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Forwards & Futures Pricing

Investment Assets Consumption Assets
Held by significant numbers of people purely for investment purposes Held primarily for consumption.
e.g. Gold, Silver e.g. Copper, Oil

Short Selling

  • Selling securities you don't own
  • Mechanism
    1. Broker borrows security from another client (B).
    2. Sells them in the market in the usual way.
    3. At some stage, buy the security to replace it in the B's account.
      • Must pay dividends/other benefits.
      • May pay small fee for borrowing the securities.
Example
  • 100 shares with price $100
  • Close the short position in 3 months (price then = $90)
  • During 3 months, dividend of $3/share paid
Solution
  • Sells in the market in usual way
    • Say, sold for \(F_{0}=\)$100
  • Buy the security from the market and replace in the client's account
    • Say, \(S_{0} =\) $90
    • in three months \(T = \dfrac{3}{12} = 0.25\) years
  • Must pay dividends and other benefits to the original owner (client)
    • Say, Dividend/Income = \(\$3\)
  • Total profit = \(100 - 90 - 3 - 0 = 7\) per share = $\(700\) from shorting
    • If there was borrowers fee, this would come down.
  • If I bought 100 shares, loss \(= 90 - 100 + 3 = -7\) per share = $\(700\)

Notation

  • \(S_{0}:\) Spot price today
  • \(F_{0}:\) Futures/forward price today
  • \(T:\) Time until delivery date
  • \(r:\) Zero coupon Risk-free interest rate (continuous compounding) p.a.

Arbitrage Examples

Example 2 (Arbitrage Opportunity, profit from rising price)
  • \(S_{0} = 40\) (non-dividend paying stock)
  • \(T = \dfrac{3}{12}\) (3 months)
  • \(F_{0} = 43\)
  • \(r = 5\%\) p.a.

Mechanism

When futures price is high. Borrow money and buy the share \(\implies\) Sell the share at a lower price

  • Now
    1. Borrow \(40\) at 5%pa
    2. Buy the security
    3. Short a forward contract to sell one share in 3 months1
  • Three months later
    1. Deliver share and receive \(43\)
    2. Loan repayment = \(40e^{ 0.05 \times 0.25 }=40.50\)
    3. Profit = \(43-40.50 = 2.50\)
Example 3 (Arbitrage Opportunity, profit from falling price)
  • \(S_{0} = 40\)
  • \(T = \dfrac{3}{12}\)
  • \(F_{0}=39\)
  • \(r = 5\%\)

Mechanism

When futures price is low, short the share (Borrow share and sell) \(\implies\) Buy the share at a lower price

  • Now
    1. Short one share2
    2. Invest proceeds \(40\) @ \(5\%\)
    3. Long a forward contract to buy one share in 3 months
  • Three months later
    1. Value of investment is now \(40 e^{ 0.25 \times 0.05} = 40.50\)
    2. Buy the share in forward contract @ 39
    3. Profit = \(40.50 - 39 = 1.50\)

Realization

  • To enter into a forward contract, no money is required. It is just an agreement
  • To enter into a futures contract, we need to deposit a margin amount from which market-to-market adjustments would be made. Helps prevent default risk.

The Forward Price

  • If the spot price for an investment asset that provides no income is \(S_{0}\),
  • and the forward price for a contract deliverable in \(T\) years is \(F_{0}\).

Then,

\[ F_{0} = S_{0}e^{ rT } \]

where, \(r\) is the \(T\)-year risk-free rate of interest.

Scenarios

If Short Sales are NOT Possible

Formula still works for investment asset (for an investor who is holding the asset to immediately sell it and buy forward contracts when \(F_{0} \lt S_{0}\)).

\[ F_{0} = S_{0}e^{ rT } \]

When Investment asset provides a Known Income

\[ F_{0} = (S_{0}-I)e^{ rT } \]

where \(I\) is the fixed income during the life of the contract.

When Investment asset provides a Known Yield

\[ F_{0} = S_{0} e^{ (r-q)T } \]

where \(q\) is the average yield during life of contract (continuous compounding).

Valuing a Forward Contract

Let the delivery price be \(K\) (= the original delivery price3).

  • Value for long forward = \((F_{0}-K)e^{ -rT }\) (as we buy the asset at \(K\) but then we can enter a forward contract with price \(F_{0}\), which is a gain)
  • Value for short forward = \((K-F_{0})e^{ -rT }\) (We sell the asset and gain \(K\) now, and we can enter a forward contract by paying \(F_{0}\))

Forward vs Futures Prices

  • When maturity and asset price are same
    • Forward price = Futures price
    • except Eurodollar futures
  • When interest rates are uncertain, they differ slightly
    • Interest rate and asset price:
      • strong positive correlation \(\implies\) futures \(\gt\) forward price
      • strong negative correlation \(\implies\) futures \(\lt\) forward price

Stock Index

Interpretation
  • Look at it like an investment asset paying dividend yield
\[ F_{0} = S_{0}e^{(r-q)T} \]
  • The index should represent an investment asset (changes in index correspond to value of tradable portfolio)
Index Arbitrage
  • Simultaneous trades in futures & different stocks (use computer)
  • Not always possible, the no-arbitrage relation doesn't hold but simultaneous trades are not possible

Futures and Forwards on Currency

Foreign currency = security providing a yield. Yield = Foreign risk-free rate, \(r_{f}\) and the current spot exchange rate \(S_{0}\). and \(r\) is the domestic risk-free rate

\[ F_{0} = S_{0}e^{ (r-r_{f})T } \]

where \(F_{0}\) is the locked-in exchange rate after the \(T\)-period maturity. This relationship holds because, say you have 1000 units of foreign currency. You can

  • Convert it immediately to \(1000S_{0}\) now \(\to\) valued \(1000S_{0}e^{ rT }\) after \(T\)
  • Enter a forward contract now, so you have \(1000e^{ r_{f}T }\) after \(T\) \(\to\) clear the contract to get \(1000F_{0}e^{ r_{f}T }\) dollars in the future (\(F_{0}\) was decided upon)
  • Hence, \(1000S_{0}e^{ rT }= 1000F_{0}e^{ r_{f}T }\)

Consumption Assets

Storage

  • Storage = -ve income.
  • If \(u =\dfrac{\text{storage cost}}{\text{time}}\) as a \% of asset value.
\[ F_{0} \leq S_{0}e^{ (r+u)T } \]
  • If \(u = PV(\text{storage cost})\)
\[ F_{0} \leq (S_{0}+U)e^{ rT } \]

Cost of Carry

  • Cost of carry, \(c =\text{storage cost} +\text{interest cost}-\text{income earned}\)
  • Investment asset, \(F_{0}=S_{0}e^{ cT }\)
  • Consumption asset, \(F_{0}= S_{0}e^{ (c-y)T }\) where \(y=\text{convenience yield}\)

Futures Price & Future \(E(S_{T})\)

  • Suppose \(k\) is the expected return required by investor in an asset
  • Invest \(F_{0}e^{ -rT }\) at a risk-free rate (enter into long futures), to create \(S_{T}\) cash inflow at maturity.
  • Meaning: I have the long position, so I have to buy the asset at \(F_{0}\) at time \(T\). For that, I need money. So I invest \(F_{0}e^{ -rT }\) now, that will grow into \(F_{0}\) in the future. I buy and posses the asset that way. Now, I can sell that asset at the expected spot price at maturity, \(E(S_{T})\). I value the asset at \(S_{T}\) since my expected return required is \(k\).
\[ F_{0}e^{ -rT } e^{kT} = E(S_{T}) \implies F_{0} = E(S_{T}) e^{ (r-k)T } \]

  1. Again, to enter a short position means to agree to sell (a share that you own) @ specified price on a specified date. ↩

  2. Here, it refers to short selling (unlike the previous example, where we enter a short position in a forwards contract). So note that short selling \(\neq\) short position ↩

  3. Suppose we bought this forward 3 months ago, and had agreed at \(K\) (i.e. at that point in time, \(F_{0,\text{3mo. ago}} = K\)). Where as \(F_{0}\) is the delivery price if we get into a similar contract today. The interpretation may however be wrong. Please check. ↩