Chap 6 - Derivatives and Risk-Neutral Valuation

Question 1

What is traded in the organized markets ?

Answer 1

Organized markets:
* Products: Futures, options * Standard contracts * Exchange rules * Clearing house

Question 1

What is a derivative product ?

Answer 1

Definition :
* Financial instrument whose value depends on the price of an underlying asset
(real or financial instrument).

Contract types:
* Forward
contracts
* Options
* Swaps
* FRAs
* Etc

Question 2

What is traded in the over the Counter (OTC) markets ?

Answer 2

Over the Counter (OTC) markets :
* Products: Forwards, swaps, exotic options * Non-standard contracts between two
parties
* No clearing house * 70% of transactions

Question 3

What is a foward ?

Answer 3

Definition :
The investor undertakes to buy (sell) a specified quantity of the underlying at a specified
price on a specified date.

Foward price = S0e^rt

How it works :
* Over-the-counter (OTC) contracts
* No exchange before maturity
* Forward price set at t=0 so that contract value = 0
* Delivery or cash settlement (more common)

Question 4

What is the formula for a foward price with dividends ?

Answer 4

Forward price in
discrete compounding : F = (S0 - I)e^rt

I = Present value of dividends

Question 5

What is the formula for the continuous forward price ?

Answer 5

F = S0^e(r-q)t

q = Average annual dividend rate, under continuous compunding

Question 6

What are some disavantages of using forwards ?

Answer 6

Disadvantages of Forward :

• Difficult to exit a Forward position (customized contract)
• Impossible to cancel the contract
• Possible to offset the position with a 2nd contract, but difficult to do
• Credit risk
• Delivery risk

Solution: Futures contracts

Question 7

What is a future ?

Answer 7

Definition :
* The investor undertakes to buy (sell) a specified quantity of the underlying at a specified
price on a specified date.

Future price = S0e^rt

How it works
* Standardized contract on an organized market
* Contract usually closed before maturity
* Marking to Market (MTM)

Question 8

What are some advantages of using futures and it’s biggest disadvantage ?

Answer 8

Advantages of Futures :

Market standardization and clearing house:
* Increases contract liquidity (entry and exit)
* Makes it easy to compare prices
* Reduces default risk (Marking to Market, margin replenishment)

Disadvantage of futures :
* No customized contracts

Question 9

What is a Initial margin ?

Answer 9

Initial margin :

• Future market participants must have collateral to take long or short
  positions.
• Represents a percentage of the asset purchase price

Question 10

What is a Maintenance margin ?

Answer 10

Maintenance margin :

• Minimum margin to be maintained
• If the value of collateral falls below the minimum margin - Margin Call

Question 11

What are the two types of options ?

Answer 11

1- Call
* the right, but not the obligation, to buy the underlying at a given price during a
predetermined period.

2- Put
* the right, but not the obligation, to sell the underlying at a given price during a
predetermined period.

Question 12

What is the formula of the payout from a call ?

Answer 12

Call = Max( ST-K, 0 )

Question 13

What is the formula of the payout from a put ?

Answer 13

Put = Max( K-ST,0)

Question 14

What are the Black and Scholes formulas ?

Answer 14

In certanity : C = S0 - Ke^-rt

Uncertanity : C= S0*N(d1) - Ke^-rt * N(d2)

With N(d1) and N(d2) representing probabilistic adjustments for uncertainty

Question 15

What is Put-Call parity ?

Answer 15

Put-Call parity

Let’s assume we have the following two portfolios:
* Portfolio A = A European call and a quantity of money equivalent to 𝐾𝑒(–rt)
* Portfolio B = A European put and a stock
Both portfolios are worth at expiration:
max( 𝑆T,𝐾)
Both portfolios should have the
same value today:

𝑐 + 𝐾𝑒(–rt) = 𝑝 + 𝑆0

Question 16

Example of how to determine a bond price to prevent arbritage :

Answer 16

Nine-month riskless securities trade for $97,000, and 12-month riskless securities sell for P (both with $100,000 face values and zero coupons). A forward
contract on a three-month, riskless, zero-coupon bond, with a $100,000 face value and a delivery of nine months, specifies a forward price of $99,000. What is the arbitrage-free price of the 12-month zero-coupon security (i.e., P)? The 12-month bond offers a ratio of terminal wealth to investment of ($100,000/P). The nine-month bond reinvested for three months using the forward contract offers a 12-month wealth ratio of ($100,000/$97,000) ($100,000/$99,000).
Setting the two wealth ratios equal and solving for P generates P=$96,030.
The 12-month bond must sell for $96,030 to prevent arbitrage.

Question 17

In a risk-neutral world (in which investors do not require risk premiums for bearing risks), the forward price will be driven toward equaling the expected spot price because any other relationship would allow trading that offered abnormal expected return (note that due to the assumption of risk neutrality there would be no
concern regarding risk).

Answer 17

True

Question 18

What is a reference rate and some examples ? (US and EU)

Answer 18

A reference rate is a market rate specified in contracts such as a forward contract that fluctuates with market conditions and drives the magnitude and direction
of cash settlements. (LIBOR, Euribor, SOFR)

Question 19

The term notional principal is used to indicate that the principal amount is not actually exchanged, but rather serves to scale the size of the rate-related payments.

Answer 19

True

Question 20

Application on how to find the FRA rate on a perfect market:
FT−t = [(T × rT) − (t × rt)] / (T − t)

A three-year riskless security trades at a yield of 3.4%, whereas a forward contract on a two-year riskless security that settles in three years trades at a forward rate of 2.4%. Assuming that the rates are continuously compounded, what is the no-arbitrage yield of a five-year riskless security? Inserting 3.4% as the shorter-term rate in Equation 6.2 and 2.4% as the left side of Equation 6.2, the longer-term rate, RT, can be solved as 3.0%, noting that T=5 and t=3. Note that earning 3.0% for five years (15%) is equal to the sum of earning 3.4% for three years (10.2%) and 2.4% for two years (4.8%).
The rates may be summed due to the assumption of continuous compounding.

Question 21

When actual market prices deviate from arbitrage-free prices,
investors may use skill-based strategies that attempt to earn superior profits by anticipating that relative prices will tend to revert toward their arbitrage-free levels. Relative value hedge fund strategies (discussed in Chapter 17) are examples of such
strategies.

Answer 21

True That

Question 22

What is a commodity swap ?

Answer 22

A commodity swap is a portfolio of commodity forwards. Typically, the settlement times are equally spaced. For example, an oil refinery might regularly need to purchase crude oil. Rather than bear the risk of fluctuating oil prices, the refinery may decide to lock in the purchase price of the oil by entering various forward contracts to purchase the oil at prespecified prices (i.e., to swap cash for oil). Instead of entering into a series of separate forward contracts, the refinery may enter into a single swap that calls for quarterly or monthly exchanges through time at prices set at the initiation of the swap.

Question 23

In summary, for forward contracts on financial securities, the slope and curvature of the term structure of forward prices (the forward curve) are driven entirely by the relationship between the underlying security’s dividend yield and the riskless interest rate (both of which may vary in T). The forward curve will be flat when r=q, upward sloping when r>q, and downward sloping when q>r.

Answer 23

True

Question 24

What is cost of carry (Carrying cost) ?

Answer 24

A cost of carry (or carrying cost) is any direct financial difference between maintaining a position in the cash market and maintaining a position in the forward market. For example, being physically long wheat
requires storage (and generates storage costs), whereas being long a forward contract on wheat does not require physical storage.

Question 25

What is a convinience yield ?

Answer 25

A convenience yield (y), is the
economic benefit that the holder of a physical inventory (e.g., a commodity) receives from directly holding the inventory rather than having a long position in a forward contract on the physical assets.

Question 26

Is it possible to get information of future prices from the shape of foward interest and the foward prices ?

Answer 26

Forward prices on financial assets do not reveal information on future price
changes beyond the information already contained in spot prices. For example, it is a mistake to interpret the shape of forward interest rate curves as containing information beyond what is in the term structure of spot rates because there is a one-to-one
relationship between the two structures. Similarly, expected spot prices do not drive the spread between forward prices and spot prices for financial assets.

Question 27

What is a marginal market participant ?

Answer 27

The marginal market participant to
a derivative contract is any entity with individual costs and benefits that make the
entity indifferent between physical positions and synthetic positions.

Question 28

What is marked-to-market ?

Answer 28

The term marked-to-market means that the side of a
futures contract that benefits from a price change receives cash from the other side
of the contract (and vice versa) throughout the contract’s life. The cash exchanges
resulting from positions being marked-to-market are intended to cause each side of
the derivative to have a zero market value at the end of each day.

Question 29

What is crisis at maturity ?

Answer 29

The importance of the marking-to-market process is to avoid the counterparty
risk known as the crisis at maturity. A crisis at maturity is when the party owing
a payment is forced at the last moment to reveal that it cannot afford to make the
payment or when the party obligated to deliver the asset at the original price is forced
to reveal that it cannot deliver the asset.

Question 30

If a Initial Margin is 50 000$ and the trader losses money beyond the maintenance margin, down to 20 000$, explain the processus for a margin call:

Answer 30

The initial collateral of
$50,000 falls to a remaining margin balance of $20,000.
The trader receives a margin call, since the remaining margin is less than the
maintenance margin requirement. The amount of the margin call is $30,000 to
bring the margin back to the initial margin requirement.

Question 31

What is Rolling contracts ?

Answer 31

Rolling contracts refers to the process of closing positions in short-term
futures contracts and simultaneously replacing the exposure by establishing similar positions with longer terms.
In other words, to maintain an exposure in the forward market, it is necessary to close a position in one contract as it approaches or reaches settlement and open a new position in a contract with the same underlying commodity but with a longer time to
settlement.

Question 32

What is a covered call ?

Answer 32

A covered call combines being long an asset with being short a call
option on the same asset. Note from the diagrams that a covered call has the same
net risk exposure as a naked put.

Question 33

What is a protective put ?

Answer 33

A protective put combines being long an asset with a long position in a put
option on the same asset. Note from the diagrams that a protective put has the same
net risk exposure as a call option.

Question 34

What is an option spread ?

Answer 34

An option spread (1) contains either call options or put options (not both),
and (2) contains both long and short positions in options with the same underlying
asset. Option spreads contain options that differ with regard to strike price, expiration date, or both. Option spreads based on differences only in expiration date are
termed calendar spreads, or horizontal spreads. The option spreads that differ
only by strike price are often referred to as vertical spreads. Diagonal spreads
differ by both expiration date and strike price.

Question 35

What is an option combination ?

Answer 35

An option combination contains both calls and puts on the same underlying asset.

Question 36

What is an option straddle ?

Answer 36

An option straddle is
a position in a call and put with the same sign (i.e., long or short), the same underlying asset, the same expiration date, and the same strike price.

Question 37

What is an option strangle ?

Answer 37

An option strangle is
a position in a call and put with the same sign, the same underlying asset, the same expiration date, but different strike prices.

Question 38

How does syntethic positions work ?

Answer 38

Consider an option combination with a single call option and a single put option with the same underlying asset, the same time to expiration, but opposite signs. If the strike prices of the call and put are the same, the combination is a synthetic position in the underlying asset. If the call option is the long position and the put is short, the result is a synthetic long position in the underlying asset. If the call option is the short position and the put is long, the result
is a synthetic short position in the underlying asset.

Question 39

How is it possible to create a risk reversal strategy ?

Answer 39

A long out-of-the-money
call combined with a short out-of-the-money put on the same asset and with the same expiration date is termed a risk reversal.

Question 40

What is an option collar ?

Answer 40

An option collar uses positions in options to limit the upside and downside exposure of a position to a price or rate, with the most common example being when the
risk of a price or rate is limited by establishing a long position in a put option with a strike price K1 and a short position in a call option with a strike price or rate of K2, in which K1 < K2.