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