September 2011 Q4

Hi,

I have some trouble understanding the role of managing costs in the question. The question says that the continuously compounded costs of managing the asset are 100x% (let's say 100x% instead of x% as in the question to make calculations easier!) of the value of the asset.

Now, if there is no further income from the asset, the value of the asset after T years will be S(0)*e(-xT) where S(0) is value of the asset at time 0.

However, according to the solution if there is an income y from the asset at time 0, its accumulation at time T is given as y*e^((x+r)T).

Does the managing cost play a reverse role in the income from the asset? That is to say the managing costs declines the value of the asset while it increases the rate of accumulation of any income from the asset.

Does this mean that there are no managing costs like commission, fund management etc on any income from the asset?

Thanks in advance.

 

Hi,

In general in arbitrage pricing questions, management costs are treated in exactly the same way as income, except that because they are payable and not receivable, you need to "change the sign" around.

So, in this (tricky) question, continuously compounded management costs are paid uniformly through time and are funded by selling a small amount of asset each instant of time. The amount of the asset held therefore declines continuously and exponentially through time. So, in order to end up with exactly one unit of asset at at time T (to replicate the payoff from the forward) you need to start off with exp(x*T) assets at time 0, which are then worth S0*exp(x*T).

So, at time t = 1/2, when you receive the income, you have exp(x*(T-1/2)) units of asset left and receive a cash income of y*exp(x*(T-1/2)). This then accumulates up at the risk-free rate to give a cash sum of y*exp((r+x)*(T-1/2)) at time T.

Phew!

 

starting with exp(xT) units of asset at time 0 how come and we end up with 1 at time T? Thanks