8: Optionstotal. On the other hand, if one year from now the share price equals $40, then the call options we sold
will be worthless, and we will have no cash outflow. In either case, the total cash inflow from the portfolio
will be $40.
Because this portfolio pays $40 in one year, no matter what happens, we call it a perfectly hedged
portfolio. The value of h is called the hedge ratio because it tells us what combination of shares and calls
results in a perfectly hedged position.^13
Step 2: Calculate the Present Value of the Portfolio
Because the portfolio, which consists of one share and two short call options, pays $40 for certain
next year, we can say that the portfolio behaves like a risk-free bond (in technical terms, the portfolio
is a ‘synthetic’ risk-free bond). The second step requires us to calculate the present value of the
portfolio. Because we already know that the risk-free rate equals 4%, the present value of the
portfolio equals:
$40
1.04PV== $38.46
It is crucial at this step to understand the following point. Buying one share and selling two
calls yields the same future payoff as buying a risk-free, zero-coupon bond with a face value of
$40. Because both of these investments offer, with certainty, $40 in one year, they should both
sell for the same price today. That is the insight that allows us to determine the option’s price in
the next step.
Step 3: determine the Price of the Option
If a risk-free bond paying $40 in one year costs $38.46 today, then the net cost of buying one share of the
Financial Engineers enterprise and selling two call options must also be $38.46. Why? Both investment
strategies offer the same future cash flows, so they must both sell for the same price. Therefore, to
determine the price of the option, we need to write down an expression for the cost of our hedged
portfolio and set that expression equal to $38.46.
From the information given in the problem, purchasing one share costs $55. Partially offsetting this
cost will be the revenue from selling two call options. Denoting the price of the call option, C, we can
calculate the total cost of the portfolio as follows:
Total portfolio cost = $55 – 2C = $38.46
Solving for C, we obtain a call value of $8.27.
At this point, it is worth reviewing what we’ve accomplished. We began with an assumption aboutthe future movements of the underlying shares. Next, given the type of option we wanted to value and its
characteristics, we calculated the payoffs of the option for each of the two possible future share prices.
Given those payoffs, we discovered that, by buying one share and selling two calls, we could generate
a certain payoff of $40 in one year. Because the present value of that payoff is $38.46, the net cost of
buying the share and selling the calls must also equal $38.46. That implies that we received revenue of
$16.54 from selling two calls, or $8.27 per call. The following example repeats the process to value an
identical put option on the same underlying shares.
13 The hedge ratio can be defined as the ratio of calls to shares in a perfectly hedged portfolio (the definition we use here) or as the ratios of
shares to calls. In this example, the hedge ratio equals either 22:1 (using our definition) or 21:2 (using the alternative definition). Either way,
the hedge ratio defines the mix of options and shares that results in a perfectly hedged portfolio.
hedge ratio
A combination of shares and
options that results in a risk-
free payoff