Derivatives - Options Ch 20 & 21

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Derivatives OptionsChapters 20 & 21Finance 367

DerivativesDerivatives securities or just Derivatives are securities whose prices are derived from the prices of other securities. Also called Contingent Claims because their payoffs are contingent on the prices of other securities.

Derivatives can be powerful tools for hedging, but also for speculation.

Major types of derivatives:OptionsFuturesSwaps

2015 J. David Miller

Option ContractsAn option is a right or obligation to receive or deliver an underlying security at a specified price on or before a specified date.

Call option gives the right to buy (bullish)Put option gives the right to sell (bearish)

Strike or Exercise Price is the price set for calling (buying) or putting (selling) an asset. This is a factor in determining whether the option has any intrinsic value.

The cost of buying the option is called the Premium. The premium is quoted as the price per share. Usually 100 opitons per options contract.


Option Contracts (contd)The person who sells the option to someone else is called the option writer. You would say that a person writes an option.

The person buying the option will pay the premium to the writer of the option.

The option premium can be thought of as the fee that the buyer pays the writer for giving them the option.

All option contracts must have both a buyer and a writer.


Option Contract ExampleIt is currently April 1. An investor purchases a Call Option on IBM stock that will expire on June 1. The premium on the option is $3 and the Strike Price or Exercise Price is $75.

This investor now has the right to purchase 100 shares of IBM stock from the option writer for $75 per share. The investor has this option until June 1.

If the investor does not exercise (or sell) the contract on or before June 1, it will expire and become worthless.


Option Contracts ValueThe value of an option consists of two parts, the Intrinsic Value and the Time Value.

Intrinsic Value exists when the market price exceeds the strike price (for a call option) or is less than the strike price (for a put option).

For example, the market price is $80 and the Strike Price is $75. The option buyer will exercise the option and buy 100 shares for $75 per share and then sell them at the market price of $80 per share.

An option that has intrinsic value is said to be in the moneyAn option does not have intrinsic value is said to be out of the moneyOption where Strike Price = Market Price is at the money


Option Contracts Value (contd)The Time Value of an option contract is a value that represents the time until the option expires. The longer until expiration, the greater the time value of an option.

Option Premium = Intrinsic Value + Time ValueOption contracts can be traded up to expiration.Expiration dates are normally on the Saturday following the third Friday of the exercise Month

American Options can be exercised before or at expiration.European Options can only be exercised at expiration. 2015 J. David Miller

Options TradingA large percentage of options are traded on organized exchanges. These are standardized contracts and provide standardized maturities.

Other the counter options may be tailored to the needs of a trader, but at a higher cost compared to exchange traded options.

The Chicago Board Options Exchange and the International Securities Exchange Options Clearing Corp. guarantee performance.

Options traded on these exchanges are traded directly with the Option Clearing Corp. The OCC effectively matches option buyers with option writers to eliminate its risk. 2015 J. David Miller

Options Trading (contd)Because the exchange guarantees the performance of the contracts, it requires the option writer to provide margin.

More margin money is required if the option being sold is in the money, because it is more likely that the option will be exercised.

The margin requirements can be satisfied if the option writer owns the underlying shares and they are being held by a broker. 2015 J. David Miller

Options Quotes


Other Listed OptionsAn Index Option is a call or put option based on a stack market index such as the S&P 500 or New York Stock Exchange Index. There are options that cover many broad market indexes as well as industry specific indexes.

The call and put writer does not have to deliver the index or the stocks in the index, if the option is exercised. Cash settlement is used. The writer pays the buyer the difference between the index value price and the exercise price.


Other Listed Options (contd)Futures Options holder has right to buy or sell a futures contract using a futures price as the exercise price. Cash settlement is used.

Foreign currency options quoted in dollar per unit of foreign currency.

Interest Rate options traded on fixed income securities like Treasuries, CDs. Options on several interest rate futures are also traded. 2015 J. David Miller

Corporation Issued OptionsCompanies can issue securities whose value is derived from the value of the stock price. While these are not derivatives, they have many similarities.

Warrant long term option to buy stock at a fixed price

Right short term option to buy at a fixed price

Bond Call Option option written by the bond investor (who demands a higher coupon and yield compared to an equivalent straight bond) and is held by the issuer (Company)

Convertible Bond option can be exercised to trade the bond for shares of the issuing company 2015 J. David Miller

Value of Call Options at ExpirationThe value of an call option = ST X if ST > X = 0 if ST < or = X

Where ST is the Stock Price and X is the Strike Price

The payoff to the holder of call options cannot be negative because the option will not be exercised when the stock price is less than the strike price. 2015 J. David Miller

Call Option with $80 Exercise PriceThe value at expiration to a call holder with $14 premium

Income 2015 J. David Miller

Long Call ExampleAn investor purchases a July Call on Dow Chemical stock with a strike (exercise) price of $85. The investor pays a premium of $1.37. If Dow Chemical rises to $91.37 and the investor thinks that it will go no higher? What will be her profit?

Exercise the call and buy the stock at $85 per share. Sell the stock in the market for $91.37.

Profit = $91.37 $85 - $1.37 = $5 per share 2015 J. David Miller

Long Call Example IIAn investor purchases a July Call on Dow Chemical stock with a strike (exercise) price of $85 per share. The investor pays a premium of $1.37 per share.

At what price for Dow Chemical stock, will the buyer of the option break even?

Strike Price + Premium = $85 + $1.37 = $86.37

Exercise option and buy stock at $85.00. Sell at market for $86.37.

Profit = $86.37 - $85 - $1.37 = 0 2015 J. David Miller

Call Option with $80 Exercise PriceThe value at expiration to a call writer with $14 premium


Value of Put Options at ExpirationThe value of an put option = 0 if ST > or = X = X - ST if ST < X

Where ST is the Stock Price and X is the Strike Price

The payoff to the holder of put options cannot be negative because the option will not be exercised when the stock price is greater than the strike price. 2015 J. David Miller

Put Option with $80 Exercise PriceThe value at expiration to a put holder with $14 premiumIncome


At price of $0 on horizontal axis, the option is worth $80 on vertical axis. As price increases, (moving toward right on horizontal axis), the value of the option decreases. At 80-14, the profit on the vertical axis is $0.20Long Put ExampleAn investor purchases a March Put on IBM stock with a strike (exercise) price of $120. The investor pays a premium of $2.15 per share. If IBM falls to $112.45 and the investor thinks that it will not fall anymore, what should he do and what is his profit?

Buy the stock at $112.45 at the market and exercise the put option to sell the stock for $120.

Profit = $120 $112.45 - $2.15 = $5.40 2015 J. David Miller

Long Put ExampleAn investor purchases a March Put on IBM stock with a strike (exercise) price of $120. The investor pays a premium of $2.15. If IBM rises to $128 just before expiration, what should the investor do and what will be his profit?

The stock price is above the strike price, so the investor will lose money if he exercises the option. He will let the option expire worthless.

Loss = $2.15 for the premium paid. 2015 J. David Miller

Short Put ExampleAn investor believes that Microsoft stock will not go down in the next several months. She writes a May Put on Walgreens stock with a strike (exercise) price of $100. The investor receives a premium of $2.75 per share. If Walgreens rises to $108 what will be the profit (loss) to the writer of the option?

The stock price is above the strike price, so the option will expire worthless. The investors profit will be the premium of $2.75


Short Put Example IIAn investor believes that Microsoft stock will not go down in the next several months. She writes a May Put on Walgreens stock with a strike (exercise) price of $100. The investor receives a premium of $2.75.

If the stock price fell to $89 and the buyer of the option exercised, what will be the gain(loss) to the option writer? Option seller will buy the stock for $100 from the purchaser and then sell it at market for $89.

The gain will be $89 - $100 + $2.75 = -$8.25 or a loss of $8.25 per share. 2015 J. David Miller

Options vs. Holding StockBuying call options is a bullish strategy and buying put options is a bearish strategy. Writing puts is also a bullish strategy, whereas writing calls is a bearish strategy.

Buying a call option can be viewed as substitute for the purchase of the shares of a stock.

A comparison of bullish strategies will make this clearer.

Because purchasing one call option contract allows us to control the equivalent of 100 shares of stock, we have several investment choices to make.


Options vs. Holding Stock IISuppose we believe that a $90 stock is undervalued and it will rise over the next 6 months. However, it could also fall in price, so we must consider our investment options. We have $9,000 to invest. We are considering a six-month maturity call option with exercise price of $90 that sells for $10.

Strategy A: Invest entirely in stock. Buy 100 shares of stock at $90 per share

Strategy B: Invest entirely in at-the-money options. Buy 900 calls, each selling for $10. This is 9 contracts.

Strategy C: Purchase 100 call options for $1,000. Invest the remaining $8,000 in 6-month T-bills to earn 2% interest. 2015 J. David Miller

Options vs. Holding Stock IIIWhen the option expires in six-months, the different investment options will be worth:

Each of these portfolios involves the same $9,000 initial investment.Note: these are the payout values but not your profit/loss values. You must consider premiums to find your P&L.


Options vs. Holding Stock IVThe rates of return on the three investment options are:

Note: these are the payout values but not your profit/loss values. You must consider premiums to find your P&L.


Options vs. Holding Stock VNote:The call option acts like a leveraged investment in the stock

The option can act as insurance. Strategy C cannot be worth less than $8,160, but has upside potential

While options can be used to speculate, investors can use them to reduce risk.


Call and Put Comparison Max GainMax LossBuy CallUnlimitedPremiumSell (write) CallPremiumUnlimitedBreakevenStrike Plus PremiumMax GainMax LossBuy PutStrike Price less PremiumPremiumSell (write) PutPremiumStrike Price less PremiumBreakevenStrike Less Premium 2015 J. David Miller

Options StrategiesBecause calls and puts can be combined with stock, an unlimited number of combinations with different payoffs can be created.

Some investment strategies involve combining one call or put with stock while others involve buying both calls and puts along with stock.

Combinations can also be created using different exercises prices and maturities to change the risk reward characteristics.

The use of strategies to limit the risk of a portfolio is called risk management.


Protective PutIf you want to invest in a stock, but are unwilling to take losses beyond a specific point, you might consider investing in the stock in combination with a put option.

The combination of the investment in the stock and the put option limits downside risk, while retaining most of the profit potential.

The value of the protective put investment can be calculated as


Protective Put Payoff Chart

In combination 2015 J. David Miller

Protective Put vs. Stock InvestmentThis shows the return on a Protective Put Strategy versus investment in stock alone


Protective Put ExampleAn investor purchases shares of AMD for $75 per share. She believes that AMD will go up in price, but wants to limit her loss to $10 per share. To hedge her risk, she purchases a put option on AMD stock with an exercise price of $67 per share. She pays the option premium of $2 per share.

If the AMD stock price rises to $85 per share and she sells her stock, what will her profit (loss) be?

Because the stock price is higher than the option strike price, she will let the option expire.$85 sales price - $75 purchase price - $2 option premium equals $8 per share. 2015 J. David Miller

Protective Put Example IIAn investor purchases shares of AMD for $75 per share. She believes that AMD will go up but wants to limit her loss to $10 per share. To hedge her risk, she purchases a put option on AMD stock with an exercise price of $67 per share. She pays the option premium of $2 per share.

If the AMD stock price falls to $60 per share, what will she do and what will her profit (loss) be?

She will exercise her option to sell the stock for $67 per share. $67 sales price - $75 purchase price - $2 option premium equals -$10. Her loss will only be $10 per share. 2015 J. David Miller

Covered CallA covered call position is the purchase of a share of stock with the simultaneous sale of a call on that stock. The sale of a call is also called writing an option.

The option that is written is covered because the potential obligation to deliver the stock is covered by the stock held in the portfolio. The strike price on the option will be set at the level the investor is willing to sell the stock.

Writing an option without an offsetting stock position is called naked option writing.

The payoff of a covered call


Covered Call Payoff ChartIn combination

Some investment managers will write a covered call rather than selling a stock so they can boost income from the premium. They forfeit capital gains if the stock price rises above the exercise price, but if they planned to sell the stock at the exercise price, the written call guarantees the stock sale will occur, while adding to their profit with the premium received.Profit 2015 J. David Miller

Covered Call ExampleJulia purchased a share of Intel stock last year for $40 per share and now believes that the stock will reach $60 per share in the next month. She plans to sell at that price and wants to make an additional profit on her sale. Because Julia is willing to sell at that price, she writes a call option with a strike price of $60. She receives $3 in option premium per share.

If the Intel stock price rises to $65 per share and the purchaser of the option exercises, what will Julias profit (loss) be?

$60 sales price when option is exercised - $40 purchase price + $3 option premium received equals $23 profit per share. If Julia did not write the option and sold the stock at $60, she would have made a $20 profit per share. 2015 J. David Miller

StraddleA straddle position is one that allows the investor to profit based on how much the price of the underlying security moves, regardless of the direction of the price move.

A put and a call are purchased on a single stock, with the same strike price and time until expiration.

The investor profits if the stock price moves a great deal away from the strike price. However, the direction does not matter. The intent is to profit from or hedge the volatility of the underlying security.


Long Straddle Payoff ChartIn combination

A short straddle can be created by writing both a call and put 2015 J. David Miller

Short Straddle Payoff ChartA short straddle can be created by writing both a call and put at the same exercise price.PayoffProfitST0Profit 2015 J. David Miller

Straddle Payoffs Max GainMax LossLong StraddleUnlimited on profitable side less both premiumsBoth PremiumsMax GainMax LossShort StraddlePremium on both sidesUnlimited on either side less both premiums 2015 J. David Miller

Straddle ExampleJohn believes that Allied Waste Inc. stock is going to be extremely volatile in the next month because the of the trouble the company is having. The stock is currently trading at $10. John believes that the stock could either go down or up a great deal depending on whether the company can get the financing required to stay in business. To profit from this volatility, John buys a call option and a put option on Allied Waste both with a strike price of $10. He pays $1 premium on the call option and a $2 premium on the put.

If the stock price rises to $19, what will his profit (loss) be?

He exercises the call option and buys the shares from the option writer for $10 per share and then sells them at market for $19. He allows the put option to expire worthless. His profit is $19 sale price - $10 purchase price - $1 call option premium - $2 put option premium = $6 per share. 2015 J. David Miller

Straddle Example IIJohn believes that Allied Waste Inc. stock is going to be extremely volatile in the next month because the of the trouble the company is having. The stock is currently trading at $10. John believes that the stock could either go down or up a great deal depending on whether the company can get the financing required to stay in business. To profit from this volatility, John buys a call option and a put option on Allied Waste both with a strike price of $10. He pays $1 premium on the call option and a $2 premium on the put.

If the stock price rises to $11 just before the options expire, what will his profit (loss) be?

He exercises the call option and buys the shares from the option writer for $10 per share and then sells them at market for $11. He allows the put option to expire worthless. His return is $11 sale price - $10 purchase price - $1 call option premium - $2 put option premium = $2 loss per share. 2015 J. David Miller

Long Strangle Payoff ChartPayoffProfitST0ProfitA strangle is similar to a straddle in that both a call and a put are used. However, with a strangle, the exercise price on the two options is not the same. The exercise prices that are chosen with a strangle are both out of the money. For a long strangle, you would buy one out-of-the-money call and one buy out-of-the-money put.

A short version of this strategy also exists. 2015 J. David Miller

SpreadsA bullish spread position is one where a call is purchased and a call is written on the same underlying security but at different strike prices.

The investor profits from a rise in the price of the underlying security, but only up to a point. The difference in the strike prices determines the level of profits that are possible. The income from the call that was written helps offset the cost of the call that was purchased.


Spreads Payoff ChartIn combination

A bearish spread can be created using put options rather than call options 2015 J. David Miller

Spread ExampleJeff believes that the price of True Religion Jeans stock, currently trading at $57, will go up in price but only to $67 in the next two months. He buys a call option with a strike price of $57 and writes a call at $67. The premium on the call he purchased is $1.50 per share and the premium he receives from the call he writes is $1 per share.

If the stock rises to $70 per share before expiration and the purchaser of the call that Jeff wrote exercises, what will Jeff do and what will his profit (loss) be?

Jeff will exercise his option to purchase shares at $57 and sell them to the purchaser of the call option he wrote for $67. His profit is $67 sale price - $57 purchase price - $1.50 call option Jeff purchased premium + $1 option premium he wrote = $9.50 per share. 2015 J. David Miller

CollarsCollars are very rare among speculators but are common among investors who already have stock in a company. Collars allow an investor to sell their stock at a predetermined range of prices while also preventing or limiting the loss from a fall in the stock price.

Collars involve the selling of a call option at one stock price and using the proceeds to purchase at put option at a lower price. The cost to the investor is essentially zero as the option premiums cancel each other out.

This strategy is known as a collar because it brackets the value of a portfolio between two bounds.


Collar ExampleJudy is an executive at IBM and has recently been awarded a significant amount of IBM stock, which is currently trading at $100. She believes that IBM stock will go up in the next three months, but knows that technology stocks can fall quickly if bad news is released. Judy cannot afford to lose this award, but she would also like to try to get $10 more per share for her stock ($110). She does not want to sell for anything less than $90.

She sells a call option with a strike price of $110 for a $5 premium. Using the $5 she buys a put option with a strike price of $90. Her cost in the trade is $0.

Regardless of what happens to the price of IBM stock, Judy will receive between $90 and $110 if she decides to sell her shares before the options expire.


Collar Example GraphImage from http://web.streetauthority.com/terms/options/6.asp


A Note on OptionsThere are many more options strategies which we wont discuss here. They have names such as butterflies, condors, and strangles. A good source for more information is the Options Industry Council. http://www.optionseducation.org/strategy/

Options have a value of their own which depends on both the intrinsic value and the time value of the option.

While many options are not exercised until close to their maturity, profits can be made on options WITHOUT EXERCISING THEM.

Options can be sold to other investors for prices which are determined according to complex valuation models. We will look at those models next.


Other Strategies (NOT ON TEST)

Married Put 2015 J. David Miller

Options ValuationOption valuation can be a highly quantitative issue, but we are going to start by identifying the features of an option that affect market value.

We will follow with a simple pricing model, followed by the famous/infamous Black-Scholes Model.

We will conclude with a discussion of the applications of options to risk management.


Intrinsic and Time ValueThe intrinsic value of an option is the gain that could be attained by immediate exercise of an in-the-money option.

For call options, this is market price of stock exercise price

For put options this is exercise price market price of stock

The time value is the difference between an options price and its intrinsic value.

Premium of an option = Time Value + Intrinsic Value


Intrinsic and Time Value IIMost of an options time value is a type of volatility value. The holder of the option can choose not to exercise the option but hold it until expiration, it has the potential to create a profit.

As the stock price rises on a call option, it becomes more likely that the option will be exercised. As it becomes more likely to be exercised, its volatility value decreases as a percentage of overall value.


57Determinants of Option Value1- Stock price an increase in the price of the stock increases the value of a call option and decreases the value of a put option.

2- Exercise price an increase in the exercise price decreases the value of a call option and increases the value of a put options.

3- Stock price volatility - an increase in the volatility of the stock increase the value of both a call and a put option


Determinants of Option Value4- Time to expiration an increase in the time to expiration increases the value of a call option and increases the value of a put option

5- Interest rate an increase in interest rates increases the value of a call option and decreases the value of a put option

6- Stock dividend a higher dividend payout lowers the value of a call option and may raise the value of a put option


Put / Call ParityIn many cases, put prices can be derived from the call prices. Prices of European put and call options are linked together in an equation known as the put-call parity relationship.

Where C = call premiumP = put premium S0 = dollars initially invested in stockX = exercise priceRf = interest rate on debt T = time to expiration (as a fraction of a year)

This formula tells us that difference between the call and put values is equal to the difference between the stock price and the present value of the exercise price.


Put / Call Parity ExampleWhat is the approximate put premium when the exercise price on an option is $100, the stock price is $105 and the call premium is $20? Assume that the interest rate on debt is 11.1% per year and the time to expiration is one year.

C P = 105 (100 / (1.111))20 - P = 105 - 9020 - P = 15P = 5


Put / Call Parity MispricingWe can use the Put/Call Parity relationship to determine if options are priced correctly. If we find that an option is incorrectly priced, we can profit from it based on the assumption that the prices will move back into parity.

The strike price on a call option with one year until maturity is $100, the stock price is $105 and the call premium is $20 when interest rates are 11.1%. The premium on a put option with the same strike price and maturity is $4. How can we profit from this?

We found on the previous example that the price of the put should be $5. This means that the put option is underpriced, therefore we should purchase the put option. When the prices return to parity, we can sell it to make $1 profit.


Put / Call Parity Mispricing ExampleThe strike price on a call option with one year until maturity is $75, the stock price is $70 and the call premium is $5 when interest rates are 10%. The premium on a put option with the same strike price and maturity is $4. How can we profit from this and what will be the profit if prices return to parity?

5 - p = 70 - (75 / (1.1))P = 3.18

The put should be priced at $3.18, but is selling for $4. This put is overpriced, so we should sell the put. When it returns to parity, we can buy it back for $3.18 and make a profit of $4 - $3.18 = $0.82


Binomial Option PricingWe can begin to get an understanding of call option pricing by using a relatively simple, but powerful pricing model called binomial pricing.

Suppose a stock currently sells for $100 per share and the price will either increase by a factor of u=1.2 (u=up)to $120 or fall by a factor of d = .9 (d=down) or $90

We can draw a tree showing what could happen

$100$120$90InitialEnd of period 2015 J. David Miller

Binomial Option PricingNow suppose a call option on the stock has a strike price of $110 with one year to expiration. The interest rate is 10%. At the end of the year, the payoff to the holder of the call option will either be zero (if it falls) or $10 (if it goes up) based on the performance of the stock.

For this option, in the best case scenario we receive $10.How much should this option cost?

C0$10$0InitialEnd of period 2015 J. David Miller

Replicating PortfolioIn Finance, when an investment is difficult to value or doesnt have a clear value, one way we attempt to value these assets is to create a portfolio with the same return as the difficult to value asset.

We can then value the assets in the replicating portfolio to get an idea of the value of the difficult to value assets.

This is the first way we will value the derivative.


Binomial Option PricingTo find the value of the previous option, compare it to a portfolio consisting of one share of stock and borrowing the PV of the future stock price where the option has no value.

Buy a share of stock for $100 and borrow $81.82 = (PV of $90 at 10% or 90/1.1. $90 will be repaid in one year)The cash outlay to create this portfolio = -100 + 81.82 = -$18.18 or a $18.18 outflow.

If the stock goes up to $120, then you pay back the loan of $90, so you have ($120-$90) = $30OrStock goes down to $90, then you pay back the loan of $90 and you have ($90-$90) = $0.


Loan value to be repaid one year in the future will be $90. The PV of $18.18 is the cost to you now to create the portfolio.67Binomial Option PricingIf the stock goes up to $120, then you pay back the loan of $90, so you have (120-90) = $30OrStock goes down to $90, then you pay back the loan of $90 and you have (90-90) = $0.

The $30 payoff is three times the $10 option payoff, so three options would replicate the payoff of the portfolio. The three options to one share is called the hedge ratio.

Payoff from three calls equal one share of stock, so one call = $6.06

(buy stock) -$100.00(take out loan) + 81.82stock = $120pay back loan of $90 netting $30$30 times one-third = $10stock = $90pay back loan of $90netting $0 -$18.18One-third = -$6.06

Stock Price 2015 J. David Miller

Binomial Option Pricing VA portfolio made up of one share of stock and writing 3 call options is perfectly hedged. (the portfolio value is the same whether the stock rises or falls)

An investor has formed a riskless portfolio with a payout of $90, so the value must be equal to the PV of $90 or $82.18 which in turn should equal $100-3C = 81.82; C = 6.06

Stated another way: What should the option sell for today?

Stock Value$120 $90 less obligation from 3 calls written-30$0 $90 $90 $18.18$30$0C$10$0PortfolioCall Option 2015 J. David Miller

A Second Method of ValuationWe can find the number of shares that can be hedged or protected by one option. The formula for the hedge ratio can be written as:

where Cu or Cd refers to the call options value when the stock goes up or down and uS0 and dS0 are the stock prices in the two states.

The hedge ratio is the ratio of the possible swings in the possible end-of-period values of the option and the stock.

If an investor writes one option and holds H shares of stock, the value of the portfolio will be unaffected by the stock price.


Hedge Ratio ExampleWe have one share of stock selling for $60. In one year, the stock will either be selling for $78 (u=1.3) or $54 (d=.9). The interest rate is 8%. A call option on this stock has a strike price of $72. What is the hedge ratio and how much should each option cost?

If stock goes up, the option will generate $6 (Cu=$6) in income and if the stock price falls, it will generate $0 (Cu=$0). = ($6-$0) / ($78 - $54) = 1/4

So the hedge ratio is . It will take four options to hedge the portfolio.


Hedge Ratio ExampleWe want to create a portfolio of stock and options that will have the same value regardless of the changes in the stock price. So whether the stock price is ends up at $78 or down at $54, our portfolio value should be the same.

We know that we will need 4 options for every 1 share of stock. Lets create a portfolio that will have the value of $54 regardless of changes in the stock price.

$60$78$54C$6$0PortfolioCall Option 2015 J. David Miller

Hedge Ratio ExampleWe want to create a portfolio of stock and options that will have the same value regardless of the changes in the stock price. So whether the stock price is ends up at $78 or down at $54, our portfolio value should be the same.

We know that we will need 4 options for every 1 share of stock. Lets create a portfolio that will have the value of $54 regardless of changes in the stock price.

Buy one share for $60 and write 4 options at price C. We want the value of our portfolio one year from today to equal $54, so today it should equal the present value of $54 or $54/1.08 = $50. We can now determine how much the options should cost to make this portfolio work.

$60 4C = 54/1.08. C=$2.5


Hedge Ratio ExampleLets see what happens with our portfolio if the stock price goes up or down.

We buy one share for $60 and sell 4 options for $2.50 each. Our cost to create this is $60 - (4 x $2.50) = $50.

If the stock ends at $54, then our one share is worth $54 and the options we sold wont be exercised. Our portfolio is worth $54.

If the stock ends at $78, then our one share is worth $78 and the option buyer will exercise the options at $72. Our cost is $6 per option times 4 options = $24. The value of our portfolio is $78 - $24 = $54.

This proves that our portfolio was perfectly hedged and that the options were worth $2.50 a piece.


In Search of a Simplified ProcessIf you want to think of process of creating the replicating portfolio as a formula.

1) Find the Hedge Ratio.

2) Create an equality by setting (the current stock price minus the numbers you sold) equal to (the present value of the stock in its down state).

3) Solve for the value of the options you sold.

Using the previous example:$60 4C = 54/1.08C=$2.5


Generalizing Two-State ApproachIf it seems unrealistic to only have two possible values for a stock a year in the future, we can increase the realism by breaking the time period into multiple parts, with a new step for each unit of time.

Over a six month period a stock could increase by 10% or decrease by 5%. The stock initially selling for $100 could follow these possible paths over period of a year.

$100$110$95$121$104.50$90.256 Months6 Months 2015 J. David Miller

Generalizing Two-State ApproachThere are now three possible end-of-year values for the stock and three for the option

We could use Cuu and Cud to find the value of Cu. Using Cud and Cdd we then find the value for Cd. Then we find C using Cu and Cd.

CCuCdCuuCudCdd 2015 J. David Miller

Two-State ExampleRisk Free Rate = 5% over 6 months (10.25% annual rate). Assume X=$110. When the stock price is 121, Cuu is $121-$110 or 11 (value at expiration). When the stock is at $104.50, Cud is worth zero. Value Cu.

Hedge Ratio = (cuu cud) / (uuSo udSo) = (11 - 0) / (121 - 104.5) = 2/3

So for every 2 shares of stock, I need to write 3 call options.

CCuCdCuuCudCdd10011095121104.5090.25 2015 J. David Miller

Two-State Example6 month Risk Free Rate = 5%. Assume X=$110. When the stock price is 121, Cuu is $121-$110 or 11 (option value at expiration). When the stock is at $104.50, Cud is worth zero. Value Cu.

Buy 2 shares of stock at uSo and write 3 options. If stock ends at 121, then our portfolio value equals (2 x 121) = 242 minus (3 options cost of $11 per option = 33) = $209If stock ends at 104.50, the our portfolio value equals (2 x 104.5) = 209 minus (3 options which wont be exercised = 0) = $209

This portfolio must have current market value of $209.

CCuCdCuuCudCdd10011095121104.5090.25at price 104.5at price 121 Buy two shares at uSo = $110 2 *104.5 = 209 2*121=242 Write three calls at price of Cu 0 3*11=-33 209 209 2015 J. David Miller

Two-State ExampleThis portfolio must have current market value of $209. Since the risk free rate over the 6-month period is 5%, then the cost of the portfolio is 2 *110 3cu = 209/1.05 Cu = $6.984.

We now have the value for Cu

Then find Cd using the same process. Once we have the value for Cu and Cd, it becomes a simple one-stage problem to find the value of C. (Like our previous example)

CCuCdCuuCudCdd 2015 J. David Miller

Black-Scholes Option ValuationThe Black-Scholes Pricing Formula is another method for valuing an option which uses the stock price, the strike price, the risk-free rate, the time to expiration and the standard deviation of the stock return. The formula for European-style call option is:

Dividend paying stock

Non-Dividend paying stock


What does it all mean?The formula looks scary, but can be straightforward with one simplification.

N(d) is the probability that the call option will expire in the money. If d1 and d2 are = 1, which means that the option will expire in the money, the formula simplifies to:

which means that the option cost is a function of the stock price, the strike price, the risk free rate and time until maturity.


What does it all mean? IIOn the other hand, if d1 and d2 are = 0, this means that the option will expire out of the money. The formula simplifies to C0 = 0, which makes sense because options that expire out of the money are worthless.

N(d) is a probability which fits a cumulative distribution


Black-Scholes Hedge RatiosThe Black-Scholes hedge ratio for a call = N(d1)

The Black-Scholes hedge ratio for a put = N(d1)-1

Lets work through an example.


Black-Scholes ExampleWhat is the current call option value for an option that will expire in three months according to the Black-Scholes Formula if the underlying stock price is currently $100, the exercise price is $95, the 3-month interest rate is 10% and the standard deviation of the returns of the stock is 0.50?

T = .25 d1 = [ln(100/95) + (0.10 + (0.5)2/2)*0.25] / (0.5* (0.25)1/2) =(0.0513 + 0.0563) / 0.25 = .43

d2 = 0.43 - (0.5* 0.251/2) = 0.18

We must use the cumulative probability table to look up the probabilities for d1 and d2


Black-Scholes Example IIYou might have to estimate a little.d1 =.43d2 =.18N(d1) = 0.6664N(d2) = 0.5714

We must use the cumulative probability table to look up the probabilities for d1 and d2


Black-Scholes Example IIIN(d1) = 0.6664N(d2) = 0.5714

C0 = 100 * 0.6664 95e-0.10*0.25 * 0.5714=66.64 95 * 0.9753 * 0.5714= 66.64-52.94= 13.70The call option premium is $13.70


Using Black-ScholesThe hedge ratio, also known as delta (), is the number of shares of stock that can be hedged by holding one option.

Recall that the formula for calculating the hedge ratio is:

The call hedge ratio will be positive and less than 1 while the put hedge ratio will be negative and less than 1 in absolute value.

Call is the number of shares of the underlying stock that the investor can hedge with a written call option.Put is the number of shares of the underlying stock that the investor can hedge with an owned put option


Volatility and the VIXMany times, investors will use Black-Scholes with known stock prices and known option prices in the market to solve for the standard deviation, , necessary for the observed option price to be consistent with the model. They are solving for the implied volatility. Basically, how much volatility is the market expecting based on the pricing of the option.

One measure of implied volatility is actually traded by itself in the markets. The Chicago Board Options Exchange Volatility Index, call the VIX, is a popular measure of the implied volatility of the S&P 500 index options.

The VIX is a weighted blend of prices for a range of options on the S&P 500 index.


Volatility and the VIX (contd)The VIX is often referred to as the Fear Index, because it represents one measure of the markets expectation of volatility over the next 30 day period.

High values correspond to a more volatile market and therefore more costly options.

The VIX can be used to help measure the markets expectations of volatility in the short term future.

You can track the VIX using Yahoo! Finance. VIX Yahoo Finance Ticker is ^VIX (^ is shift+6 on most keyboards)

There is even an ETF that you can use to make money or help hedge the volatility of your portfolio. Its ticker is VXX. 2015 J. David Miller

Using Hedge Ratio in Another WayWe can use the hedge ratio to tell us how sensitive option prices are to movements in the underlying stock price.

The hedge ratio is .6 and an investor writes 100 options and holds 60 shares of stock.

(The hedge ratio is the number of shares of stock that can be hedge by holding one option. 0.6 shares can be hedged by holding one option. If the investor multiplies each side by 100, 60 shares can be hedged by 100 options)

A $1 increase in the stock price would result in a $60 gain in the stock. Loss on the options would be 100 * .6 = $60

This is a hedged position where total wealth is unchanged by changes in the stock price.


Using the Hedge Ratio ExampleThere are two portfolios. The hedge ratio on some call options is .6. Which portfolio has more dollar exposure to IBM price movements? Use the hedge ratio (H).

Portfolio 1: 750 IBM Calls + 200 Shares of IBMPortfolio 2: 800 IBM shares

Each option changes in value by H dollars for each dollar change in the stock price. So if H=.6 then the 750 IBM calls are equivalent to 750 * .6 = 450 shares in terms of their response to the stock price of IBM.

Portfolio 1 has the equivalent of 450 + 200 shares = 650 shares of IBM. Portfolio 2 has dollar movements equivalent to 800 shares of IBM.Portfolio 2 has more dollar sensitivity to price.


Option ElasticityAnother important concept is option elasticity, which is the percentage increase in an options value given a 1% increase in the value of the underlying security.

While the options in the previous example have less Dollar Sensitivity than a share of stock, that does not mean that they are less volatile in the rate of return.

For a stock selling for $120 and a hedge ratio of .6, an option with an exercise price of $120 may sell for $5. If the stock price increases to $121, the call price would be expected to increase by only $0.60 to $5.60.

However, the percentage increase in the option value is $0.60 / $5.00 = 12%, while the percentage stock price increase is only $1 / $120 = 0.83%.


Option Elasticity (contd)The percentage increase in the option value is $0.60 / $5.00 = 12%, while the percentage stock rise increase is only $1 / $120 = 0.83%.

Option elasticity measures the sensitivity of change in option price to change in the stock price.

So for the example above, the option elasticity would be 12% / 0.83% = 14.4%

For every 1% increase in the stock price, the option price increases by 14.4%.


Option Elasticity ExampleA share of Microsoft (Ticker: MSFT) stock is selling for $75 and its hedge ratio is .5. A call option with exercise price $75 is selling for $4. If the stock price increases to $77, the call price would be expected to increase by how much? For a 1% increase in the stock price, by what percentage will the option price increase?

Step 1) Find the dollar change. Hedge ratio of .5 means that for a $1 change in the stock price, the option will increase by $0.50. The stock changed by $2, so the option price increased by $1 to $5.

Step 2) The option elasticity = ($1/$4)/($2/$75) = (25%)/(2.6667%) = 9.375. So a 1% increase in stock price will cause a 9.375% increase in the price of the option.


Documents

Derivatives - Options Ch 20 & 21