Table 10.3: Working Backwards in the Tree
Tải bản đầy đủ - 0trang
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
Step 3:Work Backwards in the Tree to Get
the Current Option Value
23
• 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
24
Step 3:Work Backwards in the Tree to Get
the Current Option Value
25
• 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
27
• 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
29
• 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