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## PowerPoint Slideshow about 'Options and Corporate Finance' - paul

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Presentation Transcript

Key Concepts and Skills

- Understand the options terminology
- Be able to determine option payoffs and pricing bounds
- Understand the five major determinants of option value
- Understand employee stock options
- Understand the various managerial options
- Understand the differences between warrants and traditional call options
- Understand convertible securities and how to determine their value

Chapter Outline

- Options: The Basics
- Fundamentals of Option Valuation
- Valuing a Call Option
- Employee Stock Options
- Equity as a Call Option on the Firm’s Assets
- Options and Capital Budgeting
- Options and Corporate Securities

Option Terminology

- Call
- Put
- Strike or Exercise price
- Expiration date
- Option premium
- Option writer
- American Option
- European Option

Stock Option Quotations

- Look at Table 14.1 in the book
- Price and volume information for calls and puts with the same strike and expiration is provided on the same line
- Things to notice
- Prices are higher for options with the same strike price but longer expirations
- Call options with strikes less than the current price are worth more than the corresponding puts
- Call options with strikes greater than the current price are worth less than the corresponding puts

Option Payoffs – Calls

- The value of the call at expiration is the intrinsic value
- Max(0, S-E)
- If S<E, then the payoff is 0
- If S>E, then the payoff is S – E
- Assume that the exercise price is $35

Option Payoffs - Puts

- The value of a put at expiration is the intrinsic value
- Max(0, E-S)
- If S<E, then the payoff is E-S
- If S>E, then the payoff is 0
- Assume that the exercise price is $35

Work the Web Example

- Where can we find option prices?
- On the Internet, of course. One site that provides option prices is Yahoo Finance
- Click on the web surfer to go to Yahoo Finance
- Enter a ticker symbol to get a basic quote
- Follow the options link
- Check out “symbology” to see how the ticker symbols are formed

Call Option Bounds

- Upper bound
- Call price must be less than or equal to the stock price
- Lower bound
- Call price must be greater than or equal to the stock price minus the exercise price or zero, whichever is greater
- If either of these bounds are violated, there is an arbitrage opportunity

A Simple Model

- An option is “in-the-money” if the payoff is greater than zero
- If a call option is sure to finish in-the-money, the option value would be
- C0 = S0 – PV(E)
- If the call is worth something other than this, then there is an arbitrage opportunity

What Determines Option Values?

- Stock price
- As the stock price increases, the call price increases and the put price decreases
- Exercise price
- As the exercise price increases, the call price decreases and the put price increases
- Time to expiration
- Generally, as the time to expiration increases both the call and the put prices increase
- Risk-free rate
- As the risk-free rate increases, the call price increases and the put price decreases

What about Variance?

- When an option may finish out-of-the-money (expire without being exercised), there is another factor that helps determine price
- The variance in underlying asset returns is a less obvious, but important, determinant of option values
- The greater the variance, the more the call and the put are worth
- If an option finishes out-of-the-money, the most you can lose is your premium, no matter how far out it is
- The more an option is in-the-money, the greater the gain
- You gain from volatility on the upside, but don’t lose anymore from volatility on the downside

Employee Stock Options

- Options that are given to employees as part of their benefits package
- Often used as a bonus or incentive
- Designed to align employee interests with stockholder interests and reduce agency problems
- Empirical evidence suggests that they do not work as well as anticipated due to the lack of diversification introduced into the employees’ portfolios
- The stock just is not worth as much to the employee as it is to an outside investor

Equity: A Call Option

- Equity can be viewed as a call option on the company’s assets when the firm is leveraged
- The exercise price is the value of the debt
- If the assets are worth more than the debt when it comes due, the option will be exercised and the stockholders retain ownership
- If the assets are worth less than the debt, the stockholders will let the option expire and the assets will belong to the bondholders

Capital Budgeting Options

- Almost all capital budgeting scenarios contain implicit options
- Because options are valuable, they make the capital budgeting project worth more than it may appear
- Failure to account for these options can cause firms to reject good projects

Timing Options

- We normally assume that a project must be taken today or foregone completely
- Almost all projects have the embedded option to wait
- A good project may be worth more if we wait
- A seemly bad project may actually have a positive NPV if we wait due to changing economic conditions
- We should examine the NPV of taking an investment now, or in future years, and plan to invest at the time that produces the highest NPV

Example: Timing Options

- Consider a project that costs $5000 and has an expected future cash flow of $700 per year forever. If we wait one year, the cost will increase to $5500 and the expected future cash flow increase to $750. If the required return is 13%, should we accept the project? If so, when should we begin?
- NPV starting today = -5000 + 700/.13 = 384.16
- NPV waiting one year = (-5500 + 800/.13)/(1.13) = 578.62
- It is a good project either way, but we should wait until next year

Managerial Options

- Managers often have options after a project has been implemented that can add value
- It is important to do some contingency planning ahead of time to determine what will cause the options to be exercised
- Some examples include
- The option to expand a project if it goes well
- The option to abandon a project if it goes poorly
- The option to suspend or contract operations particularly in the manufacturing industries
- Strategic options – look at how taking this project opens up other opportunities that would be otherwise unavailable

Warrants

- A call option issued by corporations in conjunction with other securities to reduce the yield
- Differences between warrants and traditional call options
- Warrants are generally very long term
- They are written by the company and exercise results in additional shares outstanding
- The exercise price is paid to the company and generates cash for the firm
- Warrants can be detached from the original securities and sold separately

Convertibles

- Convertible bonds (or preferred stock) may be converted into a specified number of common shares at the option of the bondholder
- The conversion price is the effective price paid for the stock
- The conversion ratio is the number of shares received when the bond is converted
- Convertible bonds will be worth at least as much as the straight bond value or the conversion value, whichever is greater

Valuing Convertibles

- Suppose you have a 10% bond that pays semiannual coupons and will mature in 15 years. The face value is $1000 and the yield to maturity on similar bonds is 9%. The bond is also convertible with a conversion price of $100. The stock is currently selling for $110. What is the minimum price of the bond?
- Straight bond value = 1081.44
- Conversion ratio = 1000/100 = 10
- Conversion value = 10*110 = 1100
- Minimum price = $1100

Other Options

- Call provision on a bond
- Allows the company to repurchase the bond prior to maturity at a specified price that is generally higher than the face value
- Increases the required yield on the bond – this is effectively how the company pays for the option
- Put bond
- Allows the bondholder to require the company to repurchase the bond prior to maturity at a fixed price
- Insurance and Loan Guarantees
- These are essentially put options

Quick Quiz

- What is the difference between a call option and a put option?
- What is the intrinsic value of call and put options and what do the payoff diagrams look like?
- What are the five major determinants of option prices and their relationships to option prices?
- What are some of the major capital budgeting options?
- How would you value a convertible bond?

Options - Outline

- Derivative Security
- Definitions
- Examples
- Buy a Call
- Buy a Put
- Write a Covered Call
- Pricing
- factors on which price depends
- Short-cut to approximate price
- Additional topics
- Using Options as Hedge
- 12 ways to buy S& P 500

Derivative Instruments

- Some contracts give the contract holder either the obligation or choice to buy or sell a financial asset.
- the price of those contracts derive its value from the price of the underlying financial asset
- so, these contracts are called derivative instruments
- options
- futures, forwards
- swaps, caps, floors, collars

Derivatives

- Reasons to use
- reduce risk
- change nature of one’s financial exposure
- reduce transaction costs
- classified on basis of underlying asset, index or currency to which linked
- some standardized contracts are traded on exchanges while others are customized, OTC
- history - 1670 in Amsterdam; 1848 CBT, 1865 futures, 1972 currency futures; 1975 financial futures
- 1900s Put & Call Dealers Association, 1973 CBOE, 1975 AMEX & PHLX

Definitions - Options

- buy = own
- write = sell
- option price = option premium
- exercise price = strike price
- expiration date = maturity
- call
- put
- American
- European
- covered vs. naked
- Exchange-traded vs. OTC (dealer)

Options

- Buy an option- own an option
- have Right (but NOT obligation) to buy or sell an item at a predetermined price (strike price) until some future date
- write or sell an option - then if the owner (holder) of option wants to exercise it, you are obligated to sell or buy the item
- if have a mortgage on your home, you own an option - the right to prepay your mortgage.
- When signed contract to take mortgage - bought option -right to borrow at that interest rate, but if rates fell, could walk away & only lose commitment fee; nonrefundable deposit to lock in price & deal on home sale - buyer can walk away & lose only deposit.

Options vs. Futures Contracts

- The option at the strike price exists over the period of time (if American) , at a given date (if European).
- The buyer of an option pays the seller (writer) a premium which the writer keeps - whether or not the option is ever exercised.- in futures no party compensates the other
- The option does not have to be exercised by buyer; it can be sold, if it has a market value, before expiration date.- in futures both are obligated to perform.
- Option buyer sets price at which can acquire underlying but can take advantage of more favorable price; in futures - price is locked in
- Gains and losses are unlimited with futures contracts; with options the buyer can lose only the premium and the commission paid.
- With futures both put up margin, with options- only writer posts margin; writer position is marked to market; both in futures

Calls and Puts

- Call option -- buyer has the option to buy an item at the strike price.
- Put option -- buyer has the option to sell an item at the strike price.

It is vital to know who has the Right to transact vs. who may be Obliged to transact in order to determine the direction of currency flows at expiry

16 -33

Covered and Naked Options

- Covered option -- writer either owns the security involved in the contract or has limited his or her risk with other contracts.
- Naked option -- writer does not have or has not made provision to limit the extent of risk.

American vs European Options

American - the option can be exercised at any time before expiration or maturity

The option may be exercised at any time up to and including the expiry date and the funds will be transferred with spot value from exercise

European - the option can only be exercised on the expiration or maturity date

The option may only be exercised on the expiry date and the funds will be transferred on the value date

Option Terminology

- Buy - Long
- Sell - Short
- Call
- Put
- Key Elements
- Exercise or Strike Price
- Premium or Price
- Maturity or Expiration

Market and Exercise Price Relationships

In the Money - exercise of the option would be profitable

Call: market price>exercise price

Put: exercise price>market price

Out of the Money- exercise of the option would not be profitable

Call: market price<exercise price

Put: exercise price<market price

At the Money - exercise price and asset price are equal

Call Option Example (pretend it is Nov 2002)

- Suppose you pay $3 for an Dec 50 call on XYZ stock.
- Since each option is to buy 100 shares of stock, you would pay $300.
- XYZ Dec 50 $3
- 100 x $3 = $300
- Stock Expiration Strike Premium
- Month Price
- By paying $300, the premium, you have the right to purchase 100 shares of XYZ at any time (American) before the option expires on the third Friday in December (12/20/02) at $50 per share. (Can buy an option up to 3 years but most are for 3 months or less.)
- Like the buyer concluding he/she wants to purchase the house, the option buyer can exercise the option and purchase the stock at $50 per share if the stock trades above $50 per share (e.g., $60).

illustration

- Stock price
- 60 - ---
- 50- ___________________ strike price
- 40 - expiration date
- If the stock falls below $50, call option buyer has no obligation to purchase the stock; he can let the option expire worthless and the only loss is the $300 option premium (analogous to the deposit on the home).

Summary, so far

- Discussed premium, strike price, expiration month
- what are reasons to buy a call
- Leverage
- Limitation of risk to premium paid
- Buying call options gives you the ability to participate in stock movement by paying only option premium
- Leverage - suppose buy 100 shares at $50 per share; pay $5000. If XYZ stock rises to $60 per share, may be able to sell stock for $6000 (excluding commissions & taxes) - profit = $1000 or 20%

Leverage

- Suppose instead of buying stock, you purchase an XYZ call for $300.
- Then if the stock rises to $60 per share, pay $5000 and immediately sell for $6000.
- Make $1000 Profit
- less 300 Option Premium
- $ 700 Profit on $300 investment
- or 233 percent - that’s leverage
- By buying the stock option can participate in the price movement for three months.
- Of courseleverage works both ways. If the stock price falls, can have a loss on the options.
- But the most you can lose is the premium you paid of $300.

How is option profitable?

- Suppose the stock rises to $52.
- Then you can buy for $50 and sell for $52, making $200 on the stock transaction.
- However, since you paid $300 premium for the option, your net position is a $100 loss.
- If you had bought the stock (without buying the option), then you would have made $200. Buying the option was like buying an insurance policy.
- For the call option purchase to be profitable, the stock price must rise enough in value to cover the cost of the option premium plus any transaction costs.
- What if the stock price instead of rising a small amount, actually fell (as the stock market did through much of 2002) to e.g., $43 per share.
- Then have a loss of $7 per share, or $700 on 100 shares.

Risk

- If XYZ is trading below $50, you will not exercise the option, but will let it expire worthless and forfeit the premium of $300. Hence there is some risk with options.
- With stock ownership get full upside and downside potential. With option ownership, you can participate in the potential profit, but the loss is limited to the option premium paid.
- Stock price Stock profit Option profit
- 80 + 30 + 27
- 70 + 20 + 17
- 60 + 10 + 7
- 50 0 - 3
- 40 - 10 - 3
- 30 - 20 - 3
- 20 - 30 - 3
- 10 - 40 - 3
- Do you have to exercise the call? No. Can sell the call because options are traded on an exchange. Can sell for < or > price paid, depending on where the stock is trading & how long until expiration.

Put Options

- If the market is falling, there are options to provide protection. The put option is similar to the call option in that we have an exercise price, an expiration month and a premium.
- If you buy a put, you have the right to sell a stock at a particular price before a particular date.
- For example, you may buy an Dec 45 put on ZYX for $1.
- Stock Expiration month Strike price Premium
- ZYX June 45 $1
- 100 x $1 = $100
- If ZYX falls, say to $40, prior to the third Friday in Dec., then the option has a value of approximately $5, representing the difference between the strike price and the current market price. In this example, owning the put gives the holder the right to sell the stock at $5 above the market price. If the market is above $45, allow the option to expire and lose only the premium that you paid.

Using puts as protection

- Suppose you bought the stock at $50 and the price has risen to $65. You could sell it and take a profit, or if you are worried about the next market “correction” but would like to hold the stock if the market does not drop, you could buy a put at $60 or $65 and then exercise that put if the market falls. If the market continues to rise, you can hold onto the stock. The gains on the stock will be reduced by the premium paid for the put. But the most money that you can lose is the premium that you paid. You can think of the purchase of the option as analogous to the purchase of term insurance.
- If you want to sell the option before it expires, you can because it is traded on an exchange during trading hours. But if you sell it, you no longer have the protection.

Writing covered calls

- A holder of a stock can sell calls against the stock he/she owns; this is known as covered call writing. The buyer of the call you wrote has the right to purchase the stock at the strike price.
- The seller has the obligation to sell the stock at the strike price should the buyer exercise.
- The holder pays a premium for this right; seller receives this premium:
- Buyer Seller
- right to buy obligation to sell
- pays premium receives premium
- Therefore limits the appreciation seller can receive
- Let us look at an example. You bought XYZ stock at $40; it has now risen to $48.
- Suppose you sell a $50 Dec. call at $3, then you agree to sell 100 shares at $50.

Covered call example

- You bought XYZ stock at $40; it has now risen to $48.
- Suppose you sell a $50 Dec. call at $3, then you agree to sell 100 shares at $50.
- If the stock price remains below $50 from now until Dec. 20, you will keep the $300 premium and still hold the stock. Once the option expires, you are no longer obligated to sell the stock. If the stock price drops, then you have reduced your loss by the $300 premium you received. But if the stock price rises above $50 and the buyer exercises, then you must sell at $50. Since you already received the $300 premium, you effectively are selling at $53. But remember no matter how much the stock price has risen above $50, you are obligated to sell at $50.
- If you no longer want to be obligated to sell the stock, you can buy back the call. But the price you pay to buy the call may be higher than the original premium paid to sell the call. If the price you pay to buy is > price at which you sold, you have a loss on the options transaction.

Covered call writing

- Why would someone buy the calls that you are writing?
- They may be more optimistic about the stock potential or have a different strategy or different objective.
- With all the strikes available, how know at which one to sell?
- The higher the strike price, the less premium you will receive, but also the less likely that you have to sell the stock.
- With low strike, receive more premium, providing greater downside protection, but also more likely obligated to sell.
- Choosing the strike price depends on your outlook for the stock.
- Will you still collect dividends if sell calls against stock you own?
- Yes, so long as you still own the stock on the ex-dividend date.
- What are appropriate market conditions for covered call writing?
- One where market is trending moderately higher. If continually sell calls against stock that you own in such a market, you will get a greater consistency of return. In exchange for that consistency, you give up the right to benefit if the stock makes a big upside move.

Option valuation

- The option premium can be thought of as representing the Expected Value (EV) of the Pay-Off at Maturity, discounted to the present
- It is simply a function ofProbability
- The Expected Value(EV) of an event is defined as being...

EV = The Pay-off X The Probability of of an Event achieving it

- The Option Premium can be thought of in 3 different ways
- Expected value of the pay-out (e.g., examples of lottery ticket price, dice games – see notes to this slide and slides at end of this ppt presentation)
- Intrinsic (ITM) plus extrinsic (time) value
- intrinsic – how in the money the strike is, relative to the market;intrinsic value can never be negative ( > 0)
- extrinsic – time until expiration, how far OTM it is plus volatility and the relationship between short term interest rates and dividend on the underlying stock
- Estimated costs of hedging

Option

- premium (price pay) depends on:
- strike (exercise) price-
- market price (market - strike) = intrinsic value (intrinsic value = economic value of exercising immediately)
- time until expiration = time value
- short term interest rates
- volatility
- anticipated cash payments on the underlying (div.)
- price of call option < stock price but never < 0 or payoff to immediate exercise. If stock is worthless, option is worthless. As stock price grows very large, option price approaches stock price minus PV of exercise price

Option Pricing

- Effect of an increase of the factor on
- Factor Call Price Put Price
- current price of underlying + -
- strike price - +
- time to expiration of option + +
- expected price volatility + +
- short-term interest rate + -
- anticipated cash payments - + (dividends)

Examples of stock market price & option premia for call options

5/5-5/9/99 AT&T Stock Price Strike Price Premium

- In-the Money 61 7/8 60 4 5/8
- At the money 60 60 3 1/2
- Out of the money 59 60 2

CISCO Stock price 12.50

11/12/02 CISCO Exp. month Strike (exer.price) Prem.

- In-the Money 12.50 Nov. 10 2.53
- At the Money 12.50 Nov. 12.50 0.24
- Dec. 12.50 0.80
- Out of the Money 12.50 Dec. 15 0.05
- Jan. 15 0.25

At the money- the strike price of the option is equal to the market price

In the money - the strike price of the option is more favorable than the market price

Out of the money - the strike price of the option is less favorable than the market price

Effect of implied volatility

- As implied volatility rises:
- The distribution of possible outcomes broadens
- The options’ value rises
- Delta tends towards 50% - i.e. the certainty of exercise/non-exercise decreases for all options
- Higher implieds increase the amount of extrinsic value in the option price, but the option will have higher theta (and vice versa for lower implied volatility)

EURO - Historic vs. Implied Volatility

Short cut-If not have Black Scholes pricing model & tables

- For companies with average stock price volatility, and an average dividend yield, the Black-Scholes option value is typically 33 percent of the price of the optioned stock on the date of the grant (e.g., $33 for a $100 stock). The option price percentage may be above 50 percent for highly volatile stocks with low to no dividends or 25 percent or less for less volatile stocks with higher dividend yields.
- Companies are giving stock options to employees as a benefit. For our simple example, say you receive 10-year options on stock that has a market value of $1000 today. Assuming that the stock will grow 10 percent a year (based on historical performance) over the decade, the optioned shares will be worth approximately $2600 [$1000 x (1.10)10] after 10 years. Then subtract the $1000 price you paid to buy the stock by exercising the option from the $2600 value, yielding a $1600 gain. Next add the 10-year Treasury yield (approximately 5.5 percent) to an equity risk premium of 7.5% (obtaining 13%). Finally find the present value of the $1600 gain by using the 13% discount rate [1600/(1.13)10]. That works out to $471 or 29%. (This example is in Business Week, July 22, 1996, p. 87.)

12 ways to buy S&P 500 index

- Buy every one of the 500 stocks in the index
- buy one futures contract on the S&P 500
- negotiate a forward contract on S&P 500
- buy a call option on S&P 500
- buy all 500 stocks + buy a put option - same benefit as buying a call- ensure against drop; benefit from rise
- buy a bond convertible into S&P 500
- buy a structured note with an interest rate tied to the return on the S&P 500
- buy an equity -linked CD tied to the S&P 500
- buy a GIC with same linkage
- enter into an equity swap-pay LIBOR & receive rate of return on S&P 500
- buy a UIT that holds S&P 500 example SPDR
- Buy an exchange traded fund, e.g. SPIDER

hedging

- Besides protecting the price appreciation in stock by buying a put option, options may be used to hedge:
- future issuance of a debt security
- rollover cost of commercial paper issuance
- return available on money market instruments
- principal value of fixed income portfolios
- cost of money market deposit accounts of banks
- net interest margin of banks
- return available on short-term portfolios
- principal value of long-term fixed income assets.
- Besides Hedgers, other market participants are Speculators and Arbitrageurs
- Mortgage Banks use options to hedge the mortgage pipeline.

Mortgage banks use options to hedge pipeline

- Let us return to the house that you wished to buy. After you agreed upon a price, you probably went to a bank or mortgage bank to lock-in the mortgage rate. Suppose it is May 1999.
- Suppose the mortgage banker is making two-month loan commitments at 7.0% and is charging 2 ½ points as a commitment fee. The loans can be securitized into 6.5% FNMA’s after taking out 50 basis points for servicing.
- The loans committed to in May will be closed in July and the securities delivered in August. What will the spot price of FNMA 6.5% be in August? We do not know. However, because there are 60-day forward commitments, we do have forward prices for forward delivery of the FNMA’s. The current market level for FNMA 6.5% for August delivery is 99:13.
- Suppose the mortgage banker thinks that prices might fall (rate rise) between now and August. Then he would like to lock-in the forward price of 99:13 (or he may wish to know with certainty what his future profit will be).

Mortgage bank pipeline hedge

- The simplest hedge would involve selling the product for settlement in August at the forward price of 98:27. (Or sell Treasury bond futures.) (Using data from WSJ 5/7/99, p. C16)
- But he has another concern besides rising rates and falling prices. What if rates fall? Home buyers will re-negotiate a lower mortgage rate on their loan or switch to a competing mortgage banker. Then he will have a high fall-out rate. If he had engaged in a forward sale, he would have sold a 6.5% FNMA and have no product to deliver.
- What can he do to protect himself from the fallout if rates drop?
- Then, instead of selling forward, . . . ..
- He can buy an option -- in this case a put option to sell FNMA 6.5% in August at the August strike price of 98:27. Remember this option gives him the right but not the obligation to sell the FNMA 6.5% at 98:27. If he does not have the loan to securitize, he will not sell it. Also, if the market has rallied, the price for FNMA 6.5% in August will be above 98:27. He would then have let the option expire worthless and sell his FNMA 6.5% at the new market level -- provided (of course) that his 7.0% loan has closed

Hedging mortgage pipeline

- . (For production with a low probability of realization, he can hedge with options on Treasury bond futures with transaction costs of 1/64 to 2/64; or with options on T-bonds -- on p. C 16 - CBOE Futures prices & interest options 30 year T-bonds WSJ 5/7/99.)
- His loss in this transaction is limited to the fee he paid for the option, i.e., 1 and 5/32.
- For an additional 1:05 in cost (paid for by the commitment fee the mortgage banker has received), the put option provides downside price protection in a rising rate scenario. However, in a volatile falling rate environment, the put strategy significantly outperforms a forward sale hedge by eliminating the need to pair off loan fallout.
- In contrast to the forward sale, the put option allows the mortgage banker to lock in profits on the mortgage loan pipeline over a wide range of mortgage rates and price levels. The result is a stable level of profitability on the pipeline with a minimal amount of risk.

Put-Call Parity

- Relationship between price of a call option & price of a put option on the same underlying instrument, with the same strike price & same expiration date
- Put option price - Call option price = PV(strike price) + PV(cash distribution) - Price of underlying asset
- put-call parity for European options - approx for Amer
- example- buy XYZ at a price of $50; sell a call at a premium of $3 and buy a put at a premium of $2. Strategy - long XYZ, short the call; long the put. No matter what XYZ’s price at expiration, strategy produces a profit of $1
- this strategy cannot exist in an efficient market.

Put-call parity example

- In implementing strategy to capture the $1 profit, the actions of market participants will have > 1 of the following consequences that will eliminate the $1 profit:
- price of XYZ will increase
- call option premium will drop or
- put option premium will rise
- call & put option price will tend to be equal if ignore time value of money (financing cost, opportunity cost, cash payments & reinvestment income)
- if underlying asset makes cash distributions, then: Put option price - Call option price = PV(strike price) + PV(cash distribution) - Price of underlying asset
- if put-call parity does not hold, then arbitrage opportunities exist.

Black Scholes

- EX = exercise price of option; PV(EX) is calculated by discounting by the risk free interest rate (Treasury-bill) rfPV(EX) = EXe - rf
- t = number of periods to exercise date
- P = market price of stock now s = standard deviation per period of (continuously compounded) rate of return on stock.
- Using option pricing tables, you can price the options in four steps: 1. Multiply the standard deviation of the proportionate changes in the asset’s value by the square root of time to the option’s expiration.
- 2. Calculate the ratio of the asset value to the present value of the option’s exercise price.
- 3. Look at a table that indicates Call Option Values as a Percent of Share Price. On the horizontally the table will show Share Price Divided by PV (exercise price) while on the vertical axis the table will show the Standard Deviation Times the Square Root of Time (these are the values you determined in steps 1 and 2). The option delta is in another table which has the same horizontal and vertical axes.

Pricing the Option - Black Scholes

- Black and Scholes derived a formula (used by options traders) to value the options by setting up an option equivalent by combining stock investment and borrowing. The net cost of buying the option equivalent must equal the value of the option. The number of shares that are needed to replicate one call is called the hedge ratio or option delta.
- option delta = Spread of possible option prices
- Spread of possible share prices
- Black Scholes derived a formula to compute the fair (or theoretical) price of a European call option on a non-dividend paying stock that can be interpreted as follows:
- Value of option = [delta x share price] - [bank loan]
- [N(d1) x P ] - [N(d2) x PV(EX)]
- where d1 = log [P/PV(EX)] + sr t
- sr t 2
- d2 = d1 - s r t
- N(d) =cumulative normal probability density function

Binomial Pricing Model

- Payoff from owning a call option can be replicated by purchasing the asset underlying the call & borrowing funds; price of option is < cost of creating the replicating strategy.
- Derive a one-period binomial option pricing model: construct a portfolio consisting of
- long position in a certain amount of the asset &
- a short call position in the underlying asset
- amount of underlying asset purchased is such that the position will be hedged against any change in the price of the asset at the expiration date of the option - the portfolio will produce the risk-free interest rate.
- This portfolio is called a hedged portfolio

Binomial Option Pricing Model

- Assume XYZ has a current market price of $80 & only 2 possible future values 1 year from now:
- State Price
- 1 $100
- 2 $ 70
- Assume there is a call option on XYZ with a strike price of $80 (same as market price) that expires in 1 year. Suppose an investor forms a hedged portfolio by acquiring 2/3 of a unit of XYZ & selling 1 call option. The 2/3 of unit of XYZ is the hedge ratio.
- If price of XYZ 1 year from now is $100, buyer of call will exercise it. So investor must deliver 1 unit of XYZ in exchange for strike price of $80. Since investor has only 2/3 of unit of XYZ, he must buy 1/3 of XYZ at $33 1/3 (1/3 of market price of $100). So outcome will be:
- $80 - $33 1/3 + Call option premium = $46 2/3 + Call option premium
- If price of XYZ in one year = $70, option buyer will NOT exercise; Investor will own 2/3 of 1 unit of XYZ. At price of $70, value of 2/3 of unit = $46 2/3. Outcome is then value of XYZ plus Call option premium = $46 2/3 + Call option premium.

Binomial options pricing model

- The portfolio consisting of a short position in the call option & 2/3 of unit of XYZ generates outcome that hedges changes in the price of the asset; hence, hedged portfolio is riskless. This will hold no matter what is price of call, which affects only magnitude of outcome.
- Deriving the hedge ratio
- S = current asset price
- u = 1 + % change in asset’s price if price rises in next period
- d = 1 + % change in asset’s price if price falls in next period
- r = risk free one-period interest rate (risk free rate until expiration)
- C = current price of Call option
- Cu = intrinsic value of call option if asset price rises
- Cd = intrinsic value of call option if asset price falls
- E = strike price of call option
- H = hedge ratio
- in our example, u = 1.250 = $100/$80 d = 0.875 = $70/$80
- H = 2/3

Binomial Option Pricing Model

- Investment made in hedged portfolio = cost of buying H amount of the asset minus the premium received by selling call option. Thus cost of hedged portfolio = HS - C
- Payoff of hedged portfolio at end of 1 period = value of H amount of asset purchased minus call option premium:
- uHS - Cu if asset price rises 1.250 H $80 - Cu = $100 H - Cu
- dHS - Cd if asset price falls 0/875 H $80 - Cd = $ 70 H - Cd
- no matter the state, want payoff of hedged portfolio to be the same :
- want uHS - Cu = dHS - Cd so
- H = Cu - Cd
- (u - d) S
- Cu = Max [0, (uS-E)] = Max[0, ($100 - $80)] = $20 if XYZ’s price rises
- Cd = Max[0,(dS-E)] = Max[0, ($70-$80)] = $0 if XYZ’s price falls
- so H = $20 - $0 = 2
- (1.25 - .875)$80 3

Binomial Options pricing model

- now we derive a formula for the call price. (diagrammed in figure 11-5, p. 270)
- Since hedged portfolio is riskless, the return must = the risk free rate. So one period from now, we should have: (1 + r) (HS - C)
- payoff will be same if price of XYZ rises or falls, so use uHS - Cu
- thus, (1 + r) (HS - C) = uHS - Cu
- solve for C
- C = ( 1 + r - d)( Cu ) + ( u - 1 - r)( Cd )
- ( u - d) (1 + r) ( u - d) (1 + r)
- where u = 1.250, d = 0.875, r = .10, Cu = $20, Cd = $0
- C = ( 1 - 0.10 - 0.875) $20 + ( 1.25 - 1 - 0.10 ) $0 ( 1.25 - 0.875) (1 + 0.10) (1.25 - 0.875) (1 + 0.10)
- C = $10.90

Binomial Option Pricing Model

- This approach is oversimplified, since we assume only 2 possible future states. But we can extend the procedure by making the periods smaller so that we can calculate a fair value for the option.
- The book shows the extension to a 2 period model.
- We will leave the detail of this model as well as the Black Scholes model to a future course.
- In a more advanced course, we will also discuss option strategies

Exotic options

- OTC options can be customized in any manner desired by institutional investors; the more complex options are called exotic
- alternative option - payoff that is the best independent payoff of 2 distinct assets
- out-performance option - option whose payoff is based on the relative payoff of 2 assets at the expiration date.

WSJ 5/7/99

- CBOE Interest Options p. C16
- Calls Puts
- Strike Price May Jun May Jun
- 55 3 3 1/2 5/16 1/4
- 57 1/2 1 1/16 3/4 1 5/16
- 60 1/4
- total call volume 450
- total put volume 578
- total call open interest 2,655
- total put open interest 868
- p. C16 Options on Futures

Listed Options Quotations WSJ 11/13/02 p. C14 will update with WSJ in class

- Most Active

Option Strike Exp prem. price stock call/put volume

- Intel 17.50 Jan 1.90 17.79 call 18,942
- Cisco 12.50 Nov 0.60 12.87 call 17,402
- Ph Mor. 40 Nov 2.80 37.03 put 14,251
- Index Options Trading
- S&P 500 600 Dec 0.35 889.95 put
- S&P 800 Dec 92 call
- S&P 800 Dec 9.50 put
- S&P 800 Jan 15 put
- S&P 900 Dec 24 call
- S&P 900 Dec 41 put
- LEAPS - Long Term Options - Long-term Equity AnticiPation securities that are similar to conventional options except that they are long-term options with maturities of up to 2 1/2 years.

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