**2.5
PUT OPTION VALUATION: A RISKLESS
HEDGE APPROACH
**

Consider
a portfolio of +1 stock and +k puts. The
future payoffs for this portfolio are shown in Figure 2.5.

**Figure 2.5
**

**Future Payoffs**

**The
Hedge Ratio (k)
**

For
a portfolio to be riskless, you have to choose *k*
so that the payoff in both states is equal, whether the stock moves up or down.

This
requires

which
is called the *hedge ratio*.

**The
Riskless Hedged Portfolio: Put
Options
**

The
portfolio in Figure 2.5 is known as the riskless hedged portfolio.
It is constructed from the hedge ratio *k*,
so that for every stock held long *k*
put options are bought. The
riskless (put option) portfolio is:

*S + kP
*

This
is a riskless hedge because when the terminal stock value is low, the put is
valuable. The converse is true for
a high terminal value. Setting k
equal to the hedge ratio balances the losses with the gains.

**The
Cost of the Riskless Hedge
**

The
cost of acquiring this portfolio today is*
S + kP*. Since

the
portfolio end-of-period payoff is a certain amount.
Thus, using the risk-free interest rate, you can equate the cost of
acquisition today to the present value of the certain end-of-period payoff. The end-of-period payoff can be defined by either the up- or
downtick, because both are the same. Let
this be fixed as the realized downtick value:

By
substituting for* k*, you can solve for
the value of the put option *P*.

This
gives you the price of the put option as a function of the current stock price,
the future stock values, the strike price, and the risk-free interest rate.

For
example, when *X *= 30, *Su* = 40, *Sd* = 20, and
one plus risk-free interest rate *r *=
1,

this
gives* S +*2*P* = 40 so *P*
= (40-*S*)/2,

An
alternative approach to the European option valuation problem is to create a
synthetic call option by constructing an appropriate portfolio consisting of the
stock and the bond. You can see
this approach in the next topic, __Option
Valuation: A Synthetic Option
Approach__.