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Presentation of Presentation of ProceedingsProceedings PaperPaper

Application of the Option Market Paradigm

to the Solution of Insurance Problems

Author’s Response to Stephen Mildenhall Discussion

Michael G. Wacek

CAS Annual MeetingNovember 14, 2005

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Mildenhall CriticismsMildenhall 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

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Wacek Mea CulpaWacek 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

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Comparison of Formulas (1.1) and (1.3) Comparison of Formulas (1.1) and (1.3)

Ct (S) = P0 ∙ N(d1) – Se-rt ∙ N(d2) (1.1)

Ct (S) = e-rt (x-S) ∙ f (x) dx (1.3)

(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

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Parameters for Numerical Illustration Parameters for Numerical Illustration of Formulasof 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)

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Numerical Illustration of Formulas Numerical Illustration of Formulas (1.1) and (1.3)(1.1) and (1.3)

Formula (1.1) [Black Scholes]: $2.47

Formula (1.3): $2.71

Obviously, $2.47≠ $2.71

Paper’s claim that (1.3) encompasses (1.1) was not “inappropriate” – it was wrong

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Formula (1.3)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)

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Pricing ParadoxPricing 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?

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Pricing Paradox AnswersPricing 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

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Analysis of Option LiabilityAnalysis 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)

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PP00e e μμtt ∙ N(d ∙ N(d11((μμ)))) – S ∙ N(d– S ∙ N(d22

((μμ))))

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

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Example – Expected Option Payoff Example – Expected Option Payoff LiabilityLiability

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

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Pricing of This LiabilityPricing 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)

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Pricing This Liability in Pricing This Liability in Different Available Asset ScenariosDifferent 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

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Case A – Underlying Asset Tradable 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

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Case A – Simple HedgeCase 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)

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Case A – Simple Hedge with NumbersCase 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

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Case A – Black-Scholes HedgeCase 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

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Case A – Black-Scholes HedgeCase 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)

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Case A – Black-Scholes Hedge Case A – Black-Scholes Hedge (continued)(continued)

Thus, call0 = P0 ∙ N(d1) – Se-rt ∙ N(d2)(1.1)

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

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Case A – Black-Scholes Hedge with Case A – Black-Scholes Hedge with NumbersNumbers

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

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Case A - CommentsCase 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

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Case A – More CommentsCase 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)

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Case B – Underlying Asset Not TradableCase 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?

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Case B – How to Price Option?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

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Case C – Tradable Proxy Asset ExistsCase 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

Possible to construct a partial hedge that results in a lower cost to fund the option liability than the Treasuries strategy

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Case C – Hedge Illustration for StockCase 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

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Case C – Hedge Illustration for Case C – Hedge Illustration for InsuranceInsurance

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)

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Case C - InsuranceCase 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?

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Analysis of PricingAnalysis 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!

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Analysis of Pricing (continued)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

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Optimal Pricing Requires Risk Optimal Pricing Requires Risk ManagementManagement

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

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Summary of Author’s ResponseSummary 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

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Final Point on the Perils of PricingFinal 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 Black-Scholes price without engaging in the optimal asset strategy

• In illiquid markets, it is a mistake to sell the option for the Black-Scholes price because the optimal asset strategy does not yield that price (must use formula (1.3) instead)

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AcknowledgmentAcknowledgment

Thank you to Stephen Mildenhall for his discussion