Figure 10.1: Payoff as a Function of the Value of the Underlying Asset at Maturity Call Option, Put Option, Underlying Asset, Risk-Free Bond
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Basic Definitions
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• Put-call parity does not rely on any particular option pricing
model. It states
• It can be derived from considering two portfolios:
• One consists of underlying asset and put option and
another consists of call option, and a cash position equal to
the discounted value of the strike price.
• Whether underlying asset price at maturity,
ends up
below or above strike price X; both portfolios will have
same value, namely
, at maturity
• Therefore they must have same value today, otherwise
arbitrage opportunities would exist
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Basic Definitions
The portfolio values underlying this argument are shown in the
following
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Basic Definitions
9
• put-call parity suggests how options can be used in risk
management
• Suppose an investor who has an investment horizon of days owns
a stock with current value St
• Value of the stock at maturity of the option is
which in the worst
case could be zero
• An investor who owns the stock along with a put option with a
strike price of X is guaranteed the future portfolio value
, which is at least X
• The protection is not free however as buying the put option requires
paying the current put option price
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Option Pricing Using Binomial Trees
• We begin by assuming that the distribution of the future price of the
underlying risky asset is binomial
• This means that in a short interval of time, the stock price can only
take on one of two values, up and down
• Binomial tree approach is able to compute the fair market value of
American options, which are complicated because early exercise is
possible
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Option Pricing Using Binomial Trees
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• The binomial tree option pricing method will be illustrated using
the following example:
• We want to find the fair value of a call and a put option with three
months to maturity
• Strike price of $900.
• The current price of the underlying stock is $1,000
• The volatility of the log return on the stock is 0.60 or 60% per year
corresponding to
per calendar day
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Step 1: Build the Tree for the Stock Price
• In our example we will assume that the tree has two steps during
the three-month maturity of the option
• In practice, a hundred or so steps will be used
• The more steps we use, the more accurate the model price will be
• If the option has three months to maturity and we are building a
tree with two steps then each step in the tree corresponds to 1.5
months
• The magnitude of up and down move in each step reflect a
volatility of
• dt denotes the length (in years) of a step in the tree
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Step 1: Build the Tree for the Stock Price
• Because we are using log returns a one standard deviation up
move corresponds to a gross return of
• A one standard deviation down move corresponds to a
gross return of
• Using these up and down factors the tree is built as seen in
Table 10.1, from current price of $1,000 on the left side to
three potential values in three months
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Table 10.1: Building the Binomial Tree Forward
from the Current Stock Price
Market Variables
D
St=
1000
Annual r =
0.05
1528.47
Contract Terms
X=
1100
B
T=
0.25
1236.31
Parameters
Annual Vol=
tree steps
0.6
2
dt=
0.125
u=
1.236311
d=
0.808858
A
E
1000.00
1000.00
C
808.86
F
654.25
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Step 2: Compute the Option Pay-off at
Maturity
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• From the tree, we have three hypothetical stock price values at
maturity and we can easily compute the hypothetical call option at
each one.
• The value of an option at maturity is just the payoff stated in the
option contract
• The payoff function for a call option is
• For the three terminal points in the tree in Table 10.1,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen