April 2022 - Paper A - Q6

[Niall1234]

Hi.

I'm just a bit confused about how you go about setting up two portfolios to prove the forward price.

In this question we have fixed income being received on the share.

I'm just not sure how you know what to put in each portfolio in order to prove the forward price.

Any advice would be appreciated.

Thanks.

 

[Fiorentinaa]

The way I like to think about it is by starting with what we will need at time T(i.e the end time). As it is a forward contract we know at time T we will buy the share ST for K. To be able to buy the share at time T we will need to hold an amount of cash at time t (start time) to be able to do this so we hold the discounted value of the strike price K, which is Ke^-rT. Lets call this portfolio A.
Portfolio B will be one share.
So at time t
Portfolio A is worth Ke^-rT
Portfolio B is worth St.
At time T
Portfolio A we use the accumulated cash which is now worth K to buy one share which is worth ST.
Portfolio B our share is now worth ST.

These are equal at time T so by no arbitrage we know they are equal at every other time period including t so we know St=Ke^-rT.

However in this example we also have two dividends. We are told we can use I to represent the Present Value of the Dividends.
By using the same Portfolios above: we don't get dividends on Portfolio A as we don't own the share until after they are paid but we do get the dividends on portfolio B and so Portfolio B is now worth ST + Ie^rT at time T
Portfolio A and B are no longer equal
To make them equal we need to alter portfolio B so it will still be equal to ST at time T.
To do this we know we need -Ie^rt so we can use -I cash at time t as this will accumulate to -Ie^rt at time T
Portfolio B at time t is now St - I
Portfolio B at time T is therefore ST +IeRT (dividends) - Iert (the accumulated cash) which = ST
And therefore it is again equal to Portfolio A

So by no arbitrage we know at time t, Ke^-rt = St-I
Rearrange to get K

Hope this helps, appreciate it's a bit long winded