1Ch20&21 – MBA 566

Options

Option BasicsOption strategiesPut-call parityBinomial option pricingBlack-Scholes Model

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Buy - Long Sell - ShortCallPut Key Elements

Exercise or Strike PricePremium or PriceMaturity or Expiration

1. Option Basics: Terminologies

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In the Money - exercise of the option would be profitable.Call: market price>exercise pricePut: exercise price>market price

Out of the Money - exercise of the option would not be profitable.Call: market price<exercise pricePut: exercise price<market price

At the Money - exercise price and asset price are equal.

Market and Exercise Price Relationship

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Examples

Tell if it is in or out of the moneyFor a put option on a stock, the current stock price=$50 while the exercise price is $47.For a call option on a stock, the current stock price is 55 while exercise price is 45.

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

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

Should we early exercise American options?

American vs. European Options

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Stock OptionsIndex OptionsFutures OptionsForeign Currency OptionsInterest Rate Options

Different Types of Options

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Notation Stock Price = ST Exercise Price = XPayoff to Call Holder

(ST - X) if ST >X

0 if ST < XProfit to Call Holder

Payoff - Purchase Price

Payoffs and Profits at Expiration - Calls

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Payoff to Call Writer - (ST - X) if ST >X

0 if ST < XProfit to Call Writer

Payoff + Premium

(pages 678 – 680)

Payoffs and Profits at Expiration - Calls

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Payoff and Profit to Call Option at Expiration

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Payoff and Profit to Call Writers at Expiration

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Payoffs to Put Holder0 if ST > X

(X - ST) if ST < X

Profit to Put Holder Payoff - Premium

Payoffs and Profits at Expiration - Puts

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Payoffs to Put Writer0 if ST > X

-(X - ST) if ST < X

Profits to Put WriterPayoff + Premium

Payoffs and Profits at Expiration - Puts

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Payoff and Profit to Put Option at Expiration

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Straddle (Same Exercise Price) – page 686Long Call and Long Put

Spreads - A combination of two or more call options or put options on the same asset with differing exercise prices or times to expiration.Vertical or money spread:Same maturityDifferent exercise priceHorizontal or time spread:Different maturity dates

2. Option Strategies

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Value of a Protective Put Position at Expiration

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Protective Put vs Stock (at-the-money)

Page 684

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Value of a Covered Call Position at Expiration

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Intrinsic value - profit that could be made if the option was immediately exercised.

Call: stock price - exercise pricePut: exercise price - stock price

Time value - the difference between the option price and the intrinsic value.

Option Values

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Figure 21.1 Call Option Value before Expiration

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Restrictions on Option Value: Call

Value cannot be negativeValue cannot exceed the stock valueValue of the call must be greater than the value of levered equityC > S0 - ( X + D ) / ( 1 + Rf )T

C > S0 - PV ( X ) - PV ( D )

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If the prices are not equal arbitrage will be possible

3. Put Call Parity

0(1 )Tf

XC S Pr

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Stock Price = 110 Call Price = 17Put Price = 5 Risk Free = 5%Maturity = 1 yr X = 105

117 > 115Since the leveraged equity is less expensive, acquire

the low cost alternative and sell the high cost alternative

Put Call Parity - Disequilibrium Example

0(1 )Tf

XC S Pr

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100

120

90

Stock Price

C

10

0

Call Option Value X = 110

4. Binomial Option Pricing: Text Example

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Alternative PortfolioBuy 1 share of stock at $100Borrow $81.82 (10% Rate)Net outlay $18.18PayoffValue of Stock 90 120Repay loan - 90 - 90Net Payoff 0 30

18.18

30

0Payoff Structureis exactly 3 timesthe Call

Binomial Option Pricing: Text Example

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18.18

30

0

C

30

0

3C = $18.18C = $6.06

Binomial Option Pricing: Text Example

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Generalizing the Two-State Approach

See page 725, get hedge ratio, then compute call price.

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Co = SoN(d1) - Xe-rTN(d2)

d1 = [ln(So/X) + (r + 2/2)T] / (T1/2)

d2 = d1 + (T1/2)

whereCo = Current call option value.

So = Current stock price

N(d) = probability that a random draw from a normal dist. will be less than d.

5. Black-Scholes Option Valuation

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X = Exercise pricee = 2.71828, the base of the natural logr = Risk-free interest rate (annualizes continuously

compounded with the same maturity as the option)

T = time to maturity of the option in yearsln = Natural log functionStandard deviation of annualized cont.

compounded rate of return on the stock

Black-Scholes Option Valuation

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So = 100 X = 95

r = .10 T = .25 (quarter)= .50d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2)

= .43 d2 = .43 + ((5.251/2)

= .18

Call Option Example

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Co = SoN(d1) - Xe-rTN(d2)Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70Implied VolatilityUsing Black-Scholes and the actual price of the

option, solve for volatility.Is the implied volatility consistent with the

stock?

Call Option Value

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Black-Scholes Model with Dividends

The call option formula applies to stocks that pay dividends.One approach is to replace the stock price with a dividend adjusted stock price.Replace S0 with S0 - PV (Dividends)

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Put Value Using Black-Scholes

P = Xe-rT [1-N(d2)] - S0 [1-N(d1)]

Using the sample call dataS = 100 r = .10 X = 95 g = .5 T = .2595e-10x.25(1-.5714)-100(1-.6664) = 6.35

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Determinants of Call Option Values

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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 ElasticityPercentage change in the option’s value given a 1% change in the value of the underlying stock.

Using the Black-Scholes Formula

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Figure 21.9 Call Option Value and Hedge Ratio

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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.

Portfolio Insurance

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Solution

Synthetic Put Selling a proportion of shares equal to the put option’s delta and placing the proceeds in risk-free T-bills (page 763-764)Additional problem

Delta changes with stock pricesSolution: delta hedgingBut increase market volatilitiesOctober 19, 1987

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Hedge Ratios Change with Stock Prices

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Hedging On Mispriced Options

Option value is positively related to volatility:If an investor believes that the volatility that is implied in an option’s price is too low, a profitable trade is possible.Profit must be hedged against a decline in the value of the stock.Performance depends on option price relative to the implied volatility.

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Hedging and Delta

The appropriate hedge will depend on the delta.Recall the delta is the change in the value of the option

relative to the change in the value of the stock.Example (page 767):

Implied volatility = 33% Investor believes volatility should = 35%Option maturity = 60 daysPut price P = $4.495Exercise price and stock price = $90Risk-free rate r = 4%Delta = -.453