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Table 10.2: Computing the Hypothetical Option Payoffs at Maturity

# Table 10.2: Computing the Hypothetical Option Payoffs at Maturity

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Step 3:Work Backwards in the Tree to

Get the Current Option Value

18

Stock price at B = \$1,236.31 and at C = \$808.86

We need to compute a option value at B and C

Going forward from B the stock can only move to either D or E

We know the stock price and option price at D and E

We also need the return on a risk-free bond with 1.5 months to

maturity

• The term structure of government debt can be used to obtain this

information

• Let us assume that the term structure of interest rates is flat at 5%

per year

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Step 3:Work Backwards in the Tree to

Get the Current Option Value

19

• Key insight is that in a binomial tree we are able to construct a

risk-free portfolio using stock and option

• Our portfolio is risk-free and it must earn exactly the risk-free rate,

which is 5% per year in our example

• Consider a portfolio of 1 call option and ∆ B shares of the stock

• We need to find a ∆ B such that the portfolio of the option and the

stock is risk-free

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Step 3:Work Backwards in the Tree

to Get the Current Option Value

20

• Starting from point B we need to find a ∆ B so that

• which in this case gives

• which implies that

• So, we must hold one stock along with the short position

of one option for the portfolio to be risk-free

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

21

Table 10.3: Working Backwards in the Tree

Market Variables

St=

1000

D

Annual r =

0.05

Contract Terms

X=

T=

1528.47

628.47

0.00

900

0.25

Parameters

Annual Vol=

tree steps =

dt=

u=

d=

RNP =

Stock is black

Call is green

Put is red

0.6

2

0.125

1.236311

0.808858

0.461832

B

1236.31

341.92

0.00

A

1000.00

181.47

70.29

E

1000.00

100.00

0.00

C

808.86

45.90

131.43

F

654.25

0.00

245.75

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Step 3:Work Backwards in the Tree

to Get the Current Option Value

22

• The value of this portfolio at D (or E) is \$900 and the portfolio

value at B is the discounted value using the risk-free rate for 1.5

months, which is

• The stock is worth \$1,236.31 at B and so the option must

be worth

• which corresponds to the value in green at point B in Table

10.3

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

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Table 10.2: Computing the Hypothetical Option Payoffs at Maturity

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