Put call parity - dividend paying shares. October 2010 Q8

Hi

On question 8 of Oct 2010, we are asked to consider a share which pays dividend D at time T'. We also consider c_t and p_t that are both written on the share and mature at time T, where T> T'.

The solution presents portfolio B set up at current time t as: one put + one share less a borrowing of D.

Please could someone help - I have the following questions:
1. Is the "one share" dividend paying or non dividend paying?
2. If the "one share" is dividend paying, then why do we need to borrow D? Would the dividend paying share not provide for D itself at time T'.
3. On the other hand Portfolio A at time t contains one call + cash lump sum of Ke^-rT. Why is there no explicit allowance for D in this case? Are we only to make an explicit allowance in one of either portfolio A or B - and if so, does it matter which it is?

Thanks

 

In a put call parity, we have:
Put(t) + Share(t) = Call(t) + Ke^-r(T-t)

Let's take a simplified example of a dividend paying share, where the share goes ex dividend exactly a few milliseconds before expiry (so you will get price of share at T + "D" on expiry date). Let's say you also borrow D*exp^-r(T-t) right now where you pay out exactly "D" time T.

In this case, using the put call parity equation,

If S(T) > K, 0 + (Share(T) + D) - D = (S(T) - K) + K => S(T) = S(T)
If S(T) < K, (K-S(T)) + (S(T) + D) - D = 0 + K => K = K

So the portfolio in both sides replicate each other.

Their present value is:
P(t) + S(t) + D*exp^-r(T-t) = C(t) + Ke^-r(T-t)

Hope this helps.

 

In cases like this, wIth a lump sum dividend that is not a proportion of the share price, you always:


(1) assume that the dividend is reinvested at the risk-free force of interest
(2) adjust the corresponding arbitrage pricing formula without dividends by deducting the present value of the dividend from the current share price.

So, in this case, the present value of the dividend paid at T' is D*exp[-r*(T' - t)]

and so put-call parity becomes:

C(t) + K*exp[-r(T-t)] = P(t) + S(t) - D*exp[-r(T'-t)]

To show this, perhaps the two simplest portfolios to use are:

A: call + cash (equal to discounted present value of strike price)
B: put + share - cash (equal to discounted present value of dividend)

For Portfolio B, the accumulated values of the dividend and the minus cash item then cancel each other out at maturity, so that both portflios provide a payoff of max[ST, K] at time T, ie the same payoffs as put-call parity without dividends.