金融衍生品定价理论(期权定价)2

i.e.
ct Ke
r (T t )
St .
Proof of Theorem 2.2 cont. cont.
Now consider a European call option c Since V (c) ( S - K ) 0 , T T and By Theorem 2.1 when t<T
hold different assets
Investment
At time 0, invest S
When t=T, Payoff = ST S0 Return = ( ST S0 ) / S0
For a risky asset, the return is uncertain,

.
Proof of Theorem 2.2
lower bound of
consider

ct (upper leaves to ex.)
two portfolios at t=0:
1 1 E call Option+Bond B of Ke rT 2 1 share
VT (1 ) VT (c) VT ( Ke rT ) ( ST - K ) ( Ke rT )e rT ST , ST K , ( ST - K ) K K , ST K ;
there holds call-put parity
ct Ke
r (T t )
pt St
Proof of Theorem 2.3
2 portfolios when t=0 rT 1 c Ke , 2 p S when t=T
VT (1 ) VT (c) VT ( Ke rT ) ( ST K ) K max K , ST ,

VT ( 2 ) VT ( p) VT ( S ) ( K ST ) ST max K , ST .

Proof of Theorem 2.3 cont.
So that
VT (1 ) VT ( 2 )
By Corollary 2.1
Vt (1 ) Vt ( 2 ), t T ,
Proof of Corollary 2.1
0, Vt (1 ) Vt ( 2 ). In the same way
Vt (1 ) Vt ( 2 )
Then
源自文库
Vt (1 ) Vt ( 2 ), t [0, T ]
Corollary has been proved.
Prob VT (c) 0 Prob ST - K 0 0,
Vt (c) 0,
i.e. ct 0. Together with last inequality, 2.2 proved.
Theorem 2.3
For European Option pricing,
VT ( c ) E[1 r (T t )] 0,
*
Proof of Theorem cont.
It follows
Prob VT ( c ) 0 Prob VT (1 ) VT ( 2 ) 0 0
There is an Arbitrage Opportunity, Contradiction!
i.e.,
S is a random variable
A Portfolio
a risk-free asset B
n risky assets
a portfolio B i Si ,
i 1
Si Sit , i 1,...n
n
, 1 ,...n is called a investment strategy
Proof of Theorem cont.
r – risk free interest rate, at t=T
VT ( c ) VT (1 ) VT ( 2 ) {E / Bt* }VT ( B) Then * * VT ( B) Vt* ( B)[1 r (T - t )] Bt* [1 r (T - t )] From the supposition
Considerc 1 2 B Then VT ( c ) VT ( B) 0
By Theorem, for t [0, T ],
Vt (c ) Vt (1 ) Vt ( 2 ) Vt ( B) 0
Namely Vt (1 ) Vt ( 2 ) Vt ( B)
VT (1 ) VT ( 2 ),
& Prob{VT (1 ) VT ( 2 )} 0
t [0, T ),
Vt (1 ) Vt ( 2 ).
Proof of Theorem
Suppose false, i.e., t * [0, T ), s.t.Vt* (1 ) Vt* ( 2 ) Denote E Vt ( 2 ) Vt (1 ) 0
T * (0, T ], s.t. V0 () 0,VT * () 0
and Probability Prob VT * ( ) 0 0.


Arbitrage Free Theorem
Theorem 2.1 the market is arbitrage-free in time [0, T], 1 , 2 are any 2 portfolios satisfying
Strike Price
Assumptions
1. The market is arbitrage-free
2. All transactions are free of charge
3. The risk-free interest rate r is a
constant 4. The underlying asset pays no dividends
* *
B is a risk-free bond satisfying Bt* Vt* ( B)
Construct a portfolio c at t t *
c =1 2 + E / Bt* B


Vt* ( c ) Vt* (1 ) Vt* ( 2 ) {E / Bt* }Vt* ( B) 0

Proof of Theorem 2.2 cont.
At t=T,
and
VT (1 ) ST VT ( 2 ),
Prob VT (1 ) VT ( 2 ) Prob K - ST 0 0.
By Theorem 2.1
t [0, T ], Vt (1 ) Vt ( 2 ),
American Option v.s. European Option
For an American option and a European
option with the same expiration date T and the same strike price K, since the American option can be early exercised, its gaining opportunity must be >= that of the European option. Therefore
Theorem 2.2
For European option pricing, the
following valuations are true:
( St - Ke
( Ke
r (T t )
r (T t )
) ct St ,
r (T t )
St ) pt Ke
i.e. call-put parity holds
Theorem 2.4
For American option pricing,
if the market is arbitrage-free, then t [0, T ]
Ct ( St K ) Pt ( K St )
Corollary 2.1
Market is arbitrage free
if portfolios 1 & 2
satisfying
VT (1 ) VT ( 2 ),
then for any t [0, T ],
Vt (1 ) Vt ( 2 ).
Proof of Corollary


Proof of Theorem 2.4
Take American call option as example. Suppose not true, i.e.,t [0, T ) s.t Ct St - K At time t, take cash Ct to buy the American call
Notations
St

ct
pt Ct Pt
K T r
------ the risky asset price, ------ European call option price, ------ European put option price, ------ American call option price, ------ American put option price, ------ the option's strike price, ------ the option's expiration date, ------ the risk-free interest rate.
Chapter 2 Arbitrage-Free Principle
Robert C. Merton
Financial Market
Two Kinds of Assets
Risk

free asset
asset
Bond Stocks Options ….
Risky

Portfolio – an investment strategy to
Option Pricing
European Option Pricing
Call-Put Parity for European Option
American Option Pricing
Early Exercise for American Option
Dependence of Option Pricing on the


option and exercise it, i.e., to buy the stock S with cash K, then sell the stock in the stock market to receive S t in cash. St Ct K 0, Thus the trader gains a riskless profit instantly. But this is impossible since the market is assumed to be arbitrage-free. Therefore, Ct (St - K ) must be true. Pt (K St ) can be proved similarly.
on time t, wealth:
, i portion of the cor. Asset
Vt () t t Bt it Sit
i 1
n
Arbitrage Opportunity
Self-financing - during [0, T]
no add or withdraw fund Arbitrage Opportunity - A self-financing investment,