1 Option Pricing And Insurance Pricing August 15, 2000

2 Overview Options And Option Pricing Insurance Pricing Waceks Paper Mildenhalls Review

3 Options Defined Call Option Put Option Put-Call Parity No Arbitrage Pricing

4 Call Option Right To Buy Security Strike Price, K Expiration Time, t –Exercisable Only At t At Expiration –C 0 =Max[0, S-K] What Is C t ?

5 Put Option Right To Sell Security Strike Price, K Expiration Time, t –Exercisable Only At t At Expiration –P 0 =Max[0, K-S] What Is P t ?

6 Put-Call Parity Call Plus Present Value Strike = Put + Stock –At Expiration, Both Sides Are The Same –So Portfolios Must Have Same Value We Will Concentrate On Call Price, C

7 No Arbitrage Pricing Derive Boundary For Call Option Price One Year Case Assume C<S-K/(1+r) S The Current Stock Price Show Risk Free Profit Exists Conclude That C>=S-K/(1+r) More Generally C>=S-Ke -rt

8 Suppose C<S-K/(1+r) If S>K, Plug S-K/(1+r) For C And Net CF>0 If S K So We Have A Boundary For C C>=S-K/(1+r)

9 Boundary For C

10 Black Scholes Formula

11 Black Scholes Pricing Extension Of No Arbitrage Pricing Requires Market For Underlying Security Risk Priced In Underlying Market Stock Price Distribution –Assume Continuous % Change Distribution –Result Is Lognormal Stock Price Distribution Option Price = f[Stock Price] Dividends Make A Hash Of Math

12 Insurance Pricing Actuarial Estimation Of Expected Loss Principal Use Of Distributions –Per Occurrence Excess Expected Loss –Aggregate Excess Expected Loss –Prove Risk Transfer Risk Premium From The Insurance Market Pure Supply And Demand At Any Time Black/Scholes Has No Practical Use

13 Waceks Paper Black Scholes Discussion Product Design By Analogy

14 Waceks Black Scholes Discussion Observes –Lognormal Stock Price –Option Price = Discounted Excess Pure Premium –Suggests That Risk Premium Is Missing Problems With Footnotes 1 and 2 –1. Price or Premium Does Not Include Risk? –2. Risk Neutral And No Arbitrage Pricing Are The Same?

15 Waceks Bull Cylinder Reinsurance Cylinder –Bull: Long A Call, Short A Put –Bear: Long A Put, Short A Call –If K P < S < K C at expiration either position is worthless. Insurance Companies Are Short The Losses –As Losses Go Up, Insurance Profits Go Down Bull Cylinder Reinsurance –Small (Or Zero) Initial Premium –Low Losses Leave Put In The Money Insurer Pays Additional Premium –High Losses Leave Call In The Money Insurer Recovers From Reinsurer –Retrospective Rating Backwards

16 Waceks Reinsurance Call Options Reinsurance Is A Security Insurers Want To Price Stability Reinsurers Could Offer Call Option –Embedded In Reinsurance Contract –Sold Seperately –Catastrophe Example: 1.46% For Call At 30% With Current Price of 20% Result Is Current Expense Of 21.46% Multi-Year Pricing As Form Of Reinsurance Call

17 Mildenhalls Review Black Scholes Sets Price Including Risk Works Out No Arbitrage For Binomial Case Shows That Actual Price Follows Model