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CFAI Flashcards > Derivatives > Flashcards

Derivatives Flashcards

1
Q

Forward rate is less than spot rate (carry arbitrage)

A

Under carry arbitrage, when interest rates fall, forward prices decline.

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2
Q

Carry benefits vs. cost

A

Benefits (dividends, foreign interest, bond coupon payments)

represented by lowercase gamma

Costs (zero for financial instruments, but waste, storage and insurance on commodities)

Represented by lowercase theta

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3
Q

Value of a Forward Contract at Expiration

A

V(Long) = Spot - Forward (buying forward contracts)

V(Short) = FWD contract - Spot price (selling forward contracts)

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4
Q

Arbitrageur rules

A
  1. Borrow funds to purchase the underlying asset on a forward for futures contract

And invest proceeds from short selling transactions at the risk-free interest rate

  1. Don’t take price risk. Only focus is market price risk of the underlying or derivative.
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5
Q

FV of spot price using continuous compounding rate on a forward contract

A

FV(S0) = S0(underlying) x e^(rate x T)

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6
Q

FV spot price using annual compounding on forward contract

A

FV(S0) = S0 x (1+r)^T

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7
Q

Cash flows on Forward Contract

A

Done in parallel (not sequential)

@T=0

  1. Borrow funds to purchase underlying asset (+) plus financing using RFR
  2. Purchase underlying asset (-)

@ T=1

  1. Sell forward contract for delivery of underlying asset at T=1
    FWD = FV(S0) - no arbitrage
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8
Q

Market FWD Price > Carry Model FWD Price

A
  1. Sell forward contract
  2. Borrow arbitrage profit at T=0
    And pay off loan at contract expiration
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9
Q

Reverse Carry Arbitrage

A

Opposite transactions than Carry Model. Relevant when:

Market FWD Price < Carry Model FWD Price

  1. Buy forward contract
  2. Short the underlying
  3. Lend the short sale proceeds
  4. Borrow arbitrage profit
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10
Q

Valuation of Forward Contract

A

Forward value is the prevent value of the difference in forward contract prices PV(new - old)

Vt(T) = PV of differences in forward prices

Vt(T) = PVt,T[Ft(T) - F0(T)]

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11
Q

FV of underlying adjusted for carry cash flows

A

FV(0,T) = S0 + theta - gamma

I.e. spot price + carry costs - carry benefits

As carry benefits increase, forward price decreases (carry benefits lessen the burden of “carrying” the underlying by reducing the cost to carry the asset)

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12
Q

Continuous Dividend Yield

A

Assumption that dividends accrue continuously over the period in question rather than on specific discrete dates

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13
Q

Interest rate compounded annually equivalence as continuously compounded

A

(1+r)^T-t = ln(1+r)

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14
Q

Future value of the underlying adjusted for carry (continuous dividend yield)

A

FWD(T) = Spot x e^(r+theta-gamma)

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15
Q

Price of an equity forward contract

A

Forward Price=

S0 - PVD) x (1+r)^(T-t/365

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16
Q

Value of an Equity Forward Contract

A

The value of the long position in a forward contract on a stock at time t =

[S - PVD] - [FP / (1+r)^(T-t)

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17
Q

Price of a Fixed Income Forward Contract

A

Forward Price =

(S - PVC) x 1+r^(T-t)

Note: for treasury bonds, coupons are paid semi-annually

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18
Q

Value of a Fixed Income Forward Contract

A

Forward Price Value =

[S-PVC] - FP / (1+r^T-t)

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19
Q

Option valuation key assumptions

A
  1. Replicating instruments are identifiable and investable
  2. No market frictions such as transaction costs and taxes
  3. Short selling is allowed
  4. Underlying instrument follows a known statistical distribution
  5. Borrowing/Lending at RFR
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20
Q

Value of a call option at expiration

A

Call Value =

Max (0, MP - Strike)

Note: Unlimited upside and downside limited to zero

21
Q

Value of Put Option at expiration

A

Value of Put option =

Max(0, Strike - Market Price)

Note: Upside limited (price doesn’t fall below zero)

22
Q

Assumptions of the BSM model

A
  1. Implies that the continuously compounded return is normally distributed
  2. Price of underlying instrument moves smoothly from value to value
  3. Underlying instrument is liquid
  4. Continuous trading is available
  5. Short selling permitted
  6. No market frictions
  7. No arbitrage
  8. All options are European
  9. RFR is known and constant/borrowing and lending allowed at RFR
  10. Volatility of return on the underlying is known and constant
  11. Yield on the underlying is continuous known and constant yield at an annualized rate
23
Q

BSM Model

A

PV of the expected option payoff at expiration

Described as having two components: stock and bond

24
Q

Stock and Bond Components - Call Option

A

SN(d1) = stock component

e^(-rT) * XN(d2) = bond component

Call option = stock component - bond component

= SN(d1) - e^(-rT) * XN(d2)

25
Q

Stock and Bond Components - Put Options

A

Stock component =

SN(-d1)

Bond component =
e^(-rT)*XN(-d2)

Bond - Stock

e^(-rT)*XN(-d2) - SN(-d1)

26
Q

Replicating strategy cost (Call Option)

A

Purchase underlying stock by borrowing (zero-coupon bond)

Ns(price of stock) + Nb(price of bond)

Where Ns = N(d1)
Where Nb = -N(d2)

Where price of zero-coupon bond B = e^(-rT)X

27
Q

Replicating strategy cost (Put option)

A

Ns(stock) + Nb(bond)

Where Ns = -N(-d1)
Where Nb = N(-d2)

Price of a zero coupon bond B = e^(-rT)X

28
Q

Determine contracts rate on Forward Rate Agreement

A

E.g. 3 x 5 FRA

Where 90-day LIBOR is 5%
Where 150-day LIBOR is 6%

Step 1. De-annualized each rate

.05 x (90/360)
.06 x (150/360)

Step 2. Calculate the period rate

[Deannualized 150-day LIBOR/ Deannualized 90-day LIBOR] - 1

Step 3. Annualize the period rate

FRA Price x {360/[150-90]}

29
Q

Option Gamma

A

change in a given option’s delta for small change in stock’s value (i.e. rate of change of delta as stock changes)

**Change in delta / change in value of underlying

Measure of the curvature in the option price in relationship to the stock price

Gamma of call = gamma of put

Always non-negative, gamma is zero on a stock

Buying options will always increase net gamma

Considered analogous to Convexity, in that it measures LARGE changes in delta (second-order effect)

30
Q

Delta

A

Most fundamental risk of an option is sensitivity to the price of the underlying.

= change in value of option / change in value of the underlying

Call delta ranges from a value of 0 to 1
Put delta ranges from a value of 0 to -1

Considered analogous to duration in that it measures SMALL changes

31
Q

Vega

A

An important sensitivity measure for options; reflects the relationship between the option price and the volatility of the underlying

Change in value of option / change in volatility of underlying

32
Q

Duration of Option (Delta)

A

Delta can be used to approximate the new price of an option as the underlying changes

C + 🔺C x 🔺S (call option)
P + 🔺P x 🔺S (put option)

33
Q

Convexity of Option (Gamma)

A

Using delta and gamma, the new call price is:

C + 🔺C x 🔺S + {[call gamma (🔺S)^2] / 2}

34
Q

Delta Neutral Hedge

A

Determine the number of option contracts needed to create a delta neutral hedge =

of shares in stock / delta on option

35
Q

Delta Put Option

A

Delta of call option -1

36
Q

Forward Rate Agreement (FRA)

A

An OTC forward contract in which the underlying is an interest rate on a deposit

Fixed receiver - short FRA
Floating Receiver - long FRA

No initial exchange of cash flows & FRA value is 0 @ initiation

37
Q

What is advanced set?

A

Reference interest rate that is set at the time the money is deposited to determine the interest accrued over the life of the deposit.

This convention is almost always used because most issuers and buyers of the financial instrument want to know the rate on the instrument while they have a position in it.

38
Q

Advance Set, Advanced Settled

A

The convention used for settling FRAs. The settlement of interest accrued is paid at the time of FRA expiration

39
Q

Formula for calculating interest accrued on bank deposit (aka LIBOR spot)

A

Notional Principal x (1+LIBOR(actual/360)

40
Q

How to compute accrued interest for fixed income forward or futures contract?

A

Accrued interested =

Accrual period x periodic coupon payment

OR AI = [(days since last coupon payment / days between coupon payments) * coupon amount]

41
Q

What is the purpose of a conversion factor for a fixed income futures contract?

A

Fixed income futures contracts halve more than one bond that can be delivered by the seller.

Bonds trade at different prices based on maturity and stated coupon

Conversion factor is an adjustment used to make all deliverable bonds roughly equal in price

42
Q

When settling a futures contract, what is cheapest-to-deliver?

A

Seller will deliver the bond that is least expensive after adjusting bonds with a conversion factor.

43
Q

Quoted price of a bond vs. Full Price

A

Quoted price is also known as the clean price and doesn’t include the interest accrued since the last coupon date

Full Price is also known as the dirty price and includes the interest accrued since the last coupon date

44
Q

How do you solve for the quoted futures price based on carry arbitrage model?

A

Quoted Futures Price = Conversion factor adjusted future value of the underlying adjusted for carry =

(1/CF) * [((1+r)^T)* (S0 + AI(0) - AI(T) - FVCI]

45
Q

Define a currency swap

A

Two counterparties agree to exchange future interest payments in different currencies using a fixed or floating interest rate

Requires the exchange of notional amounts at both initiation and expiration

46
Q

Define interest rate swap and explain how it is derived

A

One party agrees to pay floating and receive fixed or vice-versa

Derived from the LIBOR curve

47
Q

Define the steps to calculate the swap fixed rate

A

Step 1. Calculate the DFs for each period

DFs = 1 / (1 + (LIBOR x days/360))

Step 2. Calculate the periodic SFR

SFR(periodic) = 1 - final DF / sum of DFs

Step 3. Calculate the annual SFR

SFR(annual) = SFR(periodic) x # of settlement periods

48
Q

What is the formula to calculate the MV of an IRS?

A

Value to the payer =

Sum of DF x (SFRnew - SFRold) x (days / 360) x Notional principal

Where days = # of settlement periods

49
Q

Valuation of a currency swap

A

Value of swap (currency A)

= NP x {SFR(periodic) x Sum of PV Discount Factors + Final DF}

Value of swap (currency B translated to currency A)

=[NP x {SFR(periodic) x Sum of PV Discount Factors + Final DF}] * exchange rate

VA - VB

CFAI Flashcards (10 decks)

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