Ch 12 questions

[Adithyan]

Hi!

I am unable to understand why increase in dividends increases the value of put option. Pg13

Also with respect to the attached, I dont understand how at time T the "net holding of shares is 0". The person is actually paying K to get the share right? pg17

Could you please tell me what the value of the contract at a particular time t means?

Can you tell me how do you find is the initial value of the contract for question in the attachment.

Also help me with last expression on the attachment. what is St - Ke^-r(T-t). pg11

 

[Anna Bishop]

Hi Adithyan, I've put some responses below for you. Hope the examples help with understanding.
Anna

Dividends and value of the put option

Suppose you own a share and you are thinking of selling it. Here are two choices;
1) Sell the share on the open market today
2) Buy a put option to sell the share in 3-months' time
If dividends increase, what would you rather do?
Personally, I would prefer 2) rather than 1). I would like to benefit from the higher dividends before selling. This therefore increases the value of the put option.

Chapter 12 Page 17

At time 0, the Core Reading tells us that we have sold a share. What it doesn't make explicit is that we have 'short' sold the share. Short selling means borrowing a share worth S(0) from someone else and then selling it for S(0) in cash.

At time, T, we honour the forward contract by paying K and receive a share worth S(T). However, the person we borrowed the share from at time 0 will want it back! So although we receive a share worth S(T) under the forward, we then give it back to the person we borrowed it from. Thus the net shareholding is 0.

Values of forward contracts at time 0

Here's an example:
S(0) = 12, r = 0.05, T = 5

The forward price is set to be K = S(0)exp(rT) = 12exp(0.05 x 5) = 15.41.
The forward price is the amount of cash that will be paid in five years' time to buy the share.

The value of the contract is the expected present value of the future payoffs.

The long party expects to receive a share worth S(T) at time T and to pay K at time T.
The expected present value of these payoffs at time 0 to the long party is:
V(0) = S(0) - Kexp(-rT) = 12 - 15.41exp(-0.05 x 5) = 0

The short party expects to receive K at time T and to sell a share worth S(T) at time T.
The expected present value of these payoffs at time 0 to the short party is:
V(0) = Kexp(-rT) - S(0) = 15.41exp(-0.05 x 5) - 12 = 0

We can see that the value of the forward contract is worth 0 to both parties at time 0. That is deliberate, otherwise there would be an arbitrage opportunity.

Values of forward contracts at subsequent time periods

Over time, the share price may not actually grow in line with the risk-free force of interest and hence the value of the contract to each party changes.

Returning to the above example, let's say the share price falls at time 1 to S(1) = 10.

Remember that the value of the contract is the expected present value of the future payoffs.

The long party expects to receive a share worth S(T) at time T and to pay K at time T.
The expected present value of these payoffs at time 1 to the long party is:
V(1) = S(1) - Kexp(-r(T-1)) = 10 - 15.41exp(-0.05 x 4) = -2.62

The short party expects to receive K at time T and to sell a share worth S(T) at time T.
The expected present value of these payoffs at time 1 to the short party is:
V(1) = Kexp(-r(T-1)) - S(1) = 15.41exp(-0.05 x 4) - 10 = 2.62

In this example, because the share price has fallen at time 1, the forward contract has a negative value to the long party and a positive value to the short party. This is logical, since the long party has agreed to pay 15.41 at time 5, which, in light of the fall in the share price, is now more than what the share is expected to be worth at time 5.

Last expression

V(t) = S(t) - Ke^-r(T-t) is the value of the forward contract to the long party at time t. It is the expected present value of the future payoffs. S(t) is the expected present value of S(T) and Ke^-r(T-t) is the expected present value of the cash K at time T.

 

[Adithyan]

Thanks a ton for an elaborate reply. Just a small quick question.

If dividends increase do you mean put option price increases and hence one holds it during the tenure of rising dividends to reap the benefits of increasing the dividends?

Aren't we forgoing the dividends by holding put option.. is that fine? Increased income doesn't reach my hand as I will be holding only the put option right?

 

[mugono]

Thanks a ton for an elaborate reply. Just a small quick question.

If dividends increase do you mean put option price increases and hence one holds it during the tenure of rising dividends to reap the benefits of increasing the dividends?

Aren't we forgoing the dividends by holding put option.. is that fine? Increased income doesn't reach my hand as I will be holding only the put option right?

A put option gives the buyer the right but not the obligation to sell stock at some point in future (T).

A put option has the following payoff: max(K - F, 0)
F = Forward price; K = strike price
The forward price will (already) include the market's dividend expectation over the life of the contract.

All else equal, if the market suddenly revised its dividend expectations (e.g. because the company announced an unexpected upward revision in its dividend during the life of the contract) this would cause the price of the forward (F) to fall.

Why? Because the market would be expecting an even bigger drop to the stock price when the stock goes ex-dividend. As a result, the price of the put option would increase.


Finally, you asked: 'aren't we forgoing the dividends by holding the put option?'

Only stocks (and not options) accrue the actual dividend. The correct answer is that the dividend is embedded within the price of the put option. You could 'harvest it' by simiply purchasing the stock at the open on the ex-dividend date when the stock price would have dropped by the amount of the dividend. Have you made money? Probably not.

If you did make money you would have found yourself an arbritrage opportunity.

These ideas are fundamental in being able to master the option / derivative landscape.

I've intentionally tried to be succinct here. The ideas can take time to embed.

Happy to discuss.

 

[Anna Bishop]

If dividends increase do you mean put option price increases

Yes

and hence one holds it during the tenure of rising dividends to reap the benefits of increasing the dividends?

Yes, you could do. This would be an example of speculation. For this to work, you would have to pre-empt the increase in dividends before the rest of the market, otherwise the increase would already be reflected in the price of the put option (as Mugono mentions above). So you would buy the put option before the market expects dividends to rise and then sell the put option once the market has adjusted its expectations.

Aren't we forgoing the dividends by holding put option.. is that fine? Increased income doesn't reach my hand as I will be holding only the put option right?

It depends! You may actually own the underlying share and have bought the put option for hedging reasons, ie you want to lock into the price at which you can sell the share in three months' time. If you own the underlying share then the increased income will reach your hand!

 

[Adithyan]

If dividends increase do you mean put option price increases

Yes

and hence one holds it during the tenure of rising dividends to reap the benefits of increasing the dividends?

Yes, you could do. This would be an example of speculation. For this to work, you would have to pre-empt the increase in dividends before the rest of the market, otherwise the increase would already be reflected in the price of the put option (as Mugono mentions above). So you would buy the put option before the market expects dividends to rise and then sell the put option once the market has adjusted its expectations.

Aren't we forgoing the dividends by holding put option.. is that fine? Increased income doesn't reach my hand as I will be holding only the put option right?

It depends! You may actually own the underlying share and have bought the put option for hedging reasons, ie you want to lock into the price at which you can sell the share in three months' time. If you own the underlying share then the increased income will reach your hand!

Can I understand this in the most simplest of terms as increase in dividends reduces the capital value of share and hence value of put option increases?

 

[mugono]

Can I understand this in the most simplest of terms as increase in dividends reduces the capital value of share and hence value of put option increases?

Replace ‘share’ with the forward price... then yes.

The share price will drop in future - the day the stock trades ex its dividend.

 

[Anna Bishop]

Hi Adithyan

We have certainly opened a can of worms here have we not?

I agree with Mugono about what happens to the put option price when a share goes ex-dividend.

Just before a share pays a dividend, it goes 'ex-dividend'. This means that, if Person A sells the share to Person B on this date or shortly after, then Person A (not B) will get the next dividend. This is because there is not enough time for the company paying the dividends to change over the payment details to Person B. Now, because Person B is not going to get this next dividend, she does not want to pay for this dividend when she buys the share. Hence, the price of the share tends to drop on the ex-dividend date, by the amount of the dividend.

So, on the ex-dividend date:
- the share price, St, falls
- the intrinsic value of the put option, max(K-St, 0) increases
- the put option price increases.

However, I think there is more to it than this! Having reread Chapter 12 and also the relevant section in a couple of text books, we are meant to be considering the effect of a change of one of the factors, eg St, K, sigma, r, q, t in isolation to the rest of the factors. Perhaps in a more theoretical rather than practical sense?

If this question came up in the Subject CM2 exam, the examiners would not be expecting you to discuss the share going ex-dividend.

The Core Reading, on Chapter 12, page 13, assumes that you own a share and that you've bought a put option (for hedging purposes) to be able to sell your share at a later date at a known price.

In this situation, if dividends increase then, ignoring all impacts on other variables, there is increased (time) value to having the put option. This is because, by holding the share and deferring its sale (via the put option), you benefit from the increased dividend income relative to if you sold the share today for cash.

This is the angle I would encourage you to take in the exam as this is what has gained the marks in the past!
Not easy is it? But thank you for asking the questions!
Anna

 

[Adithyan]

At the outset I would like to thank you for taking so much of efforts to help me understand the concept. I am now able to get a better picture of what you are trying to convey. It was that I was missing on the aspect that the person is holding both the share and the put option.

Just to ensure I got the point right, As a holder of both the share and the put option, I benefit from both the dividend increase and increase in Time value of the option.

 

[Anna Bishop]

Assuming you hold the share, you definitely benefit from the increase in the dividend (on your share)!

Whether or not you benefit from the increase in the value of the put option depends on your reasons for holding the put option.

If you are holding the put option with the intention of exercising it at maturity and selling your share, then you won't benefit from an increase in the put option value. At maturity, you either exercise your option by selling your share, worth S(T) in return for cash of K, or you don't exercise the option and the payoff is 0. You don't get anything for the put option itself at this point!

If you have bought the put option for more speculative reasons then there is nothing stopping you buying a put option at price p(t) and then writing a put option at price p(t') a few days later to benefit from any increase in value. However, by holding and writing a put option on the same underlying asset with the same strike and maturity, your payoff at maturity is:

max(K - S(T),0) - max(K - S(T),0) = 0

So it very much depends on what you are trying to achieve - are you hedging (ie trying to protect the value at which you can sell your share) or are you speculating?

 

[Adithyan]

Thank you Anna for a detailed reply. It was indeed very helpful! Sorry for the delay in my response.

 

[Sandor Kelemen]

Hello all,

Can I ask on top of the above quite instructive discussion for a similar explanation however for the effect on option price in case of the isolated interest rate movement?

The core reading states: An increase in the risk-free rate of interest will result in a higher value for a call option because the money saved by purchasing the option rather than the underlying share can be invested at this higher rate of interest, thus increasing the value of the option.

So we can compare a call option to buying immediately the share itself. In what "bloody" sense can we do this comparison? I think I fully miss something here.

Note: Having the Garman-Kohlhagen formulae in hand all this analysis turns into rigorous partial derivative math. However, I am quite fighting with these first principles.

Side question: not having any model on share price (as the case in chapter 12). How should one interpret the share price volatility? I think it is a bit vague concept there.

Thanks in advance for your help.

Sándor

 

[Anna Bishop]

Hi Sandor

Maybe I think of these things too simplistically ... but this is often a useful approach for the exam!

Re the call option and the interest rate movement, I give myself two choices. I want to buy a share. I can either:

1) buy the share now
2) buy a call option to buy a share at some point in the future (eg in 3-months time).

If interest rates increase, then I would prefer the second approach (assuming that I am not concerned about when I get the share). The second approach means that I can keep my money in the bank and earn the higher interest rate. This increases the desirability of the second approach, increases the demand for the call option and increases its price.

Re volatility of the share price, I interpret this as the standard deviation of returns.

Anna