CFA L2 Derivatives Flashcards

Question

Example: Calculating the quoted futures price of a Treasury bond futures contract There is a 1.2-year Treasury bond futures contract. The cheapest-to-deliver (CTD) bond is a 7% T-bond with exactly 10 years remaining to maturity and a quoted price of $1,040 with a conversion factor of 1.13. There is currently no accrued interest because the bond has just paid a coupon. The annual risk-free rate is 5%. The accrued interest on the bond at maturity of the futures contract will be $14.

Answer 1

Full price = $1,040 (clean price) + $0 (accrued inerest) Semi-annual Coupon = $1,000 * 0.07 * 0.5 = $35 1.2 years to maturity - 0.5 = 0.7 & 1.2 years to maturity - 1 year = 0.2 FVC = 35 * (1.05)^.7 + 35 * (1.05)^.2 = $71.56 QFP = [ $1,040 * (1 + 0.05)^1.2 - $71.56 - $14 ] ÷ 1.13 = $900.13

Answer 2

Deposits in large banks outside the United States denominated in U.S. dollars.

Answer 3

The lending rate on dollar-denominated loans between banks. It is quoted as an annualized rate based on a 360-day year. ## Footnote * This is a benchmark rate, whereas the Fed Funds rate is simply a target rate * The CFA referes to SOFR as MRR

Answer 4

An OTC contract between parties that determines the rate of interest to be paid on an agreed-upon date in the future. The long position is effectively borrowing money (long the loan, with the contract price being the interest rate on the loan). If the floating rate at contract expiration > the rate specified in the forward agreement, the long position in the contract can be viewed as the right to borrow at below-market rates and will receive a payment from the short. If the floating rate at the expiration date < the rate specified in the forward agreement, the short will receive a cash payment from the long. ## Footnote * FRAs are always cash settled. * FRAs pay off at the date of expiry (the start of the notional borrowing period), * The price of a FRA is always quoted as an interest rate

Answer 5

Long position: Pay fixed and receive floating Short position: Pay floating and receive fixed

Answer 6

The forward price in an FRA is actually a forward interest rate. There are 3 things to keep in mind when pricing and valuing FRAs: 1. MRR rates in the Eurodollar market are add-on rates and are always quoted on a 30/360 day basis in annual terms. So if it's quoted at 6%, the unannualized rate is 6 * (30/360) = 0.5% 2. The long position in an FRA benefits when the rate increases. 3. The payoff of the FRA is INDEPENDENT of the payoff of the underlying loan.

Answer 7

1. De-annualize the MRR 2. [ (1 + long rate) ÷ (1 + short rate) ] - 1 3. Re-annualize the rate: #2 * (360 ÷ # of days of the contract)

Answer 8

4% * (30/360) = 0.33% 5% * (120/360) = 1.67% So, now we need to calculate the actual rate on a 90 day loan from month 1 to month 4 (day 30 to day 120): [ (1 + .0167) ÷ (1 + .0033) ] - 1 = 0.0133 No we need to re-annualize: 0.0133 * (360/90) = 5.32% → this is the no arbitrage forward that that makes the values of the long an short positions in the FRA zero at initiation.

Answer 9

1. [ (new rate of the # of days in the contract) - the contract rate ] * notional principal. 2. Take the PV of #1

Answer 10

1. 0.057 * (20/360) = 0.0031667 2. 0.059 * (110/360) = 0.0180278 3. ((1 + 0.0180278) ÷ (1 + 0.0031667)) - 1 = 0.0148 4. 0.0148 * (360/90) = .0593 5. [ (.0593 * (90/360) - (.0532 * (90/360)) ] * $1,000,000 = $1,514.20 6. ($1,514.20) ÷ [ 1+ (.059 * (110/360)) ] ## Footnote * Remember that if the rate in the future is less than the FRA rate, the long is "obligated to borrow" at above-market rates and will have to make a payment to the short. If the rate is greater than the FRA rate, the long will receive a payment from the short.

Answer 11

Essentially forward contracts that trade on exchanges. A clearinghouse guarantees that traders in the futures market will honor their obligations. To safeguard the clearinghouse, the exchange requires both sides of the trade to post margin and settle their accounts on a daily basis.

Answer 12

The process of adjusting the margin balance in a futures account each day for the change in the value of the contract from the previous trading day, based on the settlement price. Futures contracts have no value at contract initiation. Unlike forward contracts, futures contracts do not accumulate value changes over the term of the contract. Since futures accounts are marked to market daily, the value after the margin deposit has been adjusted for the day's gains and losses in contract value is always zero. The futures price at any point in time is the price that makes the value of a new contract equal to zero. The value of a futures contract strays from zero only during the trading periods between the times at which the account is marked to market. ## Footnote * Between the mark-to-market account adjustments, the contract value is calculated just like that of a forward contract; it is the difference between the price at the last mark-to-market and the current futures price, (i.e. the futures price on a newly issued contract).

Answer 13

Current futures price − Previous mark-to-market price ## Footnote * If the futures price increases, the value of the long position increases. The value is set back to zero by the mark-to-market at the end of the mark-to-market period.

Answer 14

(1) At initation, the forward/future price should be the same: S0 * (1 + Rf)^T = $1,800 * (1 + .05)^(60/365) = $1,814.49 (2) $1,814.49 * .1 = $181.45 (3) $1,820 * (1.05)^(59/365) = $1,834.41 → $1,834.41 - $1,814.49 = $19.92 added to margin account (4) $1,805 * (1.05)^(58/365) = $1,819.05 → $1,819.05 - $1,834.41 = -$15.36 subtracted from the margin account

Answer 15

A derivative wherer one party agrees to pay floating and receive fixed and the other party pays fixed and receives floating. At initiation, the PV of the floating rate payments is equal to the PV of the fixed rate payments. The fixed rate is the **swap rate**. If market interest rates increase during the life of the swap, the fixed rate payer will benefit and vice versa for falling rates. | The value of a swap at initation is zero ## Footnote * Interest rate swaps settle over multiple periods, whereas FRAs are just for one period.

Answer 16

The swap fixed rate is derived from the MRR curve corresponding to the swap tenor. For instance, if there's a 2-year semiannal interest rate swap, the swap fixed rate underlying this swap will be determined based on the MRR rates corresponding to the four settlement dates of this swap. We must first calculate each of the discount factors (Zs) for each settlement date. We can then calculate the periodic swap fixed rate (SFR).

Answer 17

Z = 1 ÷ [ 1 + (MRR * (days ÷ 360)) ] | However many decimal places the question uses, use that in calculations

Answer 18

(1 - final discount factor) ÷ (sum of discount factors) ## Footnote * To calculate the annual swap fixed rate, multiply this result by the # of settlement periods.

Answer 19

(1) [ 1 ÷ (1 + (.030 * 90/360)) ] = 0.993 [ 1 ÷ (1 + (.035 * 180/360)) ] = 0.983 [ 1 ÷ (1 + (.040 * 270/360)) ] = 0.971 [ 1 ÷ (1 + (.045 * 360/360)) ] = 0.957 ( 1 - 0.957) ÷ (0.993 + 0.983 + 0.971 + 0.957) = 0.011 * 100 = 1.1% = Periodic SFR. Annual SFR = 1.1% * 4 = 4.4% (2) 0.011 * $5,000,000 = $55,000

Answer 20

Value to the payer = ΣZs * (SFRnew - SFRold) * (days/360) * notional principal ## Footnote * For ΣZs, you must calculate all of the z-factors that have already happened (ex: if valuing a swap after 180 days, take the 90 day Z and the 180 day Z) and recalculate them based on the new SFR. * **Value to the receiver = -1 * value to the payer**

Answer 21

The int. rates used to price currency swaps are the swap rates calculated from each currency's term structure. ***Principal amounts are exchanged at initiation*** based on the exchange rates at inititation and periodic payments are based on each currency's fixed rate.

Answer 22

(1) The fixed rate on a 1-year quarterly UK interest rate swap is: (1 - 0.93458) ÷ (0.99010 + 0.97561 + 0.95694 + 0.93458) * 4 = 6.78% (2) $5,000,000 × £0.50 per $ = £2,500,000. At the initiation of the swap, we would exchange £2,500,000 for $5,000,000. We would pay 1.1% quarterly on the $5,000,000 notional principal ($55,000) and receive 1.7% on £2,500,000 quarterly (£42,500). At the end of one year, we would exchange the original principal amounts.

Answer 23

1. After however many days, calculate the Zs of the new interest rates 2. Multiply the Zs by the sum of notional principals of their respective currencies and any remaining periodic payments 3. Both of the #2s need to be converted into the same currency 4. Subtract the two values of #3 ## Footnote * Each CF corresponds to a settlement date * Since CFs are in different currencies, we must adjust one using the current exchange rate

Answer 24

An exchange of FCFs between two parties where one party pays a fixed rate plus any negative equity return and the other party pays any positive return on equity. ## Footnote * Nothing is paid at initiation

Answer 25

Fixed-rate payer: Pays the fixed rate plus any negative equity return and receives positive equity return Equity return payer: Pays any positive return on equity and receives a fixed rate payment plus any negative equity return ## Footnote * Equity return payer is bearish, whereas the fixed-rate payer is bullish

Answer 26

1. Calculate the value of the pay fixed side per $ of notional principal: DF of days remaining until expiration * (principal + coupons) 2. #1 * notional principal 3. Notional principal * (new value of index ÷ starting value of index) 4. #3 - #2

Answer 27

Value of the fixed side (per $1 notional) = PV of coupons + PV of principal * (0.0605 ÷ 4) * (0.99010 + 0.97363 + 0.95541 + 0.93567) + 0.93567 = 0.9940 * 0.9940 * $10MM = $9.94MM * The value of the $10MM investment after 30 days = $10MM * (996/985) = $10,111,675 * Value of the swap after 30 days to the fixed rate payer = $10,111,675 - $9.94MM = $171,675

Answer 28

value of swap = (–0.013 − 0.008) × $1,000,000 = –$21,000

Answer 29

An options valuation model based on the idea that, over the next period, the value of an asset will change to one of two possible values (binomial). In this model, the underlying will increase or decrease in value based on one of two possible values: U (up factor) or D (down factor). The spread between U and D will impact the option value. To construct a binomial model, we need to know the beginning asset value, the size of the two possible changes, and the probability of each of these changes occurring. ## Footnote * With interest rate trees in fixed income, the up or down scenario was 50/50 at each path, however this is not the case with the binomial model in derivatives.

Answer 30

U = up move factor (1 + % up) - going to go up by some %. D = size of the down move - **(1 - %)** S+ = stock price if an up move occurs S- = stock price if a down move occurs πU = probability of an up move πD = probability of a down move: **(1 - πU)**

Answer 31

(1 + Rf - D) ÷ (U - D)

Answer 32

False, they are risk-neutral probabilities

Answer 33

How analysts calculate the value of an option from this binomial tree. Calculation: 1. Calculate the payoff of the option at maturity in both the up-move and down-move states. 2. Calculate the expected value of the option in one year as the probability-weighted average of the payoffs in each state. 3. Discount the expected value back to today at the Rf ## Footnote * All binomial models calculate value as the expected outcome discounted back to the PV at the appropriate interest rate.

Answer 34

πU = (1 + 0.07 - 0.75) ÷ (1.33 - 0.75) = 55% πD = (1 - 0.55) = 45% Payoff in U scenario = $40 - $30 (exercise price) = $10 Payoff in D scenario = $0 The expected value of the option in one period: ($10 * 0.55) + ($0 * 0.45) = $5.50 The value of the option today, after discounting at the risk-free rate of 7%, is: $5.50 ÷ 1.07 = $5.14

Answer 35

An alternative method to the binomial method to value a put option. This only applies to European options. Put-call parity states that the value of a fiduciary call is equal to the value of a protective put. Formula: S0 + P0 = C0 + PV(X)

Answer 36

Long a call option + investment in a zero-coupon riskless bond with par value equal to the strike price Formula: C0 + PV(X)

Answer 37

long a stock and long a put option Formula: P0 + S0

Answer 38

Synethic call: C0 = P0 + S0 - PV(X) Synthetic stock: S0 = C0 - P0 + PV(X) Synthetic put: P0 = C0 - S0 + PV(X) Synthetic bond: PV(X) = P0 - C0 + S0

Answer 39

S0 + P0 = C0 + PV(X) P0 = C0 + PV(X) - S0 P0 = 8 + (60 ÷ 1.04) - 62 = $3.69 ## Footnote * Any other price than $3.69 allows for risk-free profit

Answer 40

1. Calculate the stock values at the end of 2 periods 2. Calculate the 3 possible option payoffs at the end of 2 periods. 3. Calculate the expected option payoff at the end of 2 periods (t = 2) using the up- and down-move probabilities. 4. Discount the expected option payoff (t = 2) back one period at the Rf to find the option values at the end of the first period (t = 1). 5. Calculate the expected option value at the end of one period (t = 1) using up- and down-move probabilities. 6. Discount the expected option value at the end of one period (t = 1) back one period at the Rf to find the option value today (t = 0).

Answer 41

πU = (1 + 0.07 - 0.80) ÷ (1.25 - 0.80) = 0.6 πD = 1 - 0.6 = 0.4 S++ = $50 * (1.25)^2 = 78.125 C++ = 78.13 - 45 = 33.13 S+-/S-+ = $50 * (1.25) * 0.8 = 50 C+- = 50 - 45 = 5 S-- = $50 * 0.80 = 32 C-- = $32 - 45 = 0 payoff S+ = $50 * 1.25 = 62.5 **C+ = [ (πU * C++) + (πD * C+-) ] ÷ (1 + Rf)** = [ (0.6 * 33.13) + (0.4 * 5) ] ÷ (1.07) = 20.45 S- = $50 * 0.80 = 40 **C- = [ (πU * C+-) + (πD * C--) ] ÷ (1 + Rf)** = [ (0.6 * 5) + (0.4 * 0) ] ÷ (1.07) = 2.80 **C = [ (πU * C+) + (πD * C-) ] ÷ (1 + Rf)** = [ (0.6 * 20.45) + (0.4 * 2.80) ] ÷ (1.07) = 12.51 ## Footnote * S++ = value of stock in two upward movement scenario * C++ = payoff of call option in two upward movement scenario

Answer 42

Intrinsic value: The actual value of the strike price versus the current market price Time value: The amt by which the option premium exceeds the exercise value. ## Footnote * Time value = 0 at expiration if option is at or out of the money. * Time value = exercise value at expiration if option is in the money.

Answer 43

True Interest on early payoff < time value lost ## Footnote * American call options that do have carry benefits are worth more than Euro call options.

Answer 44

False, it will be worth more. Interest on early payoff > time value lost

Answer 45

πU = (1 + 0.03 - 0.80) ÷ (1.30 - 0.80) = 0.46 πD = (1 - 0.46) = 0.54 S++ = $50(1.30)^2 = $84.50 P++ = $0 S+-/S-+ = $50 * 1.30 * 0.80 = $52 P+- $0 S-- = $50 * (0.80)^2 = $32 P-- $18 S+ = $50 * 1.30 = $65 P+ = [ (0.46 * $0) + (0.54 * $0) ] ÷ 1.03 = 0 S- = $50 * 0.80 = $40 P- = [ (0.46 * 0) + (0.54 * $18)) ÷ 1.03 = $9.44 P = [ (0.46 * 0) + (0.54 * $9.44)) ÷ 1.03 = $4.95

Answer 46

πU = (1 + 0.03 - 0.80) ÷ (1.30 - 0.80) = 0.46 πD = (1 - 0.46) = 0.54 S++ = $50(1.30)^2 = $84.50 P++ = $0 S+-/S-+ = $50 * 1.30 * 0.80 = $52 P+- $0 S-- = $50 * (0.80)^2 = $32 P-- $18 S+ = $50 * 1.30 = $65 P+ = [ (0.46 * $0) + (0.54 * $0) ] ÷ 1.03 = 0 S- = $50 * 0.80 = $40 P- = Option will be exercised for ($50 - $40) = $10 P = [ (0.46 * 0) + (0.54 * $10)) ÷ 1.03 = $5.24 | Since P- is deep in-the-money, it'll be exercised. ## Footnote * *With American options, we must always be on the lookout for a situation where the option is in-the-money. In these cases, the options will be exercised and the becomes out C or P values.*

Answer 47

The change in the call value given a change in the stock price. The result of this can be used to set up a hedged portfolio. Calculation: h = ( (C+) - (C-) ) ÷ ( S+ - S- ) OR ( (P+) - (P-) ) ÷ ( S+ - S- )

Answer 48

C0 = (h * S0) + [ (-h * S+) + C+ ] ÷ (1 + Rf) OR C0 = (h * S0) + [ (-h * S-) + C- ] ÷ (1 + Rf)

Answer 49

πU = (1 + 0.07 - 0.75) ÷ (1.333 - 0.75) = 0.5489 πD = 1 - 0.5489 = 0.4511 S+ = $30 * 1.333 = $40 C+ = $40 - $30 = $10 S- = $22.5 C- = 0 C = [ (.5489 * $10) + (0.4511 * 0) ] ÷ (1.07) = 5.13 h = ($10 - $0) ÷ ($40 - $22.5) = 0.5714 C0 = (0.5714 * $30) + [ (-0.5714 * $40) + $10 ] ÷ 1.07 = 5.13 * Since we know the option's market price is $6.50, there is risk-free arbitrage. And our cash and carry trade principal tells us to long the stock in the spot market when an option is overpriced and short the option in a futures contract. * Let's say we short 100 call options in futures contracts, then we need to long (h * 100) = 57.14 shares. * Net portfolio cost = (57.14 * $30) - (100 * 6.50) = $1,064 * We will borrow $1,064 at the Rf and have to repay ($1064 * 1.07) = $1,138.48 at T = 1. * Since at T0, the market can move up (S+) or down (S-), we must determine the value in both scenarios * Portfolio after up move = (57.14 * $40) - (100 * $10) = $1,286 * Portfolio after down move = (57.14 * $22.5) - (100 * 0) = $1,286 * Profit in 1 year = $1,286 - $1,138.48 = $147.52 * PV of profit = $147.52 ÷ 1.07 = $137.87

Answer 50

Payoff of interest rate call: [ notional principal * (Max(0, MRR - exercise rate)) ] Payoff of interest rate put: [ notional principal * (Max(0, exercise rate - MRR)) ]

Answer 51

(1) (.107383 - .055): Will be called (.071981 - .055): Will be called (.048250 - .055): Won't be called (2) 1iUU: [$1MM * (.107383 - .055) ] = $52,383 1iUL: [$1MM * (.071981 - .055) ] = $16,981 1iLL: [$1MM * (.048250 - .055) ] = 0 (3) C+ = [ ($52,383 * 0.50) + ($16,981 * 0.5) ] ÷ 1.057883 = $32,784 C- = [ ($16,981 * 0.5) + ($0 * 0.5) ] ÷ 1.038800 = $8,173.37 C = [ ($32,784 * 0.5) + ($8,173.37 * 0.5) ] ÷ 1.03 = $10,882

Answer 52

Long FRA = long interest rate call + short interest rate put Long cap = series of interest rate calls - Useful for hedging floating rate liabilities - allows to pay fixed while receiving MRR Long floor = series of interest rate puts - Useful for hedging floating rate assets - allows to pay MRR while receiving fixed Payer swap = long cap + short floor

Answer 53

An options valuation model that values options in continuous time and uses the no-arbitrage condition. - Binomial models can deal w/ Euro and American options. BSM can only handle Euro. ## Footnote * The binomial model uses discrete time

Answer 54

Formula: C0 = [ (S0 * e^(σ * T) * N(d1)) ] - (Xe^(-r * T) * N(d2)) - Essentially taking the PV of S0 & then subtracting the PV of the exercise price P0 = [ (X * e^(-r * T)) * N(-d2) ] - [ S0 * N(-d1) ] - Won't be asked to value a stock w/ these formulas. - d1 = [ ln(S ÷ X) + (r + (σ^2 ÷ 2)) * T ] ÷ (σ * √T) -d2 = σ * √T - T = time to option expiration - r = continuously compounded Rf - S0 = current asset price - σ = annual volatility of asset returns - N(*) = cumulative standard normal probability -N(-X) = (1 - N(X)

Answer 55

1. The underlying asset price follows a lognormal distribution 2. The Rf is constant and known 3. The volatility of the underlying asset is constant 4. Markets are frictionless (no taxes, transaction costs) 5. Continuously compounded dividend yield is constant 6. Options are European (no early exercise) | Binomial modes can deal with Euro and American options. BSM only Euro.

Answer 56

1. A call can be thought of as a leveraged stock investment where an investor longs N(d1) units and shorts (e^(-r * T) * X * N(d2)) of borrowed funds. 2. A portfolio that replicates a put option consists of a long position (X * e^(-r * T) * N(-d2) bonds and a short position in (S0 * N(-d1)) stocks.

Answer 57

S0 = $50 X = $45 σ = 25% Rf = 3% T = 6/12 = 0.5 C0 = $7 P0 = $1 To replicate a call we long the stock and short the bond Long 0.779 shares * $50 = $38.95 Borrow $56 * e^(-0.03 * 0.5) =* 0.723 = $32.05 Net cost = $6.90 - compared to the market price of call - $7 - this results in a $0.10 profit. If it were a put: N(-d1) = 1 - 0.779 = 0.221 N(-d2) = 1 - 0.723 = 0.277 To replicate a put, we long the bond and short the stock Long 0.277 shares: $45 * e^(-0.03 * 0.5) * 0.277 = $12.28 Short 0.221 shares: $50 * 0.221 = $11.05 Net cost = $12.28 - $11.05 = $1.23

Answer 58

Formula: C0 = [ (S0 * e^(-δ * T) * N(d1)) ] - (Xe^(-r * T) * N(d2)) - Essentially taking the PV of S0 and removing the PV of the dividend & then subtracting the PV of the exercise price P0 = [ (X * e^(-r * T)) * N(-d2) ] - [ S0 * e^(-δ * T) * N(-d1)]

Answer 59

P0 + (S0 * e^(-δ * T) = C0 + (e^(-r * T) * X)

Answer 60

C0 = [ S0 * e^(-r(B) * T) * N(d1) ] - [ Xe^(-r(P) * T) * N(d2) ] - S0 is the spot exchange rate versus in the other BSM models it's the stock price - - Essentially here we are take the spot exchange rate discounted at the base currency and subtracting it by the exercise exchange rate discounted at the price currency's interest rate. P0 = [ Xe^(-r(P) * T) * N(-d2) ] - [ S0 * e^(-r(B) * T) * N(-d1) ] - r(B) = continuously compounded base currency interest rate - r(P) = continuously compounded price currency interest rate - For currencies, the carry benefit IS NOT dividends, but rather INTEREST EARNED on the foreign currency.

Answer 61

Used to price European options on forwards and futures. - The Black Model ignores MTM w/ futures contracts Formula: C0 = e^(-r^(c)_f) * [ F_T * N(d1) - X * N(d2) ] - Don't be intimidated: All this is doing is substituting PV(F_T) for S0 in the BSM. - d1 = [ ln(FT ÷ X) + (σ^2 ÷ 2) * T ] ÷ (σ * √T) -d2 = d1 - (σ * √T) - σ = standard deviation of returns on the futures contract - F_T = futures price ## Footnote * When using the Black Model, ensure that the payoff is discounted at the maturity of the rate underlying the FRA (when looking at f(j,k), do not discount at j, but rather at (j + k). * Given an interest rate option is an option on a FRA, call options will gain in value when interest rates rise and put options will fall in value. * The accrual period needs to be factored in when valuing the option. This is because quoted rates are annual rates but in reality, the time between the FRA expiration and the maturity of borrowing and lending may not be one year. The accrual period can be viewed as a fraction of a year.

Answer 62

1. The value of a call option on futures = value of a portfolio w/ a long futures position (PV of futures price * N(d1)) and a short bond position (PV of X * N(d2)) 2. The value of a put = value of a portfolio w/ long bond and short futures position 3. Value of a call can also be thought of as the PV of the difference between futures price and X.

Answer 63

False, FRAs generally use a 30/360 convention, options on FRAs use an actual/365 convention.

Answer 64

1. A long interest rate call & short interest rate put (w/ exercise rate = current FRA rate) can be used to replicate a long FRA (pay fixed receive floating) 2. If exercise rate = the current FRA rate, a short interest rate call and long interest rate put can be combined to replicate a short FRA position (pay floating receive fixed) 3. A series of interest rate calls w/ different maturities but same exercise price can be combined to form an interest rate cap. 4. A series of interest rate puts w/ different maturities but same exercise price can be combined to form an interest rate floor. 5. If the exercise rate on a cap and floor is same, a long cap and short floor can be used to replicate a payer swap. 6. A short cap and long floor can replicate a receiver swap. 7. If the exercise rate on a floor and a cap are set equal to a market swap fixed rate, the value of the cap will be equal to the value of the floor.

Answer 65

An option that gives the holder the right but not the obligation to enter into an interest rate swap.

Answer 66

The right to enter into a specific swap at some date in the future at a predetermined rate as the fixed-rate payer. As interest rates increase, this instrument becomes more valuable. The holder of a payer swaption would exercise it and enter into the swap if the market rate is greater than the exercise rate at expiration.

Answer 67

The right to enter into a specific swap at some date in the future as the floating-rate payer at the rate specified in the contract. As interest rates decrease, this instrument gains value. The holder of a receiver swaption would exercise if market rates are less than the exercise rate at expiration.

Answer 68

(AP) * PVA * [ SFR * N(d1) - (X * N(d2)) ] * NP ## Footnote * AP = (1 ÷ # of settlement periods per year in the underlying swap) * SFR = current market swap fixed rate * X = exercise rate specified in the payer swapation * NP = notional principal of the underlying swap * d1 = [ ln(SFR ÷ X) + (σ^2 ÷ 2) * T ] ÷ (σ * √T) * d2 = d1 - (σ * √T)

Answer 69

(AP) * PVA * [ X * N(-d2) - SFR * N(-d1) ] * NP

Answer 70

A receiver swap can be replicated using a long receiver swaption and a short payer swaption with the same exercise rates A payer swap can be replicated using a long payer swaption and short receiver swaption with the same exercise rates. A long callable bond can be replicated using a long option-free bond plus a short receiver swaption.

Answer 71

Greek letters that describe sensivity factors w/ the relationship between each input in the BSM (asset price, exercise price, asset price volatility, time to expiration, and the Rf) and the option price.

Answer 72

The relationship between changes in asset prices and changes in option prices. Delta is the △ in the price of an option for a 1 unit △ in the price of an underlying stock. Call option deltas are positive because as the underlying asset price increases, call option value also increases. Conversely, the delta of a put option is negative because the put value falls as the asset price increases. Formulas: Delta_c = e^(-δ * T) * N(d1) Delta_p = e^(-δ * T) * N(-d1) △C = Delta_c * △S △P = Delta_p * △S | Delta is the hedge ratio ## Footnote * △C and △P are approximations. They wil be close for small changes in stock prices but less accurate for large changes in stock prices. * If the asset doesn't pay dividends, delta _c = N(d1) and delta_p = N(-d1) * Delta is the slope of the prior-to-expiration curve. * Delta < 1 * The delta of an at-the-money call option is typically close to 0.5. * As the option moves further into the money and as the expiration date approaches, the delta of a put option moves closer to –1.

Answer 73

The relationship between changes in volatility and changes in option prices. Calls are positively related (call price increases as volatility of the underlying increases) to vega and puts and also positively related. ## Footnote * When the option is at the money, changes in volatility will have the greatest effect on the option value- meaning that vega gets larger as the option gets closer to being at-the-money.

Answer 74

The relationship between changes in the Rf and changes in option prices. Calls are positively related to Rho (call price increases as Rf increases) and puts are negatively related.

Answer 75

The relationship between changes in T (time to expiration) and changes in option prices. As a call option approaches maturity, the time value gets closer to 0 (calls are negatively related to theta). This is called **time decay**. As a put option approaches maturity, the time value also gets closer to 0 (puts are also negatively related to theta). ## Footnote * 1 exception: W/ Euro puts that are deep in-the-money w/ high Rfs, and are approaching maturity. The put value may increase as the option approaches maturity.

Answer 76

The exercise price of a call options (the higher X the lower the call value) is negatively related to calls and positively related to puts.

Answer 77

False, call deltas do range from 0 to e^(-δ * T) but the further out-the money the option goes, delta approaches 0 and the further in-the money the option goes, delta approaches e^(-δ * T) if there are dividends or 1 if there are no dividends. Put deltas from from e^(-δ * T) to 0. The further out-the money a call option goes, delta approaches 0 and the further in-the money the option goes, delta approaches e^(-δ * T) if there are dividends or 1 if there are no dividends.

Answer 78

The rate of change in delta as the underlying stock price changes. Long positions in calls and puts have positive gammas. For example, a gamma of 0.04 implies that a $1.00 increase in the price of the underlying stock will cause a call option's delta to increase by 0.04, making the call option more sensitive to changes in the stock price. Gamma is highest for at-the-money options. Deep in-the-money or deep out-of-money options have low gamma. Including gamma improves the precision with which change in option value would be estimated. Formulas: △C = Delta_c * △S + 0.5 gamma * △S^2 △P = Delta_p * △S + 0.5 gamma * △S^2 ## Footnote * To decrease gamma in a portfolio, people should use shorts. * Change in delta/change in the price of the underlying * The higher the gamma, the greater the change in delta given a change in stock price.

Answer 79

Combines a long position in a stock w/ a short position in a call option so that the value of the portfolio does not change as the stock price changes. ## Footnote * The delta hedged asset position is only risk free for a very small change in the value of the underlying stock. The delta-neutral portfolio must be continually rebalanced to maintain the hedge; for this reason, it is called a dynamic hedge. As the underlying stock price changes, so does the delta of the call option, and thus too the number of calls that need to be sold to maintain a hedged position. Hence, continuously maintaining a delta-neutral position involves significant transaction costs.

Answer 80

(# of shares hedged) ÷ delta of call option ## Footnote * Since delta < 1, we will always need more calls than shares

Answer 81

-1 * [ (# of shares hedged) ÷ delta of put option ]

Answer 82

(1) 60,000 ÷ .6 = 100,000 options (2) Total change in value of stock position = 60,000 * $1 = $60,000 Total change in value of option position = 100,000 * -0.60 = -$60,000 Total change in portfolio value = $60,000 - $60,000 = $0

Answer 83

The risk that the stock price might abruptly "jump," leaving an otherwise delta-hedged portfolio unhedged. Ex: If the stock price falls abruptly, the loss in the long stock position will not equal the gain in the short call position. ## Footnote * If the assumptions of the BSM hold, changes in stock price will be continuous rather than abrupt, and hence there will be no gamma risk * When an option's gamma is higher, a delta hedge will perform more poorly over time.

Answer 84

The standard deviation of continuously compounded asset returns that is "implied" by the market price of the option. Since volatility isn't observable, we can calculate implied volatility by using the other four inputs to the BSM model and the market price of the option, and solving for volatility. ## Footnote * Traders often use implied volatilities to gauge market perceptions. For instance, implied volatilities in options with different exercise prices on the same underlying may reflect different implied volatilities (a violation of the BSM assumption of constant volatility).

Answer 85

That volatility of the call is between 40% and 45%

Answer 86

Recall vega, volatility of a call is postively correlated with the option's value. Therefore, if volatility rises so will the price. Thus, the trader should long the call option.

Answer 87

Contango describes the situation where the futures price exceeds the spot price, and there is not such thing as reverse contango.

Answer 88

A pay fixed receive floating swap

Answer 89

"N(d2) is interpreted as the risk-neutral probability that a call option will expire in the money. Similarly, N(-d2) is the risk-neutral probability that a put option will expire in the money." For a swaption, it's the risk-neutral probability that a payer swaption will expire in the money. N(-d2) in this case is the risk-neutral probability that a receiver swaption will expire in the money. | It's essentially the probability of exercise

Answer 90

False, the date of expiration for an FRA is the start of the notional borrowing period

Answer 91

False. The value of a forward contract comes from the difference between the forward contract price and the market price for the underlying asset. ## Footnote * A forward contract has only one price, which applies to both the long and to the short.

Answer 92

C. Interest rate options pay one period after exercise. Options expiring on settlements at t = 1,2,3, will mimic the uncertain swap payments at t = 2,3,4.

Answer 93

True, if it didn't there would be arbitrage

Answer 94

False, early exercise of in-the-money American options on futures is sometimes worthwhile because the immediate mark to market upon exercise will generate funds that can earn interest. It is never worthwhile for options on forwards because no funds are generated until the settlement date of the forward contract.

Answer 95

False, when a call option is deep out-of-the-money, the slope of the at expiration curve is close to zero, which means the delta will be close to zero.

Answer 96

False, long forward contract positions increase in value when the forward price increases.

CFA L2 Derivatives Flashcards

(131 cards)