• Compare consumption and investment assets
• Discuss short selling
• Explain cash-and-carry arbitrage and reverse cash-and-carry arbitrage
• Employ arbitrage principles to derive forward price formulas and the value of a forward contract on a security that provides:
  • No income
  • A known cash income
  • A known dividend yield
• Compare forward and futures prices
• Derive formulas for futures prices:
  • Futures on currencies • Stock index futures • Index arbitrage • Futures on commodities
• Define and discuss cost of carry & convenience yield
• Compare futures prices and expected spot prices

• Investment assets are held by a significant number of investors purely for investment purposes.
  • Examples: stocks, bonds, gold, silver.
• Consumption assets are held primarily for consumption or manufacturing input.
  • Examples: agricultural goods (corn, wheat), copper, crude oil.
• Crucial Difference in Pricing:
  • Arbitrage arguments can be used to determine exact forward and futures prices of investment assets.
  • Arbitrage arguments provide only an upper bound for consumption assets:
    • Ownership of the physical commodity provides benefits – convenience yield.
    • Short selling of physical commodities is not always possible.
    • Some commodities cannot be stored easily or at all (e.g., electricity).

• Short selling involves selling securities you do not currently own:
  • Your broker borrows the securities from another client and sells them in the spot market in the usual way.
  • At some future stage, you must buy the securities back (short covering) so they can be replaced in the account of the original lender.
  • You must pay all dividends and other distributions that the owner of the borrowed securities would ordinarily have received.
  • You are required to maintain a margin account with the broker to protect against default risk.
• Why short sell?
  • Speculation: Profit from an anticipated decline in the asset price.
  • Arbitrage: Exploit relative mispricing when the forward contract is underpriced (reverse cash-and-carry).

• Setup:
  • In April: An investor shorts 500 IBM shares at $120.
    → The investor’s broker borrows 500 shares and sells them in the spot market for $120.
  • In May: IBM pays a dividend of $1 per share.
  • In July: The investor buys the shares back for $100 and closes out the short position.
    → The broker buys 500 shares at $100 to return to the original lender.
• The Outcome:
  • The investor receives: $500\times \mathrm{\$}120=+\mathrm{\$}60,000$ in April.
  • The investor must pay: $500\times \mathrm{\$}1=-\mathrm{\$}500$ in May (dividend reimbursement).
  • The investor pays: $500\times \mathrm{\$}100=-\mathrm{\$}50,000$ in July to close out.
  • Net Gain (ignoring time value of money): $$500\times (\mathrm{\$}120-\mathrm{\$}1-\mathrm{\$}100)=500\times \mathrm{\$}19=\mathrm{\$}9,500$$

• $S_0$: spot price of the asset at time $t=0$.
• $S_t$: spot price of the asset at time $t$.
• $F_0$: forward price at time $t=0$:
  • The delivery price agreed in a contract negotiated at $t=0$ for delivery at maturity $T$.
• $F_t$: forward price at time $t$:
  • The delivery price in new forward contracts negotiated at time $t$ for delivery at $T$.
• $f_0$: value of the forward contract at initiation ($t=0$). Always $f_0=0$.
• $f_t$: value at time $t$ of a forward contract originally negotiated at $t=0$ for delivery at $T$.
  • Note: $F_t$ (price) and $f_t$ (value) are fundamentally different concepts!
• $r$: risk-free interest rate per annum with continuous compounding.

Let the market-quoted forward price be $F^{\ast }$. Compare with theoretical no-arbitrage price:

• If $F^{\ast }>S_0{e}^{rT}$ (Forward Overpriced):
  • Arbitrageur executes Cash-and-Carry Arbitrage (see slide 1.25): Borrow $S_0$ at rate $r$, buy the asset at $S_0$, and sell forward at $F^{\ast }$. At $T$, deliver the asset for $F^{\ast }$, repay loan $S_0{e}^{rT}$. Profit locked in = $F^{\ast }-S_0{e}^{rT}>0$.
• If $F^{\ast }<S_0{e}^{rT}$ (Forward Underpriced):
  • Arbitrageur executes Reverse Cash-and-Carry Arbitrage (see slide 1.26): Short sell the asset at $S_0$, invest proceeds in risk-free deposit at $r$, and buy forward at $F^{\ast }$. At $T$, deposit matures to $S_0{e}^{rT}$, buy asset via forward at $F^{\ast }$ to close short. Profit locked in = $S_0{e}^{rT}-F^{\ast }>0$.
• Theoretical Forward Price:
  $$F_0=S_0{e}^{rT}\text{or at time }t:F_t=S_t{e}^{r(T-t)}$$
  Examples: Non-dividend-paying stocks and zero-coupon bonds.

Suppose $I$ is the present value of all known cash income received during the life of the contract, and $I_t$ is the present value at time $t$ of the income still to be received:

• If $F^{\ast }>(S_0-I)e^{rT}$ (Overpriced):
  • Arbitrageur borrows $S_0$ to buy the asset and enters a short forward contract.
  • The income received during the contract life is invested at rate $r$ and helps repay the loan at $T$.
• If $F^{\ast }<(S_0-I)e^{rT}$ (Underpriced):
  • Arbitrageur short sells the asset for $S_0$, deposits amount $(S_0-I)$ at $r$, and enters a long forward contract.
  • Income payments on the borrowed asset are paid out of the remaining short proceeds.
• Theoretical Forward Price:
  $$F_0=(S_0-I)e^{rT}\text{and at time }t:F_t=(S_t-I_t)e^{r(T-t)}$$
  Examples: Dividend-paying stocks with known dividend amounts and coupon-bearing bonds.

• Market Setup:
  • Spot price of the stock: $S_0=\mathrm{\$}50$.
  • Quoted 10-month forward price: $F^{\ast }=\mathrm{\$}52$.
  • Risk-free interest rate: $r=8\mathrm{\%}$ p.a. (continuously compounded) for all maturities.
  • Expected dividends: $\mathrm{\$}0.75$ per share in 3 months, 6 months, and 9 months.
• Is there an arbitrage opportunity?
  • Step 1: Calculate PV of dividends ($I$): $$I=0.75e^{-0.08\times 3/12}+0.75e^{-0.08\times 6/12}+0.75e^{-0.08\times 9/12}$$ $$I=0.75(0.98020+0.96079+0.94176)=0.75\times 2.88275=\mathrm{\$}2.162$$
  • Step 2: Calculate theoretical forward price: $$F_0=(S_0-I)e^{rT}=(50-2.162)e^{0.08\times 10/12}=47.838\times 1.06894=\mathrm{\$}51.14$$
  • Conclusion: Quoted price $F^{\ast }=\mathrm{\$}52>F_0=\mathrm{\$}51.14$.
    The forward contract is overpriced by $\mathrm{\$}52-\mathrm{\$}51.14=\mathrm{\$}0.86$.

Cash-and-Carry Arbitrage Walkthrough

• Now ($t=0$):
  • Borrow $\mathrm{\$}50$ at $8\mathrm{\%}$ p.a. to buy one share in the spot market.
  • Enter into a short forward contract to sell the share at $\mathrm{\$}52$ in 10 months.
• In 3, 6, and 9 months:
  • Receive the $\mathrm{\$}0.75$ dividend each time and immediately invest at $8\mathrm{\%}$ until month 10.
• In 10 months (at maturity):
  • Repay the initial loan: $\mathrm{\$}50e^{0.08\times 10/12}=\mathrm{\$}53.45$.
  • Deliver the share under the forward contract for $+\mathrm{\$}52.00$.
  • Collect the compounded dividend investments: $$0.75e^{0.08\times 7/12}+0.75e^{0.08\times 4/12}+0.75e^{0.08\times 1/12}=\mathrm{\$}2.31$$
  • Net Arbitrage Profit: $$\text{Profit}=\mathrm{\$}52.00+\mathrm{\$}2.31-\mathrm{\$}53.45=\mathrm{\$}0.86\text{ per share}$$

Suppose $q$ is the average continuous dividend yield of the asset during the contract life:

• Example – Stock Index:
  • A stock index represents a portfolio paying dividends throughout the year.
  • Dividends can be modeled as a continuous yield $q$.
• Crucial Condition for Index Futures:
  • For this formula to hold, the index must represent an investment asset.
  • In other words, changes in the index must correspond to changes in the value of an actually tradable portfolio of securities.
  • Counter-example: The Nikkei index viewed as a dollar number does not represent an investment asset.

Consider a forward contract negotiated at time $0$ with delivery price $F_0$ at maturity $T$:

• The value of a long position at time $t$ is: $$f_t=(F_t-F_0)e^{-r(T-t)}$$
• The value of a short position at time $t$ is: $$f_t=(F_0-F_t)e^{-r(T-t)}$$
• It reflects the present value of the gain or loss from closing out the position at time $t$.
• At initiation ($t=0$): $f_0=0$ because the delivery price is set to $F_0$.
• Worked Example: $S_0=\mathrm{\$}400$, $T=6$ months (0.5 yr), $r=5\mathrm{\%}$.
  • At $t=0$: $F_0=400e^{0.05\times 0.5}=\mathrm{\$}410.13$, and $f_0=0$.
  • After 2 months ($t=2/12$, remaining $\tau =4/12$), stock rises to $S_t=\mathrm{\$}450$: $$F_t=450e^{0.05\times (4/12)}=\mathrm{\$}457.56$$ $$f_t=(457.56-410.13)e^{-0.05\times (4/12)}=47.43\times 0.98347=\mathrm{\$}46.65$$

Core Insight: Past dividends paid out do not enter subsequent valuation; only the present value of future remaining dividends affects subsequent forward price and contract value!

• Constant or Predictable Interest Rates:
  • When interest rates are constant or known functions of time, forward prices and futures prices are theoretically identical: $F_0={\text{Futures}}_0$.
• Uncertain Interest Rates & Daily Settlement:
  • Positive correlation between asset price and interest rates: As $S$ rises, long futures holders receive daily cash gains invested at higher interest rates. Holders of long forward positions are not affected by this uncertainty. Thus, futures price > forward price.
  • Negative correlation: Forward price > futures price.
  • In practice, for short/medium maturities, these differences are very small → assume $\text{Futures Price}=\text{Forward Price}$.
• Is the Value of a Forward Contract the Same as Futures? – NO!
  • Futures: Profit/loss is realized daily through marking to market. Value is reset to zero daily.
  • Forward: All profit/loss is locked until maturity $T$. Value at $t$ is the present value of the change: $f_t=(F_t-F_0)e^{-r(T-t)}$.

• When $F^{\ast }>S_0{e}^{(r-q)T}$ (Futures Overpriced):
  • Arbitrageur buys the underlying index stocks and enters a short position in index futures contracts (Cash-and-carry).
  • Often done by institutions with low transaction costs or corporations holding short-term money market funds.
• When $F^{\ast }<S_0{e}^{(r-q)T}$ (Futures Underpriced):
  • Arbitrageur enters a long position in index futures contracts and sells (or short sells) the underlying stocks (Reverse cash-and-carry).
  • Frequently executed by large pension funds that can sell physical shares and invest in cash.
• Execution & Market Realities:
  • Index arbitrage requires simultaneous trades in hundreds of stocks and index futures; automated program trading is standard.
  • Black Monday (19 October 1987, when the US stock market dropped by 30%): Massive order imbalances caused computer trading delays. Cash and futures decoupled, breaking the theoretical parity relationship.

Core Principle: The terminal index price $S_T$ cancels out completely! The locked-in profit at maturity is exactly the mispricing: $F_0-F^{\ast }=408.08-405=\mathrm{\$}3.08$ per contract unit (or $\mathrm{\$}3.08e^{-0.10\times 1/3}=\mathrm{\$}2.98$ in present value terms), with zero initial capital and zero market risk.

• Notation:
  • $S_0$: spot exchange rate – price in domestic currency of 1 unit of foreign currency.
  • $F_0$: forward exchange rate – price in domestic currency of 1 unit of foreign currency.
  • $r$: risk-free interest rate in domestic currency.
  • $r_f$: risk-free interest rate in foreign currency.
• Foreign Currency as an Income-Yielding Asset:
  • Holding foreign currency earns interest at the foreign rate $r_f$.
  • Therefore, foreign currency is analogous to an investment asset paying a continuous dividend yield of $q=r_f$:
  $$F_0=S_0{e}^{(r-r_f)T}\text{and at time }t:F_t=S_t{e}^{(r-r_f)(T-t)}$$
• Quotation Conventions:
  • This is not the way spot and forward rates are quoted in exchanges.
  • Exchange rates are quoted as foreign currency per dollar, except for British pound, euro, Australian dollar and New Zealand dollar.

Two equivalent strategies to hold money from time $0$ to time $T$:

S₀ is the spot exchange rate; F₀ is the forward exchange rate

By no-arbitrage, both riskless strategies must yield the identical dollar outcome:

• If $F^{\ast }<S_0{e}^{(r-r_f)T}$ (Forward Exchange Rate Too Low):
  • Borrow foreign currency at rate $r_f$.
  • Convert borrowed foreign currency to domestic currency at spot rate $S_0$.
  • Invest domestic currency at domestic rate $r$.
  • Enter a long forward contract to buy foreign currency at rate $F^{\ast }$ at maturity $T$ and repay the foreign currency loan.
  • Guaranteed domestic currency profit locked in at $T$.
• If $F^{\ast }>S_0{e}^{(r-r_f)T}$ (Forward Exchange Rate Too High):
  • Borrow domestic currency at domestic rate $r$.
  • Convert to foreign currency at spot rate $S_0$.
  • Invest foreign currency at foreign rate $r_f$.
  • Enter a short forward contract to sell the foreign currency at rate $F^{\ast }$ (converting to domestic currency) at maturity $T$ and repay the domestic currency loan.
  • Guaranteed domestic currency profit locked in at $T$.

• Investment Commodities (e.g., Gold, Silver):
  • Held for investment; storage costs can be treated as negative income.
  • If storage cost is proportional to price ($u$ per annum): $$F_0=S_0{e}^{(r+u)T}\text{or at time }t:F_t=S_t{e}^{(r+u)(T-t)}$$
  • If storage cost has present value $U$ ($U_t$ is the present value at time $t$ of the storage costs still to be paid): $$F_0=(S_0+U)e^{rT}\text{or at time }t:F_t=(S_t+U_t)e^{r(T-t)}$$
• Consumption Commodities (e.g., Oil, Copper, Agricultural Goods):
  • If $F^{\ast }>(S_0+U)e^{rT}$, cash-and-carry arbitrage works (borrow, buy commodity, store, sell futures).
  • If $F^{\ast }<(S_0+U)e^{rT}$, reverse cash-and-carry FAILS because:
    • Commercial inventory holders will not sell physical stock due to operational convenience.
    • Outside arbitrageurs cannot easily short-sell physical crude oil or grain.
  • Therefore, the theoretical relation is an upper bound: $$F_0\le S_0{e}^{(r+u)T}\text{or}{F}_0\le (S_0+U)e^{rT}$$ At time $t$: $F_t\le S_t{e}^{(r+u)(T-t)}$, $F_t\le (S_t+U_t)e^{r(T-t)}$.

• Setup:
  • Spot price of oil: $S_0=\mathrm{\$}70$ per barrel.
  • Quoted 1-year futures price: $F^{\ast }=\mathrm{\$}75.80$.
  • Interest rate: $r=5\mathrm{\%}$ p.a. with continuous compounding for all maturities.
  • Storage costs: $\mathrm{\$}1.40$ per barrel per annum, payable quarterly in advance ($\mathrm{\$}0.35$ per quarter at months 0, 3, 6, 9).
• Is there an arbitrage opportunity?
  • Step 1: Calculate PV of storage costs ($U$): $$U=0.35+0.35e^{-0.05\times 3/12}+0.35e^{-0.05\times 6/12}+0.35e^{-0.05\times 9/12}$$ $$U=0.35\times (1+0.98758+0.97531+0.96319)=0.35\times 3.92608=\mathrm{\$}1.374$$
  • Step 2: Calculate theoretical upper bound forward price: $$F_{\text{theoretical}}=(S_0+U)e^{rT}=(70+1.374)e^{0.05\times 1}=71.374\times 1.05127=\mathrm{\$}75.03$$
  • Conclusion: Quoted price $F^{\ast }=\mathrm{\$}75.80>\mathrm{\$}75.03$.
    The futures contract is overpriced by $\mathrm{\$}75.80-\mathrm{\$}75.03=\mathrm{\$}0.77$ per barrel.

Cash-and-Carry Arbitrage Execution

• Now ($t=0$):
  • Borrow $\mathrm{\$}(70+1.374)=\mathrm{\$}71.374$ at $5\mathrm{\%}$ p.a.
  • Buy 1 barrel of oil at spot price $\mathrm{\$}70$ and pay first quarter storage of $\mathrm{\$}0.35$.
  • Enter into a short futures contract to sell oil at $\mathrm{\$}75.80$ in 1 year.
• In 3, 6, and 9 months:
  • Pay quarterly storage costs using the borrowed reserve funds (invest $\mathrm{\$}1.374-\mathrm{\$}0.35=\mathrm{\$}1.024$ in three instalments over 3, 6 and 9 months to finance the quarterly storage cost).
• In 1 year (at maturity):
  • Repay the loan: $\mathrm{\$}71.374e^{0.05\times 1}=\mathrm{\$}75.03$.
  • Deliver oil under the futures contract and receive $+\mathrm{\$}75.80$.
  • Arbitrage Profit: $$\text{Profit}=\mathrm{\$}75.80-\mathrm{\$}75.03=\mathrm{\$}0.77\text{ per barrel}$$
• Critical note: If futures were quoted below $\mathrm{\$}75.03$, reverse cash-and-carry is NOT always available for consumption assets like oil!

The cost of carry $c$ summarizes financing costs, storage costs, and income earned:

• $c=r$ for a non-dividend paying stock.
• $c=r-q$ for a stock index providing dividend yield $q$.
• $c=r-r_f$ for a currency with foreign interest rate $r_f$.
• $c=r+u$ for a commodity with proportional storage cost $u$.
• For Investment Assets: $F_0=S_0{e}^{cT}$ (at time $t$: $F_t=S_t{e}^{c(T-t)}$).
  • The futures contract typically trades at a premium to the level of the underlying asset. This difference is referred to as the cost of carry.
  • As the underlying asset can be purchased and held for future delivery, the futures price should reflect the cost of carrying the asset to the maturity of the futures contract.
  • Futures prices trade higher than spot prices: $F_0>S_0$ whenever $c>0$ – Contango.
• For Consumption Assets: $F_0\le S_0{e}^{cT}$ (at time $t$: $F_t\le S_t{e}^{c(T-t)}$).
  • Due to convenience yield, futures often trade lower than spot – Backwardation.

• Definition: The convenience yield $y$ measures the economic benefits of holding the physical commodity rather than holding a futures contract:
  $$F_0{e}^{yT}=S_0{e}^{cT}⟹F_0=S_0{e}^{(c-y)T}=S_0{e}^{(r+u-y)T}$$ $$\text{or with dollar storage: }{F}_0=(S_0+U)e^{(r-y)T}$$ $$\text{and at time }t:F_t=S_t{e}^{(c-y)(T-t)},F_t=(S_t+U_t)e^{(r-y)(T-t)}$$
• Why hold physical inventory?
  • Maintain production flow during supply chain disruptions.
  • Take advantage of local temporary shortages or unexpected customer demand.
• Market Expectations & Forward Curve Shape:
  • If shortage is expected → physical commodity is scarce → $y$ is high → $F_0<S_0$ (Backwardation).
  • If surplus is expected → warehouse inventories are high → $y$ is low → $F_0>S_0$ (Contango).

• Market Conditions (Forward Curve Shape Today):
  • Contango: Futures price > current spot price ($F_0>S_0$).
  • Backwardation: Futures price < current spot price ($F_0<S_0$).
• Market Expectations (Futures Price vs Expected Future Spot $E[S_T]$):
  • Normal Contango ($F_0>E[S_T]$):
    • Keynes and Hicks argument: Hedgers hold net long positions (purchasers seeking price caps; expect losses on average).
    • Speculators hold short positions and earn an expected positive return.
    • Asset return is negatively correlated with the broader market (systematic risk is negative).
  • Normal Backwardation ($F_0<E[S_T]$):
    • Keynes and Hicks argument: Hedgers hold net short positions (commodity producers seeking price floors; pay insurance premium).
    • Speculators hold long positions and expect to profit: $E[S_T]-F_0>0$.
    • Asset return is positively correlated with the broader market (positive systematic risk).