Description
forwards and futures contract in detail with the help of examples. It also explains types of traders and types of orders, , it compares forwards contracts and futures contracts, determination of future prices. It also contains arbitrage and various hedging strategies using futures.
Derivatives
Forwards and Futures
Forwards Futures
Derivatives management Framework
Individual Components Options Swaps
Derivatives Markets
Exchange traded
Traditionally exchanges have used the open-outcry system, but increasingly they are switching to electronic trading Contracts are standard there is virtually no credit risk
Who Trade Derivatives?
Market-makers and dealers stand ready to buy or to sell: they take the other side of a transaction for whomever wants to trade
Although they accommodate customers, market-makers and dealers make money on transaction fees, and try not to take excessive risk
Over-the-counter (OTC)
A computer- and telephone-linked network of dealers at financial institutions, corporations, and fund managers Contracts can be non-standard and there is some small amount of credit risk
Arbitrageurs trade to take advantage of prices that are temporarily wrong Speculators get extra leverage in betting on future movements in the price of an asset. Hedgers use the instruments to reduce risk associated with price of an asset.
Derivatives Market in India
Derivatives trading started with the promulgation of the Securities Laws (Amendment) Ordinance, 1995, which withdrew the prohibition on options in securities. SEBI set up L.C. Gupta committee in 1996 to develop appropriate regulatory framework for derivatives The committee submitted its report in 1998 and recommended derivatives as ‘securities’. SEBI set up J.R. Verma committee to recommend measures for risk containment in DM The SCRA was amended in 1999 to include derivatives as securities and developed regulatory framework for derivatives
Derivatives Products in India
Index futures Stock futures Index options Stock options Commodity futures Interest rate futures Currency futures Credit derivatives
Forward / Futures Contracts
Forward / Futures contract is an agreement between the seller and the buyer whereby the seller is obligated to deliver the asset on a specified date and the buyer is obligated to pay a specified price upon delivery Forward contracts are similar to futures except that later trade in the over-the-counter market
Forwards and Futures
2.7
Forward Contract Example
Consider two parties (A and B) enter into a forward contract on 13 July, 2010 where, A agrees to deliver 1000 stocks of Unitech to B, at a price of Rs. 100 per share, on 29th July, 2010 (the expiry date). In this contract, A, who has committed to sell 1000 stocks of Unitech at Rs. 100 per share on 29th July, 2010 has a short position and B, who has committed to buy 1000 stocks at Rs. 100 per share is said to have a long position.
Physical Settlement
A has to deliver 1000 shares to B. If not available then he has to purchase from market and deliver Profit/loss depends on settlement (closing) price on July 29, 2010.
If settlement price > 100 (futures price) If settlement price = 100 (futures price) If settlement price < 100 (futures price)
Physical settlement incurs huge transaction costs
Cash Settlement
Cash settlement does not involve actual delivery or receipt of the security Each party either pays (receives) cash equal to the net loss (profit) arising out of their respective position in the contract Profit and loss position in both the cases is same except for the transaction costs involved in the physical settlement
Features of Forward / Futures Contracts
Systematic obligation for the both buyer and the seller FC can take positive, zero and negative values When FC first written, value is zero
Forward to Futures
The important drawback of the forward contract is default/counter party risk This arises because of lack intermediary who could ensure that both parties honor their obligations Answer is futures contract and mediator is stock exchange clearing house
Futures Contracts
The asset The contract size Delivery arrangements:
What can be delivered, Where it can be delivered, & When it can be delivered
Delivery months Price quotes Price limits and position limits
Daily Settlement and Margins
The operation of Margins
Margin account Initial margin Marking to market Maintenance margin
Example of a Futures Trade
An investor takes a long position in 2 August Nifty 50 futures contracts on July 7
contract size is 50 Nifty (Approx. Rs.2,00,000). futures price is Rs.4500 margin requirement is Rs.11,250/contract maintenance margin is Rs.9,000/contract
The clearing house and clearing margins Credit risk and collateralization in OTC markets
Other Key Points About Futures
They are settled daily Closing out a futures position involves entering into an offsetting trade Most contracts are closed out before maturity
Types of traders and types of orders
Types of orders
Market order Limit order Stop-loss order
Types of traders
Commission brokers and locals
Hedgers, arbitrageurs and Speculators
Questions
When a new trade is completed what are the possible effects on the open interest? Can the volume of trading in a day be greater than the open interest?
Patterns of Futures Prices
Convergence of Futures Prices
Forward Contracts vs Futures Contracts
FORWARDS Private contract between 2 parties FUTURES Exchange traded Standard contract Range of delivery dates Settled daily Contract usually closed out prior to maturity
Virtually no credit risk
Futures Price Spot Price Futures Price
Non-standard contract
Spot Price
Usually 1 specified delivery date Settled at end of contract
Time
Time
Delivery or final cash settlement usually occurs
Some credit risk
(a)
(b)
Assumptions
No transaction costs Same tax rates on all net trading profits Same rate of interest for borrowing as well as lending No arbitrage opportunities
Determination of Futures Prices
Short Selling
Short selling involves selling securities you do not own Your broker borrows the securities from another client and sells them in the market in the usual way At some stage you must buy the securities back so they can be replaced in the account of the client You must pay dividends and other benefits the owner of the securities receives
Notation for Valuing Futures and Forward Contracts
S0: Spot price today F0: Futures or forward price today T: Time until delivery date r: Risk-free interest rate for maturity T
BoI: An Arbitrage Opportunity?
Suppose that:
The spot price of BoI stock is Rs.390 The quoted 1-month futures price of BoI is Rs.425 The 1-month interest rate is 5% No income or storage costs for BoI stock
BoI: Another Arbitrage Opportunity?
Suppose that:
The spot price of BoI stock is Rs.390 The quoted 1-month futures price of BoI is Rs.390 The 1-month interest rate is 5% No income or storage costs for BoI stock
Is there an arbitrage opportunity? Is cash and carry possible?
Is there an arbitrage opportunity? Is reverse cash and carry possible?
The Forward Price of Gold
If the spot price of BoI is S and the futures price is in T period is F, then F = S (1+r )T where r is risk-free rate of interest. In our examples, S=390, T=1, and r=0.05 so that F = 390(1+0.05) = 409.50
When Interest Rates are Measured with Continuous Compounding
F0 = S0erT
This equation relates the futures price and the spot price for any investment asset that provides no income and has no storage costs
Example
A one year long forward contract on a nondividend paying stock is entered into when the stock price is Rs. 400 and the risk-free rate of interest is 10% per annum with continuous compounding.
What is the forward price? Six months later, price of the stock is Rs.450 and the risk-free interest rate is still 10%. What is the forward price?
When an Investment Asset Provides a Known Income
10 months futures contract on a stock with a spot price of 50. Dividend payment of 0.75 after 3 months, 6 months and 9 months. Risk free interest rate is 8% per annum for all the maturities. What should be the futures price?
When an Investment Asset Provides a Known Income
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract
When an Investment Asset Provides a Known Yield
6-month FC on an asset which provides income 2% of the asset price once during a 6-month period. The risk-free rate is 10% per annum. The asset price is 20. What should be the futures price?
When an Investment Asset Provides a Known Yield
Stock Index
Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore
F0 = S0 e(r–q )T
where q is the average yield during the life of the contract (expressed with continuous compounding)
F0 = S0 e(r–q )T
where q is the average dividend yield on the portfolio represented by the index during life of contract
Index Arbitrage
When F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures When F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index
Example
Assume that the risk-free interest rate is 9% per annum and the dividend yield on a stock index varies throughout the year. In Feb, May, Aug, and Nov dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31, 2010 is 4000. What is the futures price for a contract deliverable on Dec 31, 2010?
Example
Suppose that the risk-free interest rate is 10% per annum and the dividend yield on a stock index is 4% per annum. The index is standing at 4000, and the futures price for a contract deliverable in four months is 4050. What arbitrage opportunities does this create?
Valuing a Forward Contract
Long FC entered sometime ago. S=25, r = 10%, T = 6-month to maturity, and K = 24 (Initial futures price), then what is the value of contract?
Valuing a Forward Contract
Suppose that K is settlement price in a futures contract and F0 is futures price that would apply to the contract today The value of a long futures contract, ƒ, is ƒ = (F0 – K )e–rT Similarly, the value of a short futures contract is (K – F0 )e–rT
Currency Futures Price
Currency futures price is derived from Interest Rate Parity (IRP) principle
F0 = S0 e(rh–rf)T
IRP F/S = (1+rh )/(1+rf )
Currency Futures Price
Let us assume that one year interest rates in US and India are say 7% and 10% respectively and the spot rate of USD in India is Rs. 44. One year forward exchange rate should be F = 44*e(0.10-0.07)*1=45.34
Futures on Commodities
1 year futures contract that provides no income. Its costs Rs.2 per unit to store the asset and the payment is made at the end of the year. Spot price is Rs.450 and riskfree rate is 7% per annum. What is the futures price?
Futures on Commodities
In absence of storage costs and income, the forward price, F0 = S0 erT If U is the PV of the storage costs, then the forward price is, F0 = (S0 + U)erT If the storage costs made are proportional to the price of the commodity, then forward price is, F0 = S0 e(r+u )T
Example
The current price of gold is Rs.18000 per 10g. The storage costs are Rs.100 per 10g per year payable quarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of gold for delivery in nine months.
Futures on Consumption Assets
Futures on Consumption Assets
Convenience yield – the benefits from holding the physical assets In presence of convenience yield, y, forward price is, F0eyT = (S0+U )erT If the storage costs per unit are a constant proportion, u, of the spot price, then the forward price is, F0eyT = S0 e(r+u )T =
F0 ? S0 e(r+u )T
where u is the storage cost per unit time as a percent of the asset value. Alternatively,
F0 ? (S0+U )erT
where U is the present value of the storage costs.
S0 e(r+u-y)T
The Cost of Carry
The cost of carry, c, is the storage cost plus the interest costs less the income earned For an investment asset F0 = S0ecT For a consumption asset F0 ? S0ecT The convenience yield on the consumption asset, y, is defined so that F0 = S0 e(c–y )T
Forward vs. Futures Prices
Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different: A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse
Payoff Profile: Long on Futures
profit Long position
Hedging Strategies Using Futures
0
S180(Rs./$) 100 F180(Rs./$) = 47.0900
loss
Payoff Profile: Short on Futures
profit
Long & Short Hedges
A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price A short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price
If you agree to sell anything in the future at a set price and the spot price later falls then you gain.
0 F180(Rs./$) = 47.0900
S180(Rs/$)
If you agree to sell anything in the future at a set price and the spot loss price later rises then you lose. Short position
Long Hedge
A copper fabricator knows it will require 1000kgs of copper on November 25 to meet a certain demand. The spot price of copper is Rs.140/kg and the August futures price is Rs.145/kg
Hedging strategy? Gain/loss if copper price will be Rs.150/kg on expiry day?
Short Hedge
An oil producer has negotiated a contract to sell 1000 barrels of crude oil on Aug 27. Spot price of crude oil is $55 per barrel and Aug futures price is $54.
Appropriate hedging strategy? Gain/loss if spot price expiry day is $52?
Long Hedge
Short Hedge
Arguments in Favor of Hedging
Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables
Arguments against Hedging
Shareholders are usually well diversified and can make their own hedging decisions It may increase risk to hedge when competitors do not.
What happens if
It is Sept 23. A company knows that it will need to purchase 1000 barrels of crude oil on Nov 10, 2010. There is no futures contract for Nov as they are quarterly contracts. The current Dec oil futures price is $56 per barrel.
Spot price on Nov 10 = $58 Dec futures price on Nov 10 = $57
What happens if
It is Sept 23. An Indian company expects to receive $1000 at the end of November. Supposing November futures are not available in India. The Dec futures price for the USD-INR is Rs.46.5.
Spot price on Nov 30 = Rs.45.0 Dec futures price on Nov 30 = Rs.45.5
Basis Risk
Basis is the difference between spot & futures Basis risk arises because
hedged asset not same as underlying uncertainty about the date of buying or selling of underlying futures contracts close out before delivery
Long Hedge
Suppose that F1 : Initial Futures Price F2 : Final Futures Price S2 : Final Asset Price You hedge the future purchase of an asset by entering into a long futures contract Effective price paid with hedging = S2 + (F1 – F2) = F1 + Basis If basis strengthens, the hedger’s position worsens and vice-versa
Short Hedge
Suppose that F1 : Initial Futures Price F2 : Final Futures Price S2 : Final Asset Price You hedge the future sale of an asset by entering into a short futures contract Price Realized from hedging = S2+ (F1 – F2) = F1 + Basis If basis strengthens, the hedger’s position improves and vice-versa
Choice of Contract
Choose a settlement month that is as close as possible to, but later than, the end of the life of the hedge When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. This is known as cross hedging.
Cross Hedging: Optimal Hedge Ratio
Ratio of the size of the position taken in futures contracts to the size of the exposure. Proportion of the exposure that should optimally be hedged is
Optimal Hedge Ratio
An airline expects to purchase 2 million gallons of jet fuel in 1 month and decides to use heating oil futures for hedging. ?S = 0.0263, ?F= 0.0313, and ? = 0.928 Optimal hedge ratio (h) = 0.78
?
?S ?F
where ?S is the standard deviation of ?S, the change in the spot price during the hedging period, ?F is the standard deviation of ?F, the change in the futures price during the hedging period ? is the coefficient of correlation between ?S and ?F.
Optimal Number of Contracts
The number of futures contracts required is given by N* = h*NA / QF NA = size of position being hedged QF = size of one futures contract
Optimal Number of Contracts
Continuing with airlines example, each heating oil contract traded is on 42,000 gallons of heating oil, then Optimal number of contracts (N) = 37.14 or approx 37
Hedging Using Index Futures
To hedge the risk in a portfolio the number of contracts that should be shorted is
? P A
Example
Value of Nifty is 4900 Value of Portfolio is Rs.500000 Beta of portfolio is 1.5 One futures contract is 50 times the index Futures price is 5000 What position in futures contracts on the Nifty is necessary to hedge the portfolio?
where P is the value of the portfolio, ? is its beta, and A is the value of the assets underlying one futures contract
Example Cont.
Suppose the index turns out to be 4500 in 3 months and the futures price is 4600, estimate the overall gain/loss from taking short futures position. Risk free interest rate is 5% per annum Index dividend yield is 3% per annum
Reasons for Hedging an Equity Portfolio
Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.) Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outperform the market.)
Rolling The Hedge Forward
We can use a series of futures contracts to increase the life of a hedge Each time we switch from 1 futures contract to another we incur a type of basis risk
Backwardation and Contango
When futures price is below the spot price, the situation is normal backwardation When futures price is above the spot price, the situation is known as contango
doc_188452451.pdf
forwards and futures contract in detail with the help of examples. It also explains types of traders and types of orders, , it compares forwards contracts and futures contracts, determination of future prices. It also contains arbitrage and various hedging strategies using futures.
Derivatives
Forwards and Futures
Forwards Futures
Derivatives management Framework
Individual Components Options Swaps
Derivatives Markets
Exchange traded
Traditionally exchanges have used the open-outcry system, but increasingly they are switching to electronic trading Contracts are standard there is virtually no credit risk
Who Trade Derivatives?
Market-makers and dealers stand ready to buy or to sell: they take the other side of a transaction for whomever wants to trade
Although they accommodate customers, market-makers and dealers make money on transaction fees, and try not to take excessive risk
Over-the-counter (OTC)
A computer- and telephone-linked network of dealers at financial institutions, corporations, and fund managers Contracts can be non-standard and there is some small amount of credit risk
Arbitrageurs trade to take advantage of prices that are temporarily wrong Speculators get extra leverage in betting on future movements in the price of an asset. Hedgers use the instruments to reduce risk associated with price of an asset.
Derivatives Market in India
Derivatives trading started with the promulgation of the Securities Laws (Amendment) Ordinance, 1995, which withdrew the prohibition on options in securities. SEBI set up L.C. Gupta committee in 1996 to develop appropriate regulatory framework for derivatives The committee submitted its report in 1998 and recommended derivatives as ‘securities’. SEBI set up J.R. Verma committee to recommend measures for risk containment in DM The SCRA was amended in 1999 to include derivatives as securities and developed regulatory framework for derivatives
Derivatives Products in India
Index futures Stock futures Index options Stock options Commodity futures Interest rate futures Currency futures Credit derivatives
Forward / Futures Contracts
Forward / Futures contract is an agreement between the seller and the buyer whereby the seller is obligated to deliver the asset on a specified date and the buyer is obligated to pay a specified price upon delivery Forward contracts are similar to futures except that later trade in the over-the-counter market
Forwards and Futures
2.7
Forward Contract Example
Consider two parties (A and B) enter into a forward contract on 13 July, 2010 where, A agrees to deliver 1000 stocks of Unitech to B, at a price of Rs. 100 per share, on 29th July, 2010 (the expiry date). In this contract, A, who has committed to sell 1000 stocks of Unitech at Rs. 100 per share on 29th July, 2010 has a short position and B, who has committed to buy 1000 stocks at Rs. 100 per share is said to have a long position.
Physical Settlement
A has to deliver 1000 shares to B. If not available then he has to purchase from market and deliver Profit/loss depends on settlement (closing) price on July 29, 2010.
If settlement price > 100 (futures price) If settlement price = 100 (futures price) If settlement price < 100 (futures price)
Physical settlement incurs huge transaction costs
Cash Settlement
Cash settlement does not involve actual delivery or receipt of the security Each party either pays (receives) cash equal to the net loss (profit) arising out of their respective position in the contract Profit and loss position in both the cases is same except for the transaction costs involved in the physical settlement
Features of Forward / Futures Contracts
Systematic obligation for the both buyer and the seller FC can take positive, zero and negative values When FC first written, value is zero
Forward to Futures
The important drawback of the forward contract is default/counter party risk This arises because of lack intermediary who could ensure that both parties honor their obligations Answer is futures contract and mediator is stock exchange clearing house
Futures Contracts
The asset The contract size Delivery arrangements:
What can be delivered, Where it can be delivered, & When it can be delivered
Delivery months Price quotes Price limits and position limits
Daily Settlement and Margins
The operation of Margins
Margin account Initial margin Marking to market Maintenance margin
Example of a Futures Trade
An investor takes a long position in 2 August Nifty 50 futures contracts on July 7
contract size is 50 Nifty (Approx. Rs.2,00,000). futures price is Rs.4500 margin requirement is Rs.11,250/contract maintenance margin is Rs.9,000/contract
The clearing house and clearing margins Credit risk and collateralization in OTC markets
Other Key Points About Futures
They are settled daily Closing out a futures position involves entering into an offsetting trade Most contracts are closed out before maturity
Types of traders and types of orders
Types of orders
Market order Limit order Stop-loss order
Types of traders
Commission brokers and locals
Hedgers, arbitrageurs and Speculators
Questions
When a new trade is completed what are the possible effects on the open interest? Can the volume of trading in a day be greater than the open interest?
Patterns of Futures Prices
Convergence of Futures Prices
Forward Contracts vs Futures Contracts
FORWARDS Private contract between 2 parties FUTURES Exchange traded Standard contract Range of delivery dates Settled daily Contract usually closed out prior to maturity
Virtually no credit risk
Futures Price Spot Price Futures Price
Non-standard contract
Spot Price
Usually 1 specified delivery date Settled at end of contract
Time
Time
Delivery or final cash settlement usually occurs
Some credit risk
(a)
(b)
Assumptions
No transaction costs Same tax rates on all net trading profits Same rate of interest for borrowing as well as lending No arbitrage opportunities
Determination of Futures Prices
Short Selling
Short selling involves selling securities you do not own Your broker borrows the securities from another client and sells them in the market in the usual way At some stage you must buy the securities back so they can be replaced in the account of the client You must pay dividends and other benefits the owner of the securities receives
Notation for Valuing Futures and Forward Contracts
S0: Spot price today F0: Futures or forward price today T: Time until delivery date r: Risk-free interest rate for maturity T
BoI: An Arbitrage Opportunity?
Suppose that:
The spot price of BoI stock is Rs.390 The quoted 1-month futures price of BoI is Rs.425 The 1-month interest rate is 5% No income or storage costs for BoI stock
BoI: Another Arbitrage Opportunity?
Suppose that:
The spot price of BoI stock is Rs.390 The quoted 1-month futures price of BoI is Rs.390 The 1-month interest rate is 5% No income or storage costs for BoI stock
Is there an arbitrage opportunity? Is cash and carry possible?
Is there an arbitrage opportunity? Is reverse cash and carry possible?
The Forward Price of Gold
If the spot price of BoI is S and the futures price is in T period is F, then F = S (1+r )T where r is risk-free rate of interest. In our examples, S=390, T=1, and r=0.05 so that F = 390(1+0.05) = 409.50
When Interest Rates are Measured with Continuous Compounding
F0 = S0erT
This equation relates the futures price and the spot price for any investment asset that provides no income and has no storage costs
Example
A one year long forward contract on a nondividend paying stock is entered into when the stock price is Rs. 400 and the risk-free rate of interest is 10% per annum with continuous compounding.
What is the forward price? Six months later, price of the stock is Rs.450 and the risk-free interest rate is still 10%. What is the forward price?
When an Investment Asset Provides a Known Income
10 months futures contract on a stock with a spot price of 50. Dividend payment of 0.75 after 3 months, 6 months and 9 months. Risk free interest rate is 8% per annum for all the maturities. What should be the futures price?
When an Investment Asset Provides a Known Income
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract
When an Investment Asset Provides a Known Yield
6-month FC on an asset which provides income 2% of the asset price once during a 6-month period. The risk-free rate is 10% per annum. The asset price is 20. What should be the futures price?
When an Investment Asset Provides a Known Yield
Stock Index
Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore
F0 = S0 e(r–q )T
where q is the average yield during the life of the contract (expressed with continuous compounding)
F0 = S0 e(r–q )T
where q is the average dividend yield on the portfolio represented by the index during life of contract
Index Arbitrage
When F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures When F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index
Example
Assume that the risk-free interest rate is 9% per annum and the dividend yield on a stock index varies throughout the year. In Feb, May, Aug, and Nov dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31, 2010 is 4000. What is the futures price for a contract deliverable on Dec 31, 2010?
Example
Suppose that the risk-free interest rate is 10% per annum and the dividend yield on a stock index is 4% per annum. The index is standing at 4000, and the futures price for a contract deliverable in four months is 4050. What arbitrage opportunities does this create?
Valuing a Forward Contract
Long FC entered sometime ago. S=25, r = 10%, T = 6-month to maturity, and K = 24 (Initial futures price), then what is the value of contract?
Valuing a Forward Contract
Suppose that K is settlement price in a futures contract and F0 is futures price that would apply to the contract today The value of a long futures contract, ƒ, is ƒ = (F0 – K )e–rT Similarly, the value of a short futures contract is (K – F0 )e–rT
Currency Futures Price
Currency futures price is derived from Interest Rate Parity (IRP) principle
F0 = S0 e(rh–rf)T
IRP F/S = (1+rh )/(1+rf )
Currency Futures Price
Let us assume that one year interest rates in US and India are say 7% and 10% respectively and the spot rate of USD in India is Rs. 44. One year forward exchange rate should be F = 44*e(0.10-0.07)*1=45.34
Futures on Commodities
1 year futures contract that provides no income. Its costs Rs.2 per unit to store the asset and the payment is made at the end of the year. Spot price is Rs.450 and riskfree rate is 7% per annum. What is the futures price?
Futures on Commodities
In absence of storage costs and income, the forward price, F0 = S0 erT If U is the PV of the storage costs, then the forward price is, F0 = (S0 + U)erT If the storage costs made are proportional to the price of the commodity, then forward price is, F0 = S0 e(r+u )T
Example
The current price of gold is Rs.18000 per 10g. The storage costs are Rs.100 per 10g per year payable quarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of gold for delivery in nine months.
Futures on Consumption Assets
Futures on Consumption Assets
Convenience yield – the benefits from holding the physical assets In presence of convenience yield, y, forward price is, F0eyT = (S0+U )erT If the storage costs per unit are a constant proportion, u, of the spot price, then the forward price is, F0eyT = S0 e(r+u )T =
F0 ? S0 e(r+u )T
where u is the storage cost per unit time as a percent of the asset value. Alternatively,
F0 ? (S0+U )erT
where U is the present value of the storage costs.
S0 e(r+u-y)T
The Cost of Carry
The cost of carry, c, is the storage cost plus the interest costs less the income earned For an investment asset F0 = S0ecT For a consumption asset F0 ? S0ecT The convenience yield on the consumption asset, y, is defined so that F0 = S0 e(c–y )T
Forward vs. Futures Prices
Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different: A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse
Payoff Profile: Long on Futures
profit Long position
Hedging Strategies Using Futures
0
S180(Rs./$) 100 F180(Rs./$) = 47.0900
loss
Payoff Profile: Short on Futures
profit
Long & Short Hedges
A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price A short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price
If you agree to sell anything in the future at a set price and the spot price later falls then you gain.
0 F180(Rs./$) = 47.0900
S180(Rs/$)
If you agree to sell anything in the future at a set price and the spot loss price later rises then you lose. Short position
Long Hedge
A copper fabricator knows it will require 1000kgs of copper on November 25 to meet a certain demand. The spot price of copper is Rs.140/kg and the August futures price is Rs.145/kg
Hedging strategy? Gain/loss if copper price will be Rs.150/kg on expiry day?
Short Hedge
An oil producer has negotiated a contract to sell 1000 barrels of crude oil on Aug 27. Spot price of crude oil is $55 per barrel and Aug futures price is $54.
Appropriate hedging strategy? Gain/loss if spot price expiry day is $52?
Long Hedge
Short Hedge
Arguments in Favor of Hedging
Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables
Arguments against Hedging
Shareholders are usually well diversified and can make their own hedging decisions It may increase risk to hedge when competitors do not.
What happens if
It is Sept 23. A company knows that it will need to purchase 1000 barrels of crude oil on Nov 10, 2010. There is no futures contract for Nov as they are quarterly contracts. The current Dec oil futures price is $56 per barrel.
Spot price on Nov 10 = $58 Dec futures price on Nov 10 = $57
What happens if
It is Sept 23. An Indian company expects to receive $1000 at the end of November. Supposing November futures are not available in India. The Dec futures price for the USD-INR is Rs.46.5.
Spot price on Nov 30 = Rs.45.0 Dec futures price on Nov 30 = Rs.45.5
Basis Risk
Basis is the difference between spot & futures Basis risk arises because
hedged asset not same as underlying uncertainty about the date of buying or selling of underlying futures contracts close out before delivery
Long Hedge
Suppose that F1 : Initial Futures Price F2 : Final Futures Price S2 : Final Asset Price You hedge the future purchase of an asset by entering into a long futures contract Effective price paid with hedging = S2 + (F1 – F2) = F1 + Basis If basis strengthens, the hedger’s position worsens and vice-versa
Short Hedge
Suppose that F1 : Initial Futures Price F2 : Final Futures Price S2 : Final Asset Price You hedge the future sale of an asset by entering into a short futures contract Price Realized from hedging = S2+ (F1 – F2) = F1 + Basis If basis strengthens, the hedger’s position improves and vice-versa
Choice of Contract
Choose a settlement month that is as close as possible to, but later than, the end of the life of the hedge When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. This is known as cross hedging.
Cross Hedging: Optimal Hedge Ratio
Ratio of the size of the position taken in futures contracts to the size of the exposure. Proportion of the exposure that should optimally be hedged is
Optimal Hedge Ratio
An airline expects to purchase 2 million gallons of jet fuel in 1 month and decides to use heating oil futures for hedging. ?S = 0.0263, ?F= 0.0313, and ? = 0.928 Optimal hedge ratio (h) = 0.78
?
?S ?F
where ?S is the standard deviation of ?S, the change in the spot price during the hedging period, ?F is the standard deviation of ?F, the change in the futures price during the hedging period ? is the coefficient of correlation between ?S and ?F.
Optimal Number of Contracts
The number of futures contracts required is given by N* = h*NA / QF NA = size of position being hedged QF = size of one futures contract
Optimal Number of Contracts
Continuing with airlines example, each heating oil contract traded is on 42,000 gallons of heating oil, then Optimal number of contracts (N) = 37.14 or approx 37
Hedging Using Index Futures
To hedge the risk in a portfolio the number of contracts that should be shorted is
? P A
Example
Value of Nifty is 4900 Value of Portfolio is Rs.500000 Beta of portfolio is 1.5 One futures contract is 50 times the index Futures price is 5000 What position in futures contracts on the Nifty is necessary to hedge the portfolio?
where P is the value of the portfolio, ? is its beta, and A is the value of the assets underlying one futures contract
Example Cont.
Suppose the index turns out to be 4500 in 3 months and the futures price is 4600, estimate the overall gain/loss from taking short futures position. Risk free interest rate is 5% per annum Index dividend yield is 3% per annum
Reasons for Hedging an Equity Portfolio
Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.) Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outperform the market.)
Rolling The Hedge Forward
We can use a series of futures contracts to increase the life of a hedge Each time we switch from 1 futures contract to another we incur a type of basis risk
Backwardation and Contango
When futures price is below the spot price, the situation is normal backwardation When futures price is above the spot price, the situation is known as contango
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