.. slideconf:: :slide_classes: appear ============================================================================== Options ============================================================================== Call Options ============================================================================== A call option is a contract that allows the buyer the *option* to purchase an asset for a pre-specified price on or before a particular date. It has the following ingredients: .. rst-class:: to-build - Underlying asset: The asset that may be bought or sold when the option is exercised. .. rst-class:: to-build - Maturity (exercise) date: The date at which the contract expires. .. rst-class:: to-build - Strike price: The pre-specified price at which the underlying can be purchased. Call Options ============================================================================== The buyer of the option is *not obligated* to buy the underlying asset at the strike price. .. rst-class:: to-build - The *buyer* has the *option* to buy. .. rst-class:: to-build - The *seller* of the call option is *obligated* to sell the underlying if the buyer wants to exercise the option. .. rst-class:: to-build - If the price of the underlying asset is above the strike price on the maturity date, the buyer will exercise. Why? .. rst-class:: to-build - If the price of the underlying asset is below the strike price on the maturity date, the buyer will not exercise. Why? Call Options ============================================================================== The call option has no downside risk for the buyer. .. rst-class:: to-build - The buyer is better off if the underlying asset price rises. .. rst-class:: to-build - If the underlying asset price falls, the buyer doesn't lose anything. Call Options ============================================================================== However, the seller of the option *only* faces downside risk. .. rst-class:: to-build - The seller is worse off if the underlying asset price rises. .. rst-class:: to-build - If the underlying asset price falls, the seller doesn't gain anything. .. rst-class:: to-build The seller must be compensated for taking the risk of having to sell the underlying for a low price. .. rst-class:: to-build - 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 \$321.84 and an option contract with a strike price of \$320.00 and maturity date of March 15th 2013 cost \$5.30. .. rst-class:: to-build - If the price of Chipotle is less than \$320.00 on March 15th, the option will not be exercised. .. rst-class:: to-build - If the price is \$325.00 on March 15th, the option holder (buyer) will exercise the contract. .. rst-class:: to-build - The gain to the buyer will be \$5.00. Call Option Example ============================================================================== - Remember that the contract cost \$5.30, so the buyer has a net loss of \$0.30. .. rst-class:: to-build - If the price on March 15th is greater than \$325.30, the buyer will have a net gain. Put Options ============================================================================== A put option is a contract that allows the buyer the option to sell an asset (the underlying) for a pre-specified price (the strike) on or before a particular date (the maturity date). .. rst-class:: to-build - The buyer of the put benefits when the price of the underlying asset falls below the strike. .. rst-class:: to-build - The buyer of the option can buy the asset at the market price and sell it at the higher strike price (to the writer of the put option). .. rst-class:: to-build - If the price of the asset rises above the strike, the buyer will not exercise the option and has no downside loss. Put Options ============================================================================== - The put is an *option* (not an *obligation*) for the buyer to sell the asset at the strike price. .. rst-class:: to-build - The writer of the put is under *obligation* to buy the asset whenever the buyer chooses to exercise the option. Put Option Example ============================================================================== Consider again Chipotle stock which sold for \$321.84 on March 8th 2013. .. rst-class:: to-build - A put option with a strike price of \$320.00 and a maturity date of March 15th 2013 costs \$3.30. .. rst-class:: to-build - If the price of the stock is above \$320.00 on March 15th, the option will not be exercised. Put Option Example ============================================================================== - Suppose the price is below \$320.00 on March 15th: \$315.00. .. rst-class:: to-build - The buyer of the put will exercise the contract, buying the stock for \$315.00 on the market and selling to the put writer for \$320.00. .. rst-class:: to-build - The gross profit would be \$320.00 - \$315.00 = \$5.00. .. rst-class:: to-build - The net profit would be: \$5.00 - \$3.30. Moneyness ============================================================================== An option is .. rst-class:: to-build - *In the money* when its strike price would produce profits for the holder. .. rst-class:: to-build - *Out of the money* when exercise would be unprofitable. .. rst-class:: to-build - *At the money* when the strike price is equal to the asset price. .. rst-class:: to-build The moneyness can be determined at any time, as if the option were exercised at that instant. American vs. European ============================================================================== - An American option allows the buyer to exercise the contract on or before the maturity date. .. rst-class:: to-build - A European option only allows exercise on the maturity date. .. rst-class:: to-build - Since an American option encompasses all of the possibilities of a European option, it should always be more valuable and cost more. .. rst-class:: to-build - As the name denotes, virtually all options traded in the U.S. are of the American flavor. Notation ============================================================================== We use the following notation: .. rst-class:: to-build .. math:: T = \text{Maturity date} .. rst-class:: to-build .. math:: S_t = \text{Underlying asset price at time } t .. rst-class:: to-build .. math:: X = \text{Strike Price} .. rst-class:: to-build .. math:: C_t = \text{Value of a call option at time } t .. rst-class:: to-build .. math:: P_t = \text{Value of a put option at time } t Call Option Payoff (Buyer) ============================================================================== The payoff to a call option holder (buyer) at expiration is .. rst-class:: to-build .. math:: C_T = \begin{cases} S_T - X, & \text{if } S_T > X \\ 0, & \text{if } S_T \leq X. \end{cases} .. rst-class:: to-build - If the asset price is above the strike, the holder can buy the underlying for :math:`X` and immediately sell it for :math:`S_T`, yielding a profit of :math:`S_T-X`. .. rst-class:: to-build - 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. .. rst-class:: to-build - If the option is purchased at time :math:`t` for a price of :math:`C_t`, the net payoff to the holder at expiration is .. rst-class:: to-build .. 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 ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Options/bod34698_1502_lg.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/Options/bod34698_1502_lg.jpg :width: 6in Call Option Payoff (Seller) ============================================================================== On the flip side, the gross payoff to the call option writer at expiration is .. rst-class:: to-build .. math:: C_T & = \begin{cases} X - S_T, & \text{if } S_T > X \\ 0, & \text{if } S_T \leq X. \end{cases} .. rst-class:: to-build The net payoff is .. rst-class:: to-build .. math:: C_T & = \begin{cases} X - S_T + C_t, & \text{if } S_T > X \\ C_t, & \text{if } S_T \leq X. \end{cases} Call Option Payoff (Seller) ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Options/bod34698_1503_lg.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/Options/bod34698_1503_lg.jpg :width: 6in Put Option Payoff (Buyer) ============================================================================== The gross payoff to put option holders at expiration is .. rst-class:: to-build .. math:: P_T & = \begin{cases} 0, & \text{if } S_T > X \\ X - S_T, & \text{if } S_T \leq X. \end{cases} .. rst-class:: to-build - If the underlying asset price is below the strike, the holder can purchase it for :math:`S_T` and immediately resell for :math:`X`, yielding a profit of :math:`X-S_T`. .. rst-class:: to-build - 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 .. rst-class:: to-build .. math:: P_T & = \begin{cases} -P_t, & \text{if } S_T > X \\ X - S_T - P_t, & \text{if } S_T \leq X. \end{cases} Put Option Payoff (Buyer) ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Options/bod34698_1504_lg.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/Options/bod34698_1504_lg.jpg :width: 6in Speculation and Hedging ============================================================================== Options can be used for both speculation and hedging. .. rst-class:: to-build - Suppose you have \$10,000 available for investment. .. rst-class:: to-build - A share of stock costs \$100. .. rst-class:: to-build - An option with a strike price of \$100 and six-month maturity costs \$10. .. rst-class:: to-build - 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. .. rst-class:: to-build - Strategy A: Invest entirely in stock, buying 100 shares at the current price of \$100. .. rst-class:: to-build - Strategy B: Invest entirely in at-the-money options, buying 10 call contracts (each for 100 shares) selling for \$1000 a piece. .. rst-class:: to-build - Strategy C: Purchase 100 call options (1 contract) for \$1,000 and invest the remaining \$9,000 in the risk-free asset, which will yield a total of :math:`\$9,000\times1.03 = \$9,270` at the end of the six months. Speculation and Hedging ============================================================================== The values of the three strategies are: :math:`\qquad` .. image:: /_static/Options/table1.png :width: 8in :align: center Speculation and Hedging ============================================================================== The returns to the three strategies are: :math:`\qquad` .. image:: /_static/Options/table2.png :width: 8in :align: center Speculation and Hedging ============================================================================== From these tables we see two features of options. .. rst-class:: to-build - Options offer leverage. .. rst-class:: to-build - For the all-option portfolio, the return plummets to -100\% when the stock price is below the strike. .. rst-class:: to-build - 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. .. rst-class:: to-build - The mixed portfolio has limited downside loss: the investor can't lose more than -7.3\%. .. rst-class:: to-build - 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` .. ifslides:: .. image:: /_static/Options/bod34698_1505_lg.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/Options/bod34698_1505_lg.jpg :width: 6in Protective Put ============================================================================== A protective put strategy consists of simultaneously purchasing a share of stock and a put option on that stock. .. rst-class:: to-build - This limits the potential downside loss of the stock while leaving the potential gains intact. :math:`\qquad` .. rst-class:: to-build .. image:: /_static/Options/table3.png :width: 4in :align: center Protective Put ============================================================================== The put acts as insurance against loss. .. rst-class:: to-build - Comparing the net payoff of the protective put with the strategy of holding stock alone shows that the former comes at a cost. .. rst-class:: to-build - This is the insurance premium. Protective Put ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Options/bod34698_1506_lg.jpg :width: 3in :align: center .. ifnotslides:: .. image:: /_static/Options/bod34698_1506_lg.jpg :width: 4in Protective Put ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Options/bod34698_1507_lg.jpg :width: 5.5in :align: center .. ifnotslides:: .. image:: /_static/Options/bod34698_1507_lg.jpg :width: 5in Covered Call ============================================================================== A covered call strategy consists of simultaneously purchasing a share of stock and writing a call option on that stock. .. rst-class:: to-build - It doesn't eliminate downside loss (like the protective put). .. rst-class:: to-build - It covers the obligation to deliver the stock for less than its market value if the stock price is above the strike. .. rst-class:: to-build - The call writer is charging a premium (the call price) in order to forsake the upside gain of holding the stock. .. rst-class:: to-build .. image:: /_static/Options/table4.png :width: 5in :align: center Covered Call ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Options/bod34698_1508_lg.jpg :width: 3in :align: center .. ifnotslides:: .. image:: /_static/Options/bod34698_1508_lg.jpg :width: 4in Straddle ============================================================================== A straddle consists of purchasing call and put options for the same asset and strike price. .. rst-class:: to-build - It is a bet on volatility. .. rst-class:: to-build - The straddle holder will earn (gross) positive returns if the stock price moves up or down, but nothing if it remains at the strike. :math:`\qquad` .. rst-class:: to-build .. image:: /_static/Options/table5.png :width: 4in :align: center Straddle ============================================================================== So why doesn't everyone hold straddles? .. rst-class:: to-build - Because the investor must pay for both contracts. .. rst-class:: to-build - The investor doesn't earn a *net* return unless the stock price moves enough to compensate for the initial outlay. Straddle ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Options/bod34698_1509_lg.jpg :width: 3in :align: center .. ifnotslides:: .. image:: /_static/Options/bod34698_1509_lg.jpg :width: 4in Spread ============================================================================== A spread is a combination of two or more options (both calls or both puts) on the same stock with different strikes. .. rst-class:: to-build - Some of the options are purchased while others are sold. .. rst-class:: to-build - A money spread is the simultaneous purchase and sale of options with different strikes. .. rst-class:: to-build - A time spread is the simultaneous purchase and sale of options with different maturities. Bullish Spread ============================================================================== A bullish spread: :math:`\qquad` .. rst-class:: to-build .. image:: /_static/Options/table6.png :width: 7in :align: center Bullish Spread ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Options/bod34698_1510_lg.jpg :width: 3in :align: center .. ifnotslides:: .. image:: /_static/Options/bod34698_1510_lg.jpg :width: 4in 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. .. rst-class:: to-build - This strategy eliminates downside losses below the strike of the put and also upside gains beyond the strike of the call. .. rst-class:: to-build - 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. .. rst-class:: to-build - Purchase a call option with strike price :math:`X`. .. rst-class:: to-build - Purchase a T-bill (lend at the risk-free rate) with a face value equal to the call strike price, :math:`X`, and the same maturity date as the call. :math:`\qquad` .. rst-class:: to-build .. image:: /_static/Options/table7.png :width: 4in :align: center Put Call Parity ============================================================================== The payoffs in the previous table are identical to those for the protective put. .. rst-class:: to-build - Hence, the cost of the protective put strategy should be equal to the cost of the call plus bonds strategy (why?!!!). .. rst-class:: to-build - This fact is known as the *Put-Call Parity Relationship*. .. rst-class:: to-build - Mathematically, it can be expressed as: .. rst-class:: to-build .. math:: C_0 + \frac{X}{1+r_f} & = S_0 + P_0. .. rst-class:: to-build - 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 .. rst-class:: to-build - An asset currently sells for \$110. .. rst-class:: to-build - A call option with strike :math:`X = \$105` and 1-year maturity sells for \$17. .. rst-class:: to-build - A put option with strike :math:`X = \$105` and 1-year maturity sells for \$5. .. rst-class:: to-build - The risk-free interest rate is 5\% per year. .. rst-class:: to-build - Does the parity relationship hold? Put Call Parity Example ============================================================================== .. math:: C_0 + \frac{X}{1+r_f} & \stackrel{?}{=} S_0 + P_0. .. rst-class:: to-build .. math:: \$117 = \$17 + \frac{\$105}{1.05} & \neq \$110 + \$5 = \$115. .. rst-class:: to-build - The relationship doesn't hold. .. rst-class:: to-build - How might an investor take advantage of the situation? Put Call Parity Example ============================================================================== :math:`\qquad` .. image:: /_static/Options/table8.png :width: 7in :align: center