CFA L2 Derivatives

Question 1

Long vs short positions in a forward contract

Answer 1

Long position: The party that agrees to buy the asset at some point in the future

Short position: The party that agrees to sell the asset at some point in the future

Question 2

True or false: Forward contracts have margin accounts where money must be deposited at inception?

Answer 2

False, margin accounts exist with FUTURES.

Question 3

Price of a forward contract

Answer 3

The price of a forward contract DOES NOT refer to the price to purchase a contract. There is no price that either party pays at the contract’s inception. The price of the forward contract refers to the forward price of the underlying. The price is often quoted as an interest rate or exchange rate but can be quoted in $ or €. The forward price MUST prevent riskless arbitrage in frictionless market, and is thus the price that makes the values of both the long and the short positions zero at contract initiation.

Calculation: Forward price (FP) = S0 * (1 + Rf)^T
OR
S0 = FP ÷ (1 + Rf)^T

• The forward price is the future value of the spot price adjusted for any periodic payments expected from the asset.

Question 4

No-arbitrage principle

Answer 4

The idea that there should be no riskless profit from combining forward or futures contracts w/ other instruments.

Question 5

Assumptions of no-arbitrage principle

Answer 5

1. Transaction costs are zero
2. There are no restrictions on short sales or on the use of short sale proceeds
3. Borrowing and lending can be done at unlimited amounts at the Rf

Question 6

Example: No-arbitrage forward price

Consider a 3-month forward contract on a zero-coupon bond w/ a face value of $1,000 that is currently quoted at $500, and suppose that the annual Rf is 6%. What is the price of the forward contract under the no-arbitrage principle?

Answer 6

T = 3 ÷ 12

FP = $500 * (1.06)^.25 = $507.34

The $507.34 is the price agreed upon today to be paid in 3 months time.

Question 7

Cash and Carry Arbitrage

Answer 7

This is a market-neutral strategy combining the purchase of a long position in an asset such as a stock or commodity in the spot market, and shorting a position in a futures contract on that same underlying asset. When a contract is overpriced, an arbitrageur will take a short position and vice versa.

Ex: If a bond trades at $510 but the no-arbitrage price is $507.34, an arbitrageur will borrow $500 at the Rf, buy the bond for $510, and enter into a short position (selling the asset) in a forward contract. When the price drops to $500 in future, the loan will be repaid ($507.34) but the arbitrageur will receive $510 from the contract for a profit of $2.66.

This is used when an asset is overpriced

• This is considered riskless arbitrage
• Recall from L1, oftentimes if the security that’s being shorted pays dividends, the arbitrageur must pay the holder the dividends during the short period.
• Ex: If the answer in #6 DOES NOT equal what the market has the forward contract priced, we use cash and carry arbitrage.

Question 8

Reverse cash and carry arbitrage

Answer 8

A market-neutral strategy combining a short position in an asset and a long futures position in that same asset. An arbitrageur can use this when an asset is underpriced.

Ex: If a bond trades at $502 but the no-arbitrage price is $507.34, an arbitrageur will sell the bond for $500 today, invest the proceeds at the Rf, and enter into a long position in a forward contract. Then, in the future when the bond price increases, the arbitrageur will pay the $502 but will receive $507.34 from the principal and interest for a $5.34 profit.

This is considered riskless arbitrage

Question 9

Value of a long forward contract during the life of the contract

Answer 9

Vt = St - [ FP ÷ (1 + Rf)^(T - t) ]

• T = maturity date
• t = current date

Question 10

Value of long forward contract at expiration

Answer 10

S_T - FP

Question 11

Forward price of an equity derivative contract w/ discrete dividends

Answer 11

FP of equity security = (S0 - PVD) * (1 + Rf)^T
OR
FP of equity security = [ S0 * (1 + Rf)^T ] - FVD

• PVD = PV of dividend
• FVD= FV of dividend
• For equity contracts, use a 365-day basis for calculating T (ex: if it is a 60-day contract, T = 60 / 365).

Question 12

Value of a forward contract w/ an underlying that’s a dividend-paying stock

Answer 12

Vt = (St - PVDt) - [ FP ÷ (1 + Rf)^(T - t) ]
OR
(FPt - FP) ÷ (1 + Rf)^t

• (T-t) = time to maturity
• If result is positive, it’s a gain for the long and a loss for the short. If result is negative, it’s a gain for the short and a loss for the long.

Question 13

True or false: In a forward contract, the long loses when the price of the underlying increases and the short gains when the price of the underlying increases?

Answer 13

False, in a forward contract, the long gains when the price of the underlying increases and the short gains when the price of the underlying decreases.

Question 14

Benefits of carry

Answer 14

Interim CFs (ex: dividends or coupons). These benefits reduce the FP and offset costs of carry (ex: Rf).

Question 15

Price of an equity index forward contract

Answer 15

Rather than taking the PV of every dividend in the index, we can make the calculation as if the dividends are paid continuously.

FP of an equity index = S0 * e^(Rf_c - δ^c)
OR
(S0 * e^(-δT)) * e^(Rf_c * T)

• δ^c = continuously compounded dividend yield
• Rf^c = continuously compounded risk-free rate

Question 16

How to calculate the continuously compounded Rf (Rf^c)

Answer 16

ln(1 + Rf)

Ex: The Rf compounded annually at 5% = ln(1.05)

Question 17

Example: Calculating the price of a forward contract on an equity index

The value of the S&P 500 Index is 1,140. The continuously compounded risk-free rate is 4.6% and the continuous dividend yield is 2.1%. Calculate the no-arbitrage price of a 140-day forward contract on the index.

Answer 17

FP = 1140 * e^( (0.046 - 0.021) * 140/365)

Question 18

Forward price on a fixed income derivative

Answer 18

Same as for an equity derivative w/ dividends except now we use coupon payments.

Calculation: FP on a fixed income security = (S0 - PVC) * (1 + Rf)^T
OR
S0 * (1 + Rf)^T - FVC

* PVC = PV of coupon
* FVC = FV of coupon

Question 19

Value of a fixed income derivative

Answer 19

(St - PVCt) - (FP ÷ (1 + Rf)^(T - t))

Question 20

Bond futures contracts

Answer 20

Derivatives that obligate the contract holder to purchase/sell a bond on a specified date at a predetermined price. Bond futures contracts often allow the short an option to deliver any of several bonds, which will satisfy the delivery terms of the contract. This is called a delivery option and is valuable to the short. Each bond is given a conversion factor that is used to adjust the long’s payment at delivery so the more valuable bonds receive a larger payment. Bond prices are quotes as clean prices, and at settlement the buyer pays the clean price + accrued interest = full price.

Question 21

Accrued interest calculation

Answer 21

(days since the last coupon payment ÷ days between coupon payments) * coupon amount

Question 22

Bond futures price calculation

Answer 22

[ (full price) * (1 + Rf)^T - FVC - Accrued interest_T ]

Question 23

Quoted bond futures price

Answer 23

This is how to adjust the forward pricing formula to account for the short’s delivery option

Bond futures price ÷ conversion factor
OR
[bond futures price) * (1 ÷ conversion factor)

Question 24

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 24

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

Question 25

Eurodollar deposit

Answer 25

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

Question 26

Secured overnight funding rate (SOFR)

Answer 26

The lending rate on dollar-denominated loans between banks. It is quoted as an annualized rate based on a 360-day year.

• This is a benchmark rate, whereas the Fed Funds rate is simply a target rate
• The CFA referes to SOFR as MRR

Question 27

Forward rate agreement (FRA)

Answer 27

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.

• FRAs are always cash settled.

Question 28

What does the long vs short position commit to paying and receiving in an FRA?

Answer 28

Long position: Pay fixed and receive floating

Short position: Pay floating and receive fixed

Question 29

Price of an FRA

Answer 29

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 quotes 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.

Question 30

Steps to calculating the price of an FRA

Answer 30

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)

Question 31

Example: Calculating the price of an FRA

Calculate the price of a 1x4 FRA. The current 30-day MRR is 4% and 120-day MRR is 5%.

Answer 31

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.

Question 32

How to value an FRA at expiration

Answer 32

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

Question 33

Example: valuing an FRA after initiation

Value a 5.32% 1 × 4 FRA with a principal amount of $1 million 10 days after initiation if the 120 ay MRR = 6%, 110-day MRR = 5.9%, 30-day MRR = 5.8%, and 20-day MRR = 5.7%.

Answer 33

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)) ]

• 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.

Question 34

Futures contract

Answer 34

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.

Question 35

Mark-to-market

Answer 35

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.

Question 36

Value of futures contract

Answer 36

Current futures price − Previous mark-to-market price

• 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.

Question 37

MTM example:

The spot price of gold at initiation is $1,800, and customer A enters in to one 60-day forward contract and one 60-day futures contract. Assume the Rf is 5%.

(1) What is the forward and future price at initiation
(2) Assuming the initial margin requirement is 10%, what must Customer A deposit into their margin account at initiation?
(3) If at the end of day 1, the spot price of gold rises to $1,820, what amount is added/subtracted to customer A’s margin account?
(4) Now, assume at the end of day 2, the spot price of gold decreases to $1,805, what amount is added/subtracted to customer A’s margin account?

Answer 37

(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

Question 38

True or false: The price of a forward contract is constant while the value fluctuates over time, but the price of a futures contract fluctuates over time while the value at the end of each day is zero?

Answer 38

True

Question 39

Interest rate swap

Answer 39

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

• Interest rate swaps settle over multiple periods, whereas FRAs are just for one period.

Question 40

How to compute the swap rate

Answer 40

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).

Question 41

How to calculate the discount factor (Z)

Answer 41

Z = 1 ÷ [ 1 + (MRR * (days ÷ 360)) ]

However many decimal places the question uses, use that in calculations

Question 42

How to calculate the periodic swap fixed rate (SFR)

Answer 42

(1 - final discount factor) ÷ (sum of discount factors)

• To calculate the annual swap fixed rate, multiply this result by the # of settlement periods.

Question 43

Example: Calculating the fixed rate on a swap with quarterly payments

Annalized MRR spot rates are:
90-day = 3%
180-day = 3.5%
270-day = 4%
360-day = 4.5%
Notional principal = $5,000,000

Calculate:
(1) The fixed rate in % terms
(2) The quarterly fixed payments in $

Answer 43

(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

Question 44

True or false: After the initiation of an interest rate swap, the swap will take on a positive or negative value as interest rates change?

Answer 44

True

Question 45

How to calculate the value of an interest rate swap to a payer after initiation

Answer 45

Value to the payer = ΣZs * (SFRnew - SFRold) * (days/360) * notional principal

• 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

Question 46

How to price currency swaps

Answer 46

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.

Question 47

Example: Pricing a currency swap

Assume that the fixed rate on a 1-year quarterly $5,000,000 interest rate swap is 4.4%. The comparable set of U.K. rates are:
90-day - annualized rate = 4% - Z = 0.99010
180-day - annualized rate = 5% - Z = 0.97561
270-day - annualized rate = 6% - Z = 0.95694
360-day - annualized rate = 7% - Z = 0.93458

Assume the current exchange rate is £0.50 per $ OR $2 per £.

(1) What is the fixed rate on the £ swap in annual terms is?
(2) What is the notional £ principal amount of the swap?

Answer 47

(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.

Question 48

How to calculate the value of a currency swap after initiation

Answer 48

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

• Each CF corresponds to a settlement date
• Since CFs are in different currencies, we must adjust one using the current exchange rate

Question 49

Equity swaps

Answer 49

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.

Question 50

Who pays what in an equity swap

Answer 50

Equity return payer: Pays any positive return on equity and receives a fixed rate payment plus any negative equity return

Fixed-rate payer: Pays the fixed rate plus any negative equity return and receives positive equity return

• Equity return payer is bearish, whereas the fixed-rate payer is bullish

Question 51

How to value an equity swap

Answer 51

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

Question 52

Equity swap example:

A $10 million principal, one-year, quarterly settlement equity swap has a fixed rate of 6.05%. The index at inception is 985. After 30 days have passed, the index stands at 996 and the term structure of MRR is 6%, 6.5%, 7%, and 7.5% for terms of 60, 150, 240, and 330 days, respectively. The respective discount factors are 0.99010, 0.97363, 0.95541, and 0.93567. Calculate the value of the swap to the fixed-rate payer on Day 30.

Answer 52

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

Question 53

Equity swap example 2:
An investor is the Stock A returns payer (and Stock B returns receiver) in a $1 million quarterly-pay swap. After one month, Stock A is up 1.3% and Stock B is down 0.8%. Calculate the value of the swap to the investor.

Answer 53

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

Question 54

Binomial Model

Answer 54

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.

• 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.

Question 55

Binomial Model notation

Answer 55

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)

Question 56

πU calculation

Answer 56

(1 + Rf - D) ÷ (U - D)

Question 57

True or false: The up- and down-move probabilities are actual probabilities of up- or down-moves?

Answer 57

False, they are risk-neutral probabilities

Question 58

Expectations approach

Answer 58

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

• All binomial models calculate value as the expected outcome discounted back to the PV at the appropriate interest rate.

Question 59

Example: Calculating the call option value with a one-period binomial tree

Calculate the value today of a one-year call option on a stock that has an exercise price of $30. Assume that the periodically compounded (as opposed to continuously compounded) Rf is 7%, the current value of the stock is $30, the up-move factor is 1.333, and the down move factor is 0.75.

Answer 59

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

Question 60

True or false: The forward price on a fixed income derivative is the spot price + net cost of carry?

Answer 60

True

Question 61

Put-call parity

Answer 61

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)

Question 62

Fiduciary Call

Answer 62

Long a call option + investment in a zero-coupon riskless bond with par value equal to the strike price

Formula: C0 + PV(X)

Question 63

Protective put

Answer 63

long a stock and long a put option

Formula: P0 + S0

Question 64

Synthetic call vs synthetic stock vs synthetic put vs synthetic bond

Answer 64

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

Question 65

Example: Put-Call Parity

A 1-year call option on the common stock of Firm A w/ an exercise price of $60 is trading for $8. The current stock price is $62. The Rf is 4%. What is the price of the put option implied by put-call parity?

Answer 65

S0 + P0 = C0 + PV(X)
P0 = C0 + PV(X) - S0
P0 = 8 + (60 ÷ 1.04) - 62 = $3.69

• Any other price than $3.69 allows for risk-free profit

Question 66

How to value an option using a 2-period binomial model

Answer 66

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).

Question 67

Example: Valuing a call option on a stock with a two-period model

Suppose you own a stock currently priced at $50 and that a two-period European call option on the stock is available with a strike price of $45. The up-move factor is 1.25 and the down-move factor is 0.80. The risk-free rate per period is 7%. Compute the value of the call option using a two-period binomial model.

Answer 67

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

• S++ = value of stock in two upward movement scenario
• C++ = payoff of call option in two upward movement scenario

Question 68

Intrinsic value vs time value

Answer 68

Intrinsic value:

Time value: The amt by which the option premium exceeds the exercise value.

• Time value = 0 at expiration if option is at or out of the money.
• Time value = exericse value at expiration if option is in the money.

Question 69

True or false: American call options that do not have carry benefits have the same value as European options?

Answer 69

True

Interest on early payoff < time value lost

Question 70

True or false: An American put (the right to sell) that is deep in-the-money (close to zero) may be worth less than the same European put?

Answer 70

False, it will be worth more.

Interest on early payoff > time value lost

Question 71

Example: Value of a European put option

Consider a stock currently trading at $50. The periodically compounded interest rate is 3%. Suppose that U = 1.3 and D = 0.80. Calculate the value of a two-period European-style put option on the stock that has an exercise price of $50. Also, determine if early exercise would make economic sense.

Answer 71

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

Question 72

Example: Value of an American put option

Consider a stock currently trading at $50. The periodically compounded interest rate is 3%. Suppose that U = 1.3 and D = 0.80. Calculate the value of a two-period European-style put option on the stock that has an exercise price of $50. Also, determine if early exercise would make economic sense.

Answer 72

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.

• 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.

Question 73

True or false: If the market price of an option differs from the value calculated w/ a binomial model, that indicates arbitrage?

Answer 73

True

Question 74

Hedge ratio

Answer 74

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- )

Question 75

How to value an option using the hedge ratio

Answer 75

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

Question 76

Example: Calculating arbitrage profit

A one-year call option on a $30 stock (exercise price = $30) is valued at $5.13. Given that the Rf is 7%, and the up-move factor and down-move factor are 1.333 and 0.75, respectively, if the market price of the call option is $6.50, illustrate how this opportunity can be exploited to earn an arbitrage profit. Assume we trade 100 call options.

Answer 76

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

Question 77

How to calculate the payoff of an interest rate option

Answer 77

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

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

Question 78

Example:

What is the value of a European call option with an exercise rate of 5.5% and a notional principal of $1MM, given:
i = MRR = 3%
i1U = MRR = 5.7883%
i1L= MRR = 3.8800%
i1UU = MRR = 10.7383%
i1UL = MRR = 7.1981%
i1LL = MRR = 4.8250%

Answer 78

(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

Question 79

Equivalencies in interest rate derivatives:

Answer 79

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

Question 80

The Black-Scholes-Merton (BSM) Model

Answer 80

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.

• The binomial model uses discrete time

Question 81

BSM Model w/o dividends formula

Answer 81

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)

Question 82

Assumptions of BSM:

Answer 82

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
5. Continuously compounded dividend yield is constant
6. Options are European

Binomial modes can deal with Euro and American options. BSM only Euro.

Question 83

Interpretations of the BSM Model

Answer 83

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.

Question 84

Example: BSM

Stock of firm A trades at $50. The return volatility is 25% and the continuously compounded Rf is 3%. Calls and puts w/ a strike price of $45 and expiring in 6 months are trading at $7 and $1, respectively. If N(d1) = 0.779 and N(d2) = 0.723, calculate the value of the replicating portfolios and any arbitrage profits on both options

Answer 84

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

Question 85

BSM Model w/ dividends formula

Answer 85

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)]

Question 86

Put-Call Parity if the stock pays dividends

Answer 86

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

Question 87

BSM Model for Currency Options

Answer 87

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.

Question 88

The Black Model

Answer 88

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

Question 89

Interpretations of the Black Model

Answer 89

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.

Question 90

True or false: FRAs and options on FRAs generally use a 30/360 convention?

Answer 90

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

Question 91

Equivalencies in interest rate derivative contracts

Answer 91

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.

Question 92

Swaption

Answer 92

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

Question 93

Payer swapation

Answer 93

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.

Question 94

Receiver swapation

Answer 94

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.

Question 95

True or false: A swaption is equivalent to an option on a series of CFs (annuity), one for each settlement date of the underlying swap, equal to the difference between the exercise rate on the swaption and the market swap fixed rate?

Answer 95

True

Question 96

Value of a payer swaption using the Black model

Answer 96

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

• 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)

Question 97

Value of a receiver swaption using the Black model

Answer 97

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

Question 98

Swapation equivalencies

Answer 98

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.

Question 99

The greeks

Answer 99

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.

Question 100

Delta

Answer 100

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

• △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

Question 101

Vega

Answer 101

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.

Question 102

Rho

Answer 102

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.

Question 103

Theta

Answer 103

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).

• 1 exception: W/ Euro puts that are deep in-the-money w/ high Rfs, and are approaching maturity. The put value may increase as theoption approaches maturity.

Question 104

Relationship between the exercise price and option price according to BSM

Answer 104

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.

Question 105

True or false: Call deltas range from 0 to e^(-δ * T). The further out-the money a call option goes, delta approaches e^(-δ * T) and the further in-the money the option goes, delta approaches 0?

Answer 105

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.

Question 106

Gamma

Answer 106

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

• The decrease gamma in a portfolio, people should use shorts.

Question 107

True or false: Call and put options on the same underlying asset with the same exercise price and time to expiration will have equal gammas?

Answer 107

True

Question 108

Dynamic hedging/delta-neutral portfolio/delta-neutral hedge

Answer 108

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.

• 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.

Question 109

Number of short call options needed to delta hedge

Answer 109

(# of shares hedged) ÷ delta of call option

• Since delta < 1, we will always need more calls than shares

Question 110

Number of long put options needed to delta hedge

Answer 110

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

Question 111

Example: Dynamic Hedging

Investor A owns 60,000 shares of Firm A’s stock w/ a market price of $50 per share. A call option on a single share has a strike price of $4 and has a delta of 0.60.
(1) What is the # of calls necessary to create a delta-neutral hedge?
(2) What is the effect on portfolio value of a $1 increase in the price of firm A?

Answer 111

(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

Question 112

Gamma risk

Answer 112

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.

• 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

Question 113

Implied volatility

Answer 113

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.

• 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).

Question 114

Example: Implied volatility

The market price of 3-month 45 firm A call is $7.25. Given:
If volatility = 25%, value of call = 6.11
If volatility = 30%, value of call = 6.44
If volatility = 35%, value of call = 6.79
If volatility = 40%, value of call = 7.17
If volatility = 45%, value of call = 7.59

When you see 45, that’s the X

Answer 114

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

Question 115

Example: Implied volatility

Calls of Firm A stock trade at an implied volatility of 22%. A trader estimates that the future volatility will actually be closer to 25%. To capitalize on the situation, what should the trader do?

Answer 115

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.

Question 116

Contango

Answer 116

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

Question 117

Plain vanilla interest-rate swap

Answer 117

A pay fixed receive floating swap

Question 118

How can N(D1) and N(D2) be interpreted?

Answer 118

“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.”

Question 119

True or false: The date of expiration for an FRA is the end of the notional borrowing period?

Answer 119

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