Futures

In this lesson, we learned about futures contracts and how they are similar and different from forward contracts.

The problems with forward contracts are:

• They are not organized through an exchange
• As a consequence, there is no price transparency for supply and demand to accurately price contracts
• They must personalized between two parties - if there is no party willing to take the opposite side, there cannot be a contract
• Default risk of the counterparty

So we need an intermediary to solve these problems.  That intermediary is futures contracts.  Futures contracts are recorded and traded daily on exchanges with profits/losses every day.  New contracts can be organized through the exchange, so there is no need for personalized contracts between two parties that know each other. Futures contracts can also be written on many different assets - commodities, stocks, stock indices, and volatility of the markets (VIX). 

To buy/sell a future contract, individuals must open a margin account and deposit an initial margin into the account - around 5-10% of the contract value and add money to the margin account if the balance in the account drops below the necessary margin percentage. 

Pros and cons of futures contracts:

Pros	Cons
High leverage = high profit	High leverage = high risk
Very liquid	Linear functions - only linear payoffs can be hedged
Wide variety of underlying assets	May not be flexible enough

To price futures, we use the Martingale Pricing Theory.  With deterministic interest rates: the forward price is equal to the futures price. At maturity, the futures price F(t) is equal to the spot price of the underlying asset at that time: S(T) = F(T)


We then went through an example of hedging using futures.

A baker needs 500,000 bushels of wheat in 3 months.
Hedging strategy - buy 100 futures contracts each for 5000 bushels of what maturing in 3 months.

On December 1st:

• Futures position at maturity: F(T) - F(0) = S(T) - F(0)
• Buy in the spot market: S(T)
• Effective cash flow: S(T) - F(0) - S(T) = -F(0)

The price is effectively fixed at F(0), the price at the beginning of the three months.  

The only cash flows associated with this are the extra cash that must be put into the margin account for margin calls. 

This is an example of a perfect hedge.  In real life, perfect hedges are not always possible for a variety of reasons:

• T might not be a futures expiration date.
• N might not correspond to an integer number of futures contracts.
• A futures contracts on the underlying asset might not be available.
• The futures market for that asset might be illiquid.
• The payoff P(T) might be nonlinear.

Basis = the spot price - futures price

In a perfect hedge, the basis = 0 at time T

Basis risk: basis ≠ 0 at time T

Basis risk arises if the futures contract is on a related but different asset, or if the futures contract expires at a different time.

Example: hedging with basis risk

A taco company needs 500,000 bushels of kidney beans in 3 months.
Problem: no futures market for kidney beans.

Hedging strategy - buy y futures contracts each for 5000 bushels of soybeans maturing in 3 months.
Soybean prices are correlated with kidney bean prices

On December 1st:

• Futures position at maturity: [F(T) - F(0)]y 
• Buy kidney beans the spot market: P(T)
• Effective cash flow: C(T) = y[F(T) - F(0)] + P(T)

P(T) ≠ yF(T) for any y - perfect hedge is impossible.

Since we cannot get a perfect hedge, we can minimize the variance of the cash flow:

Cash flow = C(T) = y[F(T) - F(0)] + P(T)

Now minimize y:

The optimal number of futures contracts is: