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

Assumptions

Black Scholes is very complete pricing Paradigm

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

Insurance Interpretations

Aggregate Stop Loss Likened Call Option

How do you price an Aggregate Stop Loss?

em+s2/2-r'T F((ln(1/k)+(m+s2))/s) - e-rTk F((ln(1/k)+m)/s)

An Actuarial Call Option Price

Black Scholes assumes stock prices follow a Geometric Brownian Motion

ln(ST/S0) ~ N((m-s2/2)T, sT1/2)

Estimating Parameters

e(m-r) T S0 F(ln(S0/k)+(m+s2/2)T/sT) - e-rT k F(ln(S0/k)+(m-s2/2)T/sT)

Why Black Scholes is Surprising

Compare Black Scholes result with formula above

Act: e(m-r) T S0 F(ln(S0/k)+(m+s2/2)T/sT) - e-rT k F(ln(S0/k)+(m-s2/2)T/sT)
BS: S0 F(ln(S0/k)+(r+s2/2)T/sT) - e-rT k F(ln(S0/k)+(r-s2/2)T/sT)

Implies

Interpretation

How can this be?

Risk in finance is an element of stochastic behavior in future prices

Return above risk free rate required when future value contains element of risk

KEY POINT: Between underlying stock, bond and option there is only one source of stochastic behaviour (uncertainty)

Basis of Option Pricing Paradigm

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 Black-Scholes 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

Fine Print on No-Arbitrage

No-Arbitrage means "no free lunch"

Financial markets allow long and short positions

A geometric Brownian motion stock price process allows arbitrage possibilities

Questions and Answers

Appendix

Full Discussion


Formulae typeset using LaTeX and translated from TeX by TTH, version 2.34.