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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

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



7



• 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



8



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



10



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



11



• 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



12



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



13



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



14



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



15



• 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



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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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