Podcast
Questions and Answers
What is the primary difference between investment assets and consumption assets in the context of forward and futures contracts?
What is the primary difference between investment assets and consumption assets in the context of forward and futures contracts?
- Investment assets are primarily held for consumption, while consumption assets are held for investment purposes.
- Investment assets are tangible, while consumption assets are intangible.
- Consumption assets are always more liquid than investment assets.
- Arbitrage arguments can determine forward and futures prices for investment assets, but not for consumption assets. (correct)
Explain the concept of 'short selling' shares of a stock.
Explain the concept of 'short selling' shares of a stock.
- Selling shares one owns with the intent of buying them back immediately.
- Borrowing shares from a broker, selling them, and repurchasing them later to return to the lender. (correct)
- Buying shares on margin with the expectation that the price will increase rapidly.
- Purchasing shares with the intention of holding them long-term.
An investor shorts 500 shares at $120, pays a $1 dividend per share, and closes the position at $100. What is the net gain/loss, ignoring borrowing fees?
An investor shorts 500 shares at $120, pays a $1 dividend per share, and closes the position at $100. What is the net gain/loss, ignoring borrowing fees?
- Loss of $9,500
- Profit of $9,500 (correct)
- Loss of $10,500
- Profit of $10,500
How does a margin account function in short selling?
How does a margin account function in short selling?
What was the key restriction introduced by the 'alternative uptick' rule following the abolishment of the original uptick rule?
What was the key restriction introduced by the 'alternative uptick' rule following the abolishment of the original uptick rule?
What are the key assumptions often made when determining forward and futures prices?
What are the key assumptions often made when determining forward and futures prices?
In the notation used for forward and futures contracts, what does 'r' typically represent?
In the notation used for forward and futures contracts, what does 'r' typically represent?
If the forward price of a non-dividend-paying stock is too high, describe the arbitrage strategy to exploit the mispricing.
If the forward price of a non-dividend-paying stock is too high, describe the arbitrage strategy to exploit the mispricing.
What is the formula for the forward price (F0) of an investment asset with price (S0) that provides no income, with risk-free rate (r) and time to maturity (T)?
What is the formula for the forward price (F0) of an investment asset with price (S0) that provides no income, with risk-free rate (r) and time to maturity (T)?
What strategy would an arbitrageur employ if the forward price is less than $S_0e^{rT}$?
What strategy would an arbitrageur employ if the forward price is less than $S_0e^{rT}$?
A zero-coupon bond has a current price of $930 and matures in 1 year. If the 4-month risk-free rate is 6%, what is the 4-month forward price?
A zero-coupon bond has a current price of $930 and matures in 1 year. If the 4-month risk-free rate is 6%, what is the 4-month forward price?
When an investment asset provides a known income, which variable needs to be subtracted from the spot price (S0) in the forward price formula?
When an investment asset provides a known income, which variable needs to be subtracted from the spot price (S0) in the forward price formula?
What is the formula for the forward price (F0) of an asset with price (S0), income (I), risk-free rate (r), and time to maturity (T)?
What is the formula for the forward price (F0) of an asset with price (S0), income (I), risk-free rate (r), and time to maturity (T)?
A bond is priced at $900 with a coupon payment of $40 expected in 4 months. The 4-month and 9-month rates are 3% and 4%. If the forward price is $910, what arbitrage opportunity exists?
A bond is priced at $900 with a coupon payment of $40 expected in 4 months. The 4-month and 9-month rates are 3% and 4%. If the forward price is $910, what arbitrage opportunity exists?
Consider a 10-month forward contract on a stock at $50. Dividends of $0.75 are paid after 3, 6, and 9 months. If the risk-free rate is 8%, what is the forward price?
Consider a 10-month forward contract on a stock at $50. Dividends of $0.75 are paid after 3, 6, and 9 months. If the risk-free rate is 8%, what is the forward price?
If an asset provides a known yield, what impact does this have on the forward price?
If an asset provides a known yield, what impact does this have on the forward price?
What is the formula for the forward price (F0) when an asset provides a known yield (q), with spot price (S0), risk-free rate (r), and time to maturity (T)?
What is the formula for the forward price (F0) when an asset provides a known yield (q), with spot price (S0), risk-free rate (r), and time to maturity (T)?
An asset priced at $25 provides a 2% yield in 6 months. If the risk-free rate is 10%, what is the 6-month forward price?
An asset priced at $25 provides a 2% yield in 6 months. If the risk-free rate is 10%, what is the 6-month forward price?
Explain the formula for the value (f) of a long forward contract, where F0 is the current forward price, K is the delivery price, r is the risk-free rate, and T is the time to maturity.
Explain the formula for the value (f) of a long forward contract, where F0 is the current forward price, K is the delivery price, r is the risk-free rate, and T is the time to maturity.
A forward contract has 6 months to maturity. The risk-free rate is 10%, the stock price is $25, and the delivery price is $24. What is the value of the forward contract?
A forward contract has 6 months to maturity. The risk-free rate is 10%, the stock price is $25, and the delivery price is $24. What is the value of the forward contract?
Under what condition are forward and futures prices theoretically the same?
Under what condition are forward and futures prices theoretically the same?
Explain how the correlation between the price of the underlying asset (S) and interest rates affects the relationship between forward and futures prices.
Explain how the correlation between the price of the underlying asset (S) and interest rates affects the relationship between forward and futures prices.
What is the formula for the futures price (F0) of a stock index, given the index value (S0), risk-free rate (r), dividend yield (q), and time to maturity (T)?
What is the formula for the futures price (F0) of a stock index, given the index value (S0), risk-free rate (r), dividend yield (q), and time to maturity (T)?
Describe how index arbitrage works if the futures price is too high.
Describe how index arbitrage works if the futures price is too high.
If the December futures settlement price of the S&P 500 is less than the June price, what does this reveal about r and q?
If the December futures settlement price of the S&P 500 is less than the June price, what does this reveal about r and q?
Explain why CME's Nikkei 225 futures contract is described as a quanto.
Explain why CME's Nikkei 225 futures contract is described as a quanto.
What is the interest rate parity relationship formula between F0 and S0, US risk-free rate r, foreign risk-free rate rf, and time to maturity T?
What is the interest rate parity relationship formula between F0 and S0, US risk-free rate r, foreign risk-free rate rf, and time to maturity T?
If 2-year rates in Australia and the US are 3% and 1% respectively, and the spot rate is 0.9800 USD/AUD, what is the 2-year forward rate?
If 2-year rates in Australia and the US are 3% and 1% respectively, and the spot rate is 0.9800 USD/AUD, what is the 2-year forward rate?
How can a foreign currency be considered an asset providing a known yield to a US investor?
How can a foreign currency be considered an asset providing a known yield to a US investor?
What is the major difference when considering futures prices of commodities that are investment assets versus consumption assets?
What is the major difference when considering futures prices of commodities that are investment assets versus consumption assets?
If U represents the present value of storage costs, how is the forward price $F_0$ adjusted for commodity assets?
If U represents the present value of storage costs, how is the forward price $F_0$ adjusted for commodity assets?
An asset has a spot price of $450 and storage costs of $2 per unit, paid at year-end. If the risk-free rate is 7%, what is the 1-year futures price?
An asset has a spot price of $450 and storage costs of $2 per unit, paid at year-end. If the risk-free rate is 7%, what is the 1-year futures price?
Define the 'convenience yield' for a commodity.
Define the 'convenience yield' for a commodity.
How is futures price expressed if storage costs are proportional to the spot price?
How is futures price expressed if storage costs are proportional to the spot price?
Define the 'cost of carry'.
Define the 'cost of carry'.
When is it usually optimal for a party with a short position in a futures contract to deliver as early as possible?
When is it usually optimal for a party with a short position in a futures contract to deliver as early as possible?
According to Keynes and Hicks, how does hedging pressure affect futures prices?
According to Keynes and Hicks, how does hedging pressure affect futures prices?
When there is positive systematic risk in an asset underlying a futures contract, how should the futures price relate to the expected future spot price?
When there is positive systematic risk in an asset underlying a futures contract, how should the futures price relate to the expected future spot price?
Which of the following is the most accurate distinction between investment assets and consumption assets?
Which of the following is the most accurate distinction between investment assets and consumption assets?
In a short selling transaction, which action does the investor take first?
In a short selling transaction, which action does the investor take first?
An investor shorts 1,000 shares at $50. A dividend of $0.50 per share is paid out, and the position is closed when the share price is $40. Ignoring any borrowing fees or interest, what is the investor's net profit or loss?
An investor shorts 1,000 shares at $50. A dividend of $0.50 per share is paid out, and the position is closed when the share price is $40. Ignoring any borrowing fees or interest, what is the investor's net profit or loss?
Why is a margin account required in short selling?
Why is a margin account required in short selling?
The 'alternative uptick' rule introduced by the SEC in 2010 applies when a stock price has decreased by more than 10% in one day. Which of the following best describes the restriction imposed by this rule?
The 'alternative uptick' rule introduced by the SEC in 2010 applies when a stock price has decreased by more than 10% in one day. Which of the following best describes the restriction imposed by this rule?
When determining forward and futures prices, what is the primary reason for assuming that all market participants can borrow and lend at the same risk-free rate?
When determining forward and futures prices, what is the primary reason for assuming that all market participants can borrow and lend at the same risk-free rate?
In the context of forward and futures pricing, what is the significance of 'T' in the standard notation?
In the context of forward and futures pricing, what is the significance of 'T' in the standard notation?
If the current forward price of an asset is significantly higher than its expected future spot price, what action would an arbitrageur take to exploit this mispricing?
If the current forward price of an asset is significantly higher than its expected future spot price, what action would an arbitrageur take to exploit this mispricing?
What adjustment must be made to the forward price formula for an investment asset when the asset provides a known income?
What adjustment must be made to the forward price formula for an investment asset when the asset provides a known income?
If the forward price ($F_0$) of an investment asset is less than $S_0e^{rT}$ (where $S_0$ is the spot price, r is the risk-free rate, and T is time to maturity), what arbitrage strategy would be most profitable?
If the forward price ($F_0$) of an investment asset is less than $S_0e^{rT}$ (where $S_0$ is the spot price, r is the risk-free rate, and T is time to maturity), what arbitrage strategy would be most profitable?
Consider a 6-month forward contract on a zero-coupon bond with a current price of $800. If the 6-month risk-free rate is 4%, what is the nearest forward price according to the formulas outlined?
Consider a 6-month forward contract on a zero-coupon bond with a current price of $800. If the 6-month risk-free rate is 4%, what is the nearest forward price according to the formulas outlined?
Which of the following adjustments is required to the standard forward price formula when pricing a forward contract on an asset that provides a known cash income?
Which of the following adjustments is required to the standard forward price formula when pricing a forward contract on an asset that provides a known cash income?
A bond is priced at $1100 with a coupon payment of $50 expected in 3 months. The risk-free rates for 3 months and 6 months are 4% and 5% respectively. What is the approximate forward price for a six month forward contract?
A bond is priced at $1100 with a coupon payment of $50 expected in 3 months. The risk-free rates for 3 months and 6 months are 4% and 5% respectively. What is the approximate forward price for a six month forward contract?
A stock is trading at $75. Dividends of $1.00 are expected in 2 months, 5 months and 8 months. The risk-free rate of interest is constant at 7% per annum. What is the theoretical forward price for a 9-month forward contract?
A stock is trading at $75. Dividends of $1.00 are expected in 2 months, 5 months and 8 months. The risk-free rate of interest is constant at 7% per annum. What is the theoretical forward price for a 9-month forward contract?
What effect does a known yield from an asset have on its forward price?
What effect does a known yield from an asset have on its forward price?
An asset is priced at $100, and is expected to provide a known yield of 5% per annum. The continuously compounded risk-free rate over the investment horizon is 7% per annum. What is the forward price for a 1-year contract?
An asset is priced at $100, and is expected to provide a known yield of 5% per annum. The continuously compounded risk-free rate over the investment horizon is 7% per annum. What is the forward price for a 1-year contract?
Given a long forward contract with a delivery price of K, a current forward price of $F_0$, risk-free rate r, and time to maturity T, what does the formula $f = (F_0 - K)e^{-rT}$ represent?
Given a long forward contract with a delivery price of K, a current forward price of $F_0$, risk-free rate r, and time to maturity T, what does the formula $f = (F_0 - K)e^{-rT}$ represent?
Consider a forward contract with three months to maturity. The risk-free rate of interest is 6%. Given that the delivery price is $10, and the spot price is $12, what is the value of the forward contract?
Consider a forward contract with three months to maturity. The risk-free rate of interest is 6%. Given that the delivery price is $10, and the spot price is $12, what is the value of the forward contract?
Under what market conditions would forward and futures prices be closest?
Under what market conditions would forward and futures prices be closest?
How does a strong positive correlation between the price of an underlying asset and interest rates typically affect futures prices compared to forward prices?
How does a strong positive correlation between the price of an underlying asset and interest rates typically affect futures prices compared to forward prices?
What key assumption is made when using the formula $F_0 = S_0e^{(r-q)T}$ to determine the futures price of a stock index?
What key assumption is made when using the formula $F_0 = S_0e^{(r-q)T}$ to determine the futures price of a stock index?
An index is currently trading at $1,500. The risk free rate is estimated to be 6% and the dividend yield on the index is 2.5%. A trader observes that the futures price for a contract expiring in three months is more than $1,513.20. Which of the following is the correct action to take to exploit the arbitrage?
An index is currently trading at $1,500. The risk free rate is estimated to be 6% and the dividend yield on the index is 2.5%. A trader observes that the futures price for a contract expiring in three months is more than $1,513.20. Which of the following is the correct action to take to exploit the arbitrage?
If the futures price of a stock index for December is lower than the futures price contract for June, what does this indicate about the relationship between the risk-free rate (r) and the dividend yield (q)?
If the futures price of a stock index for December is lower than the futures price contract for June, what does this indicate about the relationship between the risk-free rate (r) and the dividend yield (q)?
What is a notable characteristic of CME's Nikkei 225 futures contract that classifies it as a 'quanto'?
What is a notable characteristic of CME's Nikkei 225 futures contract that classifies it as a 'quanto'?
What relationship is described by the interest rate parity when applied to forward and spot exchange rates?
What relationship is described by the interest rate parity when applied to forward and spot exchange rates?
Suppose the 1-year interest rates in Canada and the US are 4% and 2%, respectively, and the spot rate is 1.2500 USD/CAD. What is the 1-year Forward Exchange Rate?
Suppose the 1-year interest rates in Canada and the US are 4% and 2%, respectively, and the spot rate is 1.2500 USD/CAD. What is the 1-year Forward Exchange Rate?
From the perspective of a U.S. investor, in what way can a foreign currency be considered an investment asset providing a known yield?
From the perspective of a U.S. investor, in what way can a foreign currency be considered an investment asset providing a known yield?
What is the primary difference in analyzing futures prices between commodities that are investment assets and those that are consumption assets?
What is the primary difference in analyzing futures prices between commodities that are investment assets and those that are consumption assets?
How does accounting for storage costs impact the forward price ($F_0$) of a commodity with spot price $S_0$?
How does accounting for storage costs impact the forward price ($F_0$) of a commodity with spot price $S_0$?
Suppose that a commodity has a spot price of $750 and storage costs of $5 are payable at year end. Assuming a risk-free rate of 5%, what is the futures price for delivery in one year?
Suppose that a commodity has a spot price of $750 and storage costs of $5 are payable at year end. Assuming a risk-free rate of 5%, what is the futures price for delivery in one year?
What does 'convenience yield' represent in the context of commodity futures?
What does 'convenience yield' represent in the context of commodity futures?
When storage costs are proportional to the spot price of a commodity, how is the relationship between the spot price ($S_0$) and futures price ($F_0$) expressed?
When storage costs are proportional to the spot price of a commodity, how is the relationship between the spot price ($S_0$) and futures price ($F_0$) expressed?
In financial terms, what does 'cost of carry' generally encompass?
In financial terms, what does 'cost of carry' generally encompass?
Under what circumstances is it typically optimal for a party holding a short position in a futures contract to deliver as early as possible within the allowed delivery period?
Under what circumstances is it typically optimal for a party holding a short position in a futures contract to deliver as early as possible within the allowed delivery period?
According to Keynes and Hicks, how does 'hedging pressure' from participants who tend to hold short positions affect futures prices relative to expected future spot prices?
According to Keynes and Hicks, how does 'hedging pressure' from participants who tend to hold short positions affect futures prices relative to expected future spot prices?
If the systematic risk in an asset underlying a futures contract is positive, how should the futures price relate to the expected future spot price?
If the systematic risk in an asset underlying a futures contract is positive, how should the futures price relate to the expected future spot price?
What is a key difference between determining the price of futures contracts for investment assets versus consumption assets?
What is a key difference between determining the price of futures contracts for investment assets versus consumption assets?
An investor borrows shares of stock X and immediately sells them in the market. Later, the investor purchases the same number of shares of stock X to return them. What strategy is the investor employing?
An investor borrows shares of stock X and immediately sells them in the market. Later, the investor purchases the same number of shares of stock X to return them. What strategy is the investor employing?
An investor shorts 100 shares of a stock at $75 per share. The broker requires a 50% margin. How much must the investor deposit in the margin account?
An investor shorts 100 shares of a stock at $75 per share. The broker requires a 50% margin. How much must the investor deposit in the margin account?
Under the 'alternative uptick' rule, what condition must be met before a stock can be shorted after its price has decreased by more than 10% in one day?
Under the 'alternative uptick' rule, what condition must be met before a stock can be shorted after its price has decreased by more than 10% in one day?
Why is the assumption that all market participants can borrow and lend at the same risk-free rate important in forward and futures pricing?
Why is the assumption that all market participants can borrow and lend at the same risk-free rate important in forward and futures pricing?
In the forward price formula, $F_0 = S_0e^{rT}$, where $S_0$ is the spot price and $r$ is the risk-free rate, what does 'T' represent?
In the forward price formula, $F_0 = S_0e^{rT}$, where $S_0$ is the spot price and $r$ is the risk-free rate, what does 'T' represent?
If the forward price of an asset is significantly lower than what is justified by its spot price and carrying costs (interest and storage), what arbitrage strategy would be most appropriate?
If the forward price of an asset is significantly lower than what is justified by its spot price and carrying costs (interest and storage), what arbitrage strategy would be most appropriate?
Which action would an arbitrageur take if the current forward price is higher than the present value of the expected future spot price?
Which action would an arbitrageur take if the current forward price is higher than the present value of the expected future spot price?
How does a known cash income from an asset affect its forward price?
How does a known cash income from an asset affect its forward price?
If you observe that $F_0 < (S_0 - I)e^{rT}$, where $I$ is the present value of income from the asset, what arbitrage strategy should you employ?
If you observe that $F_0 < (S_0 - I)e^{rT}$, where $I$ is the present value of income from the asset, what arbitrage strategy should you employ?
A stock is priced at $80. Dividends of $1 are expected in 1 month and 4 months. The risk-free rate is 6%. Using the formula outlined, which expression correctly calculates the future price for five month forward contract?
A stock is priced at $80. Dividends of $1 are expected in 1 month and 4 months. The risk-free rate is 6%. Using the formula outlined, which expression correctly calculates the future price for five month forward contract?
How does a known yield on an asset affect its forward price?
How does a known yield on an asset affect its forward price?
Given the formula $f = (F_0 - K)e^{-rT}$ for the value of a long forward contract, what does 'K' represent?
Given the formula $f = (F_0 - K)e^{-rT}$ for the value of a long forward contract, what does 'K' represent?
What condition regarding interest rates is generally required for forward and futures prices to be theoretically the same?
What condition regarding interest rates is generally required for forward and futures prices to be theoretically the same?
In the context of futures contracts on stock indices, what does the dividend yield (q) represent?
In the context of futures contracts on stock indices, what does the dividend yield (q) represent?
What action is appropriate if $F_0 > S_0e^{(r-q)T}$?
What action is appropriate if $F_0 > S_0e^{(r-q)T}$?
According to interest rate parity, what is the relationship between spot and forward exchange rates?
According to interest rate parity, what is the relationship between spot and forward exchange rates?
From the perspective of a U.S. investor, what can a foreign currency be considered when analyzing forward prices?
From the perspective of a U.S. investor, what can a foreign currency be considered when analyzing forward prices?
How are storage costs typically treated when determining the forward price of a commodity?
How are storage costs typically treated when determining the forward price of a commodity?
Flashcards
Investment Asset
Investment Asset
Assets held for investment purposes by at least some traders (e.g., stocks, bonds, gold, silver).
Consumption Asset
Consumption Asset
Assets held primarily for consumption, not investment (e.g., copper, crude oil, corn).
Short Selling
Short Selling
Selling an asset that is not owned by borrowing it and selling it in the market, later buying it back to return it.
Uptick Rule
Uptick Rule
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T (Time to Delivery)
T (Time to Delivery)
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S0 (Spot Price)
S0 (Spot Price)
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F0 (Forward/Futures Price)
F0 (Forward/Futures Price)
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r (Risk-Free Rate)
r (Risk-Free Rate)
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Arbitrage Pricing
Arbitrage Pricing
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Forward Price Formula (No Income)
Forward Price Formula (No Income)
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I (Present Value of Income)
I (Present Value of Income)
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Forward Price Formula (Known Income)
Forward Price Formula (Known Income)
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q (Known Yield)
q (Known Yield)
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Forward Price Formula (Known Yield)
Forward Price Formula (Known Yield)
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Contract Valuation
Contract Valuation
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K (Delivery Price)
K (Delivery Price)
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f (Value of Forward Contract)
f (Value of Forward Contract)
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F0 (Current Forward Price)
F0 (Current Forward Price)
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Value of Forward Contract Formula
Value of Forward Contract Formula
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f = S0 - Ke^(-rT)
f = S0 - Ke^(-rT)
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f = S0 - I - Ke^(-rT)
f = S0 - I - Ke^(-rT)
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f = S0e^(-qT) - Ke^(-rT)
f = S0e^(-qT) - Ke^(-rT)
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Daily Settlement Impact
Daily Settlement Impact
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Spot Exchange Rate (S0)
Spot Exchange Rate (S0)
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Forward/Futures Rate (F0)
Forward/Futures Rate (F0)
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rf (Foreign Rate)
rf (Foreign Rate)
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Interest Rate Parity
Interest Rate Parity
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Storage Costs (U)
Storage Costs (U)
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Borrowing gold.
Borrowing gold.
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Commodities that can provide income.
Commodities that can provide income.
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Futures Price with Storage
Futures Price with Storage
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Storage Cost (u)
Storage Cost (u)
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Convenience Yield (y)
Convenience Yield (y)
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Futures Price with Convenience Yield
Futures Price with Convenience Yield
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Cost of Carry (c)
Cost of Carry (c)
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Delivery Option
Delivery Option
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Expected Spot Price
Expected Spot Price
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Normal Backwardation
Normal Backwardation
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Contango
Contango
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C>Y should occur when
C>Y should occur when
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Study Notes
- Forward prices and futures prices are closely related to the spot price of the underlying asset.
Forward vs. Futures Contracts
- Forward contracts are simpler to analyze because payment occurs only at maturity.
- Forward and futures prices usually converge when maturities are the same, allowing results for forwards to often apply to futures.
Investment Assets vs. Consumption Assets
- Investment assets (e.g., stocks, bonds, gold, silver) are held for investment purposes by at least some traders.
- Consumption assets (e.g., copper, crude oil, corn, pork bellies) are primarily held for consumption, not investment.
- Arbitrage arguments can determine forward and futures prices for investment assets, but not for consumption assets.
Short Selling
- Short selling involves selling an asset that is not owned, and it's possible for some investment assets.
- To short sell, an investor borrows shares through a broker and sells them, later repurchasing them to close the position.
- Profit is made if the stock price decreases; a loss occurs if it increases.
- Short sellers must pay any income (dividends, interest) that would normally be received on the shorted securities.
- Investors maintain a margin account with the broker as collateral, earning interest on the balance.
- Proceeds from the asset sale typically form part of the initial margin.
- Regulations like the uptick rule (now an alternative uptick rule) can restrict short selling, especially during high market volatility.
Assumptions and Notation
- No transaction costs for market participants.
- All market participants are subject to the same tax rate on net trading profits.
- Market participants can borrow and lend at the same risk-free interest rate.
- Market participants exploit arbitrage opportunities.
- These assumptions need only apply to key market participants like large derivatives dealers.
Notation
- T = Time until delivery date (in years)
- S0 = Current price of the underlying asset
- F0 = Forward or futures price today
- r = Risk-free interest rate (continuous compounding) for maturity at the delivery date
Forward Price for an Investment Asset
- For an investment asset providing no income, the forward price is: F0 = S0e^(rT).
- Arbitrage prevents forward prices from deviating from this formula, where higher forward prices lead to buying the asset and shorting forward contracts, and lower forward prices lead to shorting the asset and entering long forward contracts.
Implications of Not Being Able to Short Sell
- If short sales are not possible, the forward price can still be determined by the actions of investors holding the asset purely for investment.
Known Income
- When an investment asset provides a predictable cash income (present value = I): F0 = (S0 - I)e^(rT).
- Arbitrage prevents deviations from this formula, where higher forward prices lead to buying the asset and shorting a forward contract, and lower forward prices lead to shorting the asset and taking a long position in a forward contract.
Known Yield
- When the asset provides a known yield (q) per annum with continuous compounding: F0 = S0e^((r-q)T)
Valuing Forward Contracts
- The value of a forward contract (f) is generally: f = (F0 - K)e^(-rT) where K is the delivery price.
- This is derived by considering a portfolio of buying an asset for K and selling it for F0 at time T, resulting in a certain payoff of F0 - K discounted at the risk-free rate
- For an investment asset providing no income: f = S0 - Ke^(-rT)
- For an investment asset providing a known income with present value I : f = S0 - I - Ke^(-rT)
- For an investment asset providing a known yield at rate q: f = S0e^(-qT) - Ke^(-rT)
Forward Prices vs. Futures Prices
- In theory, forward and futures prices should be the same when the short-term risk-free interest rate is constant, or a known function of time
- When interest rates vary unpredictably, forward and futures prices can differ.
- The difference is usually small and negligible for contracts lasting a few months.
- Factors like taxes, transaction costs, margin requirements, counterparty default risk, and liquidity can also cause differences.
Futures Prices of Stock Indices
- A stock index acts as an investment asset paying dividends.
- If q is the dividend yield rate, the futures price is: F0 = S0e^((r-q)T).
- If F0 > S0e^((r-q)T), buy stocks underlying index & short futures
- If F0 < S0e^((r-q)T), short stocks underlying index & long futures.
- These are known as index arbitrage strategies.
Forward and Futures Contracts on Currencies
- Spot price (S0) is the current price, in USD, of one unit of foreign currency.
- The forward/futures price is: F0 = S0e^((r-rf)T)
- Variable rf is the foreign risk-free interest rate, and r is the domestic risk-free interest rate.
- This is the interest rate parity formula
Futures on Commodities
- First consider futures prices on investment assets like gold and silver.
- Then, consider futures prices of consumption assets.
Income and Storage Costs
- Storage costs can be treated as negative income.
- If U is the present value of all storage costs, net of income, during the life of a forward contract, the future price is: F0 = (S0 + U)e^(rT)
- If storage costs are proportional to commodity price (rate = u), the futures price is: F0 = S0e^((r+u)T)
Consumption Commodities
- Typically provide no income but can have significant storage costs.
- F0 ≤ (S0 + U)e^(rT)
Convenience Yields
- Users of a consumption commodity may value physical ownership over futures for production or local shortages.
- Convenience yield (y) is defined as the benefit of holding the physical asset.
- Equation given known storage costs: F0e^(yT) = (S0 + U)e^(rT)
- Equation given proportional storage costs: F0 = S0e^((r + u -y)T)
- Greater chance of shortages is related to a higher convenience yield.
The Cost of Carry
- Measures storage cost + interest to finance asset - income earned on asset.
- For investment asset: F0 = S0e^(cT).
- For consumption asset: F0 = S0e^((c-y)T). Where 'c' is the cost of carry and 'y' is the convenience yield.
Delivery Options
- Party with short position in futures contract can deliver anytime during a period.
- If futures price is increasing with maturity (c > y), deliver early.
- If futures prices are decreasing with maturity (c < y), deliver late.
Futures Prices and Expected Future Spot Prices
- The expected spot price is the market's average opinion of an asset's future spot price.
- John Maynard Keynes and John Hicks argued that normal backwardation results when hedgers hold short positions, speculators hold long positions, and the futures price is below the expected spot price.
- Conversely, the futures price will be above the expected spot price if hedgers hold long positions while speculators hold short positions.
Risk and Return
- Higher investment risk generally requires a higher expected retrun.
- Systematic and nonsystematic risk.
The Risk in a Futures Position
- The discount rate used for expected cash flow equals investor required return.
- Investment valuation equation: F0 = E(ST)e^((r-k)T)
- Return depends on its systematic risk.
- If returns from the asset are uncorrelated with the stock market: F0 = E(ST)
- If the returns from this asset are positively correlated with the stock market: F0 < E(ST)
- If the retunr from the asset are negatively correlated with the stock market: F0 > E(ST)
Normal Backwardation and Contango
- Normal Backwardation: Futures price is below expected future spot price
- Contango: Futures price is above the expected future spot price.
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