TOC
Using Forward & Futures in Hedging
Hedging, Speculation and Arbitrage
The key aspects of the three types of transaction include riskiness, original and underlying risk exposure and profitability.
Type | Risk Exposure | Possession | Profitability |
Hedger | Basis Risk | Underlying Asset & Derivatives | Expect No Profit |
Speculator | Position | Derivatives | High risk, High return |
Arbitrager | Risk free | No Position | Usually Low return |
Keynes and Hicks argument:
Hedgers will be prepared to accept negative returns on average because of the benefits of hedging whereas speculators require positive returns on average.
For example, if hedgers hold long positions and speculators hold short positions, the future price () will tend to be higher than the expected future spot price (), thus the speculators are tend to make a profit.
Notation
- : the delivery price for a contract. is negotiatied at time 0, and is negotiatied at time with .
- : the delivery date.
- : the price of underlying asset at time 0, and at time
- (or ): the forward price at time 0, and (or ) at time . Both are for delivery at time
- A contract with has zero value at time 0. Another contract with has zero value at time . But is likely to have a non-zero value.
Long & Short Hedges
A long forward/futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price
A short forward/futreus hedge is appropriate when you know you will sell an asset in the future and want to lock in the price
Hedging: Advantages and Disadvantages
Advantages:
- Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exhcange rates, and other market variables.
- Bankruptcy is costly. Bankruptcy cost is a deadweight loss to society created by market inefficiency.
Disadvantages:
- Shareholers can make their own heding decisions and are usually well diversified
- It may increase risk to hedge when competitors do not
Basis Risk
Basis
Basis is the difference between the spot & futures price:
if
In practice, it is possible that at maturity because:
- It may be difficult to arbitrage between and
- Spot & Futures are often traded in different markets and locations
- Consider also Basis risk
Basis Risk
Basis risk arises because of the uncertainty about the basis:
- Mismatch of asset to be hedged () & asset underlying the futures contract ()
- We should choose the contract whose futures price is most highly correlated with the asset price
- Cross hedge: when direct hedge is not possible
- Mismatch of date when the asset to be hedged is bought/sold () and delivery month of the futures ()
- Basis risk increases as the difference between hedge expiration and delivery month increases
- Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge
- Liquidity is larger for short-maturity contracts β Rolling forward
- Uncertainty on the date when the asset to be hedged is bought/sold ()
Cash settled futures or forwards are more likely to expose to basis risk. If the futures or forwards deliver the asset you want, there is no basis risk.
is itself a variable, but less uncertain than alone. Basis risk is higher in cross hedge. Basis risk is normally much smaller than unhedged risk.
Hedging mechanism for futures: Long hedge
- You need some gold at time
- You buy a gold futures contract at time that mature at time
- You unwound your position at time
Cash inflow at time = =
Basis risk arises because we do not know at time β It can improve or worsen the position of the hedger.
Hedging mechanism for futures: Short hedge
- You want to sell gold at time
- You sell a gold futures contract at time that matures at time
- You unwound your position at time
Cash inflow at time = =
Basis risk arises because we do not know at time β It can improve or worsen the position of the hedger.
Rolling forward - Stack & roll
Sometimes the expiration date of the hedge is later than the delivery dates of all the futures contracts in the market that can be used.
The hedger must then roll the hedge forward.
- Close out one futures contract and take the same position in a futures contract with a later delivery.
- Hedges can be rolled forward many times.
For futures, we can use a series of futures to increase the life of a hedge. Each time we switch from one futures contract to another we incur a type of basis risk.
Example
Rolling: Contango v.s. Backwardation
In contango, approaches from above. Therefore, if we can figure out a contango trend, we should short futures and long spots which end up with an arbitrage strategy.
In backwardation, approacahes from bottom. Therefore, if we can figure out a backwardation trend, we should short spots and long futures which end up with an arbitrage strategy.
Note:
- Contango is when the futures price is above the expected future spot price
- Backwardation is when the futures price is below the expected future spot price
Rolling the hedge forward: Selecting a contract
Nearest to maturity contract was chosen for liquidity. Futures price can be unfavourably costly (big bid-ask spread) if the market is thin.s
Rolling of the contract is typically done at least a few days before the contract expiry instead of at maturity. Derivative contract prices at maturity are often erratic and subject to market manipulation.
Optimal Hedge Ratio & Cross Hedging
Optimal Hedge Ratio in Futures hedge
Hedging is to manage and reduce the risk. However, we do not always need to hedge all the spot assetsβ units.
Notations:
- : Hedge portfolio, a portfolio of the spot asset and the futures contracts
- : Hedge ratio, number of futures contracts bought/sold relative to the number of units of the spot asset bought/sold
- : Optimal hedge ratio, the hedge ratio that minimizes the variance of the returns of the hedge portfolio
At time , is the spot price and is the futures price of the same asset.
- Enter the hedge at time , portfolio value (Apparently, if there is not basis risk, the optimal hedge ratio should be 1).
- Hold the hedge portfolio from to . Change in the portfolio value from to is:
- The variance of is:
Find the optimal hedge ratio which is
The first order condition yields:
Therefore,
Actually, the equation is the same with the OLS results.
Assuming that and are linearly related, estimate the optimal hedge ratio through the regression:
then the optimal hedge ratio is equal to the regression coefficient:
Cross Hedging
Cross hedging occurs when the asset underlying the futures contract is different from the asset whose price is being hedged.
The cross hedge is effective only if (if is large, the hedging error is large)
When , then , the correlation
- If and , then . The futures mirrors the spot perfectly, and the hedge is perfect. The implication here is that there is no basis risk
- If and , then . There still have a perfect hedge: short the future contract
- If , then minimum varaince hedge above suggests no hedging at all
Example
Hedging Using Index Futures
When we want to lock the value of our portfolio, we can short index futures (since we are the selling set under this circumstance)
Number of contracts that should be shorted is (based on the spot price):
is equal to CAPM according to their definition
- ,
- We can choose Future Market and Market, thus . Therefore,
Example
When we hedge this amount, we have reduce the beta of the portfolio to zero. If we want to change the beta to using index hedge.
- When the hedger needs to take a short position in contracts (reduce beta)
- When the hedger needs to take a long position in contracts (increase beta)
Example:
Interest Rate Quotations & Compounding
Most common compounding frequency: Annual, monthly and daily
- Continuous compounding is computationally the most convenient
- Continuous compounding is not used in practice, but is key in modelling
In an efficient market, one cannot make any money simply by having a different compounding frequency: The interest rate offered is adjusted to reflect the compound frequency
: Annualised continuously compounded rate
: Equivalent rate with compounding times per year
Example
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