Derivatives Markets Flashcards

Question

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

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

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.

Answer 5

Short Term Interest Rate. | Contract length is typically 3 months.

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

The variation margin.

Answer 10

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

Answer 11

By the end of the next day, before close.

Answer 12

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.

Answer 13

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.

Answer 14

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.

Answer 15

The Commodity Futures Trading Commission (CFTC)

Answer 16

Front running.

Answer 17

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.

Answer 18

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.

Answer 19

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.

Answer 20

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).

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

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

Answer 26

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)

Answer 27

A hedge that completely eliminates risk.

Answer 28

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.

Answer 29

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.

Answer 30

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

Answer 31

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.

Answer 32

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.

Answer 33

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.

Answer 34

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

Answer 35

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.

Answer 36

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.

Answer 37

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.

Answer 38

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.

Answer 39

It increases it.

Answer 40

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

Answer 41

Inter Continental Exchange London Inter Bank Offer Rate.

Answer 42

Sterling Over Night Index Average

Answer 43

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.

Answer 44

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

Answer 45

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.

Answer 46

1. The underlying asset of the futures contract. | 2. The choice of the delivery month.

Answer 47

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.

Answer 48

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.

Answer 49

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.

Answer 50

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

Answer 51

F = P(1+rT)

Answer 52

F = P(1+r)^T

Answer 53

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

Answer 54

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.

Answer 55

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.

Answer 56

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.

Answer 57

F = P(1+r)^-T

Answer 58

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

Answer 59

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

Answer 60

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

Answer 61

We discount each cash flow at the appropriate zero rate.

Answer 62

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).

Answer 63

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

Answer 64

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

Answer 65

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.

Answer 66

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.

Answer 67

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.

Answer 68

The Bid-Offer Average

Answer 69

**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.

Answer 70

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)

Answer 71

In V(swaps), it is the Bond.

Answer 72

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 ```

Answer 73

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

Answer 74

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.

Answer 75

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.

Answer 76

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.

Answer 77

The Bid-Offer Average

Answer 78

**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.

Answer 79

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)

Answer 80

In V(swaps), it is the Bond.

Answer 81

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. ```

Answer 82

1. Time value | 2. Intrinsic Value

Answer 83

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.

Answer 84

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))

Answer 85

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.

Answer 86

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.

Answer 87

Call option writer is selling the right to buy shares. | Put option write is selling the write to sell shares.

Answer 88

Call option holder is buying the write to buy shares. | Put option holder is buying the right to sell shares.

Answer 89

Call relates to the right to buy | Put Relates to the right to sell

Answer 90

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.

Answer 91

+ 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)

Answer 92

- 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.

Answer 93

+ 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.

Answer 94

+ 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.

Answer 95

+ 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.

Answer 96

C ≤ S((0)) | Otherwise you would buy the stock and short calls on it without any risk to one's own wealth.

Answer 97

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

Answer 98

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.

Answer 99

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).

Answer 100

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.

Answer 101

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.

Answer 102

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

Answer 103

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

Answer 104

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))

Answer 105

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.

Answer 106

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

Answer 107

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.

Answer 108

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.

Answer 109

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.

Answer 110

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

Answer 111

They are known as spreads. 1. Bull 2. Bear 3. Box 4. Butterfly

Answer 112

These are known as combinations. 1. Straddle 2. Strip 3. Strap 4. Strangle

Answer 113

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 2. 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.

Answer 114

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.

Answer 115

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

Answer 116

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.

Answer 117

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.

Answer 118

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.

Answer 119

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.

Answer 120

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

Answer 121

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)

Answer 122

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

Answer 123

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.

Answer 124

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.

Answer 125

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.

Answer 126

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.

Answer 127

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.

Answer 128

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.

Answer 129

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

Answer 130

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.

Answer 131

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.

Answer 132

Φ(0, 0.5) | With the standard deviation being the root of .5

Answer 133

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.

Answer 134

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

Answer 135

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

Answer 136

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).

Answer 137

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).

Answer 138

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

Answer 139

Brownian Motion.

Answer 140

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.

Answer 141

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.

Answer 142

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.

Answer 143

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.

Answer 144

To indicate that we are operating in continuous time.

Answer 145

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

Answer 146

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.

Answer 147

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.

Answer 148

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.

Answer 149

A geometric Brownian Motion Formula.

Answer 150

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

Answer 151

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

Answer 152

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.

Answer 153

Space for P.14.

Answer 154

Space for P.14.

Answer 155

Space for P.14.

Answer 156

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

Answer 157

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

Answer 158

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

Answer 159

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.

Answer 160

(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**/

Answer 161

slide 20/25

Answer 162

slide 20/25

Answer 163

slide 20/25

Answer 164

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

Answer 165

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

Answer 166

GO OVER NOTES.

Answer 167

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.

Answer 168

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.

Answer 169

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.

Answer 170

d1 = [ln (s0/E) + 0.5*(r + (σ^2/ 2)]/ σ√T S0: the current price of the stock. E: Strike Price.

Answer 171

d2 = d1 - σ√T

Answer 172

c = S0 N(d1) - Ee^(-rt)N(d2) S0: the current price of the stock. E: Strike Price.

Answer 173

p = Ee^(-rt)N(-d2) - S0 N(-d1) S0: the current price of the stock. E: Strike Price.

Answer 174

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

Answer 175

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.

Answer 176

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 )

Answer 177

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)

Answer 178

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).

Answer 179

Δ((put)) to = N(d1) - 1

Answer 180

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.

Answer 181

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.

Answer 182

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 ```

Answer 183

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.

Answer 184

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).

Answer 185

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.

Answer 186

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.

Answer 187

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.

Answer 188

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

Answer 189

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

Answer 190

θ((call)) - θ((put)) = -rEe^(-rT).

Answer 191

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.

Answer 192

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.

Answer 193

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

Answer 194

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).

Answer 195

N'(d1) = [1/ (√2π] e^[-d1^2)/ 2]

Answer 196

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

Answer 197

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

Answer 198

Γ = 𝛿^2 π / 𝛿 S^2

Answer 199

𝛿π/𝛿S Where π is the value of the portfolio.

Answer 200

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.

Answer 201

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.

Answer 202

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.

Answer 203

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.

Answer 204

Jump risk.

Answer 205

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.

Answer 206

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.

Answer 207

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.

Answer 208

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.

Answer 209

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.

Answer 210

θ + rSΔ + 1/2σ^2 S^2 Γ = rπ

Answer 211

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.

Answer 212

ν = S0 (√T) N'(d1)

Answer 213

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).

Answer 214

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

Derivatives Markets Flashcards

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