Two essays in financial product pricing

We explore the optimal withdrawal strategy adopted by the rationalpolicyholder that maximizes the expected discounted value of the cash flows generatedfrom holding this variable annuity p

ZONG JIANPING

(M.Sc., National University of Singapore)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE

2011

I would like to thank my supervisor, A/P Dai Min, who gave me the opportunity

to work on such an interesting research project, paid patient guidance to me, gave meinvaluable help, constructive and inspiring suggestion

My sincere thanks go to all my department-mates and my friends in Singaporefor their friendship and so much kind help I am also grateful to national university ofSingapore for providing scholarship and enjoyable environment for living and studying

I would like also to dedicate this work to my families, especially my parents for theirunconditional love and support

Finally I would also wish to appreciate my wife Mrs Li Ling who is always ing and encouraging me

support-Zong Jianping August 2011

ii

2.1 Introduction 3

2.2 Model formulation 7

2.2.1 A Static Model of GMWB 7

2.2.2 A Dynamic Model of GMWB 12

2.3 Pricing behaviors and optimal withdrawal policies 31

2.4 Conclusion 39

3 HJM Model for Non-Maturing Liabilities 48 3.1 A General Framework for HJM Model 48

iii

3.1.1 HJM Model under Forward Measure 54

3.1.2 Cross Currency HJM Model 55

3.2 Gaussian HJM Model 56

3.2.1 The Pricing of Caps and Floors 59

3.2.2 The pricing of European Swaptions 62

3.3 LGM2++ As HJM Two-Factor Model 67

3.3.1 The Pricing of Caps and Floors under LGM2++ Model 69

3.3.2 The pricing of European Swaptions under LGM2++ Model 70

3.3.3 Monte Carlo Simulation of LGM2++ Model 71

3.3.4 Numerical Examples 75

3.4 A New HJM Two-Factor Model (HJM2++) 81

3.4.1 Monte Carlo Simulation of HJM2++ Model 84

3.4.2 Numerical Examples 87

3.5 Gaussian HJM model for Non-Maturing Liabilities 91

3.5.1 Literature Review on Non Maturity Deposit 91

3.5.2 Model Assumption 92

3.5.3 Cash Flow of Non-Maturity Deposit 93

3.5.4 Modeling of Deposit Volume, Deposit Rate and Market Rate 95

3.5.5 Closed-Form Solution of Jarrow and Devender with LGM2++ 96

3.5.6 Model Implementation 99

3.5.7 Numerical Results 104

4 Conclusion 108 4.1 GMWB 108

4.2 Non-Maturing Deposit 108

al withdrawal rate is set and no penalty is imposed when the policyholder chooses towithdraw at or below this rate Subject to a penalty fee, the policyholder is allowed towithdraw at a rate higher than the contractual withdrawal rate or surrender the policyinstantaneously We explore the optimal withdrawal strategy adopted by the rationalpolicyholder that maximizes the expected discounted value of the cash flows generatedfrom holding this variable annuity policy An efficient finite difference algorithm usingthe penalty approximation approach is proposed for solving the singular stochastic con-trol model Optimal withdrawal policies of the holders of the variable annuities withthe guaranteed minimum withdrawal benefit are explored We also construct discretepricing formulation that models withdrawals on discrete dates Our numerical tests showthat the solution values from the discrete model converge to those of the continuous model.

v

In the second chapter we develop HJM model for non-maturing deposit valuation Westart from general HJM framework and derive some useful lemmas for HJM model Later

we introduce two special two-factor gaussian HJM model: LGM2++ model and HJM2++model Exact simulation scheme in both risk-neutral and forward measure is developedfor pricing purpose Numerical results for caps/floors and swaptions show that our exactsimulation is quite close to analytical price Then we introduce two deposit volume anddeposit rate model for non-maturity deposits We develop exact simulation scheme usingLGM2++ as market rate model Numerical results for price and Greeks of non-maturingdeposit are compared in both risk-neutral and forward measure

value with varying values of the penalty parameter λ and penalty charge k. 272.2.5 Examination of the rate of convergence of the Crank-Nicholson scheme forsolving the penalty approximation model with quarterly withdrawal frequency 302.2.6 The dependence of the fair value of the GMWB annuity on the withdraw-

al frequency per year The annuity value obtained using the continuouswithdrawal model (frequency becomes ∞) is close to that corresponding to

monthly withdrawal (frequency equal 12) The differences in annuity valueswith and without the reset provision are seen to be small 31

2.3.1 Impact of the GMWB contractual rate g, penalty charge k and equity volatility σ of the account on the required insurance fee α (in basis points) with r = 5% . 32

vii

3.3.1 The Cap and Floor price by Monte Carlo Simulation and Analytical formula

under LGM2++ Model where the simulation is done under risk-neutral

measure 763.3.2 The Payer/Receiver Swaption price by Exact Monte Carlo Simulation and

Analytical formula under LGM2++ Model where the simulation is done

under risk-neutral measure 773.3.3 The Payer/Receiver Swaption price by Approximation and Analytical for-

mula under LGM2++ Model 783.3.4 The Cap Implied Volatility Surface by Analytical formula under LGM2++

Model 803.3.5 The ATM Swaption Volatility Surface by Approximate formula under L-

GM2++ Model 803.4.1 The Cap and Floor price by Monte Carlo Simulation and Analytical formula

under HJM2++ Model where the simulation is done under risk-neutral

measure 883.4.2 The Payer/Receiver Swaption price by Exact Monte Carlo Simulation and

Approximation formula under HJM2++ Model where the simulation is done

under risk-neutral measure 893.4.3 The Cap Implied Volatility Surface by Analytical formula under HJM2++

Model 903.4.4 The ATM Swaption Implied Volatility by Analytical formula under L-

GM2++ Model 903.5.1 Part of DBS Group Balance Sheet from 2001 to 2010 (in Billion SGD) 913.5.2 NPV, Duration, Average life and IRPV01 of Deposit where the simulation

is done under risk-neutral measure The standard error of NPV is also

included in parenthesis 106

3.5.3 NPV, Duration, Average life and IRPV01 of Deposit where the simulation

is done underforward measure The standard error of NPV is also included

in parenthesis 1063.5.4 The bucket IRPV01 of Deposit where the simulation is done under risk-

neutral measure 1063.5.5 The bucket IRPV01 of Deposit where the simulation is done underforward

measure 107

2.3.1 Plot of the optimal withdrawal boundary in the (W, A)-plane at t = ∆t The left boundary intersects the A-axis at A = − G

rln(1− k) and in the red

region optimal withdrawal can be either G or 0 . 35

2.3.2 Plot of the optimal withdrawal boundary in the (W, A)-plane at t = ∆t The left boundary intersects the A-axis at A = − G

rln(1− k) and in the red

region optimal withdrawal can be either G or 0 . 38

2.3.3 Plot of the optimal withdrawal boundary in the (W, A)-plane at t = ∆t The left boundary intersects the A-axis at A = − G

rln(1− k) and in the red

region optimal withdrawal can be either G or 0 . 39

2.3.4 Optimal Withdrawal Boundary at time t = 0 Model Parameters are G =

7, r = 5%, σ = 0.2, α = 523b.p., k = 5%. 40

2.4.1 The characteristic lines are given by t + A

G = ξ0 for varying values of ξ0 For

ξ0> T , the characteristic lines intersect the right vertical boundary: t = T ;

and for ξ0 ≤ T , the characteristics lines intersect the bottom horizontal

boundary: A = 0 . 462.4.2 The continuation region lies in the region (shaded part) {(t, A) : A ≤

Chapter 1

Introduction

Financial product pricing is a one of the most important and challenging topics in financialindustry In this thesis we study two quite important financial products: guaranteedminimum withdrawal benefit (GMWB) and non-maturing deposit

GMWB is an insurance rider on variable annuity policies It allows the policy holder

to withdrawal a fixed percentage of the total annuity premium regardless of the ment performance However the insurance company charges annual insurance fee on suchbenefit In chapter 2, we shall formulate the pricing problem of GMWB and study itsoptimal withdrawal strategy

invest-Non-maturing deposit (e.g checking and savings deposit) has no stated terminationdate The bank customer has the right to withdrawal or deposit any amount of cash atany time Banks (especially commercial banks) count on core deposits as a stable source

of funds for their lending base However valuing non-maturity deposit is not a simple taskwithout market comparison benchmark In chapter 3, we try to introduce HJM model toestimate its value and interest rate sensitivity

2

on preset future dates The variable payments would depend on the performance ofthe reference portfolio, thus the policyholders are provided with the equity participation.Variable annuities are attractive to investors not only because of the tax-deferred feature.

In addition, they also offer different types of benefits, such as guaranteed minimum deathbenefit, guaranteed minimum accumulation benefit, guaranteed minimum income benefit

In recent years, variable annuities with the guaranteed minimum withdrawal benefits

3

(GMWBs) have attracted significant attention and sales These benefits allow the cyholders to withdraw funds on an annual or semi-annual basis There is a contractualwithdrawal rate such that the policyholder is allowed to withdraw at or below this ratewithout a penalty The GMWB promises to return the entire initial investment, thus theguarantee can be viewed as an insurance option More precisely, even when the personalaccount (investment net of withdrawal and proportional insurance fees) of the policy-holder falls to zero prior to the policy maturity date, the insurer continues to providethe guaranteed withdrawal amount until the entire original premium is paid out If theaccount stays positive at maturity, the whole remaining balance in the account is paid

poli-to the policyholder at maturity Therefore, the poli-total sum of cash flows received by thepolicyholder is guaranteed to be the same or above the original premium deposit (notaccounting for the time value of the cash flows) Under the dynamic setting of the policy,the policyholder is allowed to withdraw at a rate higher or lower than the contractual rate

or in a finite amount or even surrender instantaneously, according to his best economicadvantage The annuity contract may include the following clause that serves to discour-age excessive withdrawal When the policyholder withdraws at a higher rate than thecontractual withdrawal rate, the guarantee level is reset to the minimum of the prevailingguarantee level and the account value For example, suppose the policyholder decides towithdraw $10, which is higher than the contractual withdrawal amount $7 Suppose thecurrent guarantee level is $80 while the personal account is $60, then the guarantee level

drops to min($80, $60) − $10 = $50 after the withdrawal of $10 In addition, there is a

percentage penalty charge applied on the excessive portion of the withdrawal amount.There has been much research devoted to the pricing and hedging of variable annuities

and insurance policies with various forms of embedded options For hedging strategies,

Coleman et al (2006) suggest risk minimization hedging for variable annuities under

both equity and interest rate risks Milevsky and Posner (2001) use risk neutral optionpricing theory to value the guaranteed minimum death benefit in variable annuities Chuand Kwok (2004) and Siu (2005) analyze the withdrawal and surrender options in variousequity-linked insurance products Milevsky and Salisbury (2006) develop the pricing model

of variable annuities with GMWB under both static and dynamic withdrawal policies.Under the static withdrawal policies, the policyholders are assumed to behave passivelywith withdrawal rate kept fixed at the contractual rate and to hold the annuity to maturity

In their dynamic model, policyholders are assumed to follow an optimal withdrawal policyseeking to maximize the annuity value by lapsing the product at an optimal time Sincethe withdrawal is allowed to be at a finite rate or in discrete amount (infinite withdrawalrate), the pricing model leads to a singular stochastic control problem with the withdrawalrate as the control variable

In this chapter, we would like to study the nature of GMWB in variable annuitiesbeyond the results reported by Milevsky and Salisbury (2006) We provide a rigorousderivation of the singular stochastic control model for pricing variable annuities withGMWB using the Hamilton-Jacobi-Bellman equation Both cases of continuous and dis-crete withdrawal of funds are considered The valuation of the variable annuities product

is performed under the risk neutral framework, assuming the underlying equity portfolio

is tradeable or the holder is a risk neutral investor Our pricing models do not includemortality factor since mortality risk is not quite crucial in guaranteed minimum with-drawal benefit riders Also, we have assumed deterministic interest rate structure since

interest rate plays its influence mainly on discount factors in pricing the guaranteed imum withdrawal benefit This is different from equity fluctuation, where it has muchmore profound impact on the optimal withdrawal policy We assume the policyholder to

min-be fully rational in the sense that he chooses the optimal dynamic withdrawal strategy so

as to maximize the expected discounted value of the cash flows generated from holdingthe annuity policy In our pricing formulation, we manage to obtain a set of parabolicvariational inequalities that govern the fair value of the variable annuity policy with theGMWB The constraint inequalities are seen to involve the gradient of the value function

By extending the penalty method in the solution of optimal stopping problems as

pro-posed by Forsyth and Vetzal (2002) and Dai et al (2007), we propose an efficient finite

difference scheme following the penalty approximation approach to solve for the fair value

of the annuities The numerical procedure of using the penalty approximation approachrepresents a nice contribution to the family of numerical methods for solving singular s-tochastic control problems (Kumar and Muthuraman, 2004; Forsyth and Labahn, 2006)

In addition, we design the finite difference scheme that allows for discrete jumps acrossdiscrete withdrawal dates for solving the discrete time withdrawal model

The chapter is organized as follows In the next section, we consider a static GMWB

pricing model assuming the passive policy holder withdrawals a fixed rate G throughout

the term of contract In section 2 we derive the singular stochastic control model thatincorporates the GMWB into the variable annuities pricing model We start with theformulation that assumes continuous withdrawal, then generalize the model to allow for adiscrete withdrawal on specified dates We outline the numerical approach using the finitedifference scheme with penalty approximation for solving the set of variational inequalities

of the pricing formulation Numerical tests were performed that serve to illustrate therobustness of the proposed numerical schemes for both the continuous and discrete models.

In Section 4, we analyze the optimal withdrawal behaviors of the policyholders We alsoexamine the impact of various parameters in the singular stochastic control pricing model

on the fair insurance fee to be charged by the insurer for provision of the guarantee Asummary and concluding remarks are presented in the last section

2.2.1 A Static Model of GMWB

The static model poses a sub-optimal withdrawal strategy which may significantly reducethe value of GMWB Int this subsection we shall formulate static continuous and discretetime pricing model for GMWB

Static Continuous Withdrawal Model

Let S tdenote the value of the reference portfolio of assets underlying the variable annuity

policy, before the deduction of any proportional fees Taking the usual assumption on

the price dynamics of equity in option pricing theory, the evolution of S t under the risk

neutral measure is assumed to follow

dS t = rS t dt + σS t dB t , (2.2.1)

where B t represents the standard Brownian motion, σ is the volatility and r is the riskfree

interest rate Let F t be the natural filtration generated by the Brownian process B t and

α be the proportional annual insurance fee paid by the policyholder.

Let W t denote the value of the personal variable annuity account After deducting the

proportional insurance fees α and withdrawal rate G, the dynamics of W t follows

dW t = [(r − α)W t − G] dt + σW t dB t if W t > 0 (2.2.2)

Once W t hits the value 0, it stays at this value thereafter Let w0be the initial account

value of the policy, which is the same as the premium paid up front When the personal

account value stays positive at maturity T , the remaining balance is paid back to the policyholder at T

Assume that w0 = 100 dollars The typical GMWB guarantees the policyholder to

withdraw g = 7% of either the investment account or the outstanding guaranteed drawal benefit annually The maturity of GMWB is usually T = 1/g In this section, we assume that the policyholder can withdraw G = w0g dollars continuously per annum.

with-By Ito’s formula, we have

By observing the above expression, it can be seen that if W t < 0 ever reaches 0, W t will

be negative later on So the solution to equation (2.2.1) is simply given by

Both Monte Carlo method and finite difference (FD) method are implemented to compute

P (w, t) For Monte Carlo method, we take M = 10, 000 paths with the same antithetic

paths and time steps N t = 100 For FD method, we take N w = 10, 000 and N t = 1000

Table (2.2.1) shows the computational results

Let V (w, t) denote the fair price of GMWB at time t Remember that the policyholder is entitled to receive the remaining investment account W T and periodic income flow Thematurity value of the periodic income flow is

The no-arbitrage price of GMWB satisfies

r

(

1− e rT)

= w0 Milevsky and Salisbury evaluates V1(w, t) as a Quanto Asian Put(QAP).

We argue that this decomposition is not necessary from the computational point of view

By Feynman-Kac theorem, V1(w, t) solves

The computational results of fair insurance fee α are shown in Table (2.2.2).

Static Discrete Withdrawal Model

Suppose the withdrawal is only allowed at time t i , i = 1, · · · , N and the corresponding

withdrawal amount is G(t i ), i = 1, · · · , N Assume the last withdrawal date coincides

with the maturity of GMWB, i.e t I = T The present value of the periodic income flow

becomes

I

∑

i=1 G(t i )e −rt i

At time t ̸= t i , i = 1, · · · , N, the investment account in the risk-neutral world follows

dW t = (r − α)W t dt + σW t dB t if W t > 0

Table 2.2.1: GMWB Probability of Ruin within 14.28 years (40 b.p insurance fee)

σ r Monte Carlo Method(S.D.) Finite Difference Method

Table 2.2.2: The impact of the GMWB rate and the volatility of the investment account

on the fair insurance fee α where r = 5%

Guarantee rate, g(%) Maturity (years),T = 1/g σ = 0.2 σ = 0.3

At time t = t i , i = 1, · · · , N, the investment account W t jumps to (W t − G)+ Let V1(w, t)

be the fair value at time t of the remaining value of investment account at time T Similar

to the continuous withdrawal case, the fair insurance fee α solves

V1(w0, 0) +

I

∑

i=1 G(t i )e −rt i = w0

where V1(w, t) satisfies the following PDE

of the GMWB is more complicated than the standard American option problems The

reason is that American options do not exit when it is exercised at some time t while the

policyholder of GMWB has to optimally choose the withdrawal rate or amount at eachwithdrawal date

Mathematically, it is more convenient to construct the pricing model of the annuitypolicy that assumes continuous withdrawal In actual practice, withdrawal of discrete

amount occurs at discrete time instants during the life of the policy In this subsection,

we start with the construction of the dynamic continuous model by assuming continuouswithdrawal The more realistic scenario of discrete withdrawal will be considered after-wards In our singular stochastic control model for pricing the GMWB, the discretionarywithdrawal rate is the control variable Some of the techniques used in the derivation ofour pricing model are similar to those used in the singular stochastic control model pro-posed by Davis and Norman (1990) in the analysis of portfolio selection with transactioncosts

Dynamic Continuous Withdrawal Model

The most important feature of the GMWB is the guarantee on the return of premium viawithdrawal, where the accumulated sum of all withdrawals throughout the policy’s life is

the premium w0 paid up front (not accounting for the time value of the cash flows)

We let A t denote the account balance of the guarantee, where A t is right-continuous

with left limit, non-negative and non-increasing {F t } t ≥0-adaptive process At initiation,

A0 equals w0; and the withdrawal guarantee becomes insignificant when A t hits 0 As

withdrawal continues, A t decreases over the life of the policy until it hits the zero value

By the maturity date T , A t must become zero To derive the continuous time pricing

model, we first consider a restricted class of withdrawal policies in which A tis constrained

to be absolutely continuous with bounded derivatives, that is

A t = A0−

∫ t

0

Penalty charges are incurred when the withdrawal rate γ exceeds the contractual drawal rate G Supposing a proportional penalty charge k is applied on the portion of γ above G, then the net amount received by the policyholder is G + (1 − k)(γ − G) when

with-γ > G Let g denote the percentage withdrawal rate, say, g = 7% means 7% of premium

is withdrawn per annum We then have G = gw0

Let W tdenote the value of the personal variable annuity account, then its dynamics follows

dW t = (r − α)W t dt + σW t dB t + dA t , for W t > 0. (2.2.4)

Once W t hits the value 0, it stays at this value thereafter Let w0 be the initial account

value of the policy, which is the same as the premium paid up front When the personal

account value stays positive at maturity T , the remaining balance is paid back to the policyholder at T

Let f (γ) denote the rate of cash flow received by the policyholder as resulted from the

continuous withdrawal process, we then have

The policyholder receives the continuous withdrawal cash flow f (γ u) over the life of the

policy and the remaining balance of the personal account at maturity Based on theassumption of rational behavior of the policyholder that he chooses the optimal withdrawalpolicy dynamically so as to maximize the present value of cash flows generated from holding

the variable annuity policy and under the restricted class of withdrawal policies as specified

by Eq (2.2.3), the no-arbitrage value V of the variable annuity with GMWB is given by

where T is the maturity date of the policy and expectation E t is taken under the risk

neutral measure conditional on W t = W and A t = A Here, γ is the control variable that

is chosen to maximize the expected value of discounted cash flows Using the standardprocedure of deriving the Hamilton-Jacobi-Bellman (HJB) equation in stochastic control

problems (Yong and Zhou, 1999), the governing equation for V is found to be

The function h(γ) is piecewise linear, so its maximum value is achieved at either γ =

0, γ = G or γ = λ Recall that we place a sufficiently large upper bound λ for γ, namely,

0≤ γ ≤ λ It is easily seen that

For the general case where A tis allowed to be discontinuous (instantaneous withdrawal of

finite amount), the no-arbitrage value V of the variable annuity with GMWB is given by

To obtain V (W, A, t) from V (W, A, t), we allow the upper bound λ on γ to be infinite It

is well known that Eq (2.2.9) is a penalty approximation to Eq (2.2.10) (Friedman, 1982)

Taking the limit λ → ∞ in Eq (2.2.9), we obtain the following linear complementarity

formulation of the value function V (W, A, t):

One can follow a similar argument presented in Zhu (1992) to show that the value

function V (W, A, t) defined in Eq (2.2.10) is indeed the generalized solution to the HJB

equation (3.3.5) subject to the auxiliary conditions presented below To complete theformulation of the pricing model, it is necessary to prescribe the terminal condition at

time T and boundary conditions along the boundaries: W = 0, W → ∞ and A = 0.

Note that it is not necessary to prescribe the boundary condition at A = w0 due to the

hyperbolic nature of the variable A in the governing equation (3.3.5).

• At maturity, the policyholder takes the maximum between the remaining guarantee

withdrawal net of penalty charge and the remaining balance of the personal account

• When either A = 0 or W → ∞, the withdrawal guarantee becomes insignificant.

The value of the annuity becomes W e −α(T −t) The discount factor e −α(T −t) arises

due to discounting at the rate α as a proportional fee at the rate α is paid during

the remaining life of the annuity

• When W = 0, the equity participation of the policy vanishes The pricing

formu-lation reduces to a simplier optimal control model with no dependence on W Let

V0(A, t) be the value function of the annuity when W = 0, which is the solution to

the following linear complementarity formulation [considered as a reduced version of

Eq (3.3.5) with no dependence on W ]:

In summary, the auxiliary conditions of the linear complementarity formulation (3.3.5) aregiven by

As a remark, Milevsky and Salisbury (2006) have derived a similar dynamic control modelthat allows for dynamic withdrawal rate adopted by the policyholder However, theirformulation is not quite complete since it does not contain time dependency in the valuefunction Also, there is no full prescription of the auxiliary conditions associated withtheir pricing formulation.

Construction of finite difference scheme

The numerical solution of the singular stochastic control formulation in Eqs (2.2.10) and(2.2.13) poses a difficult computational problem Instead of solving the singular stochasticcontrol model directly, we solve for the penalty approximation model (2.2.9) in which the

allowable control is bounded In our numerical procedure to solve for V (W, A, t), we apply

the standard finite difference approach to discretize the penalty approximation tion (2.2.9) Since the governing equation (2.2.9) is a degenerate diffusion equation with

formula-only the first order derivative of A appearing, upwind discretization must be used to deal

with the first order derivative terms in the differential equation This technique serves to

avoid excessive numerical oscillations in the calculations when the penalty parameter λ

assumes a large value

To avoid truncating the domain of W , we we can take the following transformation

The transformed final and boundary conditions become

¯

v(ξ, 0, t) = ξe −α(T −t)

We will now discretize equation (2.2.15) We first divide the spatial domain [0, 1] × [0, A0]

into small subdomains using lines ξ i = i∆ξ, A j = j∆A where ∆ξ = 1/M, ∆A = A0/N

and M, N are positive integers Let ¯ v i,j t denote ¯v(ξ i , A j , t) Consider the following

dis-cretization scheme at nodal (ξ i , A j , t)

t i+1,j − 2¯v t

i,j+ ¯v t i −1,j

t+1 i+1,j − 2¯v t+1

t i+1,j − ¯v t

P m − LONG

]+

= 0

where LON G stands for

(1− ξ i)2

P m

[(1− θ)¯

t i+1,j − ¯v t

i,j + θ¯ v i,j t+1

]+

[(1− θ)¯

t i,j − ¯v t i,j −1

]

∆A

[(1− θ)¯v t

i,j −1 + θ¯ v i,j t+1 −1

]

In matrix notation, we have

j − f1

]+

= 0(2.2.16)

where A, B, C are matrices of size (M −1)×(M −1), V t

j , V j t+1 , f0, f1, β are column vectors

0,j + θ¯ v t+1 0,j

]

0

.0

(a M −1 + b M −1)

[(1− θ)¯v t

0,j + θ¯ v t+1 0,j

]

0

.0

γ M −1

[(1− θ)¯v t

We quit Newton’s iteration if

so-Eq (2.2.9) is expected to have a linear growth at infinity, the strong comparison principle

holds in the sense of viscosity solution [Crandal et al (1992); Barles et al (1995)] As

a consequence, by virtue of the result established by Barles and Souganidis (1991), one

can establish the convergence of the fully implicit scheme (corresponding to θ = 1) to the viscosity solution of Eq (2.2.9) when the penalty parameter λ is taken to be sufficiently

large and the step sizes in the numerical schemes become vanishingly small Due to thelack of monotonicity property, the analytic proof of convergence of the Crank-Nicholsonscheme cannot be established in a similar manner We resort to numerical experiments totest for convergence of the Crank-Nicholson scheme

In Table (2.2.3), we list the numerical results obtained from the Crank-Nicholson schemeusing varying number of time steps and spatial steps The values of the model parameters

used in the calculations are: G = 7, σ = 0.2, α = 0.036, k = 0.1, r = 0.05, T = 14.28, w0 =

100 and λ = 106 Let N t , N W and N A denote the number of time steps and number of

spatial steps in W and A, respectively The apparent convergence of the numerical

solu-tion is revealed in Table (2.2.3) where the “Iterasolu-tions” column means the total iterasolu-tion

is used in non-linear algebraic equations The Newton type iteration is very fast wherenormally 2 or 3 interations are needed for each non-linear equation We expect to have aquadratic rate of convergence of the numerical solution using the Crank-Nicholson scheme

such that the numerical error is reduced by a factor of 1/4 when the number of time steps

and number of spatial steps are doubled Our numerical results show that the actual rate

of convergence is slightly slower than the expected rate This may be attributed to theupwind treatment of the first order derivative terms in the numerical scheme

We also examine the convergence of the numerical solution to the penalty approximation

model (2.2.9) with varying values of λ to the annuity value of the continuous model The

numerical results shown in Table (2.2.4) were obtained using the Crank-Nicholson scheme

with N t = 512, N W = 1024, N A = 1024 We choose two different values of k and all the

other model parameters are taken to be the same as those used to generate the numericalresults in Table (2.2.3) The apparent convergence of the numerical solution to the penaltyapproximation model is revealed when the penalty parameter increases to a sufficientlyhigh value

Dynamic Discrete Withdrawal Model

Consider the real life situation where discrete withdrawal amount is only allowed at time

t i , i = 1, 2, · · · , N Here, t0 denotes the time of initiation and the last withdrawal date

t N is the maturity date T Let the discrete withdrawal amount at time t i be denoted by

γ i Since the account balance of the withdrawal guarantee A t remains unchanged within

Table 2.2.3: Examination of the rate of convergence of the Crank-Nicholson scheme forsolving the penalty approximation model.

Table 2.2.4: Test of convergence of the numerical approximation solution to the annuity

value with varying values of the penalty parameter λ and penalty charge k.

the interval (t i−1 , t i ), i = 1, 2, · · · , N, the annuity value function V (W, A, t) satisfies the

following differential equation which has no dependence on A:

∂V

∂t +LV = 0, t ∈ (t i −1 , t i ), i = 1, 2, · · · , N. (2.2.17)

The updating of A t only occurs at the withdrawal dates Upon withdrawing an amount γ i

at t i , the annuity account drops from W t to max(W t − γ i , 0), while the guarantee balance

drops from A t to A t − γ i The jump condition of V (W, A, t) across t i is given by

that for f (γ) in Eq (2.2.5) The auxiliary conditions for V (W, A, t) remain the same

as those stated in Eq (2.2.13), except that the boundary value function V0(A, t) under

discrete withdrawal is governed by

∂V0

∂t − rV = 0, t ̸= t i , i = 1, 2, · · · , N,

V0(A, t −) = max

0≤γ i ≤A {V0(A − γ i , t+) + bf (γ i)}, t = t i , i = 1, 2, · · · , N,

The above formulation resembles that of the pricing models of discretely monitored path

dependent options Here, A serves the role as the path dependent variable, which is updated whenever the calendar time sweeps across a fixing date To solve for V (W, A, t)

under the discrete withdrawal model, we apply standard finite difference technique to

discretize Eq (2.2.17) The guarantee balance A is updated on those time steps that

correspond to fixing dates In our numerical calculations, we assume a finite set of discrete

values that can be taken by γ i at fixing date t i According to Eq (2.2.18), we choose γ i

such that V (max(W − γ i , 0), A − γ i , t i) is maximized This is plausible since we know the

values of V at all discrete points of (W, A) in the computational domain.

Reset provision on the guarantee level

The GMWB annuity may contain the reset provision on the guarantee level that serves

as a disincentive to excessive withdrawals beyond G After the guarantee balance A t and

account W t are debited by the withdrawal amount γ i at time t i, the guarantee balance is

reset to min(A t , W t)− γ i if γ i > G While it is not straightforward to incorporate this

reset provision into the continuous withdrawal model, it is relatively easy to modify thejump condition (2.2.18) to include the provision in the discrete withdrawal model Withthe reset provision, the new jump condition becomes

The auxiliary conditions remain the same as those of the non-reset case, except that the

jump condition used in the calculation of V0(A, t) has to be modified as follows:

At first we want to check the convergence of our numerical solution The convergence test

is shown in the table (2.2.5) The order of convergence is roughly around 2

Table 2.2.5: Examination of the rate of convergence of the Crank-Nicholson scheme forsolving the penalty approximation model with quarterly withdrawal frequency

Ta-in our calculations are the same as those used Ta-in Tables 2.2.3 and 2.2.4 Consistent withobvious financial intuition, the tabulated results reveal that the annuity value increaseswith higher frequency of withdrawal per year Also, the annuity value obtained from thecontinuous withdrawal model using the penalty approximation is seen to be very close

to that obtained from the discrete withdrawal model with monthly withdrawal

(compar-ing 93.4194 with 93.346 and 101.045 with 100.965) The apparent agreement of annuity

values serves to verify the consistency between the continuous and discrete models Thedifferences in annuity values with and without the reset provision are seen to be small (seeTable (2.2.6)).

Table 2.2.6: The dependence of the fair value of the GMWB annuity on the withdrawalfrequency per year The annuity value obtained using the continuous withdrawal model(frequency becomes ∞) is close to that corresponding to monthly withdrawal (frequency

equal 12) The differences in annuity values with and without the reset provision are seen

2.3 Pricing behaviors and optimal withdrawal policies

Insurance companies charge proportional insurance fee (denoted by α in our pricing

mod-el) to compensate for the provision of the GMWB rider There have been concerns in theinsurance industry that the fee rate has been charged too low due to sales competition.Milevsky and Salisbury (2006) warn that current pricing of products sold in the market

is not sustainable They claim that the GMWB fees will eventually have to increase orproduct design will have to change