Podcast
Questions and Answers
Which of the following is the primary reason for analyzing forward contracts before futures contracts?
Which of the following is the primary reason for analyzing forward contracts before futures contracts?
- Forward contracts are more commonly traded than futures contracts.
- Forward contracts involve daily settlement, simplifying their analysis.
- The results obtained from forward contracts are usually not applicable to futures contracts.
- Forward contracts have a single payment at maturity, making them easier to analyze. (correct)
Under what condition are forward and futures prices of an asset expected to be very close?
Under what condition are forward and futures prices of an asset expected to be very close?
- When the risk-free interest rate is highly volatile.
- When the contracts are traded on different exchanges.
- When the maturities of the two contracts are the same. (correct)
- When the underlying asset is a consumption asset.
Which of the following is a critical distinction when considering forward and futures contracts?
Which of the following is a critical distinction when considering forward and futures contracts?
- The difference between investment assets and consumption assets. (correct)
- The volume of trading in the contracts.
- The exchange on which the contracts are traded.
- The regulatory oversight of the contracts.
What is the defining characteristic of an investment asset in the context of forward and futures contracts?
What is the defining characteristic of an investment asset in the context of forward and futures contracts?
Which of the following assets is classified as a consumption asset?
Which of the following assets is classified as a consumption asset?
Why can arbitrage arguments be used to determine the forward and futures prices of investment assets, but not consumption assets?
Why can arbitrage arguments be used to determine the forward and futures prices of investment assets, but not consumption assets?
What does 'shorting' an asset involve?
What does 'shorting' an asset involve?
If an investor instructs a broker to short 500 shares of company X, what action will the broker take?
If an investor instructs a broker to short 500 shares of company X, what action will the broker take?
Under what condition, relative to the risk-free rate (r), will the return from an underlying asset (k) result in a futures price (F0) that is an unbiased estimate of the expected future spot price (E[ST])?
Under what condition, relative to the risk-free rate (r), will the return from an underlying asset (k) result in a futures price (F0) that is an unbiased estimate of the expected future spot price (E[ST])?
What is the relationship between the futures price (F0) and the expected future spot price (E[ST]) when the asset underlying the futures contract has positive systematic risk?
What is the relationship between the futures price (F0) and the expected future spot price (E[ST]) when the asset underlying the futures contract has positive systematic risk?
Which scenario typically leads to a situation where the futures price overstates the expected future spot price?
Which scenario typically leads to a situation where the futures price overstates the expected future spot price?
In the context of futures contracts, what condition defines a market exhibiting 'normal backwardation'?
In the context of futures contracts, what condition defines a market exhibiting 'normal backwardation'?
In a market characterized by 'contango', what relationship exists between the futures price and the expected future spot price?
In a market characterized by 'contango', what relationship exists between the futures price and the expected future spot price?
An asset's return is positively correlated with the stock market. How does this correlation typically influence the expected return of investors on stocks underlying a stock index?
An asset's return is positively correlated with the stock market. How does this correlation typically influence the expected return of investors on stocks underlying a stock index?
Assume an asset displays a return negatively correlated with the stock market. Based on this, which statement accurately describes the relationship between the futures price (F0) and the expected future spot price (E[ST])?
Assume an asset displays a return negatively correlated with the stock market. Based on this, which statement accurately describes the relationship between the futures price (F0) and the expected future spot price (E[ST])?
In a scenario where the dividends provide a return of q and the expected return of investors on stocks underlying an index is generally more than the risk-free rate r, how does the expected increase in the index compare to r - q?
In a scenario where the dividends provide a return of q and the expected return of investors on stocks underlying an index is generally more than the risk-free rate r, how does the expected increase in the index compare to r - q?
According to the information provided, how does a higher foreign interest rate relative to the domestic interest rate typically affect the futures settlement prices of currencies with maturity?
According to the information provided, how does a higher foreign interest rate relative to the domestic interest rate typically affect the futures settlement prices of currencies with maturity?
On May 14, 2013, short-term interest rates for the Japanese Yen, Swiss Franc, and Euro were lower than those of the US Dollar. Based on the provided text, what impact did this have on their futures prices with increasing maturity?
On May 14, 2013, short-term interest rates for the Japanese Yen, Swiss Franc, and Euro were lower than those of the US Dollar. Based on the provided text, what impact did this have on their futures prices with increasing maturity?
What action would an arbitrageur take if the forward price ($F_0$) is greater than the spot price ($S_0$) adjusted for the risk-free rate and time ($e^{rT}$), i.e., $F_0 > S_0e^{rT}$?
What action would an arbitrageur take if the forward price ($F_0$) is greater than the spot price ($S_0$) adjusted for the risk-free rate and time ($e^{rT}$), i.e., $F_0 > S_0e^{rT}$?
On May 14, 2013, the short-term interest rates for the Australian dollar were higher than in the United States dollars. How did this interest rate differential relate to the futures settlement prices of the Australian dollar?
On May 14, 2013, the short-term interest rates for the Australian dollar were higher than in the United States dollars. How did this interest rate differential relate to the futures settlement prices of the Australian dollar?
According to the context, what is the interpretation of a foreign currency in the context of investment?
According to the context, what is the interpretation of a foreign currency in the context of investment?
In the context of forward contracts, what does 'marking to market' refer to?
In the context of forward contracts, what does 'marking to market' refer to?
What is the effect of storage costs, net of income, on the forward price of a commodity?
What is the effect of storage costs, net of income, on the forward price of a commodity?
Which of the following factors can cause differences between forward and futures prices in practice, despite theoretical models?
Which of the following factors can cause differences between forward and futures prices in practice, despite theoretical models?
In a forward contract, what remains constant throughout the life of the contract?
In a forward contract, what remains constant throughout the life of the contract?
What is the key characteristic that defines an asset as an 'investment asset' in the context of futures pricing?
What is the key characteristic that defines an asset as an 'investment asset' in the context of futures pricing?
Why might the risk of counterparty default be lower for futures contracts compared to forward contracts?
Why might the risk of counterparty default be lower for futures contracts compared to forward contracts?
What is the value of a forward contract (f) at the time it is first entered into?
What is the value of a forward contract (f) at the time it is first entered into?
Under what condition is it generally reasonable to assume that forward and futures prices are the same?
Under what condition is it generally reasonable to assume that forward and futures prices are the same?
How are storage costs typically treated in the context of forward pricing models?
How are storage costs typically treated in the context of forward pricing models?
How is the value of a long forward contract (f) calculated?
How is the value of a long forward contract (f) calculated?
Which of the following is an exception to the rule that futures and forward contracts can be assumed to be the same?
Which of the following is an exception to the rule that futures and forward contracts can be assumed to be the same?
An asset provides a known yield rather than a known cash income. How is this yield typically measured?
An asset provides a known yield rather than a known cash income. How is this yield typically measured?
What is the relationship between the risk-free rate (r), storage costs per annum as a proportion of the spot price (u), the spot price (S0), and the forward price (F0)?
What is the relationship between the risk-free rate (r), storage costs per annum as a proportion of the spot price (u), the spot price (S0), and the forward price (F0)?
In the context of stock index futures, what is the underlying investment asset typically considered to be?
In the context of stock index futures, what is the underlying investment asset typically considered to be?
What does 'U' represent in the context of the formula $F_0 = (S_0 + U)e^{rT}$?
What does 'U' represent in the context of the formula $F_0 = (S_0 + U)e^{rT}$?
What does the variable 'q' represent in the context of forward contracts?
What does the variable 'q' represent in the context of forward contracts?
An investor shorts 500 shares of a company at $80 per share. Later, they cover the position at $70 per share. A dividend of $0.50 per share was paid out during the short sale. Ignoring borrowing fees, what is the investor's net profit or loss?
An investor shorts 500 shares of a company at $80 per share. Later, they cover the position at $70 per share. A dividend of $0.50 per share was paid out during the short sale. Ignoring borrowing fees, what is the investor's net profit or loss?
In the formula for determining stock index futures prices, what does 'q' represent?
In the formula for determining stock index futures prices, what does 'q' represent?
What is the formula for the forward price ($F_0$) when the asset provides a known yield (q)?
What is the formula for the forward price ($F_0$) when the asset provides a known yield (q)?
Which of the following best describes the purpose of a margin account in short selling?
Which of the following best describes the purpose of a margin account in short selling?
A commodity has significant storage costs but provides no income. How does this affect its classification?
A commodity has significant storage costs but provides no income. How does this affect its classification?
How does short selling allow an investor to profit from a declining stock price?
How does short selling allow an investor to profit from a declining stock price?
If the actual futures price of an asset is less than the theoretical futures price, what action can an investor take to potentially improve their return if they already own the asset?
If the actual futures price of an asset is less than the theoretical futures price, what action can an investor take to potentially improve their return if they already own the asset?
What makes the CME Nikkei 225 futures contract a 'quanto'?
What makes the CME Nikkei 225 futures contract a 'quanto'?
If short sales are not possible, what action will investors who own the asset take if $F_0 < S_0e^{rT}$?
If short sales are not possible, what action will investors who own the asset take if $F_0 < S_0e^{rT}$?
According to Example 5.7, the September settlement price for the Australian dollar is about 0.6% lower than the June settlement price. What annual rate of decrease does this indicate?
According to Example 5.7, the September settlement price for the Australian dollar is about 0.6% lower than the June settlement price. What annual rate of decrease does this indicate?
What is the primary reason an investor might be 'forced to close out' a short position?
What is the primary reason an investor might be 'forced to close out' a short position?
A trader observes that the futures price for a stock index is significantly higher than what is predicted by the formula $F_0 = S_0e^{(r-q)T}$. What arbitrage strategy is most appropriate?
A trader observes that the futures price for a stock index is significantly higher than what is predicted by the formula $F_0 = S_0e^{(r-q)T}$. What arbitrage strategy is most appropriate?
What is the formula to calculate the forward price ($F_0$) of an investment asset that provides a known cash income (I)?
What is the formula to calculate the forward price ($F_0$) of an investment asset that provides a known cash income (I)?
A pension fund owns an indexed portfolio of stocks. How can they potentially benefit from index arbitrage if the futures price (F0) is less than $S_0e^{(r-q)T}$?
A pension fund owns an indexed portfolio of stocks. How can they potentially benefit from index arbitrage if the futures price (F0) is less than $S_0e^{(r-q)T}$?
An investor initiates a short sale of 1000 shares at $50 per share. The initial margin requirement is 50%. How much must the investor deposit in the margin account?
An investor initiates a short sale of 1000 shares at $50 per share. The initial margin requirement is 50%. How much must the investor deposit in the margin account?
A stock currently priced at $60 is expected to pay dividends of $1 every quarter for the next year. If the risk-free rate is 5% per annum, compounded continuously, which adjustment should be made to account for the dividends when calculating the forward price?
A stock currently priced at $60 is expected to pay dividends of $1 every quarter for the next year. If the risk-free rate is 5% per annum, compounded continuously, which adjustment should be made to account for the dividends when calculating the forward price?
In the absence of storage costs and income, what is the formula for the forward price (F0) of a commodity that is an investment asset, given the spot price (S0), risk-free rate (r), and time to maturity (T)?
In the absence of storage costs and income, what is the formula for the forward price (F0) of a commodity that is an investment asset, given the spot price (S0), risk-free rate (r), and time to maturity (T)?
What market conditions are necessary for a trader to effectively execute index arbitrage?
What market conditions are necessary for a trader to effectively execute index arbitrage?
If the storage costs (net of income) are proportional to the price of the commodity, how are they treated in the context of futures pricing?
If the storage costs (net of income) are proportional to the price of the commodity, how are they treated in the context of futures pricing?
How are dividends on shorted stocks handled?
How are dividends on shorted stocks handled?
An investor holds a short forward contract to sell an asset for a delivery price of $K$ at time $T$. If the current forward price for the same asset is $F_0$, what is the value of the short forward contract?
An investor holds a short forward contract to sell an asset for a delivery price of $K$ at time $T$. If the current forward price for the same asset is $F_0$, what is the value of the short forward contract?
An investor is considering a 1-year futures contract on an investment asset with a spot price of $450. The risk-free rate is 7% per annum. It costs $2 per unit to store the asset, with the payment being made at the end of the year. What is the present value of the storage costs (U)?
An investor is considering a 1-year futures contract on an investment asset with a spot price of $450. The risk-free rate is 7% per annum. It costs $2 per unit to store the asset, with the payment being made at the end of the year. What is the present value of the storage costs (U)?
Consider an asset with a current price of $30. It is expected to provide a yield of 6% per annum. If the risk-free rate is 10% per annum, what is the approximate forward price for a 4-month contract?
Consider an asset with a current price of $30. It is expected to provide a yield of 6% per annum. If the risk-free rate is 10% per annum, what is the approximate forward price for a 4-month contract?
What is the role of the SEC's 'alternative uptick' rule introduced in February 2010?
What is the role of the SEC's 'alternative uptick' rule introduced in February 2010?
During 'Black Monday' in October 1987, what factor hindered the effective execution of index arbitrage?
During 'Black Monday' in October 1987, what factor hindered the effective execution of index arbitrage?
According to the assumptions, what is the relationship between borrowing and lending rates for market participants?
According to the assumptions, what is the relationship between borrowing and lending rates for market participants?
What was the relationship between futures prices and the underlying index for most of October 19, 1987?
What was the relationship between futures prices and the underlying index for most of October 19, 1987?
A 9-month forward contract exists on a commodity. The current spot price is $75, and storage costs are estimated to be $3 per month, payable at the end of each month. If the risk-free rate is 6% per annum, compounded continuously, what is the approximate forward price?
A 9-month forward contract exists on a commodity. The current spot price is $75, and storage costs are estimated to be $3 per month, payable at the end of each month. If the risk-free rate is 6% per annum, compounded continuously, what is the approximate forward price?
In the context of short selling, what does 'covering the position' refer to?
In the context of short selling, what does 'covering the position' refer to?
An arbitrageur identifies that the forward price of a non-dividend-paying stock is significantly higher than the spot price adjusted for the risk-free interest rate. Which action would they take to realize risk-free profit?
An arbitrageur identifies that the forward price of a non-dividend-paying stock is significantly higher than the spot price adjusted for the risk-free interest rate. Which action would they take to realize risk-free profit?
Consider a 6-month futures contract on a stock index. The current index value ($S_0$) is 1500, the continuously compounded risk-free interest rate (r) is 6% per annum, and the dividend yield (q) is 2% per annum. Calculate the theoretical futures price ($F_0$).
Consider a 6-month futures contract on a stock index. The current index value ($S_0$) is 1500, the continuously compounded risk-free interest rate (r) is 6% per annum, and the dividend yield (q) is 2% per annum. Calculate the theoretical futures price ($F_0$).
A company enters into a forward contract to purchase a foreign currency six months from now at a delivery price of 1.40 USD per EUR. At the time of delivery, the spot price is 1.45 USD per EUR. What is the profit or loss for the company on this contract?
A company enters into a forward contract to purchase a foreign currency six months from now at a delivery price of 1.40 USD per EUR. At the time of delivery, the spot price is 1.45 USD per EUR. What is the profit or loss for the company on this contract?
What is the significance of key market participants (like large derivatives dealers) in the context of the assumptions?
What is the significance of key market participants (like large derivatives dealers) in the context of the assumptions?
If the December futures settlement price of the S&P 500 is higher than the June settlement price, what does this indicate about the relationship between the short-term risk-free rate (r) and the dividend yield (q)?
If the December futures settlement price of the S&P 500 is higher than the June settlement price, what does this indicate about the relationship between the short-term risk-free rate (r) and the dividend yield (q)?
A zero-coupon bond is currently priced at $800. The risk-free interest rate is 4% per annum. What is the theoretical forward price for delivery in 6 months?
A zero-coupon bond is currently priced at $800. The risk-free interest rate is 4% per annum. What is the theoretical forward price for delivery in 6 months?
If the forward price of an asset is less than its spot price compounded at the risk-free rate, what strategy should an arbitrageur employ?
If the forward price of an asset is less than its spot price compounded at the risk-free rate, what strategy should an arbitrageur employ?
An investor shorts 200 shares when the price is $50 and the borrowing fee is $1 per share. They close the position when the price is $40. What is the investor's net profit/loss considering the borrowing fee?
An investor shorts 200 shares when the price is $50 and the borrowing fee is $1 per share. They close the position when the price is $40. What is the investor's net profit/loss considering the borrowing fee?
What adjustment must be made when estimating the dividend yield (q) for a stock index futures contract?
What adjustment must be made when estimating the dividend yield (q) for a stock index futures contract?
What impact did temporary bans on short selling have in 2008?
What impact did temporary bans on short selling have in 2008?
What is the primary reason that the forward price of an investment asset is generally higher than its spot price?
What is the primary reason that the forward price of an investment asset is generally higher than its spot price?
How did Joseph Jett exploit a flaw in Kidder Peabody's system when trading strips?
How did Joseph Jett exploit a flaw in Kidder Peabody's system when trading strips?
An investor with a short position is NOT compensated by the broker for which of the following?
An investor with a short position is NOT compensated by the broker for which of the following?
What is the effect of transaction costs on the execution of arbitrage opportunities?
What is the effect of transaction costs on the execution of arbitrage opportunities?
A stock is currently trading at $60. The risk-free rate is 6% per annum. A forward contract expires in 4 months. What is the no-arbitrage forward price?
A stock is currently trading at $60. The risk-free rate is 6% per annum. A forward contract expires in 4 months. What is the no-arbitrage forward price?
What critical factor was Kidder Peabody's computer system not taking into account when calculating profits from Jett's strip trades?
What critical factor was Kidder Peabody's computer system not taking into account when calculating profits from Jett's strip trades?
How does the introduction of the 'alternative uptick' rule impact short selling strategies, particularly during periods of significant stock price decline?
How does the introduction of the 'alternative uptick' rule impact short selling strategies, particularly during periods of significant stock price decline?
A commodity provides storage cost. How does that affect the forward price?
A commodity provides storage cost. How does that affect the forward price?
An investor observes that the market forward price for gold is below the calculated theoretical forward price. What action should they take to exploit this arbitrage opportunity?
An investor observes that the market forward price for gold is below the calculated theoretical forward price. What action should they take to exploit this arbitrage opportunity?
What adjustment needs to be made to the basic forward price formula, (F_0 = S_0e^{rT}), when dealing with an asset that provides a known income?
What adjustment needs to be made to the basic forward price formula, (F_0 = S_0e^{rT}), when dealing with an asset that provides a known income?
Assume you short sell a stock and simultaneously buy a forward contract on the same stock. What is the primary risk you are trying to eliminate or hedge?
Assume you short sell a stock and simultaneously buy a forward contract on the same stock. What is the primary risk you are trying to eliminate or hedge?
A trader notices the forward price of corn is below the spot price. Risk-free rate is positive. What could explain this, assuming no arbitrage opportunities?
A trader notices the forward price of corn is below the spot price. Risk-free rate is positive. What could explain this, assuming no arbitrage opportunities?
How does the introduction of futures contracts on individual stocks impact market efficiency and arbitrage opportunities?
How does the introduction of futures contracts on individual stocks impact market efficiency and arbitrage opportunities?
Consider an asset with a spot price of $100. The risk-free rate is 5% per annum, and the storage costs are $2 per year, payable at the end of the year. What is the theoretical forward price for delivery in one year?
Consider an asset with a spot price of $100. The risk-free rate is 5% per annum, and the storage costs are $2 per year, payable at the end of the year. What is the theoretical forward price for delivery in one year?
What actions are required to realize arbitrage profit if the formula (F_0 = S_0e^{rT}) does not hold?
What actions are required to realize arbitrage profit if the formula (F_0 = S_0e^{rT}) does not hold?
In the context of forward contracts, what is the primary assumption made about market participants that allows for the derivation of the forward price equation, even when short sales are not possible?
In the context of forward contracts, what is the primary assumption made about market participants that allows for the derivation of the forward price equation, even when short sales are not possible?
Consider an asset with no storage costs or income. If the forward price ($F_0$) is greater than the spot price compounded at the risk-free rate ($S_0e^{rT}$), what arbitrage strategy can an investor employ to profit?
Consider an asset with no storage costs or income. If the forward price ($F_0$) is greater than the spot price compounded at the risk-free rate ($S_0e^{rT}$), what arbitrage strategy can an investor employ to profit?
An investor holds an asset and observes that the forward price ($F_0$) is less than the spot price compounded at the risk-free rate ($S_0e^{rT}$). What arbitrage strategy should the investor undertake?
An investor holds an asset and observes that the forward price ($F_0$) is less than the spot price compounded at the risk-free rate ($S_0e^{rT}$). What arbitrage strategy should the investor undertake?
A coupon-bearing bond is priced at $900. A forward contract to purchase this bond matures in 9 months. A coupon payment of $40 is expected in 4 months. The 4-month and 9-month risk-free interest rates are 3% and 4% per annum, respectively. If the forward price is $910, what action should an arbitrageur take?
A coupon-bearing bond is priced at $900. A forward contract to purchase this bond matures in 9 months. A coupon payment of $40 is expected in 4 months. The 4-month and 9-month risk-free interest rates are 3% and 4% per annum, respectively. If the forward price is $910, what action should an arbitrageur take?
An investor is analyzing a coupon-bearing bond with a current price of $900. A forward contract on this bond matures in 9 months. A coupon payment of $40 is expected in 4 months. The 4-month and 9-month risk-free interest rates are 3% and 4% per annum, respectively. If the forward price is $870, what strategy yields an arbitrage profit, assuming shorting is possible?
An investor is analyzing a coupon-bearing bond with a current price of $900. A forward contract on this bond matures in 9 months. A coupon payment of $40 is expected in 4 months. The 4-month and 9-month risk-free interest rates are 3% and 4% per annum, respectively. If the forward price is $870, what strategy yields an arbitrage profit, assuming shorting is possible?
What is the generalized formula for the forward price ($F_0$) of an investment asset that provides a known income with a present value of $I$ during the life of the forward contract, given the spot price $S_0$, risk-free rate $r$, and time to maturity $T$?
What is the generalized formula for the forward price ($F_0$) of an investment asset that provides a known income with a present value of $I$ during the life of the forward contract, given the spot price $S_0$, risk-free rate $r$, and time to maturity $T$?
A stock pays a known dividend. How does the present value of the dividend income affect the forward price of a contract on that stock?
A stock pays a known dividend. How does the present value of the dividend income affect the forward price of a contract on that stock?
A forward contract on a commodity is being evaluated. Which of the following factors would be relevant in determining the theoretical forward price?
A forward contract on a commodity is being evaluated. Which of the following factors would be relevant in determining the theoretical forward price?
Consider a scenario where the cost of carry (storage costs less income earned) is positive. How would this cost of carry typically affect the forward price of an asset?
Consider a scenario where the cost of carry (storage costs less income earned) is positive. How would this cost of carry typically affect the forward price of an asset?
When shorting a bond, what initial steps does an investor undertake, and how do they manage coupon payments during the term of the short position?
When shorting a bond, what initial steps does an investor undertake, and how do they manage coupon payments during the term of the short position?
For an investment asset with a known income, what adjustment must be made to the spot price when calculating the forward price?
For an investment asset with a known income, what adjustment must be made to the spot price when calculating the forward price?
In the context of arbitrage opportunities, which of the following best describes the expected adjustment in forward prices to align with the spot price and risk-free rate?
In the context of arbitrage opportunities, which of the following best describes the expected adjustment in forward prices to align with the spot price and risk-free rate?
How does the ability (or inability) to short sell an asset affect the derivation of forward price equations, and what assumption compensates for the absence of short selling?
How does the ability (or inability) to short sell an asset affect the derivation of forward price equations, and what assumption compensates for the absence of short selling?
If an arbitrageur borrows $900 to buy a bond and shorts a forward contract, and the coupon payment has a present value of $39.60, how is the borrowing typically structured to exploit a mispriced forward contract?
If an arbitrageur borrows $900 to buy a bond and shorts a forward contract, and the coupon payment has a present value of $39.60, how is the borrowing typically structured to exploit a mispriced forward contract?
When an investor shorts a bond to realize $900, and a coupon payment of $40 is due, how should the initial proceeds be managed to cover the coupon payment due in 4 months?
When an investor shorts a bond to realize $900, and a coupon payment of $40 is due, how should the initial proceeds be managed to cover the coupon payment due in 4 months?
A trader holds a long forward contract to purchase a commodity. If the price of the commodity increases, what happens to the value of the forward contract, and how is the gain typically realized?
A trader holds a long forward contract to purchase a commodity. If the price of the commodity increases, what happens to the value of the forward contract, and how is the gain typically realized?
In a scenario where the asset price, S, is strongly negatively correlated with interest rates, how are forward and futures prices generally related?
In a scenario where the asset price, S, is strongly negatively correlated with interest rates, how are forward and futures prices generally related?
What is the key difference in profit realization between forward and futures contracts that can cause confusion on a foreign exchange trading desk?
What is the key difference in profit realization between forward and futures contracts that can cause confusion on a foreign exchange trading desk?
A company enters a short forward contract on an asset. How is the value of this contract affected if the forward price of the asset decreases?
A company enters a short forward contract on an asset. How is the value of this contract affected if the forward price of the asset decreases?
Using equation (5.5), how does an increase in the risk-free interest rate, r, affect the value, f, of a forward contract, all other factors remaining constant?
Using equation (5.5), how does an increase in the risk-free interest rate, r, affect the value, f, of a forward contract, all other factors remaining constant?
What is the impact of daily settlement in futures contracts when the price of the underlying asset is positively correlated with interest rates?
What is the impact of daily settlement in futures contracts when the price of the underlying asset is positively correlated with interest rates?
Consider a long forward contract on an investment asset. How does the present value of known income from the asset affect the value of the forward contract?
Consider a long forward contract on an investment asset. How does the present value of known income from the asset affect the value of the forward contract?
How does increasing the time to maturity, $T$, typically impact the value of $f$ in a long forward contract, assuming all other variables remain constant and interest rates are positive?
How does increasing the time to maturity, $T$, typically impact the value of $f$ in a long forward contract, assuming all other variables remain constant and interest rates are positive?
If a systems error on a trading desk causes a forward trader to miss out on a small profit compared to a futures trader, what is the primary reason for this discrepancy?
If a systems error on a trading desk causes a forward trader to miss out on a small profit compared to a futures trader, what is the primary reason for this discrepancy?
According to equation (5.7), how does an increase in the known yield rate, $q$, affect the value of the forward contract, $f$?
According to equation (5.7), how does an increase in the known yield rate, $q$, affect the value of the forward contract, $f$?
What assumption is made when valuing a long forward contract on an asset, according to the text?
What assumption is made when valuing a long forward contract on an asset, according to the text?
Suppose two traders take offsetting positions: one in a long forward contract and the other in short forward contract. What is the net change in value between these positions if the price of the underlying asset increases.
Suppose two traders take offsetting positions: one in a long forward contract and the other in short forward contract. What is the net change in value between these positions if the price of the underlying asset increases.
A trader believes that the price of an asset will be strongly negatively correlated with interest rates. To capitalize on this belief while minimizing immediate cash outlay, which position should the trader take?
A trader believes that the price of an asset will be strongly negatively correlated with interest rates. To capitalize on this belief while minimizing immediate cash outlay, which position should the trader take?
An investor holds a short forward contract on an asset. If, at maturity, the spot price S is higher than the delivery price K, what is the investor's payoff?
An investor holds a short forward contract on an asset. If, at maturity, the spot price S is higher than the delivery price K, what is the investor's payoff?
Consider an investment asset that provides no income. According to equation (5.5), what are the implications for the forward price if the risk-free rate (r) is assumed to be zero?
Consider an investment asset that provides no income. According to equation (5.5), what are the implications for the forward price if the risk-free rate (r) is assumed to be zero?
What action should an arbitrageur take if the futures price ($F_0$) is greater than the expected spot price with storage costs ($S_0 + U$) compounded at the risk-free rate ($r$) over time $T$, as expressed by $F_0 > (S_0 + U)e^{rT}$?
What action should an arbitrageur take if the futures price ($F_0$) is greater than the expected spot price with storage costs ($S_0 + U$) compounded at the risk-free rate ($r$) over time $T$, as expressed by $F_0 > (S_0 + U)e^{rT}$?
When $F_0 < (S_0 + U)e^{rT}$ for an investment asset, what arbitrage strategy will investors likely undertake?
When $F_0 < (S_0 + U)e^{rT}$ for an investment asset, what arbitrage strategy will investors likely undertake?
Why might the spot-futures parity relationship, $F_0 = (S_0 + U)e^{rT}$, not hold true for consumption assets?
Why might the spot-futures parity relationship, $F_0 = (S_0 + U)e^{rT}$, not hold true for consumption assets?
If storage costs are expressed as a proportion ($u$) of the spot price, which of the following inequalities typically applies to a consumption commodity?
If storage costs are expressed as a proportion ($u$) of the spot price, which of the following inequalities typically applies to a consumption commodity?
An oil refiner values having crude oil in inventory more than a futures contract for crude oil. What is this benefit of holding the physical asset referred to as?
An oil refiner values having crude oil in inventory more than a futures contract for crude oil. What is this benefit of holding the physical asset referred to as?
If the dollar amount of storage costs is known (present value $U$), how is the convenience yield ($y$) defined in relation to the futures price ($F_0$) and spot price ($S_0$)?
If the dollar amount of storage costs is known (present value $U$), how is the convenience yield ($y$) defined in relation to the futures price ($F_0$) and spot price ($S_0$)?
When storage costs are a constant proportion ($u$) of the spot price, how is the futures price ($F_0$) expressed in terms of spot price ($S_0$), risk-free rate ($r$), convenience yield ($y$), and time to maturity ($T$)?
When storage costs are a constant proportion ($u$) of the spot price, how is the futures price ($F_0$) expressed in terms of spot price ($S_0$), risk-free rate ($r$), convenience yield ($y$), and time to maturity ($T$)?
For investment assets, what is the value of the convenience yield ($y$) and why?
For investment assets, what is the value of the convenience yield ($y$) and why?
If futures prices for a commodity decrease as the maturity of the contract increases, what does this pattern typically suggest about the convenience yield ($y$) during that period?
If futures prices for a commodity decrease as the maturity of the contract increases, what does this pattern typically suggest about the convenience yield ($y$) during that period?
What does the convenience yield primarily reflect regarding the commodity market?
What does the convenience yield primarily reflect regarding the commodity market?
How do high inventories of a commodity typically affect the convenience yield?
How do high inventories of a commodity typically affect the convenience yield?
What does the 'cost of carry' (c) primarily measure in the context of futures pricing?
What does the 'cost of carry' (c) primarily measure in the context of futures pricing?
For a non-dividend-paying stock, what is the cost of carry ($c$)?
For a non-dividend-paying stock, what is the cost of carry ($c$)?
For a consumption asset, how is the futures price ($F_0$) related to the spot price ($S_0$), cost of carry ($c$), convenience yield ($y$), and time to maturity ($T$)?
For a consumption asset, how is the futures price ($F_0$) related to the spot price ($S_0$), cost of carry ($c$), convenience yield ($y$), and time to maturity ($T$)?
What complication is introduced into the determination of futures prices when the party with the short position has the option to deliver at any time during a certain period?
What complication is introduced into the determination of futures prices when the party with the short position has the option to deliver at any time during a certain period?
When futures prices are an increasing function of time to maturity, which of the following actions is typically optimal for the party with the short position?
When futures prices are an increasing function of time to maturity, which of the following actions is typically optimal for the party with the short position?
If futures prices are decreasing as time to maturity increases, what is generally the optimal strategy for the party with the short position, and how should futures prices be calculated?
If futures prices are decreasing as time to maturity increases, what is generally the optimal strategy for the party with the short position, and how should futures prices be calculated?
According to Keynes and Hicks, if hedgers tend to hold short positions and speculators hold long positions, what relationship between the futures price and the expected spot price is likely to emerge?
According to Keynes and Hicks, if hedgers tend to hold short positions and speculators hold long positions, what relationship between the futures price and the expected spot price is likely to emerge?
In modern financial theory, what type of risk is considered most relevant when determining the required return on an investment, and why?
In modern financial theory, what type of risk is considered most relevant when determining the required return on an investment, and why?
Considering a speculator who takes a long position in a futures contract, what is the significance of systematic risk in determining the expected return of this investment?
Considering a speculator who takes a long position in a futures contract, what is the significance of systematic risk in determining the expected return of this investment?
What does the formula $F_0 = E(S_T)e^{(r-k)T}$ imply about the relationship between the current futures price ($F_0$) and the expected spot price at time T ($E(S_T)$), considering risk and the risk-free rate?
What does the formula $F_0 = E(S_T)e^{(r-k)T}$ imply about the relationship between the current futures price ($F_0$) and the expected spot price at time T ($E(S_T)$), considering risk and the risk-free rate?
What is the primary reason most futures contracts are closed out before their maturity date, despite the importance of understanding delivery terms?
What is the primary reason most futures contracts are closed out before their maturity date, despite the importance of understanding delivery terms?
In the context of futures contracts, what does it imply if the benefits from holding an asset (including convenience yield and net of storage costs) are less than the risk-free rate?
In the context of futures contracts, what does it imply if the benefits from holding an asset (including convenience yield and net of storage costs) are less than the risk-free rate?
Assume it's July and the November futures price for wheat is $6.00 per bushel. If market participants expect the spot price of wheat in November to be $6.50 per bushel, what does this imply about market expectations?
Assume it's July and the November futures price for wheat is $6.00 per bushel. If market participants expect the spot price of wheat in November to be $6.50 per bushel, what does this imply about market expectations?
If an investment has negative systematic risk, how does this typically affect the required return for an investor?
If an investment has negative systematic risk, how does this typically affect the required return for an investor?
A speculator takes a long position in a gold futures contract that lasts for one year. They simultaneously invest the present value of the futures price in a risk-free asset. At the end of the year, they use the proceeds to buy gold and immediately sell it at the spot price. If the spot price is lower than expected, what is the most likely outcome for the speculator?
A speculator takes a long position in a gold futures contract that lasts for one year. They simultaneously invest the present value of the futures price in a risk-free asset. At the end of the year, they use the proceeds to buy gold and immediately sell it at the spot price. If the spot price is lower than expected, what is the most likely outcome for the speculator?
An oil producer is hedging their production by taking a short position in oil futures. According to Keynes and Hicks, how might this hedging activity influence the relationship between the current futures price and the expected future spot price of oil?
An oil producer is hedging their production by taking a short position in oil futures. According to Keynes and Hicks, how might this hedging activity influence the relationship between the current futures price and the expected future spot price of oil?
Which of the following best describes how nonsystematic risk should influence an investor's required return in a well-diversified portfolio?
Which of the following best describes how nonsystematic risk should influence an investor's required return in a well-diversified portfolio?
Suppose the cost of storing a commodity is significant, but the convenience yield (benefit of holding the commodity for direct use) is low. How would this likely affect the futures price relative to the expected spot price?
Suppose the cost of storing a commodity is significant, but the convenience yield (benefit of holding the commodity for direct use) is low. How would this likely affect the futures price relative to the expected spot price?
If the investor's required return, k, is less than the risk-free rate, r, what does this imply about the systematic risk associated with the asset underlying a futures contract, and how is this reflected in the futures price, $F_0 = E(S_T)e^{(r-k)T}$?
If the investor's required return, k, is less than the risk-free rate, r, what does this imply about the systematic risk associated with the asset underlying a futures contract, and how is this reflected in the futures price, $F_0 = E(S_T)e^{(r-k)T}$?
During the stock market crash of October 1987, what factor significantly contributed to the breakdown of the traditional linkage between stock indices and stock index futures?
During the stock market crash of October 1987, what factor significantly contributed to the breakdown of the traditional linkage between stock indices and stock index futures?
What is the primary role of arbitrageurs in the context of stock index and stock index futures?
What is the primary role of arbitrageurs in the context of stock index and stock index futures?
How is index arbitrage typically implemented when dealing with indices that include a large number of stocks?
How is index arbitrage typically implemented when dealing with indices that include a large number of stocks?
What is the key technological component typically used to implement index arbitrage?
What is the key technological component typically used to implement index arbitrage?
What does the interest rate parity relationship describe in the context of forward and futures contracts on currencies?
What does the interest rate parity relationship describe in the context of forward and futures contracts on currencies?
According to the interest rate parity, if the risk-free interest rate in a foreign country is higher than that in the US, how would the forward price of the foreign currency typically relate to its spot price?
According to the interest rate parity, if the risk-free interest rate in a foreign country is higher than that in the US, how would the forward price of the foreign currency typically relate to its spot price?
In the context of currency forward contracts, what is the implication of the absence of arbitrage opportunities?
In the context of currency forward contracts, what is the implication of the absence of arbitrage opportunities?
If the 2-year interest rates in стране X and the United States are 5% and 2% respectively, and the current spot exchange rate is 1.20 USD per X, what action might an arbitrageur consider if the 2-year forward exchange rate is significantly higher than the rate suggested by interest rate parity?
If the 2-year interest rates in стране X and the United States are 5% and 2% respectively, and the current spot exchange rate is 1.20 USD per X, what action might an arbitrageur consider if the 2-year forward exchange rate is significantly higher than the rate suggested by interest rate parity?
What is the effect of arbitrageurs actions when they discover that the forward rate is lower than the rate suggested by the interest rate parity?
What is the effect of arbitrageurs actions when they discover that the forward rate is lower than the rate suggested by the interest rate parity?
How does the possibility of earning interest on a foreign currency impact the pricing of forward contracts?
How does the possibility of earning interest on a foreign currency impact the pricing of forward contracts?
What is the correct interpretation of (S_0) in the context of forward and futures contracts on currencies, as described?
What is the correct interpretation of (S_0) in the context of forward and futures contracts on currencies, as described?
What is the implication of a negative result when calculating ((r - r_f)T) in the interest rate parity equation?
What is the implication of a negative result when calculating ((r - r_f)T) in the interest rate parity equation?
In the equation (F_0 = S_0e^{(r - r_f)T}), what does 'T' represent?
In the equation (F_0 = S_0e^{(r - r_f)T}), what does 'T' represent?
Which of the following scenarios would most likely trigger an arbitrage opportunity, assuming no transaction costs?
Which of the following scenarios would most likely trigger an arbitrage opportunity, assuming no transaction costs?
What is the correct expansion of $e^{rT}$?
What is the correct expansion of $e^{rT}$?
Flashcards
Forward Contract
Forward Contract
Agreement to buy/sell an asset at a future date at a price agreed upon today.
Futures Contract
Futures Contract
Agreement to buy/sell an asset at a future date with daily settlement of gains/losses.
Investment Asset
Investment Asset
Assets held for investment purposes (e.g., stocks, bonds, gold).
Consumption Asset
Consumption Asset
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Short Selling
Short Selling
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Spot Price
Spot Price
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Forward/Futures Price
Forward/Futures Price
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Arbitrage
Arbitrage
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Closing a Short Position
Closing a Short Position
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Short Selling Profit/Loss
Short Selling Profit/Loss
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Forced Close-Out
Forced Close-Out
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Fee for Lending Shares
Fee for Lending Shares
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Income Responsibility (Short)
Income Responsibility (Short)
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Margin Account (Short)
Margin Account (Short)
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Additional Margin
Additional Margin
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Uptick Rule
Uptick Rule
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Alternative Uptick Rule
Alternative Uptick Rule
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Temporary Bans on Short Selling
Temporary Bans on Short Selling
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Market Assumptions
Market Assumptions
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T
T
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S0
S0
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r
r
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Forward Price
Forward Price
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Investment Asset (no income)
Investment Asset (no income)
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Arbitrage Strategy (High Forward Price)
Arbitrage Strategy (High Forward Price)
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Arbitrage Strategy (Low Forward Price)
Arbitrage Strategy (Low Forward Price)
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Forward Price Formula (No Income)
Forward Price Formula (No Income)
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Spot Price (S0)
Spot Price (S0)
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Risk-Free Interest Rate (r)
Risk-Free Interest Rate (r)
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Time to Maturity (T)
Time to Maturity (T)
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Cost of Carry
Cost of Carry
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Zero-Coupon Bond
Zero-Coupon Bond
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Arbitrage Profit
Arbitrage Profit
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Shorting an Asset
Shorting an Asset
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Continuous Compounding
Continuous Compounding
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Bank's Misreported Profit
Bank's Misreported Profit
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Short Sale
Short Sale
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Investment Asset Holders
Investment Asset Holders
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Arbitrage Strategy (F0 > S0erT)
Arbitrage Strategy (F0 > S0erT)
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Arbitrage Strategy (F0 < S0erT)
Arbitrage Strategy (F0 < S0erT)
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Forward Price Adjustment
Forward Price Adjustment
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Forward Contract on Income-Bearing Asset
Forward Contract on Income-Bearing Asset
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Arbitrage with High Forward Price (Income Asset)
Arbitrage with High Forward Price (Income Asset)
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Arbitrage with Low Forward Price (Income Asset)
Arbitrage with Low Forward Price (Income Asset)
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No Arbitrage Condition
No Arbitrage Condition
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Present Value of Income (I)
Present Value of Income (I)
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Forward Price Formula (Known Income)
Forward Price Formula (Known Income)
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Variables in Forward Price (Known Income)
Variables in Forward Price (Known Income)
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Strategy Without Shorting (Income Asset)
Strategy Without Shorting (Income Asset)
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Time Value of Money
Time Value of Money
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Arbitrage Condition (Cash Income)
Arbitrage Condition (Cash Income)
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Arbitrage Strategy (F0 > S0e^(rT))
Arbitrage Strategy (F0 > S0e^(rT))
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Arbitrage Strategy (F0 < S0e^(rT))
Arbitrage Strategy (F0 < S0e^(rT))
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Known Yield
Known Yield
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Continuous Compounding Yield (q)
Continuous Compounding Yield (q)
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Forward Price Formula (Known Yield)
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Yield Conversion
Yield Conversion
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Marking to Market
Marking to Market
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Delivery Price (K)
Delivery Price (K)
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Current Forward Price (F0)
Current Forward Price (F0)
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Value of Forward Contract (f)
Value of Forward Contract (f)
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Value Formula
Value Formula
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Short Forward Contract
Short Forward Contract
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No Systematic Risk
No Systematic Risk
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Positive Systematic Risk
Positive Systematic Risk
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Negative Systematic Risk
Negative Systematic Risk
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F0 = E(ST)
F0 = E(ST)
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F0 < E(ST)
F0 < E(ST)
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F0 > E(ST)
F0 > E(ST)
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Normal Backwardation
Normal Backwardation
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Contango
Contango
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Risk-Free Rate (r)
Risk-Free Rate (r)
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Current Stock Price (S0)
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Forward Price (F0)
Forward Price (F0)
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Forward Contract Value (No Income)
Forward Contract Value (No Income)
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Forward Contract Value (Income I)
Forward Contract Value (Income I)
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Forward Contract Value (Yield q)
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Known Yield (q)
Known Yield (q)
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Futures Contract Settlement
Futures Contract Settlement
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Forward Contract Settlement
Forward Contract Settlement
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Forward vs. Futures Prices
Forward vs. Futures Prices
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S Positively Correlated
S Positively Correlated
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Forward vs. Futures Price Factors
Forward vs. Futures Price Factors
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Default Risk: Futures vs. Forwards
Default Risk: Futures vs. Forwards
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Forward/Futures Price Assumption
Forward/Futures Price Assumption
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F0 Symbol
F0 Symbol
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Stock Index Futures Purpose
Stock Index Futures Purpose
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Stock Index as Investment Asset
Stock Index as Investment Asset
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Dividend Yield (q)
Dividend Yield (q)
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F0 = S0e^(r-q)T formula
F0 = S0e^(r-q)T formula
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Quanto
Quanto
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Futures Price Increase Rate
Futures Price Increase Rate
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Index Arbitrage (1)
Index Arbitrage (1)
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Index Arbitrage (2)
Index Arbitrage (2)
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Index Arbitrage: pension fund
Index Arbitrage: pension fund
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Program Trading
Program Trading
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Index Arbitrage: execution
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Index Arbitrage
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S0 (Currency)
S0 (Currency)
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F0 (Currency)
F0 (Currency)
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Interest Rate Parity
Interest Rate Parity
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Interest Rate Parity Formula
Interest Rate Parity Formula
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Currency Arbitrage Strategy
Currency Arbitrage Strategy
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Foreign Risk-Free Interest Rate (rf)
Foreign Risk-Free Interest Rate (rf)
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Hedging Currency Risk
Hedging Currency Risk
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Currency Arbitrage Profit Strategy
Currency Arbitrage Profit Strategy
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Interest Rate Differential
Interest Rate Differential
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Arbitrage and Formula (5.9)
Arbitrage and Formula (5.9)
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Japanese Yen Quote
Japanese Yen Quote
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r > rf (Futures)
r > rf (Futures)
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rf > r (Futures)
rf > r (Futures)
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Foreign Currency Yield
Foreign Currency Yield
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Gold Lease Rate
Gold Lease Rate
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Storage Costs
Storage Costs
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U (Storage Costs)
U (Storage Costs)
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Consumption Commodity
Consumption Commodity
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F0 = (S0 + U)e^(rT)
F0 = (S0 + U)e^(rT)
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u (Storage Costs per Annum)
u (Storage Costs per Annum)
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F0 = S0e^(r+u)T
F0 = S0e^(r+u)T
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F0 = S0e^(rT)
F0 = S0e^(rT)
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Arbitrage Opportunity (F0 > Expected)
Arbitrage Opportunity (F0 > Expected)
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Arbitrage Strategy (F0 > Expected)
Arbitrage Strategy (F0 > Expected)
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Arbitrage Opportunity (F0 < Expected)
Arbitrage Opportunity (F0 < Expected)
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Arbitrage Strategy (F0 < Expected)
Arbitrage Strategy (F0 < Expected)
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Futures Price (Consumption Asset)
Futures Price (Consumption Asset)
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Storage Costs (Proportion)
Storage Costs (Proportion)
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Futures Price, Proportional Storage(Consumption Asset)
Futures Price, Proportional Storage(Consumption Asset)
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Convenience Yield
Convenience Yield
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Convenience Yield Formula (Dollar Storage)
Convenience Yield Formula (Dollar Storage)
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Convenience Yield Formula (Proportional Storage)
Convenience Yield Formula (Proportional Storage)
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Futures Price with Convenience Yield
Futures Price with Convenience Yield
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Convenience Yield (Investment Assets)
Convenience Yield (Investment Assets)
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Futures Price (Cost of Carry and Convenience Yield)
Futures Price (Cost of Carry and Convenience Yield)
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Delivery Period
Delivery Period
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Expected Spot Price
Expected Spot Price
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Keynes and Hicks Theory
Keynes and Hicks Theory
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Systematic Risk
Systematic Risk
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Risk and Return
Risk and Return
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Risk Premium
Risk Premium
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Risk-free Rate
Risk-free Rate
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Required Return
Required Return
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Investment Return
Investment Return
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Early Delivery Strategy (Futures)
Early Delivery Strategy (Futures)
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Late Delivery Strategy (Futures)
Late Delivery Strategy (Futures)
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c < y Scenario
c < y Scenario
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c > y Scenario
c > y Scenario
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Study Notes
- Forward prices are related to the spot price of the underlying asset.
- Forward contracts are easier to analyze than futures because there is no daily settlement.
- The forward price and futures price of an asset are usually very close when the contract maturities are the same.
- This is convenient because results obtained for forwards are usually also true for futures.
Investment Assets vs. Consumption Assets
- Investment assets are held for investment purposes by at least some traders (e.g., stocks, bonds, gold, silver).
- Consumption assets are held primarily for consumption and not normally for investment (e.g., copper, crude oil, corn).
- Arbitrage arguments determine the forward and futures prices of an investment asset, but this is not possible for consumption assets.
Short Selling
- Short selling, or "shorting", involves selling an asset that is not owned; possible for some investment assets.
- An investor instructs a broker to borrow and sell shares, closing the position later by repurchasing the shares.
- Profit occurs if the stock price declines, loss if it rises.
- The investor must cover dividends or interest normally received on the shorted securities.
- Example: Shorting 500 shares at $120, dividend of $1/share, closing at $100/share yields a net gain of $9,500.
- Cash flows from a short sale are the mirror image of purchasing and selling shares.
- A margin account is required with the broker as a guarantee against price increases in the shorted asset.
- Initial margin is required, and additional margin may be needed if the asset price increases.
- Interest is usually paid on the balance in margin accounts.
- Regulations on short selling have evolved, including the uptick rule (now an alternative uptick rule) and occasional temporary bans.
- Alternative Uptick Rule: Restrictions on short selling when a stock's price decreases by more than 10% in one day.
Assumptions and Notation
- Assumptions:
- No transaction costs for market participants.
- Same tax rate on all net trading profits.
- Ability to borrow and lend money at the same risk-free interest rate.
- Market participants take advantage of arbitrage opportunities.
- T: Time until delivery date in years.
- S0: Price of the underlying asset today.
- F0: Forward or futures price today.
- r: Zero-coupon risk-free interest rate per annum with continuous compounding.
- Participants in derivatives markets traditionally used LIBOR as a proxy for the risk-free rate.
Forward Price for an Investment Asset
- For an investment asset providing no income: F0 = S0 * e^(rT).
- If F0 > S0 * e^(rT): An arbitrageur can buy the asset and short forward contracts.
- If F0 < S0 * e^(rT): An arbitrageur can short the asset and enter into long forward contracts.
- A long forward contract and a spot purchase both lead to asset ownership at time T.
- The forward price is higher than the spot price due to the cost of financing the spot purchase during the contract life.
- Example: A 4-month forward contract for a zero-coupon bond priced at $930 with a 6% risk-free rate gives a forward price of $948.79.
What If Short Sales Are Not Possible?
- Deriving the formula does not require short selling the asset
- Only requires that there are market participants who hold the asset purely for investment
- If the forward price is too low, they will find it attractive to sell the asset and take a long position in a forward contract.
- If F0 > S0erT , an investor can adopt the following strategy:
- Borrow S0 dollars at an interest rate r for T years.
- Buy 1 unit of the asset.
- Short a forward contract on 1 unit of the asset.
- At time T , the asset is sold for F0. An amount S0erT is required to repay the loan at this time and the investor makes a profit of F0 S0erT.
- If F0 < S0erT. In this case, an investor who owns the asset can:
- Sell the asset for S0.
- Invest the proceeds at interest rate r for time T.
- Take a long position in a forward contract on 1 unit of the asset.
- As in the non-dividend-paying stock example considered earlier, we can expect the forward price to adjust so that neither of the two arbitrage opportunities we have considered exists.
- This means that the relationship in equation (5.1) must hold.
Known Income
- For an investment asset providing known income with a present value of I: F0 = (S0 - I) * e^(rT).
- If F0 > (S0 - I) * e^(rT): An arbitrageur can buy the asset and short a forward contract.
- If F0 < (S0 - I) * e^(rT): An arbitrageur can short the asset and take a long position in a forward contract.
- Example: 10-month forward contract on a $50 stock, dividends of $0.75 expected after 3, 6, and 9 months, and an 8% risk-free rate gives a forward price of $51.14.
Known Yield
- When the asset underlying a forward contract provides a known yield rather than a known cash income.
- Formula: F0 = S0 * e^((r - q)T), where q is the average yield per annum, with continuous compounding.
- Example: 6-month forward contract on an asset priced at $25, providing income equal to 2% of the asset price and a 10% risk-free rate gives a forward price of $25.77.
Valuing Forward Contracts
- f = (F0 - K) * e^(-rT), where K is the delivery price, and f is the value of the forward contract today.
- Value of a (long) forward contract to buy an asset for K at time T must be (F0 K)e rT.
- Similarly, the value of a (short) forward contract to sell the asset for K at time T is (K F0)e rT.
- Valuing a long forward contract on an asset can be done by making the assumption that the price of the asset at the maturity of the forward contract equals the forward price F0.
- The payo! at time T is F0 K, which has a present value of (F0 K)e rT.
- Alternatively, we can value a short forward contract on the asset by assuming that the current forward price of the asset is realized.
- Investment asset with no income: f = S0 - K * e^(-rT).
- Investment asset with known income I: f = S0 - I - K * e^(-rT).
- Investment asset with known yield q: f = S0 * e^(-qT) - K * e^(-rT).
Are Forward Prices and Futures Prices Equal?
- When the short-term risk-free interest rate is constant, the forward price is theoretically the same as the futures price.
- When interest rates vary unpredictably, forward and futures prices are no longer the same.
- If the underlying asset price and interest rates are positively correlated, the futures price will tend to be slightly higher than the forward price.
- If the asset price and interest rates are negatively correlated, the forward price will tend to be slightly higher than the futures price.
- Theoretical differences between forward and futures are small enough to ignore for short term contracts.
- Factors such as taxes, transactions costs, margin requirements, and counterparty default risk can also affect this.
Futures Prices of Stock Indices
- A stock index is usually regarded as the price of an investment asset that pays dividends.
- In the formula usually the dividends provide a kown yield rather than a known cash income.
- Futures price formula: F0 = S0 * e^((r - q)T), where q is the dividend yield rate which increases at rate r - q with the contract's maturity.
- Dividend yield on an index varies throughout the year.
- The dividends used for estimating q should be those for which the ex-dividend date is during the life of the futures contract.
- Index arbitrage:
- If F0 > S0 * e^((r - q)T): Buy the stocks and short futures contracts.
- If F0 < S0 * e^((r - q)T): Short the stocks and take a long position in futures contracts.
Forward and Futures Contracts on Currencies
- S0 is the current spot price in US dollars of one unit of the foreign currency
- F0 is the forward or futures price in US dollars of one unit of the foreign currency.
- rf is the value of the foreign risk-free interest rate when money is invested for time T.
- r is the risk-free rate when money is invested for T in US dollars.
- Currency futures price formula: F0 = S0 * e^((r - rf)T).
- Identical to Equation 5.3 with q replaced by rf
- A foreign currency can be regarded as an investment asset paying a known yield.
- The yield is the risk-free rate of interest in the foreign currency.
- Example: In 2013, short-term Japanese yen, Swiss franc, and euro interest rates were lower than short-term US interest rates, and futures prices increased with maturity.
Futures on Commodities
- Gold and silver are examples of commodities that are investment assets.
- It is required that some individuals hold it for investment purposes and that these individuals be prepared to sell their holdings and go long forward contracts
- Gold owners such as central banks charge interest in the form of what is known as the gold lease rate when they lend gold.
Income and Storage Costs
- Gold can provide income to the holder and also has storage costs.
- Forward price formula without storage costs and income: F0 = S0 * e^(rT).
- With storage costs net of income (U): F0 = (S0 + U) * e^(rT).
- Storage costs are negative income.
- Example: One-year futures contract on an investment asset with $2 storage cost, spot price of $450, and 7% risk-free rate gives a futures price of $484.63.
Consumption Comodities
- For consumption assets, it is known that they usually provide no income, but can have significant storage costs.
- If the storage costs incurred at any time are proportional to the price of the commodity, they can be treated as a negative yield.
- Formula: F0 = S0 * e^((r + u)T), where u denotes the storage costs per annum.
- Formula: F0 <= (S0 + U) * e^(rT).
- The equality isn't necessarily maintained due to the convenience yield provided by the commodity. Fulfilling needs that futures contracts can't solve.
- Equation: F0 = S0 * e^((r + u - y)T).
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Description
Explore the differences between forward and futures contracts, focusing on investment versus consumption assets. Understand the role of arbitrage in pricing investment assets, and the implications of shorting an asset in the market. Questions cover key concepts related to contracts.
