equity related products futures and options

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Prof. Dr. Rainer StachuletzCorporate FinanceBerlin School of Economics slide no.: 1

Prof. Dr. Rainer StachuletzCorporate FinanceBerlin School of Economics

Equity Related Products

Futures and Options

Professor Dr. Rainer Stachuletz

Corporate Finance

Berlin School of Economics

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Futures

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Spot – Future - Parity

Today, one (theoretical) Index-Future is sold at 4.090 € (1€ per Index-point). Long and Short-positions can be described by a profit and loss diagram:

4090

Index

Profit

Loss

Long Future = Buyer

Short Future = Seller

If you are Long-Future, then you may claim for delivery of „one index“ at a price of 4090 € at the maturity of the index-future. That means, if the index at delivery is quoted at more than 4090, you will win from your futures position.

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You hold an Index-Portfolio, currently valued at 5,500 € (1 Index-point = 1 €). If the annual risk free rate rf is at 3.5 % and the expected dividends on your Index portfolio are at 100 € (d = 100/5,500) , an Index – Future with one year to maturity has a fair price of:

€40.592,5F

0182,0035,01500,5F

dr1SF

0

0

F00

To prevent our Index-Portfolio from losses, we could hedge the price risk by taking a short – future position (selling a future at 5,592.40).

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Assets Payoff1 Payoff2 Payoff3 Payoff4 Payoff5

Stock Portfolio

+4500,00

+5000,00 +5500,00 +6000,00

+6500,00

Dividends +100,00 +100,00 +100,00 +100,00 +100,00

Short Future

+1092,40

+592,40 +92,40 -407,60 -907,60

Total +5692,40

+5692,40 +5692,40 +5692,40

+5692,40

The total expected payoffs from your portfolio will depend on the future state of the environment (see below payoffs 1-5). A decreasing stock market will be compensated by profits from the short future position, increasing stock prices will be outbalanced by losses due to payment obligations from the future.

Loss

Profit

5692,40

Index

Shor

t Fu

ture

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Spot – Future - ParityAssets Payoff1 Payoff2 Payoff3 Payoff4 Payoff5

Stock Portfolio

+4500,00

+5000,00

+5500,00

+6000,00

+6500,00

Dividends +100,00

+100,00 +100,00 +100,00 +100,00

Short Future +1092,40

+592,40 +92,40 -407,60 -907,60

Total +5692,40

+5692,40

+5692,40

+5692,40

+5692,40

Initially you have paid 5,500 € for your stock portfolio. Taking the short future position, the final outcome of your portfolio will be 5,692,40 €, whatever the stock price will be, i.e. you will earn 192,40 which equals 3.5%. Obviously, this profit is riskless:

dr1SFrS

SDFF00F

0

00 Spot-Future-

Parity

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Rising future prices will – due to arbitrage trading - induce rising spot prices. For example, a future traded at 6,000 € is clearly overpriced, when the stock portfolio remains unchanged at 5,500 €. In this case, „smart“ traders will make arbitrage profits of 407,50 € per contract and bring back the market to equilibrium:

Action t0 t1

Borrow money at rF

(3,5%)+ 5,500.00 - 5,692.50

Buy/Sell Stock Portfolio - 5,500.00 + Stock

Sell/Buy Future at 6,000

0 + 6,000.00 - Stock

Total 0 + 307,50Note, that the arbitrage profit equals the difference between a fair- and mispriced future (6,000 – 5,592,40) plus Dividends. Higher Future prices will lead to massivly increased demand at spot markets until spot prices and futures are back to equilibrium.

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Spot – Future – ParityFinancial Market Stability

• Spot Markets and Future (Forward) Markets are interlinked.

• Mispriced spot or future market instruments will affect both markets

• Future market speculations that drive futures prices will also drive spot market prices due to arbitrage trading (et vice versa)

• Speculation on futures markets, resulting in higher future prices will induce higher spot market prices due to arbitrage trading. Finally this may result in spot market bubbles that jeopardizes the allocation mechanism of real goods markets.

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Options

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Economic Benefits Provided by Options

Derivative securities are instruments that derive their value from the value of other

assets.

Derivatives include options, futures, and swaps.

Options and other derivative securities have several important economic functions:

• Help bring about a more efficient allocation of risk; • Save transactions costs…sometimes it is cheaper to

trade a derivative than the asset underlying it, and• Permit investment strategies that would not otherwise

be possible.

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Options Vocabulary

Call option • Gives the holder the right to

purchase an asset at a specified price on or before a certain date

Put option • Gives the holder the right to sell

an asset at a specified price on or before a certain date

Strike price or exercise price: the price specified for purchase or sale in an option

contract

American or European option

• American options allow holders to exercise at any point prior to expiration

• European options allow holders to exercise only on the expiration date

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Options Vocabulary

Neither trade usually has any connection to the underlying firm.

Long position • The buyer of an option has a

long position, and has the right to exercise the option.

Short position

• The seller (or writer) of an option has a short position, and must fulfill the contract if the buyer exercises.

• As compensation, the seller receives the option premium.

Can trade options on an exchange (such as CBOE) or in the over-the-counter market.

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Moneyness of Options

Call Put

S>X In-the-money Out-of-the-money

S=X At-the-money At-the-money

S<X Out-of-the-money

In-the-money

S = current stock price

X = strike price

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Option Quotations

32.50

32.50

27.50

27.50

Strike

3.831.55May30.00

3.240.85April30.00

1.233.91May30.00

0.673.26April30.00

PutCallExpire

sOpti-Tech

In-the-money callsOut-of-the-money puts

In-the-money putsOut-of-the-money calls

Option quotations

• The price per share for an option contract, which is a contract to buy or sell 100 shares of the underlying stock.

• CBOE options expire on the third Saturday of the expiration month.

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Intrinsic and TimeValue of Options

Intrinsic value

• For in the money options: the difference between the current price of the underlying asset and the strike price (S-X for calls and X-S for puts).

• For out of the money options: the intrinsic value is zero.

Time value• The difference between an

option’s intrinsic value and its market price (premium)

• Consider the May call with $27.50 strike price from previous table:

•Intrinsic value = $30.00 - $27.50 = $2.50•Time value = $3.91 - $2.50 = $1.41

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Payoff Diagrams

Shows value of an option on the expiration date

Y-axis plots exercise value or “intrinsic value”

X-axis plots price of underlying asset

Use payoff diagrams

for:

Long and short positions

Gross and net positions (the net positions subtract the option

premium)

Payoff: the price of the option at expiration date

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Long Call Option Payoffs

Payoff

at

Exp

irati

on

-8stock price75 83

slope = 1

Net payoff

Payoff

x = $75, premium = $8

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Short Call Option Payoffs

x = $75, premium = $8

Payoff

at

exp

irati

on

+8

stock price

75

slope = -1

83

Net payoff

Payoff

• Both long and short positions have zero net payoff at a price of $83• On net basis, buyer of the call makes a profit when the price exceeds $

83; seller of the call makes a profit when price is below $83.

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Long Put Option PayoffsPayoff

at

exp

irati

on

-7

Price of stock

75

75

68

68

Net payoff

Payoff

x = 75, premium = $7

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Short Put Option Payoffs

x = 75, premium = $7Payoff

at

exp

irati

on

7

Stock price7568

-75

Net payoff

Payoff

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Portfolios of Options

Look at payoff diagrams for combinations of options rather than just one

Shows the range of potential strategies made possible by options

Some positions can be a form of portfolio insurance.

Some strategies allow investor to speculate on the volatility (or lack thereof) of a stock rather

than betting on which direction it will move.

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Portfolio Containing 1 Call and 1 Put (Long

Straddle)Call x = 30, premium = $4.5, Put x = 30, premium = $3.5

30

3822

-8

Net payoff

Payoff

• Buy a put and a call on the same stock at the same strike price and the same expiration date•Profits come with large price

changes in either direction.•Positive net payoff if the price

rises above $38 or falls below $22

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-20

-10

0

10

20

30

40

50

60

70

80

40 60 80 100 120 140 160 180 200

Option Strategies (Straddle)

Stock Price 40 60 80 100 120 140 160 180 200Long Call (Profit / Loss) -10 -10 -10 -10 -10 10 30 50 70Long Put (Profit / Loss) 75 55 35 15 -5 -5 -5 -5 -5Straddle (Profit/Loss) 65 45 25 5 -15 5 25 45 65

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Option Strategies (Strangle)

Strangle - Long call and long put (at different exercise prices) Strategy for profiting from high volatility

Share Price

Pos

itio

n V

alue

Strangle

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Option StrategiesSynthetic Long Future

Synthetic Long Future(Long Call & Short Put)

Pos

itio

n V

alue

Share Price

Long Call

Short Put

Exercise Price (Strike)

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Synthetic Short Future(Short Call & Long Put)

Pos

itio

n V

alue

Share PriceShort Call

Long Put

Exercise Price (Strike)

Option StrategiesSynthetic Short Future

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Option Strategies(Short Butterfly)

Stock Price 40 60 80 100 120 140 160 180 200Long Call (Profit / Loss) -10 -10 -10 -10 -10 10 30 50 70Long Put (Profit / Loss) 75 55 35 15 -5 -5 -5 -5 -5Short Call (Profit/Loss) 5 5 5 5 5 5 5 -15 -35Short Put (Profit/Loss) -37 -17 3 3 3 3 3 3 3Butterfly 33 33 33 13 -7 13 33 33 33

-60

-40

-20

0

20

40

60

80

100

40 60 80 100 120 140 160 180 200

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Other Option Portfolio Payoffs

Now look at portfolios containing options, stocks, and bonds:

Looking at these payoffs will help lead us to an important option pricing relationship:

put-call parity.

Construct portfolios that include options, stocks and bonds:

Stock and put options Bond and call options

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Gross Payoff of Stock + Put

x

x

Stock price

Payoff

at

expir

ati

on $X = strike price of put

• Position allows investor to profit if stock price rises above $X. • If stock price falls below $X, portfolio provides protection: put

option allows investor to sell at a price no lower than $X.

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Gross Payoff of Bond + Call

x

x

stock price

$X = strike price of call and face value of bond

Payoff

at

exp

irati

on

• The bond assures a minimum payoff of $X• The call allows for a higher payoff if the stock price rises

This payoff diagram and the preceding one are identical!

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Put-Call Parity

Future payoffs of “stock+put” are identical to payoffs of “bond+call” provided:

• Put and call have same exercise price and expiration date;

• Underlying stock pays no dividends during life of options;

• Put and call are European options;• Bond is risk-free, zero-coupon, price at maturity =

strike (X),• Bond matures when options expire.

If two assets A and B, have same future payoffs with certainty, then they should

sell for the same price nowPrice of put + price of stock = Price of

call + price of bondP + S = C + B

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Factors Affecting Option Prices(holding other factors equal)

Price of underlying

asset

• Asset price and call price are positively related.

• Asset price and put price are negatively related.

Time to expiration

• More time usually makes options more valuable.

Strike price• Higher X means higher put price;

lower X means higher call price.

Interest rate

• Calls: higher rate means higher call value.

• Puts: higher rate reduces put value.

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Evaluation Framework

Assume that a stock is currently quoted at 150 €. If nothing happens over the coming year, the stock‘s price will also be at 150 € in one year. The one-year risk free rate is at 10%.

Under this assumptions, a Call – Option, maturing one year from now with a strike of 120 € is easy to value:

1 year

Stock – price t0: 150 €

Stock – price t1: 150 €Strike Call t1: 120 €

= 120 X 2,7184 – 0,10Strike Call t0 (PV): 108,58 €

The Intrinsic Value of the Call (X-S) is 41,42 € !!

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Evaluation Framework

Under this conditions, the Call Option Pricing Model is like:

10,0

r

CALL

7184,212015042,41

eXSPV

This price is a fair price, as it does not allow to gain risk-free profits from arbitrage-trading:

One could also borrow money to buy the stock now. At a risk free rate of 10% p.a. the cost of borrowing over one year will add to ((150€ x (2,7184 0,10) - 1)) 15.7764 €.

The other way round – borrowing money to buy the option - and one year later the stock leads to the same borrowing costs: Borrowing of 41,42 € at 10% means to pay back 41,42 x 2,7184 0,10 = 45,7764 € after one year. Netted with the profit from the option‘s exercise at a strike of 120 € (150 € - 120 € = 30 €), the costs add to 15.7764 €.

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Determinants of Option Prices

Variable Direction Option Price

Strike ...the higher the strike

...the smaller the price

Term to Exercise

...the longer the duration

...the higher the price

Price of the Underlying



Interest Rate ...the higher the rate


Volatility of the Underlying

...the higher the volatility


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Price & Value Chart

Strike

Intrinsic Value

Premium

Option PriceIntrinsic ValueTime Value

Stock Price

Option Price

Time Value

in the money

at the money

Out of the money

„Greeks“ show the sensitivity of the option price referring to:

= Delta:Call Price and Spot Price= Gamma:Delta

= Theta:Call Price and Time to Expiration

= Kappa / Vega:Call Price and Volatility

= Rho:Call Price and Int. Rate

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Factors Affecting Option PricesVolatility

Suppose a stock now worth $40 might increase or decrease in value by $10:

Call option with X = $40 will pay $10 or $0.

Now suppose a stock worth $40 might increase or decrease in value by $20:

Call option with X = $40 will pay $20 or $0.

The 2nd call option is more valuable…upside is better, downside the same as the 1st option.

equity related products futures and options

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