Derivatives Flashcards

Question

Short put (BULLISH/ neutral) Intrinsic value Value at expiration Profit Maximum profit Maximum loss Breakeven

Answer 1

(0, X - ST) -pT - pT + p0 p0 X - p0 ST* = X - p0

Answer 2

* The premium = Intrinsic Value + Time Value - Time value is positive but declines to zero at expiry

Answer 3

- Long call/put option: Profit = Value - Premium

Answer 4

which means that profit for the long creates an equal loss for the short and vice versa

Answer 5

Reference Asset ------ Protection Buyer (Swap premium) ------ Protection Seller ( Cash payment should default occur) * Credit event - Failure to pay (default) - Bankruptcy - Debt restructuring * Reference asset - May be an individual bond, an index of issuers or a special purpose entity with a debt portfolio, e.g. loans, bonds, mortgages * Payout - Cash-settled: Investor receives bond value less recovery rate * Naked CDS - Investor buys CDS without owning an underlying bond is seeking to gain from an increase in the credit spread of an underlying bond

Answer 6

* Derivative instruments provide users with the opportunity to allocate, transfer and/or manage risk without trading the underlying * Cash/Spot market prices for financial instruments and commercial goods and services are an important source of information for the decision to buy or sell * In many instances issuers and investors face a timing difference between an economic decision and the ability to transact in the cash market. Derivatives can be used to bridge this gap Examples Issuers - A retailer my await a shipment of goods priced in a foreign currency before selling domestic currency to make the payment - An issuer may want to lock in the cost of future debt in advance of the maturity of an existing bond Investors - An investor may want to capitalize on a market view but not have the cash in hand to transact in the cash market - In anticipation of receiving a future dividend an investor may decide today how he/she will reinvest the proceeds in the future * Derivative instruments serve as a price discovery function beyond the underlying cash market * Future prices reveal information about the direction of cash markets in the future - Equity market participants monitor equity index futures prices prior to the stock market opening to get an indication of the direction of cash markets in early trading - Analysts use interest rate futures markets to understand investor expectations of a central bank interest rate increase or decrease at a future meeting - Commodity futures prices are a gauge of supply and demand dynamics between producers, consumers, and investors across maturities Derivative instruments offer a number of operational advantages to cash or spot market transactions 1. Transaction costs – commodity derivatives eliminate the need to transport, insure, and store a physical asset in order to take a position 2. Increased liquidity – derivative markets typically have greater liquidity as a result of the reduced capital required to create a position 3.Upfront cash requirements – initial margin on futures and premiums on options are relatively low 4. Short positions – it is easier to take a short position since there is no need to find a cash asset owner who is willing to lend the underlying asset for a period of time NOTE: derivative markets also make it less costly to exploit mispricing/arbitrage opportunities

Answer 7

Greater potential for speculative use: High degree of implicit leverage for some derivative strategies may increase likelihood of financial stress Lack of transparency: Derivatives add portfolio complexity and may not be well understood by stakeholders Basis risk: Potential divergence between the expected value of a derivative instrument versus an underlying or hedged transaction Liquidity risk: Potential divergence between the cash flow timing of a derivative instrument versus an underlying or hedged transaction Counterparty credit risk: Particularly when trading OTC with no central counterparty Destabilization and systemic risk: Excessive risk taking and use of leverage in derivative markets may contribute to market stress, as in the 2008 financial crisis

Answer 8

Cash flow Absorbs variable cash flow of floating rate asset or liability (forecasted transaction) * Interest rate swap to a fixedrate for floating-rate debt * FX forward to hedge forecasted sales

Answer 9

Fair value Offsets fluctuations in fair value of an asset or liability * Interest rate swap to a floating rate for fixed-rate debt * Commodity future to hedge inventory

Answer 10

Net investment Designated as offsetting the FX risk of the equity of a foreign operation * Currency swap * Currency forward

Answer 11

Investors use derivatives to: * Replicate a cash market strategy * Hedge a fund’s value against market movements * Modify or add exposure Examples * Forward commitments - Using copper forwards to create exposure to the price of copper with little initial outlay * Contingent claims - Using a call option to create a long exposure to an asset with very little outlay - Using a covered call to modify the return to a long position in a stable market

Answer 12

* Arbitrage trades can be undertaken when two assets produce the same results but trade at different prices, i.e. the ‘law of one price’ does not hold, - Buy the underpriced asset, sell the overpriced asset - Arbitrageur receives money at the start of the holding period * An arbitrage trade can also be executed if an asset with a known future price does not trade at the present value of its future price * Derivatives derive their prices from underlying assets - Can be used to hedge the underlying - All risk is eliminated * Position will earn the risk-free rate - Derivatives are priced on a risk-free basis * Called ‘risk-neutral’ pricing

Answer 13

* A strategy in which a derivative’s cash flow stream may be recreated using a combination of long or short positions in the underlying asset and borrowing or lending cash * It is typically used to mirror or offset a derivative position when the ‘law of one price’ holds

Answer 14

The relationship between spot and forward prices when there are no benefits or costs (such as dividends or storage costs) associated with the asset is given by: F0(T) = S0(1+r)T (1) or F0(T) = S0erT (2) Where r is the opportunity cost of holding the asset * Equation (1) Is used when S0 is the price of an asset such as a share * Equation (2) is used when we have portfolios, such as equity portfolios, and for FX forwards

Answer 15

F0(T) = [S0 – PV0(I) + PV0(C)](1+r)T (1) or F0(T) = S0e^(r+c-i)T (2) Where: * r is the opportunity cost of holding the asset * (C, c) is the cost of ownership. Commodities, for example, have costs of ownership such as storage, transportation, insurance * (I, i) is the benefits of ownership such as dividends on shares or coupons on bonds

Answer 16

Opportunity and other costs > Benefits: F0(T) > S0 Opportunity and other costs < Benefits: F0(T) < S0 Opportunity and other costs = Benefits: F0(T) = S0

Answer 17

forward rate = S0e(r$-r£)T NOTE: We say that USD is at a discount since we get more USD for GBP1 in 6 months than we do today. However, we say that the FX Forward price is at a premium. Here £ is the base currency and $ is the price currency

Answer 18

rprice > rbase F0(T) > S0 Discount Premium

Answer 19

rprice < rbase F0(T) < S0 Premium Discount

Answer 20

rprice = rbase F0(T) < S0 Neither a premium nor a discount Neither a premium nor a discount

Answer 21

* Represents non-monetary advantage of holding the asset * Primarily associated with commodities - Due to difficulty in shorting the asset or tight supplies * Benefits increase the current price

Answer 22

* Cost of storage, insurance, potential spoilage (for soft commodities) and opportunity cost of money invested * Reduce the current price

Answer 23

Equity Non-dividend paying stock None Risk-free rate

Answer 24

Equity Dividend paying stock, Equity index Dividend, Dividend yield Risk-free rate

Answer 25

FX Market exchange rates None Difference between the countries’ risk-free rates

Answer 26

Commodities Soft/hard commodities, Commodity index Convenience yield Risk-free rate, Storage cost

Answer 27

Interest Rates Bonds Interest income Risk-free rate

Answer 28

Credit Single reference entity, Credit indexes Credit spread Risk-free rate

Answer 29

* There is no requirement for an initial cash outlay * The contracts have an initial value of zero * The contract value after initiation, can be positive or negative * The contract price is a fixed price or rate for the underlying at a later date, not an amount to be paid at the start

Answer 30

* A forward contract is a delayed delivery sales contract where the seller (short) agrees to deliver the underlying asset to the buyer (long) in exchange for an amount that is fixed today and paid at the expiration of the forward contract Price: Fixed price or interest rate agreed Pricing: Determination of the fixed price (arbitrage theory) Value: Amount paid/received to assume a forward contract Valuation: Determination of value at any time

Answer 31

F0(T) = [S0 – PV0(I) + PV0(C)] (1+r)T * F0(T) is the forward price agreed to at time 0 for a forward contract expiring at time T * S0 is the spot price of the underlying asset at contract initiation * PV0(I) is the present value of any cash flows on underlying during life of contract * PV0(C) is the present value of any other costs incurred (e.g. storage and insurance) other than the opportunity cost * r is the risk-free borrowing and lending rate, expressed on an EAR * T is the time from contract initiation to expiration

Answer 32

Vt(T) = [St – PVt(I) + PVt(C)] - PV[F0(T)] Value of a forward to the buyer at any time t = * Price of underlying at time t * - PV at time t of remaining cash flows on the underlying * + PV at time t of any other costs * - PV at time t of forward contract price * At initiation V0 = 0 During a contract’s life the Vt can be + or - since the forward price is fixed but the underlying is not * At expiration VT = ST – F0(T)

Answer 33

Vt(T) = St - F0(T)/(1+r)(T-t)/T

Answer 34

* Agreement to pay or receive, on an agreed future date, the difference between a fixed interest rate agreed at the outset and a market reference rate (MRR) prevailing on a given future date, for a given future period (the FRA run) * Underlying asset - Interest payment made in a specified currency * Payments are based on a notional amount and paid at the start of the FRA run * No exchange of principal takes place * NOTE: the fixed rate is the implied forward rate for the FRA run and is calculated in a similar way to the previous examples Long FRA * Pay fixed rate, receive floating rate * Makes a profit when MRR Short FRA * Receive fixed rate, pay floating rate * Makes a profit when MRR

Answer 35

* Futures contracts * Standardized terms * Traded on a futures exchange * More heavily regulated * Contracts are marked to market on a daily basis (margining) - Payment by loser to the winner * Exchanges - Provide a credit guarantee - Backed by a clearing house * Transfers the cash payments on a daily basis * Cash flows - Forwards – Gains/Losses paid at expiration of the contract * MTM can be positive or negative, i.e. counterparty credit risk exists - Futures – Receive/Pay interim cash via margining, i.e. MTM is zero

Answer 36

* Forward Rate Agreements trade OTC - The long agrees to pay a fixed rate and receive a MRR - The price of the contract is the fixed rate - The long makes money if the MRR rises * Short term interest rate futures trade on exchanges - The price of the contract is: 100 – (100 x MRR) - The long makes money if the MRR goes down - MTM is zero since the margining process transfers cash payments on a daily basis Contract type: Interest rate futures Gains from rising MRR: Short futures contract Gains from falling MRR: Long futures contract Contract type: Forward rate agreement Gains from rising MRR: Long FRA: Fixed-rate payer Gains from falling MRR: Short FRA: Floating rate payer

Answer 37

Uncorrelated – forwards and futures prices will be the same - Positively correlated – then a long position in a futures contract is more desirable * When gains are made as prices and rates rise, the gain is reinvested at higher interest rates * When losses are made as prices and rates fall, the loss is made at lower interest rates * Futures prices > Forward prices - Negatively correlated * Futures prices < Forward prices

Answer 38

* Swap and A Series of Forward Contracts - Agreement between two parties to exchange a series of future cash flows * Series of forward contracts - At least one of the cash flows is determined by a later outcome - No payment upfront - Both forwards and swaps involve counterparty credit risk * Differences between a Swap and a Series of Forwards Swap Multiple settlements Payment at end of period Fixed rate is constant for each period Series of forward contracts Each forward has a single settlement Payment at the start of the period Fixed rate is typically different for each period

Answer 39

The par swap rate is the fixed rate that equates the PV of all future expected floating rates to the PV of the fixed cash flows If we wanted to find the three-year swap rate, we use the implied forward rates to solve for s3 The three-year swap rate identified in the previous example was 3.78% * This implies that an investor would be indifferent to - Paying the fixed swap rate and receiving the respective forward rates or - Receiving the fixed swap rate and paying the respective forward rates * Note that although the value of the swap at t = 0 is zero some individual forward exchanges might have positive or negative values - E.g. in one year’s time the fixed rate receiver will * Receive the fixed swap rate of 3.78% * Pay the initial floating rate of 2.46% Value of fixed payments = Value of floating payments * Receiver swap - Treat a receive fixed swap as buying a fixed-rate bond and issuing a floating-rate bond - Receiving the fixed swap rate and paying the respective forward rates * Payer swap - Treat a pay fixed swap as buying a floating-rate bond and issuing a fixed-rate bond - Receiving the respective floating swap rates and paying the respective fixed rate * Swaps can be used to change a company’s interest rate exposure - A company that is borrowing at a floating rate can convert this into a fixed rate borrowing by entering into a pay fixed swap

Answer 40

* The value of a swap at inception is zero (V0(T) = 0) * For a forward with one settlement, the value of the forward for the long is - VT(T) = ST – F0(T) where ST is the spot price and F0(T) is the forward price * For a swap with periodic settlements, the current MRR is the spot price and the fixed swap rate, SN, is the forward price - The periodic settlement value = (MRR - SN) x Notional amount x Period * The value of the swap on any settlement date equals the current settlement value as above plus the present value of all other remaining future swap settlements - Generally the MRR is set at the beginning of each period and the fixed versus floating difference is exchanged at the end of each period

Answer 41

* A bond’s duration is a measure of interest rate risk - The higher the duration the riskier the bond or a bond portfolio * When you are long a fixed rate bond you receive fixed coupons * When you enter into a receive fixed swap you receive fixed payments * Since being long a fixed-rate bond and entering into a receive fixed swap involve receiving fixed payments then doing either will increase the duration in a portfolio * To reduce the bond portfolio duration you would short bonds or enter into a pay fixed swap

Answer 42

* Options will only be exercised at maturity if the payoff is positive. That is: - (ST – X) > 0 for a call - (X – ST) > 0 for a put - If not exercised the option expires worthless and the buyer’s loss equals the premium * Before maturity (t PV(X): St – PV(F0(T)) = Max (0, St – PV(X)) * Note that here we are ignoring the option premium

Answer 43

* The time value of an option is equal to the difference between the current option price and the option’s current exercise value (payoff) - ct = Max (0, St - X/(1+r)(T-t)) + Time Value - pt = Max (0, X/(1+r)(T-t) - St) + Time Value * Time value is generally positive but declines to zero at maturity * The process is known as time value decay

Answer 44

Payoff for European Options at Maturity * cT = Max (0, ST – X) * pT = Max (0, X – ST) Buyer’s profit for European Options at Maturity * Profit for call = Max (0, ST – X) – c0 * Profit for put = Max (0, X – ST) – p0 * Forward buyer - Pays nothing up front - Unlimited profits and loss limited to forward price * Option buyer - Pays premium - Unlimited profits (for a call) and loss limited to premium

Answer 45

* Maximum value of a European call: St * Maximum value of a European put: X

Answer 46

Replicating a call option at contract inception * Borrow X at the risk-free rate * Use the proceeds to buy the asset at a price of S0 At maturity there are two possible outcomes * Exercise (ST > X): sell the underlying at ST and use the proceeds to repay X * No exercise (ST < X): No settlement is required * A key difference between replicating forwards and options is that with options we borrow a proportion of X based on the likelihood of exercise (hedge ratio) at time T. The replicating transaction has to be adjusted as the likelihood of exercise changes

Answer 47

Replicating a put option at contract inception * Sell the asset short at a price of S0 * Lend the proceeds at the risk-free rate At maturity there are two possible outcomes * Exercise (ST < X): purchase the underlying at ST from the proceeds of the riskfree loan * No exercise (ST > X): No settlement is required * Again, a key difference between replicating forwards and options is that with options we lend a proportion of X based on the likelihood of exercise (hedge ratio) at time T. The replicating transaction has to be adjusted as the likelihood of exercise changes

Answer 48

Call: incr, Put: decr Call: decr, Put: incr Call: incr Put:incr,decr Call: incr Put:decr Call: incr Put:incr Call: incr, decr Put:incr, der

Answer 49

c0 + X/(1+rf)^T = p0 + S0

Answer 50

Buy call Buy bond

Answer 51

Buy put Buy underlying asset

Answer 52

Long put + Long underlying + Short bond p0 + S0 - X / (1 + r)T

Answer 53

Long call + Short underlying + Long bond c0 - S0 + X / (1 + r)T

Answer 54

Long call + Long bond + Short put c0 + X / (1 + r)T- p0

Answer 55

Long put + Long underlying + Short call p0 + S0 - c0

Answer 56

Long underlying + Short Call S0 - c0 = X / (1 + r)T - p0

Answer 57

c0 + X/(1+rf)^T = p0 + F0(T)/(1+rf)^T

Answer 58

Buy Forward contract Buy bond Buy put

Answer 59

* Assume that at time t=0 a firm has borrowed capital in the form of a zero-coupon bond with face value D * The market value of the firm’s assets, V0, is equal to the PV of its debt plus its equity, E0 - V0 = E0 + PV(D) * When the firm’s debt matures at time T, the assets are distributed amongst the shareholders and debtholders with two possible outcomes 1. Solvency (VT > D): the firm can return capital to both its shareholders and its debtholders 2. Insolvency (VT < D): shareholders receive nothing and debtholders receive less than D (i.e. VT) * Unlike with Put-Call Parity here the bond is risky since debtholders only receive D if the firm is solvent - Debtholders will demand a premium similar to a put option premium from shareholders in order to assume the risk of insolvency Shareholders have unlimited upside and limited downside * Debtholders have limited upside whilst principal and interest payments are at risk if the firm becomes insolvent * Payoff profiles are: - Shareholder payoff is max(0, VT - D) which is the same payoff as a long call with strike D * Note that a Long Call is equivalent to a Long Asset plus a Long Put - Debtholder payoff is min(VT,D) * Note that this equates to being long the bond with face value of D and short a put with strike D on the firm value VT * Using S0 + p0 = c0 + PV(X) and substituting V0 for S0 and D for the risk free bond, X, we get V0 + p0 = c0 + PV(D) * From which we can see that V0 = c0 + PV(D) - p0 * Shareholders have a position with a payoff similar to a long call * Debtholders have a long risk-free bond plus a short put on the firm’s assets - The short put can be interpreted as the credit spread on the firm’s debt or the premium above the risk-free rate that shareholders must pay to debtholders to assume the risk of insolvency - From a debtholder perspective, the more valuable the sold put the more credit risk is present in the firm’s debt

Answer 60

Over a given period of time an asset’s price will go up from S0 to S1 + or go down from S0 to S1- * The gross return will be either - Ru = S1 +/S0 > 1 - Rd = S1-/S0 < 1 h = (c1+ – c1-)/ (S1+ – S1-) This is known as the hedge ratio V1+ = hS1+ – c1+ V1- = hS1- – c1- * The return V1+ / V0 = V1- / V0 must equal (1 + risk-free rate) - hS0 – c0 = V1+ / (1 + r)

Answer 61

u= S1+ / S0 and d=S1+/S0 pi = (1+r-d)/u-d c0 = (pic1+ + (1-pi)c1-) / (1+r) π and 1 - π are not the probabilities of an up and down movement but are the probabilities that would arise if we equate the portfolio’s value with the discounted future value using the risk-free rate as a discount factor

Answer 62

Step 1 Use the factors to estimate the next two possible prices of the asset Step 2 Use the asset prices to derive the next two possible option values Step 3 Compute the risk neutral probability Step 4 Discount the expected value of the option at the risk-free rate

Answer 63

Stir - payoff linear Fra - profit greater than loss - convexity

Answer 64

Fixed Detived from implied forward rates and spot rates

Answer 65

Fixed Detived from implied forward rates and spot rates