We saw in the previous section that a protective put portfolio, comprising a stock position and a put option on that position, provides a payoff with a guaranteed minimum value, but with unlimited upside potential. This is not the only way to achieve such protection, however. A call-plus-bills portfolio also can provide limited downside risk with unlimited upside potential.
Consider the strategy of buying a call option and, in addition, buying Treasury bills with face value equal to the exercise price of the call, and with maturity date equal to the expira- tion date of the option. For example, if the exercise price of the call option is $100, then each option contract (which is written on 100 shares) would require payment of $10,000 upon exercise. Therefore, you would purchase a T-bill with a maturity value of $10,000.
More generally, for each option that you hold with exercise price X, you would purchase a risk-free zero-coupon bond with face value X.
Examine the value of this position at time T, when the options expire and the zero- coupon bond matures:
S T # X S T . X Value of call option 0 ST2 X
Value of riskless bond X X
TOTAL X ST
If the stock price is below the exercise price, the call is worthless, but the riskless bond matures to its face value, X. The bond therefore provides a floor value to the portfolio. If
the stock price exceeds X, then the payoff to the call, S T 2 X, is added to the face value of the bond to provide a total payoff of S T . The payoff to this portfolio is precisely identical to the payoff of the protective put that we derived in Table 20.1 .
If two portfolios always provide equal values, then they must cost the same amount to establish. Therefore, the call-plus-bond portfolio must cost the same as the stock-plus-put portfolio. Each call costs C. The riskless zero-coupon bond costs X /(1 1 r f ) T . Therefore, the call-plus-bond portfolio costs C 1 X /(1 1 r f ) T to establish. The stock costs S 0 to pur- chase now (at time zero), while the put costs P. Therefore, we conclude that
C1 X
(11rf)T 5S01P (20.1) Equation 20.1 is called the put-call parity theorem because it represents the proper relationship between put and call prices. If the parity relation is ever violated, an arbi- trage opportunity arises. For example, suppose you collect these data for a certain stock:
Stock price $110
Call price (1-year expiration, X 5 $105) $ 17 Put price (1-year expiration, X 5 $105) $ 5 Risk-free interest rate 5% per year
We can use these data in Equation 20.1 to see if parity is violated:
C1 X
(11rf)T 5? S01P 171 105
1.05 5? 11015 1172115
This result, a violation of parity—117 does not equal 115—indicates mispricing. To exploit the mispricing, you buy the relatively cheap portfolio (the stock-plus-put position repre- sented on the right-hand side of the equation) and sell the relatively expensive portfolio (the call-plus-bond position corresponding to the left-hand side). Therefore, if you buy the stock, buy the put, write the call, and borrow $100 for 1 year (because borrowing money is the opposite of buying a bond), you should earn arbitrage profits.
Let’s examine the payoff to this strategy. In 1 year, the stock will be worth S T . The $100 borrowed will be paid back with interest, resulting in a cash outflow of $105. The written call will result in a cash outflow of S T 2 $105 if S T exceeds $105. The purchased put pays off $105 2 S T if the stock price is below $105.
Table 20.5 summarizes the outcome. The immediate cash inflow is $2. In 1 year, the various positions provide exactly offsetting cash flows: The $2 inflow is realized with- out any offsetting outflows. This is an arbitrage opportunity that investors will pursue on a large scale until buying and selling pressure restores the parity condition expressed in Equation 20.1 .
Equation 20.1 actually applies only to options on stocks that pay no dividends before the expiration date of the option. The extension of the parity condition for European call options on dividend-paying stocks is, however, straightforward. Problem 12 at the end of
the chapter leads you through the extension of the parity relationship. The more general formulation of the put-call parity condition is
P5C2S01PV(X)1PV(dividends) (20.2) where PV(dividends) is the present value of the dividends that will be paid by the stock during the life of the option. If the stock does not pay dividends, Equation 20.2 becomes identical to Equation 20.1 .
Notice that this generalization would apply as well to European options on assets other than stocks. Instead of using dividend income in Equation 20.2 , we would let any income paid out by the underlying asset play the role of the stock dividends. For example, European put and call options on bonds would satisfy the same parity relationship, except that the bond’s coupon income would replace the stock’s dividend payments in the parity formula.
Even this generalization, however, applies only to European options, as the cash flow streams from the two portfolios represented by the two sides of Equation 20.2 will match only if each position is held until expiration. If a call and a put may be optimally exercised at different times before their common expiration date, then the equality of payoffs cannot be assured, or even expected, and the portfolios will have different values.
Table 20.5
Arbitrage strategy Immediate
Cash Flow
Cash Flow in 1 Year Position S T , 105 S T $ 105
Buy stock 2110 ST ST
Borrow $105/1.05 5 $100 1100 2105 2105 Sell call 117 0 2( ST 2 105)
Buy put 25 105 2 ST 0
TOTAL 2 0 0
Example 20.6 Put-Call Parity
Let’s see how well parity works using real data on the IBM options in Figure 20.1 . The January expiration call with exercise price $130 and time to expiration of 44 days cost
$2.18 while the corresponding put option cost $4.79. IBM was selling for $127.21, and the annualized short-term interest rate on this date was .2%. No dividends will be paid between the date of the listing, December 2, and the option expiration date. According to parity, we should find that
P5C1PV(X)2S01PV(dividends) 4.7952.181 130
(1.002)44/3652127.2110 4.7952.181129.972127.21
4.7954.94
So parity is violated by about $.15 per share. Is this a big enough difference to exploit?
Probably not. You have to weigh the potential profit against the trading costs of the call, put, and stock. More important, given the fact that options trade relatively infrequently, this deviation from parity might not be “real,” but may instead be attributable to “stale” (i.e., out-of-date) price quotes at which you cannot actually trade.