Futures: Pricing and Hedging Strategies: Futures: Pricing and Hedging Strategies
Lecture Notes for FIN 353
Yea-Mow Chen
Department of Finance
San Francisco State University
I. Futures Market Structure: I. Futures Market Structure 1. A futures contract is an agreement that the buyer (seller) will accept (make) delivery of a particular asset on a future date at a price pre-determined today.
Example: Entering a December gold futures at $350/oz entitles the futures buyer to purchase 100 oz. of gold on the December maturity date for $350/oz, disregard the spot market price of gold in December. If the spot price on the maturity date is $370/oz, the buyer still pays $350/oz for the gold, making a $20/oz profit. On the other hand, if the spot price in December is $340/oz, the buyer would be losing $ 10/oz profit.
I. Futures Market Structure: I. Futures Market Structure Some points to know:
By entering into a futures contract, you know today the price you will pay for the purchase of gold in December ($350/oz). In such a way, you have “lock in” the price for a future transaction.
At the time of contract agreement, the contract is typically designed to be of zero value to either the buyer or the seller. Therefore none of the buyer or the seller will have to pay to the other party for the contract.
Both the buyer and the seller, however, will have to deposit into a margin account, which is used to guarantee the fulfillment of the contract.
I. Futures Market Structure: I. Futures Market Structure 2. When the forward contract is for a standardized amount of a carefully defined asset for delivery on a specific date and subject to the terms and conditions established by the organized market on which is traded, it becomes a future contract. Financial futures are standardized in
1) assets being exchanged;
2) settlement dates;
3) face value; and
4) price quotation.
I. Futures Market Structure: I. Futures Market Structure Assets being exchanged:
The underlying asset: 91-day T-bills for a T-bill futures; T-bonds with 20 years to maturity and an 8% coupon rate for a T-bond futures; and .999 contents for a gold futures.
2) Settlement dates:
March, June, September, and December cycle for most stock index futures
Face value:
$1m for a T-bill futures
$100,000 for a T-bond futures
(Index)*($500) for a S&P 500 index futures
Price quotation:
% of discount for short-term futures contracts, such as 8% discount on a T-bill futures
% of par for long-term futures contracts, such as T-bond futures are quoted at 96-24
I. Futures Market Structure: I. Futures Market Structure 3. Most markets prescribe daily settlement of any gains and losses on the futures contract to minimize the risk of default at its maturity.
This is also called Marking-to-market because, with daily settlement for any gains or losses, the value of the futures position is kept to equal to the current market value.
4. The existence of organized futures markets provides a secondary market for the trading of contracts before maturity. In fact most of contracts are offset before they become mature.
I. Futures Market Structure: I. Futures Market Structure 5. Types of Orders:
Market Order
Limited Order
Stop-loss order
Spread order
I. Futures Market Structure: I. Futures Market Structure 7. Forward Contracts vs. Futures Contracts:
II. Trading Mechanism: II. Trading Mechanism Three steps to a futures trading:
1. Agreeing To Trade: creates long and short positions.
The role of the Futures Clearing Corporation: The clear house is critical to the trading of futures because it settles and guarantees the contracts. After a contract is agreed to, the clearing house puts itself between buyer and seller and, in effect, becomes the party to whom delivery is made and from whom delivery is taken.
II. Trading Mechanism: II. Trading Mechanism 2. Margin requirements: initial margin and maintenance margin.
Initial Margin
Contract Exchange Multiple Speculator Hedger
______________________________________________________________________ S&P500 CBOE $500 $22,000 $9,000
NYSE Index NYSE 500 9,000 4,000
Major Market Index AMSE 250 21,000 7,500
Value Line Index KCBT 500 7,000 5,000
______________________________________________________________________
II. Trading Mechanism: II. Trading Mechanism EX : Suppose an investor purchases one December 1999 gold futures at $400/oz and the initial margin 2,000/contract and maintenance margin is $1,500/contract. The margin account is marked on a daily basis (daily settlement). The following table summarizes the changes in the margin account until the close of the contract.
II. Trading Mechanism: II. Trading Mechanism
II. Trading Mechanism: II. Trading Mechanism 3. Offsetting Contracts: The majority of futures contracts are offset before maturity. This is because it is costly to take delivery.
III. Leverage with Futures: III. Leverage with Futures On futures trading, the only out-of-pocket payment is the margin deposited as a security performance bond. No payments are required for the contract, nor for the underlying assets until the settlement of the contract. This provides an opportunity for leverage.
The gold futures buyer is leveraging his/her $2,000 initial margin into a contract to buy 100 oz of gold in the future, which amounts to $40,000 in today's value. This provides 20 times leverage as compare to buying gold in the spot market. This leverage, however, increases the return volatility. It only takes a small change on the gold price to wipe out the $2,000 initial investment.
III. Leverage with Futures: III. Leverage with Futures Example : Initial investment required on the gold futures = $2,000
Initial investment required for a spot market purchase
= $40,000
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING I. Commodity Futures Prices and the Cost of Carry:
A. Two important characteristics of futures prices:
1. The futures price of a commodity or asset, F, is greater than the spot price, P; i.e., F P.
2. The futures price rises as the time to maturity increases.
These characteristics reflect the cost of carry for a futures contract and illustrate a critical arbitrage relation.
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING Ex: Suppose that the spot price of No. 2 Wheat in a Chicago warehouse is 300 cents per bushel, the yield a one-month T-bill is 6%, and the cost of storing and insuring one bushel of wheat is 4 cents per month. What is the price of a futures contract that has one-month to maturity?
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING Two ways to have wheat in one month:
1. Purchase a one-month wheat futures contract at $F/bushel:
Costs in one month = $F
2. Purchase in spot today and carry it over for one month:
Costs in one month = (300¢ + 4¢)*(1 + 6%/12) = 305.5¢
For the two alternatives to be indifferent, two costs would have to be the same, i.e.,
$F = 305.5¢ or
F = P + C
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING This is an equilibrium condition. It this is not true, then market adjustments will bring back the equilibrium.
If F > P+C, a trader could make a riskless profit by taking a long position in the asset and a short position in the futures contract.
If F < P+C, the arbitrage strategy would be to buy the futures and sell the commodity short.
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING B. Two implications on the movements of futures prices:
1. The convergence of the futures price to the spot price is implied by the cost of carry relation.
2. The convergence of the futures price to the cash price at expiration of the futures contract.
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING C. Determinants of the Basis (Risk):
1. The Convergence of the Future Price to the Cash Price; If the future position is unwinded prior to contract maturity, the return from the futures position could differ from the return on the asset due to the basis risk.
2. Changes in Factors Affecting the Cost of Carry; the most significant determinant of the cost of carrying is the interest rate. As the interest rate increases, the opportunity cost of holding the asset rises, so the cost of carry- and therefore the basis-rises.
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING 3. Mismatches between the Exposure Being Hedged and the Futures Contract Being Used as the Hedge:
In a cross-hedge, there is an additional source of basis risk. Basis results not only from differences between the futures price and the prevailing spot price of the deliverable asset, but also from differences between the spot Prices of the deliverable asset and the exposure being hedged. Major factors responsible for variation in the basis for a cross-hedge:
(1) Maturity mismatch
(2) Liquidity differences
(3) Credit Risk Differences
4. Random Deviations from the Cost-of-Carry Relation: "White noises", but there are canceled out in the long run.
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING II. Futures Prices and Expected Futures Spot Prices
The expectation model states that the current futures price is equal to the market's expected value of the spot price at period T:
Ft = E(PT)
If this model is correct, a speculator can expect neither to gain nor to lose from a position in the futures market:
E(Profit)= E(PT)- FT= 0
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING EX: Suppose that at time period 0, a speculator purchases a futures contract at a price of F, and posts 100% margin in the form of riskless securities.
At contract maturity T, the value of the margin account will have grown to F0* (1+rf)
At maturity, the value of the futures contract itself will be: (PT - F0).
VI. FINANCIAL FUTURES PRICING: VI. FINANCIAL FUTURES PRICING The actual rate of return the speculator will earn is
(1+rf)F0 + (PT - F0) (PT - F0)
r = --------------------------- -1 = rf + --------
F0 F0
The expected rate of return r is
E(PT) - F0
E(R) = rf + ------------- = rf
F0
If the expectation model is correct.
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device Hedging Principle:
To hedge against falling interest rates or rising prices (on a spot position), take a long position by buying futures
To hedge against rising interest rates or falling prices (on a spot position), take a short position by selling futures
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device Long Hedging:
Spot price Spot position value Futures position value gain loss value 0
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device I. Long Hedge
A long hedge is chosen in anticipation of interest rate declines and requires the purchase of interest rate futures contract. If the forecast is correct, the profit on the hedge helps to offset losses in the cash market.
Example: In April 2005, the manager of a money market portfolio expects interest rates to decline. New funds, to be received & invested in 90 days, will suffer from the drop in yields. The manager expects an inflow of $10m in July. The discount yield currently available on 91-day T-bills is 10%, and the goal is to establish a yield of 10% on the anticipated funds.
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device Cash Market Futures Market
___________________________________________________________________
April: T-bill discount yield at 10 % April: buy 10 T-bill Contracts for
Price of 91-day T-bills, September delivery at 10% discount
$10m par = $9,747,222 yield. Value of contracts = $9,750,000
July : T-bill discount yield at 8% July: Sell 10 Sept. T-bill contracts
Price of 91-day T-bills, at 8% discount yield.
$10m par = $9,797,778 Value of contracts = $9,800,000
_______________________________________________________________________
Opportunity Loss Gain = $50,000
= $50,556
Effective Discount Yield with the Hedge
$10,000,000- ($9,797,778- $50,000) 360
= -------------------------------------------- * ---- = 9.978%
$10,000,000 91
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device Even if the expectation on future interest rates for the cash market is incorrect, the position is still hedged. The cost is that the potential profitable opportunities in the cash market is foregone.
EX; Assume the T-bill discount yield rises to 12 %, instead of declining to 8% as expected.
Cash Market Futures Market
_______________________________________________________________________ April: T-bill discount yield at 10 % April: Buy 10 T-bill Contracts for
Price of 91-day T-bills, September delivery at 10% discount
$10m par = $9,747,222 yield. Value of contracts= $9,750,000
July : T-bill discount yield at 12% July: Sell 10 Sept. T-bill contracts
Price of 91-day T-bills, at 12% discount yield.
$10m par = $9,696,667 Value of contracts = $9,700,000
_______________________________________________________________________ Opportunity gain Loss = $50,000
= $50,555
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device Long speculation: Instead of expecting new funds to arrive & invest in July, the manager could speculate on the direction of interest rates. If he/she speculates on a declining interest rate, and the speculation is materialized:
Cash Market Futures Market
_______________________________________________________________________
April: T-bill discount yield at 10 % April: buy 10 T-bill Contracts for
Price of 91-day T-bills, September delivery at 10% discount
$10m par = $9,747,222 yield. Value of contracts= $9,750,000
July: T-bill discount yield at 8% July: Sell 10 Sept. T-bill contracts
Price of 91-day T-bills, at 8% discount yield.
$10m par = $9,797,778 Value of contracts = $9,800,000
_______________________________________________________________________
Gain = $50,000
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device If he/she speculates on a declining interest rate, but market rate rises in September instead:
Cash Market Futures Market_______________________________________________________________________
April: T-bill discount yield at 10% April: Buy 10 T-bill Contracts for
Price of 91-day T-bills, September delivery at 10% discount
$10m par = $9,747,222 yield. Value of contracts= $9,750,000
July: T-bill discount yield at 12% July: Sell 10 Sept. T-bill contracts
Price of 91-day T-bills, at 12% discount yield.
$10m par = $9,696,667 Value of contracts = $9,700,000
_______________________________________________________________________
Loss = $50,000
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device II. Short Hedge:
A short hedge is chosen in anticipation of interest rate increases and requires the sale of interest rate futures. If the forecast is correct the profit on the hedge helps to offset losses in the cash market.
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device Example: A saving institution in April 2005 wants to hedge $5m in short-term CDs whose owners are expected to roll them over in 90 days. If market yields go up, the thrift must offer a higher rate on its CDs to remain competitive, reducing the net interest margin. If the CD rare rises from 7% to 9%, the interest cost will increase by $25,000 for the 3-month period. The asset/liability manager can reduce these by the sale of T-bill futures contracts.
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device Cash Market Futures Market
_______________________________________________________________________ April.: CD rate = 7% April.: Sell 5 Sept. T-bill contracts at
interest cost on $5m 3-month 7% discount yield
interest costs = $87,500 Value of contract : $4,912,500
July: CD rate = 9% July: Buy 5 Sept. T-bill contracts at
interest cost on $5m 3-month 9% discount
deposits Value of contracts = $4,887,500
= $112,500
______________________________________________________________________
Opportunity Loss = $25,000 Gain = $25,000
Net result of hedge = 0
$112,500 -$25,000 360
Effective CD Rate = ---------------------- * ----- = 7%
5,000,000 90
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device Basis Risk Using the Long Hedge Example
Long Hedge Example as previously stated:
Cash Market Futures Market
___________________________________________________________________
April: T-bill discount yield at 10 % April: buy 10 T-bill Contracts for
Price of 91-day T-bills, September delivery at 10% discount
$10m par = $9,747,222 yield. Value of contracts = $9,750,000
July : T-bill discount yield at 8% July: Sell 10 Sept. T-bill contracts
Price of 91-day T-bills, at 8% discount yield.
$10m par = $9,797,778 Value of contracts = $9,800,000
_______________________________________________________________________
Opportunity Loss = $50,556 Gain = $50,000
Effective Discount Yield with the Hedge
$10,000,000- ($9,797,778- $50,000) 360
= -------------------------------------------- * ---- = 9.978%
$10,000,000 91
V. Financial Futures: Interest Rate Futures as a Hedging Device: V. Financial Futures: Interest Rate Futures as a Hedging Device Revised Example: Rather than using T-bill contract for hedging, a long-term T-bond futures contract is used for hedging which is price at 96-12. If the T-bill rate drops to 8% in September as expected, the T-bond futures will have it price increased to 98-16.
Cash Market Futures Market
_______________________________________________________________________ April: T-bill discount yield at 10 % April: Buy 100 T-bond Contracts for
Price of 91-day T-bills, September delivery at 96-12 which
$10m par = $9,747,222 gives the value of contracts = $9,637,500
July: T-bill discount yield at 8% July: Sell 100 Sept. T-bond contracts
Price of 91-day T-bills, at 98-16 for a value $9,850,000
$10m par = $9,797,778
_______________________________________________________________________ Opportunity Loss Gain = $212,500
= $50,556
VI. Macrohedging with Futures for a Financial Institution: VI. Macrohedging with Futures for a Financial Institution Suppose a FI's balance sheet structure is as follows: Assets = $100m, Liabilities = $90m, and equity $10m. The average duration of assets and liabilities is 5 and 3 years, respectively. If interest rates are expected to rise from 10% to 11%, then:
E = (DA - kDL) * A * (R/1+R)
= - (5 - .9 * 3) * $100m * (.01/1.1)= - $2.09m
VI. Macrohedging with Futures for a Financial Institution: VI. Macrohedging with Futures for a Financial Institution The manager's objective is to fully hedge the balance sheet exposure by constructing a futures position to make a gain to just offset the loss of $2.09m on equity.
When interest rates rise, the price of futures contracts falls. The sensitivity of the price of a futures contracts depends on the duration of the deliverable bond underlying the contract, or:
F/F = - DF * (R/1+R), or
F = - DF * F * (R/1+R)
= - D *(NF* PF)* (R/1+R)
VI. Macrohedging with Futures for a Financial Institution: VI. Macrohedging with Futures for a Financial Institution Fully hedging can be defined as selling sufficient number of futures contracts so that the loss of net worth on the balance sheet is just offset by the gain from off-balance-sheet selling of futures:
F = E
which implies:
N F = [(DA - kDL) * A] / DF * PF
= [(5-.9*3)*$100m]/(9.5*$97,000)
= 249.59 contracts
if a T-bond futures contract is used for hedging. The futures is quoted $97 per $100 of face value for the benchmark 20-yr., 8% coupon bond that has a duration of 9.5 yrs.
VI. Macrohedging with Futures for a Financial Institution: VI. Macrohedging with Futures for a Financial Institution Suppose instead of using the 20-yr. T-bond futures to hedge it had used the 3-month T-bill futures that has a price of $97 per $100 par value and a duration of.25 yrs. Then
NF = (5 - .9$3)$1 00m/.25*$97,000 = 948.45 contracts
In general fewer T-bond contracts need to be sold because of its greater interest rate sensitivity. This suggests a simple transaction cost basis, the FI might normally prefer to use T-bond futures.
VI. Macrohedging with Futures for a Financial Institution: VI. Macrohedging with Futures for a Financial Institution The Problem of Basis Risk:
Because spot bonds and futures on bonds are traded in different markets, the shift in yields (R/1+R) affecting the value of the on-balance-sheet cash portfolio may differ from the shift (RF/1+RF) in yields affecting the value of the underlying bond in the futures contracts; i.e., spot and futures prices or values are not perfectly correlated. To take this basis risk into account:
E = -(DA - kDL) * A * (R/1+R)
F = - DF * (N F*P F ) * (R F /1+R F)
VI. Macrohedging with Futures for a Financial Institution: VI. Macrohedging with Futures for a Financial Institution Setting : E = F , we have
N F = [(DA - kDL) * A] / DF * PF * b,
Where b =(R/1+R)/ (RF/1+RF)
where b measures the degree to which the futures price yields move more or less than spot price yields. For example, if b = 1.1, this implies that for every 1% change in discounted spot rate (R/1+R), the implied rate on the deliverable bond in the futures market moves by 1.1%.
NF = (5 -.9*3) * $100m/9.5*497,000* 1.1
= 226.9 contracts
VII. Risks In Futures Transactions: VII. Risks In Futures Transactions 1. Basis risk: The "basis" is difference between the spot price of an instrument and the price of that asset in the futures market. Basic risk results from the fact that this price relationship may change overtime. However this basis risk is stable and predictable.
2. Related-Contract Risk: Hedges can also fail because of default in the contract being hedged.
VII. Risks In Futures Transactions: VII. Risks In Futures Transactions 3. Manipulation Risk: Most manipulation involved "short Squeezes"' whereby an individual of group tries to make in difficult on impossible for short sellers in the futures markets to liquidate their contracts through delivery of acceptable commodities. The "short" will have to buy back their contracts as inflated prices.
4. Margin Risk: An illiquid individual can also encounter difficulty by hedging in the futures markets if the future prices moves adversely and the individual must constantly pose more maintenance margin funds.