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   :slide_classes: appear


==============================================================================
Options
==============================================================================

Options
==============================================================================
Options are contracts that give the option to buy or sell an asset on
or before a specific date at a specific price.

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- A *call* option is an option to buy.

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- A *put* option is an option to sell.

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- A *European* option can only be exercised at maturity.

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- An *American* option can be exercised any time prior to maturity.


Terminology
==============================================================================
- Underlying asset: The asset that may be bought or sold when the
  option is exercised.

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- Maturity (exercise) date: The date at which the contract expires.

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- Strike (exercise) price: The pre-specified price at which the
  underlying can be bought or sold.

Underlying Assets
==============================================================================
Common underlying assets include:

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- Common stock.

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- Foreign currency.

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- Stock indices.

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- Volatility indices. 

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- Futures contracts.

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Options are written on many other underlying *assets*.

Options Exchanges
==============================================================================
Many options are exchange traded.

- `Chicago Board Options Exchange <http://www.cboe.com/>`_ (CBOE).

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- `International Securities Exchange <https://www.ise.com/>`_ (ISE).

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- `Nasdaq PHLX <http://www.nasdaqtrader.com/micro.aspx?id=phlx>`_.

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- `BATS <https://batsoptions.com/>`_.
  
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- `NYSE MKT <https://www.nyse.com/markets/nyse-mkt>`_ (formerly
  American Stock Exchange, or AMEX).

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- `NYSE Arca <https://www.nyse.com/markets/nyse-arca>`_.

Exchange-Traded Options
==============================================================================
Exchanges serve to standardize contracts on popular options.

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- Expiration dates.

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- Strike prices.

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- Class - call or put.

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- American or European.

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- Size of options contract.

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- Size of underlying.

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- Margin requirements.

VIX Options Specs
==============================================================================
  .. image:: Options/optionsSpecs.png
     :width: 8in
     :align: center

Implications of Options
==============================================================================
The buyer of a call (put) option is *not obligated* to buy (sell) the
underlying asset at the strike price.

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- The *buyer* has the *option* to buy (sell).

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- The *seller* of the call (put) option is *obligated* to sell (buy)
  the underlying if the buyer wants to exercise the option.

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- If the price of the underlying asset is above (below) the strike
  price on the maturity date, the buyer will exercise. Why?

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- If the price of the underlying asset is below (above) the strike
  price on the maturity date, the buyer will not exercise. Why?

Options as Insurance
==============================================================================
Options have no downside risk for the buyer.

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- The buyer of a call (put) is better off if the underlying asset
  price rises (falls).

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- If the underlying asset price falls (rises), the buyer doesn't lose
  anything.

Obligation of Sellers
==============================================================================
However, the seller of an option *only* faces downside risk.  

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- The seller of a call (put) is worse off if the underlying asset
  price rises (falls).

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- If the underlying asset price falls (rises), the seller doesn't gain
  anything.

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The seller must be compensated for taking the risk of having to sell
(buy) the underlying for a low (high) price.

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- The buyer pays a *premium* to purchase the option contract.

Call Option Example
==============================================================================
On March 8th 2013, stock for Chipotle Mexican Grill (CMG) sold for
:math:`\smash{\$321.84}` and an option contract with a strike price of
:math:`\smash{\$320.00}` and maturity date of March 15th 2013 cost
:math:`\smash{\$5.30}`.

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- If the price of Chipotle is less than :math:`\smash{\$320.00}` on
  March 15th, the option will not be exercised.

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- If the price is :math:`\smash{\$325.00}` on March 15th, the option
  holder (buyer) will exercise the contract.

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- The gain to the buyer will be :math:`\smash{\$5.00}`.

Call Option Example
==============================================================================
- Remember that the contract cost :math:`\smash{\$5.30}`, so the buyer
  has a net loss of :math:`\smash{\$0.30}`.

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- If the price on March 15th is greater than :math:`\smash{\$325.30}`,
  the buyer will have a net gain.

Put Option Example
==============================================================================
Consider again Chipotle stock which sold for :math:`\smash{\$321.84}`
on March 8th 2013.

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- A put option with a strike price of :math:`\smash{\$320.00}` and a
  maturity date of March 15th 2013 costs :math:`\smash{\$3.30}`.

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- If the price of the stock is above :math:`\smash{\$320.00}` on March
  15th, the option will not be exercised.

Put Option Example
==============================================================================
- Suppose the price is below :math:`\smash{\$320.00}` on March 15th:
  :math:`\smash{\$315.00}`.

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- The buyer of the put will exercise the contract, buying the stock
  for :math:`\smash{\$315.00}` on the market and selling to the put
  writer for :math:`\smash{\$320.00}`.

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- The gross profit would be :math:`\smash{\$320.00}` -
  :math:`\smash{\$315.00}` = :math:`\smash{\$5.00}`.

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- The net profit would be: :math:`\smash{\$5.00}` -
  :math:`\smash{\$3.30}`.

Moneyness
==============================================================================
An option is  

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- *In the money* when its strike price would produce profits for the
  holder.

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- *Out of the money* when exercise would be unprofitable.

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- *At the money* when the strike price is equal to the asset price.

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The moneyness can be determined at any time, as if the option were
exercised at that instant.

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

.. math::

   \begin{align}
   T & = \text{Maturity date} \\
   S_t & = \text{Underlying asset price at time } t \\
   X & = \text{Strike Price} \\
   C_t & = \text{Value of a call option at time } t \\
   P_t & = \text{Value of a put option at time } t.
   \end{align}
 
Call Option Payoff (Buyer)
==============================================================================
The payoff to a call option holder (buyer) at expiration is 

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.. math::

  C_T = \begin{cases} S_T - X & \text{if  } S_T > X \\ 0 &
  \text{if  } S_T \leq X. \end{cases}

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- If the asset price is above the strike, the holder can buy the
  underlying for :math:`\smash{X}` and immediately sell it for
  :math:`\smash{S_T}`, yielding a profit of :math:`\smash{S_T-X}`.

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- If the asset price is below the strike, the option is worthless.

Call Option Payoff (Buyer)
==============================================================================
The payoffs above did not account for the cost of the option.

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- If the option is purchased at time :math:`\smash{t}` for a price of
  :math:`\smash{C_t}`, the net payoff to the holder at expiration is

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.. math::

   C_T = \begin{cases} S_T - X - C_t, & \text{if  } S_T > X \\ -C_t, &
   \text{if  } S_T \leq X. \end{cases}

Call Option Payoff (Buyer)
==============================================================================
  .. image:: Options/bod34698_1502_lg.jpg
     :width: 8in
     :align: center

Call Option Payoff (Seller)
==============================================================================
On the flip side, the gross payoff to the call option writer at
expiration is

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.. math::

   \begin{align}
   C_T & = \begin{cases} X - S_T, & \text{if  } S_T > X
   \\ 0, & \text{if  } S_T \leq X. \end{cases}
   \end{align}
   
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The net payoff is

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.. math::

   \begin{align}
   C_T & = \begin{cases} X - S_T + C_t, & \text{if  } S_T > X
   \\ C_t, & \text{if  } S_T \leq X. \end{cases}
   \end{align}
   
Call Option Payoff (Seller)
==============================================================================
  .. image:: Options/bod34698_1503_lg.jpg
     :width: 8in
     :align: center

Put Option Payoff (Buyer)
==============================================================================
The gross payoff to put option holders at expiration is 

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.. math::

   \begin{align}
   P_T & = \begin{cases} 0, & \text{if  } S_T > X
   \\ X - S_T, & \text{if  } S_T \leq X. \end{cases}
   \end{align}

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- If the underlying asset price is below the strike, the holder can
  purchase it for :math:`\smash{S_T}` and immediately resell for
  :math:`\smash{X}`, yielding a profit of :math:`\smash{X-S_T}`.  

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- If the asset price is above the strike at expiration, the option is
  worthless.

Put Option Payoff (Buyer)
==============================================================================
The *net* payoff to put option holders is

.. math::

   \begin{align}
   P_T & = \begin{cases} -P_t, & \text{if  } S_T > X
   \\ X - S_T - P_t, & \text{if  } S_T \leq X. \end{cases}
   \end{align}

Put Option Payoff (Buyer)
==============================================================================
  .. image:: Options/bod34698_1504_lg.jpg
     :width: 8in
     :align: center

Speculation and Hedging
==============================================================================
Options can be used for both speculation and hedging. 

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- Suppose you have :math:`\smash{\$10,000}` available for investment.  

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- A share of stock costs :math:`\smash{\$100}`.

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- An option with a strike price of :math:`\smash{\$100}` and six-month
  maturity costs :math:`\smash{\$10}`.

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- You can lend money (purchase the risk-free asset) at a rate of 3\%
  for the next six months.

Speculation and Hedging
==============================================================================
Consider three strategies.  

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- Strategy A: Invest entirely in stock, buying 100 shares at the
  current price of :math:`\smash{\$100}`.

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- Strategy B: Invest entirely in at-the-money options, buying 10 call
  contracts (each for 100 shares) selling for :math:`\smash{\$1000}` a
  piece.

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- Strategy C: Purchase 100 call options (1 contract) for
  :math:`\smash{\$1,000}` and invest the remaining
  :math:`\smash{\$9,000}` in the risk-free asset, which will yield a
  total of :math:`\smash{\$9,000\times1.03 = \$9,270}` at the end of
  the six months.

Speculation and Hedging
==============================================================================
The values of the three strategies are:

.. image:: Options/table1.png
   :width: 8in
   :align: center

Speculation and Hedging
==============================================================================
The returns to the three strategies are:

.. image:: Options/table2.png
   :width: 8in
   :align: center

Speculation and Hedging
==============================================================================
From these tables we see two features of options. 

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- Options offer leverage.
    
  - For the all-option portfolio, the return plummets to -100\% when
    the stock price is below the strike.  

  - The return rockets to numbers that are much greater than simply
    holding the stock when the stock price increases above the
    strike.

Speculation and Hedging
==============================================================================    
- Options offer insurance.  
    
  - The mixed portfolio has limited downside loss: the investor can't
    lose more than -7.3\%.

  -  It also has limited upside gains: if the stock price is above the
     strike, its returns are always below the portfolio comprised of
     only stock.

Speculation and Hedging
==============================================================================
:math:`\qquad`

  .. image:: Options/bod34698_1505_lg.jpg
     :width: 8in
     :align: center

Protective Put
==============================================================================
A protective put strategy consists of simultaneously purchasing a
share of stock and a put option on that stock.  

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- This limits the potential downside loss of the stock while leaving
  the potential gains intact.

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    .. image:: Options/table3.png
       :width: 4in
       :align: center

Protective Put
==============================================================================
The put acts as insurance against loss.  

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- Comparing the net payoff of the protective put with the strategy of
  holding stock alone shows that the former comes at a cost.  

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- This is the insurance premium.

Protective Put
==============================================================================
  .. image:: Options/bod34698_1506_sm.jpg
     :width: 3in
     :align: left

Protective Put
==============================================================================
  .. image:: Options/bod34698_1507_lg.jpg
     :width: 5.5in
     :align: center

Covered Call
==============================================================================
A covered call strategy consists of simultaneously purchasing a share
of stock and writing a call option on that stock.  

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- It doesn't eliminate downside loss (like the protective put).  

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- It covers the obligation to deliver the stock for less than its
  market value if the stock price is above the strike.  

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- The call writer is charging a premium (the call price) in order to
  forsake the upside gain of holding the stock.  

Covered Call
==============================================================================
.. image:: Options/table4.png
   :width: 5in
   :align: center

Covered Call
==============================================================================
  .. image:: Options/bod34698_1508_sm.jpg
     :width: 3in
     :align: center

Straddle
==============================================================================
A straddle consists of purchasing call and put options for the same
asset and strike price.

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- It is a bet on volatility.

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- The straddle holder will earn (gross) positive returns if the stock
  price moves up or down, but nothing if it remains at the strike.

Straddle
==============================================================================
.. image:: Options/table5.png
   :width: 4in
   :align: center

Straddle
==============================================================================
So why doesn't everyone hold straddles?  

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- Because the investor must pay for both contracts.

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- The investor doesn't earn a *net* return unless the stock price
  moves enough to compensate for the initial outlay.

Straddle
==============================================================================
  .. image:: Options/bod34698_1509_sm.jpg
     :width: 3in
     :align: center

Spread
==============================================================================
A spread is a combination of two or more options (both calls or both
puts) on the same stock with different strikes.

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- Some of the options are purchased while others are sold.

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- A money spread is the simultaneous purchase and sale of options with
  different strikes.

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- A time spread is the simultaneous purchase and sale of options with
  different maturities.

Bullish Spread
==============================================================================
A bullish spread:

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.. image:: Options/table6.png
   :width: 7in
   :align: center

Bullish Spread
==============================================================================
  .. image:: Options/bod34698_1510_sm.jpg
     :width: 3in
     :align: center

Collar
==============================================================================
An example of a collar is the purchase of a protective put for one
strike price and the sale of a call option, on the same stock, for a
higher strike.

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- This strategy eliminates downside losses below the strike of the put
  and also upside gains beyond the strike of the call.

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- In this case, the investor constrains gains and losses within a
  region close to the current price of the stock.

Protective Put Alternative
==============================================================================
A protective put eliminates the downside loss of holding stock. We
could achieve this with an alternative strategy.

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- Purchase a call option with strike price :math:`\smash{X}`. 

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- Purchase a T-bill (lend at the risk-free rate) with a face value
  equal to the call strike price, :math:`\smash{X}`, and the same
  maturity date as the call.

Protective Put Alternative
==============================================================================
.. image:: Options/table7.png
   :width: 4in
   :align: center

Put Call Parity
==============================================================================
The payoffs in the previous table are identical to those for the
protective put.

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- Hence, the cost of the protective put strategy should be equal to
  the cost of the call plus bonds strategy (why?!!!).  

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- This fact is known as the *Put-Call Parity Relationship*.

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- Mathematically, it can be expressed as:
  
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.. math::

   \begin{align}
   C_0 + X e^{-r_f T} & = S_0 + P_0.
   \end{align}
   
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- This relationship is very useful because it allows us to compute the
  value of a call option if we know the price of the corresponding
  put, and vice versa.

Put Call Parity Example
==============================================================================
Assume  

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- An asset currently sells for :math:`\smash{\$110}`.  

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- A call option with strike :math:`\smash{X = \$105}` and 1-year
  maturity sells for :math:`\smash{\$17}`.

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- A put option with strike :math:`\smash{X = \$105}` and 1-year
  maturity sells for :math:`\smash{\$5}`.

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- The continuously-compounded risk-free interest rate is
  :math:`\smash{4.879\%}` per year.

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- Does the parity relationship hold?

Put Call Parity Example
==============================================================================
.. math::

   \begin{align}
   C_0 + X e^{r_f} & \stackrel{?}{=} S_0 + P_0.
   \end{align}

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.. math::

   \begin{align}
   \$117 = \$17 + \$105 e^{-0.04879} & \neq \$110 + \$5 = \$115.
   \end{align}

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- The relationship doesn't hold.

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- How might an investor take advantage of the situation?
  
Put Call Parity Example
==============================================================================
.. image:: Options/table8.png
   :width: 7in
   :align: center