Options

Question 1

What are derivatives?

Answer 1

Derivative financial instruments or simply derivatives are instruments whose values derive or emanate from the value of one or more underlying securities

• Futures
• Forward Contracts - Options
• Swaps

Question 2

What is a call option?

Answer 2

A call option is a contract giving its owner the right to buy a fixed number of shares of a specified common stock at a fixed price at any time on or before a given date.

Question 3

What is a put option?

Answer 3

A put option is a contract giving its owner the right to sell a fixed number of shares of a specified common stock at a fixed price at any time on or before a given date.

Question 4

What is an American option?

Answer 4

If the option can be exercised any time before the maturity date it is called an American option.

Question 5

What is a European option?

Answer 5

If it is only possible to exercise it at the date of expiration, it is termed a European option.

Question 6

What are three alternative actions to options trading?

Answer 6

On any trading day an owner of an option may:
• sell it back at its concurrent market price
- canceling the position
• exercise the option (if American type (but with dividends))
• retain the option and do nothing

Question 7

To what options are standardised to?

Answer 7

• Underlying security
- e.g., a specific stock: Google Inc.
• Time to maturity
- normally standardized to three or six months.
• Date of maturity
- e.g., the third Friday of the month of expiration.
• Size of contract
- normally 100 shares, called 1 lot.
• Exercise price
- option with several different exercise prices are traded.

Question 8

Why are options standardized?

Answer 8

To facilitate well functioning secondary markets in options with

• high liquidity and
• efficient pricing

Question 9

Why trade in other underlying securities options (ex., Stock Market Indexes Exchange rates–currencies) ?

Answer 9

The prices of the underlying securities are

• very volatile
• concern four fundamental risks in an advanced economy

Question 10

Name four fundamental risks

Answer 10

1. Uncertainty about the stock market which is of essential importance to portfolio managers of intermediaries and firms.
2. Uncertainty about the exchange rates which is of pivotal importance to multinational firms and intermediaries in an environment of volatile exchange rates.
3. Uncertainty about the interest rates in an economic environment with movable and highly flexible interest rates.
4. Uncertainty about prices of raw materials and of food products which are highly volatile due to weather (exogenous) and economic (endogenous) factors.

Question 11

Why are options traded?

Answer 11

The necessity to limit risk (volatility in prices) opens up markets for derivative products which allows agents to do three basic things and profit from these activities:
• to speculate in price changes;
• to hedge the positions;
• to do arbitrage.

Question 12

What is Basis risk?

Answer 12

Basis risk comes from an imperfect match between a futures contract and position being hedged with respect to

• timing
• size of the contract
• underlying security

which may be different from the one offered by the standardised options.

Basis risk constitutes a reason for tailor-made options traded Over- the-Counter (OTC)

Question 13

Who trades tailor-made options?

Answer 13

• Large institutions like insurance companies and investment and pension funds have such special needs with respect to risk and timing when handling the underlying asset.

• Banks or investment firms issue such options to their customers - they stand the counter-party risk of defaulting customers.
- they have substantial capital resources, the necessary financial
cushion to handle the large risks involved in this trade.

Question 14

What are Payoff diagrams?

Answer 14

Payoff diagrams showing the gross value of an option at the maturity date, ignoring the initial transfer of the premium.

Question 15

What are Profit diagrams?

Answer 15

Profit diagrams showing the net gain or loss of a position in options by also accounting for the costs and gains of establishing the position.

Question 16

What is the difference between futures contract and option?

Answer 16

• The owner of a futures contract must exercise it or cancel his position.
• The owner of a call has the right not to use it if it is not in his interest.
• This difference gives the reason for the two instruments to co-exist and fulfill different functions.

Question 17

Describe Vertical price spreads

Answer 17

Strategies designed to generate profits from expectations about the change of the underlying stock price
(i) buy an option of one type (call or put)
(ii) simultaneously, write an option of the same type, of the same time to maturity
• BUT with a different exercise price.

Question 18

What are the Benefits of vertical price spreads?

Answer 18

i) Lower risk than when using only one option to speculate in price changes in the underlying stock;
- both gains and losses are limited compared to a strategy using only one option
(ii) Suitable if you want to speculate in relatively small price changes in the stock.

Question 19

What is the volatility value of an option?

Answer 19

The markets expectation about the volatility of the underlying stock price until the date of maturity is reflected in the volatility value of the option.

-> The difference between its current market value CM and its exercise value CK using the current stock price and the present value of the exercise price

Question 20

Describe Straddle

Answer 20

If you expect the volatility of the stock price to increase before date of maturity

• buy one at-the-money call and
• buy one at-the-money put
• with identical time to maturity.

Question 21

Describe Strangle

Answer 21

Strangles are like straddles but use different exercise prices for the two options, therefore, both the potential losses and potential profits are lower.

If you expect increased volatility of the stock price of the underlying equity before the date of maturity

• buy a call with higher exercise price than
• the put you also buy
• in the same underlying stock and with the same time to maturity.

Question 22

Describe Butterfly spread

Answer 22

Construction
• Use four options with the same time to maturity, but different exercise prices:
(i) Write two with the same exercise price (K2) which is in the middle of the butterfly
(ii) Buy one with a higher (K3) than (K2) and buy one with lower exercise price (K1) than (K2)
K1

Question 23

What is a Hedge?

Answer 23

Combines an option with its underlying stock in such a way that - either the stock protects the option against loss or the option protects the stock against loss.

Example:
- a covered position when writing an at-the-money call, - you own the stock you write the call for.

Question 24

What is a Spread?

Answer 24

Combines options of different exercise prices but with the same underlying stock, where some are bought and others are written.

Examples:
- Positive and negative vertical price spreads and butterfly spreads.

Question 25

What is a Combination?

Answer 25

Combines options of different types

• (puts or calls) on the same underlying stock - both are either bought
• or written.

Examples:
- Straddles and strangles

Question 26

What are Hedgers?

Answer 26

Are interested in reducing a risk they already face.

Example:

• buy a put option to hedge a long position in the underlying stock
• insures you against adverse stock price movements while still benefiting from favorable movements.

Question 27

What are Speculators?

Answer 27

While hedgers want to eliminate an exposure to movements in the price of an asset, speculators wish to take a position in the market: either they are betting that a price or a volatility will go up or they are betting that it will go down.

Example:
- Use the percentage leverage effect of an option.

Question 28

What are Arbitrageurs?

Answer 28

Lock in a riskless profit by simultaneously entering into transactions in two or more markets.

Example:
- if two identical instruments or portfolios trade at different prices, arbitrage opportunities exist and an arbitrageur explores the mispricing by locking in a riskless profit.

Question 29

The higher the stock price, the ….. premium needs to be paid

Answer 29

The higher the stock price, the LOWER premium needs to be paid as it is more harder to get into the money

Question 30

Explain Deep in-the-money call close to maturity

Answer 30

• almost no risk since it is very likely to be exercised or sold with a profit
• as percentage of money invested (premium) profit may be substantial

Question 31

Explain Deep out-of-the-money call close to maturity

Answer 31

• extremely risky since it is very likely to expire without value
• as percentage of money invested (premium) loss may be 100%

Question 32

Name two other financial instruments that can be related to European call

Answer 32

• underlying stock
  
  • > long call gives the right to buy this asset at the exercise price at the date of maturity
• risk free bond with face value equal to the exercise price
  
  • > long call like a postponed expenditure or loan repayment

Question 33

Assumption of no risk free arbitrage opportunities

Answer 33

Does not assume anything about risk preferences

- they may be risk neutral, risk averse or risk lovers • Assumes only that investors prefer more to less
 - basic and plausible restriction

Question 34

Relationship between call premium and the current price of the underlying asset

Answer 34

-> Pivotal parameter since current value of this ratio affects probability of the option ending up in-the-money at the date of maturity

(1) The higher the current stock price - the more valuable the call (more likely that the call will end up in-the-money- at the date of maturity)
(2) The less valuable the put (less likely that the put will end up in-the-money- at the date of maturity)

Question 35

Relationship between call premium and the exercise price

Answer 35

(1) The higher the exercise price - the less valuable the call (less likely that the call will end up in-the-money - at the date of maturity)
(2) The more valuable the put (more likely that the put will end up in-the-money - at the date of maturity)

Question 36

Relationship between call premium and the risk-free interest rate

Answer 36

The higher the risk free interest rate

• the more valuable the call (by not paying the exercise price today you save the interest earned on the exercise price until date of maturity)
• the less valuable the put (by not receiving the exercise price today you lose the interest earned on the exercise price until date of maturity)

Question 37

Relationship between call premium and the volatility of the price of the underlying asset

Answer 37

A higher stock price volatility increases the value of an
option
- both for a call and a put

Question 38

Relationship between call premium and the time to maturity

Answer 38

(1) A longer time to maturity intensifies the effect of the interest rate ( r) - the volatility (σ).
(2) A longer time to maturity lowers the present value of the exercise price
(3) Lower present value of exercise price
- > increases value of call
- > decreases value of put
(4) A longer time to maturity implies a higher effective volatility
- > stock prices move more over a longer time horizon
(5) THUS:

• increases value of call (both because of a lower present value of the exercise price and a higher effective volatility)
• ambiguous effect on value of put (lower present value of the exercise price lowers value of put while higher effective volatility increases value)

Question 39

Relationship between call premium and the dividend

Answer 39

A dividend payment lowers price of the underlying stock by roughly the size of the dividend -> a lower stock price

- > lower call value
- > higher put value

Question 40

General restrictions of value of call before date of maturity

Answer 40

At any time before date of maturity, the value of a European call (C) is at
least the greater of zero, and the difference between the stock price and the present value of the exercise price

Question 41

General restrictions of value of an American option

Answer 41

Never pays to exercise an American call before date of maturity if underlying stock does not pay a dividend:

• Unexercised American in-the-money call worth at least S-PV(K).
• But if exercised American in-the-money call worth only S-K.

Value of an American option is at least as large as value of a European option since it has more flexibility

• an American option can be used in the same way as a European with the added advantage of early exercise
• if underlying stock pays large dividend may be more valuable to exercise an American call early
  
  • even if underlying stock does not pay dividend may be more valuable to exercise an American put early

Question 42

General restrictions of Put-Call-Parity for European options

Answer 42

Since the two portfolios have identical cash flows at date of maturity they must have the same investment costs today; otherwise there exists a risk free arbitrage opportunity

Question 43

Describe the relation between forward and spot price

Answer 43

Forward contract is an agreement to buy or sell an asset at a certain future time for a certain price – delivery price

long (short) forward position:
-> Forward price (F) is the delivery price that will make the value of forward contract zero when entered into

Spot price equals the present value of the forward price discounted at the risk free interest rate

Question 44

Describe the relation between prices of identical options of same type

Answer 44

For two identical options of the same type (put or call) but with different exercise prices (K2 > K1):

- > for calls: C(K1) ≥ C(K2)
- > for puts: P(K2) ≥ P(K1).

Question 45

List the steps for pricing in Binomial Model

Answer 45

1. Form a perfect hedge portfolio (risk free) of three instruments (i) European call; (ii) risk free bond; and (iii) underlying stock
2. Find a pricing restriction on the relative prices of these three instruments using the Principle of no risk free arbitrage
3. Use this price restriction to determine the value of the call today (C) as a function of the prices of the underlying stock (S) and the risk free bond

Question 46

What if in the Binomial model (1) 1+r>u>d? or (2) u>d>1+r?

Answer 46

(1) no one would invest in stocks as risk free rate yield higher returns
(2) no one would invest in risk free as the worse case scenario of stocks gives higher returns than the risk free rate

Question 47

What is a hedge ratio?

Answer 47

Hedge ratio informs about how large fraction of the underlying stock is needed per written call to perfectly hedge the written call

Question 48

Definition of a Dynamic Risk Free Arbitrage Strategy

Answer 48

Construct of a portfolio of assets such that

(i) locks in a sure profit today by exploiting the mispricing in the market (ii)a dynamic adjustment of the composition of the portfolio such that:
- it has zero value at the date of maturity and - the portfolio adjustments are self-financing
- no net costs are incurred after the initial date

Question 49

Construction if a Dynamic Risk Free Arbitrage Strategy

Answer 49

(i) write a call and sell it at the market price
(ii) protect the written call by forming a perfect hedge portfolio consisting of
- a fraction of the underlying stock equal to the hedge ratio and
- a borrowed risk free amount according to general formula for the price of a call.
(iii) as the stock price changes adjust the composition of the hedge portfolio such that at each stock price
- it is a perfectly hedged to the written call.
(iv) this dynamic portfolio strategy
- does not incur any costs or gains between today and the date of maturity
- generates a risk free profit equal to the initial mispricing and the interest earned on this amount.

Question 50

Describe the Risk-neutral probabilities

Answer 50

We have NOT assumed that the investors preferences are risk neutral. Principle of no risk free arbitrage only assumes that investors prefer more to less.
Unlike the hedge ratio ∆, the risk-neutral probabilities do NOT depend on the level of the stock price

Called risk neutral probabilities

• since a risk neutral investor would value the call as its expected value and discount at the risk free interest rate
• does not demand compensation to take on risk
• Risk-neutral probabilities do not depend on the original risk adjusted probabilities q and 1-q

Question 51

Interpret C= S * B(a,n,p’) - K * R^(-n) *B (a,n,p)

Answer 51

Composition of replicating portfolio -
B (a,n,p)= hedge ratio
S * B(a,n,p’) – the value of the long position in the stock
K * R^(-n) *B (a,n,p) -the amount borrowed at the risk free rate
B (a,n,p) - the share of the present value of the exercise price that is borrowed at the risk free rate

Question 52

Name three desirable properties of the stochastic process for stock prices

Answer 52

1. The price process should be consistent with weak form of market efficiency, i.e. present stock price impounds all information contained in past prices (past history and the way in which the present price has emerged from the past are irrelevant)
2. The price process should be scale independent; investors look for proportional or percentage returns (Probability of say a 10% return should be the same independently if the stock price is at 100 or 1000)
3. Because of limited liability stock prices can never go below zero (Rules out the normal distribution)

Question 53

Name the assumptions behind the Black & Scholes Formula

Answer 53

1. The stochastic process for the stock price is lognormal with constant parameters μ and σ.
2. Short selling of securities with full use of proceeds is permitted.
3. There are no transaction costs or taxes. All securities are perfectly
   divisible.
4. No dividends during the life of the derivative security.
5. Security trading is continuous.
6. The risk free continuously compounded interest rate is constant and the same for all maturities
7. There are No Risk Free Arbitrage Opportunities.

Question 54

Explain N(d1) and N(d2)

Answer 54

N(d1): the hedge ratio;
- The same interpretation as for ∆ in the Multiplicative Binomial Model
- the fraction of one share that you invest in in the underlying stock in the replicating portfolio
N(d2): fraction of the present value of the exercise price that you borrow at the risk free interest rate
- Probability that the call will end up in-the-money in a risk-neutral world (i.e. using Martingale probability p)

Question 55

What is delta natural strategy?

Answer 55

Delta neutral is a portfolio strategy utilizing multiple positions with balancing positive and negative deltas so that the overall delta of the assets in question totals zero.

-> Value does not change when stock price changes marginally - example: the perfectly hedged portfolio is delta neutral

Question 56

Explain Implied Volatility

Answer 56

A. Market assessment of volatility of a stock changes: the implied estimate can be a useful monitoring device to
expected future volatility.
B. Comparing it to the historical estimate we may infer changes in stock price volatility: Implied volatility is the best predictor of current volatility.
C. Implied volatility of one type of option can often be used when determining the value of a different option on the same underlying stock
D. Implied volatility may differ for options with different exercise prices but with the same underlying stock – volatility smile: estimate a composite implied volatility by calculating a weighted average of the individual implied volatilities.
E. The price of an at-the-money option is far more sensitive to volatility than a deep out-of-the-money option. Hence, it provides more information about the «true» volatility of the underlying stock
- use a higher weight when calculating a weighted average implied volatility or as the sole estimator of the implied volatility.

Question 57

Name three real options and explain them

Answer 57

Option to delay investment
 E.g., undeveloped land in the hands of real estate investor; a firm that owns a patent; a natural resources company that has undeveloped reserves
Option to expand
 E.g., biotechnology; software production; entry into a growing market; R&D; mobile network license
Option to abandon
 E.g., escape clauses in contracts; customer incentives

Question 58

Key Insights from Real Options

Answer 58

 Out-of-the-money real options have value
-> Even if an investment has a negative NPV, if there is a chance it could be positive in the future, the opportunity is worth something today.
 In-the-money real options need not be exercised immediately
-> The option to delay may be worth more than the NPV of undertaking the investment immediately.
 Waiting is valuable
-> By waiting for uncertainty to resolve you can make better decisions.
 Delay investment expenses as much as possible
-> Committing capital before it is absolutely necessary gives up the option to make a better decision once uncertainty is resolved.
 Create value by exploiting real options
->The firm must continually re-evaluate its investment opportunities, including the options to delay or abandon projects, as well as to create or grow them.