Option Pricing And Insurance Pricing

1. Option PricingAnd Insurance Pricing August 15, 2000

2. Overview • Options And Option Pricing • Insurance Pricing • Wacek’s Paper • Mildenhall’s 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 • C0=Max[0, S-K] • What Is Ct?

5. Put Option • Right To Sell Security • Strike Price, K • Expiration Time, t • Exercisable Only At t • At Expiration • P0=Max[0, K-S] • What Is Pt?

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, Net CF Is Greater Than 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. Wacek’s Paper • Black Scholes Discussion • Product Design By Analogy

14. Wacek’s 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. Wacek’s Bull Cylinder Reinsurance • Cylinder • Bull: Long A Call, Short A Put • Bear: Long A Put, Short A Call • If KP < S < KC 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. Wacek’s 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. Mildenhall’s Review • Black Scholes Sets Price Including Risk • Works Out No Arbitrage For Binomial Case • Shows That Actual Price Follows Model