Chapter 10: Options Markets

Tuesday March 22, 2011

By Josh Pickrell

Options: An Introduction

• Options limit downside risk, while retaining the potential for upside gain.

• Options are used as a risk management tool.

• An option’s value is derived from some underlying asset.

Options: An Introduction

• An option contract gives its owner the right, but not the legal obligation, to conduct a transaction involving an underlying asset at a predetermined future date (the exercise date) and at a predetermined price (the exercise or strike price).– Options give the option holder the right to decide

whether or not the trade will eventually take place.– The seller of the option has the obligation to perform

if the buyer exercises the option.

Options: An Introduction

• American options may be exercised at any time up to and including the contract’s expiration date.

• European options can be exercised only on the contract’s expiration date.

• Before the expiration date, an American option and a European option on the same asset with the same strike price may have different values. – Will an American option be worth less than an European

option? Explain

Options: An Introduction

• A call option gives its owner the right to buy the underlying asset at a specific price.

• A put option gives its owner the right to sell the underlying asset at a specific price.

• For every buyer of an option, there must be a seller.– The seller of the option is also called the option

writer.

Options: An Introduction

• To acquire these rights, owners of options must buy them by paying a price called the option premium to the seller of the option.– Listed stock options contracts trade on exchanges

are normally for 100 shares of stock.– After issuance, stock options are adjusted for stock

splits but not cash dividends.

Digital Options

• A digital option has a payoff at maturity which is either a fixed amount or zero. As with any other investment, only two things can be done with an option: buy or sell.

Digital Options

• Examples:– The payoff for a digital call option is $1 if the

stock price rises to $10, and $0 if the stock price never rises to $10.

– The payoff for a digital put option is $1 if the stock price falls below $10, and $0 if the stock price fails to fall below $10.

– Suppose that we combine the put and call options. What does the position look like? Draw the diagram and write out the vector notation.

Digital Options: Vector Notation

• Write out the vector notation for the following positions: long stock, short stock, buy call, write call, buy put, write put.

• The advantage of the vector notation is that it is a very simple tool that can be used to understand how to solve problems in managing risky positions.

Spot Market

• Cash / Spot markets physical market where assets transact– Prices change frequently– Two positions can be taken in the spot markets:• A long position: purchase of an asset (e.g. stock) with

the anticipation that the price will rise.• A short position: sell an asset (e.g. stock) with the

anticipation that the price will fall and you will be able to purchase at a lower price in the future.

Derivatives Market

• The price of a derivative instrument is based on the underlying spot price of some specified asset– Derivatives enable us to make the short bet with

ease, whereas the spot market may be difficult to short.

Option Markets

• Exchange-trade or listed options are regulated, standardized, liquid, and backed by the Option Clearing Corporation for Chicago Board Options Exchange transactions.–Most options have expiration dates within two to

four months of the current date.

Option Markets

• Over the Counter (OTC) options on stocks for the retail trade all but disappeared with the growth of the organized exchanges in 1971.– There is now, however, an active market in OTC

options on currencies, swaps, and equities.– These options are largely unregulated, custom, and

involves counterparty risk.

Standard Options

• Although digital options do exist, it is more relevant to discuss the standard option where the payoff is equal to the difference between the value of the stock and the strike price.

• Draw the profit diagrams (hockey sticks) for the following six positions:– (1) long stock, (2) short stock, (3) long call, (4)

write call, (5) long put, (6) write put.

Standard Options: The Notation

• St = the price of the underlying stock at time t

• X = the exercise price of the option• T = the time to expiration

• Ct = the price of a call option at time t

• Pt = the price of a put option at time t

• Rfr = the risk free rate

Standard Options: Value

• An options value is the sum of the time value and the intrinsic value.

• The intrinsic value is defined as the value that you would receive if you immediately exercised the option.

• The value of a standard option can never be less than zero.

Standard Options - Moneyness

• Moneyness refers to whether an option is in-the-money or out-the-money.– If immediate exercise of the option would generate

a positive payoff, it is in the money.– If immediate exercise would result in a loss

(negative payoff), it is out of the money.–When the current asset price equals the exercise

price, meaning exercise will generate neither a gain nor loss, the option is at the money.

Standard Options- Moneyness

• The following describes the conditions for a call option:– In the money call option: S – X > 0– Out of the money call option: S – X < 0– At the money call option: S = X

• The following describes the conditions for a put option:– In the money put option: X – S > 0– Out of the money put option: X – S < 0– At the money call option: S = X

Option Basics

• An options intrinsic value is the amount by which the option is in the money.– An option has zero intrinsic value if it is at the

money or out of the money, regardless of whether it is a call or a put option.

– The intrinsic value of a call option:• C = max[0, S – X]

– The intrinsic value of a put option:• P = max[0, X – S]

Option Basics

• The time value of an option is the amount by which the option premium exceeds the intrinsic value and is sometimes called the speculative value of the option.

• Option value = intrinsic value + time value.

Option Basics

• At any point during the life of an option contract, its value will typically be greater than its intrinsic value.*– Because there is some probability that the stock

price will change in an amount that gives the option a positive payoff at expatriation.

– For American options the longer the time to expiration, the greater the time value and, other thing equal, the greater the option’s premium (price).

Option Basics

• Lower bound: Theoretically, no option will sell for less than its intrinsic value and no option can take on a negative value.– Call option: Ct = max [0, St – X]

– Put option: Pt = max[0, X – St]

• Upper bound:– Call option: the maximum value is at any time t is the time-t

share price of the underlying stock. (St)

– Put option: the price for an American put cannot be more than its strike price. (X)

Options: Profit Equations

• (long) Call Option: Cprofit = max(0, S – X) – premium

• (long) Put Option: Pprofit = max(0, X – S) – premium

• You only need to memorize these two equations. The short (or written) positions are simply the above equations multiplied by a negative sign.

Put-Call Parity

• The put-call parity shows that the value of a put option is related to the value of the call option.– The parity is based on the efficient market concept

that two investments with same cash flows and same risks must have the same price. (Keep this in mind as we work thorough this parity).

Put-Call Parity

• Strategy 1: buy a share of the underlying asset (S), and buy a put (P) on this stock with an exercise price of X that expires at time T.

• Strategy 2: buy a call on asset (S) with an exercise price of X that expires at time T and buy a discounted risk-free bond where the maturity is equal to X expiring at time T.

Put-Call Parity

• The Put-Call Parity:S + P = C + PV(X)

• At matuirty there are three possible outcomes: (1) S > X, (2) S < X, and (3) S = X. –What is the value of each portfolio under these

three outcomes?

Put-Call Parity

• Under every possible outcome, the two strategies yield exactly the same outcome. Remember that the efficient market hypothesis says that if two assets have the same payoff structure, then they should have the same price.

• We can use the put-call parity to create a synthetic put option and a synthetic call option.

Option Pricing: Binomial Tree

• A binomial tree is based on the assumption that asset prices will move up or down at known points in time.

• Each node in the tree represents the stock at a given price and at a given time.

Option Pricing: Black-Scholes Model

• Black and Scholes (1973) developed a mathematical formula to price options. This formula can be used to price call and put options.– Note: the original formula was developed to price

European options with no dividends.

• See the formula on Page 268.

Option Pricing

• The price of call (or put) option depends on:– stock price– time to expiration– volatility– risk-free rate– strike price– dividends

Option Strategies

• Lets look at several examples of option strategies:– Long stock + write a call (covered call)– Long stock + buy a put (protective put)– Long stock + short call + long put (costless collar)– Short call + long call (x2) (Call back spread)