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   :slide_classes: appear

==============================================================================
Forward Contracts
==============================================================================


Forward Contract Definition
==============================================================================
**Definition**: A forward contract is an agreement to exchange an
asset at a future date at a prespecified price.

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- The contract settlement date is called the *expiration date*.

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- The asset that is exchanged is called the *underlying asset*.

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- The buyer holds the *long* position.

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- The seller holds the *short* position.

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- There is no initial payment or premium.

Delivery and Settlement
=======================================================
There are two types of forward contract settlements.

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- **Delivery**: The long position pays the prespecified price to the
  short position, who delivers the asset.

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- **Cash settlement**: The long and short positions pay the net cash
  value to the other.

Forward Example
==============================================================================
Two parties contract to exchange a :math:`\smash{\$100}` bond for
:math:`\smash{\$98}` at a future date.

- If the bond is worth :math:`\smash{\$98.25}` at expiry, the short
  position pays :math:`\smash{\$0.25}` to the long position at
  expiry.

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- If the bond is worth :math:`\smash{\$97.50}` at expiry, the long
  position pays :math:`\smash{\$0.50}` to the short position at
  expiry.

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- Cash-settled forwards are often called NDFs, or nondeliverable
  forwards.

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- Usually, cash settlement is used for underlying assets that are
  difficult to exchange (think of a stock index).

Market Prices
==============================================================================
  .. image:: Forwards/wsjFutures.png
     :width: 8in
     :align: center

For current data, visit the `WSJ
<http://www.wsj.com/mdc/public/page/mdc_commodities.html?refresh=on>`_
or `CME Group
<http://www.cmegroup.com/market-data/volume-open-interest/exchange-volume.html>`_
homepages.

Early Termination
==============================================================================
Suppose one party in a forward contract wishes to terminate early.

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- She could engage in another forward contract on the opposite side.

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- Depending on market conditions, the new contract may be written at a
  new price.

Early Termination Example
==============================================================================
Suppose a trader enters a long forward contract position
to exchange a barrel of crude oil on 13 Feb 2015 and decides to
terminate the contract on 16 Feb 2015.

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- On 13 Feb, the forward price is \$52.78 per barrel.

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- On 16 Feb, the forward price is \$52.73 per barrel.

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- She can write a forward contract for \$52.73 on 16 Feb.

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- Note that she takes a \$0.05 loss and is still exposed to risk of
  default on two different contracts.

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- Alternatively, she can ask her original counterparty to accept the
  present value of \$0.05 to terminate.

Notation
==============================================================================
We will use the following notation:

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- :math:`\smash{S_0}`: Spot price of the underlying asset today.

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- :math:`\smash{F_0}`: Forward price of the underlying asset today.

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- :math:`\smash{T}`: Time until delivery.

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- :math:`\smash{r}`: Risk-free rate of interest for maturity :math:`\smash{T}`.

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Note that any units (minutes, hours, days, weeks, months, years) may
be used for :math:`\smash{T}`, but that the interest rate,
:math:`\smash{r}`, must be adjusted accordingly.

Forward Valuation
==============================================================================
The price of a forward contract with maturity :math:`\smash{T}` for an
asset with price :math:`\smash{S_0}` is:

.. math::

   \begin{align*}
   F_0 & = S_0 e^{rT}.
   \end{align*}

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- :math:`\smash{r}` is the risk-free interest rate over period
  :math:`\smash{T}`.

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- If :math:`\smash{r}` is constant, :math:`\smash{F_0}` is a
  deterministic function of the spot sprice, and has nothing to do
  with the unknown, future price of the asset.

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- :math:`\smash{e^{rT}}` is known as the *basis*.

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- **Intuition**: the foward holder must pay the holder of the spot
  contract for interest that would have been earned.

Forward Valuation Example
==============================================================================
Suppose you would like to purchase a 3-month forward contract on
Coca-Cola (`KO <http://yhoo.it/1pVyJAn>`_) stock on 1 Mar 2016. What
is the value of the forward (assuming the stock never pays dividends)?

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- Set :math:`\smash{T = 0.25}` (i.e. time units of 1 year).

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- Use `Yahoo Finance <http://yhoo.it/1Ro1Qq4>`_ to determine
  :math:`\smash{S_0 = \$43.35}`.

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- Use `Quandl <http://bit.ly/1UqSs7E>`_ to determine the (annualized)
  yield on the 3-month U.S. Treasury Bill: :math:`\smash{r =
  0.0033}`.


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Thus,

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.. math::

   \begin{align*}
   F_0 & = S_0 e^{rT} = \$43.35 e^{0.0033 \times 0.25} = \$43.39.
   \end{align*}

Forward Valuation with Income
==============================================================================
Suppose the underlying asset provides income with present value
:math:`\smash{I}`.

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- This may be a single payment or a stream of payments, all
  appropriately discounted:

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.. math::
   
   \begin{align*}
   I & = \frac{d}{1+\frac{r}{m}} +
   \frac{d}{\left(1+\frac{r}{m}\right)^2} + \cdots +
   \frac{d}{\left(1+\frac{r}{m}\right)^{mT}}.
   \end{align*}

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- This assumes :math:`\smash{m}` equally spaced payments of equal size
  during interval :math:`\smash{T}`.

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The value of a forward contract is now:

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.. math::

   \begin{align*}
   F_0 & = (S_0 - I) e^{rT}.
   \end{align*}

Forward Valuation with Yield
==============================================================================
Suppose the underlying asset provides income yield (continuously
compounded) :math:`\smash{q}`. Then:

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.. math::

   \begin{align*}
   F_0 & = S_0 e^{(r-q)T}.
   \end{align*}

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- **Intuition**: the holder of the spot contract now pays interest
  (implicitly), but earns income. The foward holder must compensate
  the spot holder for interest, net of income earned over period
  :math:`\smash{T}`.

Forward Valuation with Yield Example
==============================================================================
Reconsider the previous example for Coca-Cola stock.

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- Now assume that KO has an annualized dividend yield of 3\%.

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The forward price is

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.. math::

   \begin{align*}
   F_0 & = S_0 e^{(r-q)T} \\
   & = \$43.35 e^{(0.0033 - 0.03) \times 0.25} \\
   & = \$43.06.
   \end{align*}

Forward Valuation for Currency
==============================================================================
Suppose the underlying asset is a currency, and that the risk-free
interest rate in the foreign market is :math:`\smash{r_f}`. Then:

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.. math::

   \begin{align*}
   F_0 & = S_0 e^{(r-r_f)T}.
   \end{align*}

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- The foreign interest is income and the rate is the income yield.

Curreny Forward Example
==============================================================================
What is the value of a 6-month forward contract for Canadian dollars
(CAD) on 1 Mar 2016?

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- Set :math:`\smash{T = 0.5}` (i.e. time units of 1 year).

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- Use `Quandl <http://bit.ly/1RwxRIY>`_ to determine the spot exchange
  rate for USD/CAD: :math:`\smash{S_0 = \$1.34}`.

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- Use `Quandl <http://bit.ly/1ShZeq6>`_ to determine the (annualized)
  yield on the 3-month Canadian Treasury Bill: :math:`\smash{r_f =
  0.0047}`. We already determined that :math:`\smash{r = 0.0033}`.

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Thus,

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.. math::

   \begin{align*}
   F_0 & = S_0 e^{(r-r_f)T} = \$1.34 e^{(0.0033 - 0.0047) \times 0.5}
   = \$1.339.
   \end{align*}

Forward Valuation for Commodities 
==============================================================================
Suppose that the underlying is a physical asset that must be
stored. Then:

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.. math::

   \begin{align*}
   F_0 & = (S_0 + U) e^{rT}.
   \end{align*}

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or

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.. math::

   \begin{align*}
   F_0 & = S_0 e^{(r+u)T}.
   \end{align*}

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- :math:`\smash{U}` is the present value of storage costs.

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- :math:`\smash{u}` is the annual storage cost expressed as a fraction
  of commodity value.

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- Note that storage costs are like negative income.

Cost of Carry
==============================================================================
The foregoing compounding rates are referred to as *the cost of
carry*, :math:`\smash{c}`.

.. math::
   
   \begin{align*}
   F_0 & = S_0 e^{cT}.
   \end{align*}

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- The cost of carry includes interest rate and storage costs, minus
  income.

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- For a stock index that pays a dividend yield, :math:`\smash{c =
  r-q}`.

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- For a foreign currency, :math:`\smash{c = r - r_f}`.

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- For a commodity that provides income, :math:`\smash{c = r - q +
  u}`.