**2.8 Reconciliation of Risk-Neutral Valuation with CAPM
**

H |

ow can you
reconcile risk-neutral valuation with a model of asset price
determination such as CAPM, where you solve a relatively complex
problem to value assets?

For
the underlying stock price, S, there is no conflict because*
S *can come about from any asset pricing model.
In fact, we have simply assumed that we know S, and we do not
"determine" it in any way. For
the option price, the answer lies in the observation that we can form a
synthetic put or call option from a portfolio of the stock and the bond. As a result, all the risk from holding say a call option can
be *diversified away* using the stock
and the bond.

This
is the essence of why an option is a vehicle for *transferring
risk* from one party to another, *but it
does not change the total risk*. Any
premium the market may be paying in terms of additional return for risk is
captured by the current stock price S. Hence,
risk considerations enter the option price through S, since the option contract
does not create any additional problem of pricing risk.

The
risk-neutral valuation principle applies for certain models of stock price
movements, but must be modified for others.
For example, it does not generally apply if the stock can move up, down,
or stay the same.

To
see this, consider the payoffs to a one-period call option with *Sd<X<S<Su*.
The call has payoffs *Su-X, S-X*, and 0. If we
try to replicate this by holding n stocks and
*m *bonds, we would have to
satisfy

*n Su + m Br =
Su - X
*

*n S + m Br
= S
- X
*

and

*n Sd + m Br
= 0,
*

where
*B* is the dollar amount we invest in the risk-free bond.

This
gives us three equations with two unknowns, which generally has no solution.

The
problem is that we can no longer diversify all the risk associated with the
option using only the stock and the bond. As
a result, there is now a source of risk associated with the option contract that
must be priced by the capital market.

However,
if there is a third "independent" asset, whose payoffs are correlated
with the stock, then we could again diversify all the risk from the option.
This would let us obtain the price of the call option in terms of the
stock price, the risk-free interest rate, and the price of this third asset,
without any reference to the market price of risk.

Any
portfolio that creates a synthetic riskless bond can be valued without any
reference to the market price of risk. In
the next topic, __Put-Call Parity: European
Options,__ we analyze a synthetic
riskless security that is constructed from the stock and its underlying options.