American call option arbitrage bounds
The value of a European put-option is less than or equal to X e -rT. Since the value of the option at expiration is less than or equal to X for all outcomes, then its value at time T until expiry must be less than or equal to X e -rT.
The proof is that otherwise we could: This follows from the intersection of the upper and lower bounds for the value of a European put-option. The value of an American put-option at zero stock-price equals X.
The proof begins with the reminder that the maximum payout on expiry of a put-option is X. As soon as the stock-price becomes zero the holder of an American put-option can exercise the option and receive the maximum payout immediately, earing interest on it until the expiry date.
Because nothing is gained by holding the option its value if S is zero, at any time T before expiry, is X; that is, all price-curves pass through the point 0,X. Price-curves for put-options Here is an applet which uses the Black-Scholes formulas from a later chapter to draw price-curves time-curves , p vs S, for put-options on nondividend-paying stock.
Relate the curves to the bounds obtained above. Earlier, we introduced an applet to generate price-curves for a call-option, and an applet to generate price-curves for a European put-option. We can combine these two applets into an applet which allows us to verify that the price-curves for the two options do intersect above the point X e -rT , 0 on the S-axis. In the first, the option is exercised at some time t before the expiry time T: X e -rT X e -rT e rt At time t the value of G exceeds the value of H.
In the second case, the option is held until expiry: X e -rT X At time T the value of G is no less than the value of H. Thus, whatever case, the value of G is no less than, but could exceed, the value of H. Interpret the above two bounds on the price of an American put-option on the P vs S diagram; and see, again, that for zero S the value of P is X.
Options Arbitrage
Use the P vs S diagram to see that for sufficiently low values of S the value of an American put exceeds the value of a European put. Infer that the value of an American put always exceeds the value of a European put.
Option Price - Upper and Lower Bounds I have divided the following into parts indicated by centre headings: The proof is that otherwise we could sell a call-option, and use the premium to buy a share, with some cash left over.
At expiry we have a share to satisfy the call, if exercised, together with cash which has accrued interest.
Minimum and Maximum Values for Options
A lower bound for the value of a call-option In the preceding chapter we saw that at expiry the value of a call-option is max S-X, 0.
We now see Hull p that a lower bound for a European call-option on nondividend paying stock is max S - X e -rT , 0 , where T is the time remaining until expiry.
This lower bound gives an indication of how the price curve advances to the right as the time-to-expiry approaches zero.
American call-options An American call-option on a nondividend paying stock has the same value as the corresponding European option.
See Hull p who begins by showing that it is ' never optimal to exercise an American call option on a nondividend paying stock before expiration time '. Price-curves time-curves for a call-option The evolution of the price curve may be displayed in several ways: See the applet in this chapter. For an excellent example visit Espen Haug's Virtual Reality Option World. This would be more that enough to fulfill the put obligation which cannot exceed X.