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What is a Derivative?

- A derivative is an instrument whose value depends on, or is derived from, the value of another asset.
- Examples: futures, forwards, swaps, options, exotics…

Obs: you may jump to slide #21 to start direct with options.

How Derivatives are Used

- To hedge risks
- To speculate (take a view on the future direction of the market)
- To lock in an arbitrage profit
- To change the nature of a liability
- To change the nature of an investment without incurring the costs of selling one portfolio and buying another

Options vs. Futures/Forwards

- A futures/forward contract gives the holder the obligation to buy or sell at a certain price at a certain date in the future
- An option gives the holder the right, but not the obligation to buy or sell at a certain price at a certain date in the future

Foreign Exchange Quotes for GBP, (£) May 24, 2010

The forward price may be different for contracts of different maturities (as shown by the table)

Long position and short position

- The party that has agreed to buyhas a long position
- The party that has agreed to sell has a short position

Example

- On May 24, 2010 the treasurer of a corporation enters into a long forward contract to buy £1 million in six months at an exchange rate of 1.4422
- This obligates the corporation to pay $1,442,200 for £1 million on November 24, 2010
- What are the possible outcomes?

Svenska termer

Forwards och Terminer

• Spotkontrakt: en överenskommelse mellan två

parter att utbyta något idag för ett specificerat

pris, spotpriset. à vista marknad.

• Terminskontrakt: en överenskommelse

(skyldighet) mellan två parter att utbyta något

för ett specificerat pris, terminspriset, vid en

specifik framtida tidpunkt, lösendagen.

Price of Underlying at Maturity, ST

Profit from a Long Forward Position (K= delivery price=forward price at the time contract is entered into)Payoff diagram

K

Price of Underlying

at Maturity, ST

Profit from a ShortForward Position (K= delivery price=forward price at the time contract is entered into)K

Futures Contracts

- Agreement to buy or sell an asset for a certain price at a certain time
- Similar to forward contract
- a forward contract is traded over the counter

(OTC) (Skräddarsydd)

- a futures contract is standardized and traded on an exchange.

CME Group

NYSE Euronext,

BM&F (Sao Paulo, Brazil)

TIFFE (Tokyo)

Key Points About Futures

- They are settled daily
- Closing out a futures position involves entering into an offsetting trade
- Most contracts are closed out before maturity

Margins

- A margin is cash or marketable securities deposited by an investor with his or her broker
- The balance in the margin account is adjusted to reflect daily settlement
- Margins minimize the possibility of a loss through a default on a contract

Pricingof forward

- Guld (commodities): F = (1 + rf + s) · S0
- Finansiella tillgångar: F = (1 + rf) · S0

S0 is the spot price.

S is the storagecost

rf is risk freeinterestrate

F is the forward price

Examples of Futures Contracts

Agreement to:

- Buy 100 oz. of gold @ US$1400/oz. in December
- Sell £62,500 @ 1.4500 US$/£ in March
- Sell 1,000 bbl. of oil @ US$90/bbl. in April

Oz: ounce

Bbl: barrel

Example : An Arbitrage Opportunity?

Suppose that:

The spot price of gold is US$1,400

The 1-year forward price of gold is US$1,500

The 1-year US$ interest rate is 5% per annum

Q: What should be the 1-year forward price? Is there an arbitrage opportunity?

The Forward Price of Gold

If the spot price of gold is S and the forward price for a contract deliverable in T years is F, then

F = S (1+r )T

where r is the 1-year (domestic currency) risk-free rate of interest.

In our examples, S = 1400, T = 1, and r=0.05 so that

F = 1400(1+0.05) = 1470

Hedging Examples

- An investor owns 1,000 Microsoft shares currently worth $28 per share. A two-month put with a strike price of $27.50 costs $1. The investor decides to hedge by buying 10 contracts

Some Terminology

- Open interest: the total number of contracts outstanding
- equal to number of long positions or number of short positions
- Settlement price: the price just before the final bell each day
- used for the daily settlement process
- Trading Volume : the number of trades in one day

Forward Contracts vs Futures Contracts

FORWARDS

FUTURES

Private contract between 2 parties

Exchange traded

Non-standard contract

Standard contract

Usually 1 specified delivery date

Range of delivery dates

Settled at end of contract

Settled daily

Delivery or final cash

Contract usually closed out

settlement usually occurs

prior to maturity

Some credit risk

Virtually no credit risk

Options

The right but not the obligation…

Options

- A call option is an option to buy a certain asset by a certain date for a certain price (the strike price)
- A put option is an option to sell a certain asset by a certain date for a certain price (the strike price)

American vs. European Options

- An American option can be exercised at any time during its life
- A European option can be exercised only at maturity
- The time value will be lost when you exercise prematurely.

Option Value: Example

Option values given an exercise price of $720

What are the payoff limits for call option buyers? Sellers?

What are the payoff limits for put option buyers? Sellers?

Call Option Value

Call option value (buyer) given a $720 exercise price.

Call option value

$120

720 840

Share Price

Call Option Profit

$20 call option (buyer) given a $720 exercise price

Call option value

$100

720 840

Share Price

Call Option Value

Call option payoff (seller) given a $720 exercise price.

720 840

Call option $ payoff

$-120

Share Price

Call Option Profit

$20 call option (seller) given a $720 exercise price:

720 840

$-100

Call option $ payoff

$-120

Share Price

Call Option: Example

How much must the stock be worth at expiration in order for a call holder to break even if the exercise price is $50 and the call premium was $4?

Put Option Value

Put option value (buyer) given a $720 exercise price:

Put option value

$120

600 720

Share Price

Put Option Profit

$30 put option (buyer) given a $720 exercise price:

Put option value

$90

Share Price

600 720

Put Option Value

Put option payoff (seller) given a $720 exercise price.

Share Price

-$120

Put option $ payoff

600 720

Put Option Profit

$30 put option (seller) given a $720 exercise price.

-$90

Share Price

Put option $ payoff

600 720

Put Options: Example

What is your return on exercising a put option which was purchased for $10 with an exercise price of $85? The stock price at expiration is $81.

Option Value

- Point A -When the stock is worthless, the option is worthless.
- Point B -When the stock price becomes very high, the option price approaches the stock price less the present value of the exercise price.
- Point C -The option price always exceeds its minimum value (except at maturity or when stock price is zero).
- The value of an option increases with both the variability of the share price and the time to expiration.

Option Value

Components of the Option Price

1 - Underlying stock price

2 - Strike or Exercise price

3 - Volatility of the stock returns (standard deviation of annual returns)

4 - Time to option expiration

5 - Time value of money (discount rate)

Put-Call Parity: No Dividends

- Consider the following 2 portfolios:
- Portfolio A: call option on a stock + zero-coupon bond (or a deposit) that pays K at time T
- Portfolio B: Put option on the stock + the stock

Values of Portfolios are the same at expiration (förfalldag)

The Put-Call Parity Result

- Both are worth max(ST, K ) at the maturity of the options
- They must therefore be worth the same today. This means thatc + Ke -rT= p + S0

Suppose that

- What are the put option price?

c + Ke -rT= p + S0

p = c-S0 +Ke -rT

=3-31+30*EXP(-0,1*0,25)

= 1,259

Synthetic options

Two or more options combines together creates exotic options

Long call

Straddle

Option Value: profit diagram for a straddleStraddle - Long call and long put

- Strategy for profiting from high volatility

Position Value

Share Price

Option Value

Straddle - Long call and long put

- Strategy for profiting from high volatility

Straddle

Position Value

Share Price

An investor may take a long straddle position if he thinks the market is highly volatile, but does not know in which direction it is going to move.

Exotic options: a butterfly option

- A butterfly

x3

x2

x1

A long butterfly position will make profit if the future volatility is lower than the implied volatility.

The spread is created by buying a call with a relatively low strike (x1), buying a call with a relatively high strike (x3), and shorting two calls with a strike in between (x2).

30

20

10

Terminal

stock price ($)

70

80

90

100

0

110

120

130

-5

Long CallProfit from buying one European call option: option price = $5, strike price = $100, option life = 2 months

110

120

130

5

0

70

80

90

100

Terminal

stock price ($)

-10

-20

-30

Short CallProfit from writing one European call option: option price = $5, strike price = $100

30

20

10

Terminal

stock price ($)

0

40

50

60

70

80

90

100

-7

Long PutProfit from buying a European put option: option price = $7, strike price = $70

Terminal

stock price ($)

7

40

50

60

0

70

80

90

100

-10

-20

-30

Short PutProfit from writing a European put option: option price = $7, strike price = $70

Payoffs from OptionsWhat is the Option Position in Each Case?

K = Strike price, ST= Price of asset at maturity

Payoff

Payoff

K

K

ST

ST

Payoff

Payoff

K

K

ST

ST

Real options

- With the limited liability of the modern corporations, the shareholders´ equity can be regarded as a real option on the assets of the firm.
- The shareholder value of equity value is

max(VT −D, 0)

where VT is the value of the firm and Dis the debt repayment required.

Thus the company can be considered as a call option on the firm value V at the strike price of D.

Options on Real Assets

Real Options - Options embedded in real assets

Option to Expand

Option to Abandon

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