Essentials of Investments: Chapter 15 - Options Markets

Essentials of Investments: Chapter 15 - Options Markets Option Terminology, Market and Exercise Price Relationships, American vs European Options, Different Types of Options, Payoffs and Profits on Options at Expiration - Calls.

Chapter 15

Options Markets

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

American vs European Options

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

Different Types of Options

• Stock Options

• Index Options

• Futures Options

• Foreign Currency Options

• Interest Rate Options

Payoffs and Profits on Options at Expiration - Calls

Notation Stock Price = S T Exercise Price = X Payoff to Call Holder

( S T - X) if S T >X

0 if S T < X Profit to Call Holder

Payoffs and Profits on Options at Expiration - Calls

Payoff to Call Writer

- ( S T - X) if S T >X

Profit to Call Writer

Payoff + Premium

0

Call Writer Call Holder

Profit Profiles for Calls

Payoffs and Profits at Expiration Puts

-Payoffs to Put Holder

0 if S T > X (X - S T ) if S T < X

Profit to Put Holder

Payoff - Premium

Payoffs and Profits at Expiration Puts

-Payoffs to Put Writer

0 if S T > X -(X - S T ) if S T < X

Profits to Put Writer

Payoff + Premium

Profit Profiles for Puts

Equity, Options & Leveraged Equity Text Example

Equity only Buy stock @ 80 100 shares $8,000

Options only Buy calls @ 10 800 options $8,000

Leveraged Buy calls @ 10 100 options $1,000

Yield

Equity, Options & Leveraged Equity Payoffs

-Microsoft Stock Price

$75 $80 $100 All Stock $7,500 $8,000 $10,000

All Options $0 $0 $16,000

Lev Equity $7,140 $7,140 $9,140

Equity, Options & Leveraged Equity Rates of Return

-Microsoft Stock Price

$75 $80 $100 All Stock -6.25% 0% 25%

All Options -100% -100% 100%

Lev Equity -10.75% -10.75% 14.25%

Put-Call Parity Relationship

S T < X S T > X Payoff for

Call Owned 0 S T - X Payoff for

Put Written-( X -S T ) 0 Total Payoff S T - X S T - X

Payoff of Long Call & Short Put

Long Call

Short Put

Payoff

Stock Price Combined =

Leveraged Equity

Arbitrage & Put Call Parity

Since the payoff on a combination of a long call and a short put are equivalent

to leveraged equity, the prices must be equal.

C - P = S 0 - X / (1 + r f ) T

If the prices are not equal arbitrage will be possible

Put Call Parity - Disequilibrium Example

Stock Price = 110 Call Price = 17 Put Price = 5 Risk Free = 10.25%

Maturity = 5 yr X = 105

C - P > S 0 - X / (1 + r f ) T 17- 5 > 110 - (105/1.05)

12 > 10 Since the leveraged equity is less expensive ,

acquire the low cost alternative and sell the

Put-Call Parity Arbitrage

Immediate Cashflow in Six Months Position Cashflow S T <105 S T > 105

Option Strategies

Protective Put

Long Stock Long Put

Covered Call

Long Stock Short Call

Straddle (Same Exercise Price) Long Call

Option Strategies

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 maturity Different exercise price Horizontal or time spread

Different maturity dates

Chapter 17

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

Time Value of Options: Call

Factors Influencing Option Values:

Calls

Factor Effect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expiration increases Interest rate increases

Binomial Option Pricing:

Binomial Option Pricing:

is exactly 2 times the Call

Binomial Option Pricing:

Another View of Replication of Payoffs and Option Values

Alternative Portfolio - one share of stock and 2 calls written (X = 125)

Portfolio is perfectly hedged Stock Value 50 200 Call Obligation 0 -150

Hence 100 - 2C = 46.30 or C = 26.85

Black-Scholes Option Valuation

C o = S o e -dT N(d 1 ) - Xe -rT N(d 2 )

d 1 = [ln(S o /X) + (r – d + s 2 /2)T] / (s T 1/2 )

d 2 = d 1 - (s T 1/2 )

where

C o = Current call option value.

S o = Current stock price N(d) = probability that a random draw from a normal dist will be less than d.

Black-Scholes Option Valuation

X = Exercise price.

d = Annual dividend yield of underlying stock

e = 2.71828, the base of the nat log.

r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option.

T = time to maturity of the option in years.

ln = Natural log function

Call Option Example

Probabilities from Normal Dist.

N (.43) = 6664 Table 17.2

.43 6664 Interpolation

Probabilities from Normal Dist.

N (.18) = 5714 Table 17.2

Call Option Value

C o = S o e -dT N(d 1 ) - Xe -rT N(d 2 )

C o = 100 X 6664 - 95 e - 10 X 25 X 5714

C o = 13.70 Implied Volatility Using Black-Scholes and the actual price

of the option, solve for volatility.

Is the implied volatility consistent with the

Put Option Value: Black-Scholes

P=Xe -rT [1-N(d 2 )] - S 0 e -dT [1-N(d 1 )]

Using the sample data

P = $95e (-.10X.25) (1-.5714) - $100 (1-.6664)

P = $6.35

Put Option Valuation: Using Put-Call Parity

P = C + PV (X) - S o

= C + Xe -rT - S o Using the example data

C = 13.70 X = 95 S = 100

r = 10 T = 25

P = 13.70 + 95 e -.10 X 25 - 100

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 (d 1 ) Put = N (d 1 ) - 1

Option Elasticity Percentage change in the option’s value given a 1% change in the value of the underlying stock

Portfolio Insurance - Protecting Against Declines in Stock Value

• Buying Puts - results in downside protection with unlimited upside potential