22.C Risk-Neutral Pricing and Marginal Utility in the Binomial Model
Appendix 22.C RISK-NEUTRAL PRICING AND MARGINAL
UTILITY IN THE BINOMIAL MODEL
This appendix links the discussion in this chapter to that in Appendix 11.B.
We now see how the concepts we have just discussed arise in the binomial model. The
discussion in this section builds on Appendix 11.B. There are two assets, a risky stock and
a risk-free bond. Investors are risk-averse, there is one period and two possible outcomes:
a high stock price (the good state) and a low stock price (the bad state).
The payoff to the risky stock is S1H in the high state, which occurs with probability
H
p , and S1L in the low state, with probability p L = 1 − p H . The investor receives a marginal
utility of consumption today of U0 (C0). In the future, if the high state occurs then the
investor consumes C1H and obtains a marginal utility of U1 (C1H ). In the low state, the
investor consumes C1L and obtains a marginal utility of U1 (C1L). As we have discussed,
these consumption levels are consequences of the decisions the investor makes about how
much to consume and how to invest savings. We simply take the outcome of this decision
process for granted.
Valuing the Stock and Bond with Physical Probabilities
We can directly apply the formulas from the preceding sections to value the stock and
bond. It simplifies notation to define the marginal utility ratios χ H = U1 (C1H )/U0 (C0) and
χ L = U1 (S1L)/U0 (C0). The values χ H and χ L are the stochastic discount factors for the
high and low state. The value of the stock at time 1 will be either S1H or S1L. Using equation
(22.5), the value of the stock at time 0 is
S0 = E0
U1 (C1)
U0 (C0)
S1
=p
H
U1 (C1H )
U0 (C0)
S1H
+p
L
U1 (C1L)
U0 (C0)
S1L
= p H χ H S1H + p Lχ LS1L
(22.50)
The value of the bond at time 0 is
H
L
U1 (C1)
H U1(C1 )
L U1(C1 )
B1 = p
B1 + p
B1
B0 = E0
U0 (C0)
U0 (C0)
U0 (C0)
= p H χ H B1 + p L χ L B1
(22.51)
Equations (22.50) and (22.51) present stock and bond valuations using the stochastic
discount factor and physical probabilities.
Risk-Neutral Valuation
Using the change of measure result from section 22.3, we can pick the bond as numeraire
and define a new probability distribution. We accomplish this by multiplying and dividing
by the price ratio B1/B0 in equation (22.50):
1
2
Chapter 22. Exotic Options: II
U1 (C1)B1 S1
S 0 = B 0 E0
U0 (C0)B0 B1
H L
S
S1
B
B
1
1
1
= B0 p H χ H
+ pLχ L
B0 B 1
B0 B1
(22.52)
This valuation uses the physical probabilities, p H and p L, and marginal utilities.
Now define new probabilities p̂ H and p̂ L:
p̂ H = p H χ H
p̂ L = p Lχ L
B1
B0
B1
B0
(22.53)
(22.54)
We have to make sure that these are legitimate probabilities. They are nonnegative because
physical probabilities, marginal utilities, and bond prices are nonnegative. We can also
verify that they sum to 1:
p̂ H + p̂ L = p H χ H
=
B1
B
+ pLχ L 1
B0
B0
1 H H
p χ B1 + p L χ L B1 = 1
B0
The last equality follows from equation (22.51).
We can substitute equations (22.53) and (22.54) into equation (22.52), and using the
fact that B0/B1 = 1/(1 + r), the result is
1 H
S0 =
p̂S1 + (1 − p̂)S1L
1+ r
This is risk-neutral binomial pricing: Compute the expected stock price using the riskneutral probability, and discount the result at the risk-free rate. The risk-neutral probabilities
are the physical probabilities weighted by marginal utilities.
This example illustrates how risk-neutral pricing permits us to sidestep explicit utility
calculations while still performing a correct valuation. Risk-neutral probabilities have the
property that we obtain the correct time-0 value of a security when we compute expected
cash flows using those probabilities, without explicit utility adjustments, and then discount
that expectation at the risk-free rate. This is feasible because instead of utility-weighting
the cash flows and computing expectations, we utility-weight the probabilities, creating new
“risk-neutral” probabilities.
A final observation about this example. Note that U1 (C1H ) < U1 (C1L) (marginal utility
decreases with consumption). Because of this, p̂ H < p H : The effect of utility-weighting the
physical probabilities in equation (22.14) is to reduce the probability associated with the
high state. Assigning a lower weight to the more valuable outcome has the same effect as
using a discount rate greater than the risk-free rate, which is what we do in ordinary DCF
valuation.1
1. Problem: Verify algebraically that p̂ H < p H .
22.C Risk-Neutral Pricing and Marginal Utility in the Binomial Model
BOX
22.3: State Prices
In the example above, p̂
H /(1 +
r) is the state
price for the event that the economy does well,
and p̂ L/(1 + r) is the state price for the event that
the economy does badly.
Let QH be the price at time 0 of a security that
pays $1 when the high state occurs, and QL the
price of a security paying $1 when the low state
occurs.* Call these the high and low securities.
To understand the economics of a state price, we
can think about marginal utility. If the investor
buys an additional unit of the high security, the
additional future cash flow is $1 received when
the high state occurs. In order to pay for this
security, the investor reduces consumption at time
0 by QH , which has a utility cost of U0 (C0)QH .
In exchange, the investor receives $1 of extra
consumption in the high state, which has a utility
value of U1 (C H ) × $1. The utility given up must
equal the expected utility received, so we have
U0 (C0)QH
=p
We can rewrite this as
× U1 (C1H ) × $1
QH = p H × χ H
(22.55)
Similarly, we can write the price of the low
security as
QL = p L × χ L
(22.56)
The variables χ H and χ L are state-specific discount factors: Cash flows in the high state are
discounted by χ H and in the low state by χ L.
We can now value the bond and the stock. The
bond pays $1 in each state. Using the state prices,
we have
B0 = [p × χ H × $1] + [(1 − p) × χ L × $1]
= QH × $1 + QL × $1
(22.57)
Similarly, the value of the stock is
S0 = [p × χ H × S1H ] + [(1 − p) × χ L × S1L]
= QH × S1H + QL × S1L
(22.58)
*These are often called “Arrow-Debreu” securities, named
after Nobel prize-winning economists Kenneth Arrow and
Gerard Debreu.
Appendix 11.B presents a numerical example corresponding to this discussion.
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