chapter 17 option pricing. 2 framework background definition and payoff some features about option...

Chapter 17 Option Pricing

2

Framework

Background Definition and payoff Some features about option strategies

One-period analysis Put-call parity, Arbitrage bound, American call option

Black-Scholes Formula Price using discount factor Derive Black-Scholes differential equation

23/4/18 Asset Pricing

Background (1)

Option; Call/Put; Strike Price Expiration Date Underlying Asset European/ American Option Payoff/Profit

3

, t TC C C , t TP P P

,tS S TS

X

T

( ,0)T TCall payoff C Max S X

( ,0)T TPut payoff P Max X S


Background (2)


5

Background (3) Some Interesting Features of Options

High Beta (High Leverage)- Trading- Hedging Shaping Distribution of Returns:- OTM Put + Stock

But Short OTM Put Option and Long Index Return Distribution Extremely Non-normal The Chance of Beating the Index for one or even five

years is extremely high, but face the catastrophe risk So what kind of investments can and cannot be made

is written in the portfolio management contracts.


6

Background (4)

Strategies By combining options of various strikes, you can buy

and sell any piece of the return distribution. A complete set of option is equivalent to complete

markets. Forming payoff that depends on the terminal stock

price in any way


7

One-period analysis

The law of one price existence of a discount factor

No arbitrage existence of positive discount factor

How to pricing option Put-Call Parity Arbitrage Bounds Discount Factors and Arbitrage Bounds Early Exercise

p E mx


8

Put-call parity

Strategies (1) hold a call, write a put ,same strike price (2) hold stock, borrow strike price X

T T TP C S X T T TC P S X

In the book of John C. Hull,

max( , )

max ,

rtT

T

C Xe S X

P S S X


Put-call parity

9

According to the Law of One Price,

applying to both sides for any m,

We get ： E m

/ fP C S X R

T T TP C S X


10

Arbitrage bounds

Portfolio A dominates portfolio B

Arbitrage portfolio , 0A B m

E mA E mB

(1) 0 0

(2) /

(3)

T

fT T

T T

C C

C S X C S X R

C S C S

T T TP C S X


11

(1) 0 0

(2) /

(3)

T

fT T

T T

C C

C S X C S X R

C S C S

C

Call valueToday

Call value in here

X/Rf SStock value

today

Arbitrage bounds


12

Discount factors and arbitrage bounds

, , , , max ,0c ct t T t t TC E m x where x S X

( )t t TS E mS

1 ( )ftE mR

0m

max ( ), . .ct t TC E mx s t

This presentation of arbitrage bound is unsettling for two reasons, First, you many worry that you will not be clever enough to dream up dominating portfolios in more complex circumstances.Second, you may worry that we have not dream up all of the arbitrage portfolios in this circumstance.


13

Discount factors and arbitrage bounds

• This is a linear program. In situations where you do not know the

answer, you can calculate arbitrage bounds.(Ritchken(1985))

• The discount factor method lets you construct the arbitrage bounds

max ( ) ( ) ( ), . .c

t Tm s

s

C s m s x s s t( ) 0

( ) ( ) ( )

1 ( ) ( )

t Ts

f

s

m s

S s m s S s

s m s R


14

Early exercise? By applying the absence of arbitrage, we can never

exercise an American call option without dividends before the expiration date.

S-X is what you get if you exercise now. the value of the call is greater than this value, because you can delay paying the strike, and exercising early loses the option value

payoff

price1fR

max ,0T T TC S X S X

C

C

/ fS X R

S X


15

Black-Scholes Formula (Standard Approach Review)

S S t S z 2

2 22

1

2

f f f ff S S t S z

S t S S

ff S

S

f

f SS

22 2

2

1

2

f fS t

t S

r t

22 2

2

1

2

f f fS t r f S t

t S S

22 2

2

1

2

f f frS S rf

t S S

1: derivative

:f

shareS

Portfolio Construction:


16

Black-Scholes Formula (Standard Approach Review)

max ,0X

E V X V X g V dV

2 / 21 2

ˆ max ,0 m sE V X e N d XN d

ˆ max ,0rtTc e E S X

2ln 2m E V s

0 1 2rt rte S e N d XN d

0 1 2rtS N d Xe N d

Where:

Risk Neutral Pricing:


17

Black-Scholes Formula (Discount Factor)

Write a process for stock and bond, then use

to price the option. the Black-Scholes formula results, (1) solve for the finite-horizon discount factor

and find the call option price by taking the expectation

(2) find a differential equation for the call option and solve it backward.

0/T

0 0 0/ cT TC E x


18

Black-Scholes Formula (Discount Factor)

The call option payoff is The underlying stock follows

The is also a money market security that pays the real interest rate

In continuous time, all such discount factors are of the form:

max ,0T TC S X

dSdt dz

S

rdt

;w

d rrdt dz dw

0E dwdz


19

Method 1: price using discount factor Use the discount factor to price the option

directly:

0 00 0

max ,0 max ,,0T TT T TTC E S X S X SdF

Where and are solutions to TS T

dSdt dz

S

;w

d rrdt dz dw

ˆ max ,0rtTc e E S X


20

How to find analytical expressions for the solutions of equations of the form (17.2)

dSdt dz

S

Y Y

dYdt dz

Y

2 22

1 1 1ln

2 2Y Y Y

dYd Y dY dt dz

Y Y

2

0 0 0

1ln

2

T T T

Y Y Y td Y dt dz

20 0

1ln ln

2T Y Y Y TY Y T z z


21

Applying the Solution to (17.2)

dSdt dz

S

;w

d rrdt dz dw

2

0

1ln ln

2T

r rr T T

20

1ln ln

2TS S T T

We get:

20 0

1ln ln

2T Y Y Y TY Y T z z

Ignoring the term ofAnd Proof Later

wdw


22

0 00

max ,0TTC E S X

0

,T

TT T T

S X

S X dF S

0T

TT

S X

S X dF

Evaluate the call option by doing the integral

0 0T T

T TT

S X S X

S dF XdF

2

21 12 2

0 ( )T

r rr T T T T

S X

e S e f d

21

2

T

r rr T T

S X

X e f d


23

221

( )2

0 0 ( )T

r rr T T

S X

C S e f d

2

1

2

T

r rr T T

S X

X e f d

2

2 21 1( )

2 2

0

1

2T

r rr T T

S X

S e d

221 1

2 21

2T

r rr T T

S X

X e d

2

1

20

1

2T

rT

S X

S e d

2

1

21

2T

rT

rt

S X

Xe e d

2(1/ 2)1

2f e


24

2(1/ 2)( )11

2 a

e d a a

20

0 0

1ln ln

2X S T

rC S T

T

20

1ln ln

2rt

X S Tr

Xe TT

20

0

1ln

2S X r T

ST

20

1ln

2rT

S X r TXe

T

20

1ln ln ln

2TX S S T T

20

1ln ln

2X S T

T


25

Proof: not Affect

0 00

max ,0TTC E S X

0

,

T

TT

S X

S X dF dF

0 0

, ,

T T

T TT

S X S X

S dF dF XdF dF

2

2 21 1 12 2 2

0 ( )w w

T

r rr T T T T T

S X

e S e f d f d

2

21 1

2 2 w w

T

r rr T T T

S X

X e f d f d

dSdt dz

S

;w

d rrdt dz dw


e d

wdw

26

2

22

1122

0 ( )w w

T

r rr T TT T

S X

S e f d e f d

2

21122 w w

T

r rr T TT T

S X

X e f d e f d

Where: 2 2 21 1 1

2 2 21

2

w w w wT T T Te f d e d

21

21

2

w Te d

This is the integral under the normal distribution, with mean of and, standard variance of 1,so the integral is 1.we multiply both sides without any change.

wT


27

Method 2:derive Black-Scholes Differential Equation Guess that solution for the call option is a

function of stock price and time to expiration, C=C(S,t). Use Ito’s lemma to find derivatives of C(S,t)

21

2t S SSdC C dt C dS C dS

2 21

2t S SS SC C S C S dt C S dz


28

0 ( ) ( ) ( )t t t tE d C CE d E dC E d dC

( ) 0t t t t tE dp E p p ( ) 0tE d p

( )d p pd dp d dp

2 210

2t S SS t SCr dt C C S C S dt E C S dz

2 21

2t t S SS S

rE rdt dz C C S C S dt C S dz

( )d C Cd dC d dC

/tE d rt

d rrdt dz

2 21

2t S SS SdC C C S C S dt C S dz

2 210

2t S SS SCr dt C C S C S dt C r S dt

2 210

2t S SS SrC C C S C S C r S


t t t t t t tC E C

29

2 210

2t S SSrC C SrC C S • This is the Black-Scholes differential equation

for the option price

22 2

2

1

2

f f frS S rf

t S S

, max ,0t T TC S T S X

22 2

2

, , ,1,

2

C S t C S t C S trC S t Sr S

t S S

• This differential equation has an analytic solution, one

standard way to solve differential equation is to guess and

check, and by taking derivatives you can check that (17.7)

does satisfy (17.8).


30

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