Winter 2009 PSTAT 160A
Homework # 8
(Suggested)
Problem 1. Graph the payoff function (as a function of the underline stock S) of the following strategies,
and indicate what view of the market does this conrespond to. All options are European, and with same
maturity T. It’s quite easy if you just condidere prices above, under strikes and between strikes.
1) Buy one put with strike K(long on put)
2) Sell one put with strike K(short on put)
3) Sell one call with strike K(short on call)
4) Buy one call and one put with same strike K.
5) Buy one call at strike K1and sell a second one at a larger strike K2.
6) Buy one call at strike K1and sell a second one at a lower strike K2.
7) Buy two calls and one put with 3 different strikes (consider all possible 3 cases of which are lower
or higher)
Problem 2.
Suppose that on a market, two commodities are traded: gas (with price S1) and electricity (with price
S2). We consider the price variation for one time period, say time 0 and time 1. The interest rate is 0.1,
and we normalize the prices so that at time 0, the price for both commodities is $100. Three events are
possible:
S1goes up to 120 and S2go up to 130,with probability .2
S1stays to 100 and S2goes down to 70 with probability .5
S2stays to 100 and S1goes down to 90 with probability .3
An electricity producer needs to protect himself against a sudden rise in gas price, (maybe also a drop
of electricity price), or if the difference between gas prices an electricity is to large (bigger than its
manufacturing cost). He could need also to consider a drop of electricity price but we don’t worry about
it here.
In addition to these two commodities, there are two European options:
One Call option that allows to buy gas (S1) at time 1 for $110.
For the other option, the holder will exercise his right if the difference between S1and S2exceeds $10,
and the option pays this excess. More precisely, the payoff of the option is (S1
−S2
−10)+, that is
S1
−S2
−10 if this number is positive. Clearly, this protect against gas price being too high compare to
electricity, and this is called a spread option.
Assuming that the interest rate is 0, apply the arbitrage theorem to find the fair price for these two
options. Is there an arbitrage opportunity?
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