Introduction to Financial Derivatives

Lecture #4 on option

Jinho Bae

May 8, 2008

Ch 8. Option pricing models

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

II. Factors that affect the price of an option

I. Value of an option

• Value of an option =Option premium=Option price

• The price that an option holder pays to an option writer for the right to sell or buy an asset

• Value of an option= Intrinsic value + Time value

• When the spot price (S) exceeds the strike price (X)

Intrinsic value=S-X>0

e.g) Google call option with X=$460

Google share price S=$465

Intrinsic value=S-X=$5

I-1-1. Intrinsic value of a call option

Intrinsic value of a call option

• When the spot price (S) does not exceed the strike price (X)

Intrinsic value=0

e.g) Google call option with X=$460

Google share price S=$450

Intrinsic value=0

• Mathematical expression of intrinsic value of a call option

max(S-X, 0)• When S>X, S-X>0 take S-X • When S<X, S-X<0 take 0

Intrinsic value of a call option

valueIntrinsic value

X S

Intrinsic value of a call option

I-1-2. Intrinsic value of a put option

• When the strike price (X) exceeds the spot price (S)

Intrinsic value=X-S>0

e.g) Google put option with X=$460

Google share price S=$450

Intrinsic value=X-S=$10

Intrinsic value of a put option

• When the strike price (X) does not exceed the spot price (S)

Intrinsic value=0

e.g) Google call option with X=$460

Google share price S=$465

Intrinsic value=0

Intrinsic value of a put option

• Mathematical expression of intrinsic value of a put option

max(X-S, 0)• When X>S, X-S>0 take X-S• When X<S, X-S<0 take 0

Intrinsic value of a put option

value

Intrinsic value

X S

Relationship between intrinsic value and ITM, OTM, ATM

S>X

Call ITM

Intrinsic value >0

Put OTM

Intrinsic value=0

S=X

ATM

Intrinsic value=0

ATM

Intrinsic value=0

S<X

OTM

Intrinsic value=0

ITM

Intrinsic value >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

e.g) IBM call option with X=$100 trades at $10 IBM share price S=$106 Intrinsic value=S-X=$6 Time value= $10-$6=$4

Two elements of time value of an option

1) Time value 1: Expected payoff when holding the option until maturity

2) Time value 2: Time value associated with cash flow from selling or buying underlying asset of the option

1) Time value 1

Two scenarios of asset price movement until maturity

• Asset price moves in a favorable direction unlimited positive payoff

• Asset price moves in an unfavorable direction no or bounded loss

Expected payoff is positive.

E.g) IBM call option, X= $100, maturity=1 month

① current S=$100 (ATM)

If ST (at maturity) > $100 Payoff: ST - $100

If ST (at maturity) < $100 No loss

Expected payoff from changes in the asset price until maturity > 0

Possibilities of changes in the asset price until maturity

Price change Probability

20 increase 1/8

10 increase 2/8

0 2/8

10 decrease 2/8

20 decrease 1/8

S STProbabil

ityPayoff Expected payoff

100

1/8

2/8

2/8

2/8

1/8

② current S=$90 (OTM)

Intrinsic value=$0

If ST (at maturity) > $100 Payoff: ST - $100

If ST (at maturity) < $100 No loss

S STProbabi

lityPayoff Expected

payoff

90

1/8

2/8

2/8

2/8

1/8

Expected payoff Greater than 0. However, smaller than that for ATM. Why?

③ current S=$110 (ITM)

Intrinsic value =$10

If asset price increases above 110 Payoff increases proportionally

If asset price increases below 110, intrinsic value decreases but bounded from 10.

S STProbabil

ityPayoff Expected

payoff

110

1/8

2/8

2/8

2/8

1/8

Expected payoff Greater than 0. However, smaller than that for ATM.

Time value 1 of a call option

X SCurrent spot price

value

Time value 1

OTM ATM

Time value 1 of a put option

X SCurrent spot price

value

Time value 1

ATM OTM