The Basic Replication Argument: One Period, Two Possible End of Period Values

A Primer on

Option Pricing

(keyed to Chapter 21 in R. A. Brealey, S. C. Myers, and F. Allen, Principles of Corporate Finance, 8th ed., McGraw-Hill/Irwin, Inc., 2006; hereafter (BMA))

The Basic Replication Argument: One Period, Two Possible End of Period Values

Let a share of stock be worth $ at present and suppose that at the end of the period it could be worth $ or $, where . Let denote the risk-free rate of interest for this one period. Let the value of a call option on this stock with exercise price $, be denoted by $. Let denote the value of the call at the end of the period if the stock price is , and let denote the value of the call at the end of the period if the stock price is .

At the beginning of the period we know the price of two assets, the price of the stock and the price of a risk-free bond with face value of $1, . We also know the exercise price of the call . To value this call we need to model how the stock price might evolve over this one period. We have modeled this as the two possible values and. This information determines the possible call values and . Given what we know and given how we have modeled the stock price movements, our objective is to find the value of the call as a function of this given data (the given data are abstract now, but they are numbers we will have in any application).

To do this we will construct a portfolio of the stock and the riskless bond that will replicate the end of period payoff from the option, i.e., we will create the option synthetically as a portfolio of the stock and riskless bonds. Then, to avoid arbitrage, the option and the portfolio must have the same beginning of period value. Since we know the beginning of period value of the stock and the riskless bond, we know the value of the portfolio and hence the value of the call .

Consider a portfolio that has shares of the stock and riskless bonds each with face value of $1. So this portfolio has $ invested in stock and $ in bonds.

The end of period payoffs on the portfolio and on the option are as follows:

End of Period Stock Price

$ $

Portfolio

Option

The quantities and are determined so the payoffs on the portfolio and the option are the same. Setting these payoffs equal, we get the equations:

= ,

and

= .

Solving for in the first equation and plugging this solution into the second equation, we get that

, (1)

just as in the middle of page 567 in (BMA).

Substituting this value of delta into the expression for gives

. (2)

Notice that given in equation (1) is non-negative since we have assumed that , and this implies that . So the replicating portfolio involves a long position in the stock. This accords well with intuition since the call option gives the owner the right to buy the stock.

It turns out that the number of bonds in this replicating portfolio, given in equation (2), is negative. This means the replicating portfolio is short the riskless bond, i.e., we are levering the long position in the stock by borrowing at the risk-free rate. Thus the replicating portfolio and hence the call option is a levered long position in the stock, something that options traders had intuited long before options were valued in this way.

The cost of the replicating portfolio at the beginning of the period and the value of the call must be the same or there is an arbitrage opportunity. If is higher than the cost of the portfolio, sell the call and buy the replicating portfolio. At the end of the period, the payoffs you owe on the call you sold are the same as the payoff on the portfolio. Net end of period payoff is zero, regardless of the stock price. The beginning of period profit is thus arbitrage profit.

Similarly, if is less than the cost of the portfolio, buy the call and sell the portfolio, again making the difference in arbitrage profit.

Using equations (1) and (2),then in the absence of arbitrage, we must have that

(3)

This argument is the same as the one given on pages 566-567 in (BMA) for the call option on Amgen stock. In this Amgen example, now is November 2003, and the price of Amgen stock is $ = $60. The one period is 8 months from now and the modeled stock price moves are either a move down to $ = $45 or a move up to $ = $80. The 8 month call option on Amgen stock is written at the money so = 66, and the 8 month risk-free rate is = 0.01, or about 1.5% per annum.

The end of period value of the call option if the stock price is is = = = 0 (the call expires worthless). Similarly, end of period value of the call option if the stock price is is = = = 80 - 60 = 20. See first displayed table on p. 566 in (BMA).

We have done the hard part in figuring out how many shares of Amgen and how many bonds we need to sell to replicate the payoffs on the call. Plugging the numbers above into equation (1) above we get that

= = = ,

just as in the middle of page 567 in (BMA).

If you buy 4/7 of a share of Amgen stock and invest in

=

=

= -25.71

riskless bonds, the value of this portfolio now, and hence the value of the call , is

=

= (60)

=

=

=

= = 8.82602546,

just as in the computation at the top of page 567 in (BMA). (Check that you get this same answer for if you just plug in the data in the expression on the far right in (3)).

As indicated in the discussion on page 567 and analogous to the Black/Scholes formula on page 576 in (BMA), the term given in (1) above is the option delta or hedge ratio. For the call option this hedge ratio is a non-negative fraction so that . (When is , and when is ? The answer depends on the size of .) The absolute value of the term given in equation (2) above is the value of the bank loan. The computation of the face value of the loan and the loan proceeds in the Amgen example in footnote 2 at the bottom of page 566 in (BMA) is correct. The formula given in equation (2) above is a more formal expression of this computation.

We can make a similar argument to replicate a put option on the stock. Again, let a share of stock be worth $ now and have two possible end of period values $ and $, where . The risk-free rate is for this one period. Consider a one period put option on this stock with exercise price and value now of $. Let denote the value of the put at the end of the period if the stock price then is and let denote the value of the put at the end of the period if the stock price then is .

We know , , and now and we model the price movements over the one period by and , which determines the quantities and , respectively. Our objective is to determine as a function of this data.

To do so we will construct a portfolio of the stock and the riskless bond, whose prices now we know, that will replicate the end of period payoffs on the put option. For arbitrage to be absent it must be that the put option and the synthetic put option, the replicating portfolio, have the same price now.

Consider a portfolio that has shares of the stock and riskless bonds each with face value of $1. So this portfolio has $ invested in stock and $ in bonds.

The end of period payoffs on the portfolio and on the option are as follows:

End of Period Stock Price

$ $

Portfolio

Option

The quantities and are determined so the payoffs on the portfolio and the option are the same. Setting these payoffs equal, we get the equations:

= ,

and

= .

These equations are exactly the same as for the call option above except that pees replace the cees. Solving these equatios will yield versions of equations (1), (2), and (3) with cees replaced by pees. Therefore we do just that here. We have , (4)

just as in the top of page 569 in (BMA), , (5)

and finally

(6)

Notice that given in equation (4) is non-positive (the range is ) since we have assumed that , and this implies that . So the replicating portfolio for a put option involves a short position in the stock. This accords well with intuition since the put option gives the owner the right to sell the stock.

Note also that the put option delta equals the call option delta minus one. See footnote 5, page 569 in (BMA).

It turns out that the number of bonds in this replicating portfolio, given in equation (5), is positive. This means the replicating portfolio is long in the riskless bond, i.e., we are investing the proceeds of the short position in the stock at the risk-free rate. Thus the replicating portfolio and hence the put option is a long position in the riskless bond levered by a short position in the stock.

Let us illustrate the formulas using again the Amgen example on pages 568-569 of (BMA). The data are the same as for the call, = 60, = 45, = 80, = 60, and = 0.01. This gives the end of period value of the put if stock price moves down to of = = = 60 - 45 = 15. The end of period value of the put if the stock price moves up to is = = = 0 (the put expires worthless).

Plugging these numbers into equation (4), we get that

=

= = = -.428571429.

So the replicating portfolio is short 3/7 shares. From equation (5), the number of bonds is

=

= =

= 34.28571429.

The computation of this quantity is somewhat cloudy in (BMA). Note that $34.29 shows up in the computations in the box in the middle of page 569 in (BMA) and they say you lend $33.95 after the computation of the put delta at the top of page 569. This number is the present value = 34.28571429/1.01 = 33.94625177. They do not have equation (5) either formally in informally as in footnote 2, p. 566.

If you short 3/7 shares of Amgen and buy riskless bonds each with face value of $1, the cost of this portfolio and hence the put option is

=

=

= =

= 8.231966054,

as in the middle of page 569 in (BMA). (Check that you get the same answer if you just plug the data into the expression on the far right in equation (6)).

Exercise. Show that equation (3) can be manipulated so that

, (7)

and that equation (6) can be manipulated so that

, (8)

where is the same in equation (7) and (8) and is the risk-neutral probability of the up move in the stock price (from to ). A formula for this risk-neutral probability is given in the formula at the top of page 568 in (BMA).

Risk-Neutral Valuation

Equations (7) and (8) are examples of risk-neutral valuation. Calculation of the value of the Amgen call option using equation (7) is done on page 568 in (BMA). Calculation of the value of the Amgen put option using equation (8) is done on pages 569-570 in (BMA).

There are several examples in the rest of Chapter 21 and in Chapter 22 in (BMA) that further illustrate risk-neutral valuation. In fact the Black/Scholes formula is a risk-neutral valuation as it must be if there is no arbitrage. See the class notes “Risk Neutral Valuation” in the reading packet.

Cases and Readings in Corporate Finance. Class Notes III, by David C. Nachman. 57