Introduction to Financial Derivatives

Introduction to Financial Derivatives

Lecture #5 on option

Jinho Bae

May 27, 2008

Ch 8. Option pricing models

I. Value of an option– Intrinsic value – Time value

• Time value 1• Time value 2

II. Factors that affect the price of an option

I. Value of an option

• Value of an option= Intrinsic value + Time value

• Intrinsic value of an option– Call option

max (S-X, 0)– Put option

max (X-S, 0)

I-2. Time value of an option

• The value of an option arising from the time left to maturity

• Time value = Option premium - Intrinsic value

• Time value 1: Expected payoff from holding the option until maturity

Time value 1 of a call option

Current spot price

value

Time value 1

OTM ITM

X S

Time value 1 of a put option

value

Time value 1

ITM OTM

X S

2) Time value 2• Time value associated with cash flow arising from

option writer’s selling or buying underlying asset of the option

• Call (put) option writers buy (sell) underlying asset when they write the option

• Why do they do so?• What is the cash flow?

• Why do call option writers buy the asset when they write the option?

• When a call option is exercised, call option writer is required to sell an underlying asset to option holder

• In preparation for this, the writer buys the underlying asset

Time value 2 of a call option

Cash flow associated with the purchase of the asset

• Option writer pays the asset price• He/she gives up interest that the money could

earn until maturity • Why? • If he/she does not write the option, the writer

need not pay the price• Thus, the interest is opportunity cost of writing

option• Option writer adds the opportunity cost to the

option price

Size of the interest• Depends on the amount of the asset that is purchased• How many underlying assets should the writer buy?• The amount that is purchased depends on the probability

that the option is exercised– If the probability is low, buy only a small portion of option

positions– If the probability is high, buy a large portion of option positions– If the probability is 1, buy as many as option positions

• What determines the probability?• The current underlying asset price S relative to the

exercise price X– If S ≫ X (deep ITM), the probability is very high very large

interest– If S > X (shallow ITM), the probability is high large interest– If S < X (shallow OTM), the probability is low small interest– If S ≪X (deep OTM), the probability is very low very small

interest

Time value 2 of a call option

Time value 2

X S

Value

OTM ITM

Time value1 and 2 of a call option

Time value1Time value2

X S

Value

OTM ITM

Total time value of a call option

Time value

X S

value

OTM ITM

Characteristics of time value of a call option

1. Not symmetric around X– Bigger when ITM than when OTM– Due to asymmetry of interest cost

2. Always non-zero

Value of a call option (summary)

X S

valueTime value

Intrinsic value

• When a put option is exercised, the option writer is required to buy the asset

• In preparation for this, the writer short-sells the underlying asset when writing the option

Time value 2 of a put option

• Option writer short-sells the underlying asset

• The proceeds earn interest until maturity

• If he/she did not write the option, the writer would not earn the interest

• Option writer lowers the option price by the interest

• Time value 2 of a put option is negative

Size of the interest• Depends on the amount of the asset that is

short-sold• The amount is determined by the probability of

exercising a put option• The probability depends on S

– If S≪X (deep ITM), very high probability – If S < X (shallow ITM), high probability – If S > X (shallow OTM), low probability – If S≫X (deep OTM), very low probability

• The higher the probability, the larger portion of option positions the writer short-sells, and the larger the interest

Time value2

X

Time value 2 of a put option

S

Value

OTMITM

X

Time value1 and 2 of a put option

Time value2

Time value1

S

value

OTMITM

Total time value of a put option

Time value

X S

value

OTMITM

Characteristics of time value of a put option

1. Not symmetric around X– Bigger when OTM than when ITM– This is due to asymmetry in time value 2

2. Total time value can become negative when time value 2 exceeds time value 1 at very deep ATM

3. Overall time value of a call option is slightly larger than that of a put option

• For call options, interest is included in time value

• For put options, interest is excluded from time value

Value of a put option (summary)

X S

Value

II. Factors that affect the price of an option

1) Underlying asset price S

2) Exercise price X

3) Time left to maturity

4) Variability of underlying asset price

5) Interest rate

1) Underlying asset price S

Other things being equal,

as S increases,

• call option price rises

• Put option price falls

Underlying asset price and call option price

Value

X S

KOSPI200 C 200806 220

S c

5/26/2008 230.43 12.40

5/27/2008 233.15 14.15

Underlying asset price and put option price

Value

X S

KOSPI200 P 200806 220

S p

5/26/2008 230.43 1.27

5/27/2008 233.15 0.85(12:15 pm)

2) Exercise price X

Other things being equal,

as X increases,

• Call option price decreases since its intrinsic value decreases

• Put option price increases since its intrinsic value increases

KOSPI200 C 200806

X c

210 23.15

220 14.15

230 6.35

(as of 12:16 pm on 5/27/2008)

KOSPI200 P 200806

X p

210 0.19

220 0.86

230 3.05

(as of 12:16 pm on 5/27/2008)

3) Time to maturity

Other things being equal,

• The longer the time to maturity, the larger option prices get

• As we get closer to maturity, option prices fall

4) Variability of underlying asset price S

Other things being equal,

as S get more variable, option becomes more expensive

Variability and call option price

Low variability

High variabilityValue

X S

Variability and put option price

Value

X S

Low variability

High variability

5) Interest rate

① interest rate is related to option price through time value 2

Other things being equal, as interest rate rises,

• Interest cost rises call option price rises

• Interest earning rises put option price falls

② interest rate is related to option price through underlying asset price S

Other things being equal, as interest rate rises,

S falls

call option price falls,

put option price rises