optimal surrender policy for variable annuity guarantees

Introduction Optimal surrender boundary: GMAB case Optimal exercise boundary: Path-dependent case Conclusion References

Optimal Surrender Policy for VariableAnnuity Guarantees

Anne MacKay

University of Waterloo

January 31, 2013

Joint work withDr. Carole Bernard, University of Waterloo

Max Muehlbeyer, Ulm University

Research funded by the Hickman Scholarship of the Society of Actuariesand NSERC

Anne MacKay – University of Waterloo Optimal Surrender for VA 1/22


Outline

1 Introduction

2 Optimal surrender boundary: GMAB case

3 Optimal exercise boundary: Path-dependent case

4 Conclusion



Variable annuities and financial options

� Variable annuities: similar to mutual funds with additionalguarantees

� VA guarantees paid for via a fixed fee rate throughout the lifeof the contract (not paid upfront)

� Decreases return on fund� Impacts value of option

� VA contract can be surrendered (Knoller, Kraut, andSchoenmaekers (2011))

� Financial needs� Higher costs of opportunity� Moneyness of the option



Continuous fee and the surrender option

� Simple payoff at maturity:

max(G ,FT ) = FT + (G − FT )+

� Option paid by continuous fee set as percentage of fund:� Fee is low when option value is high� Incentive to surrender when fund value is high

� Surrender region: surrender benefit higher than payoffexpected if contract is kept



Setting

Index value St :

dSt

St= r dt + σdWt

Account value Ft based on index:

dFt

Ft= (r − c)dt + σdWt

⇒ Ft |Fs ∼ LN (log(Ft) + (r − c − σ2

2)(t − s), σ2(t − s))



Contract

� Accumulation benefit:

max(FT ,G ), G = F0egT , g < r

� Surrender benefit:e−κ(T−t)Ft



Integral representation for surrender option

� Similarities between surrender option and American option

� Can use techniques developed for American options (Kim(1990), Kim and Yu (1996), Carr, Jarrow, and Myneni (1992),Wu and Fu (2003))

� Integral can be obtained in different ways:� Finite number of surrender times (as in Kim (1990))� No-arbitrage arguments (as in Kim and Yu (1996))



Trading strategy for surrender option

� Confirm that optimal strategy is a threshold strategy

� Hold the VA below the surrender boundary Bt

� When Ft crosses Bt from below, sell the VA and invest theproceeds

� When Ft crosses Bt from above, use the investment to buythe VA

� Payoff of portfolio is payoff of VA



Gain from surrender

Proposition

The benefit associated with the surrender option between[t, t + dt] for an infinitesimal time step dt is given by

e−κ(T−t)(c − κ)Ftdt



Proof of Proposition

� Suppose surrender at t.

� Policyholder receives

e−κ(T−t)Ft = e−κ(T−t)e−ctSt = e−κT e−(c−κ)tSt

� To buy the VA at time t + dt, policyholder only needs

e−κ(T−(t+dt))Ft+dt = e−κT e−(c−κ)(t+dt)St+dt

� Investment made at t becomes

e−κT e−(c−κ)(t+dt)St+dt + e−κT e−(c−κ)tSterdt(1− e−(c−κ)dt)

= e−κ(T−(t+dt))Ft+dt + e−κ(T−t)(c − κ)Ftdt + o(dt)



Price of VA with surrender

Price of VA contract

V (t,Ft) = E [e−rt max(G ,FT )|Ft ]︸︷︷︸Maturity benefit

+

e−κT (c − κ)Ftect

∫ T

te−(c−κ)uΦ

(d1(Ft ,Bu, u, t)

)du︸︷︷︸

Surrender option

� Maturity benefit: Similar to vanilla option under Black-Scholes

� Surrender option:∫ T

te−r(u−t)

∫ ∞Bu

e−κ(T−u)(c − κ)xfFu(x |Ft)dxdu.



Optimal exercise boundary condition

� At maturity BT = GT and along the surrender boundary,V (Ft , t) = e−κ(T−t)Ft = Bt .

� Work backwards to solve for Bt

Bt =v(Ft , t) + e(Ft , t)

=e−c(T−t)Bteκ(T−t)Φ(d1(Bte

κ(T−t),GT ,T , t))

+ e−r(T−t)GTΦ(d2(Bteκ(T−t),GT ,T , t))

+ (c − κ)Bte(c−κ)t

∫ T

t

e−(c−κ)uΦ(d1(Bteκ(T−t),Bu, u, t))du



Numerical example

Assumptions:

� T = 15

� G = F0 = 100

� κ = 0, unless otherwise indicated

� c = 0.91% (fair fee for maturity benefit)

� r = 0.03

� σ = 0.2, unless otherwise indicated


0 5 10 15

100

120

140

160

180

200

Time t (Years)

Und

erly

ing

fund

pric

e F

(t)

Optimal exercise boundary, sensitivity analysis: sigma

sigma=0.15sigma=0.20sigma=0.25sigma=0.30

0 5 10 15

100

120

140

160

180

200

Time t (Years)

Und

erly

ing

fund

pric

e F

(t)

Optimal exercise boundary, sensitivity analysis: kappa

k=0k=c/4k=c/3k=c/2


Geometric average

� Consider the payoff max(GT ,YT ), where YT is the geometricaverage defined as

Yt = exp

1

t

t∫0

lnFsds

� The conditional distribution of Yu|(Yt ,Ft) for u > t is again

log-normal with mean and variance given by

Mgt =

t

ulnYt +

u − t

ulnFt +

r − c − σ2

2

2u(u − t)2

V gt =

σ2

3u2(u − t)3



Pricing formula

Theorem

Let V g (Yt ,Ft , t) denote the price at time t of the VA with guarantee GT

and a surrender benefit equal to e−κ(T−t)Yt . Then V g (Yt ,Ft , t) can bedecomposed into a European part vg (Yt ,Ft , t) and an early exercisepremium eg (Yt ,Ft , t)

V g (Yt ,Ft , t) = vg (Yt ,Ft , t) + eg (Yt ,Ft , t),

where

vg (Yt ,Ft , t) = e−r(T−t)eMgt +

Vgt2 Φ(

− ln(GT )+Mgt +V g

t√V gt

)+ e−r(T−t)GTΦ

(ln(GT )−Mg

t√V gt

),

eg (Yt ,Ft , t) = e−κT ertT∫t

eu(κ−r)eV̂u,t

2 Ytut F

u−t2u

t E [k(u,Fu, t)] du



Particularities of the path-dependent case

� Optimal surrender behaviour depends on account value Ft andgeometric average Yt .⇒ Optimal surrender surface

� To solve for optimal surrender surface, need to consider manyvalues Ft at each time t.

� To simplify calculations, assume that Bt(Ft) has the form

Bt(Ft) = max(GT e−r(T−t), at + btFt)



Numerical example

Additional assumptions:

� T = 10

� Payoff: max(YT ,GT )

� GT = F0e0.025T

� c = 0.0197


0 1 2 3 4 5 6 7 8 9 10

6080

100

120140

160180

20090

100

110

120

130

140

150

160

timeF

t

Opt

imal

Bou

ndar

y


Conclusion

� Integral representation for the surrender option

� Can retrieve optimal surrender boundary

� Can be used for path independent and path dependent payoffs

Future work:

� Use for other types of fee structures

� Consider flexible premium (as in Chi and Lin (2013))



References

Carr, P., R. Jarrow, and R. Myneni (1992): “Alternative characterizations of American put options,”Mathematical Finance, 2(2), 87–106.

Chi, Y., and S. X. Lin (2013): “Are Flexible Premium Variable Annuities Underpriced?,” working paperavailable at SSRN.

Kim, I. J. (1990): “The Analytic Valuation of American Options,” The Review of Financial Studies, 3(4), 547–572.

Kim, I. J., and G. G. Yu (1996): “An alternative approach to the valuation of American options andapplications,” Review of Derivatives Research, 1(1), 61–85.

Knoller, C., G. Kraut, and P. Schoenmaekers (2011): “On the Propensity to Surrender a VariableAnnuity Contract,” Discussion paper, Working Paper, Ludwig Maximilian Universitaet Munich.

Wu, R., and M. C. Fu (2003): “Optimal exercise policies and simulation-based valuation for American-Asianoptions,” Operations Research, 51(1), 52–66.



Thank you for your attention!


optimal surrender policy for variable annuity guarantees

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