II. THE BASIC IDEA Suppose the current price of an underlying stock is S=$50, and at the end of a period a1 t ine, it: p: IIe mut t r'it irt 2:: 4-h.'t S:=$100. A call on the stock is available with a striking price of K=$50, expiring at the end of the period. It is also possible to bor row and lend at a 25 rate of interest. The one piece of information left unfurnished is the current Price c of the call. However, if riskless profitable arbitrage is not possible, we can deduce from the given information alone what the price of the call. must be! Consider forming the following levered hedge: (l)write 3 calls at c each (3)borrow $l0 at 25%, to be paid back at end of period Table 1 gives the return from this hedge, for each possible'level of he stock price at expiration. Regardless of the outcome, the hedge exactly breaks even or the expiration date. Therefore, to prevent profitable riskless arbitrage, its current value must be zero; chet is 3c+100-40=0 The current value of the call must then be C=$20 To keep matters sir against cash dividends mple, assume for now that the call is protecte le also ignore transactions costs, margin, and
Table I. Arbitrage Table Illustrating the Formation of a Riskless Hedge Present Expiration Date Date s大=25S大=100 Write 2 calls -150 Buy 2 shares 50 200 Borrow 50 Total 一一一 If the call were not priced at $20, a sure profit would be pos- sible. In particular, if C=$25, the above hedge would yield a current cash inflow of $15 and would experience no further gain or loss in the future. On the other hand, if C=$15, then the same thing could be ac- compIished by buying 3 calls, selling short 2 shares, and lending.$4 Table 1 can be interpreted as demonstrating that an, appropri- ateiy Levered position in stock wiiz replicate the future returns of a calL, That is, if we buy shares and borrow against them in the right proportion, we can, in effect, duplicate a pure position in calis. In view of this, it should seem less surprising that all we needed to de- termine the exact value of the call was its striking price, underlying stock price, ramge of movement in the under lying stock price, and the rate of interest. What may seem more incredible is what we don't need to know: among other things, we don't need to know the probability that
the stock price wiZz rise or fall. Bulls and bears tust agree on the value of the call, relative to its underlying stock price Clearly, our numerical example has been chosen for simplicity, not reals Among other things, it gives no consideration to the existing liquid secondary market, which permits closing transactions any tine priot to expiration, and it posiTs Very tmli't i
further by setting d=u and q =5, we choose to retain this greater leve】 of generality When the call expires, we know that its contract and a rational exercise policy imply that its value must either be c E max[O,uS-KI C,三如ax[0,uS一K] with probabi1ityq C, E max[O,ds -k] with probability 1-q Suppose we form a hedge at the beginning of the period by writing one call against a shares of stock, This would cost cS-C. The buyer of the call will either retain it until expiration or exercise it immed iately. This will depend on which is higher, the retention value or the exercise value, max[O, S-K]. To find out. we will first calculate the value of the call if he retains it. If the call is unexercised. then our hedge will return aus-c, with probability q ads-C d with probability q Now, since we can choose a any way we wish, suppose we select the"neu- tral hedge ratio, that is, the a that makes the hedge riskless. We ac complish this by selecting the a which equates the dollar returns in the two possibilities:
ads =auS〓 Solving this equation, the hedge ratio a which eliminates all risk is with this hedge ratio, since the return from the hedge, ads is riskless, to prevent riskless profitable arbitrage, it must have t same return as an investment of as-c dollars in riskless borrowing or lending. Therefore, ads-C, = r(as-c Rearranging this equality and substituting for a To state this more simply, observe that defining p==u, ther 1-p- Therefore (1) c=[ (1-p)c This is the exact formula for the value of a call one perlod prior to ex piration in terms of s, K, u, d, and r The formula gives the value of the call 1f, as we assumed, the buyer does not immediately exercise it. However, it easy to see that