Derivatives Markets

Question 1

L1. Define Derivatives

Answer 1

An arrangement or product that derives its value from the value of the underlying asset.

Question 2

L1. OTC meaning and the agreement that takes place.

Answer 2

Over the counter markets. A bilateral agreement is created between the two parties covering all the transactions that will take place.

Question 3

L1. Forward Contracts?

Answer 3

An agreement to buy or sell an asset at a certain future time at a certain price.

Question 4

L1. Spot Contract?

Answer 4

An agreement to buy or sell an asset almost immediately, usually up to a few days.

Question 5

L1. What are the characteristics of a forward contract? Market, Party positions.

Answer 5

Traded in the OTC market.
Two positions for parties:
Long: Party agrees to buy the underlying asset at a future date and at a certain price.
Short position: Party agrees to sell the underlying asset a a future date at a certain price.

Question 6

L1. Forwards: Bid price and Offer price?

Answer 6

The bid price is the amount that a bank/ party will offer to buy the underlying asset for. The offer price is the price at which the counter party will be willing to buy the underlying asset for.

Question 7

L1. Define Hedging

Answer 7

Hedging against investment risk means strategically using financial instruments or market strategies to offer the risk of any adverse price movements. You are locking in a price at a given level. and preventing any further losses or gains given a change in the price of the hedged asset.

Question 8

L1. Define Futures

Answer 8

An agreement made to buy or sell an asset between two parties at a certain future date and at a certain price.

Question 9

L1. Features of Futures? (market, type of contract)

Answer 9

Usually traded on an exchange (so certain standardise features are placed on the contracts).
Futures can be used for both speculation and hedging.

Question 10

L1. What are the examples of markets that trade futures (3)?

Answer 10

1. Chicago Mercantile Exchange
2. NYSE Euronext
3. Tokyo Financial Exchange

Question 11

L1. What are the features of option contracts? (market, Types of Options, right of buyer)

Answer 11

Options contracts are traded on OTC or on exchanges and are used for both speculation and hedging.
The two types of options are Call Options and Put Options.
The option gives the holder the write to exercise the contract but this right does not have to be exercised.

Question 12

L1. Options: Call Options?

Answer 12

Gives the buyer the right to buy a certain asset (underlying asset) by or at a certain dates (maturity date) at a certain price (exercise price).

Question 13

L1. Options: Put Options?

Answer 13

The right to sell the underlying asset by the strike date for the exercise price.

Question 14

L2. What is the difference between most European and American Option?

Answer 14

American options can be exercised any time up to the maturity date, whereas, European options can only be exercised on the maturity date.

Question 15

L2. In the exchange-traded equity option market, how many shares are usually traded for one contract?

Answer 15

100

Question 16

L2. What are the key differences between a forwards and futures contract?

Answer 16

Forwards:

1. Private contract between 2 parties.
2. Non-standard contract
3. Usually 1 specified delivery date
4. Settled at end of contract
5. Delivery or final cash
6. Some credit risk

Futures

1. Exchange traded
2. Standard contract
3. Range of delivery dates
4. Settled daily
5. Contract usually closed out
6. Virtually no credit risk

Question 17

L2. What are the four positions one could hold in an options contract?

Answer 17

Buyer of call option
Seller of call option
Buyer of put option
Seller of put option

Question 18

L2. What would you expect to see with regards to the price of call options and put options as the strike price increases?

Answer 18

For call options (right to buy the underlying asset) the premium falls as the strike price increases.
For Put options (the right to sell the underlying asset), the premium increases as the strike price increases.

Question 19

L2. Key difference between futures and options with regards to the risk?

Answer 19

The risk for futures is far higher as the profit and loss are leveraged. For options the losses are limited to the amount paid for the premiums, whereas the gains are leveraged if the spot price is reached.

Question 20

L2. How would an individual with a long position in a futures contract close out their position?

Answer 20

They would take to a short position with the same delivery month for the goods at a later date. They thereby cover their position and lock in their profits or losses. A shorter of ta futures contract would do the opposite.

Question 21

L2. Futures Contract: Specify the particulars of a contract.
HINT

(Assets, Contract size, Delivery arrangements,
delivery months,
price quotes, price and position limits).

Answer 21

The asset: What is the asset that the futures investor agrees to buy or sell.
Contract Size: How many unites of the asset will be delivered at expiry.
Delivery Arrangements: Where will delivery take place.
Delivery Months: When will delivery of the asset take place.
Price Quotes: What current will the quote be in.
Price and positions limit: whether there are any.

Question 22

L2. Futures: When buying or selling futures contracts, what are some important things to note about timing?

Answer 22

Contract trade for the closest delivery months and a number of subsequent months.
Most futures contracts are closed out before delivery.

Question 23

L2. Futures: What ties the futures price to the spot price?

Answer 23

The final delivery of the underlying asset.

Question 24

L2. Futures contracts: What happens when a short position reaches the expiration date of a contract?

Answer 24

A notice of intention to deliver will be filed with the exchange. The intention confirms the grade of the asset that will be delivered and the chosen delivery locations.

Question 25

L2. Futures: What is important to note about commodities following a notice of intention to delivery?

Answer 25

The quality of commodities can differ in grades. The exchange will often set a a range of qualities of the underlying asset that can be delivered, with the price per quantity of item being adjusted following the quality being confirmed.

Question 26

L2. Futures: What are the negatives and positives of having contract sizes that are too large or small?

Answer 26

There are costs associated with each contract taken out. As such, if the contracts are too small, the costs for taking a large amount of contracts will be greater. If the contract month the other hand are too large, those investors wish to only hedge relatively small exposures will not be able to operate in the market.

Question 27

L2. Futures contracts: Locations effect on costs?

Answer 27

Sometimes, the entity shorting the contract will alter the price based on the location that the underlying asset will be delivered to.

Question 28

L2. Futures: Price limits and position limits and the purpose of these?

Answer 28

The exchange will set a daily fluctuation limit based on the previous days close. If the shift from the previous days price hits the limit, either higher or lower, then the market will close at limit up or limit down respectively. The limit has occasionally been known to change in the day. The purpose is to stop speculators having undue influence on the market.

Question 29

L2. STIR? (How long would these futures and forward contacts be traded for)?

Answer 29

Short Term Interest Rate.

Contract length is typically 3 months.

Question 30

L2. What is the process between the spot price and forward contract price converging?

Answer 30

When the Futures price is greater than the spot price, demand for the futures contracts will fall and the price will drop. Traders will take advantage of arbitrage opportunities. When the sport price is less than the futures price, companies that find it profitable to enter purchase the underlying asset will by the futures contract, leading to increased demand and the futures price rising.

Question 31

L2. What are Margin accounts used for?

Answer 31

To mitigate the risk of a entity defaulting on payment. It balances the day to day gains and losses on contracts. An initial margin requirement will be set in place by a BROKER, and if the margin falls below the maintenance level due to a depreciation in the value of a contract, the contract holder will have to add additional funds to their account to cover the losses. If they fail to oblige to this, the position will be closed.

Question 32

L2. How does the relationship between buyer and seller of a contract get settled by intermediary parties (think margin)?

Answer 32

If the value of an underlying asset drops, the long position’s broker will transfer funds to the exchange cleaning house, who will subsequently send the excess monies to the short’s broker and these funds will be available to the shorter to withdraw.

Question 33

L2. What is the extra margin supplied by an open position in a contract should a margin call be issued?

Answer 33

The variation margin.

Question 34

L2. Margin: What amount is required to be added to an account if the margin falls below the maintenance margin?

Answer 34

The account must be topped up to the level of the initial margin.

Question 35

L2. Margin: If the margin in an account falls below the maintenance margin, when is the investor required to increase the amount back to the initial margin level?

Answer 35

By the end of the next day, before close.

Question 36

L2. Futures contracts: Settlement price?

Answer 36

Previous settlement price is the price of the contract immediately before the close of the previous day. The settlement prices are used for calculating the days gains and losses.

Question 37

L2. Futures Contracts: Trading Volume and the difference between this and open interest?

Answer 37

The number of contracts traded in the respective day.
The difference between it and open interest is that open interest shows the amount of contracts outstanding, that is the number of long/ short contracts.

Question 38

L2. What are the two different patterns of futures markets that we observe and what do we call these markets?

Answer 38

When the price of a contract is an increasing function with time (cost of contract increases the more months away from delivery, we call this a NORMAL MARKET.
When the price of the futures contract is a decreasing function of time, we call this an inverted market.

Question 39

L2. Who regulates furthers markets in the US?

Answer 39

The Commodity Futures Trading Commission (CFTC)

Question 40

L2. What is the process called when an exchange, such as the CME, trades first based on orders made by the market?

Answer 40

Front running.

Question 41

L2. Define Contango. What would be the reasons for this with say, cacao.

Answer 41

A situation in which the spot or cash price of a commodity is lower than the forward price. There are costs involved in storing the product, such as warehousing costs, which lead to a higher forward price.

Question 42

L2. Define Backwardation. What is the role of arbitrageurs in ensuring spot = forward at point of delivery?

Answer 42

A situation in which the spot or cash price of a commodity is higher than the forward price.
The arbitrageurs will buy the commodity in the futures market and sell it in the spot market, as they can make a profit by buying low and selling high.

Question 43

L2. Futures: What is the information behind volume of trades?

Answer 43

Volume of trades is the number of active trades during a given period. A buyer will take a long position in x amount of contracts, with the market maker matching the long position to a seller making will to short x assets. x will add to the volume of trades. These contracts already exist, and the buyer of the long is simply opening contract with another previous buyer show now wishes to close off their position.

Question 44

L2. Futures: What is the information behind open interest?

Answer 44

Open interest indicates the number of options and futures contracts that are held by investors in active positions. These positions have not been closed off, expired or exercised. This number will reduce when holders or writers of options (or buyers and sellers of futures) close out their positions. For futures, this number will increase if new contracts have been created and decrease if positions are closed off through taking offsetting positions (shorting the asset if in a long position, which halves causes the long and short to cancel each other out).

Question 45

L2. Open Interest: Run through the scenario of 4 traders in the market, and open interest rising and falling.

Answer 45

TR1 long on contract, TR2 Short
OI: 1
TR3 Long on contract, TR2 Short
OI: 2
TR1 closes position by shorting the contract initially held. TR4 decides to close off their short position and buys the contract (long) from TR1, resulting in the closing off of the contract they originally introduced to the market.
OI falls to 1.
Note that, if TR3 had purchased the contract, the OI would remain at 2, as there are still two buyers and

Question 46

L2. Open interest: When will it increase?

Answer 46

When new contracts are created by sellers that do not hold the opposite position in the market.

Question 47

L2. Commodity Futures Trading Commission mission?

Answer 47

In the US: The mission of the Commodity Futures Trading Commission (CFTC) is to foster open, transparent, competitive and financially sound markets.

Question 48

L2. European Securities and Markets Authority mission?

Answer 48

In the EU: To enhance investor protection and promote stable an orderly financial market.

Question 49

L2. Financial Conduct Authority?

Answer 49

In the UK: We aim to make financial markets work well so that consumers get a fair deal.

Question 50

L2. STIR Contract Specifications? HINT

Description
Symbol
Unit of Trading
Delivery Date
Delivery Months
Quotation
Minimum Price Movement

Answer 50

Description: Cash settles future based on ICE Benchmark Administration Limited London Interbank Offered Rate (ICE LIBOR) rate for three months depots.
Symbol: L
Unit of Trading: Interest rate on a three month deposit of £500,000.
Delivery Months: March, June, September, December, and two serial months, such that 26 delivery months are available for trading, with the nearest three delivery months being three consecutive calendar months.
Quotation: 100.00 minus rate of interest .
Minimum Price Movement: First quarterly delivery month: Half Basis Point £6.25)
All other quarterly delivery months and all serial delivery months: One Basis Point (£12.50)

Question 51

L2. Perfect hedge?

Answer 51

A hedge that completely eliminates risk.

Question 52

L2. Hedge and Forget?

Answer 52

An assumption that we set in place to simplify or analysis of hedging. Once a hedge has been made, there is no attempt to adjust a hedge once it has been put into place.

Question 53

L2. When would you make a Short hedge on a forward contract? Relate to the pig farmer example.

Answer 53

When an entity already owns and asset and expects to sell it off at some point in the future. The entity winked take a short position in the futures market. It is when the asset is not currently owned but will be at a future date. E.g. a pig farmer who will be able to sell pigs in three months.

Question 54

L2. A long hedge?

Answer 54

Used when an entity knows that it will have to purchase an asset in the future and wants to lock in the price. now.

Question 55

L2. What is important to note about hedging with futures?

Answer 55

Most contracts get closed off prior to the delivery date, basically making the amount cash settled and then the entity with the long (short) position can buy (sell) the asset/s on the spot market without incurring storage or delivery costs.

Question 56

L2. Hedging: How could one argue against the use of hedging when investors or shareholders are concerned?

Answer 56

Hedging represents a reduction of risk to fluctuations in price. Some industries and business will have their level of profitability directly linked to the level of demand and fluctuations in their industry. If shareholders expect profits to rise when the price of an asset increases, say the price of gold when investing in gold mining companies, they may expect profits to rise inline with the gold price increase. If a hedge prevented this, profits for the business would be less and investors could be displeased. The best solution is to ensure that all directors in a company and investors are aware of whether a company will be hedging or not. Some investors will invest in gold so that profits highly correlate with the price of gold. These investors would choose not to invest in companies that hedge.

Question 57

L2. What three problems give rise to basis risk?

Answer 57

1. An asset whose price is being hedged may not be exactly the same as the asset underlying the futures contract.
2. The hedge may be uncertain as to the exact due date hat the asset may be bought or sold.
3. A hedge may require for the futures contract to be closed out before its delivery month.

Question 58

L2. The basis risk equation?

Answer 58

Basis Risk = the spot price of the asset to be hedged - the futures price of contract used.

Question 59

L2. What is an increase in the basis called and what are its implications? What is a decrease in the basis called and what are its implications.

Answer 59

An increase in the basis occurs when the Spot price increases more than the forwards price. An increase is a strengthening of the basis, whilst as decrease is known as a weakening of the basis.

Question 60

L2. For an asset that is shorted at t1 and sold at t2, what would the profit calculation be? What would the implications be if someone purchased a long contract instead.
Use S1, S2, F1, F2 and the basis rate

Answer 60

S2 + F1 - F2 = F1 + b2

The profit of the hedge would be F1 - F2
I.e, the spot price in p2 (the price you could have sold the asset for) + the difference between the contract price you shorted and the contract price you longed = the forward price at t1 + the basis at t2 (S2-f2).
The two sides of the equation are identical but the second makes use of the basis 2 value.
If a long contract was purchased, the COST of the hedge would be F1 - F2 and the price they pay for the hedge would be the same as the equation above.

Question 61

L2. What is the hedging risk related to and and what is this known as?

Answer 61

The hedging risk depends on the Basis at t2, which holds the two unknown values of the futures spot price and future forward price. This is known as the BASIS RISK. Without such a risk, we would have the perfect hedge.

Question 62

L2. What would the result on a hedger that shorted an asset if the basis strengthened?

Answer 62

Their position would improve. The difference between S2 and F2 would increase meaning that they can buy a long contract to confirm their position at a better value, further above the Spot than had been previously expected. They still might still be at a loss but they have their position straightened nonetheless.

Question 63

L2. How does cross hedging affect basis risk?

Answer 63

It increases it.

Question 64

L2. How would we differentiate between hedging and speculation?

Answer 64

If an entity has an open position in another asset and they are investing to offset risk.

Question 65

L3. What does ICE LIBOR stand for?

Answer 65

Inter Continental Exchange London Inter Bank Offer Rate.

Question 66

L3. What does SONIA stand for?

Answer 66

Sterling Over Night Index Average

Question 67

L3. Currency: When would a short lose money?

Answer 67

If the exchange rate went up, so the pound increased in value. This is because you are locked in to sell the currency at a lower rate.

Question 68

L3. When would a company have a long hedge on a futures contract?

Answer 68

When the expect to buy the underlying asset in x months.

Question 69

L3. When would a company make a profit if the short hedged a future? in terms of f2 and f1.

Answer 69

If F1 > F2, as we have agreed to sell it for F1, but if F1 > F2, we can actually sell it for a higher amount.

Question 70

L3. What are the two elements that can effect basis with regards to the type of contract being used?

Answer 70

1. The underlying asset of the futures contract.

2. The choice of the delivery month.

Question 71

L3. Why would long positions for futures contracts often close out their position prior to the delivery month?

Answer 71

As delivery is often expensive. Long positions often close off positions and purchase from their normal suppliers. Prices can also be erratic during delivery month. Therefore, If you are expecting to make purchase the product in May, you would select a forward contract that expired in June at the earliest.

Question 72

L3. What is ICE LIBOR and how is it calculated? What was it formerly known as?

Answer 72

Designed to reflect the short term funding costs of major banks active in London. It was formerly known as BBA LIBOR. It is a polled rate and banks reply to the following question: At what rate could you borrow funds, were you to do so by asking for and then accepting inter-bank offers in a reasonable market size just prior to 11am? Only the banks with significant London presence are requested to answer the question. It is a trimmed arithmetic mean, meaning the top and bottom quartiles are removed and the rest averaged.

Question 73

L3. Why will SONIA become more relevant?

Answer 73

The BoE and FCA are working to shift markets into using SONIA rather than ICE LIBOR as the primary interest rate benchmark in sterling markets for sterling derivatives and relevant financial contracts.

Question 74

L3. ICE LIBOR: How many currencies (with maturities)? reported to? How many contributors?

Answer 74

10 currencies with 15 maturities (150 daily) reported to Thomas Reuters.
6-18 contributes.

Question 75

L3. Simple Interest formula?

Answer 75

F = P(1+rT)

Question 76

L3. Yearly Compound Interest Formula?

Answer 76

F = P(1+r)^T

Question 77

L3. Frequent Compound Interest Formula?

Answer 77

F = P(1+ r/m)^mT

Question 78

L3. What is Euler’s Number used for and how is it incorporated into the Frequent Compound Interest Formula?

Answer 78

For constant compounding, it is the rate at which to multiply by when we have continuous compounding. We replace (1+m/r)^mr with (1+1/m/r)^mT so it is in the format (1+1/n)^n. We can then write the equation F = P[(1+r/m)^m/r]rT to Pe^rT.

Question 79

L3. What is the formula for the Rc in terms of Rm and what is does it show?
This will be the

Answer 79

Rc = mLn (1 + Rm/m)
This will be the rate required for the constant compounding interest that leads to the same interest rate of compounding interest rate of m periods within a year.

Question 80

L3. What is the formula for finding Rm in terms of Rc and what is does it show?

Answer 80

Rm = m(e^Rc/m - 1)
This will be the rate required during m periods of a year that leads to the same interest rate of a continuous interest rate.

Question 81

L3. Annual Discounting Formula?

Answer 81

F = P(1+r)^-T

Question 82

L3. More Frequent than Annual Discounting?

Answer 82

F = P(1+ r/m)^-mT

Question 83

L3. Continuous Discounting?

Answer 83

Pe^-rT.

Question 84

L3. Zero-Coupon Bonds?

Answer 84

Bonds in which there are no intermediate payments, instead, the principal and interest are paid together at the end of n years.

Question 85

L3. What are other names for an n year zero-coupon bond?

Answer 85

The n-year Spot rate or n-year interest rate.

Question 86

L3. How can we calculate the theoretical value of a coupon-bearing bond?

Answer 86

We discount each cash flow at the appropriate zero rate.

Question 87

L3. Zero Rate (or spot rate)?

Answer 87

For maturity T is the rate of interest earned on an investment that provides a payoff only at time T (there are no intermediate payments or coupons).

Question 88

L3. Bond Yield?

Answer 88

The discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond.

Question 89

L4 - What is a swap?

Answer 89

An agreement between to parties to exchange cash flows at a specified future time according to certain specified rules.

Question 90

L4 - Why would a company enter in to a contract in which they payed out an amount yearly at 6% of the notional amount and receive an amount based on LIBOR every 6 months?

Answer 90

If the company has an outstanding liability with a variable rate. The interest rate swap facilitates the transformation of a variable-rate liability into a fixed-rate liability.
If it believes that rates are about to go up, it would want to lock in the fixed rate now to prevent borrowing costs rising in the future.

Question 91

L4 - Swaps - For semi annual swap contract, what LIBOR rate would a company receive?

Answer 91

Half of the % amount in the last period. So at t=1.5 years, the company will receive an amount based on the LIBOR at t=1 year.

Question 92

L4 - Swaps - Who pays the FI?

Answer 92

The amount is split between the two parties. One party will pay the fixed rate and half the FI amount and the other will receive the fixed rate (not including the FI cost) minus half the FI cost. This is the same as a bid and offer amount.

Question 93

L4 - Swaps: What is the swap rate?

Answer 93

The Bid-Offer Average

Question 94

L4 - Swaps: A corporate customer wanting to transform an existing fixed liability into a floating one needs to look to which column, the bid or offer?

Answer 94

**The Offer Column.
The FI will receive a lower rate to receive LIBOR (and take a f and a higher rate to pay LIBOR. This makes sense as the FI makes a bid for LIBOR rates, and will offer a rate lower than LIBOR to pay it.
** GO OVER AND GET CORRECT.

Question 95

L4 - How would we value an interest rate swap?
From the perspective of the:
a. floating-rate payer?
b. fixed rate player?

Answer 95

The value difference between the value of a fixed-rate bond and the value of a floating-rate bond.

a. V(swap) = B(fix) - B(float)
b. V(swap) = B(float) - B(fix)

Question 96

L4 - Swaps: What is the notation ‘B’ used to signify?

Answer 96

In V(swaps), it is the Bond.

Question 97

L4 - Swaps: What is the Par value of a floating rate swap over time? (4)

Answer 97

GO OVER AND CORRECT THIS CARD. SEE SLIDE 17/20 on LECTURE 4.
1. C is the LIBOR payment at time t; PAR is the notional
value of the swap
2. We will price the
oating leg of the swap as we would price a foating-rate bond: C corresponds to the bond’s floating coupon payment at time t; and PAR corresponds to the
value at issue (face value) of the bond.
3. The coupons of a
oating-rate bond are set at the beginning
of each coupon period|this is the reset date|and paid out
at the end of the coupon period.
4. On every reset date, the coupon.
5. On every reset date, the value of the
oating-rate bond is
PAR; e.g. if PAR = 100 and LIBOR is (expressed as
percent per annum), the semi-annual coupon is C = 100=2
and the bond’s value is
(100 + C)

1 +

2
􀀀1
= 100
6. Hence, on the next reset date, the bond will be worth PAR
7. Right before that, the value of the 
oating rate bond is
PAR + C
8. At any time, the value of the 
oating rate bond is PAR + C
discounted at the appropriate rate

Question 98

L3 - SLIDE 17 onwards

Answer 98

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Question 121

L4 - What is a swap?

Answer 121

An agreement between to parties to exchange cash flows at a specified future time according to certain specified rules.

Question 122

L4 - Why would a company enter in to a contract in which they payed out an amount yearly at 6% of the notional amount and receive an amount based on LIBOR every 6 months?

Answer 122

If the company has an outstanding liability with a variable rate. The interest rate swap facilitates the transformation of a variable-rate liability into a fixed-rate liability.
If it believes that rates are about to go up, it would want to lock in the fixed rate now to prevent borrowing costs rising in the future.

Question 123

L4 - Swaps - For semi annual swap contract, what LIBOR rate would a company receive?

Answer 123

Half of the % amount in the last period. So at t=1.5 years, the company will receive an amount based on the LIBOR at t=1 year.

Question 124

L4 - Swaps - Who pays the FI?

Answer 124

The amount is split between the two parties. One party will pay the fixed rate and half the FI amount and the other will receive the fixed rate (not including the FI cost) minus half the FI cost. This is the same as a bid and offer amount.

Question 125

L4 - Swaps: What is the swap rate?

Answer 125

The Bid-Offer Average

Question 126

L4 - Swaps: A corporate customer wanting to transform an existing fixed liability into a floating one needs to look to which column, the bid or offer?

Answer 126

**The Offer Column.
The FI will receive a lower rate to receive LIBOR (and take a f and a higher rate to pay LIBOR. This makes sense as the FI makes a bid for LIBOR rates, and will offer a rate lower than LIBOR to pay it.
** GO OVER AND GET CORRECT.

Question 127

L4 - How would we value an interest rate swap?
From the perspective of the:
a. floating-rate payer?
b. fixed rate player?

Answer 127

The value difference between the value of a fixed-rate bond and the value of a floating-rate bond.

a. V(swap) = B(fix) - B(float)
b. V(swap) = B(float) - B(fix)

Question 128

L4 - Swaps: What is the notation ‘B’ used to signify?

Answer 128

In V(swaps), it is the Bond.

Question 129

L4 - Swaps: What is the Par value of a floating rate swap over time? (4)

Answer 129

GO OVER AND CORRECT THIS CARD. SEE SLIDE 17/20 on LECTURE 4.
1. C is the LIBOR payment at time t; PAR is the notional
value of the swap
2. We will price the
oating leg of the swap as we would price a foating-rate bond: C corresponds to the bond’s floating coupon payment at time t; and PAR corresponds to the
value at issue (face value) of the bond.
3. The coupons of a
oating-rate bond are set at the beginning
of each coupon period|this is the reset date|and paid out
at the end of the coupon period.
4. On every reset date, the coupon.
5. On every reset date, the value of the
oating-rate bond is
PAR; e.g. if PAR = 100 and LIBOR is (expressed as
percent per annum), the semi-annual coupon is C = 100=2
and the bond’s value is
(100 + C)

1 +

2
􀀀1
= 100
6. Hence, on the next reset date, the bond will be worth PAR
7. Right before that, the value of the 
oating rate bond is
PAR + C
8. At any time, the value of the 
oating rate bond is PAR + C
discounted at the appropriate rate.

Question 130

L4 - To Complete

Answer 130

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Question 131

L4 - To Complete

Answer 131

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Question 132

L4 - To Complete

Answer 132

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Question 133

L4 - To Complete

Answer 133

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Question 134

L4 - To Complete

Answer 134

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Question 135

L4 - To Complete

Answer 135

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Question 136

L4 - To Complete

Answer 136

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Question 137

L4 - To Complete

Answer 137

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Question 138

L4 - To Complete

Answer 138

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Question 139

L4 - To Complete

Answer 139

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Question 140

L4 - To Complete

Answer 140

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Question 141

L4 - To Complete

Answer 141

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Question 142

L4 - To Complete

Answer 142

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Question 143

L4 - To Complete

Answer 143

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Question 144

L4 - To Complete

Answer 144

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Question 145

L4 - To Complete

Answer 145

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Question 146

L4 - To Complete

Answer 146

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Question 147

L5 - In-the-money?

Answer 147

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Question 148

L5 - At-the-money?

Answer 148

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Question 149

L5 - Out-of-the-money?

Answer 149

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Question 150

L5 - Given a constant risk-free rate and a constant price variance of the non-dividend paying underlying asset, the option premium consists of two types of value? (2)

Answer 150

1. Time value

2. Intrinsic Value

Question 151

L5 - What is the Profit and Loss Equation for a call option holder?
When will the call option holder make a net profit and exercise at maturity?

Answer 151

Y((T)) = MAX (0, S((T)) - X) - C((0))

S((T)) : Stock price at maturity
X = Strike Price
C = Price (premium) of call option

((T)) : subscript T.

They will make a profit if the stock rice exceeds the exercise price plus the price of the call option. They will exercise the stock if the price at maturity is above the strike price.

Question 152

L5 - What is the Profit and Loss Equation for a call option writer?
When will a call option writer make a profit?

Answer 152

Y((T)) = C((0)) - MAX (0, S((T)) - X)

((T)) : subscript T.

S((T)) : Stock price at maturity
X = Strike Price
C = Price (premium) of call option.

A profit will be made if the stock price is below the exercise price plus the price of the call option.
S((T)) < X + C((0))

Question 153

L5 - What is the Profit and Loss Equation for a put option holder?
When will they make a profit?

Answer 153

Y((T)) = MAX (0, X - S((T))) - P((0))

((T)) : subscript T.

S((T)) : Stock price at maturity
X = Strike Price
P = Price (premium) of put option

Profit is made when the strike price is below the share price minus the price of the put option.

Question 154

L5 - What is the Profit and Loss Equation for a put option writer?
When will they make a profit?

Answer 154

Y((T)) = P((0)) - MAX (0, X - S((T)))

((T)) : subscript T.

S((T)) : Stock price at maturity
X = Strike Price
P = Price (premium) of put option

Profit is made when the strike price is higher than the share price minus the price of the put option.

Question 155

L5 - Call option writer and put option writer difference?

Answer 155

Call option writer is selling the right to buy shares.

Put option write is selling the write to sell shares.

Question 156

L5 - Call option writer and put option writer difference?

Answer 156

Call option holder is buying the write to buy shares.

Put option holder is buying the right to sell shares.

Question 157

L5 - Call option and put option difference (in terms of buying and selling the right to attain shares)

Answer 157

Call relates to the right to buy

Put Relates to the right to sell

Question 158

L5 - What is the Call Option Premium Determinants for a put option and call option premium and the effect they have on the cost of an option.
e.g. the factors that determine the price of a non dividend paying American call option.
What about the signs for American Put options?

Answer 158

C = f(S((0)), X, r, 𝜏, σ^2)

1. S((0)) (+) : Stock price at any point in time before maturation.
2. X (-): Strike Price
3. r (+):
4. 𝜏 (+): The time variant.
5. σ^2 (+):

American Put option will have the same sign for 4 and 5 but the opposite signs for 1, 2, and 3.

Question 159

L5 - Call Option Premium Determinants [ C = f(S((0)), X, r, 𝜏, σ^2) ] - What does S((0)) stand for and what are its implications? We increase one variable and keep others fixed

Answer 159

+
S((0)) : The stock price at any point prior to the maturity date - even once the option has been purchased.
When the price goes up, the likelihood of the option being in the money at maturity becomes more likely. An increase moves the stock in the direction of in the money (S((T)) - X)

Question 160

L5 - Call Option Premium Determinants [ C = f(S((0)), X, r, 𝜏, σ^2) ] - What does X stand for and what are its implications?

Answer 160

-
The Strike Price.
The bank can quote multiply different strike prices based on the time period to maturity of the option.
Strike price represent obstacles to overcome, and the further away the obstacle, the less likely it will be surpassed and the low the demand. As such, the higher the strike price, the lower the cost of an option.

Question 161

L5 - Call Option Premium Determinants [ C = f(S((0)), X, r, 𝜏, σ^2) ] - What does r stand for and what are its implications?

Answer 161

+
r: the risk free rate, usually taken as the LIBOR.
Investors are risk averse and demand a premium to take on risk.
Think in terms of investing in an asset vs investing Ain a call option. To get the same return, you can invest all your money in the asset or part of your money in the option, leaving some amount to be invested at the risk free rate. Therefore, there is an additional incentive to buy an option, so will increase investors demand for the option. thereby increasing the price.
You should be able to explain this mathematically once more lectures have been completed.

From the book: If r increases, the expected return from a stock is expected to increase as well, as the opportunity cost of investing in the stock increases. The present value of any future cash flows also decreases. This is holding all other factors equal, which would not actually occur if the risk free rate increased. This should make more sense with time, other wise use the first explanation.

Question 162

L5 - Call Option Premium Determinants [ C = f(S((0)), X, r,𝜏 , σ^2) ] - What does 𝜏 stand for and what are its implications?

Answer 162

+
𝜏
Time to expiration.
Consider two call option holders that only differ as far as the expiration date is concerned. The holder with more time has all the same options as the individual with a shorter expiration, and more. Therefore, the longer expiration must be at least worth as much as the shorter one.

Question 163

L5 - Call Option Premium Determinants [ C = f(S((0)), X, r, 𝜏, σ^2) ] - What does σ^2 stand for and what are its implications?

Answer 163

+
σ^2, the risk of the stock.
This relates to an increased variance on a normal distribution of the stock.
Volatility in stocks is positive. It gives an increased chance of the stock price passing the strike price, meaning it is more likely to result in a gain. Note that the asset can also fall in value a lot as well, but when you are below X, you are out of the money, so you still use the same amount of money. Losses are capped at the call option premium.

Question 164

L5 - What is the upper boundary condition for an option?

Answer 164

C ≤ S((0))

Otherwise you would buy the stock and short calls on it without any risk to one’s own wealth.

Question 165

L5 - Call Option Premium Determinants [ C = f(S((0)), X, r, 𝜏, σ^2) ] - What does the C stand for?

Answer 165

The price of the call option, e.g. the premium paid by the holder.

Question 166

L5 - What is the lower boundary condition for an option?

Answer 166

C ≥ MAX(0, S((0)) - Xe^(-r𝜏)

S((0)) - Xe^(-r𝜏) - The stock price at a given time and the discounted in continuous time of the strike price.

Question 167

L5 - Prove the lower boundary condition - Two Portfolios.

Answer 167

See L5 slide 19/20.
Portfolio A: Buy a European call and buy a zero-coupon bond that makes a payment of $X at maturity date.
Portfolio B: Buy a share of the stock underlying the option of Portfolio A.

Present Value of Portfolio A:
Call: C
Bond: Xe^(-r𝜏)
Total: C + Xe^(-r𝜏)

Future Value of Portfolio A when S((T)) > X:
Call: S((T)) - X
Bond: X
Total: S((T))

Future value of Portfolio B when S((T)) ≤ X:
Call: 0
Bond: X
Total: X

Present Value of Portfolio B:
S((0))

Future Value of Portfolio B when S((T)) > X:
S((T))

Future value of Portfolio B when S((T)) ≤ X:
S((T))

A Stochastically dominates portfolio B. Therefore, the cost of setting up Portfolio A should be greater than the cost of setting up B. So the premium of the call option will be greater than S((0)) - Xe^(-r𝜏).

Question 168

L5 - Why would it be better to sell an American call option to the market rather than exercising it prior to maturity?

Answer 168

This situation would only occur when the option is in the money (otherwise the value of exercising would be 0). If they exercised the option, they would receive S((t)) - X [when t < t],

As such, the value would be at least S((t)) - Xe^(-r(T-t)), as others will value the asset with the potential of it increasing in value.

Question 169

L5 - Difference between American and European options in the context of the pricing of option determinants?

Answer 169

We assume that the American Option is non-dividend paying.
European options are often dividend paying, which results in a few changes in the impact of time (a dividend could be paid during a longer expiration and not in a shorter one, reducing the value of the stock) and volatility on the cost of the option.

Question 170

L5 - What is the equation that proves from our portfolio answer that an American option is better alive than dead?

Answer 170

S((t)) - X < S((t)) - Xe^(-r𝜏)

when Xe^(-r𝜏) < 1, so r and 𝜏 are > 0.

Question 171

L5 - Extra

Answer 171

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Question 172

L5 - Extra

Answer 172

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Question 173

L5 - Extra

Answer 173

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Question 174

L5 - Extra

Answer 174

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Question 175

L6 - What is the equation for the Put-Call Parity Condition?

Answer 175

C + Ke^(-rT) = P + S((0))

Question 176

L6 - Explain the Put-Call Parity using two portfolios. See slide 4/24 for answer in full.

Answer 176

Portfolio A: Buy a European call and buy a zero-coupon bond that makes a payment of $K at time T.
Portfolio B: European put option plus one share of the underlying asset.

Think of Call-Put Parity to work through

At maturity, both are worth Max(St, K)

Portfolio A.

Present cash flows of Portfolio A:
Call: C
Bond: Ke^(-r𝜏)
Total: K + Xe^(-r𝜏)

Future cash flows of Portfolio A when S((T)) > K:
Call: S((T)) - K
Bond: K
Total: S((T))

Future cash flows of Portfolio A when S((T)) ≤ K:
Call: 0
Bond: K
Total: K

Present cash flows of Portfolio B:
Put: P
Share: S((0))
Total: S((0)) + P

Future Cash Flows of Portfolio B when S((T)) > K:
Put: 0
Share: S((T))
Total: S((T))

Future cash flows of Portfolio B when S((T)) ≤ K:
Put: K - S((T))
Share: S((T))
Total: K

It follows that, if two cash flows generate the same return in the future [ MAX(S((T)), K) ], then setting up these two portfolios must be the same to avoid the arbitrage opportunity:
C + Ke^(-rT) = P + S((0))

Question 177

L6 - For the Put-call Parity, what can be said about the state of the world when:
1. S((T)) > X
2. S((T)) ≤ X with relation to the put and call options.?
What does this mean in terms of cash flows?

Answer 177

1. S((T)) > K
   The Call option is in the money and the put option is out of the money.
2. S((T)) ≤ K
   The put option is in the money and call option the put option is out of the money.

Question 178

L6 - For the put-call parity, why do we make use cash flows rather than value to assess the two portfolios?

Answer 178

Because we disclose the original put and call option premiums and focus only on the inflows from these two streams of money.

Question 179

L6 - Put-call Parity - What is the condition that must be met to ensure no arbitrage opportunities exist?
If you recognise they do not balance, what do you do (say A < B?

Answer 179

C + Ke^(-rT) = P + S((0))
If they do not balance, you buy the instruments in the portfolio that is lower and sell the instruments that resulted in a higher value.
If A < B.
You purchase a call option (C) and short the stock [S((0))] [borrow the stock and sell it in the market - so a cash in flow] and also sell a put option for a positive cash inflow of P.

The arbitrage profit is then made by investing the money received at the interest free rate and then taken away the stock that you borrowed from the broker and pay it back.

We ignore the Zero coupon as the outcome will be the same.

Question 180

L6 - Put-call Parity [C + Ke^(-rT) = P + S((0)) ]- If portfolio A (LHS) is cheaper than portfolio B (RHS), what is the mechanism that will be undertaken if:

a) S((T)) ≤ K
b) S((T)) > K

Answer 180

If A is cheaper, we buy A and short B:

You purchase a call option (C) and short the stock [S((0))] [borrow the stock and sell it in the market - so a cash in flow] and also sell a put option for a positive cash inflow of P.

a) S((T)) ≤ K
The short put will be exercised by the counter party who sell the asset to you at K, with a cash flow of K - S((T)). You then return the asset back to the broker.

The call will not be exercised.

b) S((T)) > K
The long call will be exercised by yourself , meaning the shares will be received with a cash flow of S((T)) - K. Once you have taken delivery of the asset, it is returned to the broker who lent the stock (the shorted put option).

The put option will not be exercised.

Question 181

L6 - Put-call Parity - What is the equalising factor if there exists an arbitrage profit?

Answer 181

Prices will adjust until no arbitrage opportunity exists.

The price of the pul or put are not sufficiently far apart to prevent arbitrage, so through increasing demand, the cost of either of these will adjust to ensure no profits can be made through arbitrage.

Question 182

L6 - Types of strategies and options - What are the Four types of strategies in which you take a position in the option and the underlying asset?

Hint

Care and inverse

Answer 182

1. Covered Call
2. Inverse Covered Call
3. Protective Put
4. Inverse Protective Put

Question 183

L6 - Types of strategies and options - What are the four types of strategies in which you take a position in two or more options? What are they known as?

Hint:

Contained Animals

Answer 183

They are known as spreads.

1. Bull
2. Bear
3. Box
4. Butterfly

Question 184

L6 - Types of strategies and options - What are the four types of strategies in which you take a position in both calls and puts? What are they known as?

Hint

S&M

Answer 184

These are known as combinations.

1. Straddle
2. Strip
3. Strap
4. Strangle

Question 185

L6 - Types of strategies and options - 1. Covered Call and 2. Protected Call?

Answer 185

1. Short Call, long Stock

Short Call Position (so as the price increases, the cost to us increases as the counter party will buy the stock at the lower price from us).
We have the short call option and cover it by going long on the stock.

We cover this by going long on the stock.

However, if the price falls a lot, although the short call isn’t enacted, we make a loss on the price of the share.

S((0)) - C = Ke^(-rT) - P

A covered call is the equivalent of a short put.

This caps Gains

We have a Long Call Position and cover it by shorting the stock.
- [S((0)) - C] = -[Ke^(-rT) - P]

A long call and shorting the stock will be the equivalent of a long put.

This will cap losses.

Question 186

L6 - Types of strategies and options - 1. Protected Put and 2. Inverse Protected Put?

Answer 186

1. Protected Put:
   A long position in the asset which will be protected by a short position in the put.

This corresponds to a long call.

This will cap losses

2. Inverse Protected Put?
   A short position in the asset which will be protected but a long position in the put.

This corresponds to a short call

This will cap profits.

Question 187

L6 - Put-call Parity Condition - How can we interpret the signs of the condition?

Answer 187

Interpret signs in the Put Pull Parity condition as whether they are long or short positions

Question 188

L6 - Spreads - Bull Spread? Explain why this strategy is desirable?

Answer 188

A long and a short position in two call options of the same underlying asset at different strike price.

Bull as we are expecting the price to rise.

Long on a call with a low strike price and short on a call with a high strike price.

The low strike price call premium costs more.

This strategy is beneficial as it reduces the cost of initial transaction, as the initial cost will be the Long call cost - shot call profit.

Question 189

L6 - Spreads - Bear Spread? Why is this strategy desirable?

Answer 189

You expect the price of an asset to fall. Therefore you buy a long put option with a strike price and short on a put with a lower strike price.

You thereby reduce the price of the transaction but limit the total amount that can be made if the share drops more than K((1)). So it is chosen when you believe the price will not fall below K1.

Question 190

L6 - Spreads - Box Spread? What would you buy and sell to enact the spread if the theoretical price is greater than the market price?

Answer 190

A combination of a bull call and bear put spread, meaning you add the positions of the bull and the bear.

The result is that the total payoff is always K((2)) - K((1)).

It is a risk free spread as you will always receive this amount.

We discount the certain future payoffs e^(-rT) (K2 - K1). If the current market price is below this theoretical value, then you should buy the box spread.

if the theoretical price is higher than the market price, you would buying a call at K1, selling a call at K2, buying a put at strike price K2 and selling a put at strike price K1.

Question 191

L6 - Box Spread - What would you buy and sell to enact the spread if the theoretical price is greater than the market price?

Answer 191

if the theoretical price is higher than the market price, you would buying a call at K1, selling a call at K2, buying a put at strike price K2 and selling a put at strike price K1.

The opposite will be the case if the theoretical price is below the market price.

Question 192

L6 - Box Spread - What options does this strategy work for and why might it often fail.

Answer 192

Only works with European options and often does not work due to the number of trades involved and the costs associated with these.

Question 193

L6 - Butterfly Spreads?

Answer 193

Positions in options with three different strike prices.

Buy a European Call option at the low strike price of K1, at the higher price of K3 and selling a TWO call options at the price in the middle of K2.

The investor could also create the same effect by buying a Long put at K1 and K3, and selling TWO puts at K2.

K2 is generally close to the current stock price.

A butterfly spread leads to a profit if the stock price stays close to K2, but gives rise to a small loss if there is a significant stock price move in either direction.

It only requires a small investment initially and a profit is made if the stock at time T is between the K3 and K1 (minus / plus the cost of the call options)

Question 194

L6 - Combination strategies?

Answer 194

An option trading strategy that involves taking a position in both calls and puts on the same stock.

Question 195

L6 - Combination strategies - Straddle?

Answer 195

Buying a European call and put with the same price (close to the current price) and expiration date. - This is called a bottom straddle or straddle purchase.

A significant move in either direction will lead to ta gain.

Question 196

L6 - Combination strategies - Strips?

Answer 196

Stripping back to less value
A long position in one European call and TWO European Puts with the same strike price and expiration date.

The investor is betting that there will be a big stock price move and considers a decrease in the stock price to be more likely than an increase.

Question 197

L6 - Combination strategies - straps?

Answer 197

Strapping on for an increase.
A long position in TWO European Calls and one European Put with the same strike price and expiration date.

The investor is betting that there will be a big stock price move and considers an increase in the stock price to be more likely than a decrease.

Question 198

L6 - Combination strategies - Strangles?

Answer 198

Buy a European Put and a European call with the same expiration date and different strike prices.
The investor believes that the price will move dramatically, but wants less downside risk that with a straddle. The downside is that the price will need to move more than would be the case for a straddle.

The farther apart the prices are, the less downside risk and the farther the stock prices to move for a profit to be realised.

Question 199

L7 - What is a Markov process?

Answer 199

A particular type of stochastic process where only the current value of a variable alone is relevant for predicting the future. The past history if the variable and the way that the present has emerged are irrelevant.

Question 200

L7 - If we assume that stock prices follow a Markov process, what does this mean? What does it allow us to say?

Answer 200

Future movements in a stock price depends only on the current price on not the past, meaning probability distributions are independent.

It is a sufficient criteria to allow us to say that there is weak-form market efficiency.

Question 201

L7 - Weak-form market efficiency in the stock market meaning?

Answer 201

This asserts that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices
In other words, technical analysis does not work

Question 202

L7 - Why do we introduce the Markov assumption?

Answer 202

Once we state that stock prices are Markov, predictions of the future value are uncertain and so will be expressed in terms of probability distributions. With the Markov property, the probability distributions of the price at any particular future time is not dependent on the particular path followed by the price in the past.

Question 203

L7 - Φ(m,v) meaning? What would the probability distribution of the change in the value of the variable during 2 years?

Answer 203

A normal distribution with a mean of m and a variance of v^2.

The change in two years is the sum of two normal distributions, each with a mean of 0 and variance of 1^2. As the two normal distributions are independent the resultant distributions will be the sum of both. So the normal distribution over two years will be Φ(0,2). The standard deviation will be root 2.
So we have Φ(0,Δt) as the Wiener Process.

Question 204

L7 - If the normal distribution for a stock over a year is Φ(0,1), what would be the normal distribution over 6 months?

Answer 204

Φ(0, 0.5)

With the standard deviation being the root of .5

Question 205

L7 - For the normal distribution of stocks that follow the Markov rule, what is the rule for deciding Φ(m,n)?
Use and example of 1.5 years and 0.5 years.

Answer 205

the mean will remain the same, 0.

The variance will change. For periods of more than a year, expressed in terms of a year, it will be Tn.

1.5 years will be 1.5n.
Φ(0,1.5) - beng

0.5 years will be
Φ(0, 0.5). 1 year would be considered to be two of the half year normal distributions.

Question 206

L7 - What is the Wiener Process? What are the two Properties?

Answer 206

A type of Markov stochastic process with a mean of 0 and a variance rate of 1 per year.

Question 207

L7 - How can we get from district time to continuous time with a Markov normal distribution?

Answer 207

For very small value of t (at the limit), this will be continuous time.

Question 208

L7 - What are the two properties of a Wiener Process?

What is the ε?

Answer 208

For a variable z,
1. Δz = ε √(Δt), where ε has a standard normal distribution Φ(0, 1)[(this means that the change in z is just a proportion of the variable ε (which follows a normal distribution) multiplied by the standard deviation.]

2. The value of Δz for any two different short intervals of time, Δt, are independent.
   It follows that Δz itself has a normal distribution with
   Mean = 0
   SD = √(Δt)
   Variance = Δt

The ε is the normal distribution Φ(0, 1), so we are multiplying ε by √(Δt).

Question 209

L7 - What would the variance be for a wiener process with a small change in time? Use example with working days in a year.

Answer 209

Δt, where Δt it the fraction of time in terms of a fraction of a the overall amount of time that we are measuring. Say we are measuring in relation to working days in a year, and say there are 250, 2 days of work will be 2/250.
Δt is 1/ (number of periods during 1 time period, T).

Question 210

L7 - How can we work out N, being the number of time small time periods for a relatively long period, T?

Answer 210

N = T / Δt

For two years, with 12 time periods:
Number of periods = 2 / (1/ 12) = 2 * 12 = 24.

Question 211

L7 - What is the Generalised Wiener Process aka?

Answer 211

Brownian Motion.

Question 212

L7 - Generalised Wiener Process - Drift rate and Variance Rate with symbols

Answer 212

Drift Rate (α): The average change per unit time for a stochastic process [The average (mean)]

Variance Rate (β): The variance per unit time [The Variance]

These are used for grammatical reasons.

Question 213

L7 - Generalised Wiener Process - What is the key difference with the Wiener Process and Generalised Wiener Process?

Answer 213

Does not have a drift rate of 0 and a standard deviation = 1. It can have any numbers.

The Wiener Process is a special case of the Generalised Wiener process with the drift rate 0 and SD of 1.

Question 214

L7 - Generalised Wiener Process - Show the Equation and explain what each part means?

Answer 214

dx = α dt + β dz

dz is the Wiener Process [normal distribution Φ(0, 1)]
This drift rate would mean that the expected value of z at any future time is equal to its current value.
The variance rate (β^2) means that the variance of a change in x in a time interval of length T equals T.

The above shows the generalised Wiener Process for a variable x in terms of dz.

Question 215

L7 - Generalised Wiener Process - How do we get from the equation back to the unique Wiener Process case?

Answer 215

dx = α dt + β dz

With a normal distribution Φ(0, 1), When β dz = 0, dx/dt = the drift rate = α = 0

When with a change in time. the volatility will be β times the Wiener Process, and the SD = β per unit of time.
dz = ε√(dt).

It follows that β times a Wiener process has a variance of β^2.

Question 216

L7 - Generalised Wiener Process - Why do we swap from Δz and (Δt) to dx and dt?

Answer 216

To indicate that we are operating in continuous time.

Question 217

L7 - Generalised Wiener Process - By combining a Generalised Wiener Process and a Wiener Process equation, how can we formalise Δx for a small unit of time? What are the mean, Var and SD?

Answer 217

Combining:
1. Δz = ε √(Δt)
[the first property of a Wiener Process]

2.
dx = α dt + β dz
[A generalised Wiener Process]

Outcome:
Δx = α Δt + β ε√Δt

It follows that β times a Wiener process has a variance of β^2. (as we are increasing the volatility of a Wiener process by β times).

Mean of Δx = α Δt
SD of Δx = β √Δt
Var of Δx = β^2 Δt

Question 218

L7 - Why can we not assume that a stock follows a generalised Wiener Process, namely that it has a constant expected drift rate and constant variance?

Answer 218

We look at stocks in terms of returns, and the generalised Wiener Prices fails to capture the price of a stock. The return on a stock is independent of the price, so a 14% return would be expected on either a $10 stock or $50 stock. We therefore have to look at expected return rather than expected stock price.

Question 219

L7 - In continuous time, what would the model (equation) be for a stock price change using the Generalised Wiener Process as a base?

Explain how we attain this and what the notation means.

Answer 219

dS/ S = μdt + σdz.

OR

dS = μSdt + σSdz.

1. μ is the annualised expected drift rate in decimal form, then the expected drift rate should be μS. For a small period of time, this will be μSΔt, and as Δt tends to 0, we have μSdt.
   When dz is 0, dS = μSdt or
   dS/ S = μdt.
2. The expected drift will be in relation to the stock price, so we attain σSdz.

We then divide both sides by S.

μ is the stocks expected rate of return.

σ is the volatility of the stock price.

Question 220

L7 - In DISCRETE time, what would the model (equation) be for a stock price change using the Generalised Wiener Process as a base?

Start at continuous time and change.

Answer 220

Continuous time:

dS = μSdt + σSdz.

Discrete Time:

dS/ S = μΔt + σε√Δt.
OR
dS = μSΔt + σSε√Δt.

The first one is preferable as we want to show show the return of a stock independent of the share price.

Question 221

L7 - From the Generalised Wiener Process, what can the stock price formula, dS/ S = μdt + σdz, be referred to as?

Answer 221

A geometric Brownian Motion Formula.

Question 222

L7 - Dx?

Answer 222

The difference between Xt and X0, being the difference between X now and in one moment (infinitely small).

Question 223

L7 - Itô Process- Definition and one equation?

Answer 223

A generalised Wiener Process in which the parameter α and β are functions of the value of the underlying variable x and time.

It is used to value the price of a derivative (thereby the call and put price).

dx = α(x,t) dt + β(x,t) dz

Question 224

L7 - Itô Process - What is relevant about the α and β values?

Answer 224

The expected drift rate and variance rate of an Itô Process change over time.
Over a small interval t and t + Δt, the variable changes from x to x + Δx, where

Δx = α(x,t) Δt + β(x,t) ε √(Δt)

More specifically, the price of a stock option is a function of the underlying stock’s price and time or that the price of any derivative is a function of the stochastic variables underlying the derivative and time.

Question 225

L7 - Itô Process* Space

Answer 225

Space for P.14.

Question 226

L7 - Itô Process* Space

Answer 226

Space for P.14.

Question 227

L7 - Itô Process * Space

Answer 227

Space for P.14.

Question 228

L7 - Black-Scholes-Merton Model - What 3 concepts underlie the model and lead us to the BSM differential?

Answer 228

1. The option price and the stock price depend on the same underlying source of uncertainty
2. We can form a portfolio consisting of a positions in the stock and the option which eliminates this source of uncertainty
3. The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

Question 229

L7 - Black-Scholes-Merton Model - What assumptions must we hold in equilibrium?

Answer 229

1. The stock price is characterised by a geometric Brownian motion (constant μ and σ) and is lognormally distributed
2. Short selling is available
3. No transaction costs or taxes
4. The underlying asset is non-dividend-paying
5. No riskless arbitrage (in equilibrium)
6. Trading takes place in continuous time
7. Constant risk-free rate across all maturities

Question 230

L7 - BSM model - Lognormal distribution?

Answer 230

LnS((T)) is normally distributed. No depth gone into during lectures. A lognormal destruction can take any value between zero and infinity.

Question 231

L7 - BSM Model - How do we derive the equation with a portfolio?
What would be the value of this model?

Answer 231

We set up the portfolio such that:
-1 for the derivative (put or call)
[so short one derivative]

𝛿f/𝛿S for the shares
[Long this number of shares]

π = -f + (𝛿f/𝛿S)*S
Profit when you are in the money is negative the short + the price of the stock * quantity.

Question 232

L7 - BSM Model - How do we get from the profit function of the portfolio π = -f + (𝛿f/𝛿S)*S to proving its riskless?

Answer 232

(1) π = -f + (𝛿f/𝛿S)S
The change of Δπ in the value of the portfolio in the interval Δt is given by:
(2) dπ = -df + (𝛿f/𝛿S)dS
[The instantaneous change in each parameter]
We know df from the Itö’s Lemma, we know dS from the geometric brownian function and 𝛿f/𝛿S is just a partial derivative.

The Itô process becomes
(3) df = [𝛿f/𝛿*μS + 𝛿f/𝛿t + 1/2 * 𝛿^2f/𝛿S^2 *σ^2 * S^2]dt + 𝛿f/𝛿S * σ S dz

(4) dS = μSdt + σSdz

Therefore, as we know all the variable, it must return the risk free return, so

dπ/π = rdt

By substituting in dπ, df and dS into (5)
* MAKE SURE THE ABOVE IS CORRECT AND RUN THROUGH THE ALGEBRA/

Question 233

L7 - BSM Model - *

Answer 233

slide 20/25

Question 234

L7 - BSM Model - *

Answer 234

slide 20/25

Question 235

L7 - BSM Model - *

Answer 235

slide 20/25

Question 236

L7 - BSM Model - What is the boundary condition for a European Call option?

Answer 236

Call option:
f = max (0, S - E)

A Put option would be:
f = max (0 , K - S)

Question 237

L7 - BSM Model - What is the boundary condition for a European Put option?

Answer 237

f = max (0 , K - S)

A put option would be:
f = max (0, S - E)

Question 238

L7 - BSM Model - Go through NOTES.

Answer 238

GO OVER NOTES.

Question 239

L7 - Why would a call option theoretical price under the BSM out of the money still have a positive value?

Answer 239

Due to the time value of the option. The share price may fall, leading to a put option having the potential to be be in the money by expiry.

Question 240

L7 - BSM - Why is it relevant to note hat the price under the BSM are theoretical?

Answer 240

The price under the BSM is theoretical and therefore may not be the price in the market. Investors will have different expectations about the price of an option, which may lead to investors buying underprices options, increasing their price. Nothing would have fundamentally changed (the stock price may be the same) and very little time would have changed, but the option will change in the market.

Question 241

L7 - BSM - Given that the BSM model only generates a theoretical price, how do we account for changes in the market price of an option?

Answer 241

We choose a volatility that equates the theoretical price of the option with the market price of the option, thereby pretending that we cannot have an estimate of the volatility (which could be estimated from previous returns - As we apply the Markov Rule, this is assumed to be impossible). This will be called the ‘implied volatility’. The market determines the price of options and puts.

Question 242

L7 - BSM - d1?

Answer 242

d1 =
[ln (s0/E) + 0.5*(r + (σ^2/ 2)]/ σ√T

S0: the current price of the stock.
E: Strike Price.

Question 243

L7 - BSM - d2?

Answer 243

d2 = d1 - σ√T

Question 244

L7 - BSM - Call Option Formula?

Answer 244

c = S0 N(d1) - Ee^(-rt)N(d2)

S0: the current price of the stock.
E: Strike Price.

Question 245

L7 - BSM - Put Option Formula?

Answer 245

p = Ee^(-rt)N(-d2) - S0 N(-d1)

S0: the current price of the stock.
E: Strike Price.

Question 246

L7 - Extra

Answer 246

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Question 247

L7 - Extra

Answer 247

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Question 248

L7 - Extra

Answer 248

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Question 249

L7 - Extra

Answer 249

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Question 250

L7 - Extra

Answer 250

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Question 251

L7 - Extra

Answer 251

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Question 252

L8 - BSM - Explain the idea behind N(d#).

What is N(.)?

Answer 252

N(d#) gives us the probability that d1 will be that number or less than that number.
So we calculate d# and then attain a N(d#).

N(.) is the CDF of the standard normal

Question 253

L8 - BSM - Explain what the RHS of this equation means: c = S0 N(d1) - Ee^(-rt)N(d2).

Answer 253

This is basically the present value of what is expected to be received minus the present value of what is expected to be
paid upon exercise.

Question 254

L8 - BSM - What can we say about N(d1) intuitively?

C = S((0)) N(d1) - Ee^(-rt)N(d2).

Where is the call option premium.

Answer 254

N(d1) can be shown to be the partial derivative of the call option with respect to the asset price.

Δ = 𝛿c/𝛿S0 = N(d1)

We use the GREEK LETTER Δ((call)) for N(d1).

Delta will be a % (between 0 - 1 )

Question 255

L8 - BSM - What can we say about N(d2)?

c = S0 N(d1) - Ee^(-rt)N(d2).

Answer 255

N(d2) is the probability that the call option will expire
in-the-money.

We don’t need to prove why this is.

N(d2) = Prob(ST ≥ E)

Question 256

L8 - BSM - Explain the intuitive understanding behind the call option formula:

c = S0 N(d1) - Ee^(-rt)N(d2)..

(explain each of the above and its meaning)

Answer 256

The stock price * Δ (being how the call option price changes with respect to changes in price of the underlying asset) - the discounted strike price (E) times by N(d2) (which gives us the likelihood that the stock price at maturity will be at least equal to the strike price, so therefore it is the probability that we will end up in the money and exercise the option).

Question 257

L8 - BSM - We defined Δ((call)) = N(d1). What would Δ((call)) be for a put option?

Answer 257

Δ((put)) to = N(d1) - 1

Question 258

L8 - What can we say about Δ((call)) in a portfolio?

Answer 258

They will be positive in a portfolio for a long position and negative for a short positions. This will always be the case.

Remember, N(d1) can be shown to be the partial derivative of the call option with respect to the asset price. Intuitively, if the stock price increases, the call option price will also increase. See Graph in L8 S4.

Question 259

L8 - A bank has sold for $300k a European call option on 100k shares of a non-dividend paying stock. The data are:
The Black-Scholes value of the option is $240k meaning a theoretical profit of $60k for the bank.
Why wouldn’t the bank simply buy the shares?
How does the bank hedge its risks?

LONG answer.

Answer 259

If the bank purchased the shares, and the value of the shares fell, they would lose money. This would be known as a covered call, which is also known as a short put.

The delta of the stock will be 1 (a change in the stock price will change the portfolio stock by 1)

The delta of the option will be N(d1) * the number of shares held in the portfolio. This is the option position.

The bank will set a delta that will offset the other positions, so that the delta of the entire portfolio will be 0. This will be a delta neutral portfolio.

If we are shorting a call, we would long the shares (covered call). You don’t buy 100,000 units, but instead build up the holdings in the asset as the price of the underlying asset changes with delta.

Shorting a call would be a positive delta, and the long position would be a positive( as is always the case - there is a positive relationship between the price of the asset and the price of the call price)

If the price of the asset goes up for a delta neutral position, any fall in the value of the shorted option will be offset by a rise in the share price.

Question 260

L8 - Δ (delta) neutral portfolio -

1. If you had a long call, what would you do with the shares.
2. If you had a short call…
3. If you long a put…
4. If you short a put.

Answer 260

Call options will always have opposite directions with the shares.

1. Long call, short the shares. Δc, -Δs
2. Short call, long the shares. -Δc, Δs

Put options will be the same way as the shares.
3. Long the shares.
Δp, Δs
4. Short the shares
-Δp, -Δs

Question 261

L8 - Δ (delta) of a stock be?

Answer 261

we have defined delta as the change a variable due to a change in the stock price. The variable here is the stock, so the delta will be 1.

Question 262

L8 - Δ (delta) neutral portfolio?

Answer 262

When the delta of one poison (say an option) offsets the change in the price of the delta of another position (say a stock holding).

Question 263

L8 - θ (Theta) Definition?

What is it often referred to as?

Answer 263

The θ of the portfolio is the rate of change of a derivative’s value with respect to the passage of time ceteris paribus.

Often referred to as time decay.

Question 264

L8 - θ (Theta) - what is the sign for a call and a put and its intuitive understanding?

When would this not be the case.

THIS IS THE MOST COMPLICATE OF THE GREEK PARAMETERS.

Answer 264

Theta is usually negative for both call and puts unless it is so far out of the money, in which case it has no value. This means, ceteris paribus, that the value of a call or put option declines as time passes, which makes sense as the option will hold less value as time passes (less chance to reach or go above the strike.

The exception would be when a put is deep in the money, as the time shortening will make it even more likely that the option will be executed.

Question 265

L8 - What are the two impacts that time has on the value of a stock and how does the passage of time affect option value through these effects?

Answer 265

1. Volatility:
   - As there is less time for it to have an impact, it is negative for calls and puts.
2. Interest Saving/ Costs:
   -Interest saving for a call due to deferred exercise is reduced so the effect on the call option value is negative.
   You are only exercising your right in the future.
   - Interest opportunity cost for a put due to deferred exercise is reduced, so the effect not he put option value is positive.
   You have to hold the asset and delay the sale of it.

Question 266

L8 - Recap - How do we attain any of the greek parameters?

Answer 266

We find the partial derivative of the call or put option with respect to something.

Question 267

L8 - What would N(-d2) be in terms of N(d2)

Answer 267

N(-d2) = 1 - N(d2)

Question 268

L8 - From the Put-call Parity, what holds with regards to Theta θ?

Hint: The θ of the portfolio is the rate of change of a derivative’s value with respect to the passage of time ceteris paribus.

Answer 268

θ((call)) - θ((put)) = -rEe^(-rT).

Question 269

L8 - An investor has sold a call option to buy 2000 shares of a stock. If Δ (the slope of the graph plotting Option Price (y) against stock price (x) is 0.6, how would the investor hedge his position?

Answer 269

by buying 0.6 * 2000 = 1200 shares at that point in time.

Note: Δ would change over time so this position would have to be adjusted in line with fluctuations in the stock price.

Question 270

L8 -

1. Δ((call)) = N(d1)
2. Δ((put0) = N(d2) - 1

So how do we connect the two of these?

Answer 270

The put-pull parity condition.

By differentiating both sides of the put-call parity condition
(C + Ke^(-rT) = P + S((0)))
w.r.t. the stock price.

From the og equation,
C + Ke^(-rT) = P + S((0))

We differentiate to attain:
Δ(call) = Δ(put) + 1.

So Δ(call) - Δ(put) = 1.

Question 271

L8 - θ (Theta) - What is the formula for θ((call)) derived from the BSM formula?

Answer 271

θ(call) = [(S0 N’(d1)σ) / 2√T] - rKe^(-rt)N(d2).

Question 272

L8 - θ (Theta) - What is the formula for θ((put)) derived from the BSM formula?

How does this formula relate to the corresponding call formula.

Answer 272

θ(put) = [(S0 N’(d1)σ) / 2√T] + rKe^(-rt)N(-d2).

Note - The difference between this and the call formula is the + and the N(-d2) rather than N(d2).

Due to this, the Theta of the put exceeds the theta of the call by rKe^(-rt).

Question 273

L8 - θ (Theta) - For the θ(call) formula: [(S0 N’(d1)σ) / 2√T] - rKe^(-rt)N(d2), what is N’(d1) mathematically?

Answer 273

N’(d1) = [1/ (√2π] e^[-d1^2)/ 2]

Question 274

L8 - What is the formula for N(-d2)?

Answer 274

1 - N(d2)

Question 275

L8 - What is N’(d1)? (explain)

Answer 275

N’(d1) is not the CDF, but just the standard normal distribution function. It is the probably that σ will be this value exactly.

Question 276

L8 - Gamma Γ - Define Γ

Answer 276

The Γ of a portfolio on an underlying asset is the rate of change of the portfolio’s delta with respect to the price of the underlying asset (𝛿Δ/𝛿S0).

Remember, Δ (delta) is the partial derivative of the call option with respect to the asset price : Δ = 𝛿c/𝛿S0 = N(d1), but when we have a portfolio of options dependent on a single asset whose price is S, the delta of the portfolio will be 𝛿π/𝛿S.

This leads to the formula.
Γ = 𝛿^2 π / 𝛿 S^2

Question 277

L8 - What is the Γ (gamma) partial derivative equation?

Answer 277

Γ = 𝛿^2 π / 𝛿 S^2

Question 278

L8 - We previously notes that Δ (delta) = 𝛿c/𝛿S0 = N(d1). What would the delta of a portfolio of options or other derivatives dependent on a single asset whose price is S be?

Answer 278

𝛿π/𝛿S

Where π is the value of the portfolio.

Question 279

L8 - Gamma Γ - What is the intuitive understanding behind Gamma Γ, i.e. what happens if it changes?

Hint: Small vs large Gamma.

Answer 279

Recall: The Γ of a portfolio on an underlying asset is the rate of change of the portfolio’s delta with respect to the price of the underlying asset.

When the gamma is small, delta changes slowly, and adjustments to keep a portfolio delta neutral need to be made only relatively infrequently.

If game is highly negative or positive, delta is very sensitive to the price of the underlying asset, making it risky to leave a delta neutral portfolio unchanged for any length of time.

Gamma measures the curvature of the relationship between the option price and the stock price, being the hedging error.

Question 280

L8 - Gamma Γ - Why would a large gamma mean more risk?

related to delta

Answer 280

Look at the digram on slide 17/23.

Gamma indicates how quickly the portfolio cost changes due to a change in the underlying asset, with respect . It is the second differential of the Delta. As banks do not operate in continuous time, but rather discrete time, there will be changes in the value of the stock within this period, which will result in the cost of an option being less than the actual value if the delta were to be adjusted (as the delta will be constant before being adjusted). The large the gamma, the larger the different between the assumed call option premium and the actual premium to keep the portfolio delta neutral.

Question 281

L8 - When long on an option, what will the sign of gamma Γ be?

Answer 281

Gamma will always be positive or a long positions and negative for a short position.

It is greatest for options that are near the money.

Question 282

L8 - What is the Γ (gamma) equation?

Answer 282

Γ = N’(d1) / [S0 σ √T]]

S0: stock price at a particular point in time
σ: volatility of the underlying asset price
√T: square root of time remaining to maturity.

Question 283

L8 - what is the hedging error called when gamma rises too much?

Answer 283

Jump risk.

Question 284

L8 - How would you make a portfolio Γ (gamma) neutral?

(there is a formula for this.

2. What would you do if you have negative gamma?

Answer 284

A position in the underlying asset has a gamma of 0, so only the position in the option can be changed to alter the gammer of the portfolio.

A delta neutral portfolio has gammer equal to Γ, and a traded option has a gamma equal to Γ((T)).

The gamma in a delta neutral portfolio with the addition of a number of traded options will be:

w((T))Γ((T)) + Γ.

The position required in a traded option is therefore

w((T)) = -Γ / Γ((T)).

2. The negative sign means that you do the opposite. If you have negative gamma, it means you have shorted the call options, and need to buy call options with a positive gamma to offset the existing gamma and attain gamma neutrality.

Question 285

L8 - Why would you make a portfolio both gamma and delta neutral?

Answer 285

Gamma neutrality provides protection against larger movements in the stock price BETWEEN HEDGE REBALANCING whereas delta neutrality provides protection abasing relatively small stock price moves BETWEEN REBALANCING.

Question 286

L8 - Suppose a portfolio is delta neutral and has gamma of -3000. The delta and gamma of a particular traded call option are .62 and 1.5. How can the portfolio be made gamma neutral?

What will happen to the delta of the portfolio?

Answer 286

Portfolio
Γ = -3000
Δ = 0

This indicates that the individual is short on the stock, so will need to go long on the call in order to balance.

w((T)) = -Γ / Γ((T)).

w((T)) = -(-)3000/ 1.5 = 2000 in the call option.

The delta of the portfolio will change from 0 to 2000*.62=1240.

his means that 1240 units of the underlying asset must be sold from the portfolio to keep it delta neutral.

Question 287

L8 - How will making a portfolio gamma neutral effect the delta neutrally of a portfolio?

Answer 287

Gamma neutrality will require taken out a position an option as the gamma of the underlying asset will be 0. As such, once an investor goes long on a call option (due to negative gamma) they will then need to rebalance their portfolio by taking up the opposite position in the underlying asset, so the number of calls purchased * the Δ of the purchased call will need to need to be sold of the asset.

Question 288

L8 - The gamma equation is as follows:

Γ = N’(d1) / [S0 σ √T]]

When is the option at its highest and what happens over time?

Answer 288

Highest value is when the option is at or around the money and falls with time.

When the option is out of the money or in the money, the gamma will rise and then fall with time.

Question 289

L8 - What is the equation that shows the relationship between Delta, Gamma and Theta?

Answer 289

θ + rSΔ + 1/2σ^2 S^2 Γ = rπ

Question 290

L8 - What is ν (Vega) and what is the intuition behind its size?

Answer 290

The rate of change of the value of a the portfolio with respect to the volatility of the underlying asset.

Vega is always positive for a long position. (so negative for a short position).

So ν = 𝛿π/𝛿σ

When Vega is highly positive or negative, the portfolios value is very sensitive to small changes in volatility. When close to 0, volatility changes have relatively little impact on the value of the portfolio.

Question 291

L8 - What is the equation to calculate ν (Vega) for calls and puts?

Answer 291

ν = S0 (√T) N’(d1)

Question 292

L8 - What is the equation that will make a portfolio ν (Vega) neutral?

Answer 292

A portfolio can be made Vega-neutral by taking a position:
- ν / ν((T))

ν: the vega of the delta-neutral portfolio.
ν((T)): the Vega of the traded option that we wish to take a position in.

(the same formula as for Gamma neutrality but using ν instead).

Question 293

L8 - What is the differential equation that derivatives must satisfy?

How does this simplify to include delta, Gamma and Vega?

Answer 293

𝛿π/ 𝛿t + rS 𝛿π/ 𝛿S + 0.5 σ^2 S^2 𝛿^2π/ 𝛿S^2 = rπ

As
Δ: 𝛿π/ 𝛿t
θ: 𝛿π/ 𝛿S
ν: 𝛿^2π/ 𝛿S^2

We attain
θ + rSΔ + 1/2σ^2 S^2 Γ = rπ

Question 294

L8 - Extra space for notes.

Answer 294

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Question 295

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Answer 295

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Question 296

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Question 297

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Question 298

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Question 299

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Answer 299

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Question 300

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Answer 300

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Question 301

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Question 302

L8 - Extra space for notes.

Answer 302

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