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A Risk-Neutral Shortcut
We pointed out earlier in the chapter that the binomial
model valuation approach is arbitrage-based. We can value
the option by replicating it with shares of stock plus bor-
rowing. The ability to replicate the option means that its
price relative to the stock and the interest rate must be
based only on the technology of replication and not on risk
preferences. It cannot depend on risk aversion or the capi-
tal asset pricing model or any other model of equilibrium
risk-return relationships.
This insight—that the pricing model must be inde-
pendent of risk aversion—leads to a very useful shortcut
to valuing options. Imagine a risk-neutral economy, that
is, an economy in which all investors are risk-neutral. This
hypothetical economy must value options the same as our
real one because risk aversion cannot affect the valuation
formula.
In a risk-neutral economy, investors would not
demand risk premiums and would therefore value all
assets by discounting expected payoffs at the risk-free
rate of interest. Therefore, a security such as a call option
would be valued by discounting its expected cash flow
at the risk-free rate:
“E”(CF)
C
5
1
1 r
f
. We put the expectation
operator E in quotation marks to signify that this is not
the true expectation, but the expectation that would
prevail in the hypothetical risk-neutral economy. To be
consistent, we must calculate this expected cash flow
using the rate of return the stock would have in the risk-
neutral economy, not using its true rate of return. But
if we successfully maintain consistency, the value derived
for the hypothetical economy should match the one in
our own.
How do we compute the expected cash flow from the
option in the risk-neutral economy? Because there are no
risk premiums, the stock’s expected rate of return must
equal the risk-free rate. Call
p the probability that the
stock price increases. Then p must be chosen to equate the
expected rate of increase of the stock price to the risk-free
rate (we ignore dividends here):
“E”(S
1
) 5 p(uS) 1 (1 2 p)dS 5 (1 1 r
f
)S
This implies that p 5
1
1 r
f
2 d
u
2 d
. We call p a risk-neutral
probability to distinguish it from the true, or “objective,”
probability. To illustrate, in our two-state example at the
beginning of Section 21.2, we had u 5 1.2, d 5 .9, and
r
f
5 .10. Given these values, p 5
1
1 .10 2 .9
1.2
2 .9
5
2
3
.
Now let’s see what happens if we use the discounted
cash flow formula to value the option in the risk-neutral
economy. We continue to use the two-state example from
Section 21.2. We find the present value of the option pay-
off using the risk-neutral probability and discount at the
risk-free interest rate:
“E”(CF)
C
5
1
1 r
f
p C
u
1 (12p) C
d
5
1
1 r
f
2/3
3 10 1 1/3 3 0
5
1.10
6.06
5
This answer exactly matches the value we found using our
no-arbitrage approach!
We repeat: This is not truly an expected discounted value.
•
The numerator is not the true expected cash flow from
the option because we use the risk-neutral probability,
p, rather than the true probability.
•
The denominator is not the proper discount rate for
option cash flows because we do not account for
the risk.
•
In a sense, these two “errors” cancel out. But this is
not just luck: We are assured to get the correct result
because the no-arbitrage approach implies that risk
preferences cannot affect the option value. Therefore,
the value computed for the risk-neutral economy must
equal the value that we obtain in our economy.
When we move to the more realistic multiperiod
model, the calculations are more cumbersome, but the
idea is the same. Footnote 4 shows how to relate p to any
expected rate of return and volatility estimate. Simply set
the expected rate of return on the stock equal to the risk-
free rate, use the resulting probability to work out the
expected payoff from the option, discount at the risk-free
rate, and you will find the option value. These calculations
are actually fairly easy to program in Excel.
WORDS FROM THE STREET
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C H A P T E R
2 1
Option
Valuation
737
hedged over the coming small interval. By continuously revising the hedge position, the
portfolio would remain hedged and would earn a riskless rate of return over each inter-
val. This is called dynamic hedging, the continued updating of the hedge ratio as time
passes. As the dynamic hedge becomes ever finer, the resulting option-valuation proce-
dure becomes more precise. The nearby box offers further refinements on the use of the
binomial model.
In the table in Example 21.3, u and d both get closer to 1 ( u is smaller and d is larger) as the time interval
Δ t shrinks. Why does this make sense? Does the fact that u and d are each closer to 1 mean that the total
volatility of the stock over the remaining life of the option is lower?
CONCEPT CHECK
21.5
21.4
Black-Scholes Option Valuation
While the binomial model is extremely flexible, a computer is needed for it to be useful in
actual trading. An option-pricing formula would be far easier to use than the tedious algo-
rithm involved in the binomial model. It turns out that such a formula can be derived if one
is willing to make just two more assumptions: that both the risk-free interest rate and stock
price volatility are constant over the life of the option. In this case, as the time to expira-
tion is divided into ever-more subperiods, the distribution of the stock price at expiration
progressively approaches the lognormal distribution, as suggested by Figure 21.5 . When
the stock price distribution is actually lognormal, we can derive an exact option-pricing
formula.
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