Exotic Options
Through 100 Years to Current State of the Art
Risk Management of Investment Guarantees
Contents
The First 100 Years
Exotic Options
State of the Art
Risk Management
Milliman
100 Years
Milliman
The First 75 Years
z
Bachelier (1900)
z
Levy (1925)
z
Ito (1951)
z
Markowitz (1952)
z
Reddington (1952)
z
Sprenkle (1958)
z
Samuelson (1965)
z
Thorpe (1969)
z
Black & Scholes (73)
Milliman
Black & Scholes
z Black
& Scholes (1973)
z Merton
(1973)
z Harrison
& Kreps (1977)
Milliman
Exotic Options
Milliman
Exotic Options
z
Margrabe (1978)
POT = max(Q1 S1 (T ) − Q2 S 2 (T ),0)
z
Stulz (1982) & Rubinstein (1991)
POT = max(max(S1 , S 2 ) − X ,0)
z
Turnbull & Wakeman (1991),
Curran (1992, 1994), Yor(1993),
Zhang (1994).
Milliman
Deferred Annuities
z
Margrabe (1978)
POT = max(Q1 S1 (T ) − Q2 S 2 (T ),0)
Milliman
Deferred Annuities
z
Margrabe (1978)
POT = max(Q1 S1 (T ) − Q2 S 2 (T ),0)
GCT = max(gbT a x +T ,i (T ) − AST ,0)
Milliman
Compound Options
z
Stulz (1982) & Rubinstein (1991)
POT = max(max(S1 , S 2 ) − X ,0)
Milliman
Compound Options
z
Stulz (1982) & Rubinstein (1991)
POT = max(max(S1 , S 2 ) − X ,0)
z
Useful for valuing benefits of the form
max(S1 , S 2 , X )
= X + max(max(S1 , S 2 ) − X ,0)
z
Generalises
max(S1 , S 2 , S 3 , X )
= max(S1 , max(S 2 , S3 , X ,0),0)
Milliman
Guaranteed Annuity Options
z
Stulz (1982) & Rubinstein (1991)
POT = max(max(S1 , S 2 ) − X ,0)
Benefit = max( AST , GBT , GBT gT ax +T ,i (T ) )
GCT = max( AST , GBT , GBT gT a x +T ,i (T ) ) − AST
= GBT + max(max( AST , GBT gT a x +T ,i (T ) ) − GBT ,0) − AST
ULBenefit = max( AST , AST gT ax +T ,i (T ) )
CommuteGCT = max(AST , gbT ax+T ,i (T ) , gbT hT ) − AST
Milliman
Regular Premium Business
z
Asian Options
Options based on the average rather than the
terminal value of the underlying asset/ index
z
Regular Premium
Policyholder has an option to pay future premium
instalments -- office
⎡I
I
I ⎤
AST = (1 − w) T (1 − k ) P ⎢ T + T + L + T ⎥
I1
I T −1 ⎦
⎣ I0
AST = (1 − w) T (1 − k ) P[ S1 + S 2 + L S T ]
= (1 − w) T (1 − k ) PTS T
GCT = max(GBT − AST ,0)
= max(GBT − (1 − w) T (1 − k )TPS T ,0)
Milliman
State of the Art
Risk Management
Milliman
Economic Balance Sheet
Unhedged
Starting
Position
Total
Liabs
Total
Assets
Equity
Bonds
Asset
Backing
100
100
50
50
Asset
Backing
Value of
Guarantee
20
20
10
10
Balance
5
5
2.5
125
125
62.5
Total
z
Total
Liabs
Total
Assets
Equity
Bonds
90
90
40
50
Value of
Guarantee
25.6
18
8
10
2.5
Balance
-3.1
4.5
2
2.5
62.5
Total
112.5
112.5
50
62.5
Derivatives
Equities
down 20%
Derivatives
Scenario: equities down 20%, bonds unchanged
z20%
equity fall Æ value of guarantee increases to 25.6
ALM capital impact: £7.6. Difference between new value of
guarantees and re-valued assets that support guarantees (= 25.6–18)
z
Capital injection = £3.1 to restore solvency and £7.6 to restore
financial strength to perfectly matched position
z
Milliman
Economic Balance Sheet
OTC Hedge
Starting
Position
Total
Liabs
Total
Assets
Equity
Bonds
Asset
Share
100
100
50
50
Value of
Guarantee
20
20
Balance
5
5
2.5
2.5
125
125
52.5
52.5
Total
Derivatives
Equities
down 20%
Total
Liabs
Total
Assets
Equity
Bonds
90
90
40
50
Value of
Guarantee
25.6
25.6
Balance
4.5
4.5
2
2.5
120.1
120.1
42
52.5
Asset
Share
20
20
Total
Derivatives
25.6
25.6
Buy an OTC put option to match the liability guarantee
z In theory a perfect match: no ALM impact or capital injection; but:
z
–
–
–
–
Pricing & Cash Flow
At mercy of market: pay implied volatility of market / counterparty
Other assumptions needed to make hedge perfect
Periodic rebalancing required, incurs high transactions costs
Milliman
Replication
Put Option = (Short Position in Underlying)
+ Long Position in Risk Free
+
Position in Volatility Sensitive Asset
+ Position in Interest Sensitive Asset
Delta: Æ Protects against small immediate changes in underlying
z Gamma: Æ Protects against small immediate changes in delta
z Vega: Æ Protects against small immediate changes in Vol
z Rho: Æ Protects against small immediate changes in interest
z
Milliman
Dynamic Hedge in the Balance Sheet
Equity
Bonds
Risk Free
Options
Swaps
(24)
(24)
60
5
3
Asset
Backing
Value of
Guarantee
20
Balance
“Manufacture” the OTC derivative
z Protects against risks of:
z
Underlying assets falling
–
–
Volatility spiking
Interest rates changing
zDynamic
Î Technology requirements
Milliman
Risk Inventory
Market
¾Interest Rate
¾Credit
¾
INVESTMENT RISK
DEMOGRAPHIC
RISK
Underwriting
¾ Experience Statistics
¾ Reinsurance
¾
OPERATIONAL
RISK
Procedures and Controls
fraud
¾ Mis-selling
¾
Milliman
Dynamic Hedging: Requirements
Market consistent stochastic valuation of liability guarantees on
a policy-by-policy and daily basis
– Capital markets expertise to identify hedge portfolio
– Systems capability to dynamically monitor position frequently
– Effective control systems
–
z
z
z
z
Performance and risk attribution through analysis of profit
movements
Financial reporting requirements
Financial projection analysis – back testing of strategy to
demonstrate effectiveness of the hedge
The bottom line: a very complex task requiring specialist
actuarial, capital market and systems expertise / investment
Milliman
MG-Hedge System
structure reflects market risk management framework
IT Interface
In force Data
Actuarial
Actuarial
Liability
Liability
Trade
Trade
Positioning
Positioning
Financial
Financial
Reporting
Reporting
Valuation
Valuation
System
System
System
System
¾¾Per
Perpolicy
policy
seriatim
seriatim
¾¾Stochastic
Stochastic
IT
Interface
¾¾Live
¾¾Financial
Livemarket
marketdata
data
Financialcontrol
control
¾¾Asset
¾¾Profit
Asset
Profitmeasurement,
measurement,
management
analysis
management
analysisand
and
interface
interface
projections
projections
Asset &
Market Data
Milliman
Overview of Hedging Program
Daily
for Insurance Guarantees
Trading
Grid
Trading
Grid
Nightly
Valuation
MG-Hedge™
Monitor
Markets
Monitor
Markets
Nightly
Valuation
Trade
Execution
Weekly
Produce
P&L Report
Quarterly/Annually
Performance
Attribution
Monitor Experience
Fund Modeling
Mortality
Lapses
Fund Transfers
Financial Reporting
Hedging Strategy for New Products
Milliman
Goal of Hedging
Replicate Embedded Option So That:
Beginning of Period Guarantee Value
+
Interest
-
Claims
+
Guarantee Premiums
+
Changes due to market movements
End of Period Guarantee Value
Beginning of Period Hedge Asset Value
+
Gains on Hedge Assets
due to market movements
-
Losses on Hedge Assets
due to market movements
End of Period Hedge Asset Value
EOP Guarantee Value
¯
EOP Hedge Asset Value
Net Gain (Loss)
Balance Sheet Volatility
Allowing for Demographic & Path Dependencies
Milliman
Hedge Report
Income Statements & Projections
Income Statement Projection
Projection Year
Total Income
Premium Income
Investment Income
Fixed Income Portfolio
Futures
Options & Swaps
Total Expenses
Increase in Fair Value Liability
Interest on Debt
Pre-Tax Income
Equity Market Return
10 Year Interest Rate
1
$ 12,274.0
2
$ 46,955.1
3
$ 98,128.8
4
$ 124,681.3
$
5
7,796.7
$
668.4
$ 11,605.6
$
244.0
$ 1,864.6
$ 9,497.0
$ 9,826.8
$ 9,758.1
$
68.6
$ 2,447.2
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
2,608.6
5,188.1
13,586.5
333.8
(8,732.2)
11,270.7
11,270.7
(3,474.0)
-25%
5%
1,728.0
45,227.0
1,470.2
6,156.0
37,600.8
39,441.3
39,291.6
149.7
7,513.8
-25%
5%
3,461.3
94,667.5
4,729.2
8,421.2
81,517.1
91,189.1
91,067.3
121.8
6,939.7
-25%
4%
2,901.5
121,779.8
10,155.3
6,260.8
105,363.7
125,981.0
125,977.3
3.7
(1,299.7)
-25%
4%
15%
3%
Milliman
Hedge Reporting
P&L - Unhedged vs. Hedged
Quarterly P&L Volatility Unhedged
5,000
4,000
3,000
2,000
90th Percentile
1,000
75th Percentile
10th Percentile
1
(1,000)
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
25th Percentile
Median
(2,000)
(3,000)
(4,000)
Quarterly P&L Volatility Hedged
(5,000)
5,000
4,000
3,000
2,000
90th Percentile
1,000
75th Percentile
-
10th Percentile
1
(1,000)
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
25th Percentile
Median
(2,000)
(3,000)
(4,000)
(5,000)
Milliman
Embedded Value Reporting
Milliman
Replication
Reddington
Immunisation
Black Scholes
Dynamic Hedging
Immunises
Small immediate changes
Immunises
Small Immediate Changes
Frequent Rebalancing
Frequent Rebalancing
Duration Matching (rate of
Delta (rate of change of value
change wrt underlying interest) with respect to underlying)
Convexity Matching
(second derivative)
Gamma
(second derivative)
ADD: Vega, Rho etc
Milliman
Contacts
For further information, please contact
Gary Finkelstein
Milliman Financial Risk Management
Finsbury Tower
103-105 Bunhill Row
London EC1Y 8LZ
Ph: +44 207 847.6100
Fx: +44 (0)20 7847 6105
Email: gary.finkelstein@milliman.com
Milliman