Sample Question #3

independent1019 · #1 09-28-2007, 10:18 PM

Hi, can someone explain to me how did they get {S(1)+max(0,103-S(1)}={S(0)+15.2}. I don't really understand their explanation. Thank you!

jraven · #2 09-28-2007, 10:51 PM

We need to construct a portfolio whose payoff will be

$\text{Payoff} = \pi (1 - y\%) \max[S(T)/S(0), (1 + g\%)^T]$

But since $S(0) = 100$ , $g\% = 3\%$ and $T = 1$ , this becomes

$\text{Payoff} = \pi (1 - y\%) \max[S(1)/100, 1.03] = \frac{\pi}{100} (1 - y\%) \max[S(1), 103]$

Ok. To figure out a portfolio that will pay this, we need to split it up into the payoffs of stocks and options:

$\text{Payoff} = \frac{\pi}{100} (1 - y\%) \left[ S(1) + \max[0, 103 - S(1)] \right]$

The first term in the brackets is the payoff from a 1-year prepaid forward on the stock -- since the dividends are incorporated into the index (i.e. the stock effectively doesn't pay dividends), the price of such a forward is just the current price of the stock. On the other hand the second term is the payoff from a 1-year European put with strike price 103 -- and miraculously we are told that the price for exactly that put is 15.21. So the cost of the corresponding portfolio is

$\text{Price of portfolio} = \frac{\pi}{100} (1 - y\%) \left[ F^P_{0,1}(S) + Put(S, 103, 1 \,\text{year}) \right]$

$\text{Price of portfolio} = \frac{\pi}{100} (1 - y\%) \left[ 100 + 15.21 \right] = 1.1521 \pi (1 - y\%)$

In order for the insurance company to neither make nor lose money on this, the cost of the portfolio should be the total amount invested by the client:

$\pi = 1.1521 \pi (1 - y\%) \quad \Longrightarrow \quad y\% = 13.202%$

Hrm. This is more or less identical to the given solution, so it might not be much help... what specifically is the problem?

independent1019 · #3 09-28-2007, 11:20 PM

IC, I did't quite understand how you see that S(1) is the one year pre-paid forward price. But if that's the case, I understand it. But, How do you know that it is the one year pre-paid forward price.

jraven · #4 09-28-2007, 11:30 PM

S(1) isn't the prepaid forward price; it's the payoff at time 1 from a prepaid forward. In general the initial price of a prepaid forward is

$F^P_{0,T}(S) = S(0) e^{-\delta T}$

but in our case T=1 and there are no dividends, so this becomes

$F^P_{0,1}(S) = S(0)$

I'm going to re-edit my first post to make it clearer which equations are about payoffs (at time 1) and which are about prices (at time 0).

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