6.期权定价与动态套利均衡分析 清华大学绝版金融工程课件

CHAPTER FIVE: Options and Dynamic No-Arbitrage

A Brief Introduction of OptionsAn option is the right of choice exercised in future. The holder (buyer, or longer) of the option has a right but not an obligation to buy or sell a special amount of the asset with a special quality at a predetermined price. Call and put

Exercise price X Expiration date T American options (C and P) vs. European options ( c and p) 2

— A Brief Introduction of Options (Cont.)The payoff profiles of call and putCallLong+0 _

PutShort Long+ 0

Short+

+

ST 0 X_

X

ST_

ST 0 X_

X

ST

In-the-money, out-of-the-money, at-the-money, intrinsic value and time value3

The Basic No-Arbitrage1) 2) 3) 4) If

c t C t S t p t Xe rf T t

,

p t P t X

C t c t 0 T1 t T2 t

,

P t p t 0 C1 t C2 t ,

, then

P t P2 t 1

5)

C t max S t X ,0 c T max S T X ,0 P t max X S t ,0 p T max X S T ,0 4

The Basic No-Arbitrage (Cont.)The underlying is a non-dividend-paying stockc t max S t Xe

rf T t

,0

Suppose c t S t XeArbitrage Position

rf T t

, thenCash Flow on the expired date S T

Immediate Cash Flow

Short a stockLong an European call Long riskless security Net cash flows Arbitrage Opportunity

S t c t

max S T X ,0

XeS t Xe

rf T t

X

rf T t

c t max S T X ,0 S T X

!

0

0

The Basic No-Arbitrage (Cont.) r T t max t Xe S ,0 c t S t f

T

e

rf T t

0

c t S t

PropositionIf the period to expiration is very long, the value of an European call is almost equal to its underlying.

C t max S t Xe

rf T t

,0 max S t X ,0

PropositionAn American call on a non-dividend-paying stock should never be exercised prior to the expiration date.

C t c t

The relationship between American options and European optionsp t max Xe

rf T t

S t ,0

P t max Xe

rf T t

S t ,0

P t max X S t ,0 ? P t max X S t ,0 Conclusion:

C t c t

and

P t p t 7

The Parity of Call and Put The underlying is a non-dividend-paying stock

S t c t p t Xe rf T t

rf T t

S can be replicated by c, p and riskless security Suppose S t c t p t

XePositionBuy a share

Arbitrage!

Cash flow attime t

Cash flow at time T

Short a callLong a put Short treasury

S t c t p t

S t XS T

0X S T

S T S T X

S t X

Xe

rf T t

0 X

X

Net cash flow S t c t p t Xe rf T t

0

0

Relationship between exercise and forward priceS t Fe rf T t

F X F X F X

c t p t c t p t c t p t

Non-dividend-paying stock’s American call and putC t c t P t p t

S t C t P t Xe C t P t S t Xe

rf T t

rf T t

C t P t S t X ?9

Non-dividend-paying stock’s American call and put (Cont.) To Prove C t P t S t X t t TPosition Short a share Long an Amer. call Cash flow at Cash flow at time t when put exercised

time t

S t X S t

S t

Short an Amer. putLong treasury Net cash flow

C t P t

X S t C t P t X 0 rf T t

X S t r f t t

C t

S t X S t C t 0

Xe

Xe f

r t t

X C t

Xe fXe fr t t

r t t

C t S t

C t max S t Xe

,0 S t Xe

rf T t

0

Xe f

r t t

Xe

rf T t

0

Non-dividend-paying stock’s American call and put (Cont.)

S t X C t P t S t Xe Underlying is dividend-paying stockPresent value of a long stock forward position

rf T t

C t c t S t PV D Xe P t p t Xe r f T t

r f T t

S t PV D

Present value of dividends at time t

Present value of a short stock forward position11

Underlying is dividend-paying stockFor European call and put

S t c t p t XeFor American call and put

rf T t

PV D rf T t

Proved!

S t PV D X C t P t S t XeHow to prove it? Dividend paid

C t P t

Holds for nondividend-paying stock underlying

Please see the next page!12

Proof ofCash flow at time t

C t c t

Position

Cash flow at time t when put exercised

S t X

S t Effect of dividends PV T t D c t Long an Euro. callShort a share Short an Amer. put Long treasury

S t PV T t D ct X S t

S t

X

P t

S t PV T t D ct

X c t P t S t PV t t D X

Xe fXe fr t t

r t t

0

0

Xe f

r t t

Net cash flow

0

PV T t D Xe fr t t

X c t

PV T t D r t t

Xe f

r t t

c t S t

0 Xe rf t t

c t S t PV D Xe

rf T t

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