**2.6
OPTION VALUATION: A SYNTHETIC OPTION APPROACH
**

An
alternative approach to valuing the option is to replicate the option using the
stock and bond. This replicating
portfolio is called a *synthetic option*.
The value of this replicating portfolio gives you the arbitrage-free
value of the option __directly__. You
can contrast this approach to the riskless hedge approach.
The riskless hedge portfolio is also a synthetic security.
This portfolio gives you the value of a *synthetic riskless bond* in terms of a stock and an option.
Given that you know the value of the stock and the arbitrage-free value
of the synthetic riskless bond, you can derive the value of the option.

Clearly,
the two approaches must give you the same answer but the way that you apply the
two approaches is different. In the
riskless hedge portfolio you may recall that the portfolio consists of the ratio
of +1 stock and -*k* call options (or +1
stock and +*k* put options) where *k*
is the hedge ratio. For a call
option, the hedge ratio is determined from

*S - kC* =
Constant Payoff

and
for a put option the hedge ratio is determined from:

*S + kP* =
Constant Payoff

The
hedge ratio comes out to be

for
the call, and

for
the put.

You
can interpret the constant payoff as the end-of-period value of a riskless bond *B*. As a
result, you can also determine the option price by using the stock and the
riskless bond to replicate the cash flows of the call option (put option).
By rearranging the call option example and substituting *B*
for constant payoff, you can verify that:

**Constructing a Synthetic Option Directly
**

Let
us now consider how a portfolio that replicates one call option is constructed. Suppose you buy d stocks and
borrow $*b *today, and you choose d and $*b* so
that __at the end of the period__:

and

Given
that we have two equations and two unknowns, we can choose
d and b so that:

d is the change in the call value (numerator) divided
by the change in the stock price. Therefore,
if we choose d and *b*
according to these equations, then our portfolio of d
stocks and $*b *of the risk-free asset has exactly the same cash flows as the call
option. But then the price of the
call option must equal the price of this (equivalent) portfolio, otherwise one
could make a pure arbitrage profit. This
means that the cost today is:

Now
let us consider an example where *X* =
30, *Su *= 40, *Sd* = 20, and
one plus risk-free interest rate *r*
= 1, so

which
gives d = 1/2. Therefore,
to replicate the call option payoffs, b = -10.
We again find that the cost of the synthetic call option is *C*
= (*S *- 20)/2.

In
the synthetic option equation, observe that

d measures
the sensitivity of the call price with respect to the price of the underlying
stock. It is commonly referred to
as the "delta" of the option.

We
can also value a European put option in a similar manner.
Now, we require

leading
to

and
the cost of the put is

Note
that the delta of a put option is negative since

From
either this synthetic option approach or the riskless hedge portfolio approach,
we can deduce a very general and important valuation principle known as the __Risk-Neutral
Valuation Principle__.