value of put as interest rates increase

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myzegna

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Hi, can anyone tell me what happens? I believe the value of the put will decrease?

Also, with regards to put-call parity, why does it seem that that as time increases, the value of your put option decreases? I am pretty sure that it should go up.

P = C + S + X(e^rt)

Sorry for the basic and probably dumb qns but i really can't remember!
 
Well, from memory the formula is:

call + bond = stock + put

So rearranging actually gives you:

put = call + bond - stock

So there's an error in your formula.

I also think that the bond should be X(e^-rt). i.e. the PV of the bond falls as rates and time increase.

Options arn't my best area, but if rates rise i'd expect the present value of the bond to fall, the stock to stay about the same, and therefore either the Call needs to increase in price, or the Put to fall in price - most likely a bit of both. I dare say you could prove this rigourously by going back to Black Scholes.



Edited 1 time(s). Last edit at Friday, June 29, 2007 at 06:02AM by DeadCat.
 
The illusion that the value of the put falls as time increases stems from assuming that the value of the call remains constant. It wont. It will increase too.

Infact, on reflection I think much of your confusion stems from using the put call parity relationship innapropriately.

You can use the formula once you have values for the three other variables, but if you want to calculate the value of a particular put or call you need to start by using Black Scholes. If you pump the numbers into that, you will see that the value of the call or put rises as the time span increases.
 
"Well, from memory the formula is:

call + bond = stock + put

So rearranging actually gives you:

put = call + bond - stock"

That looks good to me.^

The bond would be X/(1+r)^t...I believe.

So as the risk free rate rises the value of the bond goes down...hence the value of the put goes down.

This isn't my area of expertise either, so I may be oversimplifing. Only a hopeful level 1 pass.
 
myzegna Wrote:
-------------------------------------------------------
> Hi, can anyone tell me what happens? I believe the
> value of the put will decrease?
>
> Also, with regards to put-call parity, why does it
> seem that that as time increases, the value of
> your put option decreases? I am pretty sure that
> it should go up.
>
> P = C + S + X(e^rt)
>
> Sorry for the basic and probably dumb qns but i
> really can't remember!



Edited 1 time(s). Last edit at Friday, June 29, 2007 at 11:01AM by abacus.
 
I recall that the value of the put and the call INCREASE as the interest rate goes up. It has something to do with how the arbitrage portfolio changes as a result of higher interest rates. It is not as intuitive to me as I want it to be, but I remember trying to pound this into my brain before the exam.

The put call parity formulation misleads, because both the put and the call values increase when interest rates rise. What the put-call parity formula does tell you is that the DIFFERENCE in the put price and call price decreases if the interest rate goes up.
 
bchadwick,

How could the value of the call and put both increase as interest rates went up?

call + bond = stock + put

So:

C=S+P-B
and
P=C+B-S

With the call a larger interest rate (RFR) would lead to a smaller value for the bond. Because you are subtracting the value (short) it would lead to less subtraction or a higher value for the call.

With the put you are adding less (long) for the bond so it is less valuable.

There is no doubt that the higher interest rate decreases the PV of the bond in both cases, but with the call you are short and with the put you are long.

Where am I screwing up?



Edited 1 time(s). Last edit at Friday, June 29, 2007 at 11:22AM by mwvt9.
 
At least with European options on stocks and other similar B-S like securities, rho is negative for puts (i.e., interest rates up, put down) and positive for calls (i.e., interest rates up, call up). I'm sure if I think about it long enough I can find exceptions to this for other securities (bond options don't count as delta completely overwhelms rho).
 
mwtv9,

Here's the put call parity formulation:

* call + bond = stock + put

Rearranging

* bond - stock = put - call = difference between put and call.

The PV of the bond goes down with a higher interest rate, so assuming the stock price is constant, what happens is that the difference between the put and call price decreases.

This is formulation you used earlier:

* put = call + bond - stock

The problem with this formula is that the interest rate affects three things: 1) the put price, 2) the call price, and 3) the bond price. (probably will affect the stock price too, but let's assume that doesn't change).

The problem is that (as far as I can tell) you assumed that the call price was constant. However, if you are assuming that the price of the put option will change with interest rates, it seems unreasonable to assume that the price of a call option won't change.


Joey pointed out that the sign of rho changes with puts and calls, something I knew on test day but forgot subsequently. I remember that calls prices increase with interest rates, because this seemed counterintuitive to me, and presumably has to do with how interest rates affect the arbitrage portfolio.

Let's see... a call can be made synthetically by:

* call = delta*shares - bond

(maybe I'm wrong on this, please correct, but please don't flame)

So the value of the bond decreases with the interest rate, and the value of the call goes up. Maybe delta depends on the interest rate, though...



Edited 2 time(s). Last edit at Friday, June 29, 2007 at 11:51AM by bchadwick.
 
You can either buy a put option on the stock or short the stock itself. If you short the stock, you can immediately invest it at the risk free rate and get a return. Higher the interest rate, higher will be your incentive to short the stock instead of buying a put option. Thus the price of a put option decreases with an increase in the interest rate.
 
Thanks abacus, that's nice and clear.

Also, rereading my post, I realized that I implicitly assumed that the stock price was below the strike price, if that's not true, then the absolute value of the difference between put and call prices could increase, although technically the difference does decrease in the sense of "getting more negative."

Geez, I thought I understood this stuff at one point...
 
thanks guys. i was trying to think of it while having beer in the office! yeah my formula typed above was wrong.

Also, I think my confusion stemeed from the fact i was confusing put-call parity...

I have lost the ability to think since starting working...
 
Abacus' way of looking at this is neat - if that works for you, you should use it.

I still like put call-parity. Following DeadCat:

"So rearranging actually gives you:

put = call + bond - stock

So there's an error in your formula.

I also think that the bond should be X(e^-rt). i.e. the PV of the bond falls as rates and time increase. "

So now interest rates increase and what happens to the put?

First, the stock does nothing because there is only some probabilistic relationship between interest rates and stocks and that is not what we are talking about here. Now if this is an option on some AA preferred stock, obviously interest rate is the only thing going on and delta now overwhelms rho. We're talking rho here so stock does nothing.

Next, the bond is a zero-coupon bond. That means that it has the maximum duration for any bond of the same maturity as the option expiration. That's lots of interest rate sensitivity. Interest rates increase, bond value decreases which should imply that the put price decreases. Except as bchadwick points out, there is the issue of the call.

If the call increases in value more than the bond decreases, then the conclusion that the put decreases in value when interest rates rise is suspect. On the face of it, this must seem unlikely because the effect on an option must somehow be greater than the effect on a zero coupon bond. We know that zeroes are highly sensitive to niterest rate changes but the effect of interest rate changes on options is unclear. Right off the bat, we have to suspect that the effect on the call is less than on the zero.

To make that a bit more rigorous, think about a delta = 1 call. This call is just a stock substitute except that you are not out the money for the stock yet. It allows you to keep and earn interest on $X/share in the bank until expiration with no uncertainty about what you are going to do at expiration (that's the delta = 1 part). That means you get to own a zero which has exactly the same interest rate sensitivity as the bond. So now the interest rate sensitivity of the call and the bond cancel.

So move to a delta = 0.1 call. The chances that you exercise this are dramatically less and in fact let's say it is 0.2. This means that we could expect to have 5 independent options all with the same characteristics and keep that bank account for all 5 which in some sense means that each option has 1/5 the interest rate sensitivity of the zero. That means the zero wins. In essence, the call option is sensitive to interest rate moves in direct proportion to how likely it is to be exercised. If it is completely certain to be exercised, it has the same sensitivity to interest rates as the bond. Otherwise it has less.

Obviously, I don't go through this every time I have to come up with this but I can remember that the apparent conclusion from put-call parity is correct.
 
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