Presentation of Proceedings Paper
Application of the Option Market Paradigm
to the Solution of Insurance Problems
Author’s Response to Stephen Mildenhall Discussion
Michael G. Wacek
CAS Annual Meeting
November 14, 2005
1
Mildenhall Criticisms

Accepted paper’s main point that resemblance between call option /
XL insurance concepts can lead to useful insights

Paper overstated similarity between options and insurance

In particular, paper’s assertion that “pricing math is basically the
same” is “inappropriate”

Argues that insurance risks are diversified, financial risks are hedged,
implying different paradigms
2
Wacek Mea Culpa

Mildenhall was polite

Paper overreached in claim that my formula (1.3) is a general formula
for European call pricing that encompasses the Black-Scholes formula
(1.1)

Author’s response shows how overarching framework is same, even if
resulting formulas are different
3
Comparison of Formulas (1.1) and (1.3)
Ct (S) = P0 ∙ N(d1) – Se-rt ∙ N(d2)
(1.1)
Ct (S) = e-rt
(1.3)
(x-S) ∙ f (x) dx
(1.3) reduces to (1.1) only if underlying asset price distribution at
expiry is lognormal and the expected return equals the risk free rate r
N(z) is c.d.f. of standard normal; see response for d1 and d2
4
Parameters for Numerical Illustration of Formulas

Current stock price = $100

Option strike (exercise) price = $100

Time to option expiry = 20 days (20/365 years)

Stock expected return = 13% (annual, continuous)

Risk free return = 5% (annual, continuous)

Standard deviation of stock return (volatility) = 25% (annual)
5
Numerical Illustration of Formulas (1.1) and (1.3)

Formula (1.1) [Black Scholes]: $2.47

Formula (1.3):

Obviously, $2.47≠ $2.71

Paper’s claim that (1.3) encompasses (1.1) was not “inappropriate” – it
was wrong
$2.71
6
Formula (1.3)

Formula for p.v. pure premium of aggregate excess cover

Since Black-Scholes conditions are not present, for insurance
applications it is correct to use (1.3) rather than (1.1)
7
Pricing Paradox

If liability arises from call option on stock, use formula (1.1)

If same liability arises from aggregate excess insurance, use formula (1.3)

Why does same liability have different pricing depending on context?
8
Pricing Paradox Answers

Can be framed in terms of martingale measures and incomplete
markets theory

Author takes more tangible approach of asset – liability matching
• Within that framework the price for the transfer of a liability is
function of both the liability and its optimal matching assets
9
Analysis of Option Liability

Assume stock price at option expiry is lognormal random
variable x

Expected value of option at expiry:
E(callt) =
(x-S)f(x)dx
= E(x) ∙ N(d1(μ)) – S ∙ N(d2(μ))
= P0e μ t ∙ N(d1(μ)) – S ∙ N(d2(μ))

N(z) is c.d.f of standard normal; see response for d1(μ) and d2(μ)
10
P0e μt ∙ N(d1(μ)) – S ∙ N(d2(μ))

First term is expected market value of the assets to be sold by option
grantor to option holder at expiry

Second term is expected value of the sale proceeds from transaction
11
Example – Expected Option Payoff Liability

Same parameters as earlier example

20-day option

Po = S = $100, μ = 13%, r = 5%, σ = 25%

Expected payoff = Expected market value – Expected proceeds
= $56.40 - $53.68 = $2.72

Variance = 14.45

Standard Deviation = $3.80
12
Pricing of This Liability

Premium that market can be expected to ask for assuming liability
depends on optimal strategy available for investment of premium to
fund the liability

Assume investors / traders will find and execute optimal strategy to
force asking price in market to be no greater than the level indicated by
that strategy (“No arbitrage”)

Market premium = minimum expected value cost of acquiring assets
to fund expected value of liability at expiry and a risk charge related to
undiversifiable variability of net result

If variance can be forced to zero, risk charge is zero (B-S scenario)
13
Pricing This Liability in
Different Available Asset Scenarios

Case A – Underlying asset is tradable

Case B – Underlying asset is not tradable

Case C – Underlying asset is not tradable, but a tradable proxy exists
14
Case A – Underlying Asset Tradable

Traditional actuarial approach is to invest matching assets in
Treasuries

Easy to improve on that if liability arises from an option on a traded
stock
15
Case A – Simple Hedge


At inception, buy N(d1(μ)) shares at cost of P0 ∙ N(d1(μ))
At expiry those shares expected to be worth P0 e μt ∙ N(d1(μ)) = expected
liability

Expected proceeds at expiry = S ∙ N(d2(μ))

Borrow against those proceeds at inception: Se- rt. N(d2(μ))

Funding gap for purchase is P0 ∙ N(d1(μ)) - Se-rt ∙ N(d2(μ)) , which is
amount required from option buyer (ignoring risk charge)
16
Case A – Simple Hedge with Numbers

Buy 0.56 shares (per option) for $56.00

Borrow $53.54

Charge option premium = $56.00 - $53.54 = $2.46

Much lower than $2.71 from traditional actuarial formula (1.3)

Moreover, this asset strategy has less risk (standard deviation) than a
Treasuries strategy: $1.75 vs. $3.80

This simple hedge superior to Treasuries, but B-S found an even better
one
17
Case A – Black-Scholes Hedge

Suppose stock trades in accordance with B-S assumptions:
• Price follows geometric Brownian motion through time
• Shares continuously tradable at zero transaction costs
• Other

Black and Scholes showed optimal investment strategy is one of
dynamic asset-liability matching conducted in continuous time
18
Case A – Black-Scholes Hedge

At inception, by n0 shares at cost of P0 ∙ n0, financed by a loan of L0 and
call premium proceeds of P0 ∙ n0 - L0

An instant later, adjust number of shares to n1, (to reflect any stock
price change and infinitesimal passage of time) and amount of loan
to L1

If n0 and L0 have been chosen correctly and time interval is short
enough, the actual gain/loss in net position (value of net stock position
less value of option) is essentially zero. (Mean and variance also
essentially zero)

Repeat this procedure continuously until option expires

Black and Scholes proved n0 = N(d1) and L0 = Se-rt ∙ N(d2)
19
Case A – Black-Scholes Hedge (continued)

Thus, call0 = P0 ∙ N(d1) – Se-rt ∙ N(d2)

d1 = [ln(P0/S) + (r + 0.5σ2)t] / σ√t and d2 = d1 - σ√t

Since (1.1) does not depend on μ, the option seller employing this
strategy not only faces no process risk but also no μ -related parameter
risk
(1.1)
20
Case A – Black-Scholes Hedge with Numbers

Buy 0.5303 shares for $53.03

Borrow $50.56

Charge option premium = $53.03 - $50.56 = $2.47

Compares to $2.71 and $2.46 from “actuarial” and “simple hedge”
approaches

This asset strategy has no risk, so no risk charge needed
21
Case A - Comments

Without hedging, option seller faces expected present value liability
cost of $2.71

Market will force price of liability to $2.47

Option seller must hedge to sell option for $2.47 without facing an
expected p.v. loss of $2.71 - $2.47 = $0.24
22
Case A – More Comments

In practice, stocks are not continuously tradable at zero transaction
costs

The less liquid the stock and the greater the trading costs the less
accurate (1.1) will be in predicting the market asking price of the call
option (due to residual risk and/or expenses not considered in the
formula)
• See Esipov & Guo (2000 Michelbacher Prizewinner)
23
Case B – Underlying Asset Not Tradable

Company to go public in 20 days

Assume no “when issued” or forward market for stock prior to IPO

Stock valued at $100 today

Other parameter estimates as in Case A

How to price this option?
24
Case B – How to Price Option?

Key question is how to invest the call premium to fund expected payoff
of $2.72

Cannot invest in underlying stock

Good strategy would seem to be to invest in Treasuries (implying
formula (1.3))

Premium = $2.71 + risk charge (standard deviation = $3.80)

μ and σ parameter risk in addition

Analogous to insurance aggregate excess scenario

Not necessarily the optimal strategy
25
Case C – Tradable Proxy Asset Exists

Suppose there is a similar, already public competitor to our
soon-to-be-public company

Assume same return and volatility characteristics

Assume the stock price movements are believed to be correlated with
r = 60%.

Possible to construct a partial hedge that results in a lower cost to fund
the option liability than the Treasuries strategy
26
Case C – Hedge Illustration for Stock

Pursue same strategy as if hedging with actual underlying stock
• Buy 0.5303 shares for $53.03
• Borrow $50.56
• Charge option premium = $53.03 - $50.56 = $2.47 + risk charge λ
• Standard deviation = $3.67 vs. $3.80 for Treasuries strategy
• Lower pure premium ($2.47 vs. $2.71) and lower standard deviation
make this a superior strategy to investing in Treasuries
27
Case C – Hedge Illustration for Insurance

Suppose underlying insurance claims are 60% correlated with the
consumer price index (CPI-U)

Insurer can reduce standard deviation of net result by investing in
index-linked Treasury notes (TIPS) rather than conventional Treasuries

Monte Carlo analysis indicates $3.29 vs. $3.80

P.v. pure premium = $2.71 both strategies (conventional Treasuries and
TIPS)
28
Case C - Insurance
Food for Thought

Could an insurer reduce both its risk and its required pure premium by
identifying and investing in higher return securities that are partially
correlated with its liabilities?
29
Analysis of Pricing

In all the cases A, B and C, the expected value of the option payoff (or
aggregate excess liability) obligation at expiry is the same: $2.72

Only difference are type and tradability of assets available for
investment

It is the characteristics of the asset side of the asset-liability equation
that determine the optimal asking price for transfer of the liability!
30
Analysis of Pricing (continued)

Pricing is a function of both the liability and the nature of the assets
needed to fund it

In insurance applications, where there are usually no suitable assets
other than Treasuries available, the liability alone appear to drive the
price

However, that appears to be a special case

Actuaries should be alert to the possibility that some insurance
situations might lend themselves to asset-liability matching that
reduces the p.v. pure premium, risk, or both
31
Optimal Pricing Requires Risk Management

If the optimal asset strategy drives the pricing of the liability, it is
critical that the seller actually invests according to that strategy

If option seller believes μ = 13%, and sells call option for B-S price of
$2.47, it would be a mistake to invest proceeds in Treasuries

Expected result = $2.47 ert -$2.72 = -$0.24

Standard deviation = $3.80 (vs. zero under optimal asset strategy)

Lesson: B-S assumes optimal strategy; option seller not automatically
protected
32
Summary of Author’s Response

Author acknowledges that original claim of generality for (1.3) is wrong

However, response does show that insurance and option pricing are
governed by the same overarching pricing paradigm implied by optimal
asset-liability matching
33
Final Point on the Perils of Pricing

The dynamic asset-liability matching regimen underlying Black-Sholes
formula imposes a different burden on the seller of the option than the
more passive asset-liability matching seen in Case B and in insurance
applications
• In liquid markets, it is a mistake to sell the option for the BlackScholes price without engaging in the optimal asset strategy
• In illiquid markets, it is a mistake to sell the option for the BlackScholes price because the optimal asset strategy does not yield that
price (must use formula (1.3) instead)
34
Acknowledgment

Thank you to Stephen Mildenhall for his discussion
35