Chapter 11

Options and Other Derivative Securities

Call Option

• Gives owner privilege (or choice) to buy specified number of shares of specified asset at specified price prior to an expiration date.

• Example:– $60 December call option on Xerox common

stock gives holder of option right to purchase from writer 100 shares of Xerox at $60 anytime up until option’s expiration date in December

Put Option

• Permits owner to sell specified number of shares of specified asset at specified price prior to expiration date.

• For example:– Holder of $60 December put option on Xerox

stock has right to sell 100 shares of Xerox stock to writer of put at $60 anytime up until option’s expiration date in December

Long & Short

• Long: owns option

• Short (writer): sale of option not previous owned, thus creating new contract

Prices Associated with Options

• Premium: price of option itself

• Exercise (strike) price: price at which option can be exercised

• Price of underlying security

Relationship between Stock Price and Strike Price

• In-the-money

• At-the-money

• Out-of-the-money

In-the-Money

• Option’s strike price more favorable to option holders than current market price of underlying security– For calls: current stock price > strike price– For puts: current stock price < strike price

• Option has speculative or time value only

Out-of-the-Money

• When option’s strike price is less attractive than current market price of its underlying stock– for calls: current stock price < strike price– for puts: current stock price > strike price

• Option has no intrinsic value, but has speculative or time value based on potential stock price movements prior to option’s expiration.

At-the-Money

• When the current stock price is same as strike price

• No intrinsic value to option per se

Option Markets

• American Stock Exchange

• Chicago Board Options Exchange

• International Securities Exchange

• Pacific Exchange

• Philadelphia Stock Exchange

Options Clearing Corporation

• OCC acts as an intermediary between the two principals in every option trade– Each put and call buyer and seller is actually

contracting with the OCC, rather than directly with the opposite party to the transaction

• Writer places an order to buy an option with identical terms as one sold and OCC cancels the writer from that contract

Expiration Date & Exercise

• Date on which an option expires – Saturday following third Friday of stated month

in standard stock option contracts

• American: exercisable up till expiration

• European: exercisable only on expiration

• Bermuda: exercisable on multiple dates

Why Options Have Value

• Option may end up being in the money on or before expiration date

• Value based on variability of price of underlying security, NOT its expected return

Intrinsic Value

• The payoff obtained by exercising an option immediately– On the expiration date:

premium = intrinsic value– Prior to expiration date:

premium > or = intrinsic value

Speculative Value

• Equals difference between market price of the option and intrinsic value

• Also called time value of the option

• Speculative value approaches zero as the option approaches the expiration date– Market price of premium approaches the

intrinsic value

Example of Speculative Value(1 of 2)

• Stock trades at $50– In 6 mos., 50% probability stock price will

equal $60 & 50% probability it equals $40– Expected value of stock price in 6 months:– > 50 x $60 + .50 x $40 = $50– Expected rate of return: 0%

Example of Speculative Value (2 of 2)

• Intrinsic value of call option in 6 mos.:– If SP = $60, Call option = $10– If SP = $40, Call option = $0

• Expected intrinsic value:– > 50 x $10 + .50 x $0 = $5

• What is value today of something expected to be worth $5 in six month? Ans.: > zero!

Why Trade Options

• Three roles in investment planning– Speculation– Hedging– Arbitraging

Profit and Payoff Functions

• Profit function– Profit equals function of price of

underlying asset on expiration date– Incorporates effect of premium

• Payoff function– Payoff equals function of price of

underlying asset on expiration date– Ignores effect of premium

Profit Function for Long a Call

• Assume strike price = $50

• Premium = $8

• Long a call: pay $8 per share for the call option

• If SP closes above $50, option has value

• If SP closes below $50, option worthless

Call Option

50

NetProfit

NetLoss

Underlying AssetPrice

Break-even Point

-8

0

Types of Call Options

• Covered call: a call option written against stock that one owns

• Naked call: call option written by investor who does not own underlying asset– Risky

• Call writer is obligated to purchase the asset if the call is exercised

• No limit to how high the asset’s market price might rise

Profit Function for Long a Put

• Assume strike price = $50

• Premium = $3

• Long a put: pay $3 per share for the put option

• If SP closes below $50, option has value

• If SP closes above $50, option worthless

Put Option

NetProfit

NetLoss

50

-3

Underlying AssetPrice

Break-even Point

0

Types of Put Options

• Naked put: put option owned without any other position in asset

• Married put: put option held by investor who also owns underlying security

Combinations of Puts & Calls

• Straddle: combination put and call option on same stock at same strike price

• Spread: one option is purchased and other is sold, with each option having different exercise price or expiration date

(continued)

Figure 11-9

Price at

Exp Profit

110

105

100

95

90

5

0

-5

0

5

10

8

6

4

2

-2

-4

- 6

-8

-10

90 95 100 105 110

Profit

Loss

Price

at Exp

-5

Combinations of Puts & Calls(continued)

• Bullish spread: buy call with lower strike price, sell call with higher strike price

• Bearish spread: buy call with higher strike price, sell call with lower strike price

Bullish Spread

0

NetProfit

NetLoss

LowExercise Point

High Exercise Point

UnderlyingAsset Price

Break-even Point

Bearish Spread

0

NetProfit

NetLoss

LowExercise Point

High Exercise Point

Break-even Point

UnderlyingAsset Price

Models for Valuing Options

• Black-Scholes option pricing model

• Binomial option pricing model

• Put-call parity

Black-Scholes Model

• Assumes that a riskless hedge between an option and its underlying stock should yield the riskless return.

• Option’s value function of – stock price

– strike price

– stock return volatility

– riskless interest rate

– length of time to expiration

Five Variables in Black-Scholes Model

• Time to maturity: longer time to maturity, more valuable call

• Interest rate: higher the interest rate, more valuable the call.

• Price of underlying stock: higher the stock price, more valuable the call.

• Volatility: more volatile price of underlying stock, more valuable the call.

• Strike price: higher the strike price, less valuable the call.

Hedge Ratio

• In Black-Scholes model, ratio of number of calls written that would exactly offset stock price movement of number of shares of underlying stock held

• Small move in stock’s price would be precisely offset by change in value of option position with ratio of number of calls to number of shares of stock

• Investor theoretically holding equivalent of risk-free asset.

Binomial Option Pricing Model

• Full model mathematically complex• Simple model assumes

– price at end of period will be one of two values– alternative to call option is to borrow enough so

to buy one share of stock and just breakeven if stock closes at lower price

– Price of call option will be based on how many calls are necessary to duplicate loan strategy, and equity to set up loan

Put-Call Parity

C0 = P0 + S0 – X erf x t

 

C0 = call value

P0 = put value

rf = risk-free rate

e = 2.718 (the natural logarithmic constant)

S0 = initial stock price

X = strike price

t = time to expiration as a fraction of the year

Simple Positions and Their Synthetic Equivalents

Simple Position Synthetic Equivalent• Long Stock Long a Call & Short a Put• Short Stock Short a Call & Long a Put• Long a Call Long Stock & Long a Put• Short a Call Short Stock & Short a Put• Long a Put Short Stock & Long a Call• Short a Put Long Stock & Short a Call

Synthetic (Manufactured) Call

• Call-like position generated by combination position in underlying stock and put

• Position whose profit function is exact same shape as that of a call

• C0 = P0 + S0 – X/erfxt

Synthetic (Manufactured) Put

• Put-like position generated by combination positions in underlying stock & call option

• Position with payoff matrix similar to Put

• P0 = C0 – S0 + X/erfxt

Other Types of Options

• Stock index options– option on value of a stock index– cash settlement

• Interest rate options– option to buy or sell government securities

• LEAPS®

– options with initial maturities of up to three years

Convertible Securities

• Convertible Bonds

• Convertible Preferred Stocks

• Concepts the same

Conversion Ratio

• Conversion ratio = Par / Conversion Price

• Conversion Price defined in indenture

• Conversion ratio (or exchange ratio) = # of shares of common stock received upon conversion

Conversion Value & Premium

• Conversion Value =

Conversion Ratio x Price of Common Stock

• Conversion Premium =

Market Price of Bond – Conversion Value

• % Conversion Premium =

Conversion Premium / Conversion Value

44

Value of Bond asDebt and as Stock

Value of ComparableNonconvertible Bond(Straight-Debt Value)

Price of Stock

Conversion Value

Bond Market Value

ConversionPremium

Convertibles Always Callable

• Allows company to force conversion

• Investor must convert or sale

• After call date, no longer accrues interest or is convertible

Rates of Return on Convertibles

• Historically:– Better than non-convertibles– Worse direct ownership of equity

• Less risky than equity, riskier than straight debt