SOA Life 2006 Spring Meeting – 50TS, What is by ulf16328

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									                                   What is Black-Scholes
                                        And why should I care?

                                      SOA Spring Meeting
                                        Session 50 TS


                        Mary Hardy
                        May 25 2006




                                        Why are we here?
                  • The actuary who asked for a simple explanation
                  • The place of Black-Scholes in the actuarial
                    curriculum
                  • Common misconceptions
                  This is an unapologetic ab initio discussion




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                    Outline
                  • The Story of Derivatives (abridged)
                  • Actuarial connections
                  • The binomial version
                  • The Black Scholes Equation
                  • Black Scholes for Variable Annuities – a simple
                    case study.
                  • Common misconceptions




                                The Story of Derivatives
                  • A derivative is a contract with a value
                    determined by the price of another underlying
                    asset or commodity.
                  • Eg a payment triggered by a fall in the price of
                    corn/airline fuel/Microsoft shares below some
                    specified threshold




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                The Story of Derivatives
                  • It’s insurance against a fall in price?
                          But it’s non-diversifiable or systematic risk to
                          the seller
                          Traditional insurance techniques won’t help
                          You can buy the ‘insurance’ even if you
                          don’t have the underlying.




                                 Examples of derivatives
                  • Forward contract – Agreement to buy at a fixed
                    price at a fixed time
                  • European Call option – Option to buy at a fixed
                    price at a fixed time.
                  • European Put option – Option to sell at a fixed
                    price at a fixed time.




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                 Examples of derivatives
                 • Swaps
                        Agreement to exchange cashflows over a
                        specified period, for example
                             Swap Equity returns for fixed interest
                             Swap fixed interest for commodity delivery
                                  Oil, fuel, electricity




                         Examples in Actuarial Practice
                  • Variable annuity guarantees are put options
                  • Equity-Indexed Annuities involve call options
                  • An annuity is a pre-paid swap
                  • Level premiums for traditional insurance = swap
                  • DC pension plan with DB underpin = exchange
                    option
                  • Expense guarantee = swap




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                        Diversifiable risk
                  • 10,000 term insurance contracts
                  • Independent lives
                  • qx=0.05; each policy has 5% chance of claim
                  • Expected number of claims = 500
                  • Prob > 600 claims <10-5




                                 Non-Diversifiable Risk
                  • 10,000 pure endowment equity-linked contracts
                  • Claim arises if equity index falls below starting value
                  • Probability of claim=0.05
                  • Expected number of claims = 500
                  • 5% chance of all contracts → claims; 95% chance of
                    no claims
                  • So, Prob >600 claims =5%.




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                  Diversifiable and Non-
                                    Diversifiable Risk
                  • Traditional actuarial techniques assume diversifiable
                    risk
                  • Financial engineering addresses non-diversifiable
                    (systemic; systematic) risk
                  • Most risks are a mixture
                  • Other insurance systemic risks: mortality improvement,
                    expense inflation etc.




                            Two key elements of Black-
                                     Scholes
                  • No Arbitrage
                          No free lunch
                          Two assets offering the same cashflows with same
                          risk must cost the same amount (exceptions?)
                  • Replication
                          Two assets offering the same cashflows with same
                          risk must cost the same amount.




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                        Simple Example- Actuarial and
                              Financial pricing
                  • Suppose an actuary, Alice, values a liability of $1000
                    due in two years; she will…
                          Consider what assets the economic capital will be
                          invested in: 60% equities, 40% bonds
                          Estimate the rate of return for those assets: say 8%
                          per year
                          Discount $1000 for 2 years at 8% for value of $857.




                        Simple Example- Actuarial and
                              Financial pricing
                  • Now – the financial engineer – Fred – will…
                          Determine the asset that will exactly pay off $1000
                          in two years…
                              A zero coupon bond with face value $1000 (we
                              can strip a coupon bond if necessary)
                          Value of $1000 liability = current cost of 2-year
                          ZCB –approx $916.
                          BUY THE ZCB




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                 Arbitrage
                  • Fred offers Alice $858 at t=0 for $1000 in 2 yrs.
                  • Fred sells a 2-yr ZCB and takes in $916 i.e. $58 profit at
                    start of the deal
                  • In 2 years, Alice gives Fred $1000 (she’s an actuary, so
                    she doesn’t renege)
                  • Fred then pays the redemption on the ZCB
                  • profit at t=0, breakeven at t=2, no risk
                  • ⇒Free Lunch




                               Arbitrage and Replication
                  • Any price other than the MV of a 2-yr ZCB will
                    generate arbitrage
                  • If there is a 1000 liability in 2 years, we can
                    exactly replicate it with the 2-yr ZCB
                  • Replication mitigates risk




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                     The binomial model
                  • Simplified Market Model
                  • Discrete time step, up-down model for stock
                    price
                  • Nevertheless, can be used to develop more
                    complicated results
                  • Let the time interval → 0




                               Binomial Model Example
                  • The risk free interest rate r=0.05 is level,
                    constant compounded continuously.
                          A 1-yr ZCB costs e-r for $1 face value
                  • The stock price S(t) is a random process
                          S(0)=100; S(t) =1.1S(t-1) or 0.85S(t-1)
                         That is, the market either rises by 10% or
                         falls by 15%
                          No probabilities assumed



SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                               Binomial Model Example
                  • A 1-year put option is issued;
                  • Guarantees a price of 105 for the stock in 1 year
                  • 105 is the Strike Price
                  • So, in 1 year if S(1) is >105,
                          the option holder will not exercise,
                          The option expires with no value




                           Binomial Model Example (2)
                  • If S(1)<105
                          The option holder will sell for 105
                          The cost to the option seller is 105-S(1)
                  • Put option payoff=max(105-S(1),0) at time 1
                  • It isn’t necessary for the stock to change hands




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                 Binomial Stock Process

                                                                    S(1)=110=Su



                   S(0)=100


                                                                     S(1)=85=Sd
                  Time 0
                                                          Time 1




                                   Option seller’s payoff

                                                                    0



                   Price P
                   (Income)
                                                                     100-85=15
                  Time 0                                             (Outgo)
                                                          Time 1




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                               Binomial Model Example
                  • We can construct a portfolio at t=0 to replicate
                    the option payoff at t=1
                  • It has $k face value of 1-year ZCB and b units
                    of stock, worth bS(t) at t.
                  • We assume we can buy or sell in any quantity
                  • No transactions costs




                               Binomial Model Example
                  • Price of ZCB at t=0:                ke-r
                  • Price of stock at t=0:              bS(0)
                  • Price of ZCB at t=1:                k
                  • Price of stock at t=1: bS(1) =                        bSu or
                                                                          bSd
                  • Price of portfolio at t=0: bS(0)+ke-r
                  • Price of portfolio at t=1: bS(1)+k




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                               Binomial Model Example
                  • What portfolio will exactly pay off the option in
                    either state?
                          bSu+k=0 i.e. 110b+k=0
                          bSd+k=15 i.e. 85b+k=15
                          ⇒b= -15/25=-0.6 and k=66
                  • The value of this portfolio at t=0 is
                                –0.6S(0)+ke-r =2.78




                               Binomial Model Example
                  • So, if we short 0.6 units of equity, and buy $66
                    face value of ZCB,
                          it will cost $2.78
                         It will exactly pay off the option in either
                         state
                  • The value of the option must be $2.78.
                          Law of one price
                          Otherwise -- arbitrage




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                         Moral of the 1-period example
                  • Pricing comes from replication
                  • We never needed to construct a probability
                    model for the random S;
                  • For the general put option we have




                                              Moral (cont)
                                     S u e − r − S (0)
                    P0 = ( K − S d )
                                         Su − S d
                                                                           P0 is time zero price
                        = ( K − S d )e − r p *                             K is gteed price
                                                                           (STRIKE)
                                  Su − S (0)e r
                        where p =       *

                                    Su − S d
                        so          0 < p* < 1                   (why?)




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                   Moral (extrapolated)
                     • If Cd , Cu are the payoffs in the down and up
                       states, (here Cu=0, Cd=15)
                     • Then the price at time 0 can be written as
                     • P0=Cd e-r p*+Cu e-r (1-p*)
                     • And this is true for any derivative of S with
                       payoffs in up or down state




                                              More Moral
                 • A short way to write this is
                                    P0 =e-r E*[C]
                 • C is the payoff function (random as a function
                   of S)
                 • E[] because {p*,1-p*} looks like a probability
                   distribution for ‘down’ ‘up’ probabilities




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                              Moral (cont)
                  • Note: this looks like the expected value of the payoff
                    discounted at the risk free rate.
                  • Even though we have not actually used expectation or
                    probability anywhere
                  • p* is NOT the true probability that the stock price falls
                  • And we do not assume that the expected return on
                    stocks is r




                           The Risk Neutral Probability
                                  Distribution
                  • Using the artificial probabilities p* for down,
                    (1-p*) for up
                  • The expected value at t=1 of the stock is:
                                            Su − S0e r      S0e r − S d
                     p * S d + (1 − p*)Su =            Sd +             Su
                                             Su − S d        Su − S d
                                                 = S0e r




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                           The Risk Neutral Probability
                                  Distribution
                 • So, under this artificial probability distribution, the
                   expected return on the risky asset S is the risk free rate r
                 • Hence risk neutral distribution
                 • Important in finance pricing and hedging
                 • But, it’s not an assumption.
                 • And we do not use a probability approach.




                                                   So far…
                  • Price =cost of replicating
                  • Risk neutral distribution is an artifice to go
                    directly to price without having to work through
                    replication




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                              Two-period Stock Process

                                                                                    Suu(2)=121

                                             Su(1)=110
                                                                                    Sud(2)=93.5
                 S0=100


                                            Sd(1)=85
                                                                                    Sdd(2)=72.25




                                       Two-period option
                  • European put option
                  • Strike price 105
                  • Two year term
                  • ⇒ max(0,S(2)-105) paid at t=2




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                             Two-period Option Process

                                                                                    P2=0

                                                  P1(up)
                                                                                    P2=11.5
                    P0

                                              P1(down)
                                                                                    P2=32.75




                                    2-period option price
                  • Work back from payoff;
                  • At t=1, suppose first move was up.
                  • Then we have a 1-period option with payoff 0 (up) or
                    11.5 (down)
                  • Construct a replicating portfolio: P1(up)=ke-r+aSu(1)
                    where
                  • k+aSuu(2)=0                          ⇒a= −0.418; k=50.60
                                                     Suu(2)=121, Sud(2)=93.5
                  • k+aSud(2)=11.5                       And P1(up)=2.1324




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                    2-period option price
                  • At t=1, suppose first move was down.
                  • Then we have a 1-period option with payoff 11.5
                    (up) or 37.5 (down)
                  • Construct a replicating portfolio: P1(down)=ke-r+aSd(1)

                  • k+aSdu(2)=11.5                  Sdu(2)=93.5, Sud(2)=72.25
                                                         ⇒a= −1.0; k=105
                  • k+aSdd(2)=32.75
                                                         And P1(down)=14.879




                             Two-period Option Process

                                                                                    P2=0

                                                2.132
                                                                                    P2=11.5
                    P0

                                              14.879
                                                                                    P2=32.75




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                 Two period option price
                  • So at t=0 we construct a replicating portfolio
                      P0=ke-r+aS(0) where
                  • k+aSu(1)=2.132
                  • k+aSd(1)=14.879                      ⇒a= −0.510; k=58.232
                                                         And P0=4.391




                                                  Short cut
                  • We can use the discounted expected value of the
                    payoff under the artificial probabilities
                    p*=0.1949, 1-p*=0.8051
                                        Check: p*Sd+(1-p*)Su
                                        =100er
                  • P0=e-2r(p*2(105-72.25)+2p*(1-p*)(105-93.5))
                        =4.391




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                       The story so far…
                  • Pricing by replication/hedging using the
                    artificial, risk neutral distribution
                  • Dynamic replication – re-balance hedge at each
                    time point
                  • Self-financing – rebalancing does not require
                    any more or less cash




                            The Black Scholes Formula
                  • If we let the time unit tend to zero
                  • The stock price risk neutral process would be
                    lognormal
                  • Log(S(t)/S(t-1))~N(r-σ2/2, σ2)
                  • E*[S(t)]=S(0) etr
                  • σ is the market volatility – i.e. annualized standard
                    deviation of log-returns




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                            The Black Scholes Formula
                   • The price at t=0 of a T-year put option is:
                        P0 = − S 0 N (− d1 (0)) + Ke − rT N (− d 2 (0))
                        K = strike price
                        N () = Φ () = Standard Normal D.F.
                                  log(S t / K ) + (r + σ 2 / 2)(T − t )
                       d1 (t ) =
                                               σ T −t
                                  log( St / K ) + (r − σ 2 / 2)(T − t )
                       d 2 (t ) =
                                                σ T −t




                            The Black Scholes Formula
                  • The payoff is hedged at 0 with the replicating
                    portfolio:
                  • Ke-rT N(-d2(0)) in bonds, value K N(-d2(0)) at T
                  • And – N(-d1(0)) units of stock.




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                   Replicating Portfolio
                  • But we re-balance the hedge every instant, so
                    that the replicating portfolio at t, 0<t<T is
                  • Ke-r(T-t) N(-d2(t)) in bonds,
                  • And – N(-d1(t)) units of stock, value at t
                      – St N(-d1(t))
                  • So at t, value/hedge is
                  • Pt= – St N(-d1(t)) + Ke-r(T-t) N(-d2(t))




                                                         So
                  • In continuous time we have the same results:
                          Price is an artificial expected value
                          Price is the cost of a replicating porfolio
                          Rebalancing of the replicating portfolio is
                          costless




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                            What were the assumptions?
                  • Underlying assumptions:
                          Stock prices in real probability follow a lognormal
                          process
                          Portfolios can be rebalanced every instant
                          There are no transactions costs
                          No dividends, flat yield curve, infinitely divisible
                          units of stock…




                       Can it possibly work in practice?
                  • Yes – despite the simple nature of the
                    assumptions
                  • Used for $billions of transactions daily
                  • Banks use the theory to limit their risks
                  • Also, use Value at Risk etc for un-hedged risks




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                  Example
                  • 5-year variable annuity with guaranteed living
                    benefit; assume everyone survives.
                  • Initial premium(net of MER) 75, Guaranteed
                    payout 100
                  • r=5%; σ=15%
                  • Monthly rebalancing
                  • Initial option cost $11.68




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                        Final 8 months…




                                           Example (cont)
                  • The Black Scholes hedge exactly pays off the
                    embedded option at maturity
                  • Hedging errors are small; total PV approx -$1
                  • This used the S&P 500 returns from 1969-74




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                               Common misconceptions
                  • Black-Scholes assumes stocks increase at the risk free
                    rate
                          We make no assumption about the average rate of
                          increase of stocks
                  • Black-Scholes hedging increases risk
                          Hedging is an important tool for minimizing risk
                  • The assumptions of Black-Scholes are so simplifying it
                    can’t work in practice
                          Proven results in extreme circumstances




                                                 Summary
                  • Hedging mitigates risk;.
                  • This is how banks manage risk
                  • There is no example of hedging being more
                    risky than not hedging
                  • Actuaries invented the fore-runner to hedging =
                    immunization




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                          Black Scholes
                                               and
                                             Beyond
                                                   SOA Annual Meeting
                                                     Hollywood, FL
                                                       May 2006

                              Chris K. Madsen, GE Insurance Solutions, Copenhagen, Denmark




                 Outline

                 Setting the stage
                    – Background and perspectives
                 Case study
                    – Non-life insurance industry (where
                      volatility is great and options more
                      valuable)



                                                                                             2




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                 1
                 Why should I care?

                 Options are everywhere, and we ignore the value of
                 them at our own peril whether it is through specific
                 limits on contracts or through “vague” contract terms
                 (embedded options).
                 As actuaries, we are not trained to think in terms of
                 optionality – we are trained to to think of expected
                 outcomes.
                 There’s a reason why we think as we do (but not an
                 excuse), and there’s a reason why this is changing.

                                                                                    3




                 What would you charge?

                 You are providing a stop loss cover with a deductible
                   of $100.0. A year from now, the underlying loss
                   will be either $110.0 or $90.9 – each with 50%
                   chance of occurring. Rate of interest is 5%. What
                   is the price of your coverage?

                      A.       (-9.1)*0.50+(10)*0.50 = 0.45
                      B.       10*0.50/1.05 = 4.76
                      C.       10*0.50 = 5.00
                      D.       10*.74/1.05 = 7.03
                      E.       Cannot be solved based on information provided


                                                                                    4




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                        2
                 The point

                 Simple discounting and time-value-of-money
                 ignore the option value




                                                                                    5




                 Notation

                 Ltu    Loss payment assessment (reserve) at time t in the up
                 case (share price)
                 K      Attachment point (strike price)
                 r      Risk-free rate
                 u, d Up and down factors for example 1.1 and 1/1.1.
                 R      1+r
                 p      The “real” probability
                 q      The “risk-neutral” probability (R-d)/(u-d)
                 C P    Call option price under the “P” measure
                 C Q    Call option price under the “Q” measure


                                                                                    6




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                        3
                 Insurance Arbitrage

                 A Insurance                              B Reinsurance
                                Lu= u × L0                                            Cu=Max(uL-K,0)                           50% chance
                    L0
                                                                C0
                                Ld= d× L1                                             Cd=Max(dL-K,0)                          50% chance


                                                                              ⎡ Max(L1 − K ,0) ⎤ p ⋅ Max(L0 ⋅ u − 1,0) + (1 − p ) ⋅ Max(L0 ⋅ d − 1,0)
                                                                    C0P = Ε P ⎢                ⎥=
                                                                              ⎣      R         ⎦                          R
                   u=1.1, d=1/u, r=0.05,                            C 0P = 0.0476
                  p=50%, L0=100, K=100
                                                                             ⎡ Max(L1 − K ,0) ⎤ q ⋅ Max(L0 ⋅ u − 1,0) + (1 − q ) ⋅ Max(L0 ⋅ d − 1,0 )
                                    What was                        C0 = Ε Q ⎢
                                                                     Q
                                                                                              ⎥=
                                                                             ⎣      R         ⎦                          R
                                      your
                                                                    C 0 = 0.0703
                                                                      Q

                                    answer?



                                                                                                                                                        7




                 Insurance Arbitrage

                   At time 0, if B Re charges under the P measure, A Insurance should
                                                      C 1, u − C 1, d
                         Sell Insurance worth                           ⋅ S 0 = 0 .5238
                                                          u−d
                                             d ⋅ C1,u − u ⋅ C1, d
                         Buy Bond worth                             = 0.4535
                                                 R ⋅ (u − d )

                         Buy call option worth 0.0476

                   Net cash outlay at time 0 is hence = 0.5238 – 0.4535 - 0.0476 = 0.0227, and simple
                   calculations will show that no matter if prices goes up or down, the net payoff at time 1
                   for A insurance will be 0. Hence A Insurance has an arbitrage opportunity.

                   Arbitrage in insurance is real. It is just difficult to exploit infinitely.




                                                                                                                                                        8




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                                                            4
                 Course 8, Fall 2005
                 13 out of 120 points (most valuable question)

                  3. (13 points) LifeCo’s management is concerned by the losses arising from the dynamic hedging of the options embedded in its
                  variable annuities. An external report highlighted that the target delta is currently based on a lognormal distribution with the volatility
                  equal to the sample standard deviation of the fund investment return over the past 12 months.
                  (a) Describe the options embedded in the variable annuity product.
                  (b) Describe and compare the following models used to estimate the volatility from past data
                                 (i) sample standard deviation
                                 (ii) exponentially weighted moving average model
                                 (iii) generalized auto-regressive conditional heteroscedasticity
                  (c) Recommend ways to improve the dynamic hedging program.
                  (d) Describe strategies that can be used to minimize the model risk.
                  LifeCo is considering whether to continue its current dynamic hedging program or pursue another risk management strategy.
                  (e) Review alternative strategies for managing the embedded option exposure.
                  (f) Recommend which of these strategies would be most appropriate if the dynamic hedging strategy is discontinued. Justify your
                  recommendation.


                                    Clearly, somebody else thinks this is important

                                                                                                                                                                9




                 Why consider options now?
                 A non-life historical perspective
                        Regulated market place (~1983 and prior)
                                 Average combined ratio of 109.5% (excl. 1970)
                                 Standard deviation of combined ratio of 20.0%
                                 Price are set centrally and jointly. Reinsurers are very profitable and there is little competition
                                 on price. Pricing did not matter, as prices were generally high.

                        Market place in transition (~1984 through ~1993)
                                 The market begins to change in some locations allowing price to fluctuate. Pricing begins to
                                 matter and more actuaries are hired.
                                 Average combined ratio of 163.1%
                                 Standard deviation of combined ratio of 42.2%

                        Deregulated market place (~1994 and later)
                                 Market prices fall where they may. Actuaries price deals extensively but no coherent pricing
                                 theory exists and there is always uncertainty about the right price leaving room for parameters
                                 to be modified and pricing biased.
                                 Average combined ratio of 120.4%
                                 Standard deviation of combined ratio of 27.5%




                                                                                                                                                            10




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                                                                    5
                 Surely, we are missing something…




                   Source: http://www.iii.org/media/facts/statsbyissue/pcinscycle/




                                                                                                                          11




                 Life vs. Non-Life Insurance

                 ROE
                   25%

                   20%
                                                                                            But is life any smarter?
                   15%

                   10%

                    5%

                    0%

                   - 5%
                       1980                            1985                          1990         1995             2000
                                                               US Life Insurers                  US P&C Insurers
                  John Coomber, Swiss Re, presentation to Intl’l Insurance Society
                  Insurance data based on statutory figures. US banks data for FPK US bank universe (median).
                  Sources: A.M.Best, RAA , Swiss Re Economic Research & Consulting, FPK


                                                                                                                          12




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                               6
                 Insurance vs. Banks

                 ROE
                 25%

                 20%

                 15%

                 10%

                  5%

                  0%

                  -5%
                     1980             1985                1990                1995                  2000
                                       US Life Insurers          US P&C Insurers            US Banks
                 1982 - 03
                 Average RoE                         13.0%                           8.5%                   15.0%
                 Standard deviation                   4.4%                           4.2%                    1.4%

                                                                                                                    13




                 Identical, but different worlds (for
                 now)
                                                    “Cat bonds” is the classic
                                                     insurance securitization
                             Risk                    case. They did not take                      Risk
                                                       off mainly because (i)
                                                            they were index
                                                      transactions, so sellers
                                                      retained basis risk and
                     Investment                        (ii) reinsurance capital
                                                     was willing to be lost. It
                                                                                      (Re)insurance
                        Bank                             was cheaper to buy             Company
                                                    reinsurance than to sell a
                                                                 bond.                           Securitization?
                                  GNMA, FNMA etc
                                                      Why insurers decided
                       Investors                    that they could price this               Investors
                                                    better than markets is an
                                                    open question. Clearly, in
                                                      hindsight, they would
                                                    have been well served to
                                                       follow the financial
                                                          market pricing.                                           14




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                         7
                 A Pricing Perspective Across Worlds
                                                                                                                                                                                                                      Similarities
                                                                                                              Short Equity (sell                                                                                                                                                                              Buying Insurance
                 Fixed Income                                                            Equity (buy options)    options)                                                                                                              Selling Insurance                                                         (Commutations)
                 Credit risk only                                                        Multiple risks       Multiple risks                                                                                                           Multiple risks                                                         Multiple risks
                 Very limited                                                            Limited downside     Limited upside                                                                                                           Limited upside                                                         Limited downside
                    downside                                                             Somewhat predictable Somewhat                                                                                                                 Somewhat                                                               Somewhat
                 Predictable return                                                         return               predictable return                                                                                                       unpredictable                                                          unpredictable

                                                     Bond                                                                  Equity                                                              Short Equity                                                          Sell Insurance                                                         Commutation
                              150.0                                                                   150.0                                                                   150.0                                                                150.0                                                                  150.0




                                                                                                                                                                                                                                                                                                                                                                                     t
                              100.0                                                                   100.0                                                                   100.0                                                                100.0                                                                  100.0




                                                                                                                                                                                                                                                                                                e
                                50.0                                                                   50.0                                                                    50.0                                                                  50.0                                                                   50.0
                  Cash Flow




                                                                                                                                                                 Cash Flow




                                                                                                                                                                                                                                       Cash Flow
                                                                                         Cash Flow




                                                                                                                                                                                                                                                                                                              Cash Flow
                                 -


                                                                                        t               -                                                                       -                                                                     -                                                                      -




                                                                                    rke                                                                                                                                                                                                     ark
                                        0   1    2   3   4    5     6   7   8   9   10                         0   1   2   3   4     5     6   7   8   9   10                          0   1   2   3   4    5     6   7   8   9   10                         0   1    2   3   4    5     6   7   8   9   10                         0   1   2   3   4    5     6   7   8   9    10
                               (50.0)                                                                 (50.0)                                                                  (50.0)                                                                (50.0)                                                                 (50.0)

                              (100.0)                                                                (100.0)                                                                 (100.0)                                                               (100.0)                                                                (100.0)

                              (150.0)                                                                (150.0)                                                                 (150.0)                                                               (150.0)                                                                (150.0)




                                                                                   a                                                                                                                                                                                                       M
                                                             Year                                                                   Year                                                                   Year                                                                   Year                                                                  Year




                                                                                 eM
                                                     Bond




                                                                                                                                                                                                                                                                                        te
                                                                                                                           Equity                                                              Short Equity                                                          Sell Insurance                                                         Commutation
                              150.0                                                                  150.0                                                                   150.0                                                                 150.0                                                                  150.0




                                                                                                                                                                                                                                                                                    ple
                              100.0                                                                  100.0                                                                   100.0                                                                 100.0                                                                  100.0
                                50.0


                                                                                t
                 Cash Flow




                                                                                                       50.0                                                                    50.0                                                                  50.0                                                                   50.0
                                                                                         Cash Flow




                                                                                                                                                                Cash Flow




                                                                                                                                                                                                                                       Cash Flow




                                                                                                                                                                                                                                                                                                              Cash Flow
                                                                            ple
                                 -                                                                      -                                                                       -                                                                     -                                                                      -
                                        0   1    2   3   4    5     6   7   8   9   10                         0   1   2   3   4     5     6   7   8   9   10                          0   1   2   3   4    5     6   7   8   9   10                         0   1    2   3   4    5     6   7   8   9   10                         0   1   2   3   4    5     6   7   8   9    10
                               (50.0)                                                                 (50.0)                                                                  (50.0)                                                                (50.0)                                                                 (50.0)




                                                                                                                                                                                                                                                                                  m
                              (100.0)                                                                (100.0)                                                                 (100.0)                                                               (100.0)                                                                (100.0)




                                                                                                                                                                                                                                                       o
                              (150.0)




                                                                          m
                                                                                                     (150.0)                                                                 (150.0)                                                               (150.0)                                                                (150.0)
                                                             Year                                                                   Year                                                                   Year                                                                   Year                                                                  Year




                                                                        Co                                                                                                                                                                         Inc
                                                     Bond                                                                  Equity                                                              Short Equity                                                          Sell Insurance                                                         Commutation
                              150.0                                                                  150.0                                                                   150.0                                                                 150.0                                                                  150.0
                              100.0                                                                  100.0                                                                   100.0                                                                 100.0                                                                  100.0
                                50.0                                                                                                                                           50.0                                                                  50.0                                                     the future?
                  Cash Flow




                                                                                                       50.0
                                                                                                                                                                Cash Flow




                                                                                                                                                                                                                                       Cash Flow
                                                                                                                                                                                                                                                                                                                            50.0
                                                                                         Cash Flow




                                                                                                                                                                                                                                                                                                              Cash Flow
                                 -                                                                      -                                                                       -                                                                     -                                                                      -
                                        0   1    2   3   4    5     6   7   8   9   10                         0   1   2   3   4     5     6   7   8   9   10                          0   1   2   3   4    5     6   7   8   9   10                         0   1    2   3   4    5     6   7   8   9   10                         0   1   2   3   4    5     6   7   8   9    10
                               (50.0)                                                                 (50.0)                                                                  (50.0)                                                                (50.0)                                                                 (50.0)
                              (100.0)                                                                (100.0)                                                                 (100.0)                                                               (100.0)                                                                (100.0)
                              (150.0)                                                                (150.0)                                                                 (150.0)                                                               (150.0)                                                                (150.0)
                                                             Year                                                                   Year                                                                   Year                                                                   Year                                                                  Year



                                                Investment banks: Extensively covered, good supporting theory                                                                                                                                                    Insurance                                                              Insurance +




                                                                                                                                                                                                                                                                                                                                                                           15




                 What is Insurance?

                 Insurance is usually an option to collect triggered
                 by some occurrence
                     Financial (continuous)
                     > Losses exceed attachment point (call option) –
                       reinsurance and non-life
                     > Interest rates drop below a threshold (put
                       option)
                     Event (discrete)
                     > Death (put option on human capital)
                     > Disability (put option on human capital)
                                                                                                                                                                                                                                                                                                                                                                           16




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                                                                                                                                                                                                                                                                                         8
                 Development of loss payments
                 estimate
                 Just like a lattice

                 Anything derived from this lattice such as a
                 payment above or beyond a certain point is an
                 option and should be priced accordingly
                 This includes reinsurance, rate guarantees,
                 payment extensions etc.


                                                                                    17




                 Development of ultimate loss
                 payments estimate: Binomial lattice
                                                               L4uuuu
                                                      L3uuu
                                            L2uu
                                 L1u
                                            L2ud
                     L0
                                 L1d



                                                                                    18




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                         9
                 Insurance Arbitrage

                 Ignoring this, (Re)insurers are (indirectly) being
                 arbitraged against, as P measure is used for pricing
                 ((re)insurers have only themselves to blame).

                 Only a (more) complete market can force the Q
                 measure – eliminating arbitrage.

                 But this does not mean that you can’t protect yourself
                 in an incomplete market by reflecting Q in your
                 pricing.

                                                                                      19




                 Case: Property Casualty Insurance

                      Property and casualty insurers’ profitability has long been
                      depressed relative to the risk they assume. Could this be
                      due to failure to price under Q?

                      The underwriting cycle is usually shown as the “combined
                      ratio” over time. This is total losses plus expenses relative
                      to premium charged. Thus, any ratio over 100% suggests
                      that more was paid out in losses and expensed that was
                      charged:+ E
                             L
                      CRt = t       t
                               Pt


                                                                                      20




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                           10
                 The Non-Life Insurance Underwriting Cycle


                 Madsen & Pedersen in 2003 showed that basic
                    option pricing can help us understand the non-life
                    underwriting cycle
                 In particular, we found that the industry has a
                    tendency to under-price relative to the risk-neutral
                    price and that this explains much of the uw cycle
                                 CRt = .2027 + .08836 ⋅ CRt −1 + .3979 ⋅ CRt − 2 + .4258 ⋅ Ct −1

                                      Insurance is a call option on losses – yet when the value of that
                                       option increases, so does the combined ratio. Insurers are not
                                  reflecting the increased price of risk essentially allowing consumers to
                    Combined Ratio                                arbitrage                             Option price index


                                                                                                                         21




                 The Non-Life Insurance Underwriting Cycle


                 The significance of the positive sign on the call option price
                 baffled us!!! This suggests that the industry does worse when
                 option prices increase. In other words, when pricing should be
                 going up (reinsurance is a call option on losses), results got
                 worse.

                 Then it became apparent: We don’t price our business as
                 options. Thus, we should be able to explain the underwriting
                 cycle (and our poor results) through the spread between the
                 option price under the Q-measure (risk-neutral probability) and
                 the expected price under the P-measure (real probability).

                                                                                                                         22




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                              11
                 But how to price in an incomplete
                 market?
                 Incompleteness does not imply complete lack of
                 arbitrage as we saw in our simple example. It just
                 means it cannot be exploited by all parties ad
                 infinitum.
                 There is one universal price of risk reflected in the
                 pricing of all risky cash flows. Implied volatility is a
                 good indicator of this and we can get observable
                 market prices for this.
                 If we can apply the market price of risk to our options
                 framework, then we have a way of getting directly at
                 the right (risk-neutral) price.
                                                                                              23




                 VIX: The CBOE Volatility Index

                      The CBOE Volatility Index - more commonly referred to as "VIX" - is an
                      up-to-the-minute market estimate of expected volatility that is
                      calculated by using real-time S&P 500 (SPX) index option bid/ask
                      quotes.

                      VIX uses nearby and second nearby options with at least 8 days left to
                      expiration and then weights them to yield a constant, 30-day measure
                      of the expected volatility of the S&P 500 Index.

                      The underlying for options is an "Increased-Value" Volatility Index
                      (VXB), which is calculated at 10 times the value of VIX. For example,
                      when the level of VIX is 12.81, VXB would be 128.10.

                      The VIX began trading in 1986.


                                                                                              24




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                   12
                                                                                        These should be “good times”, since consumers are willing to
                                                                                       pay relatively more. Instead, the industry prices too low allowing
                                                                                         consumers to arbitrage, which is a direct drain on industry
                                                                                                                     capital
                                                                                                                                                              VIX Prices


                                             40.00%


                                                                             VIX
                                             35.00%
                                                                             10 per. Mov. Avg. (VIX)
                Price (Implied Volatility)




                                             30.00%




                                             25.00%




                                             20.00%




                                             15.00%




                                             10.00%
                                                      1/3/1986


                                                                  1/3/1987


                                                                                 1/3/1988


                                                                                            1/3/1989


                                                                                                       1/3/1990


                                                                                                                  1/3/1991


                                                                                                                             1/3/1992


                                                                                                                                        1/3/1993


                                                                                                                                                   1/3/1994


                                                                                                                                                                 1/3/1995


                                                                                                                                                                              1/3/1996


                                                                                                                                                                                         1/3/1997


                                                                                                                                                                                                    1/3/1998


                                                                                                                                                                                                               1/3/1999


                                                                                                                                                                                                                          1/3/2000


                                                                                                                                                                                                                                     1/3/2001


                                                                                                                                                                                                                                                1/3/2002


                                                                                                                                                                                                                                                           1/3/2003


                                                                                                                                                                                                                                                                      1/3/2004


                                                                                                                                                                                                                                                                                 1/3/2005
                                                                                                                                                                            Date



                                                                                                                                                                                                                                                                                    25




                                             The Time Series

                                              PActual ,t , LActual ,t , E Actual ,t                                                         Actual premium, losses and expenses

                                                                 LActual ,t + E Actual ,t
                                             CRt =                                                                                           Actual combined ratio
                                                                                PActual ,t
                                                      ⎛ ( X − k )+ ⎞
                                             Pt = Ε P ⎜            ⎟                                                                         Discounted expected value of future cash
                                                      ⎜ 1+ r ⎟
                                                      ⎝        t   ⎠                                                                         flows under P-measure
                                                      ⎛ ( X − k )+ ⎞
                                             Qt = Ε Q ⎜            ⎟                                                                        Discounted expected value of future cash
                                                      ⎜ 1+ r ⎟
                                                      ⎝        t   ⎠                                                                        flows under Q-measure

                                                                                       ⎛ ( X − k )+ ⎞
                                                                                       ⎜            ⎟
                                                                             VIX t
                                             QtVIX = ΕQ                                ⎜ 1+ r ⎟
                                                                                                                                             Discounted expected value of future cash
                                                                                       ⎝        t   ⎠                                        flows under Q-measure with volatility adjusted
                                                                                                                                             according to Vix index


                                                                                                                                                                                                                                                                                    26




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                                                                                                                                                                                            13
                 Comparing “Standard” Pricing with Option
                 Pricing
                                                                                   Systematic Over and Under Pricing


                                         118.0                                                                                               1.20




                                                                                                     QtVIX t
                                         116.0                                                                                               1.15
                                                                                   Pt                                                                                  Forecast
                                         114.0
                                                                                                                                             1.10
                                                                                                                                                                      from 2003
                                         112.0
                                                                                                                                             1.05
                    Combined Ratio (%)




                                                                                                                                                    Pricing Index
                                         110.0                                                                                                                        Combined Ratio

                                                                                                                                             1.00
                                                                                                                                                                      Standard Pricing
                                         108.0                                                                                                                        Option Pricing

                                                                                                                                             0.95
                                         106.0

                                                                                                                                             0.90
                                         104.0

                                                                                                                                                                    Key turning points
                                                                                                                                             0.85
                                         102.0
                                                                                                                                                                    explained
                                         100.0                                                                                               0.80
                                                 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
                                                                                           Year

                                                                                                               CRt               Source for CR: A.M.Best

                                                                                                                                                                                         27




                 Feedback to 2003 Paper


                                         Industry combined ratio is on reported results. As
                                         such it contains reserving biases, some smoothing
                                         and will tend to lag actual performance
                                         To further test our hypothesis, it would be
                                         interesting to look at a single homogeneous line of
                                         business with real reserving data without the
                                         inherent lag

                                         Published follow-up in 2005, presented at AFIR
                                         conference

                                                                                                                                                                                         28




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                                                                                              14
                                 Single line of business
                                                                         Combined Ratio Versus Pricing Methods


                                 200%                                                                                  200%


                                                                                                                 CRt
                                 180%     Combined Ratio              Actual Price                                     180%




                                 160%                                                                                  160%




                                                                                                                              Price relative to Actual Premium
                                        PActual ,t
                Combined Ratio




                                 140%                                                                                  140%



                                 120%                                                                                  120%



                                 100%                                                                                  100%



                                 80%                                                                                   80%



                                 60%                                                                                   60%
                                     70
                                     71
                                     72
                                     73
                                     74
                                     75
                                     76
                                     77
                                     78
                                     79
                                     80
                                     81
                                     82
                                     83
                                     84
                                     85
                                     86
                                     87
                                     88
                                     89
                                     90
                                     91
                                     92
                                     93
                                     94
                                     95
                                     96
                                     97
                                     98
                                     99
                                     00
                                     01
                                     02
                                     03
                                     04
                                     05
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   20
                                   20
                                   20
                                   20
                                   20
                                   20
                                                                                                                             29




                                 Single line of business
                                                                         Combined Ratio Versus Pricing Methods


                                 200%                                                                                  200%
                                               Combined Ratio
                                                                                                                 CRt
                                 180%                                                                                  180%
                                               Option Price Relative to Premium
                                               Actual Price
                                 160%                                                                                  160%
                                                                                                                              Price relative to Actual Premium




                                        PActual ,t
                Combined Ratio




                                 140%                                                                                  140%

                                                  Qt
                                 120%                                                                                  120%




                                 100%                                                                                  100%




                                 80%                                                                                   80%




                                 60%                                                                                   60%
                                     70
                                     71
                                     72
                                     73
                                     74
                                     75
                                     76
                                     77
                                     78
                                     79
                                     80
                                     81
                                     82
                                     83
                                     84
                                     85
                                     86
                                     87
                                     88
                                     89
                                     90
                                     91
                                     92
                                     93
                                     94
                                     95
                                     96
                                     97
                                     98
                                     99
                                     00
                                     01
                                     02
                                     03
                                     04
                                     05
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   20
                                   20
                                   20
                                   20
                                   20
                                   20




                                                                                                                             30




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                                                                 15
                                 Single line of business
                                                                           Combined Ratio Versus Pricing Methods


                                 200%                                                                                            200%
                                            Combined Ratio
                                                                                                                   CRt
                                 180%                                                                                            180%
                                            Option Price Relative to Premium
                                            Option Price adjusted for Price of Risk                                      VIX t
                                            Actual Price                                                                 Q
                                                                                                                         t
                                 160%                                                                                            160%




                                                                                                                                        Price relative to Actual Premium
                                        PActual ,t
                Combined Ratio




                                 140%                                                                                            140%

                                                 Qt
                                 120%                                                                                            120%




                                 100%                                                                                            100%




                                 80%                                                                                             80%




                                 60%                                                                                             60%
                                     70
                                     71
                                     72
                                     73
                                     74
                                     75
                                     76
                                     77
                                     78
                                     79
                                     80
                                     81
                                     82
                                     83
                                     84
                                     85
                                     86
                                     87
                                     88
                                     89
                                     90
                                     91
                                     92
                                     93
                                     94
                                     95
                                     96
                                     97
                                     98
                                     99
                                     00
                                     01
                                     02
                                     03
                                     04
                                     05
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   19
                                   20
                                   20
                                   20
                                   20
                                   20
                                   20
                                                                                                                                       31




                                 Why Can This Exist?

                                 Classic incomplete market (bilateral, where buyers
                                    and sellers find each other and make direct
                                    contact)
                                               Prof. Vernon Smith’s behavioral economics study: People, when
                                               given a choice, choose to stay in an incomplete market. This is
                                               “survival economics” rather than “maximization of profits”
                                               Impetus needed to move from incomplete to complete. Players prefer
                                               incomplete markets because they feel better.


                                 No recognition that insurance can be arbitraged
                                               Reinsurance coverage can be replicated by selling insurance and
                                               buying bonds. This enables arbitrage, but on a limited scale since
                                               transaction costs are high (insurance license, staff etc)

                                                                                                                                       32




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                                                                           16
                 Pricing Framework

                 We make two important changes to the status quo

                      We price under the Q measure (risk-neutral
                      through observable market prices for volatility)

                      We create the implied volatility of an insurance
                      contract and use this in our pricing



                                                                                                                     33




                 Pricing Framework

                  Traditional insurance pricing (status quo)
                               [
                      P = Ε P e − rt (Lt − K ) =
                                                     +
                                                         ]       1
                                                                 R
                                                                   ( p ⋅ C u + (1 − p) ⋅ C d )

                  Option pricing
                    C = Ε Q [e − rt (Lt − K ) ] = (q ⋅ C u                           + (1 − q ) ⋅ C d )
                                             +   1
                                                                  R
                                                     R − e − σ t Δt                           R − e −σλt       Δt
                                                                                                           *
                                              R−d
                                      q=          = σ Δt                             =
                                              u−d e t    − e −σ t               Δt
                                                                                         e   σλ* Δt
                                                                                               t
                                                                                                      −e   −σλ* Δt
                                                                                                              t


                                                                 N

                                              vixt               ∑ vix      t
                                       λ* =
                                        t                vix =   t =1

                                              vix                       N


                                                                                                                     34




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                          17
                 What If Industry Pricing Reflected
                 This
                 Single Line of Business
                                                                          Combined Ratio Under Different Pricing Assumptions


                                    250%

                                                                                                                                                             Actual CR

                                                    ROE of –12% (31% St. Dev.)                                                                               Adj CR

                                    200%                                                                                                                     Adj CR incl PoR
                   Combined Ratio




                                    150%


                                                                                                                                                                  ROE of 5% (20% St. Dev.)
                                    100%



                                                                                                             Adj CR incl
                                    50%                                   Actual CR           Adj CR            PoR
                                             Expected Value                  133.5%             119.9%           101.4%
                                             Standard Error                   31.1%              25.4%            20.2%
                                                                                                                                                             ROE of 23% (15% St. Dev.)
                                     0%
                                           1986   1987   1988   1989   1990   1991   1992   1993   1994   1995   1996   1997   1998   1999   2000   2001   2002   2003   2004   2005
                                                                                                      Underwriting Year




                                                                                                                                                                                                          35




                 Financials: Insurance (all lines) vs. Banks
                                                   Loss from lack of option pricing
                                                                                            Loss from interest rate guarantees
                 ROE
                                                                                                            Loss from equity investments
                 25%

                 20%

                 15%

                 10%

                  5%

                  0%
                                                                                       Banks appear better at managing their risk
                  -5%
                     1980                                                 1985                                    1990                      1995                                           2000
                                                                              US Life Insurers                                 US P&C Insurers                                  US Banks
                 1982 - 03
                 Average RoE                                                                              13.0%                                                       8.5%                        15.0%
                 Standard deviation                                                                        4.4%                                                       4.2%                         1.4%

                                                                                                                                                                                                          36




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                                                                                                                               18
                 When to use what

                 Financial time series lend themselves most logically to option pricing.
                 If the volatility is low relative to the loss, the q probability approaches zero
                 and the option price will approach the current price adjusted for the PV of
                 the deductible. In these cases, the expectation under Q and P will be
                 relatively close. In other words, pricing under one measure or the other is
                 less important, if the underlying is not volatile.
                 The P measure has served life better than non-life, but it still misses part of
                 the point.
                 Theory needs to guide you. In practice, you can be lucky, but it doesn’t
                 prove anything. In general, option pricing will not lead you astray. In some
                 situations (as above), it may not make much of a difference, but it will still
                 give the right answer.



                                                                                               37




                 “Real Options”

                 Even less obvious situations can be priced with option theory
                 Real options
                     > Pharmaceutical drug developments – option to
                       continue/terminate (continue as long as option value
                       exceeds necessary investment)
                     > Option to work/option to retire – retire when option to
                       work no longer has positive value (M.A.Milevsky))
                 Options are everywhere and they are never free




                                                                                               38




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                    19
                 Conclusion

                 Option pricing (using the Vix index to determine the market price of risk)
                 explains 60% of the variation in the non-life underwriting cycle

                 Ignoring option pricing can be costly

                 More publicly traded insurance linked securities will make arbitrage easier
                 forcing better pricing and supporting theory

                 Financial disciplines continue to merge

                 Thanks



                                                                                              39




SOA Life 2006 Spring Meeting – 50TS, What is Black-Scholes and Why Should I Care?
                                                                                                   20

								
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