Relationship between American Calls and Put Options (no dividends)

[Bruno]

Hi David,

The inequality for american options is (I use "X" for the strike price):

S - X ≤ C - P ≤ S - Xe^(-rT)



My question is:

Is it always a covered call (either with an american or european call) greater or equal than a protective put (constructed with an american put)?

Covered call ≥ American Protective put

C + X ≥ P + S or
c + Xe^(-rT) ≥ P + S




As per the inequality for american options, S - X ≤ C - P ≤ S - Xe^(-rT)

so rearranging the lower bound, we can arrive to my question C + X ≥ P + S

and since C = c, also c + Xe^(-rT) ≥ P + S

Is this correct?

Thanks in advance.

 

[Bruno]

Sorry my previous question contained an error. I mean fiduciary call and not covered call.

Is it always a fiduciary call (either with an american or european call) greater or equal than a protective put (constructed with an american put) ?

Fiduciary call ≥ American Protective put

C + X ≥ P + S or
c + Xe^(-rT) ≥ P + S



As per the inequality for american options, S - X ≤ C - P ≤ S - Xe^(-rT)

so rearranging, C + X ≥ P + S

and since C = c, also c + Xe^(-rT) ≥ P + S

Is this correct?

 

[David Harper CFA FRM]

Hi Bruno,

Interesting! I'd start with the fact that put-call parity sets an equality (=) if the options are European; i.e.,

put call parity says:
fiduciary Euro call = protective Euro put (on same stock, both options same strike price)

So, by put call definition, in regard to European options, they are always the same.

On the American side, I get a different result: I get the fiduciary American call must be less than or equal to (<=) protective American put.

If we take put-call parity, but replace American with European (replace c with C and p with P), then Hull 9.4 says:
S - K <= C - P <= S - K*e^(-rT)

So I actually get from this equation the different result:
C + K*e^(-rT) <= P+S; i.e., fiduciary American call must be less than or equal to (<=) protective American put

but "on the other side of the equation" I do get your result but i am unsure how to "give it a name:"

C+K >= P+S; i.e., American call plus (bond = strike) is greater than >= protective American put. But technically, I am not sure what to call this. I don't think this is a fiduciary call. I think technically the fiduciary call is discounted strike, per the put-call parity because (per the delta hedge) you are borrowing only as much as you need for the exercise in the future.

(I don't get how we can apply c=C? I've seen Hull use the c=C in the solutions book, so you have a precedent, but I don't understand why he can do that. Unless I just miss something, an American call may converge to a Euro call as maturity nears, but otherwise C >= c)

Thanks, these are great practice to understanding....

David

 

[Bruno]

Hi David,

I see your point that C + K is not a fiduciary call because the amount you will receive at expiration with the bond (strike price) is not discounted but still C + K ≥ P + S.

Sorry to keep bothering you, but would you please check the following:


The value of a fiduciary American call is (S - K) + Ke^(-rT) when the call is in-the-money and it is exercised early K and Ke^(-rT) will not cancel out. Therefore, a fiduciary American call should be equal to than a covered European Call because it does not make sense to "exercise" the fiduciary american call early.


The reason Hull uses C=c and not C ≥ c is that when an American call is exercised, it is only worth S - K. Since this value is never larger than S - Ke^(-rT) for any r and T > 0, it is never optimal to exercise early. That is, the investor can keep the cash equal to K, which would be used to exercise the option early, and invest that cash to earn interest on expiration. Since exercising the american call early means that the investor would have to forgo interest, it is never optimal to exercise an American call on nondividend-paying stock before the expiration date. Therefore C=c

Thanks

 

[Bruno]

I keep making the same mistake

The value of a fiduciary American call is (S - K) + Ke^(-rT) when the call is in-the-money and it is exercised early K and Ke^(-rT) will not cancel out. Therefore, a fiduciary American call should be equal to than a FIDUCIARY European Call because it does not make sense to “exercise” the fiduciary american call early.

 

[David Harper CFA FRM]

Hi Bruno,

Please, it's no bother...

I can't agree with any of this, sorry. But I surely may be wrong! It seems to me that both points above depend on the same idea, that C = c = S-K. If that's true, clearly you can substitue S-K = C into C + K*e^(-rT) <= P + S and get:

S - K + K*e^(-rT) <= P + S

But I cannot seem to agree with either C=c or c/C = S-K

Of course, I agree with "It is never optimal to exercise an American call on nondividend-paying stock before the expiration date" (under the restrictive assumptions)

But I see broken logic in going from that to this: C = S- K or C=c

C=S-K or c=S=K says "option value is worth only intrinsic value and zero time value." I do not see how this is possible except at the instant before expiration.

C=c says the option to early exercise (an option on an option) is worth zero, but the option (on an option) must be worth something, so I still don't see why it can be any other way but: C >= c

(I looked up the solutions manual where Hull says c=C but he does not explain. My best guess is that is a typo)

David

 

[David Harper CFA FRM]

Bruno,

I looked up the CFA texts, and indeed they say (this is quite news to me):

when the underlying makes no cash payments, c = C

So you appear to be right about that. I still can't see why C = S-K, as time value must (?) be worth nonzero...

David

 

[Bruno]

Right now, I don't have the CFA books with me. I guess you are referring to the one written by Chance. I will take a look at the book next Monday.


Of course, I agree with “It is never optimal to exercise an American call on nondividend-paying stock before the expiration date” (under the restrictive assumptions)
But I see broken logic in going from that to this: C = S- K or C=c


IMO, the problem is embedded in that statement. Do we really agree that it is never optimal optimal to exercise an American call on nondividend-paying stock before the expiration date (under the restrictive assumptions)? If we agree, I guess we can "go back" from C ≥ c to C=c.

As far as I understand it, the reason C = S - K is that you can exercise the option at any time.Therefore, you can choose to immediately "pay" the strike price for the stock. In other words, the lower bound for an American call (in-the money) must be S-K because you can exercise it right away. However, c= S - Ke^(-rT) is more valuable than S -K. Therefore, under the restrictive assumptions both C and c should trade at the same value.

I agree with you that if C=c and c = S - Ke^(-rT), it does not make sense to believe that C = S - K
(except at the instant before expiration) because, as we have been instructed, apparently it doesn't make sense to exercise an American call on nondividend-paying stock before the expiration date . Therefore, S - Ke^(-rT), should prevail over S - K.
However, if you believe that the stock price will decrease (S), the risk-free rate will decrease and the volatily will decrease, then S - K should prevail over S - Ke^(-rT) and C will be higher than c.
So we are back to the beginning. I don't see the light. Do you?


C=c says the option to early exercise (an option on an option) is worth zero, but the option (on an option) must be worth something, so I still don’t see why it can be any other way but: C >= c


I totally agree with you. The lower bound for an American call (in-the money) must be S-K because you can exercise it right away. It may be worth more for the fact that there is a probability that S, t, and volatility will decrease before expiration (penalizing c). But once again, is it true that it's never optimal optimal to exercise an American call on nondividend-paying stock before the expiration date?


Thanks

 

[David Harper CFA FRM]

Hmmm...

here are the assertions that I am more comfortable with (or rather, I have merely taken them for granted for a long while):

1. the lower bound on a Euro call is stock - discounted (strike); c >= S - X*exp[(-r)(T)]. It is easily shown that if the stock increases at the risk free rate, the PV of the future gain is this minimum value (long history; FAS 123 allowed it for private company options, calling in the 'minimum value')

2. This is consistent with: option value = intrinsic value + time value (i.e., value attached to the fact you can hold the option for longer and may get more upside).

3. If the above is true (option value = intrinsic + time value), then both c >= S-K and C >= S-K, but as shown, for a Euro call, that is an inferior lower bound, the true (true = highest possible lower bound) lower bound is c >= S - X*exp[(-r)(T)] as it credits the intrinsic value plus something for the time value.

Here is what I have not logically been convinced of:

That "it is never optimal optimal to exercise an American call on nondividend-paying stock before the expiration date (under the restrictive assumptions)" necessarily leads to c=C. (despite the fact both Hull and CFA text have it). Because C ought to be exercised like c does not convince me that c=C.

You ask, "Do we really agree that it is never optimal optimal to exercise an American call on nondividend-paying stock before the expiration date" and I think it's important this has a specific meaning. It isn't true in practice (i.e., no executive/employee option holder *ever* acted this wway) because it abstracts risk aversion and liquidity out of the picture. It is a statement about what is rationale behavior under perfect forecasting (the stock price will continue to increase, is not temporarily mis-priced) and, more importantly, restrictive assumptions (namely, the holder does not value the liquidity). What i really mean here is, the proofs above about the lower bound are proofs (they hold regardless or risk aversion and utility/liquidity prefs of holder) but this statement exists on a lower plane; ergo it does not lead to a conclusion about the lower bound.

So, my question is still: assume you will pay me $X for a Euro call (c) that expires in T years. Now instead I will ADD A FEATURE (the early exercise option) and instead sell you an American call (C). I have added a feature (a liquidty option on the option) and subtracted nothing from the instrument, how can it be otherwise than C is worth at least as much or more (>=) than c ? Even if you rationally should hold to the end, you still hold an liquidity option on the option, which is worth >0.

David

 

[Bruno]

David, I cannot agree more with you.

I would try to investigate why Hull and others state that C=c just for intellectual curiosity.
I'll keep reading around and post back when/ if i find something convincing!

best regards,

 

[Bruno]

David,

I think I've found a sound argument for C=c

Proposition: An American call option on an asset without dividends should never be exercised early—but perhaps sold. It therefore has the same price as a European call option.

Suppose that you are pretty sure that price of the underlying will drop tomorrow. The above argument shows that you should still not exercise the call option, but it might be sensible to sell the option today. If we exercised early, then we would effectively through away the put protection (against downside movements) inherent in the call option and be left with the underlying asset (recall from the European put-call parity that the call option can be thought of as a portfolio of the underlying, a put, and some cash) and also pay the strike price now instead of later—neither of which is good (and which a potential buyer of the call option would be willing to pay for).

Proof: To avoid early exercise, selling (getting C) should be more profitable than exercising (getting S - K), C > S - K.

Put-call parity for European options says c = S - K + [1 – e^(-rT)]* K + p

The sum of last two terms is positive (before the expiration date), so c > S - K. Since C ≥ c ,
we have, C > S - K;

so selling the option is always more profitable than exercising early. The reason is that early exercise throws away the put protection (p) and also the “rebate” due to later payment of the exercise price (pay K instead of the present value K*e^(-rT).