# Option Pricing And Insurance Pricing by liuhongmei

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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
– 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

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