presentation - longevity risk & capital markets solutions conference

Real World Pricing of Long Term Contracts

Eckhard PlatenSchool of Finance and Economics and Department of Mathematical Sciences

University of Technology, Sydney

Pl. & Heath (2006).A Benchmark Approach to Quantitative FinanceSpringer Finance, 700 pp., 199 illus., Hardcover, ISBN-10 3-540-26212-1.

Marquardt, Jaschke & Pl. (2008). Valuing Guaranteed Minimum Death Benefit Optionsin Variable Annuities under a Benchmark Approach. QFRC Report 221, University of Technology, Sydney

Pl. (2009). Real World Pricing of Long Term Contracts. QFRC Report 262,University of Technology, Sydney

Introduction

• variable annuitiesfund-linked

tax-deferred

guarantees

• guaranteed minimum death benefits(GMDBs)

roll-ups:

original investment accrued at a pre-defined interest rate

c© Copyright E. Platen 10 Long Term Contracts 1

• GMDB put, floating put and/or look-back put option

long maturities of contracts

standard option pricing theory ?

no obvious best choice for price:

IFRS Phase II

CFO-Forum (2008)

CRO-Forum (2008)

Solvency II

c© Copyright E. Platen 10 Long Term Contracts 2

• pricing and hedging of GMDBs

benchmark approachin

Pl. (2002, 2004), Pl.& Heath (2006)

Buhlmann& Pl. (2003)

best performing portfolio asbenchmark

doesnot require

existence of an equivalent risk neutral probability measure

exploits real trends

may provide significantly lower prices

c© Copyright E. Platen 10 Long Term Contracts 3

Financial Model

• underlying risky security (unit)

dSt = (µt − γ)Stdt + σtSt dWt

γ ≥ 0 management fee rates

Wt - Brownian motion

• savings account

dBt = rtBt dt

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• market price of risk

θt =µt − γ − rt

σt

• risky asset

dSt

St

= rt dt + σtθt dt + σt dWt

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• instantaneous portfolio return

dVt

Vt

= π0t

dBt

Bt

+ π1t

dSt

St

= rt dt + π1t σt(θt dt + dWt)

• fractions

π0t = δ0

t

Bt

Vt

, π1t = δ1

t

St

Vt

π0t + π1

t = 1

c© Copyright E. Platen 10 Long Term Contracts 6

• numeraire portfolio (NP) as benchmark

Long (1990)

V ∗ is best performing in several ways

• is growth optimal portfolio

Kelly (1956)

V ∗ = max E(log VT )

π1∗

t =µt − γ − rt

σ2t

=θt

σt

dV ∗t

V ∗t

= rt dt + θt(θt dt + dWt)

Merton (1992)

c© Copyright E. Platen 10 Long Term Contracts 7

• NP best performing portfolio

lim supT →∞

1

Tlog

(

V ∗T

V ∗0

)

≥ lim supT →∞

1

Tlog

(

VT

V0

)

global well diversified index≈ NP

Pl. (2005)

c© Copyright E. Platen 10 Long Term Contracts 8

0

10

20

30

40

50

60

70

80

90

100

1930 1940 1950 1960 1970 1980 1990 2000

time

Figure 1: Discounted S&P500 total return index.

c© Copyright E. Platen 10 Long Term Contracts 9

-2

-1

0

1

2

3

4

5

1930 1940 1950 1960 1970 1980 1990 2000

time

Figure 2: Logarithm of discounted S&P500.

c© Copyright E. Platen 10 Long Term Contracts 10

• benchmarked price

Ut = Ut

V ∗

t

U nonnegative =⇒

Ut ≥ Et

(

Us

)

t ≤ s

supermartingales (no upward trend)

Pl. (2002)

no strong arbitrage

c© Copyright E. Platen 10 Long Term Contracts 11

• benchmarked securities that form martingales are calledfair.

Ut = Et

(

Us

)

t ≤ s

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• Law of the Minimal Price

Pl. (2008)

Fair prices are minimal prices

V nonnegative fair portfolio

V ′ nonnegative portfolio

VT = V ′

T

supermartingale property

=⇒Vt ≤ V ′

t

t ∈ [0, T ],

c© Copyright E. Platen 10 Long Term Contracts 13

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1930 1940 1950 1960 1970 1980 1990 2000

time

benchmarked savings bondbenchmarked fair zero coupon bond

Figure 3: Benchmarked savings bond and benchmarked fair zero coupon bond.

c© Copyright E. Platen 10 Long Term Contracts 14

• real world pricing formulafor

Et

(

HT

V ∗T

)

< ∞

UH(t) = V ∗

t Et

(

HT

V ∗T

)

t ∈ [0, T ]

no risk neutral probability needs to exist

c© Copyright E. Platen 10 Long Term Contracts 15

• actuarial pricing formula

whenHT is independent ofV ∗T

=⇒UH(t) = P (t, T ) Et(HT )

zero coupon bond

P (t, T ) = V ∗

t Et

(

1

V ∗T

)

c© Copyright E. Platen 10 Long Term Contracts 16

• standard risk neutral pricing

Ross (1976), Harrison & Pliska (1983)

complete market

candidate Radon-Nikodym derivative

Λt =dQ

dP

∣

∣

∣

∣

At

=BtV

∗0

B0V ∗t

supermartingale =⇒

1 = Λ0 ≥ E0 (ΛT )

c© Copyright E. Platen 10 Long Term Contracts 17

=⇒

UH(0) = E

(

V ∗0

V ∗T

HT

)

= E

(

ΛT

B0

BT

HT

)

≤E(

ΛTB0

BT

HT

)

E(ΛT )

c© Copyright E. Platen 10 Long Term Contracts 18

Only in the special case whenΛT martingale

=⇒ risk neutral pricing formula:

UH(0) = EQ

(

B0

BT

HT

)

Ignores any real trend !

c© Copyright E. Platen 10 Long Term Contracts 19

• Trends ignored also when using:

stochastic discount factor, Cochrane (2001);

deflator, Duffie (2001);

pricing kernel , Constantinides (1992);

state price density, Ingersoll (1987);

NP as Long (1990)

c© Copyright E. Platen 10 Long Term Contracts 20

• Example zero coupon bond

benchmarked fair zero coupon

P (t, T ) =P (t, T )

V ∗t

= Et

(

1

V ∗T

)

martingale (no trend)

for rt - deterministic

P (t, T ) = V ∗

t Et

(

1

V ∗T

)

= exp

−T∫

t

rs ds

Et

(

V ∗t

V ∗T

)

,

V ∗t =

V ∗

t

Bt

- discounted NP

c© Copyright E. Platen 10 Long Term Contracts 21

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1930 1940 1950 1960 1970 1980 1990 2000

time

benchmarked savings bondbenchmarked fair zero coupon bond

Figure 4: Benchmarked savings bond and benchmarked fair zero coupon bond.

c© Copyright E. Platen 10 Long Term Contracts 22

=⇒ downward trend reflects equity premium

=⇒ Λ strict supermartingale (downward trend), then

P (t, T ) < exp

−T∫

t

rs ds

=Bt

BT

c© Copyright E. Platen 10 Long Term Contracts 23

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1930 1940 1950 1960 1970 1980 1990 2000

time

Radon-Nikodym derivativeTotal RN-Mass

Figure 5: Radon-Nikodym derivative and total mass of putative risk neutral measure.

c© Copyright E. Platen 10 Long Term Contracts 24

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1930 1940 1950 1960 1970 1980 1990 2000

time

savings bondfair zero coupon bond

savings account

Figure 6: Savings bond, fair zero coupon bond and savings account.

c© Copyright E. Platen 10 Long Term Contracts 25

-8

-7

-6

-5

-4

-3

-2

-1

0

1

1930 1940 1950 1960 1970 1980 1990 2000

time

ln(savings bond)ln(fair zero coupon bond)

Figure 7: Logarithms of savings bond and fair zero coupon bond.

c© Copyright E. Platen 10 Long Term Contracts 26

• discounted NP

V ∗

t =V ∗

t

Bt

dV ∗

t = V ∗

t θt (θt dt + dWt)

= αt dt +

√

αt V ∗t dWt

c© Copyright E. Platen 10 Long Term Contracts 27

• discounted NP drift

αt := V ∗

t θ2t

=⇒ volatility

θt =

√

αt

V ∗t

reflects leverage effect

c© Copyright E. Platen 10 Long Term Contracts 28

• minimal market model

Pl. (2001, 2002)

MMM

discounted NP drift

assume

αt = α0 exp{ηt}

α0 > 0

net growth rateη > 0

=⇒ MMM

c© Copyright E. Platen 10 Long Term Contracts 29

• normalized NP

Yt =V ∗

t

αt

dYt = (1 − ηYt) dt +√

Yt dWt

square root process of dimension four

Only one parameter !

c© Copyright E. Platen 10 Long Term Contracts 30

• volatility of NP

θt =1

√Yt

=

√

αt

V ∗t

leverage effect

c© Copyright E. Platen 10 Long Term Contracts 31

• Zero coupon bond under the MMM

rt - deterministic

P (t, T ) = exp

−T∫

t

rs ds

(

1 − exp

{

− V ∗t

2(ϕ(T ) − ϕ(t))

})

Explicit formula !

c© Copyright E. Platen 10 Long Term Contracts 32

0

0.2

0.4

0.6

0.8

1

1930 1940 1950 1960 1970 1980 1990 2000

time

Figure 8: Fraction invested in the savings account.

c© Copyright E. Platen 10 Long Term Contracts 33

-2e-005

-1e-005

0

1e-005

2e-005

3e-005

4e-005

5e-005

6e-005

7e-005

1930 1940 1950 1960 1970 1980 1990 2000

time

Figure 9: Benchmarked profit and loss.

c© Copyright E. Platen 10 Long Term Contracts 34

• European put option under the MMM

Hulley, Miller & Pl. (2005)

p(t, V ∗

t , T, K, r) = V ∗

t Et

(

(K − V ∗τ )+

V ∗τ

)

p(t, V ∗

t , T, K, r) = −V ∗

t χ2(d1; 4, l2)

+ Ke−r(T −t)(

χ2(d1; 0, l2) − exp {−l2/2})

c© Copyright E. Platen 10 Long Term Contracts 35

with

d1 =4ηK exp{−r(T − t)}

Btαt(exp{η(T − t)} − 1)

and

l2 =4ηV ∗

t

Btαt(exp{η(T − t)} − 1)

c© Copyright E. Platen 10 Long Term Contracts 36

χ2(x; n, l) non-central chi-square distribution function

n ≥ 0 degrees of freedom

non-centrality parameterl > 0

χ2(x; n, l) =

∞∑

k=0

exp{

− l2

} (

l2

)k

k!

1 −Γ(

x2; n+2k

2

)

Γ(

n+2k

2

)

c© Copyright E. Platen 10 Long Term Contracts 37

• putative risk neutral price

p(t, V ∗

t , T, K, r) = p(t, V ∗

t , T, K, r)+Ke−r(T −t) exp

{

− V ∗t

2(ϕ(T ) − ϕ(t))

}

risk neutral is overpricing

for V ∗t → 0 =⇒ p(t, V ∗

t , T, K, r) → 0

and

p(t, V ∗

t , T, K, r) → K e−r(T −t) > 0

risk neutral ignores trends

c© Copyright E. Platen 10 Long Term Contracts 38

0

2

4

6

8

1930 1940 1950 1960 1970 1980 1990 2000

time

ln(K*savings bond)ln(risk neutral put)

ln(put)

Figure 10: Logarithms of savings bond timesK, risk neutral put and fair put.

c© Copyright E. Platen 10 Long Term Contracts 39

Guaranteed Minimum Death Benefit (GMDB)

• payout to the policyholder

max(egτ V0, Vτ )

time of deathτ

g ≥ 0 is the guaranteed instantaneous growth rate

V0 is the initial account value

Vτ is the unit value of the policyholder’s account at time of death τ

c© Copyright E. Platen 10 Long Term Contracts 40

• insurance chargesξ ≥ 0

=⇒ policyholder’s unit value

Vt = e−ξtV ∗

t

=⇒

max(egτ V0, Vτ ) = max(egτ V0, e−ξτ V ∗

τ )

= e−ξτ max(e(g+ξ)τ V0, V ∗

τ )

c© Copyright E. Platen 10 Long Term Contracts 41

insurance company invests the entire fund valueV in the NP V ∗

=⇒ payoff

HT = GMDBT = e−ξT[

(e(g+ξ)T V ∗

0 − V ∗

T )+ + V ∗

T

]

c© Copyright E. Platen 10 Long Term Contracts 42

fair valueGMDB0 of the total claim

real world pricing formula

GMDB0 = V ∗

0 E

(

GMDBT

V ∗T

)

= e−ξT(

p(0, V ∗

0 , T, e(g+ξ)T V ∗

0 , r) + V ∗

0

)

c© Copyright E. Platen 10 Long Term Contracts 43

0 5 10 15 20 25 300.8

0.85

0.9

0.95

1

1.05

1.1

MMM

T

GM

DB

0

0 5 10 15 20 25 300.8

0.85

0.9

0.95

1

1.05

1.1

risk neutral

T

GM

DB

0

0 5 10 15 20 25 300.8

0.85

0.9

0.95

1

1.05

1.1

1.15Black Scholes

T

GM

DB

0

0 5 10 15 20 25 300.9

0.95

1

1.05

1.1

1.15MMM

T

GM

DB

0

0 5 10 15 20 25 300.9

0.95

1

1.05

1.1

1.15risk neutral

T

GM

DB

0

0 5 10 15 20 25 300.9

0.95

1

1.05

1.1

1.15Black Scholes

T

GM

DB

0g=0.025

g=0

g=0.025g=0.025

g=0g=0

Figure 11: Present value of the GMDB under the real world pricing formula (left), the risk

neutral pricing formula (middle) and the Black Scholes formula (right) forη = 0.05, α0 =

0.05, r = 0.05, ξ = 0.01 andY0 = 20.

c© Copyright E. Platen 10 Long Term Contracts 44

• lifetime τ is stochastic

GMDB0 = V ∗

0 E

(

E

(

GMDBτ

V ∗τ

∣

∣

∣

∣

Fτ

))

GMDB0 =

∫ T

0

(

p(0, V ∗

0 , t, e(g+ξ)tV ∗

0 , r) + V ∗

0

)

e−ξtfτ (t) dt

fτ (·) - mortality density

c© Copyright E. Platen 10 Long Term Contracts 45

0 20 40 60 80 100 12010

−5

10−4

10−3

10−2

10−1

100

Age x

log(

q x)

malefemale

Figure 12: German mortality data

c© Copyright E. Platen 10 Long Term Contracts 46

rational investors will lapse when embedded put-option outof the moneyGMDBs have stochastic maturityTitanic options, Milevsky & Posner (2001)

• roll-up GMDB payoff

Ht =

(1 − βt)V∗

t , if lapsed at timet,

max(egτ V0, V ∗τ ), if death occurs at timet = τ

surrender chargeβt

βt =

(8 − ⌈t⌉)%, t ≤ 7,

0, t > 7

no credit riskno accumulation phase

c© Copyright E. Platen 10 Long Term Contracts 47

20 30 40 50 60 70 801

1.05

1.1

1.15

Age x

GM

DB

0

malefemale

Figure 13: Value of the GMDB under the MMM for male and female policyholders agedx,

assuming an irrational lapsation ofl = 1%.

c© Copyright E. Platen 10 Long Term Contracts 48

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