Mechanics of Options Contracts

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Class 6 Options (Part I):

Overview of This Class: Objectives, Important Concepts and Learning Outcomes

 Learning Objectives: Discuss fundamentals and mechanics of options contracts. Define call and put options and explain the characteristics of options markets. Discuss options markets and options trading (closing position, exercising, and the expiration process). Discuss institutional and regulatory features in options markets. Discuss “no-arbitrage” properties of options prices.

 Important Concepts: Economics of Options Markets. Mechanics of Options Contracts. Options Positions (Long Call, Short Call, Long Put, and Short Put). Intrinsic Value (Moneyness) and Time Value of Options. European vs. American Options. Options Trading, Exercising, and the Expiration Process.

 Tools/Learning Outcomes: No-Arbitrage Properties of Option Price. The Put-Call Parity.

1. Types of Options  A Call option is an option (but not an obligation) to buy a

certain asset by a certain date for a certain price (the strike price).

 A Put option is an option (but not an obligation) to sell a certain asset by a certain date for a certain price (the strike price).

 Important Note: an option contract gives the buyer of the option contract the right (an option but not an obligation) to buy or sell an asset for a predetermined strike price in the future. A call option provides the buyer of the call option contract the right to buy in the future. In contrast, a put option provides the buyer of the put option contract the right to sell in the future.

 European option can be exercised only at maturity.

 American option can be exercised at any time until maturity.

 Bermudan option can be exercised at certain date until maturity (hybrid of American and European options).

 (1) Long Call

 (2) Long Put

 (3) Short Call

 (4) Short Put

 Note: (1)  (2)  (3)  (4)

 Note: Long position in Call/Put “purchases the option” to buy/sell the underlying asset in the future.

 In contrast, Short position in Call/Put option “sells that option and must accommodate to sell/buy the underlying asset if the Long Call/Put decides to exercise the option.

 Quiz: Which position is the most risky?

2. Option Positions

2.1. Profit to Long Call Options

Profit to Long Call Option

Price of Underlying at Maturity, ST

K (Strike Price in Option Contract)

A Long Call Option gives the option (but not the obligation) to buy a certain asset by a certain date for a certain price (the strike price) agreed on today.

2.2. Profit to Short Call Options

Profit to Long Call Option

Price of Underlying at Maturity, STK

(Strike Price in Option Contract)

A Short Call Option “sells the option” and has the obligation to sell the underlying asset if the Long Call decides to exercise the option.

2.3. Profit to Long Put Options

Profit to Long Put Option

Price of Underlying at Maturity, ST

K (Strike Price in Option Contract)

A Long Put Option gives the option (but not the obligation) to sell a certain asset by a certain date for a certain price (the strike price) agreed on today.

2.4. Profit to Short Put Options

Profit to Long Put Option

Price of Underlying at Maturity, STK

(Strike Price in Option Contract)

A Short Put Option “sells the option” and has the obligation to buy the underlying asset if the Long Put decides to exercise the option.

Example: 2.1. Long Call on Facebook A Long Call Option gives the option (but not the obligation) to buy a certain asset by a certain date for a certain price (the strike price) agreed on today. For example, assume Option Premium of Facebook European Call Option = $5; Strike Price (K) = $100; Maturity Date = 3 months. Profit from a long position of Facebook European Call Option is shown as follows (assume contract size = 1 share):

-5

Profit to Long Call

Stock Price, ST

K = 100

Example: 2.2. Short Call on Facebook A Short Call Option “sells the option” and has the obligation to sell the underlying asset if the Long Call decides to exercise the option.

For example, assume Option Premium of Facebook European Call Option = $5; Strike Price (K) = $100; Maturity Date = 2 months. Profit from a short position of Facebook European Call Option is shown as follows (assume contract size = 1 share):

Profit to Short Call

Stock Price, ST K = 100

Example: 2.3. Long Put on Twitter A Long Put Option gives the option (but not the obligation) to sell a certain asset by a certain date for a certain price (the strike price) agreed on today.

For example, assume Option Premium of Twitter European Put Option = $9; Strike Price (K) = $80; Maturity Date = 3 months. Profit from a long position of Twitter European Put Option is shown as follows (assume contract size = 1 share):

-9

Profit to Long Put

K = 80

Stock Price, ST

Example: 2.4. Short Put on Twitter A Short Put Option “sells the option” and has the obligation to buy the underlying asset if the Long Put decides to exercise the option. For example, assume Option Premium of Twitter European Put Option = $9; Strike Price (K) = $80; Maturity Date = 3 months. Profit from a short position of Twitter European Put Option is shown as follows (assume contract size = 1 share):

Profit to Short Put

Stock Price, STK = 80

3. Understanding Options  Options Features: Limited risk (?); Less costly; Tremendous leverage;

Trading is zero-sum game.

 Option Class: Option Series; Intrinsic Value; Time Value.

 Option Terminology: o Option Price/Premium o Call vs. Put o Exercise Price/Strike Price/Striking Price o Expiration Date

 Option Premium: Option Premium = Intrinsic Value + Extrinsic Value Intrinsic Value is also known as Exercise Value Extrinsic Value = Time Value of Option (see Section 4)

 Moneyness of Options (affects Intrinsic Value):  Case (1) In-the-Money  Case (2) At-the-Money  Case (3) Out-of-Money

3.1. Intrinsic (Exercise) Value and Moneyness  Intrinsic Value of Call Option (boundary condition or exercise

value of Call Option):

Call’s Intrinsic Value = max(S – K, 0)

 Where S = Spot Price (on exercise date); and K = Strike Price.  The Call Option is “In-the-Money” when S > K.  See also Section 2.1 for Payoff Diagram of Long Call Option.

 Intrinsic Value of Put Option (boundary condition or exercise value of Put Option):

Put’s Intrinsic Value = max(K – S, 0)

 Where S = Spot Price (on exercise date); and K = Strike Price.  The Put Option is “In-the-Money” when K > S.  See also Section 2.3 for Payoff Diagram of Long Put Option.

   

 otherwise0

for KSKS

   

 otherwise0

for KSSK

4. Extrinsic Value (Time Value) of Options

Extrinsic (Time) Value of Option T 0 Time to Expiration

Extrinsic (Time) Value of Option = Option Price  Intrinsic Value of Option

5. Organized Options Trading and Exchange The Concept of an Options Exchange (an Example):  Listing Requirements (on large stocks); for example:

– Ownership: A minimum of 7,000,000 shares of the underlying; a minimum of 2,000 holders of the underlying.

– Trading Volume: Trading volume (in all markets in which the underlying is traded) of at least 2,400,000 shares in the preceding 12 months.

 Contract Size (e.g., 100 shares);  Exercise Prices (e.g., moneyness);  Expiration Dates;  Position Limits (e.g., 13,500-75,000 contracts);  Exercise Limits (e.g., max number of options can be exercised on 5 consecutive

business days by an individual). Useful resources: CBOE website (http://www.cboe.com) and CBOE Market Statistics (http://www.cboe.com/data/AnnualMarketStatistics.aspx) - very useful source of information for your individual research assignments.

Example: the contract specifications for CBOE equity options

 Trading hours of CBOE equity options are 8:30 am - 3:00 pm Central Time (Chicago).

 CBOE equity options are American options - can be exercised any time from purchase until the option’s last trading day (usually the third Friday of the expiration month).

 Each option is for 100 shares of the underlying stock.

 If exercised, the stock share is delivered three business days later.

 The minimal tick size is $0.05 for options trading below $3 and $0.10 for others series.

 See: http://www.cboe.com/Products/EquityOptions.aspx for further examples.

6. Option Contract Features  See Jarrow and Chatterjea (2013) for the example of equity options; option

contract features the followings:

 Maturity Dates: Equity options have an assigned quarterly cycle: either the January, February, or March cycle. For a particular cycle, maturities are listed for the next two months plus two additional maturities, selected three months apart.

 Strike Prices: Strike prices for in-the-money, at-the-money, and out-of-the-money are listed. If the stock price is over $200, the strikes are issued $10 apart. Between $200 and $25, the strikes are issued $5 apart, and below $25, only $2.50 apart. Strike prices are adjusted for stock dividends and stock splits but not for cash dividends.

 Dividends and Stock Splits: When a stock splits or pays large stock dividends, the option’s strike price is adjusted on the ex-dividend day or the ex-split day. Options are not adjusted for dividends or stock distributions that do not exceed 10% of the stock’s value.

 Exercising the Option: Options holder convey intention to exercise the option to the broker. The broker submits an exercise notice to the Options Clearing Corporation (OCC), which sends the notice to clearing members holding short positions. The clearing member representing the short is obligated to sell (in the case of a call) or buy (in the case of a put) the underlying shares at the strike price.

6.1. Option Contract Features  The Expiration Process: For listed equity options, the expiration date is usually the

third Friday of the expiration month. Brokerage firms must submit exercise notices to the OCC.

 The OCC has developed exercise by exception (also called ex-by-ex) to expedite the exercise of expiring options; i.e. every contract in-the-money will be automatically exercised unless the clearing member advises OCC otherwise.

 Currently equity and index options that are in-the-money by at least 1 cent are automatically exercised. Brokers may require their customers to notify their intention to exercise (i.e., sufficient purchasing power must be in the account to make or take delivery of the shares).

 For Derivatives Trading Simulation: you choose between (1), (2), OR (3) below:

 (1) Before expiration, you may make profits from offsetting existing contracts.

 (2) You may hold your option contracts “at/until expiration”. The procedure of "exercise by exception" is featured in the StockTrak simulation: if you have an in-the-money option at expiration, the simulation will close out the position for you. At expiration, your contracts will be settled in cash.

 (3) Early exercise of options is available in the StockTrak simulation: see our documents of “Derivatives Trading Simulation” and “Weekly Learning Guideline.  See http://content.stocktrak.com/student-faq-technical-support/ and choose

“Q: HOW DO I EXERCISE OPTIONS?”

 To help your decision on early exercise of options, you may simulate any profits from early exercising options (using the Excel model provided in Blackboard). 18

Properties of Options Contracts

Notation  c = European Call option price

 p = European Put option price

 S0 = Stock price today

 K = Strike price

 T = Time to expiration (maturity)

  = Volatility of stock price/return

 C = American Call option price

 P = American Put option price

 ST = Stock price at option maturity

 D = Present value of dividends during option’s life

 r = Risk-free rate for maturity T with continuous compounding

Effects of Variables on Option Prices [IMPORTANT]

(source: Hull (2014))

c p C PVariable

S0 K T  r D

+ + – +

? ? + + + + + + + – + –

– – – +

– + – + Longer-maturity European Calls (Puts) need not be more valuable than their shorter-maturity counterparts.

Volatility always has a positive impact on the value of option prices.

1. American vs. European Options

An American option is worth at least as much as the corresponding European option:

C  c   P  p  

See also the “Superglue Argument” by Jarrow and Chatterjea (2013): European options can only be exercised once at expiration, whereas American options can be exercised early with additional flexibility.

As “more cannot be worth less,” an American option can never be worth less than the corresponding European option.

2. Lower Bound for European Call Option Prices; No Dividends

c  S0 – Ke-rT

The European Call option price (c) cannot fall below the boundary condition (the lower bound above), which is determined by the intrinsic value of the Call option. Otherwise, there will be arbitrage.

Quiz #1: Call Options - An Arbitrage Opportunity? Suppose that

c = 28 S0 = 100 K = 70 D = 0 r = 4% T = 1

Is there an arbitrage opportunity? See answer next page.

Quiz #1: Lower Bound of European Call Option Prices European call option price (c) 28 Spot price (S0) 100 Strike price (K) 70 Risk-free rate (r) 4% Time-to-maturity (T) 1 Assume no dividend. Given the above information, apply the Lower Bound for European Call and verify if there is an arbitrage opportunity. If there is an arbitrage opportunity, construct an arbitrage strategy using the above information. Answers: Apply the Lower Bound of European Call Option Price (no-arbitrage condition): c ≥ S0 – Ke

-rT There is arbitrage opportunity if c < S0 – Ke

-rT Here we find that: c = 28 which is less than S0 – Ke

–rT = 100 – 70e –0.04*1 = 32.74473926 We find arbitrage opportunity as the call price is lower than the no-arbitrage lower bound. For arbitrage strategy, arbitrageurs may long the undervalued call, and also short stock (and lend at risk-free rate). The arbitrage profit will be 4.744739259 based on the above arbitrage strategy (long call and short stock with risk-free lending).

3. Lower Bound for European Put

Prices; No Dividends

p  Ke-rT – S0 The European Put option price (p) cannot fall below the boundary condition (the lower bound above), which is determined by the intrinsic value of the Put option. Otherwise, there will be arbitrage.

Quiz #2: Put Options - An Arbitrage Opportunity? Suppose that

p = 7 S0 = 20 K = 30 D = 0 r = 4% T = 1

Is there an arbitrage opportunity? See answer next page.

Quiz #2: Lower Bound of European Put Option Prices European put option price (p) 7 Spot price (S0) 20 Strike price (K) 30 Risk-free rate (r) 4% Time-to-maturity (T) 1 Assume no dividend. Given the above information, apply the Lower Bound for European Put and verify if there is an arbitrage opportunity. If there is an arbitrage opportunity, construct an arbitrage strategy using the above information. Answers: Apply the Lower Bound of European Put Option Price (no-arbitrage condition): p ≥ Ke-rT – S0 There is arbitrage opportunity if p < Ke-rT – S0 Here we find that: p = 7 which is less than Ke-rT – S0 = 30e

–0.04*1 – 20 = 8.823683175 We find arbitrage opportunity as the put price is lower than the no-arbitrage lower bound. For arbitrage strategy, arbitrageurs may long the undervalued put, and also long stock (and borrow at risk-free rate). The arbitrage profit will be 1.823683175 based on the above strategy (long put and long stock with risk-free borrowing).

4. Put-Call Parity (with No Dividends) [IMPORTANT]

 Consider the following 2 portfolios: (i) Portfolio A: European Call + PV(strike price) in cash (ii) Portfolio B: European Put + Stock

 Both portfolios A and B above are worth MAX(ST , K ) at time T (the maturity of the options).

 Both portfolios above must therefore be worth the same today This implies the Put-Call Parity is satisfied with no arbitrage:

c + Ke-rT = p + S0 Important note: The Put-Call Parity condition applies to European options. The Put–Call Parity states that (the call price plus the present value of strike) must be equal to (the stock price plus the put price).

4.1. Applications of Put-Call Parity

 Applications and uses of the Put–Call Parity:

 (1) Put–Call parity is used for constructing trading (arbitrage) strategies.

 (2) Put–Call parity is used for synthetic creation of an option, a stock, or a bond.

Quiz #3: Arbitrage Opportunities based on Put-Call Parity Suppose that

c = 15 S0 = 100 K = 80 r = 4% T = 1 D = 0

Is there an arbitrage opportunity if put option price (p) = 5? See answer next page.

Quiz #3: Put-Call Parity European call option price (c) 15 European put option price (p) 5 Spot price (S0) 100 Strike price (K) 80 Risk-free rate (r) 4% Time-to-maturity (T) 1 Assume no dividend. Given the above information, apply the Put-Call Parity and verify if there is an arbitrage opportunity. If there is an arbitrage opportunity, construct an arbitrage strategy using the above information. Answers: Apply the “PUT-CALL Parity of European Options” (no-arbitrage condition): c + Ke –rT = p + S0 Given the above information, we find that: Portfolio A: c + Ke –rT = 15 + 80*e -0.04*1 = 91.86315513 Portfolio B: p + S0 = 5 + 100 = 105 There is arbitrage opportunity because c + Ke –rT < p + S0 (no equality of the Put-Call Parity condition). For arbitrage strategy, arbitrageurs may long Portfolio A: long positions in (c + Ke –rT) and short Portfolio B: short positions in (p + S0). The arbitrage profit will be (105 – 91.86315513) = 13.13684487 > 0 based on Long Portfolio A and Short Portfolio B (“buy low, sell high”) above.

5. Early Exercise  American options provide early exercise opportunity (flexibility) –

i.e., it is possible to exercise an American option early (before maturity).

 An exception is an American Call on a non-dividend paying stock (note: in this case, we assume that price of non-dividend paying stock will increase in the future).

 Quiz: American Call on a non-dividend paying stock should never be exercised early. Why?

 Reasons For Not Exercising an American Call Early (No Dividends):

1) No income is sacrificed.

2) Delaying the payment of the strike price.

3) Holding the call (alive) continues to provide insurance (hedging) against adverse stock price movements.

6. Extensions of Put-Call Parity

 European options; D =0

c + Ke−rT = p + S0

 American options; D = 0

S0  K < C  P < S0  Ke−rT

 European options; D > 0

c + D + Ke−rT = p + S0

 American options; D > 0

S0  D  K < C  P < S0  Ke−rT

Important Concepts and Useful Readings

 Make sure you understand the followings for this class:

 Fundamental concepts of options and mechanics of options contracts.

 Institutional characteristics of options markets.

 The “No-Arbitrage” properties of options prices.

 Economics of options markets and importance of institutional features in options markets.

 Important Readings: Jarrow and Chatterjea (2013):

 Ch. 14: 14.1, 14.2, 14.3, 14.4, 14.5, 14.7.

 Ch. 16: 16.1, 16.2, 16.3.

 Useful (Additional/Optional) Readings: Hull (7th or 8th edition):

 Ch. 9 (focus on: “Types of options”; “Option positions”; “Underlying assets”; “Specification of stock options”).

 Ch. 10 (focus on: “Factor affecting option prices”; “Upper and lower bounds for option prices”; “Put-Call parity”).

 Note: references above are equivalent to Hull (6th edition): Ch. 8, 9.