Options Markets: Introduction CHAPTER 20 Options Markets: Introduction

Options Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims because their payoffs depend on the value of other securities. Options are traded both on organized exchanges and OTC.

Option Terminology Types Options Clearing Corporation Key Elements Exercise or Strike Price Premium or Price Maturity or Expiration Types American Options - the option can be exercised at any time before expiration European Options – can only be exercised at expiration Options Clearing Corporation Guarantees contract performance

The Option Contract: Calls A call option gives its holder the right to buy an asset: At the exercise or strike price On or before the expiration date This is a “long” strategy. Exercise the option to buy the underlying asset if market value > strike.

The Option Contract: Puts A put option gives its holder the right to sell an asset: At the exercise or strike price On or before the expiration date This is a “short” strategy Exercise the option to sell the underlying asset if market value < strike.

Market and Exercise Price Relationships In the Money - exercise of the option would be profitable Call: exercise price < market price Put: exercise price > market price Out of the Money - exercise of the option would not be profitable Call: market price < exercise price. Put: market price > exercise price. At the Money - exercise price and asset price are equal

Example 20.1 Profit and Loss on a Call A January 2010 call on IBM with an exercise price of $130 was selling on December 2, 2009, for $2.18. The option expires on the third Friday of the month, or January 15, 2010. If IBM remains below $130, the call will expire worthless.

Example 20.1 Profit and Loss on a Call Suppose IBM sells for $132 on the expiration date. Option value = stock price-exercise price $132- $130= $2 Profit = Final value – Original investment $2.00 - $2.18 = -$0.18 Option will be exercised to offset loss of premium. Call will not be strictly profitable unless IBM’s price exceeds $132.18 (strike + premium) by expiration.

Example 20.2 Profit and Loss on a Put Consider a January 2010 put on IBM with an exercise price of $130, selling on December 2, 2009, for $4.79. Option holder can sell a share of IBM for $130 at any time until January 15. If IBM goes above $130, the put is worthless.

Example 20.2 Profit and Loss on a Put Suppose IBM’s price at expiration is $123. Value at expiration = exercise price – stock price: $130 - $123 = $7 Investor’s profit: $7.00 - $4.79 = $2.21 Holding period return = 46.1% over 44 days!

Different Types of Options Stock Options Index Options Futures Options Foreign Currency Options Interest Rate Options

Profit Potential Investing in Stocks Investing in Options

Payoffs and Profits at Expiration - Calls Notation Stock Price = ST Exercise Price = X Payoff to Call Holder (ST - X) if ST >X 0 if ST < X Profit to Call Holder Payoff - Purchase Price

Payoffs and Profits at Expiration - Calls Payoff to Call Writer - (ST - X) if ST >X 0 if ST < X Profit to Call Writer Payoff + Premium

Payoff and Profit to Call Options Investing in Options

Payoffs and Profits at Expiration - Puts Payoffs to Put Holder 0 if ST > X (X - ST) if ST < X Profit to Put Holder Payoff - Premium

Payoffs and Profits at Expiration – Puts Payoffs to Put Writer 0 if ST > X -(X - ST) if ST < X Profits to Put Writer Payoff + Premium

Profit Potential of a Put Option Investing in Options

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

Option versus Stock Investments Strategy A: Invest entirely in stock. Buy 100 shares, each selling for $100. Strategy B: Invest entirely in at-the-money call options. Buy 1,000 calls, each selling for $10. (This would require 10 contracts, each for 100 shares.) Strategy C: Purchase 100 call options for $1,000. Invest your remaining $9,000 in 6-month T-bills, to earn 3% interest. The bills will be worth $9,270 at expiration.

Option versus Stock Investment Investment Strategy Investment Equity only Buy stock @ 100 100 shares $10,000 Options only Buy calls @ 10 1000 options $10,000 Leveraged Buy calls @ 10 100 options $1,000 equity Buy T-bills @ 3% $9,000 Yield

Strategy Payoffs

Protective Put Puts can be used as insurance against stock price declines. Protective puts lock in a minimum portfolio value. The cost of the insurance is the put premium. Options can be used for risk management, not just for speculation.

Table 20.1 Value of Protective Put Portfolio at Option Expiration Buy a Put to protect a stock investment.

Covered Calls Purchase stock and write calls against it. Call writer gives up any stock value above X in return for the initial premium. If you planned to sell the stock when the price rises above X anyway, the call imposes “sell discipline.”

Table 20.2 Value of a Covered Call Position at Expiration

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

Table 20.3 Value of a Straddle Position at Option Expiration

Figure 20.9 Value of a Straddle at Expiration

Spreads A spread is a combination of two or more calls (or two or more puts) on the same stock with differing exercise prices or times to maturity. Some options are bought, whereas others are sold, or written. A bullish spread is a way to profit from stock price increases.

Put-Call Parity The call-plus-bond portfolio (on left) must cost the same as the stock-plus-put portfolio (on right): If the prices are not equal arbitrage will be possible. To exploit the arbitrage you buy the cheap portfolio and sell the other.

Put Call Parity - Disequilibrium Example Stock Price = 110 Call Price = 17 Put Price = 5 Risk Free = 5% Maturity = 1 yr X = 105 117 > 115 Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative

CHAPTER 21 Option Valuation

Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value

Table 21.1 Determinants of Call Option Values

Restrictions on Option Value: Call Call value cannot be negative. The option payoff is zero at worst, and highly positive at best. Call value cannot exceed the stock value. Value of the call must be greater than the value of levered equity. Lower bound = adjusted intrinsic value: C > S0 - PV (X) - PV (D) (D=dividend)

Early Exercise: Calls The right to exercise an American call early is valueless as long as the stock pays no dividends until the option expires. The value of American and European calls is therefore identical. The call gains value as the stock price rises. Since the price can rise infinitely, the call is “worth more alive than dead.”

Early Exercise: Puts American puts are worth more than European puts, all else equal. The possibility of early exercise has value because: The value of the stock cannot fall below zero. Once the firm is bankrupt, it is optimal to exercise the American put immediately because of the time value of money.

Determining Option Prices Three stages of discussion 1) Assume risk neutral investors & equal probability of stock values between two extremes (not in text) 2) Relax assumption of risk neutral investors & add uncertainty (appended by info from outside the text) 3) assume stock values are normally distributed

Pricing of options (risk neutrality & uniform probability distributions) Valuing a Call Stock pays no div and expires in 1 year Two possible future values of the stock with equal probability Probability distribution is rectangular shaped With risk neutral investors use a discount rate = RF Example Stock price 50 or 10 RF = 10% Exercise price = 30 Present Value of stock

Value of call this equation represents the probability (second part) * average value if in the money Probability distribution for the call value is rectangular

Value of put

Relationships between option values and stock values shift of stock price distribution to right Now the stock value ranges from 20 to 60 (an increase in average stock price E(VS ) = 36.36 An increase E(VC ) = 10.23 An increase E(VP ) = 1.14 A decrease

Effect of changes in variance on option values expansion of the range of possible stock prices But same expected return of stock This will isolate the effect of changes in variance New range 10 to 70 From the text example E(VS ) = 36.36 An increase E(VC ) = 12.12 An increase E(VP ) = 3.03 An increase

Conclusions about prices of options 1) as stock price inc Value of call increases & put decreases 2) as volatility of stock price increases increases the value of calls and puts 3) given change in variance, $ price change of out of money options greater than in the money 4) options more volatile in percent terms than stocks 5) out of money more volatile than in the money

Binomial Option Pricing: Text Example 120 10 100 C 90 Call Option Value X = 110 Stock Price

Two state call option pricing call X = 110 Price stock = 100 Two rates of return on the stock are possible -10% & +20% The price can then be $90 or 120$ RF = 10% Hedge an investment buy one and sell other Say buy stock and sell options How many options to sell spread for options if Stock = 120 call = 10 Stock = 90 = 0 calc end value of portfolio own 1 share and sold 3 calls If stock goes to 90, have 1 share worth 90 Owners of options will not exercise Value of portfolio = 90 If stock goes 120, have 1 share worth 120 But owners of the calls want to exercise They give you 110 each = 330 But you must buy the stock at 120 Must spend 360 and extra $30 Net value of portfolio 120 – (360 – 330) 120 - 30 = 90 Portfolio = 90 regardless of stock price Implies a RF return

Implies a RF return if RF = 10% Solve for option price that would give a RF return Could do the same thing but reverse the transaction buy calls & short the stock (Note no investment on your part) If options sell for anything other than 6.06 then get an arbitrage situation Market conditions will force the option price at 6.06 Binomial Option Model

Two state put option pricing for a single period since puts and stock price move opposite a hedge here would include a sale of both or a purchase of both Use formula for # of calls to buy for the equivalent # for puts again eventually solve for put option price example need 2 stocks for 3 puts If stock goes to 120 Have two shares worth 240 & 3 worthless puts Total value = 240 If stock goes to 90 Have two shares of stock worth 180 Three puts worth 20 each = 60 Use formula from above to solve for the value of the puts that will give Rp = RF = 10%

Expanding to Consider Three Intervals Assume that we can break the year into three intervals. For each interval the stock could increase by 20% or decrease by 10%. Assume the stock is initially selling at $100.

Expanding to Consider Three Intervals

Black-Scholes Option Valuation Co = SoN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r + 2/2)T] / (T1/2) d2 = d1 - (T1/2) where Co = Current call option value So = Current stock price N(d) = probability that a random draw from a normal distribution will be less than d

Black-Scholes Option Valuation X = Exercise price e = 2.71828, the base of the natural log r = Risk-free interest rate (annualized, continuously compounded with the same maturity as the option) T = time to maturity of the option in years ln = Natural log function Standard deviation of the stock

Example 21.1 Black-Scholes Valuation So = 100 X = 95 r = .10 T = .25 = .50 (50% per year) Thus: Black and Scholes

Call Option Value Implied Volatility Implied volatility is volatility for the stock implied by the option price. Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock?

Black-Scholes Model with Dividends The Black Scholes call option formula applies to stocks that do not pay dividends. What if dividends ARE paid? One approach is to replace the stock price with a dividend adjusted stock price Replace S0 with S0 - PV (Dividends)

Example 21.3 Black-Scholes Put Valuation P = Xe-rT [1-N(d2)] - S0 [1-N(d1)] Using Example 21.2 data: S = 100, r = .10, X = 95, σ = .5, T = .25 We compute: $95e-10x.25(1-.5714)-$100(1-.6664) = $6.35

Put Option Valuation: Using Put-Call Parity P = C + PV (X) - So = C + Xe-rT - So Using the example data P = 13.70 + 95 e -.10 X .25 - 100 P = $6.35

Using the Black-Scholes Formula Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one option Call = N (d1) Put = N (d1) - 1 Option Elasticity Percentage change in the option’s value given a 1% change in the value of the underlying stock

Portfolio Insurance Buying Puts - results in downside protection with unlimited upside potential Limitations Tracking errors if indexes are used for the puts Maturity of puts may be too short Hedge ratios or deltas change as stock values change

Hedging and Delta The appropriate hedge will depend on the delta. Delta is the change in the value of the option relative to the change in the value of the stock, or the slope of the option pricing curve. Change in the value of the option Change of the value of the stock Delta =

Example 21.6 Speculating on Mispriced Options Implied volatility = 33% Investor’s estimate of true volatility = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk-free rate = 4% Delta = -.453

Table 21.3 Profit on a Hedged Put Portfolio

Example 21.6 Conclusions As the stock price changes, so do the deltas used to calculate the hedge ratio. Gamma = sensitivity of the delta to the stock price. Gamma is similar to bond convexity. The hedge ratio will change with market conditions. Rebalancing is necessary.

Delta Neutral When you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset, your portfolio is said to be delta neutral. The portfolio does not change value when the stock price fluctuates.

Table 21.4 Profits on Delta-Neutral Options Portfolio

Empirical Evidence on Option Pricing The Black-Scholes formula performs worst for options on stocks with high dividend payouts. The implied volatility of all options on a given stock with the same expiration date should be equal. Empirical test show that implied volatility actually falls as exercise price increases. This may be due to fears of a market crash.

CHAPTER 22 Futures Markets

Futures and Forwards Forward – a deferred-delivery sale of an asset with the sales price agreed on now. Futures - similar to forward but feature formalized and standardized contracts. Key difference in futures Standardized contracts create liquidity Marked to market Exchange mitigates credit risk

Basics of Futures Contracts A futures contract is the obligation to make or take delivery of the underlying asset at a predetermined price. Futures price – the price for the underlying asset is determined today, but settlement is on a future date. The futures contract specifies the quantity and quality of the underlying asset and how it will be delivered.

Basics of Futures Contracts Long – a commitment to purchase the commodity on the delivery date. Short – a commitment to sell the commodity on the delivery date. Futures are traded on margin. At the time the contract is entered into, no money changes hands.

Basics of Futures Contracts Profit to long = Spot price at maturity - Original futures price Profit to short = Original futures price - Spot price at maturity The futures contract is a zero-sum game, which means gains and losses net out to zero.

Figure 22.2 Profits to Buyers and Sellers of Futures and Option Contracts

Figure 22.2 Conclusions Profit is zero when the ultimate spot price, PT equals the initial futures price, F0 . Unlike a call option, the payoff to the long position can be negative because the futures trader cannot walk away from the contract if it is not profitable.

Existing Contracts Futures contracts are traded on a wide variety of assets in four main categories: Agricultural commodities Metals and minerals Foreign currencies Financial futures

Trading Mechanics Electronic trading has mostly displaced floor trading. CBOT and CME merged in 2007 to form CME Group. The exchange acts as a clearing house and counterparty to both sides of the trade. The net position of the clearing house is zero.

Trading Mechanics Open interest is the number of contracts outstanding. If you are currently long, you simply instruct your broker to enter the short side of a contract to close out your position. Most futures contracts are closed out by reversing trades. Only 1-3% of contracts result in actual delivery of the underlying commodity.

Margin and Marking to Market Marking to Market - each day the profits or losses from the new futures price are paid over or subtracted from the account Convergence of Price - as maturity approaches the spot and futures price converge

Margin and Trading Arrangements Initial Margin - funds or interest-earning securities deposited to provide capital to absorb losses Maintenance margin - an established value below which a trader’s margin may not fall Margin call - when the maintenance margin is reached, broker will ask for additional margin funds

Trading Strategies Speculators Hedgers seek to profit from price movement short - believe price will fall long - believe price will rise seek protection from price movement long hedge - protecting against a rise in purchase price short hedge - protecting against a fall in selling price

Basis and Basis Risk Basis - the difference between the futures price and the spot price, FT – PT The convergence property says FT – PT= 0 at maturity.

Basis and Basis Risk Before maturity, FT may differ substantially from the current spot price. Basis Risk - variability in the basis means that gains and losses on the contract and the asset may not perfectly offset if liquidated before maturity.

Futures Pricing Spot-futures parity theorem - two ways to acquire an asset for some date in the future: Purchase it now and store it Take a long position in futures These two strategies must have the same market determined costs

Spot-Futures Parity Theorem With a perfect hedge, the futures payoff is certain -- there is no risk. A perfect hedge should earn the riskless rate of return. This relationship can be used to develop the futures pricing relationship.

Hedge Example: Section 22.4 Investor holds $1000 in a mutual fund indexed to the S&P 500. Assume dividends of $20 will be paid on the index fund at the end of the year. A futures contract with delivery in one year is available for $1,010. The investor hedges by selling or shorting one contract .

Hedge Example Outcomes Value of ST 990 1,010 1,030 Payoff on Short (1,010 - ST) 20 0 -20 Dividend Income 20 20 20 Total 1,030 1,030 1,030

Rate of Return for the Hedge

The Spot-Futures Parity Theorem Rearranging terms

Arbitrage Possibilities If spot-futures parity is not observed, then arbitrage is possible. If the futures price is too high, short the futures and acquire the stock by borrowing the money at the risk free rate. If the futures price is too low, go long futures, short the stock and invest the proceeds at the risk free rate.

Spread Pricing: Parity for Spreads

Spreads If the risk-free rate is greater than the dividend yield (rf > d), then the futures price will be higher on longer maturity contracts. If rf < d, longer maturity futures prices will be lower. For futures contracts on commodities that pay no dividend, d=0, F must increase as time to maturity increases.

Futures Prices vs. Expected Spot Prices Expectations Futures prices equals the future spot Normal Backwardation Futures price is bid down and will rise toward to a point where futures price = future spot price Contango Demand for the contract drives up the price, so over time the futures price will fall to the future spot price.

Figure 22.7 Futures Price Over Time, Special Case