 2 . 1 3 )

fh - ,

+Z � + ( _ o g ( 兄 + c) k=\

In Equation 2.11 0i includes all parameters in 0i* plus the constant shape, the

parameter r. Equation 2.12 is the density function of random variable X ~ G (ju, r).

In this parameterization, = jli and Var(X) =ij^lr. Equation 2,12 and 2.13 together

denote the conditional Gamma GLM. According to McCullagh and Nelder (1989), a

log link is used for link function and Gamma response is chosen for response

variable distribution in the intensity model. The linear predictor m^(yt-i,权i*) takes

the same forms as in the occurrence model (see Equation 2.13). Effects of yVi,

log(yt-i + c), typhoon effects and other seasonal terms are included in the conditional

mean 风.We still use AIC as model selection criteria to test the autoregressive terms

in the sinusoids.

In the Hong Kong daily rainfall series from 1947 to 2004, there are 11878 days with

positive rainfall. Following the same modeling procedure as used in estimating the

occurrence model, we first obtain the results using direct search method in MatLab

to maximize the likelihood function.

24

In Table 2.8, a 'Y' or 'N' is placed in the column where the term is included or not

included in the model. Numbers under column «i’ n! and «3 denote the number of

sinusoidal terms (y, 5 and t), in Equation 2.13 accordingly) in the model. It is worth

noting that in the intensity model, _yt-i shows a much improved fit than log(yt-i + c),

in contrast to its performance in the occurrence model. In both Table 2.7 and Table

2.8 we find that the estimated c values for some models are around exp(lO), other c

values are a lot larger than that measure, which make the 少t-i values negligible. (In

Table 2.8’ we list only the c value as it is the parameter value which help us to make

the choice between log(yt-i + c) and 少t-i. Detailed parameter estimates can be found

in Table 2.7.) Hereafter, we use 少t-i directly in the place of log(yt-i + c) for modeling

intensity. Equation 2.13 is rewritten in the following form

< ( 兄 A . / ； - , + giT, + g j ,

+ Z k A W + W]+1： [A, A (k) + S�cC, {k)]j\_, (2.14)

k=\

Here the model returns to a standard GLM form with Gamma response, log link and

covariates as in the occurrence model. We check with higher autoregressive terms to

see whether they give significant improvements in the log-likelihood estimates. We

still use AIC as model selection criteria.

Table 2.11 summarizes some of the estimation results and lists the models with the

best AIC values only. Model 1 is selected with = 4 ,叱= 1 , n] = 1，without typhoon

factors. The model includes strong seasonality effect in occurrence probability (wi =

4)，difference in the seasonality following wet and dry days («2= 1) and seasonality

25

in the effect of _yt-i = 1). The model indicates strong dependence on previous

day's occurrence _y.t-i. In contrast to its performance in occurrence models, typhoon

durations do not have a significant effect in the intensity models. We can compare

the estimate results in Table 2.9 (including typhoon effects) with the results in Table

2.10(excluding typhoon effects). Models without typhoon effects generally have

lower AIC values than models with typhoon effects. These models tend to be listed

in the lower part of Table 2.11. Models 11 and 14 illustrate the need for the

higher-order sinusoids. Models 4 and 13 indicate that such high-order sinusoids are

not necessary. Thus AIC is optimized when n\ = A, n2= =

We use the following intensity model for all subsequent analysis.

with

/? (;c I // , r ) = exp ( - rx / // X^ / / 0 ' ^ / r ( r )

for jc > 0 , / / > 0 and r = 0.6246

Applying the estimated parameters, we have

= 1.1298 + 0.77227；.,+ 0.0231;；,.,

+ [(-0.238 l)sin(27r"365) + (-1.3663) cos(2;rr/365)

+ 0.0086 sin(47rr/365) + (-0.105)cos(47r^/365)

+ [0.10521 sin(67rr/365) + 0.1297 cos(67r"365)

+ 0.0142 sin(87r//365) + (-0.0372) cos(87r,/365)]

26

+ [(-0.07578) sin(2;rr/365) + 0.46146 cos(27r"365)]yM

+ [0.0055 sin(27r"365) + 0.0134 cos(27r"365)]凡1 (2.15)

The estimated shape parameter in the Gamma distribution is r = 0.6246. To check

the conditional Gamma distribution assumption, Grunwald and Feigin (1996)

suggested a residual testing method. The Gamma distribution has the property that if

Z ~ G (1, r) and X = /uZ then X ~ G (//, r). Define the scaled data R � = X^J 广/t(yt-i’

01*). If the model is correct, should follow a G (1, r) distribution.

A test is performed on the time-series Ri. We use the built-in gamma fit function

Cgamfif function) in Matlab with a 98% confidence interval. The test result shows

that the residual Ri follows a Gamma distribution with mean jn = 1 and shape

parameter r = 0.6242, with 98% confidence interval for r equals to [0.6236, 0.6248].

As our estimate r = 0.6246 falls in the 98% confidence interval, the residual analysis

does not reveal any model inadequacies.

Furthermore, we can make a plot of the sorted R^ values versus the quantiles of a

standard G (1’ r) distribution where r = 0.6242. QQ_plot function in Matlab displays

a quantile-quantile plot of two samples. If the samples do come from the same

distribution, the plot will be linear. Figure 2.1 shows the graph for the empirical

intensity residual Rt versus theoretical Gamma distribution using QQ_plot. (The red

dotted line represents linear relationship x=y.) Although there is some deviation

from the theoretical line in the upper tail, the fit appears to be quite good. The

deviation appears for both quantiles greater than 4’ which corresponds to about the

99th percentile. Thus the conditional Gamma function fits well for rainfall intensity

data.

27

2.3 Model for Amount

The estimated Markov model for rainfall Ft can be constructed from Equation 2.10

and Equation 2.15，the estimated results of the occurrence and intensity model. The

fitted model yields a mixed density as given in Equation 2.3. This combined model

has units of mm/day while the intensity has units of mm/wet day. The analysis

incorporates dependence of the observed amount on the previous period's

occurrence and intensity. Moreover, results and conditioning can be displayed in

various ways, depending on the purpose. We can calculate the expected value of Ft

directly from Equation 2.3 as

The conditional mean is very useful for further simulation, described in Chapter 3,

where the expected payoff is discounted to estimate the value of the weather option.

28

Table 2.1: Meaning of Typhoon Warning Signals^

1. This is a stand-by signal, indicating that a tropical cyclone is centered within T “ 800 km of Hong Kong and may later affect the territory.

Strong winds are expected or blowing in Victoria Harbor, with a sustained 丄 s p e e d of 41-62 km/h (kilometers per hour). Gusts may exceed 110 kni/h.

• Winds are normally expected to become generally stronger in the harbor areas within 12 hours after the issuing of this signal.

Gale or storm force winds are expected or blowing in Victoria Harbor, with a • 8: sustained wind speed of 63-117 kni/h from the quarter indicated. Gusts may

exceed 180 km/h.

Gale or storm force winds are increasing or expected to increase significantly ‘ in strength.

Hurricane force winds are expected or blowing. Sustained wind speeds are ‘ “ r e a c h i n g upwards from 118 km/h. Gusts may exceed 220 km/h.

6 Source: Hong Kong Observatory

29

Tabl

e 2.

2: P

aram

eter

est

imat

es f

or o

ccur

renc

e m

odel

s us

ing

dire

ct s

earc

h m

etho

d.

Px.Pi

, pi,

y, S, rj

and

c as

in e

quat

ion

"i m

"=

PJ.-x

+

A l

og(y

,-i +

c) +

W

+Y

k.c,

W.

k=\

k=\

k=\

+ 从’

1 +

沾M

I 1

1 1

^ 1

1 p：：：：??*：!?：：：：：：：：：：：]

1

I I

I I

I I

“ I

Model

df

-in

T ft�

B2

B3

v. v.

v..

�c

"i’s

nu

c ni

.. nx

c I

4 11817

23642

-0.1440

1.3103 0.2681 0.09694

_：

：_ _；

__

“

”

”

“

—1

6~

1172

7 23

466

-0 1

966

1.11

01

0.25

31

0J03

98

- -

- -

0.28

90

-0.3

266

- -

“

~3

^ 6

1152

4 23

060

-0.1

263

1.23

07

0.23

08 0,10086

0.23

678

-0.5

169

- _

__

__

__

__

_ “

“ “

“_

81

17

25

2346

6 1.

3143

0.

2518

9 0.

1062

1 0.

2894

5 -0

.328

8 0.

0594

6 0.

0236

9 -

--

- “

8 11

511

2303

8 -0

.127

7 1.

2255

0.

229

|oJo

9985

|o.2

2971

1-0.

5152

10.1

0138

|-0.

0555

1 --

|

“ |

“ |

“ |

“ |

“ I

“ I

“

‘‘AIC

= -2

*lnL

+

2*d

/

30

Table 2.3 Summarized results for occurrence models using direct search method.

Model JtA log(yt-i + c) Ti T3 Ts m m m c value AIC

1 Y Y N N N O O O 0,09694 23642 2 Y Y N N N O l O 0 J 0398 23466 3 Y Y N N N 1 0 0 0.10086 23060 4 Y N N N N 0 2 0 0J0621 23466 5 Y Y N N N 2 0 0 0.09985 23038

31

Tabl

e 2.

4: P

aram

eter

est

imat

es f

or o

ccur

renc

e m

odel

s us

ing

stan

dard

GL

M (

logi

t lin

k, b

inar

y) w

ith c

= 0

.10

with

typ

hoon

sig

nals

A，

Pi, A

，y< 在

n a

nd g

as

in e

quat

ion

"I r

1

k=i

+ +

+E

kA

W

+ "�

,c,W

]log

(y,_

, +

0.10

) k=

\ k=

i +

g3

Vl

Mod

el I

df

-In

L AI

C* |

|

y93

| ya

|

g,

I I

I )-..

c y

：.

__

__

^_

_^

__

^_

__

_^

__

__

__

__

__

fi：

__

__

-_

_-

__

-_

_-

__

—

~ ^

0.162

0.42

02 -0

.093

0.H87

0.03

82 -0

.158

-Q.10

8 0-0

229

-_

__

_ ：

_

—1

^ 11

321

2268

8 -0

.095

1.1

619

0.23

01 0

.000

5 0.

0032

0.0

044

0.17

66 -

0.74

6 0.

1823

-0.

033

-_

_-

Q.15

96 0

.422

6 -0

.140

0.0

715

0.04

98 -

0.15

3 -0

.110

0.0

259

-_

_-

-0.0

13 -

0.03

9 0.

0238

0.

024

-_

_^

1~

1131

9 22

688

-0.0

93 1

.162

2 0.

2311

0.0

005

0.00

32 0

.004

4 0.

1729

-0.

745

Q.16

94

- 0-

1613

0.4

227

-0.1

33 0

.071

8 0.

0403

-0.

157

-0.1

18 0

.026

3 -

__

- -Q

.Q15

-0.

039

0.02

08

0.02

2 -0

.017

-

0^

~ ^

11327 22

692 -0.076 1.1677 0.2374 0.0005 0.0032 0.0044 0.2099 -0.646 0.1222 -0.094

—

- 0-1256 0.3343 -0.081 0.128 0.0335 -0.156 -0.111 0.0211

- 二

：

__：

：

：

：

~ ^

11326 22694 ^

^ 0.2098 -0.646 0.1222 -0.094

- -

0.125 0.3323 -0.081 0.1278 0.0344 -0.159 -0.111 0.0192 0.029 -0.036

-_

_:

__

~_

_"

__

"

：_

~ ~

11333 22700 -0.072 1.1622 0.239 0.0005 0.0032 0.0044 0.2099 -0.646 0.1221 -0.094

- -

0.1249 0.3334 -0.075 0.1208 0.0386 -0.161

- "

" "

" ”

_：

二

~ ^

1133

2 22

702

-0.0

71

1.16

12

0.23

9 0.

0005

0.0

032

0.00

44 0

.211

7 -0

.650

0.1

169

-0.0

89 0

.039

5 -0

.018

^^

^^

^^

^ ^

^ ^

^^

^^

^ _

_ _

_ _

_

1133

7 22

708

-0.0

72 1

.161

2 0.

2377

0.0

005

0.00

32 0

.004

4 0.

2142

-0.

657

0.11

64 -

0.07

.7

0.03

9 -Q

.094

0.12

22 0

.347

4 -0

.068

_

_ _

_ _

_ _

_ 二

I

~ 15

11

340

2271

0 -0

.069

1.1

544

0.23

73 0

.000

5 0.

0032

0.0

044

0.21

78 -

0.66

8 0.

0799

-0.

018

0.04

39 -

0.09

9 _

_ _

_ _

_ _

_ _

_ "

: ^

一—

—

~ 11342 22718 -0.090

1.1

604 0.2318 0.0005

0.0

032 0.0044 0.1775 -0.740 0.0693 0.0355 0.141 0.4281

- ^

^ ^

: ：：

：

：

：

二

-0.014 -0.036 -

0.007。

严二

11

15

1134

5 22

720

-0.0

90 1

.159

9 0.

2321

0.0

005

0.00

32 0

.004

4 0.

163

-0.7

54

-- -

" 0-

1465

0.4

351

0.02

1 ^

Q^

-0

^ —

—I

~ 17

11

343

2272

0 -0

.103

1.

1697

0.

227

0.00

05 0

.003

2 0.

0044

0.1

783

-0.7

62 0

.122

1 -0

.094

-

- |o

.l536

|o.4

3211

-0.0

84 |o

.l204

| --

| --

| --

| --

| --

| -

|-0-0

131-

0-04

61

-- |

—

| -

| --

8 AIC

= - 2

* In

L + 2

* 9 M

odel

1 is s

electe

d with

AIC =

21754

and c

lf= 2

1.

32

Mod

el

# -In

L A

IC"

p� h

Pi

容 i

Si

gi

Yi

yu

yix

Ji.

yu

仏

占;.c

Sa

：么

.c

5s,,

"�,:

>；

2,. rj2

,c

"丄

13

15

11346 22722 -0.078 1.1698 0.2367 0.0005 0.0032 0.0044 0.2099 -0.646 0.1222 -0.094

- --

0.1272 0.3424 -0.074 0.1319

一

”

14

15

11347 22724 -0.102 1.1644 0.2259 0.0005 0.0032 0.0044 0.2004 -0.784 0.079 -0.030

- --

0.1356 0.4451

--

- 0.005 -0.050

--

15

15

11353 22736 -0.099 1.1658 0.227 0.0005 0.0032 0.0044 0.1909 -0.781

--

--

--

0.1416 0.4522 0.0381 0.027

—-

_ I

— I 一

卜

0.0131-0.046

1 -- |

— |

- | —

1�

AIC

= -2

*ln

L +

2*

c//

33

Tabl

e 2.

5: P

aram

eter

est

imat

es f

or o

ccur

renc

e m

odel

s us

ing

stan

dard

GLM

(lo

git l

ink,

bin

ary)

with

c =

0.1

0 w

ithou

t typ

hoon

sig

nals

.

A, A

2, A

, y. s

. rj

as in

equ

atio

n

爪"G

Vi

A +

+

A l

ogO

vi

+ +

+堂

Ks

*^

,⑷

+么

,…

�]

lo

gU

-i

+0

-10)

k=

\ k=

l M

odel

df

-In

L A

IC

p� 爲

jh

yu

Tu

'fi

* JV

^i

.. ix

hj

. <$3

,1 <5

j,c

<54j

Kz

^ii

"ic

lij.

”3’c

1 23

11447

22940

-0.042

1.1492

0.2342

0.1189

-0.787

0.1959

-0.037

—

- 0.1675

0.4383

-0.144

0.0616

0.0594

-0.155

-0.109

0.0305

—

- -0.018

-0.040

0.0233

0.0253

- —

2 25

11445

22940

-0.040

1.1491

0.2353

0.1147

-0.786

0.1793

-0.041

- -

0.1696

0.4388

-0.136

0.0601

0.0477

-0.158

-0.118

0.0321

0.020

-0.039

0.0193

0.0234

-0.021

-0.006

3 21

11450

22942

-0.048

1.1546

0.2317

0.1098

-0.799

0.137

-0.100

—

- 0.1696

0.4365

-0.098

0.1112

0.0478

-0.160

-0.106

0.0277

- ~

-0.021

-0.045

- -

--

- -

4 19

11454

22946

-0.023

1.1552

0.2418

0.1633

-0.686

0.137

-0.100

- -

0.1253

0.3479

-0.084

0.1202

0.0424

-0.158

-0.110

0.026

--

--

--

5 21

11453

22948

-0.024

1.1557

0.2415

0.1633

-0.686

0.137

-0.100

—

--

0.1246

0.3459

-0.084

0.12

0.0433

-0.161

-0.109

0.0241

0.0276

-0.035

—

—

_ --

- -

6 17

11460

22954

-0.019

1.1502

0.2432

0.1633

-0.686

0.137

-0.100

- --

0.1246

0.3468

-0.078

0.1124

0.0479

-0.163

—

--

—

~

7 19

11458

22954

-0.018

1.1489

0.2432

0.1659

-0.691

0.1307

-0.093

0.0518

-0.028

0.1219

0.3518

-0.072

0.1054

-0.004

-0.134

—

—

—

--

--

—

—

8 17

11463

22960

-0.019

1.1491

0.242

0.1682

-0.697

0.131

-0.082

0.0496

-0.099

0.1213

0.3616

-0.071

0.0984

--

--

--

--

" "

' -

9 15

11466

22962

-0.016

1.1427

0.2415

0.1712

-0.708

0.0935

-0.029

0.0545

-0.104

0.1196

0.3667

--

--

--

--

--

- —

10

17

11469

22972

-0.037

1.1488

0.2357

0.119

-0.780

0.0784

0.0237

- --

0.1489

0.4425

—

- —

—

—

-0-019

-0.036

-0.009

0.0433

--

--

11

17

11470

22974

-0.051

1.158

0.2308

0.1217

-0.803

0.137

-0.101

—

- 0.1603

0.4479

-0.090

0.1131

—

—

- -0.017

-0.046

- —

12

15

11473

22976

-0.035

1.1476

0.2368

0.0996

-0.792

- -

- --

0.1566

0.4502

- --

- -

-0.028

-0.041

-0.031

0.0364

--

—

13

15

11474

22978

-0.025

1.158

0.241

0.1633

-0.686

0.137

-0.101

—

- 0.126

0.3567

-0.078

0.124

—

—

—

- "

~

14

15

11475

22980

-0.050

1.1527

0.2298

0.1431

-0.824

0.0911

-0.041

--

--

0.1433

0.4592

--

--

--

--

-0.009

-0.050

15

15

11483

22996

-0.046

1.1533

0.2308

0.1335

-0.826

- 一

—

--

0.1485

0.4707

0.0474

0.0124

—

- |

- -

- |

- | -。

-。口

| -

。厕

|

- |

-丨

-|

-

34

Table 2.6 Summarized results for occurrence models using GLM.

Model 7V1 logOt-i +Q-IO) "1 a72 "3 T, 7) AIC df 1 Y Y 2 4 1 Y Y Y 22638 21 2 Y Y 2 4 2 Y Y Y 22688 23 3 Y Y 2 4 3 Y Y Y 22688 25 4 Y Y 2 4 0 Y Y Y 22692 19 5 Y Y 2 5 0 Y Y Y 22694 21 6 Y Y 2 3 0 Y Y Y 22700 17 7 Y Y 3 3 0 Y Y Y 22702 19 8 Y Y 3 2 0 Y Y Y 22708 17 9 Y Y 3 1 0 Y Y Y 22710 15 10 Y Y 3 0 2 Y Y Y 22718 17 11 Y Y I 1 2 Y Y Y 22720 15 12 Y Y 2 2 1 Y Y Y 22720 17 13 Y Y 2 2 0 Y Y Y 22722 15 14 Y Y 2 1 1 Y Y Y 22724 15 15 Y Y 1 2 1 Y Y Y 22736 15 16 Y Y 2 4 1 N N N 22940 23 17 Y Y 2 4 2 N N N 22940 25 18 Y Y 2 4 3 N N N 22942 21 19 Y Y 2 4 0 N N N 22946 19 20 Y Y 2 5 0 N N N 22948 21 21 Y Y 2 3 0 N N N 22954 17 22 Y Y 3 3 0 N N N 22954 19 23 Y Y 3 2 0 N N N 22960 17 24 Y Y 3 1 0 N N N 22962 15 25 Y Y 3 0 2 N N N 22972 17 26 Y Y 1 1 2 N N N 22974 17 27 Y Y 2 2 1 N N N 22976 15 28 Y Y 2 2 0 N N N 22978 15 29 Y Y 2 1 1 N N N 22980 15 30 Y Y 1 2 1 N N N 22996 15

35

Tabl

e 2.

7: P

aram

eter

est

imat

es f

or in

tens

ity m

odel

s us

ing

dire

ct s

earc

h m

etho

d.

A，

A’

y, S,

rj a

s in

equ

atio

n

k=\

…�

]log

Ovi +

c)

Model

df

-In

L AI

C"

A r

ln<c)

yi..s

yi,c

72,,

yu

而-s

c �

馬’

c "i-s

1

11 307

84 61

590

-294.6

4 0.7

702

36.678

0.29

82 8.0

652 -

35.601

94.0

59 -0.

0259 -

0.1819

-

4.378

6 -11

.777

- --

2 11

3215

9 64

340

-7.39

38 1.0

516 0

.6987

0.293

7 12.

215 -0

.2410

-1.35

67 -0.

0445

-0.159

3 -0.1

123

0.4221

-

--3

9 34

790

6959

8 9.3

967

1.0491

-0.50

28 0.2

930

16.39

__ --

0-100

4 0.4

793

0.015

0 -0.0

842

- --

4 9

3702

3 74

064

1.7701

1.0

491 -0

.0141

0.293

0 43

,44 -0

.2457

-1.38

01 -

- -0.

1007

0.4793

-

5 9

37411

7484

0 -25

.313

1.0068

0.70

63 0.2

929 3

7.482

-0.318

7 -1.0

442 -

0.032

9 -0.1

797

" "

6 7

3745

7 74

928

4.046

2 0.9

972 -

0.090

9 0.2

920

3LS1

4 -0.3

173 -1

.0183

~7

9 38

490

76998

-31.33

1 0.6

699

0.749

6 0.2

840 4

3.831

-- -

-- --

|-0.35

841-0.

92981-

0.0618

1-0.15

58 -

| --

| --

| -

“AIC

= -2

*lnL

+ 2

*#

36

Table 2.8 Summarized results for intensity models using direct search method.

Model j\.i logOt.i + c) "I /i2 nj T, Ts Tn ln(c) AIC

1 Y Y 2 0 1 N N N 8.0652 61590 2 Y Y 2 1 0 N N N 12.215 64340 3 Y Y 0 I 1 N N N 16.39 69598 4 Y Y 1 1 0 N N N 43.44 74064 5 Y Y 2 0 0 N N N 37.482 74840 6 Y Y l O O N N N 31.514 74928 7 Y Y 0 2 0 N N N 43.831 76998

37

Tabl

e 2.

9: M

ode�

sele

ctio

n fo

r int

ensi

ty u

sing

sta

ndar

d G

LM (

log

link,

gam

ma)

with

typh

oon

sign

als.

A’爲

，�

3, y, S

, rj a

nd g

as

in e

quat

ion

W'C

vm

,氏)=

A

+ A

"1

+ A

且 o

gU

-i

k=\

n-t

- r

+S

Ks^

, (k)

+Wk-

. +

W+

� Jv

m

k=\

k=\

Mod

el

df

-InL

AIC

h

爲

幻

Si

Si

}\c

Yi^

Y^^

� "i’

c ^

1 18

30

757

6155

0 1.

618

- 0.

029

0.00

02

0.00

09

0.00

2 -0

.250

-1

.286

4 0.

0122

-0

.055

6 0.

1380

6 0.

098

-0.0

431

-0.0

278

-0.0

316

0.37

306

0.00

8 0.

0172

0.

6275

3

2 9

3095

4 61

926

1.32

0.

756

0.01

4 0.

0006

0.

0011

0.

0022

0.

0015

-0

.007

2 0.

6294

6 n 78

7 S

3 7

3430

0 68

614

1.32

0 0.

755

0.01

8 0.

0006

0.

0011

0.

0022

-

--

- --

-

~ 35

660

7135

8 1.

004

0.81

1 0.

022

0.00

03

0.00

1 0.

0021

-0

.248

-1

.257

1 -0

.005

6 -0

.068

0.

1589

1 0.

1177

-0

.015

5 -0

.023

9 -0

.061

3 0.

4209

6 0.

0072

0.

0122

0.

8588

9

~5

3576

6 71

570

0.99

3 0.

855

0.01

5 0.

0003

0.

001

0.00

21

-0.2

43

-1.2

489

-0.0

542

-0.1

486

0.17

326

0.12

13

-0.0

084

-0.0

21

0.00

09

0.50

941

0.03

26

0.08

51

- 0.

8632

6 ~

3579

2 71

618

0.99

9 0.

848

0.01

5 0.

0003

0.

001

0.00

21

-0.2

45

-1.2

545

-0.0

317

-0.0

873

0.17

186

0.11

46

-0.0

094

-0.0

232

-0.0

005

0.50

73

- 0.

8643

3

~7

~ 15

36

365

7276

0 1.

02

0.80

3 0.

014

0.00

04

0.00

1 0.

0021

-0.

2366

-0

.877

5 -0

.030

2 -0

.110

7 0.

1686

2 0.

1137

-0

.016

2 -0

.037

4 0.

8885

4

8 13

36400

7282

6

1.019

0.805

0.014

0.0004

0.001

0.0021 -0.2358 -0.8758 -0.0272 -0.1093 0.17256 0.1187

_!：

_

9 9

36465

72948

1.061

0.775

0.014

0.0003

0.001

0.0021 -0.2483 -0.8604

- --

”

”

“

：“__

10

11

3703

3 74

088

1.04

4 0.

787

0.01

4 0.

0003

0.

001

0.00

21 -

0.25

42

-0.8

753

-0.0

618

“；

；

；7~

3813

9

76304

1.329

0.511

0.015

0.0005

^ ^

^ ^

^ -

__

-

-0-2334 -0.7368 -0.0241 -0.0629 0.1588

0.1129

-_

_

12

9 38

199

7641

6 1.

330

0.52

0 0.

015

0.00

05

0.00

11

0.00

21

- -

--

0.24

31

-0.7

247

3858

9 77

200

1.33

0 0.

516

0.01

5 0.

0005

0-

0021

1 --

|

--

| --

|

--

| --

|

--

| -

| -

[-0.

2571

|-0.

7390

1-0.

0686

1-0.

0873

1 —

|

--

| --

|

- |o

.975

36

38

Tabl

e 2.

10: M

odel

sel

ectio

n fo

r int

ensi

ty u

sing

sta

ndar

d G

LM (

log

link,

gam

ma)

with

out t

ypho

on s

igna

ls.

Pij

i, A

，y,

r]

as in

equ

atio

n

m

A)=

A

+ Ay；-.

+ A

log

Ovi

+

c)

+Z

k,

"^

,�

+�

.

tin

- r

1 +

(k

)+

么’

cC, (/：)]/•,., +

⑷+

Vk,c

C, W

k-i

^

Mod

el

df

-InL

AIC

� yS

, Pj

A

Xl

.c yi

s Yu

y3

.s Vs

.c y4’

c 知

头,

c 7 U

s n\

,c 化”

S 12

,0

J13

16

3074

9 61

530

1.13

0 0.

772

0.02

3 -0

.238

-1.

366

0,00

9 -0

.105

0.

105

0.13

0 0.

014

-Q.0

37 -

Q.0

76

0.46

1 -

- -

- 0.

006

0.01

3 --

-

0,62

5

一 1

10

32113 64246

1.524

0.415

0.015

- -

- ~

- :

一

-0.266 -0.792 -0.017 -0.106

0.11 0.144

- ：

二

-“

細

3 4

3478

4 69

576

1.52

4 0.

668

0.01

9 -

一

—

—

一

- -

°-80

0

4 16

37

013

7405

8 1.

123

0.81

6 0.

015

-0.2

37 -

1.36

4 0.

017

-0.1

84

0.11

5 0.

138

0.01

9 -0

.032

-0.

031

0.56

7 -0

.037

0.

072

--

--

--

0.91

4

5 14

37

400

7482

8 1.

127

0.80

9 0.

015

-0.2

37 -

1.36

6 -0

.012

-0.

131

0.11

7 0.

133

0.01

7 -0

.033

-0.

029

0.56

2 -

--

--

--

- -

0.92

9

~6

37433 74878

1.524

0.669

0.014

—

—

一

- -

--

--

--

--

-0.004 -0.008

- —

0.931

7 6

3844

1 76894

1.195

0.736

0.014 -0.265 -0.920

—一

--

- -

。娟

15

3860

8 77

246

1.70

9 --

0.

029

-0.2

34 -

1.38

8 0.

021

-0.0

91

0.09

8 0.

113

-0.0

15 -

0.04

8 -0

.042

0.

415

--

--

- 0.

006

0.01

8 -

15

0.97

8

9 10

38

666

7735

2 1.

162

0.75

4 0.

014

-0.2

54 -

0.94

5 0.

002

-0.1

47

- “

--

--

- -

--

--

—

10

0.98

0

10

8 39

254

7852

4 1.

173

0.74

8 0.

014

-0.2

66 -

0.94

5 -0

.025

-0

^ ^

^ —

- -

一

- --

_

8 1.

003

^41415

82846

1.524

0.421

--

- -

：

：

- -0.288 -0.795 -0.054 -0.137

-__-

-一

__^

：

"T

l 12

41

551

8312

6 1.

16

0.75

6 0.

014

-0.2

55 -

0.94

8 -0

.001

-

0^

-0

^ „

- 一

--

一

一

--

一

12

1.08

3

Ts

^4

21

85

8438

2 1.

524

0.42

9 0

.0

15

^ -

一

I —

I

—

I —

I

—

I 一

|-

0-27

6|-0

.77o

| 一

| 一

| —

| 一

| --

| --

| 一

| 6

| 1.

108

12 AI

C 二 - 2

* In

L + 2

* #

Mode l

1 is s

electe

d with

AIC =

61530

and d

f= 16

.

39

Table 2.11 Summarized results for intensity models using GLM.

Model y't-i ^m T, T} Th n\ n^ n^ AIC df 1 Y Y . N N N 4 1 1 61530 16 2 N Y Y Y Y 4 1 1 61550 18 3 Y Y Y Y Y O 0 1 61926 9 4 Y Y N N N O 3 0 64246 10 5 Y Y Y Y Y O 0 0 68614 7 6 Y Y N N N O 0 0 69576 4 7 Y Y Y Y Y 4 1 1 71358 19 8 Y Y Y Y Y 4 2 0 71570 19 9 Y Y Y Y Y 4 1 0 71618 17 10 Y Y Y Y Y 4 0 0 72760 15 1 1 Y Y Y Y Y 3 0 0 72826 13 12 Y Y Y Y Y l 0 0 72948 9 13 Y Y N N N 4 2 0 74058 16 1 4 Y Y Y Y Y 2 0 0 74088 11 15 Y Y N N N 4 1 0 74828 14 16 Y Y N N N O 0 1 74878 6 17 Y Y Y Y Y O 3 0 76304 13 18 Y Y Y Y Y O 1 0 76416 9 19 Y Y N N N 1 0 0 76894 6 20 Y Y Y Y Y O 2 0 77200 11 21 N Y N N N 4 1 I 77246 15 22 Y Y N N N 3 0 0 77352 10 2 3 Y Y N N N 2 0 0 78524 8 2 4 Y Y N N N O 2 0 82846 8 2 5 Y Y N N N 4 0 0 83126 12 26 Y Y N N N O 1 0 84382 6

40

Figure 2.1 Diagnostics for conditionally Gamma intensity model

M | < . 1 . , , 1

m- +

m- -•

J + 十 + + + -

J

1 狼- ^ -t ； 得十

^ ^ 一 一 i 娘 - 一 . . . . . 一 I 巧 一'—

iiiiflgr"^ s. ^ M M t f W E T n , „ .,., ,1,,,, .,.,,.,•....,.,.., I.,,.. ,1 I J gs：

� f T . � 2 4 8 B i c T ^ S 14

41

Chapter 3 Contract Specifications and Option

Evaluation

3.1 The Contract

Theme park operators and outdoor event promoters (for example concerts, fairs, and

so on) are at risk to precipitation. To offset this risk they may purchase RED (Rain

Event Day) protection with notional values corresponding to lost gate receipts and

concession income. They can purchase insurance or derivative calls on REDs, which

are defined as days with 'enough' rain. Standard units of measurement are

well-established. The imperial standard, used in the US, is based on inches/tenths of

an inch, while the metric standard, used in Europe and Asia, is based on

centimeters/millimeters. Here we adopt the Asia standard with measurements of

millimeters (denoted as ‘mm，）for rainfall amounts.

1. Contract Period

Although heavy rain is not uncommon at any time of the year in Hong Kong, it

occurs most often during the summer months. Indeed, close to 80 per cent of the

annual rainfall occurs between June and October. As stated in Chapter 1，high

attendance for theme parks in Hong Kong also occurs during the same period. It is

reasonable to consider the period from June to October as the contract period,

similar to CDD options in US energy market.

42

2. Contract Type

When there is excessive rainfall, the attendance rate in theme parks drops and the

revenue decreases. When there is little or even no rainfall at a specific day, the

attendance rate maintains stable. This observation indicates a need for an upper limit

in the rainfall event; however, a lower limit is not necessary, especially when the

distribution for rainfall intensity is a distribution left-censored at zero. Therefore, a

vanilla call option on the rainfall amount (mm per day) can be a fundamental tool to

manage one day's weather risk with such a structure. In our case, theme parks look

to stabilize their revenue streams over a much longer period of time than one day,

e.g. from June to October, when the seasonality shows that excessive rainfall occurs

during the period. Therefore, a risk management tool is needed to handle not a single

day's weather risk, but a month's or a season's weather risk. A cap is designed to

provide insurance against the risk caused by the underlying weather variable rising

above a certain level for a pre-detemiined consecutive period. It is a mechanism

designed to off-write the adverse economic effects whenever the underlying exceeds

this limit. In short, a cap is a combination of many call options with successive

expiring dates with the payment at the end of the period. Instead of purchasing a

large bunch of call options expiring at different days, the management can purchase

a cap to insure the weather risk for a certain month or even a season.

An Asian option (also called an average look-back option) is also a choice for

managing weather volatility. The payoff of an Asian option depends on the average

price of the underlying rain event over a certain period of time as opposed to at each

rain event day in a cap. Asian option contracts are attractive because they tend to

cost less than ordinary American options and caps. However, to better hedge the

43

weather risk, pros and cons should be weighed by using average (Asian option) of

the underlying variable or by using day-to-day measure of the underlying variable

(cap). In this thesis the designed option contract and the simulation process are

based on the rain event cap.

In line with US contract specifications, we can define the contract as in Table 3.1.

3.2 The Monte-Carlo Simulation

A risk management program requires more than the price of a security, A security's

probable payout distribution is also necessary. The payout distribution is based on

the distribution of the underlying, the rainfall. This and the absence of a commodity

or security underlying the weather derivative is why Black-Scholes modeling is not

useful in weather risk analysis. We proposed a model that simulates time series of

rainfall amounts in future seasons in Chapter 2, using the observed historical

sequence to define the characteristics of the population from which the sequence

will be drawn. Now we will use the results to describe the payout of the weather

derivatives described in the first part of this Chapter.

Different simulation methods are suitable for different options. Benth Dahl and

Karlson (2003) proposed simulation methods for options in the commodity and

energy markets. Dwight, Gautam and David (1997) and Linetsky (1997) proposed

the Monte Carlo approach for path-dependent options. Other related works like

Broadie and Glasserman (1997), Hartinger and Predota (2003), Schoutens and

Symens (2003) all suggested different simulation methods for pricing exotic options.

44

In this study, the price of the derivative, with an underlying fitting to the model in

Chapter 2，is theoretically equal to the expected value of the possible future payoff

plus some premium. This pricing approach, described as the 'Expected Discounted

Value Approach' in Chapter 1, is the most straight-forward method; it has few

assumptions but heavily relies on the accuracy of the underlying modeling. We can

design a Monte Carlo Simulation, sample a random path for the rainfall amount R

from the contract start date to the end of the contract, calculate the payoff from the

derivative contract and repeat the steps until we find a distribution of the sample

payoff from the derivative. The mean of the sample payoffs is discounted at the

risk-free rate to obtain an estimation of the value of the derivative.

3.2.1 The Rainfall Event

The Monte Carlo Simulation for R, in the event quarter is based on the discrete-time

version of Equation 2.10 for the binomial occurrence process

where

= (-0.1003) + 1.167A,+ 0.2279 log(yt.i + 0.10)

+ 0 . 0 0 0 5 7；+ 0 . 0 0 3 2 r j + 0.00447:v

+ [0.1665 sin(27rr/365) + -0.7565 cos(27r"365)

45

+ 0.1221 siii(47rr/365) + -0.0936 cos(47r"365)]

+ [0.162 sin(2;r"365) + 0.4202 cos(27r"365)

+ (-0.093) shi(47it/365) + 0.1187 cos(47r"365)

+ 0.0382 sin(67c//365) + -0.1575 cos(67r"365)

+ -0.1076 sin(87rr/365) + 0.0229 cos(67rr/365)]y,.,

+ [(-0.0172) sin(2;r//365) + (-0.0437) cos(2;r//365)] logO,., + 0.10)

and Equation 2.15 for the intensity process with a conditional gamma distribution,

with

p(x I / / ’ r ) = exp ( - rx 广丨 / r O ) for X > 0, /V > 0 and r = 0.6246

where

/ “ •y , - i，";)=exp(<(y ,_X) )

= 1.1298 +0.7722_/•“+0.0231 凡1

+ [(-0.2381) sin(27r^/365) + (-1.3663) cos(2;rr/365)

+ 0.0086 sm(47r,/365) + (-0.105)cos(47r//365)

+ [0.10521 sin(67r,/365) + 0.1297 cos(67r"365)

+ 0.0142 sin(87r//365) + (-0.0372) cos(87rr/365)]

+ [(-0.07578) sin(2;rr/365) + 0.46146 cos(27rr/365)]y,.,

+ [0.0055 sin(27r"365) + 0.0134 cos(27r"365)]凡i

46

The first step is to predict the rainfall amount during the next period of observation

(t=T+l) given the information presently available (t=T). The occurrence and

intensity model generated in Chapter 2 are already in a finite difference form. The

simulation starts with the occurrence process. The occurrence process follows a

binomial distribution. Given the previous day's rainfall event, we can easily simulate

the probability of rainfall occurrence at time T with Equation 2.10. However the

intensity simulation is a little different. The intensity process follows a gamma

distribution. The previous day's rainfall event and typhoon duration is incorporated

to obtain the mean of the gamma distribution in Equation 2.15. After that, we

generate a random number from a gamma distribution with the shape parameter r as

the rainfall intensity at time T. This part contributes a random factor in the

Monte-Carlo simulation and eventually forms the distribution of the payoff for the

next step.

We can obtain a time-line of simulated rainfall events by repeating the above steps

for the entire contract period. This result can be used in the second step to estimate

the claim payoff.

3.2.2 The Aggregate Payoff

The option contract is a typical path-dependent option. For the rainfall cap, the gain

at maturity will be

(3.1) /=0

47

N is the notional amount per contract. We assume A^=$l for the simulation. Risk

managers can adjust the units purchased to match the park attendance fluctuation. Rt

is the rainfall amount observed at time t. K is the strike level defined in the contract.

The payoff of a cap takes place at the end of the contract period. During the contract

effective period, whenever the observed rainfall amount exceeds the strike level, the

payoff is accumulated and counted towards the maturity under the risk-neutral

probability measure. The result of a one-time simulation is a path of rainfall events

from the contract start date (/=0) to the end of the run-off period (t=T. T=30 or 31

days for monthly contracts. T=153 days for seasonal contracts). The aggregate

payoff can be simulated by accumulating the entire possible payoff at each Rain

Event Day.

Over the simulation process, the rainfall occurrence and intensity process are based

on the previous day's simulated results while the typhoon signal durations for each

day are assumed given. We take the daily average typhoon signal duration in

minutes for the past 58 years for the simulation. The discount factor is set at

6% per annum.

3.2.3 Some Simulation Results

The simulated event paths start at point zero (with no rainfall) at time t=0. For each

strike price with certain duration, we generate more than 1000 paths to form a payoff

distribution, until the differences of the final results are trivial. The simulation

results show that the price for seasonal contracts is lower than individual monthly

contracts added up together, probably because the monthly contracts provide more

48

flexibility to risk management.

As an example, we simulate a contract with the strike level at 30, 50 and 100mm per

day, for monthly contract as well as the seasonal contract. The results are listed in

Table 3.2.

We also provide a summary table for option payoff if the design cap contract is

applied to Hong Kong daily rainfall data from 1947 to 2001. The monthly and

seasonal results are listed in Table 3.3. The listed payoffs are all 55 year average,

with standard deviation of each contract type. Observing from the standard

deviations, we find that yearly rainfall fluctuations are quite obvious.

The simulation results are purely theoretical and aim to provide an idea of further

applications for theme parks in recreation industry. It should be noted that there are

many factors left uncounted, such as the transaction cost, the origination fee and

other closing costs.

3.3 Further Applications

In this part, we are trying to find a causal effect between RED and the park

attendance rate. Such a causal effect analysis is essential to hedge the weather risks

to theme parks.

Before a model analysis between the visitor flow and the rainfall event is earned out,

the summary statistics of the Ocean Park daily visitor flow from July 2000 to June

49

2003 is provided in Table 3.4 and Figure 3.1. The weather categories in Table 3.4

and Figure 3.1 are categorized by overall weather condition, not rainfall amount only.

The statistics show the inverse relationship between the weather condition and the

park attendance. Due to the high volatility of the daily park attendance, a clear

pattern of the causal effect is difficult to identify by the summary statistics only. But

they provide some useful information. Firstly there is a trend that the local visitors

are more easily affected by severe weather. Observing the percentage change among

weather categories (local visitors/total visitors) in Table 3.4’ the percentage of local

visitors decreased while the percentage of non-local visitors increased with the

worsening of weather. Secondly, the average number of daily non-local visitors does

not change much among the weather categories, except for the category “Rainy’，’

pointing to the possibility that non-local visitors may not be influenced by bad

weather conditions so much as local visitors. Other concerns for non-local visitors,

such as their staying period, other view attractions and group traveling plans should

also be taken into consideration. With the above observations we can possibly

assume that local visitor numbers are more sensitive to weather conditions. The

assumption can be verified by comparing the non-local visitor numbers with data

from Hong Kong Immigration Department regarding total number of visitors and

their average remaining time. In the following model, we use the total number of

visitors (local and non-local) to analyze the causal effect between park attendance

and rainfall amount.

Assume a linear relationship exists between the weather variable and the attendance.

A, +a + s, (3.2)

50

Where At is the park attendance and Rt is the rainfall amount at time t We can make

a comparison between the park attendance and the rainfall amount of the same time

period as specified in the option contract specifications, June to October. We can

follow the linear regression method to estimate the parameters a and |3 in Equation

3.2 and measure the goodness of fit to see whether such causal effect is significant

or not However, note that the distribution of the park attendance data is

left-censored at zero. A censored regression model, or a tobit model, can be used in

our analysis. The formulation is usually given in terms of an index function,

= * + « + ff,

4 = 0 if A； <=0 (3.3)

A丨=A: if A; > 0

In Equation 3.3 At refers to the censored data, which is the park attendance we can

observe directly from the data set. The index variable, or the latent variable A : is the

uncensored variable. If data are always censored, the mean of the index variable will

not convey much information about the distribution. Estimation of the tobit model is

usually by maximum likelihood estimation.

We apply the Ocean Park attendance number for the period from July, 2000 to June,

2003. The analysis will be more illustrative if we have more data of the park

attendance number. The following regression analysis is performed for the

consecutive three years period, as well as for the contract period (June to October)

per year for the three years. Tobit model is applied for the time period with zero park

attendance. For each of the regression analyses, a comparison in the goodness of fit

is made on the original data set and the truncated data set. At the end of the chapter,

51

the regression results, including a and (3 values and the residual values are listed in

Table 3.5. Figure 3.2 to 3.7 give visual presentations of the correlation between daily

attendance and the rainfall amount in millimeters from July, 2000 to June, 2003

yearly.

Observing the figures, the scattering indicates a negative relationship between the

rainfall amount and the park attendance. However, the relationship is weak as the

fitting is not well-observed. According to most of the research works in operational

research and leisure science studies such as Kemperman et al. (2000) and Banks

(2002), weather materially affects the park attendance, but more or less the weather

factor influence the attendance rate together with other factors. As we cannot

eliminate from the model the impact of other factors, the deviations are quite

obvious, especially with the days of very little rainfall when other factors have

played a more important role. These 'other' factors, from public holidays,

transportation fee, individual income, to government promotion, economic

conditions and other ad hoc situations, such as the SARS or avian flu outbreak, all

tends to influence the park attendance. There are already many previous works and

complicated models for the visitor incentives and park attendance control. As these

factor analyses are beyond our range of research, we are not going to use a

multi-regression model to analyze the theme park visitor flow. However, some

previous studies of theme park attendance analysis, especially the works of

Kemperman, et al. (2000), suggest that the relationship between an influence factor

and the visitor flow should be most obvious when that factor is of an extreme nature.

In our study, according to this theory, the correlation should become clearer when

there is a large amount of rainfall Therefore in the analysis we also run a regression

for the truncated rainfall amount data and the visitor flow for each of the above

52

regressions. The rainfall amount data is truncated at 0 mm (eliminating the days

without rainfall) and 10 mm to see whether there is an improvement in the goodness

of fit. If an improvement in the goodness of fit is observed, it is an indicator for the

causal effect to hold.

The regression results for a and P are listed in the Table 3.5. The Least Squares

method would estimate the relationship between the weather variable (rainfall

amount) and the park attendance by minimizing the sum of the squared errors

between predicted events and actual events. R-square values and adjusted R-square

values for each regression are also listed. R-square can take on any value between 0

and 1，with a value closer to 1 indicating a better fit. When we truncate the data,

there is a significant decrease in the number of observations. So in our analysis, the

degrees of freedom adjusted R-square is compared instead of ordinary R-square

values. This statistic adjusts the R-square based on the residual degrees of freedom.

The residual degree of freedom is defined as the number of observations minus the

number of fitted coefficients estimated from the response values. The adjusted

R-square statistic is generally the best indicator of the fit quality when you add

additional coefficients to your model or you reduce the number of observations.

The adjusted R-square statistic can take on any value less than or equal to one, with

a value closer to one indicating a better fit.

In Table 3.5，p of the regression represents the slope that is of most interest. The

slope of this relationship would be the notional, or tick, value of the weather hedge.

So the values provide some information for the risk management team about the

hedging strategy. The negative relationship is quite clear in the figures, but the

fitting does not appear to be good. In all regression analysis, the p values range from

53

-25 to -55, mostly around -40. All the R-square and adjusted R-square statistics are

far away from the standard value one. Although the fitting is not well-observed, the

regressions with truncated rainfall amount data have better adjusted R-square values

than the original data set. This is an indicator that rainfall amount truncation

improve the goodness of fit in the regression analysis.

We also suggest that the regression analysis be carried out with data truncations of

the independent variable at the option strike prices. The regression analysis is also

an estimation of the relationship between the option payoff (at notional value $1)

and the park attendance. The shortcoming of the regression is the lack of historical

park attendance number. Because observations are limited, we perform regression

analysis (or tobit model if there is zero park attendance) with rainfall amounts

truncated at 0 mm and 10 mm. The risk management team of a theme park, while

having the full range attendance data on hand, may come to a more accurate and

satisfying result. They can apply the derivative tool together with the current

admission price (which tend to fluctuate when the policy changes) to form a risk

management strategy to hedge against the adverse weather effect The correlation

test can also be carried out against RED indexes on an aggregate, average or even

critical day basis to consider the risk management tool to be used.

54

Table 3.1: Contract Specifications

OPTIONS ON FUTURES

Contract Size: $1 times the rainfall amount in millimeters per Rain Event Day" Minimum Price Increment: 1mm in rainfall amount Tick Value: %\m Seasonal Contracts Traded: June through Oct, Monthly Contracts Traded: Jun, Jul, Aug, Sep, Oct Trading Hours: Mon. - Fri. 10A.M. to 12:30P.M. 2:30A.M. to 4:00P.M. Currency: Contracts settled in HK dollars. Strike Price Interval: Monthly Contracts

+30 mm, +50 mni, +100 mm, Seasonal Contracts

+30 mm, +50 mm, +100 mm, Exercise: European Style Settlement: On the first exchange business day at least 2 calendar days after the contract month/season.

14 Records observed from four stations (the Hong Kong Observatory Headquarters, King's Park Meteorological Station, Kai Tak Airport Meteorological Office and Chek Lap Kok Airport Meteorological Office)

55

Table 3.2: Simulation Results

Strike at 30mm

June July August September October Seasonal Contracts 1261.19 Monthly Contracts 337.7586 | 275.2234 | 303.164 | 223.6847 | 95.96

Strike at 50mm

June July August September October Seasonal Contracts 554.4887

Monthly Contracts 155.7696 116.3938 131.3848 100.0746 45.63247

Strike at 100mm

June M y August September October Seasonal Contracts 134.7008 Monthly Contracts 46.4036 | 26.01982 | 33.52973 | 22.61892 | 12.80909

56

Table 3.3: Historical Payoff if seasonal and monthly contracts applied ^

June July August September October

Seasonal Contracts Strike at 30mm

Mean ‘ 682.9873

Standard Deviation 277.8519

Monthly Contracts Strike at 30mm

190.6945 133.0273 171.7764 135.4327 52.05636

Standard Deviation 163.741 125.6531 161.6923 125.4703 1 0 6 . 8 2 ^

Seasonal Contracts Strike at 50mm Mean 436.9418

Standard Deviation 210.1873

Monthly Contracts Strike at 50mm

Mean 129.2036 78.66545 107.7145 86.11455 35.24364

- S t a n d a r d Deviation 137.2445 92.84565 130.6988 93.0585 84.9400

Seasonal Contracts Strike at 100mm Mean 一 164.7109

Standard Deviation 122.3975

Monthly Contracts Strike at 100mm ^ ^ 55.27091 |"21.64727 43.97636 30.66 13.15636

Standard Deviation | 90.70543 | 48.09992 82.5722 52.80645 44.23891

Y e a r l y a v e r a g e p a y o f f listed in this table using data ranging from 1947 to 2001

57

Tabl

e 3.

4: S

umm

ary

stat

istic

s fo

r wea

ther

inf

luen

ce o

n pa

rk a

ttend

ance

fro

m J

uly

1，20

00 to

Jun

e 30，

2003

Aver

age

Dai

ly R

ainf

all

Loca

l N

umbe

r of

Visi

tors

To

urus

t Num

ber

of V

isito

rs

Tota

l N

umbe

r of

Visi

tors

Wea

ther

Cat

egor

y #

of d

ays

(in M

illim

eter

s)

Mea

n(%

) M

ax

Mitt

S.

D

Mea

n(%

) M

ax

Min

S.

D

Mea

n M

ax

Min

S.

D

Fine

54

6 0.

6944

5,

831

(63.

96)

26,9

17

725

4,38

5 3,

285

(36.

04)

18,6

64

110

1,97

9 9,

117

45,5

81

835

5,86

8

Fine

/C

loud

y 19

0 3.

0733

5,

233

(61.

91)

24,7

45

998

3,71

8 3,

220

(38.

09)

9,38

4 11

8 1,

696

8,45

3 30

,825

1,

116

4,80

4

Clo

udy

146

3.20

14

4,45

5 (5

7.42

) 20

,632

48

3 3,

959

3,30

4 (4

2.58

) 15

,770

69

2,

205

7,75

8 36

,402

57

0 5,

412

Clo

udy/

show

er

110

18.0

927

4,05

2(56

.13)

16

,114

39

5 3,

169

3,16

7(43

.87)

11

,015

83

1,

689

7,22

0 27

,129

61

5 4,

510

Show

er/R

ainy

82

39

.884

1 3,

217(

49.8

0)

10,8

33

12

2,56

5 3,

243

(50.

20)

11,0

80

4 2,

138

6,46

0 19

,379

16

4,

241

Rai

ny*

16

67.4

875

1,38

0(44

.72)

6,

379

0 1,

606

1,70

6 (5

5.28

) 3,

740

0 1,

185

3,08

6 8,

062

0 2,

364

Mon

day

Clo

sure

5

*2 d

ays

with

zer

o vi

sitor

due

to h

eavy

rai

nfal

l or

typh

oon

signa

l T8

or a

bove

in C

ateg

ory

"Rai

ny"

Rai

nfal

l Am

ount

(ra

m)

Tl(

min

) T3

(rai

n)

T8 o

r ab

ove(

min

) V

isito

r

2001

.07.

06

142.

1 0

620

820

0

2001

.07.

25

17.7

0

290

1,15

0 0

58

Figure 3.1: Summary statistics for weather influence on park attendance

Summary statistics for weather influence and daily park attendance

10,000 � 80 9 1 1 7

9,000 ^ — 腿 j 70 8,000 一 i p i to

f ~ | 7220 . 60 B 7 , 0 0 0 — : ^ ： ~ L 岂

5： � mm I - 6 , 0 0 0 — i i ： 50 I

w 5 , 0 0 0 \ 4 0 5

I 4 , 0 0 0 3 0 麗 二 JauPtK ：：疆 1,000

^ ��es^ > 办

\。\ 。\ 妒

c / 矛。》

Weather category

Local m m Tourist ：：： Average Daily Rainfall

59

Table 3.5: Regression Results for rainfall amounts and park attendance

Untruncated Truncated at 0 mm Truncated at 10 mm 200 0 a [3 a I p a | (3

7706 -37.5 6843 -24.62 ~~6928 -25.76 R-square

0.05168 I 0.05004 | 0.1596

Adjusted R-square

0.03738 0.04384 0.1272 200 1 a p a p g | (3 ~~

9218 -54.852 8713 -50.96 7888 -40.88 R-square

0.1028 I 0.1691 I 0.1735

Adjusted R-square

0.1203 0.1243 0.1601 200 2 a P a p a | (3 一

9932 -40.47 9679 -37.11 8841 -26.13 R-square

0.04411 I 0.05267 | 0.1066

Adjusted R-square

0.03778 I 0.04414 | 0.07957 ~ ~

60

Figure 3.6 Correlation between park attendance and rainfall amount in 2002

,.....,..,,..,....一 ^ I I I

* - Rainfall Amount vs Park Attendance July to Oct 2000 Linear y=-37.5*x+7706 -

I * 1 I I I

I I I I

I I I I

I I I I

I I I I

t i l l 2 4 --；--- - - - r - -；-

I I i I k i l l

也 • I I • 盆 ， i ； ；

_ _ 丨 • •

I ^ ^ r 一 - - — r … … … - - ：

* 丨 I I

凌 t ’ -•- L 1 J--

丨 I i Q态 J … … … … … —-

、 [ 卜 - i I > I I ............

1 - I .丄... n m m m

R射 t M A 咖 _ (mm)

Figure 3.3 Correlation between park attendance and rainfall amount (truncated at 10 mm) in 2000

IIIIIIIIIIK^^^ 1 I I I I ‘ I

麵 ^ ； * Rainfall Amount vs. Park Attendance July to Oct 2000 ―®----- --2 M - — — ： — — — ~ - Linear y=-25.76*)<-f6928 -

� � … f : f 1 i- : … … -份 I I I I I I I

i � i 丨 丨 ：

1 ， ' [ - - 1 - - - - - - - 1 — — i - - - - - - - ”……:……卞… • • I I • J I

5 i i ； i i ; I ^ 1 1 -I : I 1

. A I . . • ' 1 1 1 1 ； :

, 1 * * 1 I I I I I

— i - T ^ c - ^ ^ ‘ ‘ I I I I

I • I ！ ！ • I “ • I I I I I I :

I ！ . ！ I I " I I

20 4 0 60 8Q m 1 邀 m Amount at1&(mm)

61

Figure 3.4 Correlation between park attendance and rainfall amount in 2001

: . . � • -- I … I I I 丨 I I

� - - - t * * Rainfall Amount vs Park Attendance June to Oct 2001 �.:::.:.::.:.:.::.:.:.:.:_:.:.::.:.: • — Linear y=-54.852*x49218

——:——？……1——？…——？-t . • I , • ； i ‘ ‘ ‘ ‘ ‘ I , I I I I I I 1 I I I I I I , 1 * 1 I I I , 1 • 丨 丨 丨 丨 ： ： ‘

^ I S - ；- ； ； 二 J L 募 I* I I I I I I I

1 ^ ： - i i ： ： ： i ^ . r ‘； ； ； • ； ； I i f r - " 7 • r 1 r "! L-

‘ 丨 卜 丨 丨 。‘备 \ — i " “ • V - “ -； •；:~r -；—-厂十

1 1 I - 1 二 i ‘ T H ^

% ； ； • ； r ] 卜

£ 1•、如 40 sd m 1 節 m i 难 n^ifM^mrnntimm)

Figure 3.5 Correlation between park attendance and rainfall amount (truncated at 10 mm) in 2001

1 I I I I I = 3 - t 故 州 ； ‘ Rainfall Amount vs. Park Attendance June to Oct 2001

！ ——Linear y=-4Q.88*x+7888

r T ^ - j -……--： 丨--丨

麵 丨 • • ‘ • I

份 t.S- — ‘ ^ ‘ 1 g : ; 丨 丨 丨 ； ：：

I i • i I I 丨 丨 i

I , f : ； . ； i i 义 i " V " 1 “ 7 "i I 1 I - • ' > . i ： ； ： ；

* ; i i • ' i i I . I I I I •

汉经…+—厂十：：^：：^^^^：^^^^^^^^— ：--,-；-

、 。 作 ： i : - : ： ： " ： : 0 … 、 ： 1 1 -! ； -丨

‘ ！ U m m m m w m

- H-^M^W Pmmnl trunc^teii 毎 1 0 (mm)

62

Figure 3.6 Correlation between park attendance and rainfall amount in 2002

T 1 1 I I.…• I • “ I I ^ * [ Rainfall Amount vs Park Attendance June to Oct 2002

： 丨— — L i n e a r y=-40.47*x+9932 “ I I I I 1 I 1 1 I I I I I I I I I

I I I 1 I I I I

r 1 T 1 T 1 1 r-i ^ I I I I I I I I 在 I I • • • • • ‘ ： 4 ： ； ； ； ； ； ； ；

g ——f——；——i——:——:——;——-i——H 运 ： ； ： ： ： ： ： ： ： ：

I … … I … - - + - - 4 … • I … 丨 ： 二 ： ； ； ！ ；

. i 1 — 1 i 1 j i -i i-| I] m 40 m 80 ^ 100 m i 聪 m

Rainfall Amount

Figure 3.7 Correlation between park attendance and rainfall amount (truncated at 10

mm) in 2002

— I — I I I I I I I � J * Rainfall Amount vs Park Attendance June to Oct 2002 � … L i n e a r y=-26 13*x4a841

2 5 1 1 - -> 1 1 1 1~

s ： ^ - - ; … … ] - - - - - - 」 丨 … … ] - - - - - — — i — — i — — 仁 I • i » ‘ 1 I I

卷 ！ 丨 ： ！ ： 丨 ！ ：

杰 … + … + … ― … - - 叫 … … 卞 … - 卞 … 十 ^ ： 二 ； ； ； ； ； S r ： 丨 ， ： 。 丨 ； ；：

气 “； T - ； ； ； -：：：：：；：：-：1>：；：-：；：；：-：>>>：：：：>：：：：：：：：：：：：：：： A 农 I … I I I I ：：：：：：：：：：：：：：：：：：：：：：：：：：：：：：：：：：：：

： ； ^ ； ； ： » ； ；

二 — 1 — — - ； - - - - - | - ； … 4Q m m %m m m

- M^oufit trtmc•级 at 10 (mm)

63

Chapter 4 Concluding Remarks and Discussions

In this thesis we attempt to design a derivative risk management tool for the

recreation industry in Hong Kong. We consider an ARIMA model with seasonal

effects and a Markov model with transitive density. After an analysis of the weather

risks in Hong Kong, we form and develop a Markov model with transitive density

for predicting rainfall amounts. This model was first proposed by Grunwald and

Jones (2000) and it is generally used for meteorological data with a mass cluster at

zero. With some modifications the model can meet the requirements of different

geographic locations and climates. We further elaborate the model and add another

factor, the typhoon signal durations, to fit the subtropical climate in Hong Kong.

This predictive model for the rainfall amount is used in further simulation processes

for a weather cap contract. The simulation gives an evaluation of the option (cap)

contract we proposed. An Asian option based on rainfall amount costs less but

provides with lower payoff. It could be studied and applied for different needs in the

weather market.

The applications of the option as a risk management tool seem to be more

complicated. First we analyze the correlations between the weather categories and

the visitor flow. The analysis is based on total visitor numbers from Ocean Park and

the weather categories by rainfall amount. Separating the local and non-local visitors

and analyze their correlation with the weather category may reveal additional

information, as we have already found a clearer negative relationship pattern

between local visitor numbers and the rainfall event than the non-local category. We

64

further assume that a linear relationship exists between the rainfall event and the

visitor flow, which is censored at zero. A tobit model is applied, therefore the

notional amount of the weather option will be the slope of the censored linear

regression. The estimated notional amount can provide some insight for the risk

management strategy, but there are some shortcomings to this methodology.

One shortcoming is that it will be difficult to identify the complicated factors

affecting the visitor flow to theme parks without the understanding of theme park

management and operational analysis. Another shortcoming is that the consumer's

preference is partly influenced by variety-seeking and seasonality effects. A further

shortcoming is that the visitor flow data we obtained is limited. For a risk

management team of a theme park, a full history of park attendance number may

reveal more useful information to more accurately perform the weather risk

management strategy better.

The derivative risk management tool introduced in this thesis can be used together

with a further analysis of the revenue stream and other risk-control mechanisms. As

more weather risk management tools enter the market, theme park will be able to

choose among different weather indexes to perform the analysis and the weather risk

can be more easily hedged.

65

References

Aitchison J. 1955. On the distribution of a positive random variable having a discrete probability mass at the origin. Journal of the American Statistical Association 50: 901-908.

Akaike H. 1974. A new look at the statistical model identification. IEEE Transactions on Automatic Control 19: 716-723.

Baldi P.; Caraniellino L.; lovino M.G. 1999. Pricing General Barrier Options: A Numerical Approach Using Sharp Large Deviations. Mathematical Finance. 9(4). 293-321(29) Blackwell Publishing

Banks, E. 2002. Weather risk management: markets, products, and applications. Element Re Capital Products, Inc.

Barrieu, Pauline; Karoui E. Nicole. 2002. Optimal design of derivatives in illiquid markets. Quantitative Finance. 2:181-188.

Benth F. E.，Dahl L.O., Karlsen K.H. 2003. Quasi Monte-Carlo evaluation of sensitivities of options in commodity and energy markets. International Journal of Theoretical and Applied Finance. 6(8). 865-884(20) World Scientific Publishing

Broadie M.; Glasserman P. 1997. Pricing American-style securities using simulation. Journal of Economic Dynamics and Control. 21(8). 1323-1352(30) Elsevier Science

Brody, D.’ Syroka J., Zervos M. 2002. Dynamic pricing of weather derivatives, Quantitative Finance. 2: 189-198

Cao M.; Wei J. 2000. Pricing the weather. Risk Magazine, 67-70 Davis, Mark. 2001. Pricing weather derivatives by marginal value. Quantitative

Finance. 1: 305-308. Duong QP. 1984. On the choice of the order of autoregressive models: a ranking and

selection approach. Journal of Time Series Analysis 5: 145-157. Dwight Grant; Gautam Vora; David Weeks. 1997. Path-Dependent Options:

Extending the Monte Carlo Simulation Approach. Management Science 43(11): 1589-1602,

Figlewski, S; Levich, M. Richard. 2002. Risk management: the state of the art. Boston: Kluwer Academic.

Gabriel KR, Neumann J, 1962. A Markov chain model for daily rainfall in Tel Aviv. Quarterly Journal of the Royal Meteorological Society 88: 90-95.

Geman, H. 1999. Insurance and weather derivatives: from exotic options to exotic imderlyings. London : Risk

Gregory JM, Wigley TML, Jones PD, 1993. Application of Markov models to

66

area-average daily precipitation series and interannual variability in seasonal totals. Climate Dynamics 8: 299-310.

Grunwald, G.K., Jones, R.J, 2000. Markov models for time series with mixed distribution. Environmentrics 11.

Hanselman, Duane, Littlefield, Bruce. 2001. Mastering MATLAB 6: a comprehensive tutorial and reference. Upper Saddle River, NJ . : Prentice Hall

Hartinger J.; Predota M. 2003. Simulation methods for valuing Asian option prices in a hyperbolic asset price model. IMA Journal of Management Mathematics. 14(1). 65-81(17) Oxford University Press

Harvey AC. 1989. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press: Cambridge.

Hastie T, Tibshirani R. 1990. Generalized Additive Models. Chapman and Hall: London.

Hyndman RJ, Grunwald GK. 2000. Generalized additive modelling of mixed distribution Markov models with application to Melbourne's rainfall. Australian and New Zealand Journal of Statistics ’ 42(2): 145-158.

Jones RH, Brelsford WM. 1967. Time series with periodic structure. Biometrika 54：

403-408. Katz RW. 1977a. Precipitation as a chain-dependent process. Journal of Applied

Meteorology 16: 671-676.

Katz RW. 1977b. An application of chain-dependent processes to meteorology. Journal of Applied Probability 14: 598-603.

Kemperman, Astrid; Borgers, Aloys; Oppewal, Harmen;Timmermans, Harry J. P. 2000. Consumer choice of theme parks: a conjoint choice model of seasonality effects and variety seeking behavior. Leisure Sciences, 22:1-18,

Linetsky V. 1997. The Path Integral Approach to Financial Modeling and Options Pricing. Computational Economics. 11. 129-163(35) Kluwer Academic Publishers

McCullagh P, Nelder JA. 1989. Generalized Linear Models�2nd ed. Chapman and Hall: London.

Morgan, Byron J. 丁. 2000. Applied stochastic modeling. London: Arnold.

Co-published in the USA by Oxford University Press, New York

Myers, Raymond H, 2002. Generalized linear models: with applications in engineering and the sciences. Wiley-Interscience publication.

Nakamura, Shoichiro. 2002. Numerical analysis and graphic visualization with MATLAB�2nd ed. Upper Saddle River, N.J.: Prentice Hall PTR

Rebonato R.; Cooper I, 1998. Coupling backward induction with Monte Carlo

simulations: a fast Fourier transform (FFT) approach. Applied Mathematical

67

Finance. 5(2). 131-141(11) Schoutens W., Symens S. 2003. The pricing of exotic options by Monte-Carlo

simulations in a levy market with stochastic volatility. International Journal of Theoretical and Applied Finance. 6(8): 839-864(26) World Scientific Publishing

Smith JA. 1987. Statistical modeling of daily rainfall occurrences. Water Resources Research 23: 885-893.

Stem RD, Coe R. 1984. A model fitting analysis of daily rainfall data. Journal of the Royal Statistical Society, Series A 147: 1-34.

Woolhiser DA. 1992. Modeling daily precipitation - progress and problems. Statistics in the Environmental and Earth Sciences, Walden AT , Guttorp P (eds). Halsted Press: New York; 71-89.

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