Derivatives

Forwards and Futures

Forward contracts Forward contract are among the oldest and most common derivatives. For example, a forward is simply an agreement between Alice and Bob where Alice agrees to buy an asset from Bob for a specified price ("delivery price") on a specified date. Historically, this has been a common way for farmers to reduce their financial risk by ensuring the ability to sell crops at a profitable price - even if market prices fall considerably in the interim.

Note, while forwards can be useful for farmers, there are two significant issues with using it as an investor:

• At the end of the contract, you actually have to provide or receive the good.
• There can be significant risk that the buyer or seller will renege.

The solution to the first problem is to just require that each party send cash equal to the difference between the market price and the price in the contract. Such a contract is called a cash-settled forward.

The solution to the second problem is the the futures contract Futures contract.

In a futures contract, Alice and Bob both go to a trusted third-party: Carol. Carol requires Alice and Bob to both entrust to her a small amount of cash called the margin. This margin ensures neither Alice nor Bob are incentivized to walk away from the deal if it goes poorly for them. Carol determines the required margin based on expected future price volatility.

Every day, Carol will require Alice and Bob to update their margins based on price moves (a "margin call"). For instance, if the asset's price goes down 1%, Alice will be required to pony up cash equal to the price change, while Bob will receive it. In this way, by the time the contract expires, Carol can simply refund the remaining margin to both parties and the forward contract will have been replicated.

There still exists counterparty risk in a futures contract, but they are seriously mitigated:

1. Carol will go bankrupt for other reasons, skip town, etc. This risk is minimal since large financial institutions are used to facilitate these transactions.
2. The price will move in a single day than the margin size. This gives either Alice or Bob the motive to skip town. This risk is reduced by (1) the requirement of the margin call and (2) the fact that in real life Carol has information about both parties like their social security numbers, which opens up the reneging party to civil lawsuits.

Finally, it's worth mentioning some nomenclature: the person who agrees to buy the security in the future for the specified price is said to "be long" because if the security increases in price, they gain money - similar to holding the actual asset. The person who agrees to sell is said to "be short" for the same reason.

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

Selling a put option for $p$ dollars yields a return of

Adding these together yields a total profit of

Note that a one-dollar increase in the stock price yields exactly a one-dollar increase in the price of this portfolio. Remember this.

Risk-Free, Cashless Portfolios

Consider a market that includes futures contracts, call options, and put options. Now let's add three assumptions:

• The market is efficient (EMH).
• The market has zero transaction costs.
• The market price is continuous over time

The latter two assumptions imply that (a) one can buy and sell options instantaneously and (b) that the required margins by the third party (based on predicted volatility during a predetermined time period) on these derivatives are all zero.

These assumptions also imply that no risk-free portfolio can earn more than the risk-free interest rate, because if one did it would violate the EMH.

They also imply that a risk-free portfolio can't earn less than the risk-free interest rate if it includes no cash. The reason is that you can just invert this portfolio by going short instead of long, which would let you have a risk-free portfolio that does better than the risk-free interest rate. This, again, violates the EMH assumption.

In short, we have a nice lemma: any risk-free, cashless portfolio must earn the risk-free interest rate.

Delivery Price

A futures contract doesn't have a market price (like stocks) or interest rate (like bonds) as a market mechanism to ensure demand equals supply. Instead, this rationing occurs via changes in the delivery price.

This delivery price has some economic theory behind it Forward price, and we'll discuss a special case now: where the asset in question pays no dividends. One such asset (historically) was AMZN stock.

Let

• $r$ - risk-free interest rate
• $x$ - the price of AMZN now
• $y$ - the price of AMZN in one year

If I have $x$ dollars, I can choose to buy one AMZN stock. This will yield an eventual profit of $y - x$.

I can then go short on a futures contract of AMZN with delivery price $f$, which yields a profit of

Note the return from this overall portfolio is

which just equals

Note this is a risk-free, cashless portfolio that started with $x$ dollars, so we know it must earn the risk free interest rate of $r x$, so

thus

The formula can be generalized if the asset pays dividends (like most stocks) or costs money to store (like most commodities), but the general notion that forward prices reflect risk free returns holds Forward price.

Put-Call Parity

Now 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

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

The profit of our overall portfolio is

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

If we're in scenario B or scenario C, we also gain (lose) money in interest equal to

So the overall portfolio earns

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:

If we apply this to our current portfolio, we simplify to

This simplifies further down to

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:

A little simplification yields

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

1. Borrow \$100 at the risk-free interest rate and buy one share of IBM, which I sell in one year.
2. Buy one call option and sell one corresponding put option that each expire in one year.
3. 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.