P1.T3.Hull Notes - American Call Option

[Jaskarn]

Can you please explain the reason behind this statement that I read in notes.

From a mathematical standpoint, it is never optimal to execute an early exercise on an
American call option on a non-dividend paying stock. However, it can be optimal to execute
an early exercise on an American put. In general, we can say that for an American put, the
early exercise becomes more attractive as:
 Stock price (So) decreases,
 Risk-free (r) rate increases, and/or
 Volatility (s) decreases.

 

[David Harper CFA FRM]

Hi @Jaskarn (This thread is actually for errors ....) There is a good discussion here in which I learned something from Alex https://forum.bionicturtle.com/thre...ends-on-put-call-parity-hull.4616/#post-16189 e.g.,

Early exercise of an American put becomes more attractive as the risk-free rate increases and volatility decreases, because:
The early exercise of an American put option is attractive when the interest earned on the strike price is greater than the insurance element lost. When interest rates increase, the value of the interest earned on the strike price increases making early exercise more attractive. When volatility decreases, the insurance element is less valuable. This makes early exercise more attractive.

About the early exercise of an American call option, Hull writes:

"To summarize, there are two reasons an American call on a non-dividend-paying stock should not be exercised early. One relates to the insurance that it provides. A call option, when held instead of the stock itself, in effect insures the holder against the stock price falling below the strike price. Once the option has been exercised and the strike price has been exchanged for the stock price, this insurance vanishes. The other reason concerns the time value of money. From the perspective of the option holder, the later the strike price is paid out the better."

About the early exercise of American put option, Hull writes:

"It can be optimal to exercise an American put option on a non-dividend-paying stock early. Indeed, at any given time during its life, the put option should always be exercised early if it is sufficiently deep in the money.

To illustrate, consider an extreme situation. Suppose that the strike price is $10 and the stock price is virtually zero. By exercising immediately, an investor makes an immediate gain of $10. If the investor waits, the gain from exercise might be less than $10, but it cannot be more than $10, because negative stock prices are impossible. Furthermore, receiving $10 now is preferable to receiving $10 in the future. It follows that the option should be exercised immediately.

Like a call option, a put option can be viewed as providing insurance. A put option, when held in conjunction with the stock, insures the holder against the stock price falling below a certain level. However, a put option is different from a call option in that it may be optimal for an investor to forgo this insurance and exercise early in order to realize the strike price immediately. In general, the early exercise of a put option becomes more attractive as S0 decreases, as r increases, and as the volatility decreases."

Stepping back from the details, I would highlight two differences between the call and put option that, to me, inform this difference:

• Time favors the call option because the stock price is expected to drift upwards at least be the risk-free rate. A stock price that is drifting upward, ceteris paribus, is enhancing the call option value but decreasing the put option's value
• The put option has a cap on the payoff but the call does not: a deeply in the money call option can further increase in its payoff, but a deeply in the money put has a limit on further gains.
• S(0) decreasing --> implies the put is moving closer to its limit; R(f) increasing is working against the future payoff of the put b/c it implies the minimum future stock price is higher, and therefore MAX(0, K - S) is lower; volatility decreasing, as Alex wrote, makes the "insurance element" of the put less valuable. I hope that's helpful!

 

[Nicole Seaman]

@David Harper CFA FRM @Jaskarn

Please note that I moved this question out of the Study Note Errors thread, as this was not an error being pointed out.

Thanks!

Nicole