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Table 10.3: Working Backwards in the Tree

Table 10.3: Working Backwards in the Tree

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Step 3:Work Backwards in the Tree

to Get the Current Option Value



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• 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



Step 3:Work Backwards in the Tree to Get

the Current Option Value



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• At point C we have instead that



• So that

• This means we have to hold approximately 0.3 shares for

each call option we sell

• This in turn gives a portfolio value at E (or F) of

• The present value of this is

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



Step 3:Work Backwards in the Tree to

Get the Current Option Value

• At point C we therefore have the call option value



• which is also found in green at point C in Table 10.3

• Now that we have the option prices at points B and C we

can construct a risk-free portfolio again to get the option

price at point A. We get

• which implies that



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



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Step 3:Work Backwards in the Tree to Get

the Current Option Value

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• which gives a portfolio value at B (or C) of



• with a present value of

• which in turn gives the binomial call option value of

• which matches the value in Table 10.3

• Once the European call option value has been computed,

the put option values can also simply be computed using

the put-call parity

• The put values are provided in red font in Table 10.3

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



Risk Neutral Valuation

• We have constructed a risk-free portfolio that in the absence of

arbitrage must earn exactly risk-free rate

• From this portfolio we can back out European option prices

• For example, for a call option at point B we used the formula



• which we used to find the call option price at point B

using the relationship

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



26



Risk Neutral Valuation



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• Using the ∆ B formula we can rewrite CallB formula as



• where the risk neutral probability of an up move is defined

as



• RNP is termed a risk-neutral probability because the CallB

price appears as a discounted expected value when using

RNP in the expectation

• RNP can be viewed as the probability of an up move in a

world where investors are risk neutral

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



Risk Neutral Valuation

• In our example

• So that



• We can use this number to check that the new formula

works. We get



• just as when using the no-arbitrage argument

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



28



Risk Neutral Valuation



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• The new formula can be used at any point in the tree

• For example at point A we have



• It can also be used for European puts

• We have for a put at point C



• Note that whereas ∆ changes values throughout the tree,

RNP is constant throughout the tree

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



Pricing an American Option using

the Binomial Tree

• American options can be exercised prior to maturity

• This added flexibility gives them potentially higher fair market

values than European-style options

• Binomial trees can be used to price American options

• At the maturity of the option American- and European-style

options are equivalent

• But at each intermediate point in the tree we must compare

European option value with early exercise value and put the

largest of two into tree at that point



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



30



Pricing an American Option using

the Binomial Tree



31



• Let us price an American option that has a strike price of 1,100 but

otherwise is exactly the same as the European option considered

before

• If we exercise American put option at point C we get



• We have the risk-neutral probability of an up-move RNP =

0.4618 from before

• So that the European put value at point C is



• which is lower than the early exercise value $291.14

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



Pricing an American Option using

the Binomial Tree



32



• Early exercise of the put is optimal at point C as fair market value

of the American option is $291.14 at C

• This value will now influence the American put option value at

point A, which will also be larger than its corresponding European

put option value.

• Table 10.4 shows that the American put is worth $180.25 at point A

• The American call option price is $90.25, which turns out to be the

European call option price as well

• American call stock options should only be exercised early if a

large cash dividend is imminent



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



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Table 10.3: Working Backwards in the Tree

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