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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