**2****.4
CALL OPTION VALUATION: A RISKLESS
HEDGE APPROACH
**

I |

*u*> 1 be the uptick,

*d*< 1 be the downtick, and

*S*be the current stock price.

If
an uptick is realized, the end-of-period stock price is *Su*.
Otherwise, a downtick is realized, and the end-of-period stock price is *Sd*.
You may recall from topics 2.2 and 2.3, the __Riskless
Hedge Example__RHE_BIN and __Synthetic
Option Example__SOE_BIN, that in valuing the option you do not need to know
the probability of the stock moving up or down.

Let
us now consider how to formulate the general case for the one-period option
pricing problem. First, we will
need some notation.

**Notation
**

*S*
= current stock price;

*Su* = future
high stock price (call this State H) ;

*Sd* = future
low stock price (call this State L) ;

*r =*
1 + the risk-free interest rate;

*X*
= strike price;

*C*
= current price of the call option, which is to be determined.

We
will assume that u > r > d. This
is actually necessary to prevent arbitrage (if r > u, then you should sell
the stock and invest the proceeds in the risk-free asset; if d > r, you
should borrow at the risk-free rate and buy the stock).

We
start with the call option. The
terminal values of the call are:

If
both *C**u*
and *C**d*
are zero, then the call option has no value, so suppose that *C**u* > 0 and you
have a portfolio of +1 stock and -k calls.
The future payoffs from this portfolio can be depicted as follows in
Figure 2.4:

**Figure 2.4
**

**Future Payoffs**

**The
Hedge Ratio (k)
**

For
a portfolio to be riskless, we have to choose *k*
so that the payoff in both states is equal:

In
this case we have a risk-free portfolio. This
requires

which
is called the *hedge ratio*.

**The
Riskless Hedged Portfolio: Call
Options
**

The
portfolio of one stock and k calls, where k is the hedge ratio, is called the
riskless hedged portfolio. The
hedge ratio, k, tells you that for
every stock you hold, k call options must be sold.
The riskless (call option) portfolio is:

*S - kC
*

**The
Cost of the Riskless Hedge
**

The
cost of acquiring this portfolio today is*
S - kC*. Since

the
end-of-period portfolio value is known with certainty.

Since
the future value is riskless, the present value equals the future value
discounted at the risk-free interest rate.
The end-of-period payoff can be defined from either the up- or downtick,
because both are the same. So let
us fix this at the realized uptick value

**
**

By
substituting for k, we can solve for the value of the call option *C*.

**
**

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

Consider
the example, where *X* = 20,* S *= 20, *Su* = 40, *Sd*
= 10, and one plus risk-free interest rate *r* = 1, so

**
**

which
gives 2*S* - 3*C *= 20 so *C
*= (2*S*-20)/3, just as before in topic 2.2 the __Riskless Hedge Example__.

The
riskless hedge portfolio approach to pricing put options is described in the
next topic titled __Put Option Valuation: A Riskless Hedge Approach__.