Lecture 13 Hedging

TOC

Valuing a Future/Forward Contract 1. The fee of a future contract 2. The value of a future contract The essence of hedge Hedge with futures: short position Hedge with futures: long position Basis risk 1. Definition 2. Reasons for basis risk 3. Methods to reduce basis risk Hedge ratio 1. Motivation 2. Derivation of Hedeg ratio 3. Optimal number of contracts Real World Cases

Valuing a Future/Forward Contract

1. The fee of a future contract

Unlike option, which is kind of insurance and has option fee, future does not have fee because when it was signed in the origin, it was a fair game.

2. The value of a future contract

To sign a future contract (denoted as A), agreeeing to buy an asset at at time . The payoff of this bought future contract A at T is

At time t, the price in a future contract B with the same underlying asset and the same expiration date is . The payoff of future contract B at T is

At time T, a net cash flow of A relative to B: , thus at time t, the value of contract A is (since no arbitrage)

Similarly, the value of a short contract at time t is

Example

❓

Three months ago, you signed a 6-month forward contract promising to buy a stock at $41.01. Assuming the current spot price is $45 and the continuously compounding rate is 5%. What is the value of the contract today?

The value of the forward contract is i.e.

The essence of hedge

Future price has strong correlation with the underlying asset. Hedging helps us to lock the price. Using future contract to lock price is naive, however, there are many restrictions and extra cost of the delivery. Instead of delivery, closing the position in future market is more convenient.

Short hedge

Having something to sell in future and being afraid of price rising
So sell right now at the future market

Long hedge

Planning to buy in future and being afraid of price increasing
So buy right now at the future market

Hedge with futures: short position

When the hedger already owns an asset and expect to sell it at some time in the future, a short hedge is appropriate. A hedge that involves a short position in futures contracts.

Example of short hedge

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An oil producer has to sell 1m barrels of oil on Aug 15th. Suppose each future contract is for the delivery of 1,000 barrels. How many contracts do they need to write?

The company can hedge its exposure by shorting 1000 contracts

Since there is no arbitrage,

At maturity when .

Sell oil in spot market at price of 17.5 to realize $17.5 m
Close the short position in future market by taking long position ⇒ profit = $18.75m - $17.5m = $1.25m
Total profit from two markets are $17.5m + $1.25m = $18.75m

At maturity when

Sell oil sport market at price of 19.5 to realize $19.5m
Close the short position in future market by taking long position ⇒ profit = $18.75m - $19.5m = -$0.75m
Total profit from tow markets are $19.5 - $0.75 = $18.75m

👉🏻

Profit of Short Hedges When the convergence property holds (), the total income from both markets are

lock the sell price

Hedge with futures: long position

When a company knows it will have to purchase a certain asset in the future and wants to lock in a price now. A long hedge is appropriate, that is, taking a long position in a future contract.

Example of long hedges

❓

The oil producer has to buy 1m barrels of oil on Aug 15th. Suppose each future contract is for the delivery of 1,000 barrels. How many contracts do they need to write?

The company can hedge its exposure long 1000 contracts, where

At maturity,

Buy in spot market and realize the cost of $17.5
Close the long position by taking short position, profit is $17.5-$18.75=-$1.25
Total cost is $17.5+$1.25=$18.75

At maturity,

Buy in spot market and realize the cost of $19.5
Close the long position by taking short position, profit is $19.5-$18.75=0.75
Total cost is $19.5-$0.75=$18.75

👉🏻

Profit of Short Hedges When the convergence property holds (), the total income from both markets are

lock the buy price

💡

Basic Rules of Hedging:

At the beginning of the period

To sell in spot market ⇒ take short position now in future market
To buy in spot market ⇒ take long position now in future market

At the end of the period

Close the position at the end of the period
Buy as you should and sell as you should in the spot market

Basis risk

1. Definition

What if the convergence property fails ?

For a long hedge, the cost at T turns into

As long as Basis is fixed at a constant level, the price is still locked down at.

Basis risk of hedging with futures

Defining Basis as

The uncertainty associated with basis is basis risk

2. Reasons for basis risk

Mismatch of assets

The asset whose price is to be hedged may not be exactly the same as the asset underlying the futures contract (for example, we want to hedge green apples🍏, while there are only future contracts underlying red apples🍎 🙁)

Mismatch of delivery date

Since all contracts are formulated by the future Exchange Commission

Arbitrage restrictions do not hold tightly

Any conditions we assume in our hedge strage broken, there exists basis risk.

3. Methods to reduce basis risk

Find contracts whose underlying assets are most closely correlated with the assets to be hedged

Find contracts whose expiration dates are as close as possible with delivery expiration in the spot market

Hedge ratio

1. Motivation

Value of portfolio at T is then

Short position:

Long position:

In the absence of basis risk, the variance of the portfolio is zero. So one can set up a perfect hedge by fully hedging a given exposure. However, in the presence of basis risk this is not feasible - a full hedge may not be the best thing to do.

2. Derivation of Hedeg ratio

Hedge ratio: the size of the position taken in future contracts to the size of the exposure.

Notions:

: standard deviation of

: standard deviation of

: correlation between and

: hedge ratio

: variance of portfolio changes

when and are stochastic, the variance of the portfolio is

FOC of the objective is

Thus

When , we can eliminate the risk. When , we cannot reduce the risk at all.

3. Optimal number of contracts

Notions:

: Size of position to be hedged in spot market

: Size of one future contracts

: Optimal number of futures contracts for hedging

The optimal number of futures contracts for hedging is

Example

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A beef producer has committed to purchasing 200,000 pounds of live cattle on Nov. 15th. He wants to hedge with live cattle futures and has estimated that , and . The futures contract expires Dec and each one is for 40,000 pounds of cattle.

🗝️ , Optimal number of contracts

Real World Cases

SINO-OCEAN Group: take long position in contracts whose underlying asset is shippment fee of the Mediterranean Sea. However, as a shipping company, It's in the business of “selling” shippment. The behavior was kind of speculative and eventually in 2008, it ate its own bitter fruit.🤡

TSINGSHAN Group: a seller of nickel. It actually took the right position of contracts. However, the scale is too large. The price of nickel in both future and spot marker increased a lot when it had to close its position. On the one hand, it could not pay so much money to take long position in future market and close its position. On the other hand, it could not supply so much nickel in the spot market to deliver the contract.😢

Central Bank of PRC - YuanYouBao: It took the right long position in contract underlied crude oil. However, when the expiration date coming, it didn’t close the position in time. In the midnight of the last day, the price of crude oil in future market dropped to negative😮 (In fact, commodity prices are always manipulated by large financial institutions). Anyway, if the central bank chose to close the position, it would lose pretty much. If it chose to deliver the position, that is, buy crude oil at negative, it should pay all the fee including container’s cost, shippment cost etc. Considering all these cost, even though the price of crude oil is positive in spot market, the central bank could not save the loss either.