The following is a question on financial math from the financial math actuarial exam:
Earlier in the manual, the author stated that a collared stock is equivalent to a bull spread. Therefore, in this problem I bought a \$5.20 call and sold a \$5.30 call. We are selling 50,000 at the \$5.30 price and the production cost is \$4.30 Thus the profit is
$5.30 \cdot 50000 - 4.50\cdot50000 - 0.35\cdot50000(1.04) + 0.30\cdot50000(1.04) = 50000[.80 - 0.05(1.04)] = 37400$.
The author's solution states:
A collar involves buying a put and selling a call (in this case, at strike prices of \$5.20 and \$5.30 per bushel, respectively). According to the data, the net cost of this collar is \$0.35 - \$0.30 = 0.05 per bushel. The maximum revenue under this collar is \$5.30 per bushel (because of the short call). Thus, the maximum possible profit is $[(5.30 - 4.50) \cdot 50000] - [(0.05 \cdot 50000 \cdot 1.04] = 37400$.
It seems to me that the only reason why I was able to arrive at the same answer using the bull spread is because the \$5.20 call premium and put premium are the same in this problem. But they will not necessarily be the same in every problem, and aren't even the same for other entries in the table.
What exactly is going on here? It seems as if I got lucky because the \$5.20 strike price call and put premiums were equal.
My only solution is that the prices in the table were generated using the Black-Scholes model and therefore the equivalence of the bull spread method and collared stock is already "built in."
Suppose the currently expected price in six months time is $F$. So an agreement now to buy in six months at $F$ (a forward purchase) has zero current expected value.
But this should be the same as buying a call option with strike price $F$ and selling a put option with strike price $F$. This means the call and put options at strike price $F$ have equal expected values to net to zero.
Indeed if the forward price is not the strike price at which the call and put options have equal value (and there are no fees or spreads or other complications), there will an arbitrage opportunity which will tend to close any gap. It is this no-arbitrage principle, even more than expected values, which is thought to drive market pricing.
So the question having the call and put premia equal at the forward price of $\$5.20$ is reasonable. Real life can get more complicated, but not much.