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Risk analysis and hedging in incomplete markets

DISSERTATION

Presented in Partial PulIillment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

George Argesanu, M.Sc.

$* * * * *$

The Ohio State University

2004

Dissertation Committee:

Approved by

Prof Bostwick Wyman, Ph. D., Adviser

Prof Robert Brown, Ph. D.

Prof Richard Evans, FSA

Adviser

Department of Mathematics

$eggc$ Copyright by

George Argesanu

2004

ABSTRACT

Variable annuities are in the spotlight in today's insurance market. The tax de-

ferral feature and the absence of the investment risk for the insurer (while keeping

the possibility of investment benefits) boosted their popularity. They represent the

sensible way found by the insurance industry to compete with other stock market

and Iinancial intermediaries. A variable annuity is an investment wrapped with a life

insurance contract. An insurer who sells variable annuities bears two different types

of risk. On one hand, he deals with a Iinancial risk on the investment. On the other

hand there exists an actuarial (mortality) risk, given by the lifetime of the insured.

Should the insured die, the insurer has to pay a possible claim, depending on the op-

tions elected (return of premium, reset, ratchet, roll-up). In the Black-Scholes model,

the share price is a continuous function of time. Some rare events (which are rather

frequent lately), can accompany jumps in the share price. In this case the market

model is incomplete and hence there is no perfect hedging of options. I considered a

simple market model with one riskless asset and one risky asset, whose price jumps

in different proportions at some random times which correspond to the jump times

of a Poisson process. Between the jumps the risky asset follows the Black-Scholes

model. The mathematical model consists of a probability space, a Brownian motion

and a Poisson process. The jumps are independent and identically distributed. The

approach consists of defining a notion of risk and choosing a price and a hedge in

order to minimize the risk. In the dual market (insurance and financial) the risk-

minimizing strategies defined by Follmer and Sondermann and the work of Moller

with equity-linked insurance products are reviewed and used for variable annuities,

with death or living benefits.

The theory of incomplete markets is complex and intriguing. There are many in-

teresting connections between such models and game theory, while the newest and

maybe the most powerful research tool comes from economics, the utility function

(tastes and preferences).

iii

This is dedicated to my family.

ACKNOWLEDGMENTS

First of all I want to thank my advisor, Professor Bostwick Wyman, for his con-

stant help and guidance. His support and encouragement came always at the right

time.

Secondly, I want to thank Professor Boris Mityagin for his introductory classes to

the Iield of Iinancial mathematics. I have beneficiated greatly from the experience in

his classes. I also want to thank Professor Richard Evans for his kindness in opening

the door of the actuarial world and in sharing with us real insurance problems and

Professor Robert Brown for the intriguing experience in his probability class and for

accepting to be on this Committee.

I am also very grateful for the experience I had related to the annuities market

while working at Nationwide during the summer of 2002 and 2003.

And Iinally I want to thank my wife for her constant love, support, patience and

understanding and for all the help she's given me through the years. I also want

to thank my parents and generally all my family for all their efforts in bringing me

where I am today.

$v$

VITA

1975. . . . . . Born in Ramnicu Valcea, Romania

1998. . . . . . $B . S$ . University of Bucharest

2000. . . . . . $M . S$ . joint University of Bucharest and Free University of Berlin

2002. . . . . . $M . S$ . The Ohio State University

Publications

1. On the representation type of Sweedler's Hopf algebra, Mathematical Reports

$2 (52)$ , 263-268, 2000

2. Taft algebras are cyclic serial (with Csabo Szanto), Mathematica 44(67), 11-17,

2002

3. Risk analysis, ARCH 2004.1, Actuarial Research Clearing House, Risk Theory

section: 1-7, 2004

Fields of Study

Major Iield: Mathematics

Specialization: Financial Mathematics

TABLE OF CONTENTS

Page

Abstract ii

Dedication. iv

Acknowledgments. $v$

Vita vi

List of Tables ix

List of Pigures $x$

Chapters:

1. Introduction 1

1.1 Introduction and General Settings 1

1.2 Discrete time Iinancial mathematics 2

1.3 Continuous time Iinancial mathematics 7

1.4 Options in the Black-Scholes model 10

2. A product space 15

2.1 Introduction to the dual risk in unit-linked insurance products 15

2.2 Product space. 16

2.2.1 Financial backgroud 16

2.2.2 Insurance-Actuarial background 16

2.3 Combining the two markets 18

2.4 Disjoint pricing techniques 21

vii

Risk Analysis

3.1 Derivatives in incomplete markets. 23

3.1.1 Super-replication 23

3.1.2 Utility-based indifference pricing 25

3.1.3 Quadratic approaches 25

3.1.4 Quantile hedging and shortfall risk minimization 26

3.2 Description of the GMDB problem 27

3.2.1 The model 27

3.2.2 Other market models with jumps 32

3.2.3 Risk analysis 33

3.3 Game options in incomplete markets 42

3.4 Hedging insurance claims in incomplete markets 45

3.5 The combined model in the GMDB case 47

3.5.1 GMDB with return of premium 52

3.5.2 GMDB with return of premium with interest 53

3.5.3 GMDB with ratchet 53

3.6 Living benefits 57

3.6.1 VAGLB with return of premium 60

3.6.2 VAGLB with return of premium with interest 61

3.7 Discrete time analysis 62

3.7.1 Risk comparison 66

3.8 Multiple decrements for variable annuities 77

4. Results and concluding remarks 80

Appendices:

A. Kunita-Watanabe decomposition 82

Bibliography 84

viii

LIST OF TABLES

Table Page

3.1 Pricing formulas for living benefits contracts 76

LIST OF FIGURES

Figure Page

3.1 Random walk for stock price 34

3.2 Reflection principle for random walks 36

3.3 Sample stock price evolution 55

3.4 Binomial stock price process 67

3.5 Risk-neutral probabilities. 69

3.6 Call Option Price Process 71

3.7 Hedging Process. 72

3.8 Risk-minimizing trading strategy, $μ = 1$ 74

3.9 Risk-minimizing trading strategy, $μ = 0.5$ . 75

$x$

CHAPTER 1

INTRODUCTION

1.1 Introduction and General Settings

The approach throughout this thesis is based on the concept of arbitrage. It is a

remarkably simple concept and it is independent of preferences of the actors in the

Iinancial market.

The basic assumption is that everybody prefers more to less and that any increase in

consumption opportunities must somehow be paid for.

The core background for our exposition is the risk-neutral (probabilistic) pricing of

derivatives securities. A derivative (or contingent claim) is a Iinancial contract whose

value at expiration date $T$ (or expiry) is determined by the price of an underlying

Iinancial asset at time $T$ . In this chapter we discuss the basics for pricing contin-

gent claims. The general assumption of this chapter is that we are in the classic

Black-Scholes model, which means that, according to the fundamental theorem of

asset pricing, the price of any contingent claim can be calculated as the discounted

expectation of the corresponding payoff with respect to the equivalent martingale

measure.

1.2 Discrete time financial mathematics

In this section we will consider a discrete-time model.

We consider a Iinite probability space $(Ω, F, P)$ , with $| Ω |$ a Iinite number, and

for any $ω$ $\in Ω$ $P ({ω}) > 0$ . We have a time horizon $T$ , which is the terminal

date for all economic activities considered. We use a Iiltration $F$ of $σ$ -algebras

$F_{0} \subset F_{1} \subset ... \subset F_{T}$ and we take $F_{0} = {\emptyset, Ω}$ and $F_{T} = F = P (Ω)$ the power

set of $Ω$ . This Iinancial market contains $d + 1$ Iinancial assets. One is a risk-free

asset (a bond or a bank account for example) labeled 0 and $d$ are risky assets (stocks)

labeled 1 to $d$ . The prices of these assets at time $t : S_{0} (t, ω)$ , $S_{1} (t, ω)$ , $...$ , $S_{d} (t, ω)$

are non-negative and $F_{t}$ -measurable. Let $S$ $(t) = (S_{0} (t), ..., S_{d} (t))$ denote the

vector of prices at time $t$ . We assume $S_{0} (t)$ is strictly positive for all $t \in {0, 1, ..., T}$

and also assume that $S_{0} (0) = 1$ . We define $β (t) = \frac{1}{S_{0} (t)}$ as a discount factor.

We have then constructed a market model $Λ 4$ consisting of a probability space

$(Ω, F, P)$ , a set of trading dates, a price process $S$ , and the information structure $F$ .

Definition 1.2.1 A trading strategy (or dynamic portfolio) $φ$ is a $R^{d + 1}$ vector

stochastic process $φ =$ $(φ_{0} (t, ω)$ , $φ_{1} (t, ω)$ , $...$ , $φ_{d} (t, ω))_{t = 1}^{T}$ which is predictable: each

$φ_{i} (t)$ is $F_{t - 1}$ -measurable for $t \underline{>} 1$ , where $φ_{i} (t)$ denotes the number of shares of

asset $i$ held in the portfolio at time $t$ and which is to be determined on the basis of

information available before time $t$ (predictability).

Definition 1.2.2 The value of the portfolio at time $t$ is the scalar product

$V_{φ} (t) = φ (t) ċ S (t) = \sum_{i = 0}^{d} φ_{i} (t) S_{i} (t)$ $t = 1$ , 2, $...$ , $T$

and $V_{φ} (O) = φ (1) S (0)$

The process $V_{φ} (t, ω)$ is called the wealth or value process of the trading strategy

$φ$ . We call $V_{φ} (0)$ the initial investment of the investor (endowment).

Definition 1.2.3 The gains process $G_{φ}$ of a trading strategy $φ$ is given by:

$G_{φ} (t) = \sum_{x = 1}^{t} φ (x) [S (x) - S (x - 1)]$ , $t = 1$ , 2, $...$ , $T$

If we define $S - (t) = (1,$ $β (t) S_{1} (t)$ , $...$ , $β (t) S_{d} (t))$ the vector of discounted prices

we also have the discounted value process ${\bar{V}}_{φ} (t) = φ (t) \bar{S} (t)$ for $t = 1$ , 2, $...$ , $T$ and

we can see that the discounted gains process ${\bar{G}}_{φ} (t) = \sum_{x = 1}^{t} φ (x) [\bar{S} (x) - \bar{S} (x - 1)]$

reflects the gains from trading with assets 1 to $d$ only.

Definition 1.2.4 The strategy $φ$ is self-fifinancing, $φ \in Φ$ , if

$φ (t) S (t) = φ (t + 1) S (t)$ , $t = 1$ , 2, $...$ , $T - 1$

This means that when new prices $S (t)$ are quoted at time $t$ the investor adjusts

his portfolio from $φ (t)$ to $φ (t + 1)$ , without bringing in or consuming any wealth.

To prove the fundamental theorem of asset pricing we need the following results,

which are also interested and important by themselves:

Proposition 1.2.1 A trading strategy $φ$ is self Iinancing with respect to $S (t)$ if

and only if $φ$ is self-Iinancing with respect to $\tilde{S} (t)$ .

Proposition 1.2.2 A trading strategy $φ$ is self Iinancing if and only if

$\tilde{V} (t) = V_{φ} (0) + {\tilde{G}}_{φ} (t)$

The well-being of any market is given by the absence of arbitrage opportunities

arbitrage $= f r e e$ lunch).

Definition 1.2.5 Let $Φ_{0} \subset Φ$ be a set of self-Iinancing strategies. A strategy

$φ \in Φ_{0}$ is called an arbitrage opportunity or arbitrage strategy with respect to $Φ_{0}$ if

$P {V_{φ} (0) = 0} = 1$ and the terminal wealth satisfies

$P {V_{φ} (T) \underline{>} 0} = 1$ and $P {V_{φ} (T) > 0} > 0$

We say that a security market $Λ 4$ is arbitrage-free if there are no arbitrage oppor-

tunities in the class $Φ$ .

Next we introduce the notion of "risk-neutral probability" which also also central

in Iinancial mathematics:

Definition 1.2.6 A probability measure $P^{*}$ on $(Ω, F_{T})$ equivalent to $P$ is called

a martingale measure for $\bar{S}$ if the process $\bar{S}$ follows a $P^{*}$ -martingale with respect to

the Iiltration F. We denote $7^{\supset} (\bar{S})$ the class of equivalent martingale measures.

One proposition that follows quickly and is useful in proving Theorem 1.2.1 is:

Proposition 1.2.3 Let $P^{*} \in P$ $(\bar{S})$ and $φ$ a self-Iinancing strategy. Then the

wealth process $\bar{V} (t)$ is a $P^{*}$ martingale with respect to the Iiltration P.

The no-arbitrage theorem describes the necessary and sufficient conditions for

no-arbitrage and makes the connection between the real world (finacial market) and

theory (martingales):

Theorem 1.2.1 (No-Arbitrage Theorem) The market $Λ 4$ is arbitrage-free if

and only if there exists a probability measure $P^{*}$ equivalent to $P$ under which the

discounted $d$ -dimensional asset price process $\bar{S}$ is a $P^{*}$ -martingale.

The question now is how we use this theorem to price contingent claims. We start

with a definition:

Definition 1.2.7 A contingent claim $X$ with maturity date $T$ is an arbitrary

non-negative $F_{T}$ -measurable random variable.

We say that the claim is attainable if there exists a replicating strategy $φ \in Φ$

such that

$V_{φ} (T) = X$

The following theorem is the Iirst theoretical approach to pricing contingent

claims:

Theorem 1.2.2 The arbitrage price process $π_{X} (t)$ of any attainable contingent

claim $X$ is given by the risk-neutral valuation formula:

$π_{X} (t) = β (t)^{- 1} E^{*} (X β (T) | F_{t})$ for any $t = 0, 1$ , $...$ , $T$

where $E^{*}$ is the expectation taken with respect to an equivalent martingale measure

$P^{*}$

Theorem 1.2.2 says that any attainable contingent claim can be priced using the

equivalent martingale measure. So, clearly, "attainability" would be a very desirable

property of any market. So the next definition follows naturally:

Definition 1.2.8 The market $Λ 4$ is complete if every contingent claim is attain-

able.

The following theorem gives a nice characterization of a complete market:

Theorem 1.2.3 (Completeness Theorem) An arbitrage-free market $Λ 4$ is com-

plete if and only if there exists a unique probability measure $P^{*}$ equivalent to $P$ under

which discounted asset prices are martingales.

Let's summarize what we have seen so far.

Theorem 1.2.1 tells us that if the market is arbitrage-free, equivalent martingale

measures $P^{*}$ exist. Theorem 1.2.3 tells us that if the market is complete, equivalent

martingale measures are unique. Putting them together we get:

Theorem 1.2.4 (Fundamental Theorem of Asset Pricing)

In an arbitrary-free

complete market $Λ 4$ , there exists a unique equivalent martingale measure $P^{*}$

Finally, Theorem 1.2.2 gives us in the complet market setting the following:

Theorem 1.2.5 (Risk-Neutral Pricing Formula) In an arbitrage-free complete

market $Λ 4$ , arbitrage prices of contingent claims are their discounted expected values

under the risk-neutral (equivalent martingale) measure $P^{*}$

1.3 Continuous time financial mathematics

We start with a general model of a frictionless securities market where investors are

allowed to trade continuously up to some Iixed Iinite planning horizon $T$ . Uncertainty

in this Iinancial market is modeled by a probability space $(Ω, F, P)$ and a Iiltration $F$

of $σ$ -algebras $F_{t}$ , with $0 \underline{<} t \underline{<} T$ , satisfying the usual conditions of right-continuity

and completeness. We assume that $F_{0}$ is trivial and that $F_{T} = F$ .

There are $d + 1$ primary traded assets, whose price processes are given by stochas-

tic processes $S_{0}$ , $...$ , $S_{d}$ . We assume that $S =$ $(S_{0}, ..., S_{d})$ follows an adapted, right-

continuous with left-limits (RCLL) and strictly positive semimartingale on $(Ω, F, P, F)$ .

We also assume that $S_{0} (t)$ is a non-dividend paying asset which is almost surely

strictly positive and use it as a numeraire.

We denote by $M (P)$ the Iinancial market described above.

Definition 1.3.1 A trading strategy (or dynamic portfolio) $φ$ is a $R^{d + 1}$ vector

stochastic process $φ (t) = (φ_{0} (t), φ_{1} (t)$ , $...$ , $φ_{d} (t))$ , $0 \underline{<} t \underline{<} T$ which is predictable

and locally bounded.

Here $φ_{i} (t)$ denotes the number of shares of asset $i$ held in the portfolio at time t-

determined on the basis of information available before time $t$ . This means that the

investor selects his time $t$ portfolio after observing the prices $S (t$ - $)$ .

Definition 1.3.2 The value of the portfolio $φ$ at time $t$ is given by the scalar

product

$V_{φ} (t) = φ (t) ċ S (t) = \sum_{i = 0}^{d} φ (t) S_{i} (t)$ , $t \in [0, T]$

Definition 1.3.3 The gains process $G_{φ} (t)$ is defined by

$G_{φ} (t) = \int_{0}^{t} φ (u) d S (u) = \sum_{i = 0}^{d} \int_{0}^{t} φ_{i} (u) d S_{i} (u)$

Definition 1.3.4 A trading strategy is called self-fifinancing if the value process

satisfies

$V_{φ} (t) = V_{φ} (0) + G_{φ} (t)$ for all $t \in [0, T]$

We can define the discounted price process, the discounted value process and

discounted gains process with the help of the numeraire $S_{0} (t)$ .

Definition 1.3.5 A self-Iinancing strategy is called an arbitrage opportunity or

arbitrage strategy if $V_{φ} (0) = 0$ and the terminal value satisfies

$P {V_{φ} (T) \underline{>} 0} = 1$ and $P {V_{φ} (T) > 0} > 0$

Definition 1.3.6 We say that a probability measure $Q$ defined on $(Ω, F)$ is a

(strong) equivalent martingale measure if $Q$ is equivalent to $P$ and the discounted

process $\bar{S}$ is a $Q$ -local martingale (martingale).

We denote the set of martingale measures by $7^{\supset}$

Definition 1.3.7 A self Iinancing strategy is called tame if ${\bar{V}}_{φ} (t) \underline{>} 0$ for all

$t \in [0, T]$ . We denote by $Φ$ the set of tame trading strategies.

The following proposition assures us that the existence of an equivalent martingale

measure implies the absence of arbitrage.

Proposition 1.3.1 Assume $7^{\supset}$ is not empty. Then the market model contains no

arbitrage opportunities in $Φ$ .

In order to get equivalence between the absence of arbitrage opportunities and

the existence of an equivalent martingale measure we need some further definitions

and requirements.

Definition 1.3.8 A simple predictable strategy is a predictable process which

can be represented as a Iinite linear combination of stochastic processes of the form

$ψ 1_{[τ_{1}, τ_{2}]}$ where $τ_{1}$ and $τ_{2}$ are stopping times and $ψ$ is an $F_{τ_{1}}$ -measurable random

variable.

Definition 1.3.9 We say that a simple predictable trading strategy is $δ$ -admissible

if $V_{φ} (t) \underline{>} - δ$ for every $t \in [0, T]$ .

Definition 1.3.10 A price process $S$ satisfies NFLVR (no free lunch with van-

ishing risk) if for any sequence $(φ_{n})$ of simple trading strategies such that $φ_{n}$ is

$δ_{n}$ -admissible and the sequence $δ_{n}$ tends to zero, we have $V_{φ_{n}} (T) \to 0$ in probability

as $n \to \infty$ .

The following fundamental theorem of asset pricing is proved in [7]:

Theorem 1.3.1 (Fundamental Theorem of Asset Pricing-continuous time) There

exists an equivalent martingale measure for the Iinancial market model $M (P)$ if and

only if the condition NFLVR holds true.

For all the proofs of the above theorems refer to [3].

1.4 Options in the Black-Scholes model

Let us assume that we have a market with two assets, a riskless (B) and a risky

one (S). The riskless asset (bond or savings account) is modelled by the following

ordinary differential equation

$d B_{t} = r B_{t} d t$

where $r \underline{>} 0$ is an instantaneous interest rate (difffferent from the rate in the discrete

models). Without loss of generality, we set $B_{0} = 1$ and so $B_{t} = e^{r t}$ for $t \underline{>} 0$ . The risky

asset is a stock (or stock index) whose price is modelled by the following stochastic

differential equation:

$d S_{t} = S_{t} (μ d t + σ d Z_{t})$ ,

where $μ$ and $σ$ are constants and $Z_{t}$ is a standard Browninan motion. The model

is described on a probability space $(Ω, F, P)$ equipped with a Iiltration $F$ of $σ$ -

algebras $F_{0} \subset F_{1} \subset$ . . . $\subset F_{T}$ . We take $F_{t} = σ {S_{u}, u \underline{<} t}$ , and so the price

process for the stock is adapted to the Iiltration. We want to prove that there exists

a probability measure $P^{⋆}$ equivalent to $P$ , under which the discounted share price

$S_{t}^{⋆} = e^{- r t} S_{t}$ is a martingale. We have:

$d S_{t}^{⋆} = - r e^{- r t} S_{t} d t + e^{- r t} d S_{t} = S_{t}^{⋆} ((μ - r) d t + σ d Z_{t})$

and if we set $W_{t} = Z_{t} + \frac{(μ - r) t}{σ}$ , we get

$d S_{t}^{⋆} = S_{t}^{⋆} σ d W_{t}$ .

Now, recall the Girsanov theorem:

Theorem 1.4.1. Let $(θ_{t})_{0 \leq t \leq T}$ be an adapted process satisfying $\int_{0}^{T} θ_{s}^{2} d s < \infty a . s$ .

and such that the process $(L_{t})_{0 \leq t \leq T}$ defined by

$L_{t} = e x p (- \int_{0}^{t} θ_{s} d Z_{s} - \frac{1}{2} \int_{0}^{t} θ_{s}^{2} d s)$

is a martingale. Then, under the probability $P^{L}$ with density $L_{T}$ relative to $P$ , the

process $(W_{t})_{0 \leq t \leq T}$ defined by $W_{t} = Z_{t} + \int_{0}^{t} θ_{s} d s$ is a standard Brownian motion.

For a proof of Girsanov's theorem, see [19].

Using this theorem with $θ_{t} = \frac{μ - r}{σ}$ we get that there exists a probability $P^{⋆}$ under

which $(W_{t})_{0 \leq t \leq T}$ is a standard Brownian motion. Under this probability $P^{⋆}$ , the

discounted price process $(S_{t}^{⋆})$ is a martingale and

$S_{t}^{⋆} = S_{0}^{⋆} e x p (σ W_{t} - \frac{(σ)^{2} t}{2})$ .

Let us consider now a standard European call option. The option is defined by a

non-negative, $F_{T}$ -measurable random variable $H = f (S_{T}) = (S_{T} - K)_{+}$ , where $K$ is

the exercise price.

Theorem 1.4.2. In the Black-Scholes model, any option defined by a non-negative,

$F_{T}$ -measurable random variable $H$ , which is square-integrable under the probability

$P^{⋆}$ , is replicable by a trading strategy and the value at time $t$ of any replicating

portfolio is given by:

$V_{t} = E^{⋆} (e^{- r (T - t)} H | F_{t})$ .

Thus, the option value at time $t$ can be naturally defined by the expression $E^{⋆} (e^{- r (T - t)} H | F_{t})$ .

Proof: We follow [21]. Let us assume that there is an admissible strategy $(θ, η)$ repli-

cating the option. The value of the portfolio at time $t$ is given by

$V_{t} = θ_{t} B_{t} + η_{t} S_{t}$ ,

and the terminal values are equal: $V_{T} = H$ . Defining the discounted value $V_{t}^{⋆} = V_{t} e^{- r t}$

we get

$V_{t}^{⋆} = θ_{t} + η_{t} S_{t}^{⋆}$ .

The strategy is self-Iinancing and hence

$V_{t}^{⋆} = V_{0} + \int_{0}^{t} η_{u} d S_{u}^{⋆} = V_{0} + \int_{0}^{t} η_{u} σ S_{u}^{⋆} d W_{u}$ .

It can be shown that $(V_{t}^{⋆})$ is a square-integrable martingale under $P^{⋆}$ and hence

$V_{t}^{⋆} = E^{⋆} (V_{T}^{⋆} | F_{t})$ ,

and so

$V_{t} = E^{⋆} (e^{- r (T - t)} H | F_{t})$ .

So, if a portfolio $(θ, η)$ replicates the option, its value is given by the above formula.

Now it remain to show that the option is indeed replicable, i.e. there exist some

processes $(θ_{t})$ and $(η_{t})$ such that

$θ_{t} B_{t} + η_{t} S_{t} = E^{⋆} (e^{- r (T - t)} H | F_{t})$ .

The process $M_{t} : = E^{⋆} (e^{- r t} H | F_{t})$ is a $P^{⋆}$ -square integrable martingale.

Now, using the representation theorem for martingales we obtain that there exists an

adapted process $(K_{t})_{0 \leq t \leq T}$ such that $E^{⋆} (\int_{0}^{T} K_{s}^{2} d s) < \infty$ and

$M_{t} = M_{0} + \int_{0}^{t} K_{s} d W_{s} a . s$ .

for any $t \in [0, T]$ .

The strategy $φ$ $= (θ, η)$ with $η_{t} = \frac{K_{t}}{(σ S_{t}^{⋆})}$ and $θ_{t} = M_{t} - H_{t} S_{t}^{⋆}$ is self-Iinancing and its

value at time $t$ is given by

$V_{t} (φ) = e^{r t} M_{t} = E^{⋆} (e^{- r (T - t)} H | F_{t})$ .

In our case (European call), the random variable $H$ can be written as $H = f (S_{T}) =$

$(S_{T} - k)_{+}$ and we can express the option value $V_{t}$ at time $t$ as a function of $t$ and $S_{t}$

as follows:

$V_{t} = E^{⋆} (e^{- r (T - t)} f (S_{T}) | F_{t})$

$= E^{⋆} (e^{- r (T - t)} f (S_{t} e^{r (T - t)} e^{σ (W_{T} - W_{t}) - (\frac{σ^{2}}{2}) (T - t))} | F_{t})$ .

The random variable $S_{t}$ is $F_{t}$ -measurable and $W_{T} - W_{t}$ is independent of $F_{t}$ . A

standard result in probability theory allows us to write

$V_{t} = F (t, S_{t})$ ,

where

$F (t, x) = E^{⋆} (e^{- r (T - t)} f (x e^{- r (T - t)} e^{σ (W_{T} - W_{t}) - (\frac{σ^{2}}{2}) (T - t))})$ .

As $W_{t}$ is a standard Brownian motion, $W_{T} - W_{t}$ is a zero-mean normal variable with

variance $T - t$ and so,

$F (t, x) = e^{- r (T - t)} \int_{- \infty}^{\infty} f (x e^{(r - \frac{σ^{2}}{2}) (T - t) + σ y \sqrt{T - t}}) \frac{e^{- \frac{y^{2}}{2}}}{\sqrt{2 π}} d y$ .

Now $F$ can be calculated explicitly for call options. In this case $f (x) = (x - K)_{+}$ and

$F (t, x) = E^{⋆} (e^{- r (T - t)} (e^{(r - \frac{σ^{2}}{2}) (T - t) + σ (W_{T} - W_{t})} - K)_{+})$

$= E (x e^{σ \sqrt{θ} g - σ^{2} \frac{θ}{2}} - K e^{- r θ})_{+}$

where $g$ is a standard Gaussian variable and $θ = T - t$ . We define:

$d_{1} = \frac{l n (x / K) + (r + \frac{σ^{2}}{2}) θ}{σ \sqrt{θ}}$

and

$d_{2} = d_{1} - σ \sqrt{θ}$ .

Then we have,

$F (t, x) = E [(x e^{σ \sqrt{θ} g - σ^{2} \frac{θ}{2}} - K e^{- r θ}) I_{{g + d_{2} \geq 0}]}$

$= \int_{- d_{2}}^{\infty} (x e^{σ \sqrt{θ} g - σ^{2} \frac{θ}{2}} - K e^{- r θ}) \frac{e^{- \frac{y^{2}}{2}}}{\sqrt{2 π}} d y$

$= \int_{- \infty}^{d_{2}} (x e^{- σ \sqrt{θ} g - σ^{2} \frac{θ}{2}} - K e^{- r θ}) \frac{e^{- \frac{y^{2}}{2}}}{\sqrt{2 π}} d y$

$= \int_{- \infty}^{d_{2}} (x e^{- σ \sqrt{θ} g - σ^{2} \frac{θ}{2}} - \int_{- \infty}^{d_{2}} K e^{- r θ}) \frac{e^{- \frac{y^{2}}{2}}}{\sqrt{2 π}} d y$ .

Now, using the change of variable $z$ $= y + σ \sqrt{θ}$ we get

$F (t, x) = x N (d_{1}) - K e^{- r θ} N (d_{2})$ ,

where

$N (d) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{d} e^{- \frac{x^{2}}{2}} d x$ .

Similar calculations show that the price of the put option is

$F (t, x) = K e^{- r θ} N (- d_{2}) - x N (- d_{1})$ .

CHAPTER 2

A PRODUCT SPACE

2.1 Introduction to the dual risk in unit-linked insurance

products

In this chapter I will look at two different approaches in pricing guarantees in

unit-linked insurance contracts. The unit-linked insurance contracts are very popular

in many markets. The return obtained by the insured is linked to some Iinancial

index (or generally, the Iinancial market). Some of these insurance contracts have

also some kind of a death guarantee benefit.

We are therefore dealing with products that bear two different (independent) types

of risk. First of all, we can look at the Iinancial risk (related to the market). This risk

was clearly stressed during the last few years, when the major stock market indices

have dropped so much. On the other hand, the insurer deals with another type of

risk, let's call it actuarial risk, related to the possibility of death for the insured (and

hence the possibility of a claim). While the Iinancial market model might be complete

(any contingent claim is replicable by a trading strategy), the model that assumes

both risks (financial and actuarial) is incomplete.

2.2 Product space

I will start by defining the two market models, the Iinancial and the actuarial one

and then I will take a look at the product market model.

2.2.1 Financial backgroud

The starting point for a mathematical model of the Iinancial market was the pa-

per by Bachelier [1], in 1900. He suggested that a possible approch in describing

fluctuations in stock prices might be the Brownian motion. More than 60 years later,

Samuelson [27], in 1965 proposed the idea that these fluctuations can better be de-

scribed by a geometric Brownian motion, and this approach had the clear advantage

that it didn't generate negative stock prices. This approach allowed Black and Sc-

holes [4] (1973) and Merton [22] (1973) to determine the price of European options

that doesn't allow arbitrage (no profits could arise from manipulations of stocks and

options in any portfolio). The next important step is given by the work of Cox, Ross

and Rubinstein [6] (1979) who investigated a simple discrete time model (binomial)

in which the value of the stock between two trading times can only take two values.

As limiting cases (by letting the length of time intervals between trading times tend

to 0), they recovered (rediscovered) the option pricing formula by Black and Scholes.

2.2.2 Insurance-Actuarial background

The Iirst known social welfare program with elements of life insurance appeared

in the Roman Empire ("Collegia", AD 133). The Iirst primitive mortality tables were

published in 1662 by John Graunt and had only 7 age groups. The Iirst complete

mortality table is due to the astronomer Edmund Halley. The tables had been used

for computations of premiums for life insurance contracts. De Moivre suggested

methods for evaluations of life insurance products, combining interest and mortality

under simple assumptions about mortality (which are used even today, like De Moivre

Law). His assumption is basically the uniform death distribution between integral

years and so this was an important step in describing the life insurance in a continuous

time model (rather than discrete, up to that point). The modern utility theory, whose

foundations were laid by Daniel Bernoulli, argues that risk should not be measured

by expectations alone, because an important aspect is also the preference of the

individual. For instance, it could be reasonable for a poorer individual to prefer an

uncertain future wealth to another more unsure future wealth with a bigger expected

value. This is very important in insurance in general, because it explains (together

with the concept of pooling) why individuals prefer to buy insurance at a price which

exceeds the expectations of future losses.

2.3 Combining the two markets

Let $(Ω^{f}, F^{f}, P^{f})$ be a probability space. We equip this space with the Iiltration

$F^{f}$ of $σ$ -algebras $F_{0}^{f} \subset F_{1}^{f} \subset ... \subset F_{T}^{f}$ satisfying the usual conditions of right-

continuity $(F_{t}^{f} = \cap_{ε > 0} F_{t + ε}^{f})$ and completeness ( $F_{0}$ contains all $P$ -negligible events

in $F$ ). We also take $F_{0}^{f} = {\emptyset, Ω}$ and $F_{T}^{f} = F^{f}$ . Consider a $d$ -dimensional process

$X =$ $(X^{1}, X^{2}, ..., X^{d})$ which describes the evolution of the discounted prices of $d$

tradable stocks. $X (ω)$ is the path of $X$ associated with $ω \in Ω^{f}$ . A purely Iinancial

derivative is a random variabl $e$ $H^{f} \in L^{2} (P^{f}, F_{T}^{f})$ .

Let's consider now another Iiltered probability space $(Ω^{a}, F^{a}, F^{a}, P^{a})$ . The Iiltration

is right-continuous but not necessarily complete. This space carries a pure insurance

(actuarial) risk process which describes the development of insurance claims. An

insurance risk process $U$ is a random variable defined on $(Ω^{a}, F^{a})$ and $U (ω)$ is the

path associated with $ω \in Ω^{a}$ .

A pure insurance (actuarial) contract is a random variable $H^{a} \in L^{2} (P^{a}, F_{T}^{a})$ .

We are next looking at the combined model $(Ω, F, F, P)$ , which is defined as the

product space of the two individual spaces, Iinancial and actuarial. The construction

of the combined model follows Moller [23]. We let $Ω = Ω^{f} \times Ω^{a}$ and $P = P^{f} \otimes P^{a}$ .

Let's define the $σ$ -algebra $N$ generated by all subsets of null-sets fro $m$ $F^{f} \otimes F^{a}$ , that

is:

$N$ $= σ {F \underline{\subset} Ω^{f} \times Ω^{a} | \exists G \in F^{f} \otimes F^{a} : F \underline{\subset} G, (P = P^{f} \otimes P^{a}) (G) = 0}$ .

Next we define $F = (F^{f} \otimes F^{a}) ⋁ N$ and also the following Iiltrations on the product

space (extensions of the original filtrations):

$F_{t}^{1} = (F_{t}^{f} \otimes {\emptyset, Ω^{a}}) ⋁ N$ , $F_{t}^{2} = ({\emptyset, Ω^{f}} \otimes F_{t}^{a}) ⋁ N$ .

Lemma 2.1 The Iiltrations defined above:

1. Satisfy the usual conditions;

2. They are independent;

3. The Iiltration $F = (F_{t})_{0 \leq t \leq T}$ defined by $F_{t} = F_{t}^{1} ⋁ F_{t}^{2}$ satisfies the usual conditions.

Moreover, $F_{t} = (F_{t}^{f} \otimes F_{t}^{a}) ⋁ N$ .

Proof:

1. The completeness is trivial, as $N \underline{\subset} F_{t}^{1}$ and $N \underline{\subset} F_{t}^{2} \forall t \in [0, T]$ . Next we want to

show that $F^{1}$ is right-continuous. We define, for $s \in [0, T]$ ,

$D_{s} = {F_{1} \times Ω^{a} | F_{1} \in F_{s}^{f}}$ .

By definition, $σ (D_{s}) = F_{s}^{f} \otimes {\emptyset, Ω^{a}}$ , and as $D_{s}$ is also a $σ$ -algebra, we get

$D_{s} = F_{s}^{f} \otimes {\emptyset, Ω^{a}}$ .

Hence

$\cap_{ε > 0} F_{t + ε}^{1} = \cap_{ε > 0} (D_{t + ε} ⋁ N)$ $= (\cap_{ε > 0} D_{t + ε}) ⋁ N$ $= D_{t} ⋁ N$ ,

where the last equality follows from the right-continuity of $F^{f}$ . Similarly, one can

prove the fact that $F^{2}$ is right-continuous (using the right-continuity of $F^{a} .$ )

2. By definition, we need to show that $\forall F_{1} \in F_{T}^{1}$ and $\forall F_{2} \in F_{T}^{2}$ , we have:

$P (F_{1} \cap F_{2}) = P (F_{1}) \cap P (F_{2})$

Let's consider at Iirst $F_{1} = F_{1}^{f} \times O_{2}$ and $F_{2} = O_{1} \times F_{2}^{a}$ , where $F_{1}^{f} \in F^{f}$ , $F_{2}^{a} \in F^{a}$

and $O_{1} \in {\emptyset, Ω^{f}}$ , $O_{2} \in {\emptyset, Ω^{a}}$ . Then,

$P (F_{1} \cap F_{2}) = P ((F_{1}^{f} \times O_{2}) \cap (O_{1} \times F_{2}^{a})) = P ((F_{1}^{f} \cap O_{1}) \times (F_{2}^{a} \cap O_{2}))$

$= P^{f} (F_{1}^{f} \cap O_{1}) P^{a} (F_{2}^{a} \cap O_{2}) = P^{f} (F_{1}^{f}) P^{f} (O_{1}) P^{a} (F_{2}^{a}) P^{a} (O_{2})$

$= P (F_{1}) P (F_{2})$

Now, because the sets of this type generate the entire $σ$ -algebras, this shows that

they are independent.

3. A general result from probability theory shows that $F$ satisfies the usual conditions

and the equality is trivial.

2.4 Disjoint pricing techniques

Let's assume that the underlying assets (index) in the variable annuity contract

follows the classical geometrical Brownian motion, described by the following differ-

ential equation under the physical measure $P$ :

$d S_{t} = μ S_{t} d t + σ S_{t} d W_{t}$ ,

In the classical Black-Scholes model, it can be shown that there is an equivalent

martingale measure (the risk-neutral probability measure Q) under which the price

process follows the equation:

$d S_{t} = r S_{t} d t + σ S_{t} d {\tilde{W}}_{t}$ ,

where $μ$ is the expected rate of return of the asset, $σ$ is its standard deviation, $r$ is the

risk-free rate of interest (bank savings account rate) and $W_{t}$ (and ${\tilde{W}}_{t}$ respectively) is

a standard Brownian motion under $P$ (and $Q$ respectively). The difference between

the two types of pricing is given by the expected rate of return of the asset under

each probability measure ( $μ$ under the $P$ -measure for the actuarial approach and $r$

under the $Q$ -measure for the Iinancial approach). The expected loss at time $t$ is in

both cases:

$V^{P} (t, T) = E_{P} [e^{- r (T - t)} \max (K - S_{T}, 0) | F_{t}]$ ,

and

$V^{Q} (t, T) = E_{Q} [e^{- r (T - t)} \max (K - S_{T}, 0) | F_{t}]$ .

The single premium at time 0 for each type of pricing is given by:

Premium (actuarial) $= \sum_{k = 1}^{ω - x} V^{P} (0, k)_{k} p_{x} q_{x + k}$ , and

Premium (financial) $= \sum_{k = 1}^{ω - x} V^{Q} (0, k)_{k} p_{x} q_{x + k}$

The Iinancial premium is a sum of Black-Scholes put prices. The only difference

between the two formulas is that the risk-free rate in the Iinancial price model is

replaced by the expected return in the actuarial model. This leads to higher Iinancial

premium when $μ > r$ .

The Iinancial pricing approach is meaningless in the absence of a hedging strategy.

This might be seen as a disadvantage of the Iinancial approach, which is yet counter-

balanced by some clear advantages: the premium is independent of the expected rate

of return of the underlying asset (while the actuarial premium could be affected by

errors in its estimation) and the Iinancial risk is eliminated by the hedging portfolio

(strategy).

CHAPTER 3

RISK ANALYSIS

3.1 Derivatives in incomplete markets

There are several approaches for valuing and hedging derivatives in incomplete

markets. This sections provides an overview of the most important techniques.

3.1.1 Super-replication

The idea of super-hedging (or super-replication) was Iirst suggested by El Karoui

and Quenez in 1995 [8]. In this case, there is no risk (for the hedger) associated with

the derivative, as the super-hedging price is the smallest initial capital that allows

the seller to construct a portfolio which dominates the payoff process of the derivative

(option). El Karoui and Quenez showed that a super-hedging strategy exists provided

that

$\sup_{Q \in P_{e}} E_{Q} (H) < \infty$

where $7_{e}^{\supset}$ is the set of all equivalent martingale measures and $H$ is the claim. By

defining the process

${\bar{V}}_{t} = e s s . \sup_{Q \in P_{e}} E_{Q} [H | F_{t}]$

and deriving its decomposition of the form:

${\bar{V}}_{t} = {\bar{V}}_{0} + \sum_{j = 1}^{t} {\bar{φ}}_{j} ▵ S_{j}^{⋆} - C_{t}$

where $C$ is increasing and $S_{t}^{⋆}$ is the discounted price at time $t$ , it can be shown that

the initial capital that satisfies this condition is given by

$V_{0} = \sup_{Q \in P_{e}} E_{Q} (H)$

and $V_{0}$ is called the upper-hedging price of the claim $H$ . Also, the super hedging

strategy is determined by $φ_{j} = {\bar{φ}}_{j}$ .

Moller [23] used this to compute the super-hedging price and strategy for unit-linked

insurance products. Let us consider a living benefit contract, that pays $f (S_{T})$ to

survivors at time $T$ from a group of $l_{x}$ insured age $x$ . If

$P_{t}^{f} : = E_{Q} [\frac{f (S_{T})}{B_{T}} | F_{t}^{f}] = E_{Q} [\frac{f (S_{T})}{B_{T}}] + \sum_{j = 1}^{t} (y_{j}^{f} ▵ S_{j}^{⋆}$

is the no-arbitrage price of the purely Iinancial contingent claim that pays $f (T)$ at

time $T$ , the cheapest self-Iinancing super-hedging strategy is given by:

$θ_{t} = (l_{x} - N_{t - 1}) (y_{t}^{f}$ (3.1)

and

$η_{t} = l_{x} P_{0}^{f} + \sum_{j = 1}^{t} θ_{j} ▵ S_{j}^{⋆} - θ_{t} S_{t}^{⋆}$ (3.2)

and the price of the contract is $l_{x} \times P_{0}^{f}$ . Hence, the super-hedging price is given by the

number $l_{x}$ of policies sold multiplied with the price of the Iinancial contingent claim.

Therefore, it assumes that no policy-holder dies until the expiration of the contract

i.e. the survival probability to time $T$ is 1.

3.1.2 Utility-based indifference pricing

The indifference premium is a price such that the optimal expected utility among

all portfolios containing the prespecified number of options coincides with the optimal

expected utility among all portfolios without options. In other words, the buyer

(investor) is indifferent to including the option into the portfolio. This approach was

Iirst suggested by Hodges and Neuberger [16] and is now a standard concept to value

European style derivatives in incomplete markets. Let us start by considering the

so-called mean-variance utility function

$u_{β} (Y) = E [Y] - a (V a r [Y])^{β}$ ,

where $Y$ is the wealth at time $T$ and $u$ describes the insurance company's preferences

(while $a$ and $β$ are constants). An insurance company with utility function $u$ prefers

the pair $(P_{β}, H)$ (i.e. selling the contingent claim $H$ for the premium $P_{β}$ ) to the pair

$({\hat{P}}_{β}, \hat{H})$ if

$u_{β} (P_{β} - H) \underline{>} u_{β} ({\hat{P}}_{β} - \hat{H})$ .

The indifference price I $P (H)$ for $H$ is defined by

$\sup_{φ : V_{0} \overset{ˇ}{(φ) =} V_{0}} u_{β} (V_{T} (φ) + I P (H) - H) = \sup_{φ : V_{0} \overset{ˇ}{(φ) =} V_{0}} u_{β} (V_{T} (\bar{φ}))$

where $V_{0}$ is the initial capital at time 0.

3.1.3 Quadratic approaches

These techniques can be divided into two groups: (local) risk-minimization ap-

proaches, proposed by Follmer and Sondermann [12] and mean-variance hedging ap-

proaches, proposed by Bauleau and Lamberton [2]. This approach has the big advan-

tage that hedging strategies can be obtained quite explicitly.

3.1.4 Quantile hedging and shortfall risk minimization

In the quadratic approach, losses and gains are treated equally. This is not a

desirable feature and the way out is given by quantile hedging [13] or efficient hedging

[14]. The seller minimizes the expected shortfall risk subject to a given initial capital,

i.e. the seller wants to minimize $E [l (H - V_{T})^{+}]$ over all strategies $φ$ , where $V_{t} =$

$c + \int_{0}^{t} φ_{u} d S_{u}$ and $l$ is the loss function ( $l$ : $R_{+} \to R_{+}$ , increasing and convex with

$l$ (0) $= 0)$ . Although gains are not punished in this approach, they are not rewarded

either.

3.2 Description of the GMDB problem

This chapter presents a methodology for pricing the guaranteed minimum death

benefit of a variable annuity in a market model with jumps. Recent developments in

the stock market make variable annuities very attractive products from the insured

point of view, but less attractive for insurers. The insured still has the possibility of

investment benefits, while avoiding the risk of a stock market collapse. The insurer

wants to minimize its risk and yet sell a competitive product.

The Iinancial market model consists of one riskless asset and one risky asset whose

price jumps in proportions $J$ at some random times $τ$ which correspond to the jump

times of a Poisson process. The model describes incomplete markets and there is no

perfect hedging.

In the second part of the chapter, we describe a possible method of risk analysis for

binomial tree models.

3.2.1 The model

In the Black-Scholes model, the share price is a continuous function of time. Some

rare events (which are rather frequent lately), can accompany "jumps" in the share

price. In this case the market model is incomplete, hence there is no perfect hedging

of options.

We consider a market model with one riskless asset and one risky asset whose price

jumps in proportions $J_{1}$ , $J_{2}$ , $...$ , $J_{n}$ , $...$ at some random times $τ_{1}$ , $τ_{2}$ , $...$ , $τ_{n}$ , $...$ which

correspond to the jump times of a Poisson process. Between the jumps the risky asset

follows the Black-Scholes model.

The mathematical model consists of a probability space $(Ω, F, P)$ , a Brownian motion

$(W_{t})$ and a Poisson process $(N_{t})_{t \geq 0}$ with parameter $λ$ . The jumps $J_{n}$ are independent

and identically distributed on $($ -1, $\infty)$ and $(F_{t})_{t}$ is the Iiltration which incorporates all

information available at time $t$ . The price process $(S_{t})$ of the risky asset is described

as follows:

On $[τ_{j}, τ_{j + 1})$ , $d S_{t} = S_{t} (μ d t + σ d W_{t})$ i.e. Black-Scholes model;

At time $τ_{j}$ , the jump of $(S_{t})$ is given by $▵ S_{τ_{j}} = S_{τ_{j}} - S_{τ_{j}} - = S_{τ_{j^{-}}} J_{j}$ ;

In other words, $S_{τ_{j}} = S_{τ_{j}} -$ $(1 + J_{j})$ ; As defined, $(S_{t})$ is a right-continuous process.

It is straightforward to see that we have the following formula for the price process:

$S_{t} = S_{0} (\prod_{j = 1}^{N_{t}} (1 + J_{j})) e^{(μ - \frac{σ^{2}}{2}) t + σ W_{t}}$ (3.3)

A variable annuity is an investment wrapped with a life insurance contract. The

convenient tax deferral characteristic of the variable annuities makes them a very

interesting and popular investment and retirement instrument. The average age at

which people buy their Iirst variable annuity is 50. There are a few different types of

GMDB options associated with variable annuities. The most popular are:

1. Return of premium-the death benefit is the larger of the account value on the

date of death or the sum of premiums less partial withdrawals;

2. Reset-the death benefit is automatically reset to the current account value every

$x$ years;

3. Roll-up-the death benefit is the larger of the account value on the day of death

or the accumulation of premiums less partial withdrawls accumulated at a specified

interest rate (e.g. 1.5% in many 2003 contracts);

4. Ratchet (look back) -same as reset, except that the death benefit is not allowed

to decrease, except for withdrawals.

Let $ω$ be the expiry date for a variable annuity with a return of premium GMDB

option associated with $(S_{t})$ . Let $T$ be the random variable that models the future

lifetime of the insured (buyer of the contract).Then the payoff of the product is:

$P (T) =$ $i f T > ω i f T \underline{<} ω$

where $H (t) = \max (S (0), S (t)) = S (t) + \max (S (O) - S (t), O) = S (t) + (S (0) - S (t))_{+}$

Basically, the value of the guarantee at time 0 is given by the price of a put option

with stochastic expiration date. It can be shown that in discrete settings and when

the benefit is paid at the end of the year of death,

PV (GMDB) $= \sum_{m = 1}^{ω - x} m - 1 | q_{x} P (m, S_{0})$ (3.4)

where $P (m, S_{0})$ is the price of the put option with expiry $m$ and strike $S_{0}$ , in the

Black-Scholes model.

If the benefit is paid at the moment of death, then

PV (GMDB) $= \int_{0}^{\infty} f_{T} (t) P (t, S_{0}) d t$ (3.5)

where $f_{T} (t)$ is the pdf of the future lifetime random variable. Closed form expressions

can be obtained for appropriate assumptions on $T$ (constant force, UDD, Balducci

etc).

Next we want to determine the price of the put option associated with GMDB in the

market model described in the introduction, which minimizes the risk at maturity.

Suppose $E (J_{1}) < \infty$ and let ${\bar{S}}_{t} = e^{- r t} S_{t}$ for $s \underline{<} t$ . Then

$E ({\bar{S}}_{t} | F_{s}) = {\bar{S}}_{s} E (e^{(μ - r - \frac{σ^{2}}{2}) (t - s) + σ (W_{t} - W_{s})} \prod_{j = N_{s} + 1}^{N_{t}} ((1 + J_{j}) | F_{s}))$

$= {\bar{S}}_{s} E (e^{(μ - r - \frac{σ^{2}}{2}) (t - s) + σ (W_{t} - W_{s})} \prod_{j = 1}^{N_{t} - N_{s}} (1 + J_{N_{s} + j}))$

because $W_{t} - W_{S}$ and $N_{t} - N_{S}$ are independent of $F_{s}$ .

Hence

$E ({\bar{S}}_{t} | F_{s}) = {\bar{S}}_{s} e^{(μ - r) (t - s)} E (\prod_{j = N_{s} + 1}^{N_{t}} (1 + J_{j}))$

But

$E (\prod_{j = N_{s} + 1}^{N_{t}} (1 + J_{j})) = E (\prod_{j = 1}^{N_{t}} (1 + J_{j})) - E (\prod_{j = 1}^{N_{t}} (1 + J_{j}))$

and

$E (\prod_{j = 1}^{N_{t}} (1 + J_{j})) = \sum_{n = 1}^{\infty} E (\prod_{j = 1}^{n} (1 + J_{j})) P (N_{t} = n)$

$= \sum_{n = 1}^{\infty} (1 + E (J_{j}))^{n} e^{- λ t} \frac{(λ t)^{n}}{n!} = \sum_{n = 1}^{\infty} e^{- λ t} \frac{(λ t (1 + E (J)))^{n}}{n!}$

$= e^{- λ t} e^{λ t (1 + E (J))} = e^{λ t E (J)}$

$E ({\bar{S}}_{t} | F_{s}) = {\bar{S}}_{s} e^{(μ - r) (t - s)} e^{λ (t - s) E (J)}$

Hence $({\bar{S}}_{t})$ is a martingale iff $μ = r - λ E (J)$ . In our case, we want to price a put

option with strike $S_{0}$ and expiry T.

Let $f (x) = (S_{0} - x)_{x}$ . The price of the put option which minimizes the risk at time

$t$ is given by:

$E (e^{- r (T - t)} f (S_{t}) | F_{t}) = E (e^{- r (T - t)} f (S_{t} e^{(μ - r - \frac{σ^{2}}{2}) (t - s) + σ (W_{t} - W_{s})} \prod_{j = N_{t} + 1}^{N_{T}} (1 + J_{j})) | F_{t})$

$= E (e^{- r (T - t)} f (S_{t} e^{(μ - r - \frac{σ^{2}}{2}) (t - s) + σ (W_{t} - W_{s})} \prod_{j = 1}^{N_{T - t}} (1 + J_{j})))$

$= E (P (t,$ $S_{t} e^{- λ (T - t) E (J)} \prod_{j = 1}^{N_{T - t}} (1 + J_{j})))$

where $P (t, x)$ is the function that gives the price of the option for the Black-Scholes

model. As $N_{T - t}$ is Poisson with parameter $λ (T - t)$ ,

$E (e^{- r (T - t)} f (S_{t}) | F_{t}) = \sum_{n = 0}^{\infty} E (P (t,$ $S_{t} e^{- λ (T - t) E (J)} \prod_{j = 1}^{n} (1 + J_{j}))) \frac{e^{- λ (T - t)} λ^{n} (T - t)^{n}}{n!}$

Let us now assume that $J$ takes values in ${u, d}$ and $P (J = u) = p$ , $P (J = d) = 1$ $- p$ .

We will use the following:

Lemma 1: Let $N$ be Poisson with parameter $λ$ .

Let $S = \sum_{n = 1}^{N} V_{n}$ with $P (V_{n} = u) = p$ , and $P (V_{n} = d) = 1 - p$ . Then law(S) $= 1 a w (u N_{1} +$

$d N_{2})$ , where $N_{1}$ is Poisson $λ p$ and $N_{2}$ is Poisson $(λ (1 - p))$ .

Proof: One method would be to show that the two random variables have the same

moment generating function.

Another method would be to $r e$ -write $S = \sum_{n = 1}^{N} (u + (d - u) I_{n})$ , where $I_{n} = 0$ with

probability $p$ and $I_{n} = 1$ with probability l-p. So,

$S = u N + (d - u) \sum_{n = 1}^{N} I_{n} = u N_{1} + d N_{2}$ ,

because $\sum_{n = 1}^{N} I_{n}$ is Poisson $(λ (1 - p))$ . This completes the proof of the lemma.

Now,

$\prod_{j = 1}^{N_{T - t}}$ $(1 + J_{j}) = \prod_{j = 1}^{N_{T - t}} e^{l n (1 + J_{j})} = e^{Σ_{j = 1}^{N_{T - t}} l n (1 + J_{j})}$ ,

and using the lemma we have $\sum_{j = 1}^{N_{T - t}} l n (1 + J_{j})$ has the same law as

$l n (1 + u) N_{1} + l n (1 + d) N_{2}$ where $N_{1}$ and $N_{2}$ are iid with parameters $λ p$ and $λ (1 - p)$

respectively.

So, the price of the option at time $t$ is given by:

$P (t,$ $s_{0^{e^{- λ (T - t) [p u + (1 - p) d]} e^{l n (1 + u) n_{1} + l n (1 + d) n_{2) e^{- λ} \frac{λ^{k_{1} + k_{2}} p^{k_{1}} (1 - p)^{k_{2}}}{(k_{1})! (k_{2})!}}}}}$

(3.6)

where ( $y$ $= l n (1 + u)$ and $β = l n (1 + d)$ .

Replacing now the price of the put option in formula (3.2) we get the price for GMDB

paid at the end of the year of death or in formula (3.3) we get the price of the GMDB

for continuous time model, with benefit paid at the moment of death.

Most of the time, ( $y$ and $β$ are linearly independent over $Z$ , so in this case the decom-

position $α n_{1} + β n_{2}$ is unique, and the price of the option is given by:

$\sum_{n_{1} = 0}^{\infty} \sum_{n_{2} = 0}^{\infty} P (t,$ $s_{0^{e^{- λ (T - t) [p u + (1 - p) d]} e^{l n (1 + u) n_{1} + l n (1 + d) n_{2) e^{- λ} \frac{λ^{n_{1} + n_{2}} p^{n_{1}} (1 - p)^{n_{2}}}{(n_{1})! (n_{2})!}}}}}$ (3.7)

3.2.2 Other market models with jumps

The problem of the price jumps can be analyzed in other models too. Another

model could be described as follows: only onejump whose time occurance is uniformly

distributed on the contract length. Let $ω$ be the expiration date of the contract and $T_{j}$

the random variable modeling the time of occurance of the jump. Let also $T_{d}$ be the

random variable that models the lifetime of the insurer. Let's assume for simplicity

that $T_{d}$ is exponential, i.e. $f_{T_{d}} = λ e^{- λ t}$ .

The probability that the jump occurs before the death is

$P (T_{j} < T_{d}) = \int_{0}^{ω} \int_{t_{j}}^{\infty} \frac{1}{ω} λ e^{- λ t_{d}} d t_{d} d t_{j} = \int_{0}^{ω} \frac{1}{ω} e^{- λ t_{j}} d t_{j} =$

$= \frac{1}{ω} \frac{e^{- λ t_{j}}}{- λ} |_{0}^{ω} = \frac{1 - e^{- λ ω}}{λ ω}$

Let $τ$ be the random time of the jump. Then,

$f o r t \underline{>} τ f o r t < τ$ and

As in the Iirst model, the discounted price process is a martingale for specific jump

processes and the GMDB price can be found similarly.

3.2.3 Risk analysis

We focus our attention now on a binomial tree model, and for simplicity we will

assume that the price process $(S_{t})$ of a risky asset follows a simple random walk,

going up one unit with probability 1/2 and down one unit with probability 1/2. For

simplicity we will assume that $S_{0} = 0$ , using a translation of the random variable

that models the stock price. Let $τ_{N} = \inf {k \underline{>} 0 : | S_{k} | = N}$ be the Iirst time the

random walk is at distance $N$ from the origin. If we think about the stock price, $τ_{N}$

is the random time when $S_{n}$ goes up or down $N$ units, for the Iirst time. Hence, $τ_{N}$

can be interpreted as a measure of risk.

First, it is quite easy to show that the distribution of $τ_{N}$ has an exponential tail and

hence has moments of all orders. Let $T_{N} = \inf {k \underline{>} 0 : S_{k} = N}$ . If $ω \in {τ_{N} = n}$ ,

then $ω \in {T_{N} = n} \cup {T_{- N} = n}$ .

So $P (τ_{N} = n) \underline{<} P (T_{N} = n) + P (T_{- N} = n)$ $⋁ =$ $2 P (T_{N} = n)$ .

symmetry

So, $P (τ_{N} = n) \underline{<} 2 P (T_{N} = n)$ . Next we want to Iind $P (T_{N} = n)$ . To get a path that

gets to $N$ for the Iirst time at $t = n$ , we need the path to be at $N - 1$ at time $t = n - 1$

(see Figure 3.1). Hence we need to count all possible paths that are at $N - 1$ at time

$n - 1$ and which never rise above $N - 1$ before time $n - 1$ .

2 3

1 2 3 4 n-l $n$

Figure 3.1: Random walk for stock price

Generally, for $r \underline{>} k$ and using the reflection principle, the probability of a path from

$(0, 0)$ to $(n, k)$ with maximum $\underline{>} r$ equals the probability of a path from $(0, 0)$ to

$(n, 2 r - k)$ . Let us denote this probability by $P_{n, 2 r - k}$ . We have

$P$ path with $\max \underline{<} r$ ) $= P$ (path with $\max < r$ ) $+ P$ (path with $\max = r$ )

So,

$P$ path with $\max = r$ ) $= P$ (path with $\max \underline{<} r$ ) $- P$ (path with $\max \underline{<} r - 1$ )

$= 1$ $- P$ path with $\max \underline{>} r + 1$ ) $- 1$ $+ P$ path with $\max \underline{>} r$ )

$= P_{n, 2 r - k} - P_{n, 2 r + 2 - k}$

So the probability of a path from $(0, 0)$ to $(n, k)$ with maximum $r$ equals $= P_{n, 2 r - k} -$

$P_{n, 2 r + 2 - k}$ and hence the probability of a path from $(0, 0)$ to $(n - 1, N - 1)$ with

maximum $N - 1$ is $= P_{n - 1, N - 1} - P_{n - 1, N + 1}$ . Therefore

$P (T_{N} = n) = \frac{1}{2} (P_{n - 1, N - 1} - P_{n - 1, N + 1})$ .

But clearly,

$P_{n, k} = P (S_{n} = k) = C_{\frac{n n + k}{2}} 2^{- n}$

as you need $\frac{n + k}{2}$ steps up $(+ 1)$ and $\frac{n - k}{2}$ down (-1), each with probability $\frac{1}{2}$ . Then we

have

$P_{n - 1, N - 1} - P_{n - 1, N + 1} = C_{\frac{n - 1 n + N}{2} - 1} 2^{- n + 1} - C_{\frac{n - 1 n + N}{2}} 2^{- n + 1}$

Let $k = \frac{n + N}{2}$ . We get

$P_{n - 1, N - 1} - P_{n - 1, N + 1} = C_{k - 1}^{n - 1} 2^{- n + 1} - C_{k}^{n - 1} 2^{- n + 1}$

$= \frac{(n - 1)!}{(n - k)! (k - 1)!} 2^{- n + 1} - \frac{(n - 1)!}{(n - k - 1)! k!} 2^{- n + 1}$

2 3

Figure 3.2: Reflection principle for random walks

$= 2^{- n + 1} [\frac{n! k}{(n - k)! k! n} - \frac{n! (n - k)}{(n - k)! k! n}]$

$= \frac{2^{- n + 1}}{n} \frac{n!}{(n - k)! k!} (2 k - n) = \frac{N}{n} C_{\frac{n n + N}{2}} 2^{- n + 1}$ ,

as $N = 2 k - n$ . So

$P (T_{N} = n) = {\underline{N}}_{C_{n + N}^{n} 2^{- n}}$

$n$ $\bar{2}$

Recall Stirling's formula

$n! \sim \sqrt{2 π n} n^{n} e^{- n}$ , as $n \to \infty$

So,

$P (T_{N} = n) = \frac{N}{n} \frac{n!}{(\frac{n + N}{2})! (\frac{n - N}{2})!} 2^{- n}$

$\sim 2^{- n} \underline{N} \underline{n^{n} e^{- n} \sqrt{2 π n}}$

$n (\frac{n + N}{2})^{\frac{n + N}{2}} (\frac{n - N}{2})^{\frac{n - N}{2}} e^{- \frac{n + N + n - N}{2}} \sqrt{2 π \frac{n + N}{2}} \sqrt{2 π \frac{n - N}{2}}$

$\sim \frac{N}{n} \frac{1}{(\frac{n + N}{2})^{\frac{n + N}{2}} (\frac{n - N}{2})^{\frac{n - N}{2}}} \frac{1}{\sqrt{2 π n}} \frac{1}{\sqrt{\frac{n + N}{2 n} \frac{n - N}{2 n}}} 2^{- n}$

$\sim \frac{N}{n}$ $(1 + \frac{N}{n})^{- \frac{n + N}{2}} (1 - \frac{N}{n})^{- \frac{n - N}{2}} 2^{n} 2^{- n} \frac{1}{\sqrt{2 π n}} 2 (1 + \frac{N}{n})^{- \frac{1}{2}} (1 - \frac{N}{n})^{- \frac{1}{2}}$

$\sim \sqrt{\frac{2}{π n}} \frac{N}{n}$ $(1 - \frac{N^{2}}{n^{2}})^{- \frac{n}{2}} (1 + \frac{N}{n})^{- \frac{N}{2}} (1 - \frac{N}{n})^{\frac{N}{2}} (1 - \frac{N^{2}}{n^{2}})^{- \frac{1}{2}}$

Let us now consider the following setting:

$\frac{N}{\sqrt{n}} \to x$ and $\frac{N}{n} \to 0$ .

Then we have

$(1 - \frac{N^{2}}{n^{2}})^{- \frac{n}{2}} = [(1 - \frac{x^{2}}{n})^{- n}] \frac{1}{2} \sim e^{\frac{x^{2}}{2}}$

$(1 + \frac{N}{n})^{- \frac{N}{2}} = [(1 + \frac{x}{\sqrt{n}})^{\sqrt{n}}] - \frac{x}{2} \sim e^{- \frac{x^{2}}{2}}$

$(1 - \frac{N}{n})^{\frac{N}{2}} = [(1 - \frac{x}{\sqrt{n}})^{- \sqrt{n}}] - \frac{x}{2} \sim e^{- \frac{x^{2}}{2}}$

So,

$P (T_{N} = n) = \sqrt{\frac{2}{π}} \frac{N}{\sqrt{n^{3}}} e^{- \frac{N^{2}}{2 n}}$

Next we want to look directly at $P (τ_{N} = n)$ . Let $k \in N$ be Iixed, and $l$ $\in N$ such

that $- k \underline{<} - l \underline{<} l \underline{<} k$ .

Let $y_{l}^{n}$ be the probability of a path from $(0, 0)$ to $(n, \pm l)$ without passing $t h r o u g h \pm k$ .

Note: $(a, b)$ means getting to the value $b$ at time $a$ .

We have the following recurrence relations:

$1_{y_{k - 1}^{n} \frac{1}{2} y_{k - 2}^{n - 1}}^{y_{0}^{n} . = \frac{1}{2} y_{1}^{n - 1}} . ċ y_{1}^{n} y_{l}^{n} y_{k}^{n} . ċ ċ = \frac{= 1}{2} y_{k - 1}^{n - 1} = \frac{1}{2} y_{l - 1}^{n - 1} + \frac{1}{2} y_{l + 1}^{n - 1} = y_{0}^{n - 1} + \frac{1}{2} y_{2}^{n - 1}$ , for $2 \underline{<} l \underline{<} k - 2$

We want to Iind $p_{n} : = y_{k}^{n} = \frac{1}{2} y_{k - 1}^{n - 1}$ . Then we will take $k = N$ and get the distribution

of $τ_{N}$ .

Let be a column vector.

The recurrence relations can be written as:

$y^{n} = A y^{n - 1}$

where $A$ is a $(k) x (k)$ matrix:

$A =$

Let $P_{A} (t) = d e t (t I - A)$ . Let $B_{k} : = t I$ $- A$ and let $P_{k} : = d e t (B_{k})$ . Then, using the

last row of matrix $B_{k}$ ,

$d e t B_{k} = d e t$ $00 0^{∖} t_{/}$ .

$= - (- \frac{1}{2}) d e t (C) + t d e t (B_{k - 1})$

Then again, using the last row of matrix $C$ ,

$d e t (C) = - (- \frac{1}{2}) d e t (D) + (- \frac{1}{2}) d e t (B_{k - 2}) = \frac{1}{2} d e t (D) - \frac{1}{2} d e t (P_{k - 2})$ .

But $D$ has the last column 0, so $d e t (D) = 0$ . Hence $d e t (C) = - \frac{1}{2} P_{k - 2}$ and so

$P_{k} = - \frac{1}{4} P_{k - 2} + t P_{k - 1}$ (3.8)

The recurrence relation $P_{k} - t P_{k - 1} + \frac{1}{4} P_{k - 2} = 0$ has characteristic polynomial $x^{2} -$

$t x + \frac{1}{4}$ . The roots for this polynomial are $x_{1, 2} = \frac{t \pm \sqrt{t^{2} - 1}}{2}$ and so,

$P_{k} = α_{1} x_{1}^{k} + α_{2} x_{2}^{k}$ .

But $P_{1} = t$ , because $A = 0$ when $k = 1$ and $P_{2} = t^{2} - \frac{1}{2}$ .

So we can identify $α_{1} = α_{2} = 1$ . Hence:

$P_{k} = x_{1}^{k} + x_{2}^{k} = \frac{1}{2^{k}} ((t + \sqrt{t^{2} - 1})^{k} + (t - \sqrt{t^{2} - 1})^{k})$ .

Let $P (t) = (t + \sqrt{t^{2} - 1})^{k} + (t - \sqrt{t^{2} - 1})^{k} = 2^{k} P_{k} = 2^{k} P_{A} (t)$ .

Then $P (A) = 2^{k} P_{A} (A) = 0$ , by Cayley's theorem. If $P (t) = a_{0} t^{k} + ... + a_{k}$ , then

$a_{0} A^{k} + ... + a_{k} I = 0$ . (3.9)

Next, let's multiply (5) to the right by $y^{n - k}$ , which is a column vector, for $n \underline{>} k$ . We

get

$a_{0} y^{n} + a_{1} y^{n - 1} + ... + a_{k} y^{n - k} = 0$

In particular, if we read only the last line we get:

$a_{0} y_{k - 1}^{n} + a_{1} y_{k - 1}^{n - 1} + ... + a_{k} y_{k - 1}^{n - k} = 0$ , $\forall n \underline{>} k$ .

But $p_{n} = \frac{1}{2} y_{k - 1}^{n - 1}$ , so we get the recurrence:

$a_{0} p_{n} + a_{1} p_{n - 1} + ... + a_{k} p_{n - k} = 0$ , for $n \underline{>} k + 1$

Let now $Q (t) = a_{0} + ... + a_{k} t^{k}$ . Consider also the power series $S (t) = p_{0} + p_{1} t + p_{2} t^{2} + ...$

Let $Q (t) S (t) = c_{0} + c_{1} t + c_{2} t^{2} + ...$

Por $n \underline{<} k$ , $c_{n} = a_{0} p_{n} + a_{1} p_{n - 1} + ... + a_{n} p_{0}$ .

Por $n \underline{>} k + 1$ , $c_{n} = a_{0} p_{n} + a_{1} p_{n - 1} + ... + a_{k} p_{n - k} = 0$ .

But as $p_{0} = p_{1} = ... = p_{k - 1} = 0$ , we get that $c_{0} = c_{1} = ... = c_{k - 1} = 0$ and $c_{k} = a_{0} p_{k}$ .

Hence $Q (t) S (t) = c_{k} t^{k} = a_{0} p_{k} t^{k}$ and so

$S (t) = \frac{a_{0} p_{k} t^{k}}{Q (t)}$ (3.10)

We have:

$Q (t) = a_{0} + a_{1} t + ... + a_{k} t^{k}$ , and

$P (t) = a_{0} t^{k} + ... + a_{k}$ .

These two polynomials are reciprocal and

$Q (t) = t^{k} P (\frac{1}{t}) = t^{k} ((\frac{1}{t} + \sqrt{\frac{1}{t^{2}} - 1})^{k} + (\frac{1}{t} - \sqrt{\frac{1}{t^{2}} - 1})^{k})$

$(1 + \sqrt{1 - t^{2}})^{k} + (1 - \sqrt{1 - t^{2}})^{k}$

In particular, $a_{0} = Q (0) = 2^{k}$ . Also, $p_{k} = \frac{1}{2^{k - 1}}$ (one gets $t o \pm k$ after $k$ steps iff there

are $k + 1^{'} s$ or $k - 1^{'} s$ , and any of these two events happen with probability $\frac{1}{2^{k}}$ , Hence

$p_{k} = 2 \frac{1}{2^{k}})$ .

We then get $a_{0} p_{k} = 2$ , so

$S (t) = \frac{a_{0} p_{k} t^{k}}{Q (t)} = \frac{2 t^{k}}{(1 + \sqrt{1 - t^{2}})^{k} + (1 - \sqrt{1 - t^{2}})^{k}}$ .

As a conclusion, $P (τ_{N} = n)$ is the coefficient of $t^{n}$ in the Taylor series of

$S (t) = \frac{2 t^{N}}{(1 + \sqrt{1 - t^{2}})^{N} + (1 - \sqrt{1 - t^{2}})^{N}}$ (3.11)

3.3 Game options in incomplete markets

The game options are contracts which enable both the buyer and seller to stop

them at any time up to maturity, when the contract is terminated anyway. An exam-

ple is the Israeli call option, which is an American style call option with strike price

$K$ where the seller can also terminate the contract, but at the expense of a penalty

$δ_{t_{i}} \underline{>} 0$ .

To define the game contingent claim precisely, let $(Ω, F, P, (F_{t})_{t \in [0, T]})$ be a Iiltered

probability space satisfying the usual conditions of right-continuity and complete-

ness, and let $(U_{t_{i}})_{i = 0, ..., k}$ , $(L_{t_{i}})_{i = 0, ..., k}$ , $(M_{t_{i}})_{i = 0, ..., k}$ be sequences of real-valued random

variables adapted to $(F_{t_{i}})_{i = 0, ..., k}$ with $L_{t_{i}} \underline{<} M_{t_{i}} \underline{<} U_{t_{i}}$ for $i = 0$ , $...$ , $k - 1$ and

$L_{t_{k}} = M_{t_{k}} = U_{t_{i}}$ . If $A$ terminates the contract at time $t_{i}$ before $B$ exercises then $A$

should pay $B$ the amount $U_{t_{i}}$ . Similarly, if $B$ terminates the contract at time $t_{i}$ before

$A$ exercises then $B$ should pay $A$ only the amount $L_{t_{i}}$ . Finally, if $A$ terminates and

$B$ exercises at the same time, then $A$ pays $B$ the amount $M_{t_{i}}$ .

Let $S_{i}$ , $i = 0$ , $...$ , $k$ be the set of stopping times with values in $t_{i}$ , $...$ , $t_{k}$ .

For instance, if $A$ terminates the contract at the random time $σ \in S_{0}$ and $B$ exercises

at the random time $τ \in S_{0}$ , then $A$ will pay $B$ at the random time $σ \land τ$ the amount

$R (σ, τ) = U_{σ} I (σ < τ) + L_{τ} I (τ < σ) + M_{τ} I (τ = σ)$ .

Example: In the case of the Israeli call option, $L_{t_{i}} = (S_{t_{i}}^{1} - K)^{+}$ , $U_{t_{i}} = (S_{t_{i}}^{1} - K)^{+} + δ_{t_{i}}$

and $M_{t_{i}} = (S_{t_{i}}^{1} - K)^{+} + (δ_{t_{i}}) / 2$ . The game version of an American option is cheaper,

because it is a safer investment for the company that sells it. In the case of a complete

market model, the seller $A$ wants to minimize $E_{Q} (R (σ, τ))$ and the buyer $B$ wants to

maximize the same quantity, where $Q$ is the unique equivalent martingale measure.

This is equivalent to a zero-sum Dynkin stopping game, which has a unique value,

which is also the unique no-arbitrage price of the game option [20]. In incomplete

markets, this approach fails because there is more than one equivalent martingale

measure. A possible approach was suggested by Christoph Kuhn [9], in his Ph. D.

thesis and is based on utility maximization.

We consider $u_{1}$ , $u_{2}$ : $R \to R$ two nondecreasing and concave functions that cor-

respond to the utility functions of the seller respectively the buyer of the option.

If we use the game theory language, each player chooses a stopping time $σ \in S_{0}$

(respectively $τ \in S_{0}$ and a trading strategy $θ$ . The seller wants to maximize

$E_{P} (u_{1} (C_{1} - R (σ, τ) + \int_{0}^{T} θ d S_{t}))$ ,

while the buyer wants to maximize

$E_{P} (u_{2} (C_{2} + R (σ, τ) + \int_{0}^{T} θ d S_{t}))$ ,

where the random variable $C_{i} \in F_{I}$ is the exogenous endowment of the i-th player.

Definition: We say that a pair $(σ^{*}, τ^{*}) \in S_{0} \times S_{0}$ is a Nash (or non-cooperative)

equilibrium point, if for all $(σ, τ) \in S_{0} \times S_{0}$ ,

$\sup_{θ} E_{P} (u_{1} (C_{1} - R (σ^{*}, τ^{*}) + \int_{0}^{T} θ d S_{t})) \underline{>} \sup_{θ} E_{P} (u_{1} (C_{1} - R (σ, τ^{*}) + \int_{0}^{T} θ d S_{t}))$ ,

and

$\sup_{θ} E_{P} (u_{2} (C_{2} + R (σ^{*}, τ^{*}) + \int_{0}^{T} θ d S_{t})) \underline{>} \sup_{θ} E_{P} (u_{2} (C_{2} - R (σ^{*}, τ) + \int_{0}^{T} θ d S_{t}))$ .

In the case of exponential utility $(u (x) = 1 - e^{- α_{1} x})$ Nash equilibrium can be con-

structed for various trading strategies. But there are also cases (e.g. logarithmic

utility function) when no Nash equilibrium exists.

Game options might be interesting in an insurance environment too. If the seller of

a variable annuity considers that the product becomes of high risk (for instance the

death benefit is much bigger than the account value), then the insurance company

can terminate the contract and be better off with the penalty than with the risk of a

huge claim.

3.4 Hedging insurance claims in incomplete markets

We will use the mean-variance hedging approaches proposed by Follmer and Son-

dermann [12] and the work with equity-linked insurance contracts by Moller [23]

Consider a Iinancial market with 2 traded assets: a stock with stochastic price process

$S$ and a bond with deterministic price process $B$ . The Iinancial market model is given

by $(Ω^{f}, F^{f}, P^{f})$ and the price processes follow:

$d S_{t} = (y (t, S_{t}) S_{t} d t + σ (t, S_{t}) S_{t} d W_{t}$

$d B_{t} = r (t, S_{t}) B_{t} d t$ ,

where $(W_{t})_{0 \leq t \leq T}$ is a standard Brownian motion on $[0, T]$ . The probability space

is equipped with a Iiltration $F^{f}$ satisfying the usual conditions, defined by $F_{t}^{f} =$

$σ {(S_{u}, B_{u}), u \underline{<} t} = σ {S_{u}, u \underline{<} t}$ . We can also define in this setting the market price

of risk associated with $S$ , $l J_{t} = \frac{α_{t} - r_{t}}{σ_{t}}$ . In the Black-Scholes setting, the price processes

are given by

$S_{t} = S_{0} e x p ((α - \frac{1}{2} σ^{2}) t + σ W_{t})$

$B_{t} = e x p (r t)$

We say that two probability measures $P$ and $P^{⋆}$ are equivalent iff they have the same

null-sets. By definition, the following probability measure $P^{⋆}$ defined by

$\frac{d P^{⋆}}{d P} = e x p (- \int_{0}^{T} \frac{(y_{u} - r_{u}}{σ_{u}} d W_{u} - \frac{1}{2} \int_{0}^{T} (\frac{(y_{u} - r_{u}}{σ_{u}})^{2} d W_{u}) \equiv U_{T}$

is equivalent with $P$ and $S^{⋆} : = S_{t} B_{t}^{- 1}$ is a $P^{⋆}$ -martingale.

A trading strategy (or dynamic portfolio) is a 2-dimensional process $φ_{t} = (θ_{t}, η_{t})$

satisfying certain integrability conditions (indicated later) and where $θ$ is predictable

( $θ_{t}$ is $F_{t - 1}^{f}$ -measurable and $η$ is adapted to $F^{f}$ . The pair $φ_{t} = (θ_{t}, η_{t})$ is the portfolio

held at time $t (θ_{t}$ is the number of shares of the stock held at time $t$ and $η_{t}$ is the

discounted amount invested in the savings account).

Thus, the value process is given by ${\hat{V}}_{t}^{φ} = θ_{t} S_{t} + η_{t} B_{t}$ .

The trading strategy is self-Iinancing if ${\hat{V}}_{t}^{φ} = {\hat{V}}_{0}^{φ} + \int_{0}^{t} θ_{u} d S_{u} + \int_{0}^{t} η_{u} d B_{u}$ , $\forall 0 \underline{<} t \underline{<} T$ .

A contingent claim with maturity $T$ is a random variable $X$ that is $F_{T}^{f}$ -measurable and

$P^{⋆}$ -square integrable. The contingent claim is just a simple claim when $X = g (S_{t})$ ,

where $g$ : $R_{+} \to R$ . The contingent claim is attainable if $\exists φ s . t . {\hat{V}}_{t}^{φ} = X$ $P$ - $a . s$ .

If any contingent claim is attainable, the market is called complete.

The Iinancial market is now complemented with an insurance portfolio. The as-

sumption in the insurance market model are that the lifetimes of the individuals are

independent and identically distributed. We will denote by $l_{x}$ the number of persons

of age $x$ in the group. The probability space $(Ω^{a}, F^{a}, P^{a})$ describes the insurance

model. The remaining lifetimes are modelled by the random variables $T_{1}$ , $T_{2}$ , $...$ , $T_{l_{x}}$ ,

which are iid and non-negative. The hazard function is $μ_{x + t}$ and the survival func-

tion is given by $t p x : = P^{a} (T_{i} > t) = e x p (- \int_{0}^{t} μ_{x + τ} d τ)$ . Next we define a uni-variate

process $N = (N_{t})_{0 \leq t \leq T}$ describing the number of deaths in the group, by:

$N_{t} = \sum_{i = 1}^{l_{x}} I (T_{i} \underline{<} t)$ .

This process is a cadlag. We can now equip the probability space with a Iiltration

$F^{a}$ , by $F_{t}^{a} = σ {N_{u}, u \underline{<} t}$ The stochastic intensity of the counting process $N$ can be

describes as follows:

$E [d N_{t} | F_{t}^{a}] = (l_{x} - N_{t} -) μ_{x + t} d t \equiv λ_{t} d t$

3.5 The combined model in the GMDB case

We define the product space (financial $\times a c t u a r i a l$ ) as we did in the general case

in chapter 2. The Iiltration in the combined model is given by $F = (F_{t})_{0 \leq t \leq T}$ , where

$F_{t} = F_{t}^{f} ⋁ F_{t}^{a}$ . Let's say the contract between insured and insurer has the individual

liability $g_{t} = g (t, S_{t})$ at time $t$ . Overall, (for the entire portfolio), the insurer's liability

is:

$H_{T} = B_{T}^{- 1} \sum_{i = 1}^{l_{x}} g (T_{i}, S_{T_{i}}) B_{T_{i}}^{- 1} B_{T} I (T_{i} \underline{<} T)$

$= \sum_{i = 1}^{l_{x}} \int_{0}^{T} g (u, S_{u}) B_{u}^{- 1} d I (T_{i} \underline{<} u)$

which can be written with respect to the counting process $N$ as follows:

$H_{T} = \int_{0}^{T} g (u, S_{u}) B_{u}^{- 1} d N_{u}$ .

The equivalent martingale measure is not unique anymore, but $w e' 11$ only use $P^{⋆}$

defined above, which is known as the minimal martingale measure, cf. Schweizer [28]

(1991).

We introduce the deflated value process $V^{φ}$ by

$V_{t}^{φ} = {\hat{V}}_{t}^{φ} B_{t}^{- 1} = θ_{t} S_{t}^{⋆} + η_{t}$

and the space $L^{2} (P_{S}^{⋆})$ of $F$ -predictable processes $θ$ satisfying

$E^{⋆} [\int_{0}^{T} (θ)^{2} d ⟨ S^{⋆} ⟩_{u}] < \infty$ .

I this setting, an $F$ -trading strategy is a process $φ_{t} = (θ_{t}, η_{t})$ with $θ \in L^{2} (P_{S}^{⋆})$ and $η$

$F$ -adapted with $V^{φ}$ cadlag and $V_{t}^{φ} \in L^{2} (P_{S}^{⋆}) \forall t$ .

Definition 3.5.1 (Schweizer [29], 1994). The cost process associated with the strategy

$φ$ is defined by

$C_{t}^{φ} = V_{t}^{φ} - \int_{0}^{t} θ_{u} d S_{u}^{⋆}$ ,

and the risk process associated with the strategy $φ$ is defined by

$R_{t}^{φ} = E^{⋆} [(C_{T}^{φ} - C_{t}^{φ})^{2} | F_{t}]$ .

A few comments about these two processes are very interesting. First of all, the initial

cost of the portfolio is $C_{0}^{φ} = V_{0}^{φ}$ and it is tipically greater than zero, except for cases

when we start the portfolio with some short sales. $C_{t}^{φ}$ , the total cost incured in $[0, t]$

can be seen as an initial cost and the cost during $(0, t]$ .

A strategy is called mean-self-fifinancing if the cost process $C^{φ}$ is a $(F, P^{⋆})$ -martingale.

In particular, the strategy $φ = (θ, η)$ is self-Iinancing if and only if

$V_{t}^{φ} = V_{0}^{φ} + \int_{0}^{t} θ_{u} d S_{u}^{⋆}$ ,

or, in other words, if and only if the only cost associated with $φ$ is the initial cost

$(C_{t}^{φ} = C_{0}^{φ} = V_{0}^{φ}, P^{⋆} - a . s .)$

We have seen that the combined model is not complete, and hence there are con-

tingent claims which cannot be replicated by self-Iinancing trading strategies. We

will consider the next best thing, i.e. we are looking for strategies that are able to

generate the contingent claim at time $T$ , but only at some cost defined by $C_{T}^{φ}$ . Let $H$

be a $F_{T}$ -measurable random variable (the contingent claim) and we are looking for a

strategy $φ$ with $V_{T}^{φ} = H a . s$ . and cost process $C^{φ}$ . Note that the cost is not known

at time 0 (unless the strategy is self-financing).

The mean squared error is defined as the value of the risk process at time 0. Hence,

as $F_{0}$ is trivial

$R_{0}^{φ} = E^{⋆} [(C_{T}^{φ} - C_{0}^{φ})^{2} | F_{0}] = E^{⋆} [(C_{T}^{φ} - C_{0}^{φ})^{2}] = E^{⋆} [(H - \int_{0}^{T} θ_{u} d S_{u}^{⋆} - C_{0}^{φ})^{2}]$

Thus, $R_{0}^{φ}$ is minimized for $C_{0}^{φ} = E^{⋆} [H]$ .

But $E^{⋆} [H] = E^{⋆} [C_{T}^{φ}]$ and so the trading

strategy should be chosen such that $θ$ minimizes the variances $E^{⋆} [(C_{T}^{φ} - E^{⋆} [C_{T}^{φ}])^{2}]$ .

The strategy will not be unique (there is an entire class of strategies minimizing the

mean squared error).

The construction of the strategies follows Follmer and Sondermann [12]. First we

define the intrinsic process $V^{⋆}$ by $V_{t}^{⋆} = E^{⋆} [H | F_{t}]$ . Next, using the Galtchouk-Kunita-

Watanabe decomposition (see Appendix), we can decompose $V_{t}^{⋆}$ uniquely as follows:

$V_{t}^{⋆} = E^{⋆} [H] + \int_{0}^{t} θ_{u}^{H} d S_{u}^{⋆} + L_{t}^{H}$ ,

where $L^{H}$ is a zero-mean $(F, P^{⋆})$ -martingale, $L^{H}$ and $S^{⋆}$ are orthogonal and $θ^{H}$ is a

predictable process in $L^{2} (P_{S}^{⋆})$ . Follmer and Sondermann [12] proved that

Theorem 3.5.1.: An admissible strategy $φ_{t} = (θ_{t}, η_{t})$ has minimal variance

$E^{⋆} [(C_{T}^{φ} - E^{⋆} [C_{T}^{φ}])^{2}] = E^{⋆} [(L_{T}^{H})^{2}]$

if and only if $θ = θ^{H}$ .

The number of bonds held at time 0 is given by $η_{0} = E^{⋆} [H] - θ_{0} S_{0}^{⋆}$ . Follmer and

Sondermann [12] have refined the process and have found an admissible strategy

minimizing the risk process $R_{t}^{φ}$ at any time $t$ . This strategy is unique and called

risk-minimizing.

Theorem 3.5.2: There exists a unique admissible risk-minimizing strategy $φ = (θ, η)$

given by

$(θ_{t}, η_{t}) = (θ_{t}^{H}, V_{t}^{⋆} - θ_{t}^{H} S_{t}^{⋆})$ , $0 \underline{<} t \underline{<} T$ .

The risk process is given by $R_{t}^{φ} = E^{⋆} [(L_{T}^{H} - L_{t}^{H})^{2} | F_{t}]$ , and is called the intrinsic risk

process.

It is interesting to remark that an admissible risk-minimizing strategy is mean-self-

Iinancing.

We have seen that the insurer's liability is given by

$H_{T} = \int_{0}^{T} g (u, S_{u}) B_{u}^{- 1} d N_{u}$ .

Following Moller [23] the intrinsic value process of $H_{T}$ is given in this case by

$V_{t}^{⋆} = E^{⋆} [H_{T} | F_{t}] = \int_{0}^{t} g (u, S_{u}) B_{u}^{- 1} d N_{u} + E^{⋆} [\int_{t}^{T} g (u, S_{u}) B_{u}^{- 1} d N_{u} | F_{t}]$

$= \int_{0}^{t} g (u, S_{u}) B_{u}^{- 1} d N_{u} + E^{⋆} [\int_{t}^{T} g (u, S_{u}) B_{u}^{- 1} (l_{x} - N_{t})_{n - t} p_{x + t} μ_{x + u} d u | F_{t}]$

$= \int_{0}^{t} g (u, S_{u}) B_{u}^{- 1} d N_{u} + E^{⋆} [\int_{t}^{T} g (u, S_{u}) B_{t}^{- 1} e^{- j_{t}^{u} r_{τ} d τ} (l_{x} - N_{t})_{n - t} p_{x + t} μ_{x + u} d u | F_{t}]$ .

Let us define now

$F^{g} (t, S_{t}) = E^{⋆} [e^{- j_{t}^{u} r_{τ} d τ} g (u, S_{u}) | F^{f}] = E^{⋆} [e^{- j_{t}^{u} r_{τ} d τ} g (u, S_{u}) | F]$

which is the unique arbitrage-free price at time $t$ of the claim $g$ in the complete

Iinancial model with Iiltration $F^{f}$ . Using this function, the instrinsic process can be

written as:

$V_{t}^{⋆} = \int_{0}^{t} g (u, S_{u}) B_{u}^{- 1} d N_{u} + \int_{t}^{T} F^{g} (t, S_{t}) B_{t}^{- 1} (l_{x} - N_{t})_{n - t} p_{x + t} μ_{x + u} d u$

Next, we are trying to Iind

$d (B_{t}^{- 1} F^{g} (t, S_{t})) = - r (t, S_{t}) B_{t}^{- 1} F^{g} (t, S_{t}) d t + B_{t}^{- 1} d F^{g} (t, S_{t})$

$= - r (t, S_{t}) B_{t}^{- 1} F^{g} (t, S_{t}) d t + B_{t}^{- 1} (F_{t}^{g} (t, S_{t}) d t + F_{s}^{g} (t, S_{t}) d S_{t} + \frac{1}{2} F_{s s}^{g} (t, S_{t}) σ (t, S_{t})^{2} S_{t}^{2} d t)$

$= F_{s}^{g} (t, S_{t}) d S_{t}^{⋆}$ .

as $d S_{t} = S_{t}^{⋆} d B_{t} + B_{t} d S_{t}^{⋆} = S_{t}^{⋆} (r_{t} B_{t} d t) + B_{t} d S_{t}^{⋆} = S_{t} r_{t} d t + B_{t} d S_{t}^{⋆}$ and the price process

$F^{g} (t, S_{t})$ is characterized by the partial differential equation

$- r (t, S_{t}) F^{g} (t, S_{t}) + F_{t}^{g} (t, S_{t}) + r (t, S_{t}) S_{t} F_{s}^{g} (t, S_{t}) + \frac{1}{2} σ (t, S_{t})^{2} S_{t}^{2} F_{s s}^{g} (t, S_{t}) = 0$ ,

with boundary condition $F^{g} (T, S_{T}) = g (T, S_{T})$ . Following Ikeda and Watanabe [18],

the intrinsic process can be expressed as:

$V_{t}^{⋆} = V_{0}^{⋆} + \int_{0}^{t} - F^{g} (τ, S_{τ}) B_{τ}^{- 1} (l_{x} - N_{τ}) μ_{x + r} d τ$

$+ \int_{0}^{t} (g (τ, S_{τ}) B_{τ}^{- 1} - \int_{τ}^{T} B_{τ}^{- 1} F^{g} (τ, S_{τ})_{u - τ} p_{x + τ} μ_{x + u} d u) d N_{τ}$

$+ \int_{0}^{t} (\int_{τ}^{T} F^{g} (τ, S_{τ}) B_{τ}^{- 1} - T p_{x + τ} u μ_{x + u} d u) (l_{x} - N_{τ^{-}}) μ_{x + τ} d τ$

$+ \int_{0}^{t} ((l_{x} - N_{τ^{-}}) \int_{τ}^{T} F^{g} (τ, S_{τ})_{u - τ} p_{x + u} μ_{x + u} d u) d S_{τ}^{⋆}$ .

This gives the following decomposition of the intrinsic process

$V_{t}^{⋆} = V_{0}^{⋆} + \int_{0}^{t} θ_{u}^{H} d S_{u}^{⋆} + \int_{0}^{t} l /_{u}^{H} d M_{u}$

where $M_{t} = N_{t} - \int_{0}^{t} λ_{u} d u$ is the compensated counting process and

$θ_{t}^{H} = (l_{x} - N_{t^{-}}) \int_{t}^{T} F_{s}^{g} (t, S_{t})_{u - t} p_{x + t} μ_{x + u} d u$ ,

$l y_{t}^{H} = g (t, S_{t}) B_{t}^{- 1} - \int_{t}^{T} F^{g} (t, S_{t}) B_{t^{u - t}}^{- 1} p_{x + t} μ_{x + u} d u$ .

Combining this result with theorem 3.5.1. we get the following result:

Theorem 3.5.3. The unique admissible risk-minimizing strategy for the insurance

company's contingent claim is given by:

$θ_{t}^{⋆} = (l_{x} - N_{t^{-}}) \int_{t}^{T} F_{s}^{g} (t, S_{t})_{u - t} p_{x + t} μ_{x + u} d u$ ,

$η_{t}^{⋆} = \int_{0}^{t} g (u, S_{u}) B_{u}^{- 1} d N_{u} + (l_{x} - N_{t}) \int_{t}^{T} F^{g} (t, S_{t}) B_{t^{u - t}}^{- 1} p_{x + t} μ_{x + u} d u - θ_{t}^{⋆} S_{t}^{⋆}$ ,

for $0 \underline{<} t \underline{<} T$ .

The value of the insurer's portfolio can be seen as the sum of benefits set aside for

deaths already occured and the expectations of the benefits associated with future

deaths:

$V_{t}^{φ} = \int_{0}^{t} g (u, S_{u}) B_{u}^{- 1} d N_{u} + E^{⋆} [\int_{t}^{T} g (u, S_{u}) B_{u}^{- 1} d N_{u} | F_{t}]$ .

It is interesting to note that, when a death occurs at time $t$ , the reserves set by the

insurance company for the benefits are relieved by the amount

$\int_{t}^{T} F^{g} (t, S_{t}) B_{t^{u - t}}^{- 1} p_{x + t} μ_{x + u} d u$

Let us now consider a few different types of GMDB riders.

3.5.1 GMDB with return of premium

We start by analyzing a GMDB with return of premium (ROP). In this case,

the function that models the contingent claim is given by $g (u, S_{u}) = \max (S_{u},$ $K =$

$S_{0}) = K + (S_{u} - K)^{+}$ . In this case (and assuming again that the Iinancial market is

complete), $F^{g} (t, S_{t})$ can be evaluated by the Black-Scholes formula:

$F^{g} (t, S_{t}) = E^{⋆} [e^{- r (T - t)} (K + (S_{T} - K)^{+}) | F_{t}]$

$= K e^{- r (T - t)} + S_{t} Φ (z_{t}) - K e^{- r (T - t)} Φ (z_{t} - σ \sqrt{T - t})$

$= K e^{- r (T - t)} Φ (- z_{t} + σ \sqrt{T - t}) + S_{t} Φ (z_{t})$ ,

where $Φ$ is the normal cumulative distribuition and

$z_{t} = \frac{l n (\frac{s_{t}}{K}) + (r + \frac{σ^{2}}{2}) (T - t)}{σ \sqrt{T - t}}$ .

We also notice that the Iirst order partial derivative with respect to $s$ is $F_{s}^{g} (t, S_{t}) =$

$Φ (z_{t})$ and hence, using theorem 3.5.2 we get the following hedging strategy:

$θ_{t}^{⋆} = (l_{x} - N_{t^{-}}) \int_{t}^{T} Φ (z_{t})_{u - t} p_{x + t} μ_{x + u} d u$ ,

$η_{t}^{⋆} = (l_{x} - N_{t}) \int_{t}^{T} (K e^{- r (T - t)} Φ (- z_{t} + σ \sqrt{T - t}) + S_{t} Φ (z_{t})) B_{t^{u - t}}^{- 1} p_{x + t} μ_{x + u} d u$

$+ \int_{0}^{t} g (u, S_{u}) B_{u}^{- 1} d N_{u} - θ_{t}^{⋆} S_{t}^{⋆}$

3.5.2 GMDB with return of premium with interest

The next rider we can hedge is the return of premium with interest. In this case

$g (u, S_{u}) = \max (S_{u}, K e^{δ u})$ , where $δ$ is the force of interest. Furthermore,

$F^{g} (t, S_{t}) = K s^{δ u} e^{- r (T - t)} Φ (- z_{t} + σ \sqrt{u - t}) + S_{t} Φ (z_{t})$

where now $z_{t}$ is given by:

$z_{t} = \frac{l n (\frac{s_{t}}{K e^{δ u}}) + (r + \frac{σ^{2}}{2}) (T - t)}{σ \sqrt{T - t}}$ .

Finally, using theorem 3.5.2. we get the risk-minimizing strategy:

$θ_{t} = (l_{x} - N_{t^{-}}) \int_{t}^{T} u - t p_{x + t} μ_{x + u} Φ (z_{t}) d u$ ,

$η_{t} = (l_{x} - N_{t}) \int_{t}^{T} u - t p_{x + t} μ_{x + u} K e^{- (r - δ) u} Φ (z_{t} + σ \sqrt{u - t}) d u +$

$+ \int_{0}^{t} g (u, S_{u}) B_{u}^{- 1} d N_{u} - ▵ N_{t} \int_{t}^{T} u - t p_{x + t} μ_{x + u} Φ (z_{t}) S_{t}^{⋆} d u$ .

3.5.3 GMDB with ratchet

Recall, when a GMDB has a ratchet rider, there are some anniversary dates when

the death benefit can be ratcheted up. The individual liability is $g (t, S_{t}) = \max (K =$

$53$

$S_{0}$ , $S_{1}$ , $S_{2}$ , $...$ , $S_{t}$ ). We can apply the above strategy inductively, looking at periods

bewtween two consecutive anniversary dates. So, let us consider $g_{1} (t, S_{t}) = \max (S_{0}, S_{1}) =$ :

$L_{1}$ and we apply the hedging strategy for the return of premium rider on the inter-

val $[0, 1]$ . Next function is $g_{2} (t, S_{t}) = \max (L_{1}, S_{2}) = : L_{1}$ and so we get the hedging

strategy on $[1, 2]$ . The trading strategy on $[0, 1]$ is given by:

$θ_{t}^{⋆} = (l_{x} - N_{t^{-}}) \int_{t}^{1} Φ (z_{t})_{u - t} p_{x + t} μ_{x + u} d u$ ,

$η_{t}^{⋆} = (l_{x} - N_{t}) \int_{t}^{1} (K e^{- r (1 - t)} Φ (- z_{t} + σ \sqrt{1 - t}) + S_{t} Φ (z_{t})) B_{t u - t}^{- 1} p_{x + t} μ_{x + u} d u$

$+ \int_{0}^{t} g_{1} (u, S_{u}) B_{u}^{- 1} d N_{u} - θ_{t}^{⋆} S_{t}^{⋆}$

where

$z_{t} = \frac{l n (\frac{s_{t}}{K}) + (r + \frac{σ^{2}}{2}) (1 - t)}{σ \sqrt{1 - t}}$ .

The hedging strategy is extended for $t > 1$ using similar formulas. Similar formulae

can be found for other riders, using the same technique: roll-up, reset etc. It can

also be shown that the ratio $\frac{\sqrt{R_{0}}}{l_{x}}$ converges to 0 as $l_{x}$ increases, showing that this

nonhedgeable part of the claim, actually its risk, decreases with the number of con-

tracts sold. The variable annuity with a GMDB ratchet could be priced and hedged

perfectly if instead of a single premium, the contract is sold for a variable (dynamic)

premium process. Let's assume we have the following sample path for the stock price

process given by Figure 3.3 and that the death benefit is payable at the end of the

year of death. Let's assume that the death are uniformly distributed, and the max-

imum lifetime is $ω$ . Let us assume the age of the insured is $x$ . The present value of

the benefits is given by the formula:

$P V = \sum_{m = 1}^{ω - x} m - 1 | q_{x} P (m, S_{0}) + \sum_{m = 2}^{ω - x} m - 1 | q x$ $[P (m, S_{1}) - P (m, S_{0})]_{+} + ...$

$t - - 0$ $t - - 1$ $t - - 2$ $t - - 3$ $t - - 4$

Figure 3.3: Sample stock price evolution

This allows us to sell this contract for a sequence of premiums. In our example (Figure

3.3) the Iirst premium is

$P_{0} = \sum_{m = 1}^{ω - x} m - 1 | q_{x} P (m, S_{0})$ .

At time 1, the stock price (value) is above $P_{0}$ and so the death benefit is ratcheted

up and the insured has to pay (if he wants the death benefit ratcheted up) a new

premium:

$P_{1} = \sum_{m = 2}^{ω - x} m - 1 | q x$ $[P (m, S_{1}) - P (m, S_{0})]_{+}$ .

At time 2, the stock price drops, and so this doesn't affect the death benefit. The

premium at this time is 0. This process continues untill the expiration of the contract.

3.6 Living benefits

In the previous sections we have discussed benefits that are triggered by the death

of the insured. We now take a look at benefits that are paid only if the person is

alive at the end of the contract. These benefits are more expensive in the age group

they are sold for (the probability of living untill the expiration of the contract is

bigger than the probability of dieing). Their Canadian name is VAGLB (variable

anuities guaranteed living benefits) while American insurance companies call them

GMAB (guaranteed minimum accumulation benefits) or GMIB (guaranteed minimim

income benefits-when the benefits are annuitized).

We consider a setting similar to the GMDB case. But, in this case, for living benefits,

the present value of the claim is

$X = g (S_{T}) B_{T}^{- 1} (l_{x} - N_{T})$ ,

and the intrinsic value process is given by

$V_{t}^{⋆} = E^{⋆} [X | F_{t}]$

Following Moller [23] the stochastc independence between the lifetimes of the indi-

viduals and the market (i.e. between $N$ and ( $B$ , $S$ )) allows us to rewrite $V_{t}^{⋆}$ as

$V_{t}^{⋆} = E^{⋆} [(l_{x} - N_{T}) | F_{t}] B_{t}^{- 1} E^{⋆} [g (S_{T}) B_{t} B_{T}^{- 1} | F_{t}]$

$= E^{⋆} [\sum_{i = 1}^{l_{x}} I (T_{i} > T) | F_{t}] B_{t}^{- 1} E^{⋆} [g (S_{T}) B_{t} B_{T}^{- 1} | F_{t}]$

$= \sum_{i : T_{i} > T} E^{⋆} [I (T_{i} > T) | F_{t}] B_{t}^{- 1} E^{⋆} [g (S_{T}) B_{t} B_{T}^{- 1} | F_{t}]$

$= \sum_{i : T_{i} > T} τ - t p_{x + t} B_{t}^{- 1} E^{⋆} [g (S_{T}) B_{t} B_{T}^{- 1} | F_{t}]$

$= (l_{x} - N_{t})_{T - t} p_{x + t} B_{t}^{- 1} E^{⋆} [g (S_{T}) B_{t} B_{T}^{- 1} | F_{t}]$ .

Also, note that the conditional dstribution of the market price processes doesn't

depend on information about the insurance model $(F_{t}^{a})$ , and so

$E^{⋆} [g (S_{T}) B_{t} B_{T}^{- 1} | F_{t}] = E^{⋆} [g (S_{T}) B_{t} B_{T}^{- 1} | F_{t}^{f}] = F^{g} (t, S_{t})$ .

Similar arguments to the GMDB case lead us to

$V_{t}^{⋆} = (l_{x} - N_{t})_{T - t} p_{x + t} B_{t}^{- 1} F^{g} (t, S_{t})$ .

Itô formula is next applied giving us:

$V_{t}^{⋆} = V_{0}^{⋆} + \int_{0}^{t} (l_{x} - N_{u^{-}})_{T - u} p_{x + u} μ_{x + u} B_{u}^{- 1} F^{g} (u, S_{u})$ du

$+ \int_{0}^{t} (l_{x} - N_{u^{-}})_{T - u} p_{x + u} d (B_{u}^{- 1} F^{g} (u, S_{u})) + \sum_{0 < u \leq t} (V_{u}^{⋆} - V_{u}^{⋆} -)$ .

Recall from the GMDB case that

$d (B_{t}^{- 1} F^{g} (t, S_{t})) = - r (t, S_{t}) B_{t}^{- 1} F^{g} (t, S_{t}) d t + B_{t}^{- 1} d F^{g} (t, S_{t})$

$= - r (t, S_{t}) B_{t}^{- 1} F^{g} (t, S_{t}) d t + B_{t}^{- 1} (F_{t}^{g} (t, S_{t}) d t + F_{t}^{g} (t, S_{t}) d S_{t} + \frac{1}{2} F_{s s}^{g} (t, S_{t}) σ (t, S_{t})^{2} S_{t}^{2} d t)$

$= F_{s}^{g} (t, S_{t}) d S_{t}^{⋆}$ .

We also notice that

$\sum_{0 < u \leq t} (V_{u}^{⋆} - V_{u^{-}}^{⋆}) = - \int_{0}^{t} B_{t}^{- 1} F^{g} (t, S_{t}) d t$ .

Hence, we can decompose the value process of the contingent claim $X$ as

$V_{t}^{⋆} = V_{0}^{⋆} + \int_{0}^{t} θ_{u}^{X} d S_{u}^{⋆} + \int_{0}^{t} l /_{u}^{X} d M_{u}$

where $M_{t} = N_{t} - \int_{0}^{t} λ_{u} d u$ is the compensated counting process and

$θ_{t}^{X} = (l_{x} - N_{t} -) F_{s}^{g} (t, S_{t})_{T - t} p_{x + t}$ ,

$l y_{t}^{X} = - B_{t}^{- 1} F^{g} (t, S_{t})_{T - t} p_{x + t}$

for $0 \underline{<} t \underline{<} T$ .

Therefore, we have the following:

Theorem 3.6.1: The adimissible strategies minimizing the variance $E^{⋆} [(C_{T}^{φ} - E^{⋆} C_{T}^{φ})^{2}]$

are characterized by:

$θ_{t}^{⋆} = (l_{x} - N_{t} -) F_{s}^{g} (t, S_{t})_{T - t} p_{x + t}$ ,

$η_{T}^{⋆} = X - θ_{T} S_{T}^{⋆}$ .

The minimal variance can be determined [23] by use of Pubini's theorem:

$E^{⋆} [(\int_{0}^{T} l /_{u}^{X} d M_{u})^{2}] = E^{⋆} [\int_{0}^{T} (l y_{u}^{X})^{2} d ⟨ M ⟩_{u}]$

$= E^{⋆} [\int_{0}^{T} (B_{u}^{- 1} F^{g} (u, S_{u})_{T - u} p_{x + u})^{2} λ_{u} d u]$

$= \int_{0}^{T} E^{⋆} [(B_{u}^{- 1} F^{g} (u, S_{u}))^{2}] τ_{- u} p_{x + u}^{2} E^{⋆} [(l_{x} - N_{u}) μ_{x + u}]$ du

$= \int_{0}^{T} E^{⋆} [(B_{u}^{- 1} F^{g} (u, S_{u}))^{2}] τ_{- u} p_{x + u}^{2} l_{x} u p_{x} μ_{x + u} d u$

$= l_{x} T p_{x} \int_{0}^{T} E^{⋆} [(B_{u}^{- 1} F^{g} (u, S_{u}))^{2}] τ_{- u} p_{x + u} μ_{x + u} d u$ .

Recall the Galtchouk-Kunita-Watanabe decomposition for the instrinsic value pro-

cess:

$V_{t}^{⋆} = E^{⋆} [X] + \int_{0}^{t} θ_{u}^{X} d S_{u}^{⋆} + L_{t}^{X}$ ,

and we obtain

$E^{⋆} [(L_{T}^{X} - L_{t}^{X})^{2} | F_{t}] = E^{⋆} [(\int_{t}^{T} l /_{u}^{X} d M_{u})^{2} | F_{t}]$

$= E^{⋆} [\int_{t}^{T} (l y_{u}^{X})^{2} λ_{u} d u | F_{t}]$

$= \int_{t}^{T} E^{⋆} [(l y_{u}^{X})^{2} | F_{t}] E^{⋆} [(l_{x} - N_{u}) μ_{x + u} | F_{t}]$ du

$= (l_{x} - N_{u}) \int_{t}^{T} E^{⋆} [(l y_{u}^{X})^{2} | F_{t] u - t p_{x + t} μ_{x + u} d u}$ .

We have now a theorem similar to 3.5.2. for living benefits:

Theorem 3.6.2: The unique admissible risk minimizing strategy for the living ben-

efits is given by:

$θ_{t}^{⋆} = (l_{x} - N_{t -})_{T - t} p_{x + t} F_{s}^{g} (t, S_{t})$ ,

$η_{t}^{⋆} = (l_{x} - N_{t})_{T - t} p_{x + t} F_{s}^{g} (t, S_{t}) B_{t}^{- 1} - θ_{t}^{⋆} S_{t}^{⋆}$ .

for $0 \underline{<} t \underline{<} T$ .

The intrinsic risk process is given by:

$(l_{x} - N_{u}) \int_{t}^{T} E^{⋆} [(l y_{u}^{X})^{2} | F_{t] u - t p_{x + t} μ_{x + u} d u}$ .

3.6.1 VAGLB with return of premium

In this case $g (u, S_{u}) = \max (S_{u}, K = S_{0}) = K + (S_{u} - K)^{+}$ and (assuming again

that the Iinancial market is complete), $F^{g} (t, S_{t})$ can be evaluated by the Black-Scholes

formula:

$F^{g} (t, S_{t}) = E^{⋆} [e^{- r (T - t)} (K + (S_{T} - K)^{+}) | F_{t}]$

$= K e^{- r (T - t)} + S_{t} Φ (z_{t}) - K e^{- r (T - t)} Φ (z_{t} - σ \sqrt{T - t})$

$= K e^{- r (T - t)} Φ (- z_{t} + σ \sqrt{T - t}) + S_{t} Φ (z_{t})$ ,

where $Φ$ is the normal cumulative distribuition and

$z_{t} = \frac{l n (\frac{s_{t}}{K}) + (r + \frac{σ^{2}}{2}) (T - t)}{σ \sqrt{T - t}}$ .

We also notice that the Iirst order partial derivative with respect to $s$ is $F_{s}^{g} (t, S_{t}) =$

$Φ (z_{t})$ and hence, using theorem 3.6.2 we get the following hedging strategy:

$θ_{t} = (l_{x} - N_{t -})_{T - t} p_{x + t} Φ (z_{t})$ ,

$η_{t} = (l_{x} - N_{t})_{T - t} p_{x + t} F^{g} (t, S_{t}) e^{- r t} - (l_{x} - N_{t})_{T - t} p_{x + t} Φ (z_{t}) S_{t}^{⋆}$ .

$= (l_{x} - N_{t})_{T - t} p_{x + t} K e^{- r t} Φ (- z_{t} + σ \sqrt{T - t}) - ▵ N_{t T - t} p_{x + t} Φ (z_{t}) S_{t}^{⋆}$ ,

while the instrinsic risk process is given by

$R_{t}^{φ} = (l_{x} - N_{t})_{T - t} p_{x + t} \int_{t}^{T} E^{⋆} [(e^{- r u} F^{g} (u, S_{u}))^{2} | F_{t}]_{T - u} p_{x + u} μ_{x + u} d u$ .

3.6.2 VAGLB with return of premium with interest

The next rider we can hedge for VAGLB is the return of premium with interest.

In this case $g (u, S_{u}) = \max (S_{u}, K e^{δ u})$ , where $δ$ is the force of interest. Furthermore,

$F^{g} (t, S_{t}) = K s^{δ u} e^{- r (T - t)} Φ (- z_{t} + σ \sqrt{u - t}) + S_{t} Φ (z_{t})$

where now $z_{t}$ is given by:

$z_{t} = \frac{l n (\frac{s_{t}}{K e^{δ u}}) + (r + \frac{σ^{2}}{2}) (T - t)}{σ \sqrt{T - t}}$ .

Finally we also notice that the Iirst order partial derivative with respect to $s$ is

$F_{s}^{g} (t, S_{t}) = Φ (z_{t})$ and hence, using theorem 3.6.2. we get the risk-minimizing strategy:

$θ_{t} = (l_{x} - N_{t -})_{T - t} p_{x + t} Φ (z_{t})$ ,

$η_{t} = (l_{x} - N_{t})_{T - t} p_{x + t} F^{g} (t, S_{t}) e^{- r t} - (l_{x} - N_{t})_{T - t} p_{x + t} Φ (z_{t}) S_{t}^{⋆}$ .

$= (l_{x} - N_{t})_{T - t} p_{x + t} K s^{δ u} e^{- r (T - t)} Φ (- z_{t} + σ \sqrt{u - t}) + S_{t} Φ (z_{t}) e^{- r t} - (l_{x} - N_{t})_{T - t} p_{x + t} Φ (z_{t}) S_{t}^{⋆}$ .

3.7 Discrete time analysis

We consider a descrete time model, in which the Iinancial market follows the Cox-

Ross-Rubinstein model (also known as the binomial model). The model consists of

two basic securities. The time horizon is $T$ and the set of dates in the Iinancial market

model is $t = 0, 1$ , 2, $...$ , $T$ . Assume that the Iirst security is riskless (bond or bank

account) $B$ , with price process

$B_{t} =$ $(1 + r)^{t}$ , $t = 0, 1$ , $...$ , $T$ ,

i.e. the bond yields a riskless rate of return $r$ in each time interval $[t, t + 1]$ . The

second security is a risky asset (stock, or stock index) $S$ with price process

$S (t + 1)$ $=$ $w i t h p r o b a b i l i t y 1 w i t h p r o b a b i l i t y p$ $- p$

for $t = 0, 1$ , $...$ , $T - 1$ and with $0 < d < u$ , and $S_{0} \in R_{0}^{+}$ .

The Iirst task is to Iind an equivalent martingale measure, i.e. a probability measure

$Q$ which is equivalent to the physical measure and such that the discounted price

process $S^{⋆} (t) : = S (t) B^{- 1} (t)$ is a martingale with respect to $Q$ . In other words we

want to determine $q$ such that $Q ({u}) = q$ and $Q ({d}) = 1$ $- q$ and $Q$ satisfies the

above conditions. We have the following result:

Theorem 3.7.1.

1. A martingale measure $Q$ for the discounted stock price exists if and only if

$d < 1$ $+ r < u$

2. If 1. holds true, then the measure $Q$ is uniquely determined by:

$q = \frac{1 + r - d}{u - d}$

For a proof, see [3]. This theorem tells us that the Iinancial binomial market wich

satisfies the natural condition 1. is complete. Hence, we have perfect hedging of

options. It is interesting to notice that a so-called trinomial model (when the price

process has 3 different outcomes) is not complete. The natural Iiltration in this

model $F^{f}$ is given by $F_{t}^{f} : = σ {S_{1}, ..., S_{t}}$ . Let $X$ be a contingent claim, that is

a $F_{T}^{f}$ -measurable $Q$ -integrable random variable. Define the following process $W_{t} =$

$E^{⋆} [X | F_{t}^{f}]$ which is a martingale with respect to the Iiltration $F^{f}$ and the measure $Q$ .

Then we have the following representation [30]:

$W_{t} = W_{0} + \sum_{j = 1}^{t} (y_{j} ▵ S_{j}^{⋆}$ ,

where ( $y_{j}$ is predictable (i.e. $F_{j - 1}^{f}$ -measurable), for any $j = 1$ , 2, $...$ , $t$ . Now, we can

think of $W$ as the discounted value process under some strategy $φ$ . So we have

$V_{T} (φ) = X$ i.e. the terminal value of the strategy is the claim and

$V_{t} (φ) = V_{0} (φ) + \sum_{j = 1}^{t} (y_{j} ▵ S_{j}^{⋆}$ .

Now the strategy $φ = (θ, η)$ that replicates $X$ is given by $θ_{t} = (y_{t}$ while $η$ is uniquely

determined such that the strategy is also self-Iinancing by

$η_{t} = W_{0} + \sum_{j = 1}^{t} θ_{j} ▵ S_{j}^{⋆} - θ_{t} S_{t}^{⋆}$ .

As a consequence, the price of the contract should be the initial value of the self-

Iinancing strategy, which is $W_{0} = E^{⋆} [X]$ .

Consider now the second model, with a portfolio of $l_{x}$ policy-holders age $x$ (at time 0)

and let $l_{x} - N_{t}$ the number of survivors at time $t$ . The lifetime of the individuals in the

group are modelled by $T_{1}$ , $...$ , $T_{l_{x}}$ which are independent and identically distributed

random variables. Also, define $t p x : = P (T_{1} > t)$ . Let us assume that the contract is

modelled by the claim $g (S_{T})$ and has present value

$X = (l_{x} - N_{t}) g (S_{T}) B_{T}^{- 1}$

Hence we have a guaranteed living benefit contract. As in the continuous time model,

we introduce a Iiltration on the actuarial model $F^{a}$ given by

$F_{t}^{a} = σ {N_{1}, ..., N_{t}}$ .

The Iiltration in the product space is defined by $F_{t} = F_{t}^{f} ⋁ F_{t}^{a} = σ (F_{t}^{f} \cup F_{t}^{a})$ , the

smallest $σ$ -algebra containing both $F_{t}^{f}$ and $F_{t}^{a}$ .

Following [24] we define two processes related to the Iinancial market and the actuarial

market respectively. In the Iinancial market, recall that the unique price at time $t$ of

the contract with payment $g (S_{T})$ at time $T$ is given by

$P_{t}^{f} : = E^{⋆} [g (S_{T}) B_{T}^{- 1} | F_{t}^{f}] = E^{⋆} [g (S_{T}) B_{T}^{- 1}] + \sum_{j = 1}^{t} (y_{j}^{f} ▵ S_{j}^{⋆}$ .

In the actuarial market, we introduce the process

$M_{t} : = E^{⋆} [(l_{x} - N_{T}) | F_{t}^{a}] = (l_{x} - N_{t})_{T - t} p_{x + t}$ ,

the conditional expected number of survivors at time T.

The discounted value proces is

$V_{t}^{⋆} = E^{⋆} [(l_{x} - N_{T}) g (S_{T}) B_{T}^{- 1} | F_{t}] = E^{⋆} [g (S_{T}) B_{T}^{- 1} | F_{t}] E^{⋆} [(l_{x} - N_{T}) | F_{t}]$

$= E^{⋆} [g (S_{T}) B_{T}^{- 1} | F_{t}^{f}] E^{⋆} [(l_{x} - N_{T}) | F_{t}^{a}] = P_{t}^{f} M_{t}$ .

and we obtain the recurrence:

$V_{t}^{⋆} - V_{t - 1}^{⋆} = P_{t}^{f} M_{t} - P_{t - 1}^{f} M_{t - 1} = (P_{t}^{f} - P_{t - 1}^{f}) M_{t - 1} + P_{t}^{f} (M_{t} - M_{t - 1})$

$= M_{t - 1^{(y_{t}^{f}}} ▵ S_{t}^{⋆} + P_{t}^{f} ▵ M_{t}$ ,

which gives us the decomposition:

$V_{t}^{⋆} = V_{0}^{⋆} + \sum_{j = 1}^{t} M_{j - 1} (y_{j}^{f} ▵ S_{j}^{⋆} + \sum_{j = 1}^{t} P_{j}^{f} ▵ M_{j}$ .

In this decomposition, $M_{j - 1} (y_{j}^{f}$ is predictable and $\sum_{j = 1}^{t} P_{j}^{f} ▵ M_{j}$ is a martingale. In

can also be shown that $(\sum_{j = 1}^{t} P_{j}^{f} ▵ M_{j}) S_{t}^{⋆}$ is a martingale and hence, the unique

risk-minimizing strategy [24] is given by:

$θ_{t} = (l_{x} - N_{t - 1})_{T - (t - 1)} p_{x + (t - 1)^{(y_{t}^{f}}}$ ,

$η_{t} = (l_{x} - N_{t})_{T - t} p_{x + t} P_{t}^{f} - (l_{x} - N_{t - 1})_{T - (t - 1) p_{x + (t - 1)^{(y_{t}^{f}}}} S_{t}^{⋆}$ .

The cost process for the strategy is given by

$C_{t} (φ) = V_{0}^{⋆} + \sum_{j = 1}^{t} P_{j}^{f} ▵ M_{j}$ .

The risk that remains with the insurer who applies the risk-minimizing strategy can

be assesed using the variance of the accumulated costs $C_{T} (φ)$ :

$V a r^{⋆} [C_{T} (φ)] = \sum_{t = 1}^{T} E^{⋆} [(P_{t}^{f})^{2} ▵ M_{t}^{2}] = \sum_{t = 1}^{T} E^{⋆} [(P_{t}^{f})^{2}] E [▵ M_{t}^{2}]$

$= \sum_{t = 1}^{T} E^{⋆} [(P_{t}^{f})^{2}] E [V a r [▵ M_{t} | F_{t - 1}]] + V a r [E [▵ M_{t} | F_{t - 1}]]$

$= \sum_{t = 1}^{T} E^{⋆} [(P_{t}^{f})^{2}] E [V a r [_{T - t} p_{x + t} (l_{x} - N_{t}) | F_{t - 1}]]$

$= \sum_{t = 1}^{T} E^{⋆} [(P_{t}^{f})^{2}] l_{x} T p_{x T - t} p_{x + t} (1 - 1 p_{x + (t - 1)})$ .

We will next look at an example (in discrete settings) and compare the risk-minimizing

strategy with other hedging strategies discussed in 3.1.

3.7.1 Risk comparison

Let us start by looking at a numerical example. We consider a binomial tree model

with 4 trading times, $t = 0, 1$ , 2, 3. At time 0 the stock price is 100. At time 1, it can

go up to 110 or down to 80. The entire sample process is described in the following

Figure 3.4. For simplicity and without restricting the generality, we can assume that

the time span between two consecutive trading times is 1 year. The market contains

the stock and a bank account (bond) earning interest at the rate $r = . 05$ or 5 percent

per year. We will analyze a living benefit contract associated with the stock. We will

assume that the start of the contract is $t = 0$ and the expiration is $t = 3$ . We will also

assume that we are dealing with a return of premium rider, i.e. the living benefit is the

maximum between the account value at the expiration (account value $= s t o c k$ value)

and the guaranteed return of premium. For simplicity we assume that the remaining

lifetimes of the policy-holders are independent and exponentially distributed with

hazard rate (force of mortality) $μ$ . In this case, the survival probability is

$t p_{x} = e x p (- μ t)$

and so we obtain

$E [▵ M_{t}^{2}] = l_{x} T p_{x T - t} p_{x + t} (1 - 1 p_{x + (t - 1)})$

$= e^{- 2 μ T + μ t} (1 - e^{- μ})$

This allows us to determine the variance of the accumulated cost associated with the

risk-minimizing strategy:

$V a r^{⋆} [C_{T} (φ)] = l_{x} \sum_{t = 1}^{T} E^{⋆} [(P_{t}^{f})^{2}]_{T} p_{x T - t} p_{x + t} (1 - 1 p_{x + (t - 1)})$

Recall $f (S_{T}) = \max (S_{T}, K)$ , where $K = S_{0}$ . To Iind the hedging strategy, we will have

to Iind the call option price tree associated with the stock tree process (Figure 3.4).

LABELS

$\bar{Π 8}$

$\bar{Π 7}$

$\bar{Π 6}$

$\bar{Π 5}$

$\bar{Π 4}$

$\bar{Π 3}$

$\bar{Π 2}$

$\bar{Π 1}$

Figure 3.4: Binomial stock price process

We will Iind, at Iirst, the risk-neutral probability for every branch of the tree. In

other words, we want to Iind a measure $P^{⋆}$ such that

$S (t) = E^{⋆} (\frac{S (t + 1)}{1 + r})$ , $\forall t = 0, 1$ , 2

The Iirst tree is $(100 - - > (110, 80))$ and we want to Iind $p^{⋆}$ such that

$100 = E^{⋆} (\frac{S (1)}{1 + r}) = \frac{110 p^{⋆} + 80 (1 - p^{⋆})}{1.05}$ .

Solving for $p^{⋆}$ we get $p^{⋆} = 5 / 6$ . Using the same equation $S (t) = E^{⋆} (\frac{S (t + 1)}{1 + r})$ , we

can Iind the risk-neutral probabilities for every brunch of the stock price tree and

we can round up the tree in Figure 3.5. We will look at the Iinancial probability

space. Let $Ω = {1, 2, 3, 4, 5, 6, 7, 8}$ , the set of Iinal states. We also define the $σ -$

algebra $F = P (Ω)$ , the power set of $Ω$ , and so $| F | = 2^{| Ω |}$ . We also define a Iiltration

$F_{0} \underline{\subset} F_{1} \underline{\subset} F_{2} \underline{\subset} F_{3}$ where

$F_{0} = {\emptyset, Ω}$

$F_{1} = σ {{1,$ 2,3,4},{5,6,7, $8}} = {\emptyset,$ {1,2,3,4},{5,6,7,8}, $Ω}$

$F_{2} = σ {{1,$ 2},{3,4},{5,6},{7, $8}}$

$F_{2}$ has $2^{4}$ elements and Iinally, $F_{3} = F$ . To define a probability measure on $Ω$ , it is

enough to define it on the set of simple events ${ω}$ , where $ω \in {\bar{1, 8}}$ . But for any

state of the world ${ω}$ , there is a unique path from 0 to 3 and we will define the

probability of the Iinal state as the product of the conditional probabilities (given by

the risk-neutral probabilities) along the path. Hence, we get:

$P ({8}) = \frac{5}{6} \times \frac{17}{20} \times \frac{4}{5} = \frac{17}{30}$

$P ({7}) = \frac{5}{6} \times \frac{17}{20} \times \frac{1}{5} = \frac{17}{120}$

Figure 3.5: Risk-neutral probabilities

$P ({6}) = \frac{5}{6} \times \frac{3}{20} \times \frac{29}{40} = \frac{29}{320}$

$P ({5}) = \frac{5}{6} \times \frac{3}{20} \times \frac{11}{40} = \frac{11}{320}$

$P ({4}) = \frac{1}{6} \times \frac{7}{10} \times \frac{29}{50} = \frac{203}{3000}$

$P ({3}) = \frac{1}{6} \times \frac{7}{10} \times \frac{21}{50} = \frac{49}{1000}$

$P ({2}) = \frac{1}{6} \times \frac{3}{10} \times \frac{9}{10} = \frac{9}{200}$

$P ({1}) = \frac{1}{6} \times \frac{3}{10} \times \frac{1}{10} = \frac{1}{200}$ .

Next we want to Iind the price of the call option $X (0)$ at time $t = 0$ . The price is the

discounted value of the expected payoff at $t = 3$ :

$X (0) = E^{⋆} (\frac{(S (3) . - K)_{+}}{10 5^{3}}) = \frac{40 (\frac{17}{30}) + 20 (\frac{17}{120}) + 10 (\frac{29}{320}) + 15 (\frac{203}{3000})}{1.0 5^{3}} = 23.67$

To determine the option price process, we will go backwords, using again the fact

that the price at $t$ is the discounted expected payoff. The tree is given by Figure 3.6.

To determine the hedging strategy for the call option, we will use the option price

process. Let $φ_{0} (1)$ the amount of bonds held at $t = 0$ and $φ_{1} (1)$ the number of shares

of the stock at $t = 0$ that replicate the price of the call at $t = 0$ and its price at $t = 1$ .

We have the following system:

Solving, we get $φ_{1} = . 76$ and $φ_{0} = - 52.68$ so we buy.76 shares of the stock and sell

52.68 in bonds. Continuing this process, we obtain the hedging process (i.e. number

of shares process), given in Figure 3.6. Now, recall the formulas that gives us the

hedging strategy for the living benefit:

$θ_{t} = (l_{x} - N_{t - 1})_{T - (t - 1)} p_{x + (t - 1)^{(y_{t}^{f}}}$ ,

Figure 3.6: Call Option Price Process

Figure 3.7: Hedging Process

$η_{t} = (l_{x} - N_{t})_{T - t} p_{x + t} P_{t}^{f} - (l_{x} - N_{t - 1})_{T - (t - 1) p_{x + (t - 1)^{(y_{t}^{f}}}} S_{t}^{⋆}$ .

and the tree for the hedging process ( $y_{t}^{f} = φ_{1} (t)$ (Figure 3.7). Let us assume that

$μ = 1$ . Then, for one policy-holder we have

$θ_{1} = τ_{- (1 - 1) p_{x + (1 - 1)} α_{1}^{f} = e^{- 3} (. 764)} = . 038$

and

$η_{1} = τ - 1 p_{x + 1} P_{1}^{f} - τ - (1 - 1) p_{x + (1 - 1)} α_{1}^{f} S_{1}^{⋆} = e^{- 2} \times 123.6$ $- . 038 \times 100$ $= 12.92$ .

These are the number of shares (or bonds, respectively), held in the portfolio during

the period $(0, 1]$ . At time 1, these numbers will change as functions of the stock price

evolution and the lifetime of the insured.

If the insured is not alive at $t = 1$ , then $θ_{2} = 0$ and $η_{2} = 0$ .

If the insured is alive, then we have two cases. If the stock price is up (110), then

$θ_{2 T - (2 - 1) p_{x + (2 - 1)}} = α_{2}^{f} = e^{- 2} (. 912)$ $= . 123$

and

$η_{2} = τ - 2 p_{x + 2} P_{2}^{f} - τ - (2 - 1) p_{x + (2 - 1)} α_{2}^{f} S_{2}^{⋆} = e^{- 1} \times 128.7$ -.123 $\times 104.76$ $= 34.46$ .

If the stock price is down (80), then

$θ_{2} = τ - (2 - 1) p_{x + (2 - 1)} α_{2}^{f} = e^{- 2} (. 414)$ $= . 056$

and

$η_{2} = τ - 2 p_{x + 2} P_{2}^{f} - τ - (2 - 1) p_{x + (2 - 1)} α_{2}^{f} S_{2}^{⋆} = e^{- 1} \times 105.5$ -.056 $\times 104.76$ $= 32.94$ .

Continuing this algorithm, we come up with the risk-minimizing trading strategy, see

Figure 3.8. If instead of $μ = 1$ we take $μ = 0.5$ we get a different risk-minimizing

Figure 3.8: Risk-minimizing trading strategy, $μ = 1$

Figure 3.9: Risk-minimizing trading strategy, $μ = 0.5$

trading strategy (see Figure 3.9.) Next we want to determine the variance of the

accumulated cost $C_{T} (φ)$ for the two cases ( $μ = 1$ , and $μ = 0.5$ ). We have the

following:

$V a r^{⋆} [C_{T} (φ)] = l_{x} \sum_{k = 1}^{T} E^{⋆} [(P_{k}^{f})^{2}]_{T} p_{x T - k} p_{x + k} (1 - 1 p_{x + (k - 1)})$

$= \sum_{k = 1}^{T} E^{⋆} [(P_{k}^{f})^{2}] e^{- 2 μ T + μ k} (1 - e^{- μ})$

and

$V a r^{⋆} [X] = l_{x} E^{⋆} [(P_{T}^{f})^{2}] e^{- μ T} (1 - e^{- μ}) + l_{x}^{2} V a r^{⋆} [P_{T}^{f}] e^{- 2 μ T}$

Por $μ = 1$ and $l_{x} = 1$ we get $V a r^{⋆} [C_{T} (φ)] = 736.92$ and $V a r^{⋆} [X] = 739.66$ . The

quotient $\frac{V a r^{⋆} [C_{T} (φ)]}{V a r^{⋆} [X]}$ is 0.996.

Por $μ = . 5$ and $l_{x} = 1$ we get $V a r^{⋆} [C_{T} (φ)] = 2695.77$ and $V a r^{⋆} [X] = 2746.9$ . The

quotient $\frac{V a r^{⋆} [C_{T} (φ)]}{V a r^{⋆} [X]}$ is 0.981.

Generally, this quotient is an increasing function of $μ$ because

$\frac{V a r^{⋆} [C_{T} (φ)]}{V a r^{⋆} [X]} = \frac{1 - e^{- μ}}{(1 - \frac{V a r^{⋆} [P_{1}^{f}]}{E^{⋆} [(P_{1}^{f})^{2}]}) (1 - e^{- μ}) + \frac{V a r^{⋆} [P_{1}^{f}]}{E^{⋆} [(P_{1}^{f})^{2}]}}$ .

Finally, let us review the pricing techniques discussed above (Table 3.1).

Method

Price

Risk-minimizing

Mean-variance hedging

Super-hedging

$l_{x T} p_{x}$ $\times V_{0}^{f}$

$l_{x}$ $\times V_{0}^{f}$

Table 3.1: Pricing formulas for living benefits contracts

3.8 Multiple decrements for variable annuities

Let us consider now a model with two random variables: $T$ is the time until

termination from a status and $J$ , the cause of decrement. For simplicity, assume that

$J$ is a discrete random variable, and $| J | = m$ . The joint probability density function

of $T$ and $J$ is $f_{T, J} (t, j)$ . We have the following standard relations and definitions. The

probability of decrement due to cause $j$ before time $t$ is

$t q_{x}^{(j)} = P r [(0 < T \underline{<} t) \cap (J = j)] = \int_{0}^{t} f_{T, J} (s, j) d s$

Also, $τ$ means all causes and we have

$t q_{x}^{(τ)} = P r (T \underline{<} t) = F_{T} (t) = \sum_{j = 1}^{m} t q_{x}^{(j)}$

which is the probability of termination due to any cause,

$t p_{x}^{(τ)} = P r (T > t) = 1$ $- t q_{x}^{(τ)}$

which is the probability of survival with respect to all decrements and

$μ_{x}^{(τ)} (t) = - \frac{d}{d t} l o g (_{t} p_{x}^{(τ)}) = \sum_{j = 1}^{m} μ_{x}^{(j)} (t)$

is the force of mortality to all causes.

The basic type of variable annuity with multiple decrements riders is defined for only

two decrements, let us say death and invalidity. Once a decrement kicks in, the benefit

is locked and paid at the expiration of the contract $(ω)$ . So, only one decrement can

occur for each policyholder. The benefit is

$i f T > ω i f T \underline{<} ω a n d i f T \underline{<} ω a n d J = 2 J = 1$

where $H_{1}$ and $H_{2}$ are two functions that define the benefit for each decrement. As in

the single decrement case,

$N_{t} = \sum_{i = 1}^{l_{x}} I (T \underline{<} t)$

One can also define

$N_{j t} = \sum_{i = 1}^{l_{x}} I ((T \underline{<} t) \cap (J = j))$

for any $j = 1$ , $...$ , $m$ . The insurer's liability is now a function of the decrement also.

So, the present value of the benefit is:

$H_{T} = \int_{0}^{T} g (u, S_{u}) B_{u}^{- 1} d N_{u}$ .

where $g (t, S_{t})$ is the individual liability (depends on the decrement).

The intrinsic value process is given by

$V_{t}^{⋆} = E^{⋆} [H_{T} | F_{t}] =$

$= E^{⋆} [\sum_{j = 1}^{m} \int_{0}^{T} g_{j} (u, S_{u}) B_{u}^{- 1} (l_{x} - N_{u})_{u} p_{x}^{(τ)} μ_{x + u}^{(j)} d u | F_{t}]$

Similiar computations to the single decrement case lead us to the following result:

Theorem 3.8.1. The unique admissible risk-minimizing strategy for the insurance

company's contingent claim is given by:

$θ_{t}^{⋆} = (l_{x} - N_{t^{-}}) \sum_{j = 1}^{m} \int_{t}^{T} F_{s}^{(g_{j})} (t, S_{t})_{u - t} p_{x + t}^{(τ)} μ_{x + u}^{(j)} d u$ ,

$η_{t}^{⋆} = \int_{0}^{t} g_{j} (u, S_{u}) B_{u}^{- 1} d N_{j u} + (l_{x} - N_{t}) \sum_{j = 1}^{m} \int_{t}^{T} F^{g_{j}} (t, S_{t}) B_{t^{u - t}}^{- 1} p_{x + t}^{(τ)} μ_{x + u}^{(j)} d u - θ_{t}^{⋆} S_{t}^{⋆}$ ,

for $0 \underline{<} t \underline{<} T$ .

A more complicated type of variable annuity with multiple decrements riders can be

defined if the benefit is not paid at the time of the decrement occurance, but it is

just locked; if another decrement occurs after that but before the expiration of the

contract and if the benefit is larger that what was already locked, the policyholder

will lock the larger benefit instead (which is then paid at the expiration).

The risk-minimizing technique can also be adapted in this case, although the formulas

are a lot more complicated that in the multiple decrement case when the benefit is

paid at the moment of the decrement occurance.

CHAPTER 4

RESULTS AND CONCLUDING REMARKS

In this dissertation, I am mostly interested in bringing a more theoretical ap-

proach into the problem of pricing and hedging variable annuities. There wasn't

much research of this type in this area until recently. Most insurance companies

use simulations in pricing these contracts, and there is a need for better techniques

because of the size of the market (which exceeds one trillion dollars, accoring to

Moody's) and the increased volatility of the stock market. This need is even larger in

the case of incomplete markets (with no perfect hedging of options). We considered

the problem of pricing and hedging for the most common riders attached to variable

annuities and we also looked at risk-minimizing strategies and at a possible approach

for discrete models.

There are two ways one can define an incomplete market.

We are dealing with a dual market model, Iinancial and actuarial. If the Iinancial

model is incomplete, then the product market is also incomplete. We use an incom-

plete Iinancial market model, in which the stock prices jump in different proportions

at some random times which correspond to the jump times of a Poisson process. Be-

tween the jumps the risky assets follow the Black-Scholes model.

The product space is incomplete even if the Iinancial market model is complete, be-

cause the actuarial risk (mortality risk) is not hedgeable in the stock market. The

approach in this case follows the risk minimization technique defined by Follmer and

Sondermann [12] and the work of Moller [23].

Both these alternatives were analyzed and reviewed.

APPENDIX A

KUNITA-WATANABE DECOMPOSITION

One of the most important results used in pricing and hedging in incomplete

markets is the fundamental decomposition result of Kunita and Watanabe:

Theorem Al For every $M$ $\in M_{2}$ , we have the decomposition $M$ $= N + Z$ , where

$N \in M_{2}^{⋆}$ , $Z \in M_{2}$ , and $Z$ is orthogonal to every element of $M_{2}^{⋆}$ .

The proof follows [19]. Here $Λ 4_{2}$ represents the space of square-integrable martingales

and $L^{⋆}$ denotes the set of progressively measurable processes. Also, $M_{2}^{⋆}$ is the subset

of $Λ 4_{2}$ which consists of continuous stochastic integrals

$I_{t} (X) = \int_{0}^{t} X_{s} d W_{s}$ ; $0 \underline{<} t < \infty$

where $X \in L^{⋆}$ and $W$ is a Brownian motion. We have to show the existance of a

process $Y \in L^{⋆}$ such that $M$ $= I (Y) + Z$ , where $Z \in M_{2}$ has the property

$⟨ Z, I (X) ⟩ = 0; \forall X \in L^{⋆}$ (A. 1)

Such a decomposition is unique (up to indistinguishability): if $M$ $= I (Y^{1}) + Z^{1} =$

$M$ $= I (Y^{2}) + Z^{2}$ , with $Y^{1}$ , $Y^{2} \in L^{⋆}$ and both $Z^{1}$ and $Z^{2}$ satisfy (A1), then

$Z : = Z^{2} - Z^{1} = I (Y^{1} - Y^{2})$

is a continuous element of $Λ 4_{2}$ and $⟨ Z ⟩ = ⟨ Z, I (Y^{1} - Y^{2}) ⟩ = 0$ . This implies that

$P [Z_{t} = 0, \forall 0 \underline{<} t < \infty] = 1$ .

Therefore, it is enough to establish the decomposition for every Iinite time interval

$[0, t]$ ; by uniqueness we can extend it to the entire half-line $[0, \infty)$ . Pix $T > 0$ and let

$R_{T}$ be the closed subspace of $L^{2} (Ω, F_{T}, P)$ defined by

$R_{T} = {I_{T} (X); X \in L_{T}^{⋆}}$

and let $R_{T}^{⊥} d e n o t e$ its orthogonal complement. $L_{T}^{⋆}$ denotes the class of processes

$X \in L^{⋆}$ for which $X_{t} (ω) = 0$ , $\forall t > T$ , $ω \in Ω$ . Then the random variable $M_{T}$ is in

$L^{2} (Ω, F_{T}, P)$ and so it admits the decomposition

$M_{T} = I_{T} (Y) + Z_{T}$ , (A.2)

where $Y \in L_{T}^{⋆}$ and $Z_{T} \in L^{2} (Ω, F_{T}, P)$ satisfies

$E [Z_{T} I_{T} (X)] = 0; \forall X \in L_{T}^{⋆}$ . (A.3)

Let us denote by $Z$ a right-continuous version of the martingale $E (Z_{T} | F_{t})$ . Note that

$Z_{t} = Z_{T}$ , $\forall t \underline{>} T$ . We have $Z \in M_{2}$ and conditioning (A2) on $F_{t}$ we get

$M_{t} = I_{t} (Y) + Z_{t}$ ; $0 \underline{<} t \underline{<} T$ , $a . s$ . $P$ (A.4)

Now it only remains to show that $Z$ is orthogonal to every square-integrable martin-

gale of the form $I (X); X \in L_{T}^{⋆}$ . This is equivalent to showing that ${Z_{t} I_{t} (X)$ , $0 \underline{<}$

$t \underline{<} T}$ is a $F_{t}$ -martingale. This is true if $E [Z_{S} I_{S} (X)] = 0$ for every stopping time $S$

of the Iiltration ${F_{t}}$ with $S \underline{<} T$ . Finally,

$E [Z_{S} I_{S} (X) = E [E (Z_{T} | F_{S}) I_{S} (X)] = E [Z_{T} I_{T} (X_{t} (ω) 1_{{t \leq S (ω)}})] = 0$

which proves the theorem.

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