Microsoft PowerPoint - PoF.ppt

The replicating portfolioƒ 203

Consider a portfolio of stock (number of stocks =

'

) and

money market account (amount invested at the risk-free rate = B).

ƒ

If the portfolio is to replicate the payoff to the derivative, it must be that

'

0

S

1

+ (1+r) B

0

= V

, i.p. 1

(i)

'

0

S

(H) + (1+r) B 1

0

= V

(H), i.e. 1

'

0

uS

0

+ (1+r) B

0

= V

(H) and 1

(ii)

'

0

S

(T) + (1+r) B 1

0

= V

(T), i.e. 1

'

0

dS

0

+ (1+r) B

0

= V

(T). 1

... Two equations, two unknowns (

'

0

and B

). 0

Note: B

0

= X

0

• 
  

'

0

S

0

Ä

We begin with wealth X

0

and buy

'

0

shares of

stock at time zero, leaving us with a cash position B

0

= X

0

• 
  

'

0

S

. 0

The value of our portfolio of stock and money market account at time one is given by the wealth equations (i) and (ii).

Derivative securities: Options - Binomial asset pricing model