Chapter 5: Determination of Forward and Futures Prices

Investment Assets vs. Consumption Assets

• Investment asset: Held for investment purposes by significant numbers of investors (stocks, bonds, gold). Arbitrage arguments work well.
• Consumption asset: Held primarily for consumption (copper, oil, corn). Arbitrage arguments have limits — holders may be reluctant to sell when there's an arbitrage opportunity.

Short Selling

Short selling involves selling an asset you don't own:

1. Borrow the asset from someone who owns it
2. Sell it in the market
3. Later buy it back and return it to the lender

The short seller must pay any dividends or income to the lender. A portion of the proceeds may be held as collateral.

Assumptions and Notation

Key assumptions for pricing:

• No transaction costs
• Same tax rates for all market participants
• Borrowing and lending at the same risk-free rate
• Arbitrage opportunities are exploited instantly

Forward Price for an Investment Asset

For an asset providing no income:

$F_0=S_0\cdot e^{rT}$

Proof by arbitrage: If $F_0>S_0{e}^{rT}$, borrow $S_0$, buy the asset, sell it forward. Risk-free profit = $F_0-S_0{e}^{rT}$. If $F_0<S_0{e}^{rT}$, short the asset, invest the proceeds, buy it back via the forward contract.

Known Income

For an asset providing known income with present value $I$:

$F_0=(S_0-I)\cdot e^{rT}$

Example: A 10-month forward on a stock at $50.Risk-freerateis8$0.75 expected in 3, 6, and 9 months.

$I=0.75e^{-0.08\times 0.25}+0.75e^{-0.08\times 0.5}+0.75e^{-0.08\times 0.75}=2.162$

$F_0=(50-2.162)e^{0.08\times 10/12}=51.14$

Known Yield

For an asset paying a known dividend yield $q$ (continuous):

$F_0=S_0\cdot e^{(r-q)T}$

This applies to stock indices (where $q$ is the index dividend yield) and currencies.

Valuing Forward Contracts

The value of a long forward contract (not the forward price):

$f=(F_0-K)\cdot e^{-rT}$

where $K$ is the delivery price in the existing contract and $F_0$ is the current forward price for the same maturity.

Equivalently: $f=S_0-K{e}^{-rT}$ (for asset with no income).

At initiation, $F_0=K$ and $f=0$.

Forward and Futures on Currencies

A foreign currency can be treated as an asset providing a yield equal to the foreign risk-free rate $r_f$:

$F_0=S_0\cdot e^{(r-r_f)T}$

This is interest rate parity. If $r>r_f$, the forward exchange rate is higher than the spot rate (foreign currency trades at a forward premium).

Futures on Commodities

Storage Cost (as proportion)

$F_0=S_0\cdot e^{(r+u)T}$

where $u$ is the storage cost per annum as a proportion of spot price.

Convenience Yield

Holders of consumption assets may obtain convenience yield $y$ from owning the physical commodity (ability to keep production running during shortages). Then:

$F_0=S_0\cdot e^{(r+u-y)T}$

The convenience yield explains why futures prices can be below spot prices (backwardation).

No-Arbitrage Bounds for Consumption Assets

$F_0\le (S_0+U)\cdot e^{rT}$ (if storage cost $U$ is cash amount per unit)

Arbitrageurs cannot force equality because holders may not want to sell.

The Cost of Carry

The cost of carry $c$ summarizes the relationship:

$F_0=S_0\cdot e^{cT}$

Futures Prices and Expected Future Spot Prices

Two competing theories:

• Keynes and Hicks: Hedgers tend to have short positions, so speculators require compensation for taking long positions $\to F_0<E(S_T)$ (normal backwardation)
• Risk-neutral world: $F_0=E(S_T)$ (if no systematic risk; CAPM beta is zero)
• If the asset has positive systematic risk, $F_0<E(S_T)$; investors demand a risk premium in the form of lower futures prices