金融衍生品定价理论(期权定价)2

金融衍生品定价理论(期权定价)

金融衍生品定价理论(期权定价)

Chapter 2 Arbitrage-Free Principle

Robert C. Merton

金融衍生品定价理论(期权定价)

Financial Market Two Kinds of Assets Risk

free assetasset

Bond Stocks Options ….

Risky

Portfolio – an investment strategy to

hold different assets

金融衍生品定价理论(期权定价)

Investment At time 0, invest S

When t=T, Payoff = ST S0 Return = ( ST S0 ) / S0 For a risky asset, the return is uncertain,

i.e.,

S is a random variable

金融衍生品定价理论(期权定价)

A Portfolio a risk-free asset B

n risky assets

a portfolio B i Si ,i 1

Si Sit , i 1,...nn

, 1 ,... n is called a investment strategy on time t, wealth:

, i portion of the cor. Asset

Vt ( ) t t Bt it Siti 1

n

金融衍生品定价理论(期权定价)

Arbitrage Opportunity Self-financing - during [0, T]

no add or withdraw fund Arbitrage Opportunity - A self-financing investment,

T * (0, T ], s.t. V0 ( ) 0,VT * ( ) 0and Probability Prob VT * ( ) 0 0.

金融衍生品定价理论(期权定价)

Arbitrage Free Theorem Theorem 2.1 the market is arbitrage-free in time [0, T], 1 , 2 are any 2 portfolios satisfyingVT ( 1 ) VT ( 2 ),

& Prob{VT ( 1 ) VT ( 2 )} 0 t [0, T ),Vt ( 1 ) Vt ( 2 ).

金融衍生品定价理论(期权定价)

Proof of Theorem Suppose false, i.e., t * [0, T ), s.t.Vt* ( 1 ) Vt* ( 2 ) Denote E Vt ( 2 ) Vt ( 1 ) 0* *

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

金融衍生品定价理论(期权定价)

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

VT ( c ) E[1 r (T t )] 0,*

金融衍生品定价理论(期权定价)

Proof of Theorem cont. It follows

Prob VT ( c ) 0 Prob VT ( 1 ) VT ( 2 ) 0 0 There is an Arbitrage Opportunity, Contradiction!

金融衍生品定价理论(期权定价)

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 ).

金融衍生品定价理论(期权定价)

Proof of Corollary Consider c 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)

金融衍生品定价理论(期权定价)

Proof of Corollary 2.1 0, Vt ( 1 ) Vt ( 2 ). In the same wayVt ( 1 ) Vt ( 2 ) Then

Vt ( 1 ) Vt ( 2 ), t [0, T ] Corollary has been proved.

金融衍生品定价理论(期权定价)

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

Strike Price

金融衍生品定价理论(期权定价)

Assumptions1. The market is arbitrage-fre

e

2. All transactions are free of charge3. The risk-free interest rate r is a

constant 4. The underlying asset pays no dividends

金融衍生品定价理论(期权定价)

Notations St

ctpt 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.