Options
Options: Calls and Puts
A call option Option (finance) gives the buyer the right to buy an asset at a specified price ("strike price"). When the buyer uses this right they are said to have exercised the call option.
A European call option gives them this right on a specific date ("expiry date"), while an American call option gives them this right at any time before some date - note that American options are, therefore, more valuable than European ones. Both are used in practice, but they are very similar and the European options are easier to analyze, so we'll be sticking to those here.
If a call option is exercised, the profit is simply the price of the underlying asset ("spot price") minus the strike price.
For example, suppose Alice buys a European call option from Bob for \$20 that specifies 1 IBM share with a strike price of \$100 that expires at the end of the year.
This means that Alice pays Bob \$20 today for the right (but not the obligation) to buy one IBM share at the end of the year for \$100. If IBM's share price ends up at \$130, she will have netted \$10 from the trade.
Note that, as with futures contracts, the seller of a call option must typically maintain margin with a third party - though there are some exceptions if the seller also actually owns the underlying asset.
Put options likewise give the buyer the right to sell an asset at a specified price. They work similarly.
Something to note here is that buying a call option and selling a put option with the same expiry date and strike price gives you the exact same risk as owning the underling stock.
The proof is straightforward. Let $x$ be the current price of the stock, let $y$ be the future price of the stock, and let $s$ be the strike price.
Buying a call option for $c$ dollars yields a return of
$$ \max(y - s, 0) - c $$Selling a put option for $p$ dollars yields a return of
$$ p - \max(s - y, 0) $$Adding these together yields a total profit of
$$ y - c + p $$Note that a one-dollar increase in the stock price yields exactly a one-dollar increase in the price of this portfolio. Remember this.
Put-Call Parity
Let's go back to the intuition of the risk-free, cashless portfolio: imagine we're sitting around with zero dollars and about to attempt a three-part strategy.
First, we sell a put option with a strike price, $s$. This immediately gives us $p$ in cash. The profit from this trade at the end of the year is
$$ p - \max(s - y, 0) $$Next, we join a futures contract where we go short on one share of AMZN - that is we promise to sell one AMZN share in a year for the price $f$. The profit at year's end is
$$ f - y $$The profit of our overall portfolio is
$$ \left( p - \max(s - y, 0) \right) + \left( f - y \right) $$Finally, we want to buy a one-year call option with the same strike price ($s$). However, it is not obvious a priori that we have enough cash to do this. There are three scenarios:
- Scenario A: The call price equals the put price ($c=p$), which means we can afford the call and have no cash leftover afterwards.
- Scenario B: The call price is smaller than the put price ($c \lt p$), which means we can afford to buy the call and have $p - c$ left over in cash. We will lend this out at the risk-free interest rate.
- Scenario C: The call price is larger than the put price ($c \gt p$), which means we need to borrow $c - p$ of cash at the risk-free interest rate to purchase the call
The call yields a profit of
$$ \max(y - s, 0) - c $$If we're in scenario B or scenario C, we also gain (lose) money in interest equal to
$$ r(p - c) $$So the overall portfolio earns
$$ \left( p - \max(s - y, 0) \right) + \left( F - y \right) + \left( \max(y - s, 0) - c \right) + \left( r(p-c) \right) $$Our portfolio includes buying one call option and selling one corresponding put option. Recall from above that this is equivalent to owning the stock and receiving $p - c$ dollars:
$$ \left( \max(y - s, 0) - c \right) + \left( p - \max(s - y, 0) \right) = y - c + p $$If we apply this to our current portfolio, we simplify to
$$ \left( y - c + p \right) + \left( f - y \right) + \left( r(p-c) \right) $$This simplifies further down to
$$ (1+r)(p-c) + f - s $$Note that the $y$ term has disappeared, meaning the value of this portfolio no longer depends on the performance of the AMZN stock. From this we know it must be riskless. By our lemma above and the fact we started with zero cash, it follows that the portfolio must yield $0 r$ dollars:
$$ (1+r)(p-c) + f - s = 0$$A little simplification yields
$$ p-c = \left(s - f \right)/(1+r) $$This observation is called put-call parity and is close to exact in real liquid markets Put–call parity.
One obvious implication is that the price of a call option and put option are equal if and only if their strike price equals the corresponding future's delivery price.
Leverage Equivalence
Suppose I have \$0. I could
- Borrow \$100 at the risk-free interest rate and buy one share of IBM, which I sell in one year.
- Buy one call option and sell one corresponding put option that each expire in one year.
- Enter into a futures contract where I promise to buy one shares for \$100 plus the risk-free interest rate in one year.
All three of these portfolios are exactly equivalent in a perfectly liquid market.
In the real world, the first option is frequently worse than the other two, because the explicit interest rates available to you for general leverage are typically worse than the implicit interest rates available to you when exchanging options or futures.