Applications of the Option Pricing
Paradigm to Insurance
by
Michael Wacek
Discussion by
Stephen Mildenhall
CNA Re, Chicago IL
Contents
What is Black Scholes?
Black Scholes is a mathematical theorem: a statement that certain
conclusions follow from certain premises
 Determines price of European call option on a stock
 Call Option is the right, but not obligation, to buy at a predetermined
exercise or strike price
 European: can only be exercised at expiration
Assumptions
 Stock price:
 Follows a "geometric Brownian motion"
 Instantaneous
stock returns are normally distributed
 No stock dividends are payable during life of the derivative
 Continuous time trading
 Infinitely divisible securities (e.g. can buy 0.1234 a stock)
 Ability to borrow and lend at a fixed risk free rate of interest
r
 r is fixed regardless of duration
 No restrictions on short selling; full use of proceeds
 No transaction costs or taxes
 No arbitrages
 Whether Black Scholes is true in the "real world"
depends on whether the premises hold in the real world
Black Scholes is very complete pricing Paradigm
 Gives price and an explicit way to "hedge" (manufacture)
the product for the indicated price by trading in the underlying stock and
bonds
 Like an insurance pricing model plus a reinsurance structure
guaranteeing results!
 Regardless of state of reinsurance market
Black Scholes Pricing and Market Pricing
Ex Price

Intrinsic
Value

Market
Price

BS

Pct Error

Actuarial

Volume

750

169.77 
186 
181.06

2.7% 
201.99

714 
805

114.77 
135 
130.90

3.0% 
150.55

3 
890

29.77 
67 1/2 
66.90

0.9% 
82.04

10 
900

19.77 
64 
60.88

4.9% 
75.32

6 
910

9.77 
59 1/4 
55.22

6.8% 
68.93

102 
930

0 
44 1/4 
44.96

1.6% 
57.18

3,291 
935

0 
41 3/4 
42.62

2.1% 
54.46

5 
940

0 
42 1/2 
40.37

5.0% 
51.83

264 
950

0 
36 1/4 
36.12

0.4% 
46.83

14 
960

0 
31 1/2 
32.20

2.2% 
42.17

2 
990

0 
21 
22.37

6.5% 
30.20

5 
995

0 
20 
20.99

4.9% 
28.48

107 
1025

0 
11 
14.07

27.9% 
19.71

7 







 S&P 500 European Calls September 15, 1997
 S&P closed at 919.77
 Risk free rate of interest 5.12%, discount factor 0.9868
 95 days to expiration
 Actuarial pricing assumes growth rate on stocks 15%
pa (13.98% compounded continuously) discounted at risk free rate
 Standard Deviation of stock returns: 23.5% annually
 Errors +/7% over a large range except for one, very
out of the money call
 Last trade on may have been well before close
 Oh, for such a good way to price GL!
Insurance Interpretations
Aggregate Stop Loss Likened Call Option
 Agg stop "attaches" at a certain loss ratio
and pays the excess of actual losses over the attachment
 Agg stop with a limit = difference of two agg stops
with different attachments
 Call option has an "exercise" price and pays
the excess of the final stock price over the exercise price
 Ignore niceties of time and payout patterns: assume
agg stop settles in full T years from now with a single payment
 Could argue that agg stops are more like swaps than options
How do you price an Aggregate Stop Loss?
 S = aggregate loss random variable
 Current best estimate of losses at
expiration, T years from now
 k = attachment point
 Pure Premium = E[ max(Sk, 0) ]
 PV Pure Premium = e^{r'T}E[ max(Sk, 0) ] for some
discount rate r' per year
 Premium = ???
 Example
 S ~ Lognormal(m, s)
 Mean of log(S) is m and standard
deviation is s
 Selected discounting interest rate r'
 Discounted pure premium can be determined using Part 1 calculus
e^{m+s2/2r'T}
F((ln(1/k)+(m+s^{2}))/s)  e^{r¢T}k F((ln(1/k)+m)/s)
 See appendix for details
 F is the cumulative distribution function of the standard normal distribution
An Actuarial Call Option Price
Black Scholes assumes stock prices follow a Geometric Brownian
Motion
 Changes in stock prices over a short period of time are approximately
normally distributed
 DS / S = mDt + sZÖDt, where Z is a standard normal random variable
 Mean return in small time period Dt is mDt
 Variance is s^{2}Dt
 Is this reasonable? Below is a normal probability plot for
Dell stock daily returns 1/2/97 to 2/9/00
 Note problems with fit in extreme left tail
 Indicates trouble with assumptions for deeply in and
out of the money options (volatility smile)
 GBM implies that T years from now the stock price S_{T}
is lognormal and
ln(S_{T}/S_{0}) ~ N((ms^{2}/2)T,
sT^{1/2})
Estimating Parameters
 Mean m: CAPM, historical record
 Standard deviation s: Historical
record, last thirty days returns.
 Finance has the data of an actuary's dreams!
 Discounting Interest Rate r': COTOR Comprehensive
Study, Zen Enlightenment
 Can now apply our agg stop pricing formula:
 e^{r'T}E[ max(S_{T}k, 0) ] = e^{r'T}
S_{0} E[ max(S_{T}/S_{0}  k/S_{0}, 0) ]
 m ¬ (ms^{2}/2)T
 k ¬ k/S_{0}
 s ¬ sÖT
e^{(mr¢) T} S_{0 }F(ln(S_{0}/k)+(m+s^{2}/2)T/sÖT)  e^{r¢T }k F(ln(S_{0}/k)+(ms^{2}/2)T/sÖT)
 This formula appeared in a Samuelson paper published in the
1960's following the same logic used here
 Still issue of converting from discounted pure
premium to premium
 Actuarial Price in table assumes r=r' in above formula
Why Black Scholes is Surprising
Compare Black Scholes result with formula above
Act: 
e^{(mr¢) T}

S_{0 }F(ln(S_{0}/k)+(m+s^{2}/2)T/sÖT)

 
e^{r¢T } k F(ln(S_{0}/k)+(ms^{2}/2)T/sÖT) 
BS: 

S_{0 }F(ln(S_{0}/k)+(r+s^{2}/2)T/sÖT)

 
e^{rT} k F(ln(S_{0}/k)+(rs^{2}/2)T/sÖT)

Implies
 Black Scholes assumes m = r: stock
earns risk free return!!!
 Black Scholes assumes r' = r: discount at risk free rate!!!
Interpretation
 Option Interpretation: Call option price is independent of
expected appreciation (depreciation) of particular stock during contract period
 Insurance Interpretation: Agg stop price is independent of expected losses;
only volatility matters
 All cash flow discounted as though they are risk free
How can this be?
Risk in finance is an element of stochastic behavior
in future prices
 In GBM it is the term sZÖDt, creates "wiggles"
 Risk Free: locally deterministic price, like bank CD or bill in fixed interest
rate environment
Return above risk free rate required when future value
contains element of risk
 Each source of uncertainty commands a price, called market
price of risk
 Different stocks, oil futures, insurance derivatives
KEY POINT: Between underlying stock, bond and option there
is only one source of stochastic behaviour (uncertainty)
 Option and stock prices are instantaneously perfectly correlated
 Plausible: stock price and call price must move together
 Portfolio mixing the two in the right
proportions would be risk free
 Right proportion is dC/dS =: d < 1, hence "delta" hedging
 Law of one price implies gives option price
Basis of Option Pricing Paradigm
 Write an option and combine with portfolio of stocks and
bonds to remove all risk (wiggles)
 Cost of option = cost of setting up initial portfolio of stocks and bonds
provided no more cash flows required from trading
 Black Scholes paradigm shows price does equal cost of hedging portfolio
 Hence Black Scholes is a complete pricing paradigm
 Hedging portfolio "manufactures" option
 Option Pricing Paradigm does not rely on the law
of large numbers
 Obviously horrendous correlations exist in
portfolio of equities and options!
 Cost of initial portfolio independent of m, expected
return on stock
 Leads to notion of risk neutral valuation: can assume any convenient risk
preferences
 Risk neutral assumption: stocks earn risk free return hence m
= r
 Risk free portfolios earn the risk free return, hence r = r'
 Classic example of binary stock price model and explicit
hedging strategy is given in discussion
Cox and Ross
Investor's preferences and demand conditions in general
enter the valuation problem only in so far as they determine the equilibrium
parameter values. No matter what preferences are, as long as they determine
the same relevant parameter values, they will also value the option identically.
In the BlackScholes case ... the only relevant parameters for the pricing problem
are r and sigma. To solve [for the price], then, we need only find the equilibrium
solution ... in some world where preferences are given and consistent with the
specified parameter values; the solution obtained will then be preference free.
Approach hinges on existence of traded underlying security
 Does not exist with many insurance products
 Work around possible in some cases
 Two options on same underlying (whether traded or not) share
a single source of uncertainty
 Same arguments used in Black Scholes imply there must be
a relation between prices of options at different strikes and expirations
 Could possibly be used to get constraints on Cat Options
where there is a single source of uncertainty
Fine Print on NoArbitrage
NoArbitrage means "no free lunch"
 Noarbitrage implies assets with equivalent cash flows command
the same price
 Alternative risk and packaging of risks should not lower
price  unless there is less coverage
 A model with arbitrages is not viable as a model
of market prices
since
 Agents would immediately act to change
prices!
Financial markets allow long and short positions
 If combinations A and B are equivalent
but not equally priced, the overpriced combination would be shorted by investors
who would buy B with the proceeds to produce an arbitrage profit
A geometric Brownian motion stock price process allows arbitrage
possibilities
 Doubling strategy
 Restrictions on the amount borrowed or on short sales needed
 No arbitrage is a consequence of the model framework not an assumption
Questions and Answers
Appendix
Full Discussion
Formulae typeset using LaTeX and translated from TeX by T_{T}H,
version 2.34.