presentation - longevity risk & capital markets solutions conference
TRANSCRIPT
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
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• 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
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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
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• 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)
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• 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.
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• benchmarked price
Ut = Ut
V ∗
t
U nonnegative =⇒
Ut ≥ Et
(
Us
)
t ≤ s
supermartingales (no upward trend)
Pl. (2002)
no strong arbitrage
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• 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 ],
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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.
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• 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
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• 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
)
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• 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 )
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=⇒
UH(0) = E
(
V ∗0
V ∗T
HT
)
= E
(
ΛT
B0
BT
HT
)
≤E(
ΛTB0
BT
HT
)
E(ΛT )
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Only in the special case whenΛT martingale
=⇒ risk neutral pricing formula:
UH(0) = EQ
(
B0
BT
HT
)
Ignores any real trend !
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• 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)
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• 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
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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.
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=⇒ downward trend reflects equity premium
=⇒ Λ strict supermartingale (downward trend), then
P (t, T ) < exp
−T∫
t
rs ds
=Bt
BT
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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.
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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.
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-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
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• discounted NP drift
αt := V ∗
t θ2t
=⇒ volatility
θt =
√
αt
V ∗t
reflects leverage effect
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• minimal market model
Pl. (2001, 2002)
MMM
discounted NP drift
assume
αt = α0 exp{ηt}
α0 > 0
net growth rateη > 0
=⇒ MMM
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• normalized NP
Yt =V ∗
t
αt
dYt = (1 − ηYt) dt +√
Yt dWt
square root process of dimension four
Only one parameter !
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• volatility of NP
θt =1
√Yt
=
√
αt
V ∗t
leverage effect
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• 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.
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• 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})
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with
d1 =4ηK exp{−r(T − t)}
Btαt(exp{η(T − t)} − 1)
and
l2 =4ηV ∗
t
Btαt(exp{η(T − t)} − 1)
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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.
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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
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• 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 )
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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
]
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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
)
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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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