Financial Risk Management of Insurance Enterprises

Options

What is an Option Contract?• Options provide the right, but not the obligation, to buy or

sell an asset at a fixed price– Call option is right to buy

– Put option is right to sell

• Key distinction between forwards, futures and swaps and options is performance– Only option sellers (writers) are required to perform under the

contract (if exercised)

– After paying the premium, option owner has no duties under the contract

Some Terminology

• The exercise or strike price is the agreed on fixed price at which the option holder can buy or sell the underlying asset

• Exercising the option means to force the seller to perform– Make option writer sell if a call, or force writer to

buy if a put

• Expiration date is the date at which the option ceases to exist

More Terminology

• American options allow holder to exercise at any point until expiration

• European option only allows holder to exercise on the expiration date

• The premium is the amount paid for an option

A Simple Example• Suppose PCLife owns a European call option on IBM

stock with an exercise price of $100 and an expiration date of 3 months

• If in 3 months, the price of IBM stock is $120, PCLife exercises the option– PCLife’s gain is $20

• If at the expiration date the price of IBM is $95, PCLife lets the option expire unexercised

• If the price of IBM in one month is $3,000, PCLife will not exercise (Why not?)

Option Valuation Basics• Two components of option value

– Intrinsic value– Time value

• Intrinsic value is based on the difference between the exercise price and the current asset value (from the owner’s point of view)– For calls, max(S-X,0) X= exercise price– For puts, max(X-S,0) S=current asset value

• Time value reflects the possibility that the intrinsic value may increase over time– Longer time to maturity, the higher the time value

In-the-Moneyness

• If the intrinsic value is greater than zero, the option is called “in-the-money”– It is better to exercise than to let expire

• If the asset value is near the exercise price, it is called “near-the-money” or “at-the-money”

• If the exercise price is unfavorable to the option owner, it is “out-of-the-money”

Basic Option Value: Calls

• At maturity– If X>S, option expires

worthless

– If S>X, option value is S-X

• Read call options left to right– Only affects payoffs to

the right of X

Call Value at Maturity(Long Position)

X

Asset Value

Cal

l Val

ue

Basic Option Value: Calls (p.2)

• Of course, for the option writer, the payoff at maturity is the mirror image of the call option owner

Call Value at Maturity(Short Position)

X

Asset Value

Cal

l Val

ue

Basic Option Values: Puts

• At maturity– If S>X, option expires

worthless

– If X>S, option value is X-S

• Read put options right to left– Only affects payoffs to

the left of X

Put Option(Long Position)

X

Asset Value

Pu

t V

alu

e

Combining Options and Underlying Securities

• Call options, put options and positions in the underlying securities can be combined to generate specific payoff patterns

Payoff Diagram ExampleName two options strategies used

to get the following payoff

Long Straddle

-10

-5

0

5

10

15

20

25

10 20 30 40 50

Asset Value

Stra

ddle

Val

ue

Payoff Diagram Example

• Reading with calls (left to right)– Buy one call with X=10– Sell two calls with X=30– Buy one call with X=50

• Reading with puts (right to left)– Buy one put with X=50– Sell two puts with X=30– Buy one put with X=10

Determinants of Call Value• Value must be positive

• Increasing maturity increases value

• Increasing exercise price, decreases value

• American call value must be at least the value of European call

• Value must be at least intrinsic value

• For non-dividend paying stock, value exceeds S-PV(X)– Can be seen by assuming European style call

Determinants of Call Value (p.2)

• As interest rates increase, call value increases– This is true even if there are dividends

• As the volatility of the price of the underlying asset increases, the probability that the option ends up in-the-money increases

Thus, )C C S X T r

( , , , ,

Put-Call Parity• Consider two portfolios

– One European call option plus cash of PV(X)– One share of stock plus a European put

• Note that at maturity, these portfolios are equivalent regardless of value of S

• Since the options are European, these portfolios always have the same value– If not, there is an arbitrage opportunity (Why?)

Therefore, C PV X P S ( )

Fisher Black and Myron Scholes• Developed a model to value European options on stock• Assumptions

– No dividends– No taxes or transaction costs– One constant interest rate for borrowing or lending – Unlimited short selling allowed– Continuous markets– Distribution of terminal stock returns is lognormal

• Based on arbitrage portfolio containing stock and call options

• Required continuous rebalancing

Black-Scholes Option Pricing Model

C = Price of a call option

S = Current price of the asset

X = Exercise price

r = Risk free interest rate

t = Time to expiration of the option

= Volatility of the stock price

N = Normal distribution function

)()( 21 dNrtXedSNC

2/112

2/121 /])2/()/[ln(

tdd

ttrXSd

Using the Black-Scholes Model

• Only variables required– Underlying stock price

– Exercise price

– Time to expiration

– Volatility of stock price

– Risk-free interest rate

Example

• Calculate the value of a call option with– Stock price = $18– Exercise price = $20– Time to expiration = 1 year– Standard deviation of stock returns = .20– Risk-free rate = 5%

Answer

02.1

)3532)(.9512(.20)4298(.18

)3768.(20)1768.(18

3768.)1(2.1768.

1768.))1(2(.

1))2(.5.05(.)20

18ln(

)1(05.

5.2

5.

2

1

C

C

NeNC

d

d

Use of Options

• Options give users the ability to hedge downside risk but still allow them to keep upside potential

• This is done by combining the underlying asset with the option strategies

• Net position puts a floor on asset values or a ceiling on expenses

Hedging Commodity Price Risk with Options

• P/C insurer pays part of its claims for replacing copper plumbing

• Instead of locking in a fixed price using futures or swaps, the insurer wants to get a lower price if copper prices drop

• Insurer can buy call options to protect against increasing copper prices

• If copper prices increase, gain in option offsets higher copper price

Hedging Copper Prices

0

5

10

15

20

25

30

35

40

5 15 25 35

Copper Price

Pri

ce P

aid

CallCopperNet

Additional Uses of Options

• Interest rate risk

• Currency risk

• Equity risk– Market risk– Individual securities

• Catastrophe risk

Next Lecture

• Combining the building blocks with each other to create new risk management products

• Combining the building blocks with debt or equity to create hybrid securities