April 2010, Q2, put-call parity with dividends

[BeckyBoo]

Question 2 of the April 2010 paper is to prove the put-call parity when dividends are paid at a constant rate.

In the non-dividend case, I would set up 2 portfolios - A, consisting of a call and a cash holding of the strike price discounted back to current time & B, consisting of a put and a share.

Therefore, in the dividend paying case, I would expect to do something similarm, but adjusting the cash amount in portfolio A in line with the amount of dividend paid between the current time and the expiry date. (In a similar way to the exam style question on page 27 of chapter 10).

However, in the examiner's alternative "solution" they do not adjust the cash amount from what would normally be used in the non-dividend paying case. Since they do not give details of this solution though, I'm unable to work out where I'm going wrong!

Any help anyone could give would be much appreciated!

Thank you

 

[Mike Lewry]

Put-call parity with dividends

BeckyBoo,

To derive this relationship, we assume the dividends are received as a continuous stream and immediately reinvested to purchase more shares (well tiny fractions of shares, but we could scale up as much as we want to avoid this problem).

So, rather than the cash holding in the first portfolio needing to be adjusted, it's the share holding in the second portfolio. If one share is earning dividends at rate (or force) q, then with the reinvestment, we'll have exp[q(T-t)] shares at time T. You can think of this in the same way as cash earning interest at rate r: it will grow by a factor of exp[r(T-t)].

But to tie up with the payoff from the first portfolio, we need one share, so we must start with one share discounted at rate q, that is exp[-q(T-t)] shares in the second portfolio.

I hope that gives the extra info you need.

 

[Indy007]

Different portfolios

Hi, Can I just clarify a point on the correct approach to the proof.

Firstly, does the combination of portfolios matter so long as they equal the same in all states? (e.g could I assume portfolio A held a call option and went short on e-q(T-t) shares at time t and Portfolio B held one put option and borrowed Ke-r(T-t))

Secondly, would showing that arbitrage opportunities exist if the Put-call parity relationship didn't hold be sufficient for the marks (e.g. if ct + Ke-r(T-t)>pt + Ste-q(T-t), then arbitrage opportunites would exist by buying a put and a share and shorting a call and cash).

Thanks
Indy