Topic 4 Hedging with Derivatives

Question 1

Forwards

Answer 1

A forward contract is an agreement between two parties to buy or sell an asset at a certain future time for a certain future price, which is determined at contract inception. Forward contracts are traded OTC, “Over The Counter”).

The party that agrees to buy the asset in the future is said to have a long position.

The party that agrees to sell the asset in the future is said to have a short position.

The specified future date for the exchange is known as the delivery (or maturity) date.

The agreed upon price for the future sale is known as the forward price (also known as the delivery price).

Question 2

Hedging with futures

Answer 2

Purpose of hedging is to remove uncertainty. Want to choose h (hedge ratio) such that variance of daily changes is minimized.

Question 3

A farmer expects to harvest 100,000 lbs of cotton in 3 months. He wants to hedge his price risk. In theory, he could take a short forward position. But this may not be so simple. He has to find a counterparty to take the other side and then work out all the contract details. He can always trade in the futures markets, however. Suppose there are 3 month cotton futures, for delivery of 25,000 lbs, available and the current futures price is 45 cents/lb How can he use these futures to hedge against adverse price movements?

Answer 3

He could simply short 4 of these contracts now, then unwind the positions right before delivery time. Why unwind? Why not hold the contracts until maturity? If he did not unwind, then he would need to deliver the cotton as specified in the futures contract. The specified delivery location is likely very far away from his farm. Much easier for him to close out his futures positions and sell his harvest at the local market.

His gain or loss per lb from the futures position will be the difference between the futures prices when he enters and unwinds:

F₀ - F_{T - e} ≈​ F₀ - S_T

His price per lb from selling the cotton will be S_T

Therefore the effective price he receives is approximately F₀ = $.45/lb no matter what the spot price S_T turns out to be.

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Question 4

For non-financial futures, the basis is defined as:

basis = spot price of asset to be hedged - futures price of contract used

Answer 4

If the spot and the futures are the same asset, the basis should be zero at expiration. If the spot rises by more than the futures price, the basis increases, and this is called strengthening of the basis; the opposite is weakening of the basis.

Question 5

For financial futures, basis risk tends to be small. This has to do with arbitrage being relatively easy to implement, and so prices are kept “in check”. The basis risk that does exist is largely due to the unknown future risk-free interest rate and its effect on the price.

Example of strengthened basis

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Answer 5

In our previous example, suppose

◦the spot price was $.40/lb when the farmer entered the contracts

◦At some later time t < T, the spot price increased to $.48/lb and the futures price increased to $.52/lb

Then the basis was B0=-.05; at time t the basis is Bt=-.04; therefore the basis strengthened.

Note that the effective price received by the farmer at time T is approximately

ST + (F0 – FT) = F0 + (ST – FT) = F0 + BT

Since the underlying asset of the futures contract is the same as the asset he is hedging, the basis at time T is zero. But this is not always possible.

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Question 6

Asset Mismatch

Answer 6

If the asset to be hedged is not the one on which the futures contract is based, there is additional basis risk. Define S*T as the price of the asset underlying the futures at time T, and ST be the price of the asset actually being hedged.

Once again, the price being received at time T is:

◦ST + (F0-FT), but now we have to deal with S*T also:

◦F0 + (S*T-FT) + (ST - S*T) This is just adding and subtracting S*T

◦S*T - FT is the basis risk arising from the hedge and ST - S*T is the basis risk arising from the mismatch.

Question 7

Size of the Hedge

Answer 7

An important issue is how many contracts to use to create the hedge. The hedge ratio is the ratio of the size of the position taken in futures contracts divided by the size of the exposure:

HR = (size of futures position)/(size of exposure)

It is not always the case that the optimal ratio is 1.

Question 8

Change in the value of the (long or short) hedge

Answer 8

For a short hedge (i.e. long asset, short futures), the change in the value of the hedge is: ΔS - hΔF

For a long hedge it is: hΔF - ΔS

Define the following:

ΔS - change in spot price during hedge

ΔF - change in futures price during hedge

σs - standard deviation of ΔS

σF - standard deviation of ΔF

ρ - correlation coefficient between ΔF and ΔS

h - hedge ratio (the ratio of the size of the futures position to the size of the exposure).

Question 9

The variance of ax + by is given by

Answer 9

var = a2σ2x + b2σ2y + 2ρab σxσy

Applying this here: var = σ2s + h2 σ2F - 2ρhσsσF

Choose h to minimize the variance: dvar_S,F /dh = 2_hσ2_F - 2ρσ_sσ_F = 0

h* = ρσ_s/ σ_F

Note that if the two assets are perfectly correlated and σs=σ_F, then h=1.

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Question 10

This hedge ratio simply tells you how many units of the hedge instrument to use for each unit of the primary instrument. It does not take into account the size of the hedging instrument contract.

◦To determine the optimal number of contracts to enter into, you just multiply the hedge ratio by the ratio of the size of the position being hedged (NA) with the size of one unit of the futures contract you are using (QF):

Answer 10

N^*= h^*N_A/Q_F

Question 11

What is optimal hedge ratio?

For a short hedge

Answer 11

ΔS = change in spot price during hedge

ΔF = change in futures price

Variance of daily Δ