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Options Chapter 2.5 Chapter 15

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Learning Objectives Understand key terms related to options and options markets Compute payoffs and profits to option holders and writers Calculate potential profits from various options strategies Describe the put-call parity relationship

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Derivative A derivative is a security who’s value is dependent on another assets Base Asset Ex: commodity prices, bond and stock prices, or market index values Derivatives are contingent claims Their payoffs depend on the value of another securities. Options are a specific type of derivative The holder has the right, but not the obligation, to buy or sell a given quantity of an asset on (or before) a given date, at a price agreed upon today.

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**Options: Calls and Puts**

Call: The owner of a call has the right, but not the obligation, to BUY an asset in the future at the strike or exercise price Value decreases as the strike price increases Put: The owner of a put has the right, but not the obligation, to SELL an asset in the future at the strike or exercise price Value increases with the strike price Value of both calls and puts increases with time to expiration, WHY?

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**Rights and Obligations**

Buyer: Gets all the Rights Seller: Gets all the Obligations Calls Right to Buy the asset Is obliged to Sell the asset Puts Right to Sell the asset Is obliged to Buy the asset

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Futures Contracts An agreement made today regarding the delivery of an asset (or in some cases, its cash value) at a specified delivery or maturity date for an agreed-upon price (futures price) to be paid at contract maturity Long position: Take delivery at maturity Short position: Make delivery at maturity

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**Comparison Option Futures Contract**

Right, but not obligation, to buy or sell; option is exercised only when it is profitable Options must be purchased Both sides have an obligation Long position must buy at the futures price Short position must sell at futures price Futures contracts are entered into without cost

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The Option Contract The purchase price of the option is called the premium. Stock options cover 100 shares & premium is on a per share basis Sellers (or Writers) of options receive premium. If the Holder (or Buyer) exercises the option, the Writer must deliver (call) or take delivery (put) of the underlying asset.

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**Options Terminology Exercising the Option**

Using the option Strike Price or Exercise Price The price specified by the option Spot Price The market price Expiration Date The option’s maturity date European & American options Europeans can only be exercised at expiration. Americans can be exercised at any time up to expiration. BUT NEVER ARE

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Options Terminology In-the-Money: Exercising the option results in a profit Call: exercise price < market price Put: exercise price > market price At-the-Money: Exercising the option results in 0 profit exercise price = market price Out-of-the-Money: Exercising the option results in a loss Call: market price < exercise price Put: market price > exercise price

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**Call Payoff (Intrinsic Value)**

Call Payoffs Holder of the option Pays the premium at time=0 Has the right to exercise the option at time=T Notation Stock Price at T = ST T is maturity, t is any other time Exercise Price = X Don’t Exercise Exercise Call Payoff (Intrinsic Value) ST - X

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**Call Holder Payoff and Profit**

Value of the Call at T: CT = Max [ST – X, 0] What is the value of the Call at t? ST >X ST < X CT ST – X Profit ST - X - Ct -Ct

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**Payoff and Profit to Call Option at Expiration**

What is the strike price? What is the premium?

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**What is the value of a Call at t?**

At maturity CT = Max [ST – X, 0] Would you be willing to sell for St-X at t? Hint: What could happened to the stock price? Time value is the premium a rational investor will pay above an options intrinsic value This base on the likelihood that the stock price will move making the option more valuable Time Value = Option Price – Intrinsic Value

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**Call Writer Payoff and Profit**

ST >X ST < X Exercise Cost -(ST – X) Profit -(ST - X) + Ct Ct

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**Payoff and Profit to Call Writers at Expiration**

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**Calls: A Zero Sum Game Call Option If ST < X If ST > X Decision**

No exercise Exercise Option Payoff (holder) ST – X Option Profit (holder) -C (ST – X) – C Option Payoff (writer) - (ST – X) Option Profit (writer) +C C - (ST – X)

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**Profit and Loss on a Call**

A February 2013 call on IBM with an exercise price of $195 was selling on January 18, 2013, for $3.65. The option expires on the third Friday of the month, or February 15, 2013. If the price of IBM on Feb 15, 2013 is $194, what is the call worth?

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**Profit and Loss on a Call (cont.)**

Suppose IBM sells for $197 at expiration Remember: strike = $195, premium = $3.65 What is the value of the option? Call Intrinsic value = stock price-exercise price Will the option be exercised? What is the Profit/Loss on this investment? Profit = Final value – Original investment What must the price of IBM be for the option to break-even?

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Call Option Problem At time=0 you buy a call option on IBM for $ The option gives you the right to buy 100 shares of IBM stock at time=T at $65 What is the payoff to you if ST = $70? What is the payoff for the writer if ST = $70? What is the payoff to you if ST = $60? What is the payoff for the writer if ST = $60?

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**Puts Gives holder the right (but not the obligation) to sell an asset:**

At the exercise or strike price On or before the expiration date Exercise the option to sell the underlying asset if market value < strike.

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**Put Payoff (Intrinsic Value)**

Puts Payoffs Holder of the option Pays the premium at time=0 Has the right to exercise the option at time=T Notation Stock Price at T = ST Exercise Price = X Don’t Exercise Exercise Put Payoff (Intrinsic Value) X - ST

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**Put Holder Payoff and Profit**

Value of the Call at T: PT = Max [X - ST , 0] ST <X ST > X Payoff X - ST Profit X - ST - Pt -Pt

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**Payoff and Profit to Put Option at Expiration**

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**Put Writer Payoff and Profit**

ST <X ST > X Exercise Cost -(X - ST) Profit -(X - ST) + Pt Pt

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**Puts: Another Zero Sum Game**

Put Option ST < X If ST > X Decision Exercise No Exercise Option Payoff (holder) X – ST Option Profit (holder) (X-ST) - P -P Option Payoff (writer) - (X-ST) Option Profit (writer) +P - (X-ST) +P

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Profit and Loss on a Put Consider a February 2013 put on IBM with an exercise price of $195, selling on January 18, for $5.00. Option holder can sell a share of IBM for $195 at any time until February 15. If IBM sells for $196, what is the put worth?

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**Profit and Loss on a Put Suppose IBM’s price at expiration is $188.**

What is the value of the option? Put value = Exercise price- Stock price Will the option be exercised? What is the Profit/Loss on this investment? Profit = Final value – Original investment What is the HPR?

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Put Option Problem At time=0 you buy a put option on ITT stock for $ The option gives you the right to sell 100 shares of ITT stock at time=T at $50 What is the payoff to you if ST = $55? What is the payoff to the put seller if ST = $55? What is the payoff to you if ST = $45? What is the payoff to the put seller if ST = $45?

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**Option versus Stock Investments**

Could a call option strategy be preferable to a direct stock purchase? Suppose you think a stock, currently selling for $100, will appreciate. A 6-month call costs $10 (contract size is 100 shares). You have $10,000 to invest.

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**Option versus Stock Investment**

Investment Strategy Investment Equity only Buy $100 (100 shares) $10,000 Options only Buy $10 (1,000 options) $10,000 Options + Buy $10 (100 options) $1,000 T-Bills Buy 3% Yield $9,000

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Strategy Payoffs

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**Option Versus Stock Investment**

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Strategy Conclusions The all-option portfolio, B, responds more than proportionately to changes in stock value; it is levered. Portfolio C, T-bills plus calls, shows the insurance value of options. C ‘s T-bill position cannot be worth less than $9270. Some return potential is sacrificed to limit downside risk.

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Combining Options Puts and calls can serve as the building blocks for more complex option contracts Can be used to manage risk This is financial engineering Allows you to tailor the risk-return profiles to meet your client’s desires Ex: Protective Puts: Underlying asset and put are combined to guarantee a minimum valuation Put is insurance against stock price declines.

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**Protective Put at Expiration**

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**Covered Calls Purchase stock and write calls against it.**

Call writer gives up any stock value above X in return for the initial premium. If you planned to sell the stock when the price rises above X anyway, the call imposes “sell discipline.”

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**Covered Call Position at Expiration**

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**Straddle The straddle is a bet on volatility.**

Long straddle: Buy call and put with same exercise price and maturity. To make a profit, the change in stock price must exceed the cost of both options. You make money on a large price shift in either direction The writer of a straddle is betting the stock price will not change much.

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**Straddle Position at Expiration**

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Spread A spread is a combination of two or more calls (or two or more puts) on the same stock with differing exercise prices or times to maturity. Some options are bought, whereas others are sold (written) A bullish spread is a way to profit from moderate stock price increases EX. Buy a call with a strike of $20 and sell a call with a strike price of $25

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**Bullish Spread Position at Expiration**

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**Put Call Parity P –C = PV (X) - St**

This is the relation between a put and call with the same exercise price (E) and maturity It comes from replicating portfolios: The payoffs from buying a call and selling a put is the same as the payoffs from buying the stock and borrowing the PV of the exercise price P –C = PV (X) - St

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**Payoff-Pattern of Long Call–Short Put Position**

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**Portfolios Portfolio 1: Buy a call and Write a put**

Portfolio 2: Buy the stock but Borrow PV (X) Levered Equity positions If the payoffs are the same the price must be the same -C+P = -S0 + Xe-rT C-P = S0 – Xe-rT

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**Proof by Counter Example**

Assume that: Stock Price = Call Price = 14 Put Price = Risk Free = 5% Maturity = 6 months Strike Price = 105 Portfolio 1 costs = -9 Portfolio 2 costs: e-5 = -7.59 Two different costs for the sample payoffs → ABRITRAGE

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**Arbitrage Strategy Payoff**

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**Put Call Parity Example**

What is the value of a put with an exercise price of $51, if the stock is currently trading at $49. The price of the corresponding call option is $4.65. According to put-call parity, if the effective annual risk-free rate of interest is 4% and there are three months until expiration, what should be the value of the put? FYI: It doesn’t matter if compounding is monthly or quarterly

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