Option Pricing And Insurance Pricing by liuhongmei

VIEWS: 15 PAGES: 17

									   Option Pricing
And Insurance Pricing

     August 15, 2000


                        1
                 Overview
•   Options And Option Pricing
•   Insurance Pricing
•   Wacek’s Paper
•   Mildenhall’s Review




                                 2
             Options Defined
•   Call Option
•   Put Option
•   Put-Call Parity
•   No Arbitrage Pricing




                               3
                  Call Option
• Right To Buy Security                         Call Option Price At Expiration
                                                             K=40
• Strike Price, K                          70
                                           60
• Expiration Time, t


                            Option Price
                                           50
                                           40
  – Exercisable Only At t                  30

• At Expiration                            20
                                           10

  – C0=Max[0, S-K]                         0
                                                0     20      40      60     80       100
• What Is Ct?                                               Security Price



                                                                                  4
                   Put Option
• Right To Sell Security                        Put Option Price At Expiration
                                                            K=40
• Strike Price, K                          50


• Expiration Time, t                       40




                            Option Price
                                           30
  – Exercisable Only At t                  20

• At Expiration                            10

  – P0=Max[0, K-S]                         0
                                                0     20     40       60     80       100
• What Is Pt?                                               Security Price



                                                                                  5
            Put-Call Parity
        C t  Ke     -rt
                            Pt  S t
• 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


                                             6
          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
                                            7
            Suppose C<S-K/(1+r)
                 At Time=0             At Time = 1
                                 S’<=K            S’>K
Buy 1 Call          -C              0             S’-K
Short 1 Share      +S              -S’             -S’
Invest Balance     S-C         (S-C)(1+r)      (S-C)(1+r)
Net CF               0       (S-C)(1+r)-S’    (S-C)(1+r)-K

 •   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)
                                                       8
                         Boundary For C
                              Call Option Boundary
                                K=40, t=1, r=6%
               50

               40
Option Price




               30

               20

               10

               0
                    20   30      40          50          60          70   80
                                          Stock Price
                                Intrinsic Value      Minimum Value

                                                                               9
          Black Scholes Formula
Option price, C t , is :
                                            rt
                   C t  S  N[d1 ]  K  e  N[d2 ], where
                    ln[ S ]  (r  0.5σ 2 )  t
              d1        K                        ,
                              σ t
                    ln[ S ]  (r - 0.5σ 2 )  t
              d2        K                      .
                             σ t
Compare this with :
                   Ct  S - K  e   - rt


                                                         10
         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
                                                     11
          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
                                           12
           Wacek’s Paper
• Black Scholes Discussion
• Product Design By Analogy




                              13
  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?



                                                             14
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”

                                                                  15
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

                                                           16
        Mildenhall’s Review
• Black Scholes Sets Price Including Risk
• Works Out No Arbitrage For Binomial Case
• Shows That Actual Price Follows Model




                                         17

								
To top