3 Step 3: Work Backward in the Tree to Get the Current Option Value
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Further Topics in Risk Management
Table 10.3 Working backwards in the tree
Market Variables
St =
Annual rf =
1000
0.05
Contract Terms
X=
T=
900
0.25
Parameters
Annual Vol =
tree steps =
dt =
u=
d=
RNP =
0.6
2
0.125
1.23631111
0.808857893
0.461832245
D
1528.47
628.47
0.00
B
1236.31
341.92
0.00
A
1000.00
181.47
70.29
Stock is black
Call is green
Put is red
E
1000.00
100.00
0.00
C
808.86
45.90
131.43
F
654.25
0.00
245.75
Notes: We compute the call and put option values at points B, C, and A using the no-arbitrage
principle.
At point C we have instead that
1000 ·
C
− 100 = 654.25 ·
C
−0
so that
C
=
100 − 0
= 0.2892
345.75
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 1000 · 0.2892 − 100 = $189.20. The
present value of this is
189.20 · exp(−0.05 · (3/12) /2) = $188.02
At point C we therefore have the call option value
CallC = 0.2892 · 808.86 − 188.02 = $45.90
which is also found in green at point C in Table 10.3.
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227
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
1236.31 ·
A − 341.92 = 808.86 ·
A − 45.90
which implies that
A
=
341.92 − 45.90
= 0.6925
1236.31 − 808.86
which gives a portfolio value at B (or C) of 808.86 · 0.6925 − 45.90 = $514.24 with a
present value of
514.24 · exp(−0.05 · (3/12) /2) = $511.04
which in turn gives the binomial call option value of
cBin = CallA = 0.6925 · 1000 − 511.04 = $181.47
which matches the value in Table 10.3. The same computations can be done for a
put option. The values are provided in red font 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 provided earlier.
3.4 Risk Neutral Valuation
Earlier we priced options based on no-arbitrage arguments: We have constructed a
risk-free portfolio that in the absence of arbitrage must earn exactly the 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
B
=
CallD − CallE
CallD − CallE
=
SD − SE
SB u − SB d
which we used to find the call option price at point B using the relationship
CallB = SB
Using the
B
B − (SB u
B − CallD ) exp
−rf · dt
formula we can rewrite the CallB formula as
CallB = [RNP · CallD + (1 − RNP) · CallE ] exp −rf · dt
where the so-called risk neutral probability of an up move is defined as
RNP =
exp rf · dt − d
u−d
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Further Topics in Risk Management
where dt is defined as before. RNP can be viewed as a probability because the term
inside the [∗] in the CallB formula has the form of an expectation of a binomial variable. RNP is termed a risk-neutral probability because the CallB price appears as
a discounted expected value when using RNP in the expectation. Only risk-neutral
investors would discount using the risk-free rate and so RNP can be viewed as the
probability of an up move in a world where investors are risk neutral.
In our example dt = (3/12) /2 = 0.125, u = 1.2363, and d = 0.8089, so that
RNP =
exp (0.05 · 0.125) − 0.8089
= 0.4618
1.2363 − 0.8089
We can use this number to check that the new formula works. We get
CallB = [RNP · CallD + (1 − RNP) · CallE ] exp −rf · dt
= [0.4618 · 628.47 + (1 − 0.4618) · 100.00] exp (−0.05 · 0.125)
= 341.92
just as when using the no-arbitrage argument.
The new formula can be used at any point in the tree. For example at point A we
have
cBin = CallA = [RNP · CallB + (1 − RNP) · CallC ] exp −rf · dt
It can also be used for European puts. We have for a put at point C
PutC = [RNP · PutE + (1 − RNP) · PutF ] exp −rf · dt
Notice that we again have to work from right to left in the tree when using these
formulas. Note also that whereas changes values throughout the tree, RNP is constant throughout the tree.
3.5 Pricing an American Option Using the Binomial Tree
American-style options can be exercised prior to maturity. This added flexibility gives
them potentially higher fair market values than European-style options. Fortunately,
binomial trees can be used to price American-style options also. We only have to add
one calculation in the tree: At the maturity of the option American- and European-style
options are equivalent. But at each intermediate point in the tree we must compare the
European option value (also known as the continuation value) with the early exercise
value and put the largest of the two into the tree at that point.
Consider Table 10.4 where we are pricing an American option that has a strike
price of 1,100 but otherwise is exactly the same as the European option considered in
Tables 10.1 through 10.3.
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229
Table 10.4 American options: check each node for early exercise
Market Variables
St =
Annual rf =
1000
0.05
Contract Terms
X=
T=
1100
0.25
Parameters
Annual Vol =
tree steps =
dt =
u=
d=
RNP =
0.6
2
0.125
1.23631111
0.808857893
0.461832245
Stock is black
American call is green
American put is red
D
1528.47
428.47
0.00
B
1236.31
196.65
53.48
A
1000.00
90.25
180.25
E
1000.00
0.00
100.00
C
808.86
0.00
291.14
F
654.25
0.00
445.75
Notes: We compute the American option values by checking for early exercise at each point in
the tree.
If we exercise the American put option at point C we get
Max {1,100 − 808.86, 0} = $291.14
Let us now compute the European put value at this point. Using the previous method
we have the risk-neutral probability of an up-move RNP = 0.4618, so that the European put value at point C is
PutC = [RNP · PutE + (1 − RNP) · PutF ] exp −rf · dt
= $284.29
which is of course lower than the early exercise value $284.29. Early exercise of the
put is optimal at point C and the fair market value of the American option is therefore
$291.14 at this point. 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. This is because American call stock options should only be
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Further Topics in Risk Management
exercised early if a large cash dividend is imminent. In our example there were no
dividends and so early exercise of the American call is never optimal, which in turn
makes the American call option price equal to the European call option price.
3.6 Dividend Flows, Foreign Exchange, and Futures Options
In the case where the underlying asset pays out a stream of dividends or other cash
flows we need to adjust the RNP formula. Consider an underlying stock index that
pays out cash at a rate of q per year. In this case we have
RNP =
exp rf − q · dt − d
u−d
When the underlying asset is a foreign exchange rate then q is set to the interest rate
of the foreign currency. When the underlying asset is a futures contract then q = rf so
that RNP = (1 − d) / (u − d) for futures options.
4 Option Pricing under the Normal Distribution
The binomial tree approach is very useful because it is so simple to derive and because
it allows us to price American as well as European options. A downside of binomial
tree pricing is that we do not obtain a closed-form formula for the option price.
In order to do so we now assume that daily returns on an asset be independently
and identically distributed according to the normal distribution,
1
i.i.d.
Rt+1 = ln(St+1 ) − ln(St ) ∼ N µ − σ 2 , σ 2
2
Then the aggregate return over T˜ days will also be normally distributed with the mean
and variance appropriately scaled as in
1
˜ 2
Rt+1:t+T˜ = ln St+T˜ − ln(St ) ∼ N T˜ µ − σ 2 , Tσ
2
and the future asset price can of course be written as
St+T˜ = St exp Rt+1:t+T˜
The risk-neutral valuation principle calculates the option price as the discounted
expected payoff, where discounting is done using the risk-free rate and where the
expectation is taken using the risk-neutral distribution:
˜ t∗ Max S ˜ − X, 0
c = exp(−rf T)E
t+T
where Max St+T˜ − X, 0 as before is the payoff function and where rf is the risk-free
interest rate per day. The expectation Et∗ [∗] is taken using the risk-neutral distribution
Option Pricing
231
where all assets earn an expected return equal to the risk-free rate. In this case the
option price can be written as
∞
˜
c = exp(−rf T)
Max St exp(x∗ ) − X, 0 f (x∗ )dx∗
−∞
∞
˜
= exp(−rf T)
∞
St exp(x )f (x )dx −
∗
∗
Xf (x∗ )dx∗
∗
ln(X/St )
ln(X/St )
where x∗ is the risk-neutral variable corresponding to the underlying asset return
between now and the maturity of the option. f (x∗ ) denotes the risk-neutral distribu˜ 2 ).
˜ f − 1 σ 2 ), Tσ
tion, which we take to be the normal distribution so that x∗ ∼ N(T(r
2
The second integral is easily evaluated whereas the first requires several steps. In the
end we obtain the Black-Scholes-Merton (BSM) call option price
˜ St exp(rf T)
˜ (d) − X
cBSM = exp(−rf T)
˜
= St (d) − exp(−rf T)X
where
d − σ T˜
d − σ T˜
(•) is the cumulative density of a standard normal variable, and where
d=
ln (St /X) + T˜ rf + σ 2 /2
σ T˜
Black, Scholes, and Merton derived this pricing formula in the early 1970s using a
model where trading takes place in continuous time when assuming continuous trading
only the absence of arbitrage opportunities is needed to derive the formula.
It is worth emphasizing that to stay consistent with the rest of the book, the volatility
and risk-free interest rates are both denoted in daily terms, and option maturity is
denoted in number of calendar days, as this is market convention.
The elements in the option pricing formula have the following interpretation:
●
●
●
●
˜ is the risk-neutral probability of exercise.
(d − σ T)
˜ is the expected risk-neutral payout when exercising.
X (d − σ T)
˜ is the risk-neutral expected value of the stock acquired through
St (d) exp(rf T)
exercise of the option.
(d) measures the sensitivity of the option price to changes in the underlying asset
BSM
price, St , and is referred to as the delta of the option, where δ BSM ≡ ∂c∂S
is the first
t
derivative of the option with respect to the underlying asset price. This and other
sensitivity measures are discussed in detail in the next chapter.