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
•
•
•
•
•
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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